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[a,b]Cristiane Yumi London
# Probing singularities of Landau-gauge propagators with Padé approximants
Diogo Boito Attilio Cucchieri Tereza Mendes
###### Abstract
Padé approximants are employed in order to study the analytic structure of the
four-dimensional SU(2) Landau-gauge gluon and ghost propagators in the
infrared regime. The approximants, which are model independent, are used as
fitting functions to lattice data for the propagators, carefully propagating
uncertainties due to the fit procedure and taking into account all possible
correlations. Applying this procedure systematically to the gluon-propagator
data, we observe the presence of a pair of complex poles at
$p^{2}_{\mathrm{pole}}=(-0.37\pm 0.05_{\mathrm{stat}}\pm
0.08_{\mathrm{sys}})\pm\,i\,(0.66\pm 0.03_{\mathrm{stat}}\pm
0.02_{\mathrm{sys}})\,\mathrm{GeV}^{2}$, where “stat” represents the
statistical error and “sys” the systematic one. We also find a zero on the
negative real axis of $p^{2}$, at $p^{2}_{\mathrm{zero}}=(-2.9\pm
0.4_{\mathrm{stat}}\pm 0.9_{\mathrm{sys}})\,\mathrm{GeV}^{2}$. We thus note
that our procedure — which is based on a model-independent approach and
includes careful error propagation — confirms the presence of a pair of
complex poles in the gluon propagator, in agreement with previous works. For
the ghost propagator, the Padés indicate the existence of the single pole at
$p^{2}=0$, as expected. We also find evidence of a branch cut on the negative
real axis. Through the use of the so-called D-Log Padé method, which is
designed to approximate functions with cuts, we corroborate the existence of
this cut for the ghost propagator.
## 1 Introduction
Quantum chromodynamics (QCD), which is based on SU(3) Yang-Mills theory,
describes the strong interactions involving quarks and gluons. One of its
fundamental properties is color confinement, which states that no color-
charged particles can be found in isolation. Indeed, up to now, no free quarks
or gluons were observed in nature and hadronic states are all colorless.
Confinement in QCD is related to the fact that the strong coupling,
$\alpha_{s}$, runs with the energy, as predicted by the QCD $\beta$-function;
in particular, in the usual (perturbative) $\overline{\rm{MS}}$ scheme,
$\alpha_{s}$ assumes large values at low momenta, i.e. in the infrared (IR)
region, eventually diverging at the Landau pole. A complete theoretical
explanation of color confinement is, however, still lacking. As a first step
in this direction, one can try to understand the behavior of gluon and ghost
propagators, which are the theory’s fundamental degrees of freedom, in the IR
limit, i.e. going beyond the validity of perturbative QCD.
Several theoretical frameworks were proposed to explain the color confinement
mechanism (in Landau gauge), such as the Gribov-Zwanziger scenario [1, 2] and
the scaling solution of Dyson-Schwinger equations [3], to name only two of
them. Both descriptions — in their original formulations — predicted that the
gluon propagator vanishes at zero momentum and the ghost propagator has an
enhanced singularity, i.e. it is more singular than $1/p^{2}$ in the IR limit.
These properties were verified with lattice numerical simulations, using large
lattice volumes, in 2 space-time dimensions [4, 5]. On the contrary, in 3 and
in 4 dimensions [6, 4, 7, 5], lattice data showed that the ghost propagator,
$G(p^{2})$, is not enhanced, i.e. $G(p^{2}\approx 0)\sim 1/p^{2}$, and that
the gluon propagator does not go to zero at the origin. These results for the
propagators in the IR regime motivated novel theoretical models that are in
accordance with lattice simulations, such as the decoupling solution of Dyson-
Schwinger equations [8, 9], the Refined Gribov-Zwanziger scenario [10, 11, 12]
and the Curci-Ferrari model [13, 14].
Here, we report some of the results recently presented in Ref. [15], to which
we refer for further details. The aim of our work is to study — in a model-
independent way — the analytic structure of the four-dimensional SU(2)
(quenched) Landau-gauge gluon and ghost propagators in the IR regime, by using
rational approximants as fitting functions to the data from Refs. [6, 16, 5].
We note that rational approximants were recently applied by Falcão, Oliveira
and Silva [17, 18] to analyze the Landau-gauge gluon and ghost propagators in
the SU(3) case. We stress, however, that in Refs. [17, 18] the issues related
to error propagation, and in particular the uncertainties arising from the fit
procedure, are not fully discussed. In this work we propagated all the errors,
considering off-diagonal correlations when necessary. As shown in Ref. [15],
this limits considerably the number of parameters that can be determined from
our fits.
## 2 Padé Approximants
The main type of rational approximants that we will employ as fitting
functions are the standard Padé approximants (PAs). We recall that a PA
$P_{N}^{M}(z)$ is a ratio of two polynomials of orders $M$ and $N$ [19, 20],
i.e.,
$P_{N}^{M}(z)=\dfrac{Q_{M}(z)}{R_{N}(z)}=\dfrac{a_{0}+a_{1}\,z+a_{2}\,z^{2}+\dots+a_{M}\,z^{M}}{1+b_{1}\,z+b_{2}\,z^{2}+\dots+b_{N}\,z^{N}}\;,$
(1)
where we employed $b_{0}=1$ without loss of generality. The canonical method
of applying PAs is to approximate a function $f(z)$ whose Taylor series is
known. The Padé coefficients are then obtained through a matching to the
Taylor-series coefficients of $f(z)$ up to order $N+M$, upon the expansion of
the Padé. Thus, the PA will reproduce the first $N+M+1$ Taylor coefficients of
the function $f(z)$.
The most prominent advantage of employing PAs is that they allow for a
systematic and model-independent analysis. Moreover, which is especially
relevant to our work, they can reproduce analytic features of the function
they are approximating, such as poles, residues and zeros. In addition, in
some cases, the use of these approximants is validated by convergence
theorems. The most important one for this work is Pommerenke’s theorem, which
states that PAs of the sequence $P_{N}^{N+k}(z)$, for fixed $k$, applied to a
meromorphic function $f(z)$, converges to $f(z)$ for $N\to\infty$ [19, 20]. On
the other hand, this theorem predicts that the Padé can have extraneous poles,
which cannot be identified as singularities of the original function and move
away when the order of the PA is increased. There are also spurious poles that
can appear close to zeros, which are the defects or Froissart doublets [19,
20, 21] — these defects are of transient nature and should also disappear for
PAs of higher order. It is important to stress that approximants with such
defects can still be employed to study the considered function, away from
these singularities.
In this work, we are going to use the Padé approximants as fitting functions
to data, which departs from the standard Padé theory and is not supported by
convergence theorems. However, experience shows that this approach is rather
powerful and that its application to similar particle-physics problems, such
as the extraction of resonance pole positions, is very successful [22, 23, 24,
25, 26, 27, 28]. We also recall that the gluon and ghost propagators are
expected to have branch cuts in the complex plane and that, even though there
are no general theorems for Padés applied to functions with cuts, it is
observed that the approximants mimic the branch cut of the function by
accumulating interleaved poles and zeros along the cut [19, 20, 21, 29]. A
classic example in this context is the approximation of the function $\log(z)$
by PAs.
## 3 Results
The lattice data fitted in this work have been previously presented in Refs.
[6, 16, 5] where more technical details can be found. (See also Ref. [15].) In
particular, we use the data from a (symmetric) lattice with volume
$V=n^{4}=128^{4}$. The lattice parameter was taken to be $\beta=2.2$, which
leads to a lattice spacing $a$ of approximately $0.210~{}\text{fm}$,
considering $\sigma^{1/2}=0.44~{}\text{GeV}$ for the string tension [30].
Hence, the physical lattice volume is about $(27~{}\text{fm})^{4}$, which can
be essentially considered as infinite volume, and the lowest non-zero physical
momentum allowed is $p_{min}=2\,a^{-1}\sin(\pi/n)\sim 46~{}\text{MeV}$.
The ghost-propagator data are considered in terms of the unimproved lattice
momenta $p^{2}\,=\,\sum_{\mu}\,p_{\mu}^{2}$, where
$p_{\mu}=2\sin(\pi\hat{p}_{\mu}/n)$ and $\hat{p}_{\mu}=0,1,\dots,n-1$. On the
other hand, the gluon propagator $D(p^{2})$ is given in terms [31] of the
improved momenta
$p^{2}\,=\,\sum_{\mu}\,p_{\mu}^{2}\,+\,\frac{1}{12}\,\sum_{\mu}\,p_{\mu}^{4}$,
for the sake of reducing the effects due to rotational-symmetry breaking,
which are stronger at large momenta. Thus, this definition mostly affects the
values of momenta outside the IR limit.
As shown in Ref. [15], the perturbative behavior of the gluon propagator sets
in around $2.0~{}\text{GeV}$; at the same time, the ghost-propagator data are
essentially perturbative for momenta higher than $1.5~{}\text{GeV}$. Hence, in
our work we focus the Padé analysis mostly in the IR region.
### 3.1 Gluon propagator
We start by employing the Padé approximants as fitting functions to the four-
dimensional SU(2) Landau-gauge gluon propagator data. The fit parameters are
obtained through a $\chi^{2}$ minimization, taking into account off-diagonal
correlations when necessary. The fit uncertainties are calculated using four
different methods: Hessian matrix, Monte Carlo error propagation,
$\Delta\chi^{2}$ variation and linear error propagation. We checked that
results obtained with these four methods are in good agreement. The errors
presented in this work were calculated using the Hessian matrix. As for the
fit quality, it is judged by the minimum $\chi^{2}$, divided by the degrees of
freedom (dof), and by the associated $p$-value. We stress that we limited the
number of fit parameters to 7, because for Padés with higher order the
statistical uncertainty grows considerably and the fit is not meaningful
anymore. Also, the fits are performed in the region of
$\sqrt{p^{2}}<2.4~{}\mathrm{GeV}$, and we have verified that the correlation
between the data points is negligible, so that it can be disconsidered in the
calculations.
The following Padé sequences were used to fit the lattice data:
$P_{k}^{k}(p^{2})$, $P_{k}^{k+1}(p^{2})$, $P_{k+1}^{k}(p^{2})$ and
$P_{k+2}^{k}(p^{2})$. The PAs that pass all reliability tests,111See Ref. [15]
for more details. together with the lattice data, are presented in Fig. 1a,
where the white region is the fit window. Note that, except for
$P_{2}^{3}(p^{2})$, all the approximants follow the expected behavior of the
propagator in the ultraviolet region. The behavior at high energies of
$P_{2}^{3}(p^{2})$ can be explained by the fact that this Padé goes as
$a_{4}\,p^{2}$ for large $p^{2}$, with $a_{4}>0$ (albeit small). Due to the
bad behavior of this PA, we did not use it in our final estimates.
The pole position for each approximant is shown in Fig. 1b and one can notice
that the PAs have a consistent pair of complex poles. Our final value is
obtained as follows: the central value is the average of all the Padés
results, the statistical uncertainty is the largest one from the PAs, and the
systematic error is half the maximum spread between results from two different
PAs. Employing this procedure we find that the final position for the complex
poles is
$\displaystyle p_{\rm pole}^{2}=[(-0.37\pm 0.05_{\rm stat}$ $\displaystyle\pm
0.08_{\rm sys})\,\,\pm$ $\displaystyle\pm\,\,i\,(0.66\pm 0.03_{\rm stat}\pm
0.02_{\rm sys})]\,\,\mathrm{GeV}^{2}\;,$ (2)
where the first error is statistical and the second systematic. This final
value, which is indicated in gray in Fig. 1b (with the errors added in
quadrature), is in agreement with other results available in the literature
[17, 18, 5, 32]. Thus, the Padés clearly favor a pair of complex poles, with
an imaginary part inconsistent with zero, for the Landau-gauge gluon
propagator.
(a)
(b)
(c)
Figure 1: Padé approximants fitted to the Landau-gauge gluon-propagator data
and used to determine the final results for poles and zeros. In (a) we show
the comparison of the PAs (used to determine the final results) and the
lattice data; the shaded region is not included in the fits. In (b),
respectively (c), we present the poles, respectively the zeros, of each PA. In
both cases we show our final prediction in gray.
Another noticeable feature in all the PAs of Fig. 1a is the presence of a zero
on the negative real axis of $p^{2}$. Applying the same method used for the
pole, we obtain the result
$p_{\rm zero}^{2}=(-2.9\pm 0.4_{\rm stat}\pm 0.9_{\rm
sys})\,\,\mathrm{GeV}^{2}\;,$ (3)
which is shown in Fig. 1c together with the zeros of the considered
approximants.
### 3.2 Ghost propagator
We now turn to applying the Padé analysis to the Landau-gauge ghost
propagator. The method is mostly the same used for the gluon propagator but,
in this case, the correlation between the data points is significant, reaching
up to 0.75 in non-diagonal entries. It is known that fits to strongly
correlated data are problematic, since the covariance matrix has small
eigenvalues, which makes it hard to invert numerically. What is more, the
small eigenvalues generate huge numbers in the inverse matrix. This engenders
large fluctuations in the $\chi^{2}$ values, which are not statistically
meaningful, leading to biased [33] or unreliable [34] results. In order to
avoid this problem, we will employ the so-called diagonal fits [34]. In this
procedure, only the diagonal covariance matrix is used to determine the fit
parameters; the corresponding fit quality is denoted here as $Q^{2}$. All
correlations are then included in the error propagation according to the
method explained in Ref. [34]. Let us stress that, since the diagonal
covariance matrix is used, the fit quality cannot be judged in absolute terms
by the value of $Q^{2}/\mathrm{dof}$.
By employing Padé approximants to fit the lattice data of the ghost
propagator, it turns out that all the fit parameters of the PAs are extremely
large, of the order of $\mathcal{O}(10^{10})$ or higher. This is caused by a
pole very close to the origin, as can be understood from Eq. (1). We also
verify that this feature persists when considering different Padé
approximants, indicating that this pole is indeed physical. For the sake of
exploring additional analytical structures of the ghost propagator, we then
impose this pole at the origin, through the so-called partial Padé
approximants (PPAs) [19, 20], which, in this case, are expressed as
$\mathbb{P}^{M}_{N}(p^{2})=\frac{Q_{M}(p^{2})}{R_{N}(p^{2})\,p^{2}}\;,$ (4)
where $Q_{M}(p^{2})$ and $R_{N}(p^{2})$ are defined as before.
The fits were performed in the region $\sqrt{p^{2}}\leq 3.12~{}\mathrm{GeV}$
to avoid the appearance of Froissart doublets (in some PPAs) in the considered
range of momenta, which can spoil the extrapolation of the fit results beyond
the fit window. Indeed, as said above, these defects may appear, but the
approximants can still be used away from these singularities. We applied
partial Padés of the sequences $\mathbb{P}_{k}^{k}(p^{2})$,
$\mathbb{P}_{k+1}^{k}(p^{2})$, $\mathbb{P}_{k+2}^{k}(p^{2})$ and
$\mathbb{P}_{k}^{k+1}(p^{2})$ to fit the ghost-propagator data. As before, the
number of parameters of the PPAs that can be used to fit the data is limited
by the quality of the data. In particular, in this case, partial Padés up to
order 8 can be used. The resulting PPAs and the lattice data for the ghost
propagator are shown in Fig. 2a, where the gray region is not considered in
the fit. We note that $\mathbb{P}_{2}^{2}(p^{2})$ also has a pole on the
positive real axis, which is almost cancelled by a near-by zero. This
singularity is a Froissart doublet and is a transient artifact of the PPA.
The Padés also show a pole and a zero on the negative real axis of $p^{2}$,
which suggests a cut for the ghost propagator, considering that, as we already
stressed above, the Padés emulate a cut by accumulating interleaved poles and
zeros. Moreover, even though the uncertainties in the position of the poles
and zeros of PPAs of higher orders are large, it is possible to see that their
central values also present this pattern. This corroborates the existence of a
cut on the negative real axis.
(a)
(b)
(c)
Figure 2: Partial Padé approximants fitted to the Landau-gauge ghost-
propagator lattice data used to determine our final results for poles and
zeros. In (a) we show the PPAs and the lattice data; the shaded region is not
included in the fits. In (b), respectively (c), we present the poles,
respectively the zeros, of each PPA. In both cases, our final values are shown
in gray at the bottom of each plot. We recall that a pole (at about
$p^{2}=-0.30\,\mathrm{GeV}^{2}$) followed by a zero (at about
$p^{2}=-1.0\,\mathrm{GeV}^{2}$) points towards the presence of a branch cut
along the negative real axis of $p^{2}$.
The pole of each partial Padé that passes all reliability tests are shown in
Fig. 2b. From these results we can determine our final estimate, which is
calculated by the same procedure employed in the gluon-propagator analysis.
This yields
$p^{2}_{\mathrm{pole}}=(-0.30\pm 0.05_{\mathrm{stat}}\pm
0.05_{\mathrm{sys}})\,\,\mathrm{GeV}^{2}.$ (5)
In addition, a zero is located at
$p^{2}_{\mathrm{zero}}=(-1.0\pm 0.3_{\mathrm{stat}}\pm
0.4_{\mathrm{sys}})\,\,\mathrm{GeV}^{2}.$ (6)
The zeros of the PPAs together with our final value are shown in Fig. 2c. As
mentioned before, the appearance of interleaved pole and zero can be an
indication of a cut on the ghost propagator.
For the sake of better analyzing the existence of a cut on the negative real
axis, we also used the so-called D-Log Padé approximants [19, 20]. They are a
suitable alternative for functions with branch cuts because, instead of
working with the original function $f(z)$, which has a cut, one tries to
approximate a new function $F(z)$, which only has simple poles, and then
unfold the procedure. In particular, let us assume that the function we are
interested in is given by [19, 20]
$f(z)=A(z)\,\frac{1}{(\mu-z)^{\gamma}}+B(z)$, where $A(z)$ and $B(z)$ have a
simple structure and are analytic at $z=\mu$, and where $\gamma$ can be any
real number. We then construct a new function defined as [19, 20]
$F(z)=\frac{\mathrm{d}}{\mathrm{d}z}\ln{f(z)}\approx\frac{\gamma}{(\mu-z)}$.
Thus, the branch point $\mu$ of $f(z)$ turns into a simple pole in $F(z)$,
whose residue is equal to $\gamma$. Hence, by unfolding the procedure, the
D-Log Padé $\mathrm{Dlog}_{N}^{M}(z)$ of $f(z)$ is given by [19, 20]
$\mathrm{Dlog}_{N}^{M}(z)=f_{\mathrm{norm}}(0)\,\exp{\left\\{\int\mathrm{d}z^{\prime}\,\,\bar{P}_{N}^{M}(z^{\prime})\right\\}}\;,$
(7)
where the Padé $\bar{P}_{N}^{M}(z^{\prime})$ is applied to the function $F(z)$
and the constant $f_{\mathrm{norm}}(0)$ has to be adjusted in order to
reproduce the function at $z=0$, since the constant (zeroth-order) term in the
Taylor expansion of $f(z)$ is lost due to the derivative.
Clearly, in order to apply the D-log Padés to the ghost propagator $G(p^{2})$,
we need lattice data for the function $F(p^{2})$, which is the derivative of
the logarithm of $G(p^{2})$. This can be accomplished by first taking the
logarithm of the data and then calculating the derivative, through finite
differences. However, the standard formulas for the derivative require equally
spaced data points, a property not satisfied by the ghost-propagator data.
Hence, to address this problem, the logarithm of the lattice data has been
linearly interpolated, to generate data points with a fixed separation of
$\Delta p^{2}=0.035~{}\mathrm{GeV}^{2}$; later, the derivative at a given
point is determined through the usual first-order formula. Of course, this
method introduces correlations between the data points, which were thoroughly
calculated and propagated. We note that, after applying this procedure to the
ghost-propagator data, the data points have large uncertainties (in some cases
larger than 100%) with considerable statistical fluctuations.
(a)
(b)
Figure 3: (a) Lattice data for the ghost propagator and the D-Log Padés, built
from the first-order numerical derivative of the logarithm of the data. The
shaded region is excluded from the fit. (b) Comparison of the largest (real
and nonzero) pole position — from four different partial Padés sequences
$\mathbb{P}^{M}_{N}(p^{2})$ — with the branch-point position $p_{c}$ obtained
from the D-Log Padés [see Eq. (8)] shown as the gray band.
For the D-Log Padés, the region chosen for the fit is $\sqrt{p^{2}}\leq
2~{}\mathrm{GeV}$, since the errors and fluctuations are quite large for
higher momenta; moreover, the data around $\sqrt{p^{2}}=2~{}\mathrm{GeV}$ are
already in the perturbative region (see Fig. 10a in Ref. [15]). We first apply
the Padés $\bar{P}_{N}^{M}(p^{2})$ as fitting functions to the prepared data,
again employing the diagonal-fit method due to the large correlation between
the points. Afterwards, Eq. (7) is used to build the D-Log Padés belonging to
the sequences: $\mathrm{Dlog}_{k}^{k}(p^{2})$,
$\mathrm{Dlog}_{k+1}^{k}(p^{2})$, $\mathrm{Dlog}_{k+2}^{k}(p^{2})$ and
$\mathrm{Dlog}_{k}^{k+1}(p^{2})$. Due to the large uncertainties and
fluctuations of the data points, the maximum number of parameters was set to
five. The D-log-Padé results are reported in Fig. 3a, together with the
lattice data (in purple). The fits have been performed considering only the
white region. It is possible to see a good agreement with the data, even
outside the fit window. Moreover, every approximant of Fig. 3a has a
singularity of the type $(p_{c}-p^{2})^{-\gamma}$, where the branch point
$p_{c}$ is always located on the negative real axis, not too far from the pole
determined by the partial Padés [see Eq. (5)], and the multiplicity $\gamma$
is not compatible with one, which indicates that $p_{c}$ is not a simple pole.
Using these results, our final estimate for the branch-point position is
$p_{c}^{2}=(-0.12\pm 0.08_{\rm stat}\pm 0.02_{\rm sys})~{}\mathrm{GeV}^{2}\;.$
(8)
It is important to emphasize that our systematic uncertainty for $p_{c}$ may
be underestimated, due to the fact that the evaluation of the numerical
derivative introduces an additional source of error. This result, together
with the exponent $\gamma$ not being compatible with one, reinforces that the
pole and zero of the PPAs are a manifestation of a cut on the negative real
axis.
Finally, let us compare the branch point from the D-log Padés with the pole
prediction of the partial Padés. We recall that, if the cut at negative
$p^{2}$ does exist, the largest (negative) pole of the PPAs should indicate
the position of the branch point. In Fig. 3b the gray band corresponds to the
(above) branch point predicted by the D-log PAs; we also show the largest non-
zero pole position for different PPAs. As one can notice, the central values
of the pole are in better agreement with $p_{c}$ for higher-order PPAs. Also,
the final results obtained for the branch point, Eq. (8), and for the pole
extracted from the PPAs, Eq. (5), are compatible within $1.7~{}\sigma$. Thus,
these results are in good agreement and corroborate the existence of a cut on
the negative real axis for the ghost propagator.
## 4 Conclusions
In this work we applied a systematic and model-independent method to study the
analytic structure of the IR Landau-gauge gluon and ghost propagators, by
using rational approximants. In particular, Padé approximants, partial Padé
approximants and also D-log Padé approximants were employed as fitting
functions to four-dimensional SU(2) lattice data [6, 16, 5] of both
propagators. We stress that Refs. [17, 18] also applied Padé approximants to
fit SU(3) lattice propagators and our main conclusions are in agreement with
their results. Notice, however, that in our work the errors were carefully
propagated, considering all the correlations. In addition, we also estimated
the systematic uncertainty of the employed method and explored other types of
approximants: the partial Padés and the D-Log Padés.
For the gluon propagator, the Padé approximants presented evidence of a pair
of complex poles located at $p^{2}_{\mathrm{pole}}=[(-0.37\pm 0.09)\pm
i\,(0.66\pm 0.04)]\,\,\mathrm{GeV}^{2}$, where the errors are added in
quadrature. The PAs also indicated a zero on the negative real axis at
$p^{2}_{\rm zero}=(-2.9\pm 1.0)~{}\text{GeV}^{2}$.
In the ghost-propagator case, the Padés clearly show the existence of a pole
at the origin. We then imposed this information by using the so-called partial
Padés to fit the ghost-propagator data. These fits indicated the presence of a
pole, followed by a zero, on the negative real axis. Higher-order PPAs also
had additional interleaved poles and zeros along the negative real axis,
further suggesting the existence of a cut. In order to investigate this
result, we employed D-Log Padés and found a cut with branch point at
$p_{c}^{2}=(-0.12\pm 0.08)~{}\text{GeV}^{2}$. This value is compatible with
the pole obtained using partial Padés, which is another corroboration for the
existence of a cut on the negative real axis for the ghost propagator.
## 5 Acknowledgements
DB’s work was supported by the São Paulo Research Foundation (FAPESP) grant
No. 2021/06756-6 and CNPq grant No. 308979/2021-4. AC and TM acknowledge
partial support from FAPESP and CNPq. The work of CYL was financed by FAPESP
grants No. 2020/15532-1 and No. 2022/02328-2 and CNPq grant No. 140497/2021-8.
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# Kekule spin-orbit dimer phase and triplon dynamics
GiBaik Sim Department of Physics TQM, Technische Universität München, $\&$
James-Franck-Straße 1, D-85748 Garching, Germany Munich Center for Quantum
Science and Technology (MCQST), 80799 Munich, Germany
###### Abstract
We derive and study a spin-orbital model for ions with $d^{1}$ electronic
configuration on a honeycomb lattice. In this system, the directional
character of $t_{2g}$ orbital leads to extensively degenerate dimerized ground
states. We find that additional interactions from charge transfer processes
completely lift the degeneracy and stabilize the Kekule spin-orbit dimerized
phase where dimers form a kagome superlattice. For such phase, the triplon
band spectrum resembles the electronic band structure of the kagome lattice
and becomes topologically non-trivial in the presence of inter-dimer
Dzyaloshinskii-Moriya interactions. As an experimental verification of the
Kekule dimerized phase, we propose the thermal Hall experiment, which can
directly uncover the topological profile of the corresponding triplon band
spectrum.
Isotropic spin models on two (three) dimensional lattice often have long-range
ordered ground states with gapless collective excitation modes [1]. However,
this scenario fails for the geometrically frustrated cases whose classical
ground states are (sub-)extensively degenerate [2, 3, 4]. For such cases, the
ground state manifold responds to the following effects in various ways: It
can become a highly entangled quantum spin liquid state with quantum
fluctuations [5, 6], lift the degeneracy through the order by disorder
mechanism [7, 8], which often selects less entangled magnetically ordered
state, or encounter structural phase transition [9, 10], which reduces the
symmetry of the lattice.
The concept of classical degeneracy can be applied to a spin-orbital model
where the ground state manifold is formed by an extensive number of dimer
coverings [11]. Here, the degeneracy originates from the directional character
of $t_{2g}$ orbitals, which frustrate the spin interactions even on bipartite
lattices. Recently, the spin-orbital model has been applied to many materials
with dimerized ground state such as $\alpha-$RuCl3 in the presence of high
pressure [12], Ba2YMoO6 with magnetically active Mo5+ ($d^{1}$) ions on fcc
lattice [13, 14], and Li2RuO3 with Ru4+ ($d^{2}$) ions on honeycomb lattice
[15]. In fact, Li2RuO3 shows a specific crystallized dimer ground state with a
herringbone pattern which is well captured by the presence of magnetoelastic
coupling [16].
In this work, we point out an electronic degeneracy lifting mechanism that can
be relevant to $\alpha-$MX3 with M = Ti, Zr, Hf and X = F, Cl, Br where cation
ions (M) with $d^{1}$ electronic configuration occupy a honeycomb lattice
[17]. By introducing charge transfer contributions, we show that the
degeneracy of dimer covering is completely lifted and the system stabilizes
Kekule spin-orbit dimerized phase. To investigate the dynamics of the phase,
we introduce the triplon operator and investigate the corresponding triplon
band spectrum, which turns out to be topologically non-trivial even with
infinitesimal Dzyaloshinskii-Moriya interactions (DMI). We also investigate
the thermal response and find that the topological properties are directly
encoded in the thermal Hall conductivity.
Figure 1: (color online) Possible hopping processes in $\alpha-$MX3 which
includes direct hopping (green), charge transfer (blue), and cyclic exchange
(red) contribution. The magnetically active M sites with $d^{1}$ electronic
configuration are located at the center of octahedral cage. Each cage is
formed by X sites with $p^{6}$ electronic configuration. Indirect hopping
paths between M and X sites are marked as dotted curved lines.
We first introduce $\alpha-$MX3, a two dimensional edge-shared octahedral
system with two types of atoms. The honeycomb geometry is formed by
magnetically active cation ions (M) with $d^{1}$ electronic configuration.
Each M site is surrounded by an edge-shared octahedral cage, which is formed
by anion ions (X) with $p^{6}$ electronic configuration, as shown in Fig. 1.
We focus on a case with strong crystal field splitting where one electron
resides in threefold degenerate $t_{2g}$ orbitals at each M site. In the
$\alpha-$MX3 crystal structure, three distinct nearest-neighbor bonds are
formed in $yz$, $zx$, and $xy$ planes, respectively. We label such bonds and
the $t_{2g}$ orbitals with an axis $\gamma~{}(=a,b,c)$ normal to its plane:
$a$ indicates $yz$. We first consider the leading hopping process between
neighboring $t_{2g}$ orbitals due to direct hybridization which is illustrated
as a green bond in Fig. 1. In $\gamma$ plane, the hopping occurs between
$t_{2g}$ orbitals of same $\gamma$ type. The corresponding low energy
Hamiltonian for such process reads as the following form in the limit of zero
Hund’s coupling [11, 18].
$\displaystyle H_{d}\\!$ $\displaystyle=$ $\displaystyle\\!J_{d}\sum_{\langle
ij\rangle\parallel\gamma}\Big{(}\vec{S}_{i}\\!\cdot\\!\vec{S}_{j}+\frac{1}{4}\Big{)}n_{i}^{\gamma}n_{j}^{\gamma}$
(1)
where $\langle ij\rangle\\!\\!\parallel\\!\\!\gamma$ indicates $\gamma$ type
bond. Here, $\vec{S}_{i}$ are spin-$\frac{1}{2}$ operators, $n_{i}^{\gamma}$
is the number operator for orbital $\gamma$, and $J_{d}=4t_{d}^{2}/U$ where
$t_{d}$ is the amplitude of direct hopping and $U$ is the Coulomb repulsion at
a M site.
The antiferromagnetic term $H_{d}$ acts on the $\gamma$ type bonds only when
both sites involved are occupied by an electron with $\gamma$ orbital.
Accordingly, the antiferromagnetic bonds form non-intersecting linear chains
with specific orbital patterns. As the length of the chain grows, a negative
energy gain from the antiferromagnetic spin interactions increases. At the
same time, each bond gains a positive energy, $\frac{1}{4}$, from the second
term in Eq. 1. Under these two conditions, the minimum energy of the system is
achieved when all chains are dimers as described in Ref. [11]. Therefore, a
honeycomb lattice covered with hard-core dimers corresponds to an exact ground
state of $H_{d}$. Within a dimer, two electrons occupy the same orbital, which
matches the direction of the bond, and form a spin-singlet. The $H_{d}$ term
is inactive on every inter-dimer bond and thus the ground state manifold
becomes extensively degenerate: there are infinitely many different ways to
cover the lattice with dimers. In the following, we consider an additional
hopping process which results in inter-dimer interactions that completely lift
the degeneracy and stabilize a valence bond crystal state.
In addition to direct hopping between neighboring $t_{2g}$ orbitals, there
exist different exchange mechanisms: charge transfer and cyclic exchange
processes [19]. We first consider $pd$ charge transfer process,
$d^{1}_{i}\\!-\\!p^{6}\\!-\\!d^{1}_{j}\rightarrow
d^{2}_{i}\\!-\\!p^{4}\\!-\\!d^{2}_{j}$, where two holes are created on an X
site, which connects two neighboring M sites as illustrated in Fig. 1 with a
blue bond. The relative energy of a virtual state is $2U+2\Delta-9U_{p}$ where
$\Delta$ is $p\\!-\\!d$ charge transfer gap and $U_{p}$ is Coulomb repulsion
at an X site. Collecting possible charge transfer contributions, we obtain the
spin-orbital Hamiltonian
$\displaystyle H_{ct}\\!=\\!J_{ct}\sum_{\langle
ij\rangle\parallel\gamma}\\!\\!\Big{(}\vec{S}_{i}\\!\cdot\\!\vec{S}_{j}-\frac{1}{4}\Big{)}O^{\gamma}.$
(2)
where $O^{c}\\!=\\!n_{i}^{a}n_{j}^{b}\\!+\\!n_{i}^{b}n_{j}^{a}$ for a $c$ type
bond (See Section I of SM for details). Here,
$J_{ct}=2t_{s}^{2}/(2U+2\Delta-9U_{p})$ and $t_{s}$ is the amplitude of
hopping, which takes place via intermediate $p$ orbitals. Unlike the charge
transfer process, the cyclic exchange process,
$d^{1}\\!-\\!(p^{6},p^{6})\\!-\\!d^{1}\rightarrow
d^{2}\\!-\\!(p^{5},p^{5})\\!-\\!d^{2}$, involves two X sites where a hole is
created at each X site in the virtual state as shown in Fig. 1 with a red
bond. The cyclic exchange Hamiltonian is expressed as
$\displaystyle H_{ce}\\!=\\!-J_{ce}\sum_{\langle
ij\rangle\parallel\gamma}\Big{(}\vec{S}_{i}\\!\cdot\\!\vec{S}_{j}+\frac{1}{4}\Big{)}\tilde{O}^{\gamma}$
(3)
where
$\tilde{O}^{c}=\Big{(}d^{\dagger,b}_{i}d_{i}^{a}d^{\dagger,b}_{j}d_{j}^{a}+d^{\dagger,a}_{i}d_{i}^{b}d^{\dagger,a}_{j}d_{j}^{b}\Big{)}$
for a $c$ type bond and $J_{ce}=2t_{s}^{2}/(2U+2\Delta-10U_{p})$.
Figure 2: (color online) (a,b) Two different types of inter-dimer bonds,
$b_{1}$ and $b_{2}$, which are shown as green dashed lines. Each spin-orbital
dimer is represented as a thick line and is colored according to the orbital
it occupies. $b_{1}$ couple two dimers with the same orbital. Meanwhile, the
bond $b_{2}$ couple dimers with different orbitals. (c) Kekule dimerized
pattern which shows the modification of particular bonds of the honeycomb
lattice. (d) Plot of the per site energy $E_{0}$, spin correlation
$\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, and electron density, $\langle
n_{i}^{a}\rangle$ and $\langle n_{j}^{a}\rangle$ for the ground state on YC-6
cylinder. Here, $i$ and $j$ are nearest neighboring sites and connected
through $yz$ type bond as shown in (c).
$H_{ct}$ generates interactions between dimers and, as it follows, selects a
particular superstructure of dimers, a Kekule dimerization pattern. We assume
that the charge transfer term is small and treat $H_{ct}$ perturbatively.
Figs. 2(a) and 2(b) contain two different green dotted inter-dimer bonds,
$b_{1}$ and $b_{2}$, which can be formed in the ground state of $H_{d}$.
$H_{ct}$ is active only on inter-dimer bonds whose two associated dimers have
different orbital indices as $b_{2}$ shown in Fig. 2(b). At the same time,
such bonds gain negative energy, $-\frac{1}{4}$, from the last term in Eq.
(2). Accordingly, the system prefers a specific type of dimer orientation,
resulting in Kekule dimerized ground state as shown in Fig. 2(c).
To check the stability of the Kekule dimerized state, we have performed
infinite density matrix renormalization group (iDMRG) simulation for the
Hamiltonian, $H_{t}\equiv H_{d}+H_{ct}+H_{ce}$ with $J_{d}\\!=\\!1$ and
$\tilde{J}\\!\equiv\\!J_{ct}\\!=\\!2J_{ce}/3$, on YC-2$L_{y}$ cylinders with
circumference $L_{y}\\!=\\!2$ and 3, where the periodic boundary condition
(PBC) is applied along the $y$ direction (See Section II of SM for details).
To find a ground state for each $\tilde{J}$, we compared the per site ground
state energy of YC-4 cylinder with the one of YC-6 cylinder. We find that the
latter is energetically more stable at least up to
$\tilde{J}\\!\approx\\!1/2$. In Fig. 2(d), we plot the per site energy
$E_{0}$, nearest-neighbor spin correlation
$\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, and electron density in the $yz$
orbital, $\langle n_{i}^{a}\rangle$ and $\langle n_{j}^{a}\rangle$, for a
given dimerized $ij$ bond, which is $yz$ type. It indicates the stabilization
of spin-orbit Kekule dimer state and small hybridization of spin-singlet and
spin-triplet state within the dimer. Note that the geometry of YC-4 cylinder
cannot support the Kekule dimerized state. We also performed exact
diagonalization (ED) on a cluster of six sites using PBC. The result of ED
also shows stabilization of the Kekule dimer state at least up to
$\tilde{J}\\!\approx\\!1/2$ (See Section III of SM for details). Below, we
focus on a regime where direct hopping contributions dominate the system,
$\tilde{J}/J_{d}=1/8$.
Having realized the crystallized dimer ground state, we now find the
excitation spectrum by introducing auxiliary bosons on each dimer [20, 21,
22]. We define the operator $s^{\dagger,a}$ which creates the spin-singlet
state
$|s\rangle=(|\uparrow_{L}\downarrow_{R}\rangle-|\downarrow_{L}\uparrow_{R}\rangle)/\sqrt{2}$
and $t^{\dagger,a}_{1}$, $t^{\dagger,a}_{0}$, and $t^{\dagger,a}_{-1}$ which
create the spin-triplet states
$|t_{1}\rangle=|\uparrow_{L}\uparrow_{R}\rangle$,
$|t_{0}\rangle=(|\uparrow_{L}\downarrow_{R}\rangle+|\downarrow_{L}\uparrow_{R}\rangle)/\sqrt{2}$,
and $|t_{-1}\rangle=|\downarrow_{L}\downarrow_{R}\rangle$, respectively, where
$a$ labels the orbital index of dimer. Then, spin operators, which act on an
electron with orbital $a$, read:
$\displaystyle S^{+}_{i}n_{i}^{a}$ $\displaystyle=$
$\displaystyle\frac{t^{\dagger,a}_{1,i}t^{a}_{0,i}+t^{\dagger,a}_{0,i}t^{a}_{-1,i}}{\sqrt{2}}\pm\frac{s^{\dagger,a}_{i}t^{a}_{-1,i}-t^{\dagger,a}_{1,i}s^{a}_{i}}{\sqrt{2}}\;,$
$\displaystyle S^{-}_{i}n_{i}^{a}$ $\displaystyle=$
$\displaystyle\frac{t^{\dagger,a}_{-1,i}t^{a}_{0,i}+t^{\dagger,a}_{0,i}t^{a}_{1,i}}{\sqrt{2}}\mp\frac{s^{\dagger,a}_{i}t^{a}_{1,i}-t^{\dagger,a}_{-1,i}s^{a}_{i}}{\sqrt{2}}\;,$
$\displaystyle S^{z}_{i}n_{i}^{a}$ $\displaystyle=$
$\displaystyle\frac{t^{\dagger,a}_{1,i}t^{a}_{1,i}-t^{\dagger,a}_{-1,i}t^{a}_{-1,i}}{2}\pm\frac{s^{\dagger,a}_{i}t^{a}_{0,i}+t^{\dagger,a}_{0,i}s^{a}_{i}}{2}$
(4)
where $i$ indicates the position of the dimer and the upper (lower) sign is
for the left (right) spin within the dimer.
Kekule state is a product of spin-singlet at each dimer, and thus, the density
of triplon is zero. So the mean-field theory, which abandons dynamics of
singlet operators, is applicable. By replacing these operators with
expectation value, $\langle s^{\dagger}_{i}\rangle=\langle s_{i}\rangle=1$,
and neglecting high order terms, $H_{t}$ is rewritten in terms of triplon
operators in momentum space as
$\displaystyle\mathcal{H}_{\mathbf{k}}=\sum_{\mathbf{k}}\left(\\!\\!\begin{array}[]{c}\mathbf{t}_{1,{\bf
k}}^{\dagger}\\\ \mathbf{t}_{1,-\\!{\bf k}}^{\phantom{\dagger}}\\\
\mathbf{t}_{0,{\bf k}}^{\dagger}\\\ \mathbf{t}_{0,-\\!{\bf
k}}^{\phantom{\dagger}}\end{array}\\!\right)\\!\\!\left(\\!\\!\begin{array}[]{cccc}M_{1,\mathbf{k}}&N_{1,\mathbf{k}}&\mathbf{0}&\mathbf{0}\\\
N_{1,\mathbf{k}}&M_{1,\mathbf{k}}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&M_{0,\mathbf{k}}&N_{0,\mathbf{k}}\\\
\mathbf{0}&\mathbf{0}&N_{0,\mathbf{k}}&M_{0,\mathbf{k}}\end{array}\\!\\!\right)\\!\left(\\!\\!\begin{array}[]{c}\mathbf{t}_{1,{\bf
k}}^{\phantom{\dagger}}\\\ \mathbf{t}_{1,-\\!{\bf k}}^{\dagger}\\\
\mathbf{t}_{0,{\bf k}}^{\phantom{\dagger}}\\\ \mathbf{t}_{0,-\\!{\bf
k}}^{\dagger}\end{array}\\!\right)$ (17) (18)
where $\mathbf{t}_{1,{\bf k}}^{\dagger}=(t_{1,{\bf k}}^{\dagger,a},t_{{1,{\bf
k}}}^{\dagger,b},t_{{1,{\bf k}}}^{\dagger,c},t_{{-1,{\bf
k}}}^{\dagger,a},t_{{-1,{\bf k}}}^{\dagger,b},t_{{-1,{\bf k}}}^{\dagger,c})$
and $\mathbf{t}_{0,{\bf k}}^{\dagger}=(t_{{0,{\bf k}}}^{\dagger,a},t_{{0,{\bf
k}}}^{\dagger,b},t_{{0,{\bf k}}}^{\dagger,c})$. The Bogoliubov–de Gennes
Hamiltonian, $\mathcal{H}_{\mathbf{k}}$, is separated into two blocks which
make manifest the absence of terms mixing spinful $|t_{\pm 1}\rangle$ triplon
and spinless $|t_{0}\rangle$ triplon. $\mathcal{H}_{\mathbf{k}}$ can be
conveniently expressed by using 8 Gell-Mann matrices, $\lambda_{i}$, as the
basis for orbital indices $a,b$, and $c$ (See Section IV of SM for the Gell-
Mann matrix). The hopping matrices are written as
$\displaystyle M_{1,\mathbf{k}}$ $\displaystyle\\!=\\!$ $\displaystyle
J_{d}[I_{2}\\!\otimes\\!I_{3}]+J_{ct}\big{[}\cos\frac{\bm{\delta}_{1}\cdot\mathbf{k}}{2}I_{2}\\!\otimes\\!\lambda_{4}$
$\displaystyle+$
$\displaystyle\cos\frac{\bm{\delta}_{2}\cdot\mathbf{k}}{2}I_{2}\\!\otimes\\!\lambda_{1}+\cos\frac{\bm{\delta}_{3}\cdot\mathbf{k}}{2}I_{2}\\!\otimes\\!\lambda_{6}\big{]},$
$\displaystyle M_{0,\mathbf{k}}$ $\displaystyle\\!=\\!$ $\displaystyle
J_{d}[I_{3}]+J_{ct}[\cos\frac{\bm{\delta}_{1}\cdot\mathbf{k}}{2}\lambda_{4}$
(19) $\displaystyle+$
$\displaystyle\cos\frac{\bm{\delta}_{2}\cdot\mathbf{k}}{2}\lambda_{1}+\cos\frac{\bm{\delta}_{3}\cdot\mathbf{k}}{2}\lambda_{6}]$
where $\bm{\delta}_{1}$ and $\bm{\delta}_{2}$ are two primitive vectors of the
kagome superlattice and
$\bm{\delta}_{3}\\!=\\!-\bm{\delta}_{1}-\bm{\delta}_{2}$. The pairing matrices
are given as
$\displaystyle N_{1,\mathbf{k}}$ $\displaystyle=$ $\displaystyle-
M_{1,\mathbf{k}}+J_{d}\left[I_{2}\\!\otimes\\!I_{3}\right],$ $\displaystyle
N_{0,\mathbf{k}}$ $\displaystyle=$ $\displaystyle
M_{0,\mathbf{k}}-J_{d}\left[I_{3}\right].$ (20)
The dispersion can be obtained by Bogoliubov transformation and the resulting
band structure with nine triplon bands is shown in Fig. 3(a). As a result of
SU(2) spin rotation symmetry, three bands are all three-fold degenerate. At
the same time, discrete lattice symmetries give rise to the emergence of
quadratic band touching at $\Gamma$ point and linear band crossings at $K$
point resembling the electronic band structure of kagome lattice [23].
Figure 3: (color online) (a) Complete ($|t_{\pm 1}\rangle$ and
$|t_{0}\rangle$) triplon band spectrum of Kekule dimerized state without DMI
for $J_{d}=3$meV. Because of SU(2) spin rotation symmetry, each band is triply
degenerate. (b) The band structure of $|t_{1}\rangle$ triplon with $D=0.1$meV.
The out–of–plane DMI induce gap and topological property to the band
structure. The sign of Chern number, $C$, for each band is opposite for
$|t_{-1}\rangle$ triplon. (c,d) Complete triplon band spectrum for
$J_{d}\\!=\\!3$meV, $D\\!=\\!0.1$meV, and $h_{z}\\!=\\!\pm 1$T. The sign of
the Chern number for the lowest-dotted band depends on the sign of the
external magnetic field $h_{z}$. (e) Thermal Hall conductivity $\kappa_{xy}$
versus external magnetic field $h_{z}$ at three distinct temperatures. In the
calculation, we include very small in-plane DMI to gap out band touching
points so that Berry curvature $F^{(n)}_{xy}(\mathbf{k})$ is well-defined.
To investigate the effect of anisotropic terms on the triplon band structure,
we include small inter-dimer Dzyaloshinskii-Moriya interactions (DMI),
$\sum_{\langle
ij\rangle\parallel\gamma}D(S^{x}_{i}S^{y}_{j}-S^{y}_{i}S^{x}_{j})O^{\gamma}$,
which are allowed by the lattice symmetry. The effect of DMI can be encoded by
including
$M_{1,\mathbf{k}}^{D}\\!=\\!D[\cos\frac{\bm{\delta}_{1}\cdot\mathbf{k}}{2}S_{z}\\!\otimes\\!\lambda_{5}+\cos\frac{\bm{\delta}_{2}\cdot\mathbf{k}}{2}S_{z}\\!\otimes\\!\lambda_{2}+\cos\frac{\bm{\delta}_{3}\cdot\mathbf{k}}{2}S_{z}\\!\otimes\\!\lambda_{7}]$
and $N_{1,\mathbf{k}}^{D}\\!=\\!-M_{1,\mathbf{k}}^{D}$ to the spinful hopping
matrix $M_{1,\mathbf{k}}$ and pairing matrix $N_{1,\mathbf{k}}$ in Eq. (18),
respectively. The intra-dimer DMI are forbidden since the center of each
intra-dimer bond is an inversion center. The out–of–plane DMI still preserve
the U(1) spin rotation symmetry and the excitation modes of different triplon,
$|t_{1}\rangle$, $|t_{-1}\rangle$, and $|t_{0}\rangle$, completely decouple
from each other. At the same time, it gap out all the band touching points in
$|t_{\pm 1}\rangle$ triplon band structure separately and leave non-trivial
topological bands. In Fig. 3(b), we plot $|t_{1}\rangle$ triplon band spectrum
in the presence of DMI where the bottom (top) band carries the finite Chern
number 1 (-1). Although the band spectrum of $|t_{-1}\rangle$ triplon is
exactly the same as the one of $|t_{1}\rangle$ tripon, it contains opposite
topological property: bottom (top) band carries the Chern number -1 (1). As
pointed out in Ref.[24], such an opposite pattern could lead to the triplon
analog of $Z_{2}$ spin-hall insulator state with the corresponding helical
edge states at the boundary of the system. However, the $Z_{2}$ topological
phase is destroyed by infinitesimal inter-dimer in-plane DMI, which break the
U(1) and consequently pseudo time-reversal symmetry. The pseudo time-reversal
operator is defined as U(1)$\times\mathcal{T}$ with physical time reversal
operator $\mathcal{T}$. The in-plane DMI is symmetrically allowed in
$\alpha-$MX3 where a kagome plane is no longer a mirror plane since the
octahedral cage surrounding M ion is tilted [25]. Below, we neglect the in-
plane DMI and keep U(1) symmetry intact.
The external magnetic field gives a knob to induce topological transitions
through the Zeeman term, $\sum_{i,a}g_{z}\mu_{B}h^{z}S^{z}_{i}n_{i}^{a}$.
Here, $g_{z}\\!\approx\\!2$ is the Landé $g$-factor, $\mu_{B}$ the Bohr
magneton, and $h_{z}$ an applied magnetic field perpendicular to the plane. In
Figs. 3(c) and 3(d), we plot the complete triplon band spectrum with finite
DMI at two different fields. With $h_{z}\\!>\\!0$, the well-separated lowest
band is associated to $|t_{-1}\rangle$ triplon and carries the Chern number 1.
On the other hand, the lowest band is formed by $|t_{1}\rangle$ triplon with
the Chern number -1 when $h_{z}\\!<\\!0$.
The Chern number in electronic systems is directly connected to physical
observable quantities such as quantized Hall conductance. In bosonic systems
where it is not possible, the thermal Hall effect gives an alternative [26,
27]. It depends on the Berry curvature in which the bosons are thermally
populated, rather than solely depending on the Chern number. In our case,
triplons carry a thermal current perpendicular to the temperature gradient
when the temperature is above the triplon gap. The thermal Hall conductivity
of triplons is written as
$\displaystyle\kappa_{xy}=-\frac{k_{\rm B}^{2}T}{\hbar
L}\sum_{\mathbf{k},n}c_{2}[\rho(\omega_{n,\mathbf{k}})]F^{(n)}_{xy}(\mathbf{k})$
(21)
where $F^{(n)}_{xy}(\mathbf{k})$ is the Berry curvature of $n$th band at
$\mathbf{k}$, $c_{2}(\rho)\\!=\\!\int_{0}^{\rho}\ln^{2}(1+t^{-1})dt$, and
$\rho(\omega_{n,\mathbf{k}})\\!=\\!(\exp(\omega_{n,\mathbf{k}}/k_{\rm
B}T)-1)^{-1}$ is the Bose distribution function. Since $\alpha-$MX3 is quasi
two-dimensional material and the thermal conductivity of each layer is in the
same direction, we sum the contribution of each layer to calculate the
conductivity for a three-dimensional sample with a single layer thickness
$L=0.5$nm [28]. Fig. 3(e) shows the thermal Hall conductivity $\kappa_{xy}$ as
a function of the magnetic field $h_{z}$. As $h_{z}\\!>\\!0$ is turned on,
$\kappa_{xy}$ starts to increase. The increase originates from the fact that
the well-seperated lowest band, which is formed by $|t_{-1}\rangle$ triplon
and most thermally populated, carries the Chern number -1 and dominates the
Hall response. On the contrary, $\kappa_{xy}$ decreases when the field
$h_{z}<0$ is included which can be interpreted by the dominance of the lowest
$|t_{1}\rangle$ triplon band with the Chern number 1. Such an opposite
response can be a direct signature of Kekule dimerized phase in $\alpha-$MX3.
In our calculation, we focus on a low temperature regime where triplon bands
are weakly populated and neglect interaction effects.
In the present work, we demonstrate that the neglected charge transfer
contribution crystalizes the system into Kekule spin-orbit dimerized state. To
check the stability, we perform iDMRG simulation which indicates a small
hybridization of spin-singlet and triplet within each dimer. We then introduce
triplon operators and investigate the triplon band structure which is
topologically non-trivial in the presence of inter-dimer DMI. For the
experimental verification of Kekule dimer phase, we calculate the thermal Hall
conductivity, which directly encodes the topological profile of the triplon
band spectrum.
As a future direction, it would be desirable to apply the degeneracy lifting
mechanism to many valence bond crystals with $d^{1}$ ions which include
MgTi2O4 with Ti3+ on a pyrochlore lattice [29]. An essential question regards
then whether it can verify the experimentally observed helical dimerization
pattern [30]. Regarding the fact that $d^{5}$ ions give the exactly same form
of spin-orbital Hamiltonian, it would be interesting to study the effect of
charge-transfer processes for such systems. Another challenging but
interesting avenue is to take into account the triplon interaction effects
which might stabilize the topological gap [31] or induce the emergence of
triplon bound states [32].
## I Acknowledgments
The author thanks S. B. Lee, M. J. Park, H. Kee, K. Penc, J. Romhányi, and J.
Knolle for stimulating conversations, and the latter also for detailed
comments on the experimental identification. G.B.S. is funded by the European
Research Council (ERC) under the European Unions Horizon 2020 research and
innovation program (grant agreement No. 771537). The research is part of the
Munich Quantum Valley, which is supported by the Bavarian state government
with funds from the Hightech Agenda Bayern Plus. Tensor network calculations
were performed using the TeNPy Library [33].
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## II Supplementary Material
### II.1 I. Derivation of spin-orbital model for charge-transfer processes
Here, we consider the charge-transfer processes of the type
$d^{1}_{i}\\!-\\!p^{6}\\!-\\!d^{1}_{j}\rightarrow
d^{2}_{i}\\!-\\!p^{4}\\!-\\!d^{2}_{j}$ where two holes are created on an anion
ion connecting nearest-neighbor cation ions $i$ and $j$. The virtual-state
energy is $E_{v}\equiv 2U+2\Delta-9U_{p}$. Here we give details about the
derivation for bonds on the $xy$ planes for concreteness. There are four
possible electron hopping processes for this case :
$\displaystyle d_{yz}^{1}\\!-\\!p_{z}^{2}\\!-\\!d_{zx}^{1}\rightarrow
d_{yz}^{2}\\!-\\!p_{z}^{0}\\!-\\!d_{zx}^{2},\text{ as given in
Fig.~{}\ref{fig:ct3}}$ (S1) $\displaystyle
d_{yz}^{1}\\!-\\!p_{z}^{2}p_{y}^{2}\\!-\\!d_{xy}^{1}\rightarrow
d_{yz}^{2}\\!-\\!p_{z}^{1}p_{y}^{1}\\!-\\!d_{xy}^{2},$ (S2) $\displaystyle
d_{xy}^{1}\\!-\\!p_{x}^{2}p_{z}^{2}\\!-\\!d_{zx}^{1}\rightarrow
d_{xy}^{2}\\!-\\!p_{z}^{1}p_{x}^{1}\\!-\\!d_{zx}^{2},$ (S3) $\displaystyle
d_{xy}^{1}\\!-\\!p_{x}^{2}p_{y}^{2}\\!-\\!d_{xy}^{1}\rightarrow
d_{xy}^{2}\\!-\\!p_{x}^{1}p_{y}^{1}\\!-\\!d_{xy}^{2}.$ (S4)
Figure S1: A 90∘ Mi-X-Mj geometry where magnetic ions Mi and Mj interact via
an ion X. Here, two holes with opposite spins arrive at the same $p_{z}$
orbital in an ion X.
First, we focus on a case with Eq. (S1) and give details about the derivation
of effective low-energy spin-orbital Hamiltonian within the subspace where an
initial state occupies an electron at site $i(j)$ with $yz(zx)$ orbital. The
basis for the subspace is written as
$\displaystyle
S_{yz,zx}=\\{|yz_{\uparrow},zx_{\uparrow}\rangle,|yz_{\uparrow},zx_{\downarrow}\rangle,|yz_{\downarrow},zx_{\uparrow}\rangle,|yz_{\downarrow},zx_{\downarrow}\rangle\\}.$
(S5)
There are two possible hopping processes: the spin-conserving process and the
spin-flipping process. The contribution from the first one can be represented
in $S_{yz,zx}$ subspace as
$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{t_{s}^{2}}{E_{v}}\left(\\!\begin{array}[]{cccc}0&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&0\\\ \end{array}\\!\right).$ (S10)
On the other hand, the spin-flipping process can be represented in $S_{yz,zx}$
subspace as
$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{t_{s}^{2}}{E_{v}}\left(\\!\begin{array}[]{cccc}0&0&0&0\\\
0&0&-1&0\\\ 0&-1&0&0\\\ 0&0&0&0\\\ \end{array}\\!\right).$ (S15)
Here, initial states $|yz_{\uparrow},zx_{\uparrow}\rangle$ and
$|yz_{\downarrow},zx_{\downarrow}\rangle$ cannot give any contribution since
two holes with same spins cannot occupy the same orbital at an anion ion.
Next, we focus on a case where an initial state occupies an electron at site
$i(j)$ with $xy(xy)$ orbital. The basis for such subspace is written as
$\displaystyle
S_{xy,xy}=\\{|xy_{\uparrow},xy_{\uparrow}\rangle,|xy_{\uparrow},xy_{\downarrow}\rangle,|xy_{\downarrow},xy_{\uparrow}\rangle,|xy_{\downarrow},xy_{\downarrow}\rangle\\}.$
(S16)
Here, the spin-flipping process is not possible and the contribution can be
represented in $S_{xy,xy}$ subspace as
$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,xy}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S17)
The contributions associated to Eq. (S1) in other subspaces are written as
$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,yz}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,xy}=-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,yz}=-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,yz}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S18)
Now we consider terms associated with Eq. (S2). In each subspace, the
contribution is given as
$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,zx}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,xy}$ $\displaystyle=$
$\displaystyle-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,yz}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,zx}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S19)
The contribution from Eq. (S3) in each subspace is given as
$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,yz}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,zx}=-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,xy}$ $\displaystyle=$
$\displaystyle-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,xy}$ $\displaystyle=$
$\displaystyle-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S20)
The contribution from Eq. (S4) in each subspace is given as
$\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,xy}$ $\displaystyle=$
$\displaystyle-\frac{t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,yz}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{xy,zx}=-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{yz,zx}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},$ $\displaystyle\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,xy}$ $\displaystyle=$
$\displaystyle-\frac{2t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,yz}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4},~{}\big{[}\bm{H}_{ct}^{z}\big{]}_{zx,zx}=-\frac{4t_{s}^{2}}{E_{v}}\mathbb{I}_{4\times
4}.$ (S21)
Figure S2:
One also needs to take into account symmetric processes $(yz\leftrightarrow
zx)$ with another anion as illustrated in Fig. S2. Collecting all possible
charge-transfer contributions, we obtain the effective spin-orbital
Hamiltonian,
$\displaystyle H_{ct}\\!=\\!\frac{2t_{s}^{2}}{2U+2\Delta-9U_{p}}\sum_{\langle
ij\rangle\parallel\gamma}\\!\\!\Big{(}\vec{S}_{i}\\!\cdot\\!\vec{S}_{j}-\frac{1}{4}\Big{)}(n_{i}^{a}n_{j}^{b}\\!+\\!n_{i}^{b}n_{j}^{a}).$
(S22)
### II.2 II. Detail of infinite density matrix renormalization group
simulation
To perform the infinite matrix product state related simulation, we first
clarify the ordering of lattice sites, which is implemented by labeling an
integer $i$ to each lattice site. We follow a site labeling method for
honeycomb lattice on the YC-2$L_{y}$ cylinders as shown in Fig. S3. Note that
YC-4 cylinder cannot support the Kekule spin-orbit dimerized phase.
Figure S3: (color online) (a,b) Site-labeling scheme for a honeycomb lattice
on a YC-4 or YC-6 cylinder with $x$ direction length $L_{x}=2$ or 3,
respectively. The total number of sites is $2L_{x}L_{y}=$ 8 or 18,
respectively. We wrap the honeycomb lattice onto a cylinder which is periodic
along $y$ direction and infinite along $x$ direction.
To check the convergence of the ground state obtained by iDMRG, we investigate
the scaling behavior of the ground state energy per site as a function of bond
dimension $\chi$ on the YC-6 cylinder with $L_{x}=3$. Fig. S4 shows that the
per site energy $E_{0}/J_{d}$ scales linearly with $1/\chi$ indicating a
reliable convergence.
Figure S4: (color online) Per site ground state energy $E_{0}/J_{d}$ as a
function of bond dimension $\chi$ on a YC-6 cylinder with length $L_{x}$ = 3.
### II.3 III. Detail of exact diagonalization simulation
In this section, we give a detail of ED simulation on a cluster of six sites
with periodic boundary conditions as shown in Fig. S5(a). The cluster can
support the Kekule as well as the columnar spin-orbit dimerized phase and the
ground state turns out to be the Kekule phase. In Fig. S5(b), we plot the
energy per site $E_{0}$ as a function of $\tilde{J}$ for the ground state,
Kekule phase, and the first excited state. The result indicates the
stabilization of the Kekule phase up to $\tilde{J}=0.5$.
Figure S5: (color online) (a) Six sites cluster with periodic boundary
conditions which can support the Kekule phase. (b) Energy per site $E_{0}$ for
the ground state, which is the Kekule phase, and the first excited state.
### II.4 IV. Gell-Mann matrices
In this section, we give the explicit expressions of Gell-Mann matrices
$\lambda_{i}$.
$\displaystyle\lambda_{1}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&1&0\\\
1&0&0\\\
0&0&0\end{array}\right)\\!\\!,~{}\lambda_{2}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&-i&0\\\
i&0&0\\\
0&0&0\end{array}\right)\\!\\!,~{}\lambda_{3}=\\!\\!\\!\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\
0&0&0\end{array}\right)\\!\\!,~{}\lambda_{4}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 1&0&0\end{array}\right)\\!\\!,$ (S35)
$\displaystyle\lambda_{5}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&0&i\\\
0&0&0\\\
-i&0&0\end{array}\right)\\!\\!,~{}\lambda_{6}=\\!\\!\\!\left(\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\
0&1&0\end{array}\right)\\!\\!,~{}\lambda_{7}\\!=\\!\\!\\!\left(\begin{array}[]{ccc}0&0&0\\\
0&0&-i\\\
0&i&0\end{array}\right)\\!\\!,~{}\lambda_{8}\\!=\\!\frac{1}{\sqrt{3}}\\!\\!\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&-2\end{array}\right)\\!\\!.$ (S48)
|
# Construction of Multiple Constrained DNA Codes
###### Abstract
DNA sequences are prone to creating secondary structures by folding back on
themselves by non-specific hybridization among its nucleotides. The formation
of secondary structures makes the sequences chemically inactive towards
synthesis and sequencing processes. In this letter, our goal is to tackle the
problems due to the creation of secondary structures in DNA sequences along
with constraints such as not having a large homopolymer run length. In this
paper, we have presented families of DNA codes with secondary structures of
stem length at most two and homopolymer run length at most four. By mapping
the error correcting codes over $\mathbb{Z}_{11}$ to DNA nucleotides, we
obtained DNA codes with rates $0.5765$ times the rate of corresponding code
over $\mathbb{Z}_{11}$, which include some new secondary structure free and
better-performing codes for DNA based data storage and DNA computing purposes.
Siddhartha Siddhiprada Bhoi111Department of Mathematics & Computing, Indian
Institute of Technology (ISM), Dhanbad, India<EMAIL_ADDRESS>P.
Udaya222School of Computing and Information Systems, The University of
Melbourne, Parkville, Australia<EMAIL_ADDRESS>Abhay Kumar
Singh333Department of Mathematics & Computing, Indian Institute of Technology
(ISM), Dhanbad, India<EMAIL_ADDRESS>
## 1 Introduction
Deoxyribose Nucleic Acid(DNA) is nature’s way of storing genetic data and has
been the central concept in DNA computing[10]. DNA based data storage is
important due to its high storage density, capacity, and longevity [2], [3].
These advantages motivated researchers to explore the development of the
subject.
Let $S_{DNA}=\\{A,T,G,C\\}$, a set of four DNA nucleotides named Adenine (A),
Guanine (G), Cytosine(C), and Thymine (T). DNA is made from DNA strands, which
are sequences over $S_{DNA}$ and for any positive integer $n$, a DNA code
$S^{n}_{DNA}$ is a collection of DNA sequences of length $n$. For each
$X=x_{1}\cdots x_{n}\in S^{n}_{DNA}$, then the reverse of the DNA sequence
denoted by $X^{r}$ is $x_{n}x_{n-1}\cdots x_{1}$, the complement of the DNA
sequence denoted by $X^{c}$ is $x_{1}^{c}x_{2}^{c}\cdots x_{n}^{c}$, and the
reverse complement of the DNA sequence denoted by $X^{rc}$ is
$x_{n}^{c}x_{n-1}^{c}\cdots x_{1}^{c}$. The complements of the DNA nucleotides
are $T^{c}=A$, $A^{c}=T$, $G^{c}=C$, and $C^{c}=G$. DNA sequences are
synthesized physically and read by the methods of DNA synthesis and DNA
sequencing, respectively [4]. During the synthesis and sequencing, various
types of errors can occur, which bring in the importance of the study of the
minimum distance of DNA code, the property that enables the suppression of
those errors.
The purpose of DNA sequencing is to read the DNA content, i.e., to determine
the exact DNA nucleotides along with their order. This is accomplished using
the specific hybridization between the DNA sequence and its complement [5].
Hybridization is the process of complementary base pairs binding to form a
double helix. The main source of errors at the time of the sequencing process
is due to non-specific hybridization. If in a DNA code some of the DNA
sequences are similar enough among themselves or their reverse or reverse-
complement versions, then non-specific hybridization will take place. This
situation motivates authors [[5], [6], [7], [14], [15]] to design DNA codes
whose sequences are sufficiently different among themselves as well as from
their reverse and reverse complement versions. The reverse-complement
constraint is the constraint in which DNA sequences and their reverse
complement versions are far apart by at least the minimum Hamming distance of
the code.
GC content is the percentage of nitrogenous bases in a DNA molecule that are
either guanine ($G$) or cytosine ($C$). The average GC-content in human genes
ranges from $35-60$ percent across $100$-Kb fragments [8]. The variation in GC
content can lead to variation in melting temperature and stability of the DNA
strand. DNA strands with $50$ percent GC content have the highest stability,
thus GC content is kept to nearly $50$ percent for the DNA based data storage.
At the time of synthesis and sequencing of DNA sequences with a high
repetition of nucleotides, insertion, deletion, and substitution errors can
occur [3]. This high consecutive repetition of nucleotides in DNA sequences is
called a homopolymer run. A DNA sequence is free from a homopolymer of run
length $l$ if all nucleotides in any subsequence of length $l$ are not the
same. For example, the DNA sequence $ACGCCCCCGTG$ has a homopolymer of run
length $6$. In [15], the authors have discussed enumerating methods and bounds
over DNA sequences with GC constraints, homopolymer run length at most 3, and
specific Hamming distances.
In a DNA strand, the nucleic acids can have primary, secondary and tertiary
structures. The primary structure consists of a linear sequence of
nucleotides, the secondary structure is due to the base-pairing interaction of
nucleotides within an active single-stranded DNA sequence [10], and tertiary
structure is due to the position of atoms in 3D space and its steric and
geometric constraints. A DNA strand can fold back upon itself and form a loop-
like structure with itself. Because of this structure, DNA becomes slow to
react to chemical reagents in the sequencing process[13]. Therefore, to read
such DNA sequences with a secondary structure, the DNA needs to be unfolded,
which can require extra resources and energy, leading to storage inefficiency.
The authors [10] reports new DNA codes avoiding secondary structure. For some
particular cases, some constraints such as reverse and reverse-complement
constraints and fixed GC-content constraints are also discussed in [10], [6].
Benerjee et.al.[1] took an interesting approach to the code construction by
considering a certain set of DNA sequences of length two as the base alphabet
and mapping it to the field $\mathbb{Z}_{5}$. This method enabled them to
define DNA codes as an image of codes over $\mathrm{\mathbb{Z}_{5}}$. It is a
natural question to find if other algebraic structures can provide a map to
DNA codes with better parameters for DNA based data storage such that the
resulting DNA sequences avoid secondary structure and satisfy the other
necessary constraints.
After providing the required notations and preliminaries in Section II, the
rest of the letter is organised as follows. Homopolymers and Secondary
structure avoiding maps, and their properties are discussed in the next
section. Further, in Section IV, families of DNA codes are constructed, and
other families of codes like Hamming codes and Reed Solomon codes are studied.
In Section V, a comparative analysis is established, and a comparative table
of code rates and code attributes is provided.
## 2 PRELIMINARIES AND NOTATION
Let $F_{q}$ denote an alphabet of size $q$ symbols. For any positive integer
$n$, a code $C$ over the alphabet $F_{q}$ is a subset $C\subset F_{q}^{n}$ of
size $M$ and minimum distance $d$. The Hamming distance for any two sequences
of equal length $x$ and $y$ is denoted by $d_{H}(x,y)$ and defined as the
number of positions at which the corresponding symbols are different. The
Hamming distance for a code $C$ is defined as
$d_{H}(C)=\mathrm{min}\\{d_{H}(x,y):x\neq y$ $\forall$ $x,y\in C\\}$. A DNA
code $C_{DNA}$ is any code defined over the alphabet $S_{DNA}$. For any DNA
code $C_{\mathrm{DNA}}$ with minimum Hamming distance $d_{H}$, the reverse
minimum distance $d^{r}_{H}=\mathrm{min}\\{d_{H}(x,y^{r}):x\neq y^{r}$ and
$x,y^{r}\in C_{\mathrm{DNA}}\\}$. Let $X=x_{1}\cdots x_{n}\in
S_{\mathrm{DNA}}^{n}$ be a DNA sequence of length $n$, then for any integer
$i,j$ such that $1\leq i\leq j\leq n$ the continuous sequence
$x_{i},x_{i+1}\cdots x_{j}$ is called a subsequence of $X$.
We have defined free energy and interaction energy and also provided the
relation between them. We have also discussed how the free energy affect the
secondary structure. A chemically active DNA sequence $x_{1}x_{2}\cdots
x_{n}$, to gain stability forms secondary structure and it releases energy
known as free energy [13], denoted by $E_{1,n}$. The free energy of a DNA
sequence can be used to predict the secondary structure of the DNA sequence.
The free energy depends on the energy released by pairing $(x_{i},x_{j})$,
where $1\leq i<j\leq n$. This released energy is called interaction energy.
The Nussinov-Jackson folding algorithm (NJ algorithm) is used to approximately
predict the secondary structure in a given DNA sequence [12]. In the NJ
algorithm, it is assumed that the interaction energy $\mu(x_{i},x_{j})$ is
independent of all other nucleotide pairs. They only depend on the chosen
nucleotide base pair $(x_{i},x_{j})$. Commonly used values of
$\mu(x_{i},x_{j})$ are
$\mu(x_{i},x_{j})=\left\\{\begin{array}[]{ll}-5&if(x_{i},x_{j})\in\\{(C,G),(G,C)\\},\\\
-4&if(x_{i},x_{j})\in\\{(T,A),(A,T)\\},\\\
-1&if(x_{i},x_{j})\in\\{(T,G),(G,T)\\},\\\ \ \ 0&otherwise\\\
\end{array}\right.$
As NJ algorithm consider interaction energy ,$\mu(x_{i},x_{j})$, to be
independent of other nucleotides except $x_{i}$ and $x_{j}$, we can calculate
minimum free energy of a DNA sequence through a recursion. The minimum free
energy for a subsequence $x_{i}x_{i+1}\cdots x_{j}$ of a DNA sequence
$x_{1}x_{2}\cdots x_{n}$ is given as
$E_{i,j}=\mathrm{min}\\{E_{i+1,j-1}+\mu(x_{i},x_{j}),E_{i,k-1}+E_{k,j}:k=1,2,\cdots,j\\}$
with initial conditions $E_{l,l}=E_{l-1,l}=0$ for $l=1,2,\cdots,n$ [10]. A
larger negative value for free energy means a higher probability of secondary
structure in the DNA sequence. The difference in the values of the free
energies is due to the relative strength of the bonds between DNA nucleotides,
i.e., Guanine(G) and Cytosine(C) have a triple bond between them whereas
Adenine(A) and Thymine(T) have a double bond between them. So, the relative
strength of the bond between Guanine(G) and Cytosine(C) is higher than that of
Adenine(A) and Thymine(T). For a DNA sequence $x=x_{1}x_{2}\cdots x_{n}$, we
define secondary-complement sequence as $x^{sc}=x_{n}^{sc}x_{n-1}^{sc}\cdots
x_{1}^{sc}$ with $A^{sc}$ is $T$, $C^{sc}$ is $G$, $T^{sc}$ can either $A$ or
$G$ and $G^{sc}$ can either be $C$ or $T$. Note that the reverse complements
of a DNA sequence are secondary complements, but the converse is not true in
general. A DNA subsequence can have multiple secondary complements i.e.,
$(ATCA)^{sc}=TGAT$ and $(ATCA)^{sc}=TGGT$. If we take two disjoint
subsequences of a given DNA code such that one is a secondary complement of
the other, then the subsequences are called disjoint secondary-complement
subsequences of the DNA sequence. The goal of the paper is to find such DNA
sequences which are stable without secondary structure.
Let the nucleotides in a DNA sequence $x=x_{1}x_{2}\cdots x_{n}\in
S_{\mathrm{DNA}}^{n}$ be indexed from 1 to n and for any $1\leq i,j\leq n$,
$(x_{i},x_{j})$ denote the pairs contributing to the secondary structure.
These are the pairs of nucleotides that have negative interaction energy as
per NJ algorithm. For any $1\leq i_{1},j_{1},i_{2},j_{2}\leq n$, the pairs of
nucleotide $(x_{i_{1}},x_{j_{1}})$ and $(x_{i_{2}},x_{j_{2}})$ are said to be
a consecutive set of pairs of nucleotide if both $i_{1}$ and $i_{2}$ are
consecutive and $j_{1}$ and $j_{2}$ are consecutive. We define the stem length
of a DNA sequence as the number of a consecutive set of pairs of nucleotides
contributing to secondary structure and stem is defined as the consecutive set
of pairs contributing to secondary structure. We consider these since many
pairs are not possible due to chemical and stereo-chemical constraints [11].
For example, in a DNA sequence $ATTCAAAATGGATCCGTAATGGAT$,
$\\{(x_{i},x_{25-i})$, $(x_{j},x_{5+j})|i=1,2,3,4,5;j=8,9,10,11,12)\\}$ are
two disjointed secondary structures of length $5$ as shown in Fig 1. As
$(ATTCA)^{sc}=TGGAT$ and $(ATGGA)^{sc}=TCCGT$.
Lemma 1 For a given positive integer $l$, if the DNA sequence is free from a
disjoint secondary complement of length $l+1$, then there cannot exist any
secondary structure of stem length more than $l$.
Proof. It is observed from the free energy of a DNA sequence that if there
exist two disjoint secondary-complement sub-sequences $\alpha$ and $\beta$ of
length $l+1$ in the DNA sequence such that $\alpha$ = $\beta^{sc}$ then there
exists a secondary structure of stem length $l+1$. It is also observed that if
the DNA sequence is free from secondary-complement sub-sequence of length $l$,
it is also free from secondary-complement sub-sequence of length $l+k$ where
$k$ is a non-negative integer. The lemma follows from the contrapositive of
the above observations. $\square$
For example, consider the DNA sequence $X=ATTCAAAATGGATCCGTAATGGAT$ as shown
in Fig 1. The sequence $X$ have two disjointed secondary structures of length
$5$ and hence it is free from a disjoint secondary complement of length $6$.
So according to Lemma 1, $X$ can not have a secondary structure of stem length
more than $5$.
## 3 DNA Alphabets and Their Properties
The presence of secondary structure in a DNA sequence depends upon the value
of free energy of the DNA sequence and the value of free energy depends upon
the relative bond strength and bonding between some of the DNA subsequences.
By avoiding such subsequences, we can control the free energy of DNA sequence
and hence have control upon the presence of secondary structure in the DNA
sequence.
The structure of the base alphabet is extremely important to control the free
energy, which depends on the interaction between the nucleotides. If the base
alphabet is $S_{\mathrm{DNA}}$, then in the pseudo-random sequence method, it
is impossible to control the secondary structures as interactions are ensured
between every possible combination due to the nature of the sequence
generation mechanism. In [1], the authors choose a base alphabet as
$\\{AA,AC,CA,CC,TC\\}$, a subset of length two nucleotides. In this letter, we
consider the base alphabet of length $3$. We are interested in a set S to be
the set of possible base alphabets of length $3$ nucleotides which can avoid
secondary structure when DNA sequences are constructed as concatenations of
the elements in the base alphabet. The set
$S=\\{CCC,CCA,CAC,CAA,ACC,ACA,AAC,AAA,TCC,$ $CTC,TCA\\}$ with 11 elements is
considered for this letter. For any DNA code $C_{\mathrm{DNA}}\subseteq S^{n}$
of length $3n$ constructed using a bijective map $\mathbb{Z}_{11}$ to $S$. We
use $\mathbb{Z}_{11}=(0,1,2,3,4,5,6,7,8,9,\text{\@slowromancap x@})$ in our
code construction, where $\text{\@slowromancap x@}=10\mathrm{\mod}11$.
The bound on free energy on DNA codes over $S^{n}$ is obtained through
mathematical induction on the parameter $n$ for the free energy presented
below in proposition 1.
Proposition 1. For any DNA sequence in $S^{n}$, $E_{1,3n}\geq-4n$.
Proof of the above Proposition is provided in the appendices.
It should be noted that for any arbitrary DNA sequence of length $3n$, we have
$E_{1,3n}\geq-5\left\lfloor 3n/2\right\rfloor$. From Proposition 1, it is
clear that the free energy for any DNA sequences over $S$ is restricted thus
avoiding secondary structures as explained in [1].
Lemma 2 For positive integers $n$, $m$, and $l$, such that $n\geq 6$ and
$2\leq l\leq\left\lfloor m/2\right\rfloor$. The set $U_{n}=\\{a_{1}a_{2}\cdots
a_{n}:a_{i}\in U\subset S^{m}_{\mathrm{DNA}},i=1,2,\cdots,n\\}$. If each DNA
sequence in $U_{6}$ is free from l length secondary-complement sub-sequences,
then each DNA sequence in $U_{n}$ is also free from disjoint secondary-
complement sub-sequences of length more than l. Any 3n-length DNA sequence in
$S^{n}$ is free from a secondary structure of stem length more than one.
Proof. By the way of construction of $U_{6}$, DNA sequences of $U_{6}$ ensure
that concatenation of any six DNA sequences from $U$ is free from $l$ length
secondary-complement sub-sequences. So, the same stands true for n DNA
sequences from $U$. For m = 3; l = 2, the results follow. $\square$
Note that Pairing between G and C, and between G and T is not possible, thus
preventing many secondary structures. Again, if any secondary structure
exists, then it cannot have a stem length of more than one, and thus less
energy is required to break it. From Lemma 2 and Remark, it is clear that any
DNA sequences in $S^{n}$ are free from a secondary structure of stem length of
more than one.
### 3.1 Secondary Structure Avoiding Map
Define a bijective map $\phi:\mathbb{Z}_{11}\to S$ such that:
x | $\boldmath{\phi(x)}$ | x | $\boldmath{\phi(x)}$ | x | $\boldmath{\phi(x)}$
---|---|---|---|---|---
0 | CCC | 4 | ACC | 8 | TCC
1 | CCA | 5 | ACA | 9 | CTC
2 | CAC | 6 | AAC | x@ | TCA
3 | CAA | 7 | AAA | |
where $x\in\mathbb{Z}_{11}$ and $\phi(x)\in S$. For any sequence
$x=x_{1}x_{2}\cdots x_{n}\in\mathbb{Z}_{11}^{n}$, we have
$\phi(x)=\phi(x_{1})\phi(x_{2})\cdots\phi(x_{n})$. For any $x$ and $y$
$\in\mathbb{Z}_{11}$, we define distance function as
$d:\mathbb{Z}_{11}\times\mathbb{Z}_{11}\to R$ such that
$d(x,y)=d_{H}(\phi(x),\phi(y))$. For any $x$ and $y$ $\in\mathbb{Z}_{11}^{n}$,
we define $d(x,y)=\sum_{i=1}^{n}d(x_{i},y_{i})$. For any code $C$ over
$\mathbb{Z}_{11}$, $d=\mathrm{min}\\{d(x,y):x\neq y\ and\ x,y\in C\\}$.
Proposition 2. $(\mathbb{Z}_{11}^{n},d)$ and $(S^{n},d_{H})$ are isometric.
Proof. The function $\phi:(\mathbb{Z}_{11}^{n},d)\rightarrow(S^{n},d_{H})$ is
an isometric function as it preserves the distance functions $d$ and $d_{H}$
by its definition $d(x,y)=d_{H}(\phi(x),\phi(y))$ for all
$x,y\in\mathbb{Z}_{11}^{n}$. $\square$
Theorem 1. For every code $C$ with parameters $(n,M,d)$ over
$\mathbb{Z}_{11}$, there exists a DNA code $\phi(C)$ with parameters
$(3n,M,d_{H})$ such that each DNA sequence in $\phi(C)$ is free from secondary
structure of stem length more than one and the minimum Hamming distance
$d_{H}=d$.
Proof. From Proposition 2, $\phi$ is an isometric from
$(\mathbb{Z}_{11}^{n},d)$ to $(S^{n},d_{H})$. Thus, they have the same size
and Hamming distance. The proof follows from Lemma 2. $\square$
Lemma 3. $d_{H}(x,y^{c})\geq n$ for any $x,y\in S^{n}$.
Proof. It is observed that for any $x,y\in S$, $d_{H}(x,y^{c})\geq 1$. This
implies $d_{H}(x,y^{c})=\sum_{i=1}^{n}d_{H}(x_{i},y^{c}_{i})\geq n$. $\square$
Lemma 4. For a $(3n,M,d_{H})$ DNA code $C$ over $S$, if $d_{H}\leq n$ then
$C\cup C^{c}$ is a $(3n,2M,d_{H})$ DNA code where $C^{c}=\\{x^{c}:x\in C\\}$.
Proof. From the definition of complement of DNA sequence the results on the
size and length follows. For any DNA code $C_{DNA}$ over $S$,let $x,y\in
C_{DNA}$ be two DNA sequences. Then $d_{H}(x^{c},y^{c})=d_{H}(x,y)\geq d_{H}$
and from Lemma 3 $d_{H}(x^{c},y^{c})=d_{H}(x,y)\geq n$. So, for $C_{DNA}\cup
C_{DNA}^{c}$, the minimum Hamming distance is $\mathrm{min}{d_{H},n}=d_{H}$.
Hence,the result on distance follow. $\square$
Lemma 5. For any $(n,M,d)$ code $C\subset\mathbb{Z}_{11}^{n}$, $\phi(C)$
exists and if $d\leq n$ then it satisfies the reverse complement constraint.
Proof. If $x,y\in S^{n}$, $d_{H}(x,y^{rc})\geq n\geq d$ and thus the proof
follows from Proposition 2. $\square$
### 3.2 Homopolymers Deletion Map
Define a non-surjective map $f:S^{n}\to S_{\mathrm{DNA}}^{3n}$, such that for
a sequence $y\in S^{n}$, there exists a subsequence with consecutive repeated
nucleotides of length more than four, then we flip the nucleotide present in
doubly even positions of $y$ to $T$.
For $f(y)=y_{1}y_{2}y_{3}\cdots y_{3n}$ if $y_{i+1}\cdots y_{i+l}$ are of same
nucleotide with $l\geq 4$, then flip $y_{i+j}$ to $T$ if $j\mod 4=0$. For
example, $f(ACCACCCCCCCAAAAAAA)=ACCACCCTCCCTAAATAAA$. It does not contain any
homopolymers of run length more than four.
Lemma 6. For any DNA sequence $x\in S^{n}$, $f(x)$ cannot have a secondary
structure with a stem length more than two.
Proof. For any DNA sequence $x\in S^{n}$, we observe that in $f(x)$ the
flipped nucleotides varies at least by four. Thus, from Lemma 2, the stem
length of a secondary structure cannot be more than two. $\square$
Theorem 2. For every code $C$ over $\mathbb{Z}_{11}$ with parameters
$(n,M,d)$, there exists a reverse-complement DNA code $f(\phi(C))$ with
parameters $(3n,M,d_{H})$ such that each DNA sequence in $f(\phi(C))$ is free
from homopolymers of run length more than four and from secondary structure of
stem length more than two, where $\left\lceil\frac{3d}{4}\right\rceil\leq
d_{H}\leq\left\lceil\frac{3n+3d}{4}\right\rceil$.
Proof. If there exist $m$ consecutive nucleotides, then at most $\left\lfloor
m/4\right\rfloor$ nucleotides are flipped to $T$. Thus for any $x,y\in S^{n}$
$d_{H}(x,y)-\left\lfloor m/4\right\rfloor\leq d_{H}(f(x),f(y))\leq
d_{H}(x,y)+\left\lfloor m/4\right\rfloor$. The upper and lower bounds follow
from the extreme cases, i.e., $m=3n-d_{H}$ and $m=d_{H}$. $\square$
## 4 Families of new constrained DNA codes
### Family 1:
For $q=11$, consider the family of codes $C_{k}$ of length $4^{k-1}$ and
dimension ${k}$ with $2\leq k\leq 5$ with generator matrix $G_{k}$. The matrix
$G_{2}=\left(\begin{matrix}1&1&1&1\\\ 1&5&9&\text{\@slowromancap
x@}\end{matrix}\right)$
and
$G_{i+1}=\left(\begin{matrix}1_{4^{i-1}}&2_{4^{i-1}}&3_{4^{i-1}}&4_{4^{i-1}}\\\
G_{i}&G_{i}&G_{i}&G_{i}\end{matrix}\right),\text{ for}\ \ i=2,3,4.$
where $i_{j}$ is all the $i$ vectors of length $j$ for $i=2,3,4,5$. The code
meets the Griesmer bound:
$n=\sum_{i=0}^{k-1}\left\lceil\frac{d}{q^{i}}\right\rceil$.
Theorem 3. There exists a $(2^{2k-1},5^{k},3\cdot 4^{k-2})$ DNA code
$f(\phi(C_{k}))$ with $d_{H}^{r}=2^{2k-3}$.
Proof. The distance $d(0_{4},4190)=3$, where $4190=2(1_{4}+159$x@$)$ is in the
linear span of $G_{2}$. Thus $d(0_{4^{k-1}},(4190)^{k-1})=3\cdot 4^{k-2}$ for
any positive integer k. From symmetry, the minimum distance for the code
$C_{k}$ is $d(0_{4^{k-1}},(4190)^{k-1})=3\cdot 4^{k-2}$. The size and length
follow from Theorem 1. $\square$
### Hamming Codes:
Consider a q-ary Hamming code. For an integer r$(\geq 2)$, the Hamming code is
a linear code with parameters
$\left[\frac{q^{r}-1}{q-1},\frac{q^{r}-1}{q-1}-r,3\right]$ over
$\mathbb{F}_{q}$. For $q=11$ we have the following results:
Theorem 4. There exists a
$\left(\frac{3(11^{r}-1)}{10},11^{\frac{11^{r}-1-10r}{10}},3\right)$ DNA code
$\phi(H_{r})$ for Hamming code $H_{r}$ over $\mathbb{Z}_{11}$.
Proof. The Hamming code over $\mathbb{Z}_{11}$ of degree $r(\geq 2)$ is a
$\left[\frac{11^{r}-1}{10},\frac{11^{r}-1}{10}-r,3\right]$ linear single error
correcting code. The size and length of the code follow from the parameters of
$H_{r}$ and from Theorem 1. For some $x=0_{\frac{11^{r}-1}{10}}$ and
$y=0_{\frac{11^{r}-31}{10}}191$ are the codewords in $H_{r}$ such that
$d(x,y)=3$, where $0_{i}$ represents all zero vector of length i. As
$3=d_{H}^{*}\leq d$, it follows the result. $\square$
Lemma 8. For the code $H_{r}$ over $\mathbb{Z}_{11}$, there exists a
$\left(\frac{3(11^{r}-1)}{10},11^{\frac{11^{r}-1-10r}{10}},d_{H}^{*}\right)$
DNA code $f(\phi(H_{r}))$ for $r=2$, $d_{H}^{*}=3$.
Proof. The proof follows from Theorem 2 and Theorem 4. $\square$
### Reed Solomon Codes:
A q-ary Reed Solomon code is a q-ary Bose, Chaudhari, and Hocquenghem (BCH)
code of length $q-1$ generated by
$g(x)=(x-\alpha^{a+1})(x-\alpha^{a+2})\cdots(x-\alpha^{a+\delta-1}),$
where $a\geq 0$, $2\leq\delta\leq q-1$ and $\alpha$ is a primitive element of
$q$-ary field. Reed-Solomon codes are Maximum Distance Separable (MDS) codes
with parameters $[q-1,q-\delta,\delta]$ over $F_{q}$, for any $2\leq\delta\leq
q-1$. As Reed Solomon Codes have high reliability and increase in code rate
with increase in alphabet size, they are used widely for Data Storage and long
distance communication purposes. So for $\mathbb{Z}_{11}$, RS codes have the
following properties:
Theorem 5. There exists a $\left(30,11-\delta,\delta\right)$ DNA code
$\phi(RS)$ for Reed Solomon Code $RS$ over $\mathbb{Z}_{11}$.
Proof. The Reed Solomon code over $\mathbb{Z}_{11}$ is a
$\left(10,11-\delta,\delta\right)$ linear MDS code. The size and length of the
code follow from the parameters of $H_{r}$ and Theorem 1. $\square$
Lemma 9. For the family of Reed Solomon code over $\mathbb{Z}_{11}$, there
exists a single error correcting code of parameter $\left(30,8,3\right).$
Proof. The size and length of code follow from Theorem 2 and Theorem 5 for
$\delta=3$. $0020010001$ is a $RS(11)$ codeword and $d(0_{10},0020010001)=3$.
$\square$
## 5 Discussion of the results
It is observed that all DNA codes constructed in Section IV are reverse-
complement DNA codes which can also be concluded from Lemma 6 and Theorem 2.
In Theorem 3 and Theorem 4, the DNA codes are free from the secondary
structure with stem length starting from one. Similarly, the DNA codes in
Lemma 8 and Lemma 9 are free from homopolymers of run lengths more than four
and secondary structures of stem length more than two. Comparisons of
properties of single error-correcting DNA codes are made in Table 1 against
the properties, $P_{1}$ to $P_{3}$, are respectively the secondary structure
avoiding property, the homopolymers run length, and reverse complement
property, respectively. The DNA codes constructed from Reed-Solomon codes over
$\mathbb{Z}_{11}$ appear to show a comparatively better code rate than other
codes in the literature. While comparing with the codes presented in [1], for
some lengths of DNA sequences, our codes show a higher code rate. This also
opens the problem of finding appropriate algebraic structures to construct DNA
codes that can provide optimum code rates while obeying all the constraints of
DNA codes.
In this letter, we have found DNA codes of a certain length providing a higher
code rate than the DNA codes of similar error correcting capabilities provided
in the literature. For example, a single error Reed Solomon Code over our
construction of DNA codes provides a code rate of $0.46125$ as compared to the
previous construction in [1] with code rate $0.145$. Although the asymptotic
code rate DNA code from Hamming codes is a little less than from the
construction provided in [1], our construction provides a better code rate for
DNA codes of certain lengths. For the second-degree Hamming code, our
construction provides a comparatively better code rate than previous codes in
the literature. Furthermore, the map $\phi$ considered in the letter naturally
leads to increased distances over the DNA alphabet with some codewords; for
example, $0010060006$ and $0000000000$ are two $(10,8,3)$ Reed Solomon
codewords over $\mathbb{Z}_{11}$ with $d_{H}(0000000000,0010060006)=3$ but
$d_{H}(f(\phi(0000000000)),f(\phi(0010060006))=5$.
Remark: The DNA sequences constructed by the method provided in the letter
have the unique characteristic of preserving the property of homopolymers of
run length at most four and avoiding secondary structure of stem length more
than two with concatenation operation. This operation enables us to construct
families of larger lengths using concatenation operation, preserving the
stated properties. This preservation of properties under the concatenation
operation applies to the atomic families only and not to the families obtained
by including code and its complement, such as the code in the 9th row of the
table.
## 6 Conclusions
The methods discussed in the letter provide secondary structure free DNA codes
with homopolymer run length less than four. These codes have the code rates
$0.5765(\frac{log_{4}11}{3})$ times the code rate of the corresponding code
over $\mathbb{Z}_{11}$. For some specific lengths, they provide better code
rates and satisfy more constraints than the existing codes in the literature.
Although our method yields better performing codes with specific lengths, it
is an interesting open question to determine an appropriate algebraic
structure that leads to optimal codes.
$(n,M,d_{H})$ | Code Rate | $P_{1}$ | $P_{2}$ | $P_{3}$
---|---|---|---|---
DNA Codes | $log_{4}M/n$ | | |
Example 9 [9] | $0.14937$ | No | $>4$ | No
Example 2 [10] | $0.25850$ | Yes | $>4$ | No
Example 4 [10] | $0.40105$ | Yes | $>4$ | No
$(8,256,4)$ code | $0.50000$ | No | No | Yes
Table III [6] | | | |
$(8,244,4)$ code | $0.48796$ | No | No | Yes
Table III [6] | | | |
$f(\phi(H_{7}))$ | $0.58027$ | Yes | $<4$ | Yes
[1] | | | |
$f(\phi(H_{5}))$ | $0.57639$ | Yes | $<4$ | Yes
$over\mathbb{Z}_{11}$ | | | |
$C\cup C^{c}$ | $0.42865$ | Yes | $<4$ | Yes
$C=f(\phi(H_{2}))$ | | | |
[1] | | | |
$C\cup C^{c}$ | $0.494365$ | Yes | $<4$ | Yes
$C=f(\phi(H_{2}))$ | | | |
$over\mathbb{Z}_{11}$ | | | |
$f(\phi(H_{2}))$ | $0.42865$ | Yes | $<4$ | Yes
[1] | | | |
$f(\phi(H_{2}))$ | $0.48047$ | Yes | $<4$ | Yes
$over\mathbb{Z}_{11}$ | | | |
$C\cup C^{c}$ | $0.35274$ | Yes | $<4$ | Yes
$C=f(\phi(C_{2}))$ | | | |
[1] | | | |
$f(\phi(C_{2}))$ | $0.29024$ | Yes | $<4$ | Yes
[1] | | | |
$f(\phi(C_{2}))$ | $0.28828$ | Yes | $<4$ | Yes
$over\mathbb{Z}_{11}$ | | | |
$f(\phi((10,8,3)$ | $0.145120$ | Yes | $<4$ | Yes
$RS-code))$ | | | |
[1] | | | |
$f(\phi((10,8,3)$ | $0.461258$ | Yes | $<4$ | Yes
$RS-code))$ | | | |
Table 1: Comparisons of new codes with the codes in literature the
properties: $P_{1}$ is the secondary structure avoiding property, $P_{2}$ is
Homopolymer run length, $P_{3}$ is reverse complement constraint.
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## Appendices
Proposition 1. For any DNA sequence in $S^{n}$, $E_{1,3n}\geq-4n$.
Proof of the above Proposition has been provided in the appendix.
Proof. According to NJ algorithm, the minimum free energy for a subsequence
$x_{i}x_{i+1}\cdots x_{j}$ of a DNA sequence $x_{1}x_{2}\cdots x_{n}$ is given
as
$E_{i,j}=min\\{E_{i+1,j-1}+\mu(x_{i},x_{j}),E_{i,k-1}+E_{k,j}:k=i+1,\cdots,j\\}$
with initial conditions $E_{l,l}=E_{l-1,l}=0$ for $l=1,2,\cdots,n$ [10].
The commonly used value of $\mu(x_{i},x_{j})$ are
$\mu(x_{i},x_{j})=\left\\{\begin{array}[]{ll}-5&if(x_{i},x_{j})\in\\{(C,G),(G,C)\\},\\\
-4&if(x_{i},x_{j})\in\\{(T,A),(A,T)\\},\\\
-1&if(x_{i},x_{j})\in\\{(T,G),(G,T)\\},\\\ \ \ 0&otherwise\\\
\end{array}\right.$
For sequences of length $3n$ over $S^{n}$, the minimum free energy is
$E_{1,3n}=min\\{E_{2,3n-1}+\mu(x_{1},x_{3}n),E_{1,k-1}+E_{k,3n}:k=2,\cdots,j\\}.$
As the above equation is a recurrence relation depended on the interaction
energy $\mu(x_{i},x_{j})$, for $1\leq i\leq j\leq 3n$ and initial values of
free energies are $0$. Free energy of any sequence is a function of
interaction energy. As the base alphabet set $S$ does not have any element
with nucleotide $G$, the interaction energy can only be released by the
interaction of nucleotide pairs $(A,T)$ and $(T,A)$. So,for any sequence $X$
of length $3n$,
$E_{1,3n}(X)=-4\times\text{number of (A,T) pairs present in the sequence }X$
. For n=1, $E_{1,3}(CCC)=0$, $E_{1,3}(CCA)=0$, $E_{1,3}(CAC)=0$,
$E_{1,3}(CAA)=0$, $E_{1,3}(ACC)=0$, $E_{1,3}(ACA)=0$, $E_{1,3}(AAC)=0$,
$E_{1,3}(AAA)=0$, $E_{1,3}(TCC)=0$, $E_{1,3}(CTC)=0$, $E_{1,3}(TCA)=-4$.
So, for any DNA sequence over $S$, $E_{1,3}\geq-4$.
For n=2, The sequence $TCATCA$ has only two pairs of (T, A) and
$E_{1,6}(TCATCA)=-8$. So, for any DNA sequence over $S^{2}$, $E_{1,6}\geq-8$.
Let it be true for a positive integer k. Then for any DNA sequences over
$S^{k}$, $E_{1,3k}\geq-4k$. On sequences over $S^{k}$, only the sequence
$TCA\cdots TCA$ has $k$ pairs of (T, A) on it and
$E_{1,3k}\underbrace{(TCA\cdots TCA)}_{\text{k times}}=-4k$.
For n=k+1, The sequence $TCA\cdots TCA$ has the highest number of (T, A) pairs
and
$E_{1,3(k+1)}\underbrace{(TCA\cdots TCA)}_{\text{k+1 times}}=-4(k+1)$. Hence
for any DNA sequences over $S^{k}$, $E_{1,3(k+1)}\geq-4(k+1)$.
So the proposition is true for $n=k+1$, and hence by the method of
mathematical induction, the proposition is true for any positive integer $n$.
$\square$
|
Variational Phase Estimation with Variational Fast Forwarding
Maria-Andreea Filip
13-15 Hills Road, CB2 1NL, Cambridge, United Kingdom
David Muñoz Ramo
13-15 Hills Road, CB2 1NL, Cambridge, United Kingdom
Nathan Fitzpatrick
13-15 Hills Road, CB2 1NL, Cambridge, United Kingdom
Subspace diagonalisation methods have appeared recently as promising means to access the ground state and some
excited states of molecular Hamiltonians by classically diagonalising small matrices, whose elements can
be efficiently obtained by a quantum computer. The recently proposed Variational Quantum Phase Estimation (VQPE)
algorithm uses a basis of real time-evolved states, for which the energy eigenvalues can be obtained directly
from the unitary matrix $\mathbf{U} = e^{-i\mathbf{H}\Delta t}$, which can be computed with cost linear in the
number of states used. In this paper, we report a circuit-based implementation of VQPE for arbitrary molecular
systems and assess its performance and costs for the H2, H3+ and H6 molecules. We also propose using
Variational Fast Forwarding (VFF) to decrease to quantum depth of time-evolution circuits for use in VQPE. We show
that the approximation provides a good basis for Hamiltonian diagonalisation even when its fidelity to the true time
evolved states is low. In the high fidelity case, we show that the approximate unitary $\mathbf{U}$ can be diagonalised
instead, preserving the linear cost of exact VQPE.
|
Free precategories as presheaf categories
1]Simon ForestThe work of this author
was partially supported by the French ANR project PPS (ANR-19-CE48-0014).
2]Samuel Mimram
[1]Aix-Marseille Univ, CNRS, I2M, Marseille, France
[2]LIX, École Polytechnique, Palaiseau, France
Precategories generalize both the notions of strict $n$-category and
sesquicategory: their definition is essentially the same as the one of strict
$n$-categories, excepting that we do not require the various interchange laws
to hold. Those have been proposed as a framework in which one can express
semi-strict definitions of weak higher categories: in dimension 3, Gray
categories are an instance of them and have been shown to be equivalent to
tricategories, and definitions of semi-strict tetracategories have been
proposed, and used as the basis of proof assistants such as Globular.
In this article, we are mostly interested in free precategories. Those can be
presented by generators and relations, using an appropriate variation on the
notion of polygraph (aka computad), and earlier works have shown that the
theory of rewriting can be generalized to this setting, enjoying most of the
fundamental constructions and properties which can be found in the traditional
theory, contrarily to polygraphs for strict categories.
We further study here why this is the case, by providing several results which
show that precategories and their associated polygraphs bear properties which
ensure that we have a good syntax for those. In particular, we show that the
category of polygraphs for precategories form a presheaf category.
§ INTRODUCTION
Strict polygraphs.
The notion of polygraph, also known as computad, was introduced by
Street <cit.> and Burroni <cit.> as a
generalization of the notion of presentation for strict $n$-categories, thus
extending the now classical notions of presentation for groups and monoids
introduced by Dehn <cit.> and
Thue <cit.>. From an algebraic point of view, they constitute
the right notion of “free $n$-category”, in the sense that they have been
established as being the cofibrant objects in the folk model structure on the
category of $n$-categories <cit.>. They thus
allow for computing various invariants of categories, as well as showing
coherence theorems, based on the construction of resolutions (or cofibrant
replacements) of categories of interest. For this reason, one is often
interested in constructing coherent presentations of low-dimensional categories,
which are polygraphs whose underlying free category is suitably equivalent to
the original one.
In order to be able to perform practical computations, one is generally looking
for polygraphs which are as small as possible. This task can be often be
achieved by using techniques originating from rewriting
theory <cit.>, suitably generalized to this
setting, which exploits the orientation of relations in a presentation. Namely,
when the presentation is terminating and confluent, generators corresponding to
relations between relations can be found as confluence diagrams for critical
branchings. This idea originates in the works of Squier on presented
monoids <cit.> and has been
the starting point of a series of works exploring higher dimensional
rewriting <cit.>, which has since then been further generalized to
various algebraic structures such as term rewriting
systems <cit.>, algebras <cit.> or
operads <cit.>.
While polygraphs have thus been proved to be quite a useful tool, they are still
quite unsatisfactory on many aspects.
Limitations of strict polygraphs.
From a categorical point of view, strict polygraphs are adapted to strict $n$-categories, but
those are known not to be equivalent to weak $n$-categories, which are the real
objects of interest. Namely, already starting from dimension $3$, not every
tricategory is equivalent to a $3$-category: the best we can do is to strictify
associativity and unitality, and show that every tricategory is equivalent to a
Gray category <cit.> (we should underline here that this is
not the only possible partial strictification <cit.>). Following
our terminology, a Gray category is a $3$-precategory equipped with interchange
isomorphisms satisfying suitable axioms.
Another categorical defect of polygraphs is the fact that they do not form a
presheaf category. It is namely noted in <cit.> that this
cannot be the case because of “the lack of an ordering” of $2$-dimensional
(and higher) cells, since composition is commutative for $2$-cells with identity
source and target. More formally, an abstract explanation of the fact that
polygraphs do not form a presheaf category can be found in <cit.> and
an elementary proof of this fact can be found in <cit.>. One route to
solve this consists in restricting to polygraph where generators do not have
identity sources (or targets), which has successfully been explored by
Henry <cit.>. Our exploration consists here in
taking the other route and “add ordering” to morphisms.
From a rewriting point and computer science point of view, polygraphs, when
considered as rewriting systems, lack a fundamental property found in most
settings for rewriting: we expect that a finite rewriting system has a finite number of
critical branchings. This was first observed by Lafont <cit.>
and further studied by Guiraud and Malbos who showed that, because of this,
there are finite convergent $3$-polygraphs without finite derivation
type <cit.>.
From a practical perspective, this causes problems. Namely, representing the
possibly infinite families of critical branchings is a difficult challenge, even
in low dimensions <cit.>. But in fact, even providing a
concrete representation of morphisms is a challenge, because there is no
canonical representative of morphisms in free categories, up to the axioms of
strict $n$-categories.
Polygraphs for precategories.
For these reasons, it seems natural to investigate the framework of
$n$-precategories whose definition is similar to the one of strict
$n$-categories, excepting that we do not require the interchange laws to
hold. In particular, in dimension $2$, those correspond to Street's
sesquicategories <cit.>.
We have defined in <cit.> an associated
notion of polygraph and developed a theory of rewriting in this setting
(interestingly, Araújo has recently independently come up with a very
similar notion <cit.>). It seems that, in this setting, most
of the limitations mentioned above vanish.
First, we now have canonical representatives of morphisms in free
$n$-precategories <cit.>, a property which was first
observed by Makkai while studying strict
$n$-categories <cit.>, which makes them suitable for
implementing software performing computation on morphisms. For this reason, they
are also used internally in the Globular graphical proof
assistant <cit.>.
Second, a finite rewriting system has a finite number of critical branchings,
and those can be computed effectively.
Third, we have a hope of being able to deal with weak higher-categories in this
setting. Namely, we have already mentioned that Gray categories are equivalent
to tricategories and are particular $3$-precategories, and putative definitions
of semistrict $4$-categories based on $4$-precategories have been
proposed <cit.>. Note that the polygraph corresponding to a Gray
category is almost never finite, but the infinite families of generators we add
are regular enough to be dealt with in a uniform
way <cit.>.
Properties of polygraphs.
In this article, we further study of the category of $n$-polygraphs for
$n$-precategories. Most importantly, we show that they form a presheaf
category. Our proof is based on the characterization of concrete presheaf
categories given by Makkai <cit.>. Simultaneously and
independently, another proof of this result has been given by
Araújo <cit.>. We should also mention that a notion of
polygraph for weak categories has been developed and shown to be a presheaf
category in <cit.>. Our approach gives rise to much smaller
polygraphs and thus more amenable computations, although it is not entirely
clear (yet) how to encode weak $n$-categories in our setting, excepting in low
Plan of the paper.
We begin by introducing precategories and associated polygraphs
(<Ref>) and show that functors between precategories induced by
polygraphs have the important property of being Conduché (<Ref>),
which is used subsequently. Most of the remainder of the paper is devoted to
showing that polygraphs form a presheaf category. Our proof is based on Makkai's
theorem characterizing presheaf categories (recalled in <Ref>). In
order to make computations on cells in free precategories, it is useful to
consider their support (<Ref>). These allow defining and studying
polyplexes (<Ref>) which are shapes parametrizing compositions in
precategories. This finally allows us to show that polygraphs form a presheaf
category (<Ref>). As a nice by-product, we derive a parametric
adjunction together with an associated generic-free factorization for
precategories, which gives a more conceptual view of the good syntactical
properties of precategories (<Ref>). Finally, we leave two
open questions on homotopical aspects of polygraphs of precategories. First,
whether polygraphs are the cofibrant objects for a reasonable model structure on
precategories (we explain that the usual proof for strict categories does not
immediately generalize to precategories), and second, whether the presheaf
category of polygraphs is able to model homotopy types (we explain why the proof
used by Henry for regular plexes <cit.> does not adapt here)
§ PRECATEGORIES AND THEIR POLYGRAPHS
We recall here the definition of $n$as algebras over globular
sets, as well as their elementary properties. We also recall the associated
notion of polygraph, introduced in earlier
works <cit.>, which is a particular
instance of the very general notion of polygraph associated to a monad on
globular sets introduced by Batanin <cit.>.
The notion of precategory was first introduced by Street, in dimension $2$,
under the name of sesquicategory: this means a
“$1\text{\textonehalf}$-category”, since sesquicategories have more structure
than $1$-categories, but less than $2$-categories (they lack the interchange
law). The general definition of precategories was (implicitly) given by Makkai
in <cit.>, who used them to deal with the word problem
for free strict categories. Later, they were used as data structures for the
Globular proof assistant <cit.> and more recently for studying
coherent presentations of Gray categories in <cit.> and
coherence for adjunctions <cit.>.
In the following, given $n\in\N$, we write $\Nlt n$ for the subset
$\set{0,\ldots,n-1}$ of $\N$, and $\Nle n$ for $\Nlt{n+1}$.
Globular sets.
Given $n\in\N\cup \set\omega$, an globular set$n$-globular
set $(X,\csrc,\ctgt)$ (often simply denoted $X$) is the data of sets $X_k$
for $k\in \Nle n$ together with (dround)$\csrc_i,\ctgt_i$the source
and target operations of a globular setfunctions $\csrc_i,\ctgt_i \co X_{i+1}
\to X_i$ for $i \in \Nlt{n}$ as in
\[
\begin{tikzcd}
\ar[l,shift right,"\csrc_0"']
\ar[l,shift left,"\ctgt_0"]
\ar[l,shift right,"\csrc_1"']
\ar[l,shift left,"\ctgt_1"]
\ar[l,shift right,"\csrc_2"']
\ar[l,shift left,"\ctgt_2"]
\cdots
\ar[l,shift right,"\csrc_{k-1}"']
\ar[l,shift left,"\ctgt_{k-1}"]
\ar[l,shift right,"\csrc_{k}"']
\ar[l,shift left,"\ctgt_{k}"]
\ar[l,shift right,"\csrc_{k+1}"']
\ar[l,shift left,"\ctgt_{k+1}"]
\cdots
\end{tikzcd}
\]
such that
\begin{align*}
\csrc_i\circ \csrc_{i+1}&=\csrc_i\circ \ctgt_{i+1}
\ctgt_i\circ \csrc_{i+1}&=\ctgt_i\circ \ctgt_{i+1}
\end{align*}
for $i \in \Nlt{n}$.
When there is no ambiguity on $i$, we often write $\csrc$ and $\ctgt$
for $\csrc_i$ and $\ctgt_i$ respectively. An element $u$ of $X_i$ is called an
globe$i$-globe of $X$ and, for $i>0$, the globes $\csrc_{i-1}(u)$
and $\ctgt_{i-1}(u)$ are respectively called the
source!of a globesource and
target!of a globetarget and $u$. Given $n$sets $X$
and $Y$, a
morphism!of globular setsmorphism of $n$sets
between $X$ and $Y$ is a family of functions $F = (F_k\co X_k\to Y_k)_{k \in
\Nle n}$, such that
\[
\csrc_i\circ F_{i+1}=F_i\circ \csrc_i
\]
for $i \in \Nlt{n}$.
We (Glob)$\nGlob n$the category of $n$setswrite $\nGlob
n$ for the category of $n$sets. We have canonical truncation and
inclusion functors
\[
\gtruncf_n \co \nGlob{n+1} \to \nGlob n
\qqand
\gincf_n \co \nGlob n \to \nGlob{n+1}
\]
which respectively forget the $(n{+}1)$and add an empty set of
$(n{+}1)$. They organize into an adjunction $\gincf_{n+1} \dashv
\gtruncf_n$.
It is direct from definition that globular sets are the models of an
(essentially) algebraic theory, so that the category $\nGlob n$ is essentially
algebraic. In particular, it implies that it is locally finitely presentable,
complete and cocomplete <cit.>.
For $\eps\in\set{-,+}$ and $j\geq 0$ with $j \le n-i$, we define a morphism
$\csrctgt\eps_{i,j}:X_{i+j}\to X_i$ by
\[
\csrctgt\eps_{i,j}=\csrctgt\eps_{i}\circ\csrctgt\eps_{i+1}\circ\cdots\circ\csrctgt\eps_{i+j-1}
\]
called the source!iteratediterated source (target!iteratedtarget) operation when $\eps=-$ ( $\eps=+$). We generally omit the index $j$ when there is no ambiguity and
simply write $\csrctgt\eps_i(u)$ for $\csrctgt\eps_{i,j}(u)$.
Given $i,k,l\in\Nle n$ with $i < \min(k,l)$, we
(.abXkxXl)$X_k\times_i X_l$the set of pairs of $i$ $k$-
and $l$write $X_k\times_i X_l$ for the pullback
\[
\begin{tikzcd}
X_k\times_i X_l \ar[r,dotted] \ar[d,dotted]
\ar[rd,phantom,"\lrcorner",very near start]& X_l\ar[d,"\csrc_i"]\\
X_k\ar[r,"\ctgt_i"']& X_i \pbox.
\end{tikzcd}
\]
Given $p\in\N$ and $k_0,\ldots,k_p \in \Nle n$, a sequence of
globes $u_0 \in X_{k_0}, \ldots, u_p \in X_{k_p}$ is said
composable$i$ for some $i < \min(k_0,\ldots,k_p)$,
when $\ctgt_{i}(u_j) = \csrc_i(u_{j+1})$ for $j \in \Nlt{p}$.
Given $k \in \Nle n$ and $u,v \in X_{k}$, $u$ and $v$ are said
parallelparallel when ${k = 0}$
or $\csrctgt\eps_{k-1}(u) = \csrctgt\eps_{k-1}(v)$ for ${\eps \in
\set{-,+}}$. In order to avoid dealing with the side condition $k = 0$, we use
the convention that $X_{-1}$ is the terminal set $\set{\ast}$ and
that $\csrc_{-1},\ctgt_{-1}$ are the unique function $X_0 \to X_{-1}$.
Given $n\in\N \cup \set\omega$, an precategory$n$ $C$
is an $n$set (whose $k$are called $k$ in this
context) together with, for $k \in \Nlt{n}$, identity
(idabpcat)$\unitp k {}$the identity operation for
\[
\unitp{k+1} {}\colon C_{k}\to C_{k+1}
\]
for which we use the same notation conventions than the identity operations on
strict categories,
and, for $k,l \in \N^*_n$, composition (.cab)$\pcomp_i$the
composition operation for precategoriesoperations
\[
\pcomp_{k,l}\co C_k\times_{\min(k,l)-1}C_l\to C_{\max(k,l)}
\]
which satisfy the axioms below.
Given $i,k,l \in \Nle n$ with ${i = \min(k,l)}$, since the dimensions of the
cells determine the indices of the composition to be used, we often
write $\pcomp_i$ for $\pcomp_{k,l}$. In this way, we still make explicit the
most important information which is the dimension $i$ of composition.
The axioms of $n$are the following:
* for $k \in \Nlt{n}$ and $u\in C_k$,
\[
\csrc_k(\unitp {k+1}{u})=u=\ctgt_k(\unitp {k+1}{u}),
\]
* for $i,k,l \in \Nle n$ such that $i = \min(k,l) -1$, $(u,v)\in C_k\times_iC_l$, and ${\eps \in \set{-,+}}$,
\begin{align*}
\csrctgt\eps(u \pcomp_i v)
\begin{cases}
u\pcomp_i \csrctgt\eps(v)& \text{if~$k < l$,}\\
\csrc(u)&\text{if~$k=l$ and~$\eps = -$,}\\
\ctgt(v)&\text{if~$k=l$ and~$\eps = +$,}\\
\csrctgt\eps(u)\pcomp_i v&\text{if~$k>l$,}
\end{cases}
\end{align*}
* for $i,k,l \in \Nle n$ with $i = \min(k,l) -
1$, given $(u,v)\in C_{k-1}\times_{i}C_l$,
\begin{align*}
\unit u\pcomp_i v
\begin{cases}
v&\text{if~$k \le l$,}\\
\unit{u\pcomp_i v}&\text{if~$k > l$,}
\end{cases}
\shortintertext{and, given~$(u,v) \in C_k \times_i C_{l-1}$,}
\begin{cases}
u&\text{if~$l\le k$,}\\
\unit{u\pcomp_i v}&\text{if~$l > k$,}
\end{cases}
\end{align*}
* for $i,k,l,m \in \Nle n$ with $i = \min(k,l) - 1 =
\min(l,m) - 1$, and $u \in C_k$, $v \in C_l$ and $w \in C_w$ such that $u,v,w$
are $i$,
\[
\]
* for $i,j,k,l,l' \in \Nle n$ such that
\[
i = \min(k,\max(l,l')) - 1,
\qquad
j = \min(l,l') - 1
\qqtand
i < j\zbox,
\]
given $u \in C_k$ and $(v,v') \in C_l \times_j C_{l'}$ such that $u,v$ are $i$,
\[
u \pcomp_i (v \pcomp_j v') = (u \pcomp_i v) \pcomp_j (u \pcomp_i v')
\]
and, given $(u,u') \in C_l \times_j C_{l'}$ and $v \in C_k$ such that $u,v$
are $i$,
\[
(u \pcomp_j u') \pcomp_i v = (u \pcomp_i v) \pcomp_j (u' \pcomp_i v).
\]
Note that, provided that the Axioms <ref>
to <ref> are satisfied, precat:distrib can be shown
equivalent to the more symmetrical axiom
* for every $i,j,k \in \Nle{n}$ satisfying $i < j < k$, and
cells $u_1,u_2 \in C_{i+1}$, $v_1,v_2 \in C_{j+1}$ and $w \in C_k$ such
that $u_1,w,u_2$ are $i$and $v_1,w,v_2$ are $j$, we
\[
u_1 \pcomp_i (v_1 \pcomp_j w \pcomp_j v_2) \pcomp_i u_2 = (u_1 \pcomp_i
v_1 \pcomp_i u_2) \pcomp_j (u_1
\pcomp_i w \pcomp_i u_2) \pcomp_j (u_1 \pcomp_i v_2 \pcomp_i u_2)\zbox.
\]
Given a $2$ $C$ with two $2$ $\phi$ and $\psi$ as in
\[
\begin{tikzcd}
\ar[r,bend left=60,"f",""{auto=false,name=topl}]
\ar[r,bend right=60,"{f'}"'{pos=0.52},""{auto=false,name=botl}]
\ar[r,bend left=60,"g",""{auto=false,name=topr}]
\ar[r,bend right=60,"{g'}"',""{auto=false,name=botr}]
\ar[from=topl,to=botl,phantom,"\Downarrow\phi"]
\ar[from=topr,to=botr,phantom,"\Downarrow\psi"]
\end{tikzcd}
\]
there are two ways to compose $\phi$ and $\psi$ together, given by
\[
(\phi \pcomp_0 g) \pcomp_1 (f' \pcomp_0 \psi)
\qtand
(f \pcomp_0 \psi) \pcomp_1 (\phi \pcomp_0 g')
\]
that can be represented using string diagrams by
\[
\satex{wires-left-right}
\quad\qqtand\quad
\satex{wires-right-left}
\]
and these two composites are not expected to be equal in $C$. Moreover, by our
definition of precategories, there is no such thing as a valid cell $\phi
\pcomp_0 \psi$, and the string diagram
\[
\satex{wires-mid-mid}
\]
makes no sense in this setting.
Given two $n$ $C$ and $D$, a morphism!of
precategoriesmorphism of $n$-precategories (or
prefunctor@$n$$n$) between $C$ and $D$ is a
morphism of $n$sets $F\co C \to D$ such that
* $F(\unitp {k+1} u) = \unitp {k+1} {F(u)}$ for $k \in \Nlt{n}$ and $u \in C_k$,
* $F(u \pcomp_i v) = F(u) \pcomp_i F(v)$ for $i,k,l \in \Nle n$ with $i =
\min(k,l) - 1$ and ${(u,v) \in C_k \times_i C_l}$.
We (PCatn)$\nPCat n$the category of $n$write $\nPCat
n$ for the category of $n$thus defined. We have canonical
truncation and inclusion functors
\[
\pctruncf_n \co \nPCat{n+1} \to \nPCat n
\qqand
\pcincf_n \co \nPCat n \to \nPCat{n+1}
\]
which respectively forget the $(n{+}1)$and add a set of $(n{+}1)$consisting of formal identities of $n$. They organize into an adjunction
$\pcincf_{n+1} \dashv \pctruncf_n$.
The globular monad of $n$-precategories.
The above definition of $n$directly translates into an
essentially algebraic theory so that the category $\nPCat n$ is locally finitely
presentable <cit.>.
There is a forgetful functor
\[
\ccfgff_n \co \nPCat n \to \nGlob n
\]
which maps an $n$to its underlying $n$set, and this
functor is induced by the inclusion of the essentially algebraic theory of
$n$-globular sets into the one of $n$-precategories. We thus have the
following <cit.>:
The category $\nPCat n$ is locally finitely presentable, complete and
cocomplete. Moreover, the functor $\ccfgff_n$ is a right adjoint which
preserves directed colimits.
The above proposition states the existence of a functor
\[
\freealgf_n \co \nGlob n \to \nPCat n
\]
which is left adjoint to $\ccfgff_n$, sending an $n$-globular set to the
$n$-precategory it freely generates. Moreover, the functor $\ccfgff_n$ can be
shown monadic using Beck's monadicity
theorem <cit.>:
For every $n \in \N\cup\set\omega$, the functor $\ccfgff_n$ is monadic.
This shows that, for $n\in \Ninf$, $\nPCat n$ is the category of algebras for a
monad $T^n \co \nGlob n \to \nGlob n$ on $n$sets (the monad induced by
the above adjunction).
Polygraphs of precategories.
In fact, for $n \in \N$, the monads $T^n$ is adequately derived by truncation
from $T^\omega$ <cit.>, the latter being
truncable in the sense of Batatnin <cit.>. By
general arguments on globular algebras, this allows the definition of
polygraphs for the theory of precategories.
The category of $n$ $\nPol n$ (for $n$-precategories) is
defined by induction on $n$, together with a functor
\[
\freecat[n]- \co \nPol n \to \nPCat n
\]
often written $\freecat-$, which associates to an $n$-polygraph the
$n$-precategory it freely generates, as follows.
We first define $\nPol 0 = \nGlob 0$ (which is isomorphic to $\Set$) and
$\freecat[0]- = \freealgf_0$ (which is the identity functor on $\Set$). Now,
given $n \in \N$, assuming $\nPol n$ and $\freecat[n]-$ defined in dimension
$n$, we define $\nPol{n+1}$ as the pullback
\[
\begin{tikzcd}[column sep=12mm]
\nPol{n+1}
\ar[r,dashed,"\poltoglob_{n+1}"]
\ar[d,dashed,"\poltruncf_n"']
\ar[phantom,dr,very near start,"\lrcorner"]
\nGlob{n+1}
\ar[d,"\gtruncf_n"]
\\
\nPol n
\ar[r,"{\fgfalgf_n \freecat[n]-}"']
\nGlob n
\end{tikzcd}
\]
The functor $\poltruncf_n:\nPol{n+1}\to\nPol{n}$, called the
$n$-truncation functor for polygraphs, admits a left adjoint
$\polincf_{n+1}:\nPol{n}\to\nPol{n+1}$, which extends an $n$-polygraph $\P$ as
an $(n{+}1)$-polygraph with an empty set of $(n{+}1)$-generators (using the
description of polygraphs given just below).
The image $\freecat\P$ under $\freecat[n+1]-$ of an $(n{+}1)$ $\P$ is
defined as the pushout
\[
\begin{tikzcd}[column sep={16em,between origins}]
\freealgf_{n+1}\gincf_{n+1} \gtruncf_n \poltoglob_{n+1}\P
\ar[r,"(\freealgf_{n+1} \gtrunccu_n\poltoglob_{n+1})_{\P}"]
\ar[d,"\alpha_\P"']
\freealgf_{n+1}\poltoglob_{n+1}\P
\ar[d,dashed]
\\
\pcincf_{n+1}\freecat {(\poltruncf_n\P)}
\ar[r,dashed]
\freecat\P
\phar[lu,"\ulcorner",very near start]
\end{tikzcd}
\]
where $\gtrunccu_n$ is the counit of the adjunction
$\gincf_{n+1} \dashv \gtruncf_n$ and $\alpha_\P$ is the composite
\[
\alpha_\P
\qeq
\begin{tikzcd}[column sep=5em]
\freealgf_{n+1}\gincf_{n+1} \gtruncf_n \poltoglob_{n+1}\P
\ar[r,"\sim"]
\pcincf_{n+1}\freealgf_{n} \fgfalgf_n \freecat[n]-\poltruncf_{n}\P
\ar[r,"{(\pcincf_{n+1}\varepsilon_n \freecat[n]-\poltruncf_{n})_{\P}}"]
\pcincf_{n+1}\freecat {(\poltruncf_n\P)}
\end{tikzcd}
\]
where $\varepsilon_n$ is the counit of the adjunction $\freealgf_n \dashv \fgfalgf_n$.
Intuitively, $\freecat \P$ is obtained by freely generating an
$(n{+}1)$from $\freecat{(\poltruncf_n \P)}$ by attaching the
$(n{+}1)$described by $\poltoglob_{n+1}\P$.
The mapping $\P \mapsto \freecat \P$ then naturally extends to a functor
$\freecat[n]- \co \nPol {n+1} \to \nPCat n$, which concludes the inductive
definition of polygraphs of precategories. More details on this construction can
be found in <cit.>.
Since the monad of the theory of precategory is truncable, given $n \in \N$, an
$n$ $\P$ can be alternatively described as a diagram in $\Set$ of the
\[
\begin{tikzcd}[column sep=10ex,labels={inner sep=0.5pt}]
\P_0\ar[d,"\polinj0"{inner sep=2pt}]
\ar[dl,shift right,"\gsrc_0"',pos=0.3]
\ar[dl,shift left,"\gsrc_0",pos=0.3]\ar[d,"\polinj1"{inner sep=2pt}]
&\P_2\ar[dl,shift right,"\gsrc_1"',pos=0.3]\ar[dl,shift left,"\gsrc_1",pos=0.3]\ar[d,"\polinj2"{inner sep=2pt}]
\ar[dl,shift right,"\gsrc_{n-2}"',pos=0.3]
\ar[dl,shift
left,"\gsrc_{n-2}",pos=0.3]\ar[d,"\polinj{n}"{inner sep=2pt}]
&\P_{n}\ar[dl,shift right,"\gsrc_{n-1}"',pos=0.3]\ar[dl,shift left,"\gsrc_{n-1}",pos=0.3]\\
\freecat{\P_0}
\freecat{\P_1}
\ar[l,shift right,"{\csrc_0}"']
\ar[l,shift
\ar[l,shift
\end{tikzcd}
\]
where, for $i \in \Nlt{n}$, $\polinj i$ is the embedding of the $i$$\P_i$ into the set $\freecat{\P_i}$ of freely generated $i$, such that
\[
\qqtand
\ctgt_i\circ\gsrc_{i+1}=\ctgt_i\circ\gtgt_{i+1}
\]
for $i \in \Nlt{n}$. Note that the above description is the same as the original
definition of polygraphs by Burroni <cit.>, excepting that the
sets $\freecat{\P_i}$ of $i$ cells are freely generated as $i$-precategories
instead of strict $i$-categories.
By general properties on locally presentable categories, we have:
Given $n \in \Ninf$, $\nPol n$ is a locally finitely presentable category. In
particular, it is complete and cocomplete.
The $2$of locally presentable categories, right adjoints (left
adjoints) and natural transformations is closed under bipullbacks (see
<cit.>). A pullback along an
isofibration happens to be a bipullback and the pullback of $\gtruncf_n$ along
$\fgfalgf_n$ can be shown to be a left adjoint and again an isofibration.
Then, its pullback by $\freecat[n]-$, which is known (see <cit.>) to be a left adjoint, is again a left adjoint
whose domain $\nPol n$ is a locally presentable category. A more detailed
study shows that $\nPol n$ is locally finitely presentable with finite
polygraphs as finitely presentable objects. See <cit.> for the local presentability and <cit.> for the local finite presentability.
In the following, we will write $\polterm$ for the terminal object of $\nPol n$,
for $n \in \Ninf$.
§ FREE FUNCTORS ARE CONDUCHÉ
Free precategories on polygraphs enjoy useful properties, thanks to which we
have a nice syntax for morphisms in those, as we now show. It should be noted
that many those are not valid in the usual setting of polygraph for strict
categories (as opposed to precategories).
One remarkable such property of free precategories is that their cells can be
described as canonical compositions of generators, which happen to be unique for
a given cell, so that we prefer to call them normal forms. These normal
forms are adequately reflected by free functors, since the latter reflect
elementary compositions: in other words, they satisfy the analogue of the
Conduché property for strict categories <cit.>.
In addition to providing convenient tools in the proofs, we will see in
subsequent sections that these properties entail the existence universal shapes
of compositions.
*Types and contexts.
Given $m \le n \in \N$, an $n$$C$, an $m$ is a pair of
parallel $(m{-}1)$of $C$. We use the convention that there is a unique
$0$, and all pairs of $0$of a precategory are parallel. Given a
$k$$u \in C$ for some $k \ge m$, $u$ has a canonical associated $m$:
$(\csrc_{m-1}(u),\ctgt_{m-1)}(u))$. In the following, an $m$is thought of
as the type for a formal variable, which suggests defining the notion of context
(a morphism in which the variable occurs exactly once) and of substitution
(replacing the variable by a morphism).
An $m$ $E$ for an $m$$(s,t)$ is defined by induction
on $m$, together with the evaluation $E[u]$ of $E$ at a cell of $m$$(s,t)$:
* there is a unique $0$of the unique $0$, denoted $[-]$, and
the evaluation of it at a cell $u \in C$ is $u$,
* an $(m{+}1)$of type $(s,t)$ is a triple $E = (l,E',r)$ with
$l,r \in C_m$, and $E'$ an $m$of type
$(\csrc_{m-1}(s),\ctgt_{m-1}(t))$ such that $\ctgt_{m}(l) = E'[s]$ and
$E'[t] = \csrc_m(r)$, and the evaluation $E[u]$ of $E$ at a cell $u$ is
defined by $E[u] = l \comp_m E'[u] \comp_m r$.
Alternatively, an $m$$E$ can be thought of as an expression of the form
\[
l_m \comp_{m-1} ( \cdots \comp_1 (l_1 \comp_0[-]\comp_0 r_1) \comp_1 \cdots ) \comp_{m-1} r_m
\]
where the $l_i,r_i \in C_i$ are the $i$occurring in the definition of $E$
for $i \in \Nle m$, and its evaluation at a cell $u$ as the cell obtained by
replacing $[-]$ by $u$ in the above expression.
Normal forms.
We have the following normal form for the cells of free precategories:
Given $m \in \N$
and a polygraph $\P\in\oPol$, every $m$of $\freecat\P$ can be written
uniquely as
\[
E_1[g_1] \comp_{m-1} \cdots \comp_{m-1} E_k[g_k]
\]
for some unique $g_1,\cdots,g_k \in \P_m$ and $(m{-}1)$$E_1,\ldots,E_k$ of the corresponding types.
We only sketch the proof, which is detailed
in <cit.>. One can adequately orient the
axioms <ref>–<ref> of precategories in order to
obtain a terminating and locally confluent rewriting system on the formal
expressions of cells of free precategories. By standard arguments of rewriting
theory <cit.>, this gives the existence and unicity of normal
A consequence of the above theorem is that the embeddings $\polinj i \co \P_i
\to \freecat\P_i$ introduced earlier are injective. Thus, given $g \in \P_i$,
we will often omit $\polinj i$ and write $g$ for both the element of $\P_i$
and the cell of $\freecat\P_i$.
The unicity of normal forms directly entails the that the image under a free
functor of an identity is an identity (of a generator is a generator):
Let $F\co \P \to \Q \in \oPol$ be a morphism of polygraphs, $k \in \Nle n$
and $u \in \freecat\P_k$. The following hold:
* when $k > 0$, there exists a cell $u'
\in \freecat\P_{k-1}$ such that $u = \unitp k {u'}$ if and only if there
exists a cell $\tilde u' \in \freecat\Q_{k-1}$ such that $\freecat F(u) = \unitp k
{\tilde u'}$,
* there exists a generator $g \in \P_k$
such that $u = g$ if and only if there exists a generator $\tilde g \in
\Q_k$ such that $\freecat F(u) = \tilde g$.
We should also mention now that composition in free precategories is
cancellative. This does not seem to be deducible from the more general
properties developed in the next sections.
Given $\P \in \oPol$ and $u,v_1,v_2 \in \freecat \P$ such that $u \pcomp_i v_1
= u \pcomp_i v_2$ for some $i$, then $v_1 = v_2$.
Note that, by the input and output dimension conditions of $\pcomp_i$, we
necessarily have that the dimension of $v_1$ is the one of $v_2$. We do an
induction on the dimension of the resulting cell $u \comp_i v_1$ and
distinguish three cases depending on the relative dimensions of $u$, $v_1$ and
* Suppose that $u,v_1,v_2\in \freecat\P_{i+1}$. By unicity of the
decomposition of $(i{+}1)$of free precategories
(<Ref>) and its compatibility with $i$as
concatenation, we have $v_1 = v_2$.
* Suppose $u \in \freecat\P_{i+1}$ and $v_1,v_2 \in \freecat\P_{n}$ with
$n>i+1$. We reason by induction on $v_1$.
* Suppose that $v_1 = \alpha$ for some generator $\alpha \in
\P_n$. Then, by the definition of composition and the normal forms, we
have that $v_2 = \alpha$.
* Suppose that $v_1 = E_1[\alpha]$ for some generator $\alpha \in \P_n$
and $m$$E_1$ with $0 < m < n$. By the definition of composition
and the unicity of normal forms, we have $v_2 = E_2[\alpha]$. Let
$(l_j,E'_j,r_j) = E_j$ for $j \in \set{1,2}$. If $m = i+1$, then, by
unicity of normal forms, we have $u \pcomp_i l_1 = u \pcomp_i l_2$,
$E'_1[\alpha] = E'_2[\alpha]$ and $r_1 = r_2$. By the beginning of the
proof, we have $l_1 = l_2$, so that $v_1 = v_2$. Otherwise, if $m > i+1$,
then $u \pcomp_i l_1 = u \pcomp_i l_2$, $u \pcomp_i E'_1[\alpha] = u
\pcomp_i E'_2[\alpha]$ and $u \pcomp_i r_1 = u \pcomp_i r_2$. By the
different induction hypotheses, we have $l_1 = l_2$, $E'_1[\alpha] =
E'_2[\alpha]$ and $r_1 = r_2$, so that $v_1 = v_2$.
* If
$v_1 = E^1_1[\alpha_{1}] \pcomp_{n-1} \cdots \pcomp_{n-1}
E^k_1[\alpha_{k}]$, then we necessarily have
$v_2 = E^k_2[\alpha_1] \pcomp_{n-1} \cdots \pcomp_{n-1} E^k_2[\alpha_k]$
such that $u \pcomp_i E^j_1[\alpha_j] = u \pcomp_i E^j_2[\alpha_j]$. By
the previous argument, we have $E^j_1[\alpha_j] = E^j_2[\alpha_j]$, so
that $v_1 = v_2$.
* Suppose that $u \in \freecat\P_{n}$, $v_1,v_2 \in \freecat\P_{i+1}$ with
$n > i+1$. We reason by induction on $u$.
* If $u = \unit {u'}$, then we have $u' \pcomp_i v_1 = u' \pcomp_i v_2$,
so that $v_1 = v_2$ by induction.
* If $u = E[\alpha]$ for some $(n{-}1)$$E$, then let
$(l,E',r)=E$. We then have $r \pcomp_i v_1 = r \pcomp_i v_2$ so that, by
induction hypothesis, $v_1 = v_2$.
* If $u = E_1[\alpha_1] \comp_{n-1} \cdots \comp_{n-1} E_k[\alpha_k]$
for some $k \ge 1$, $\alpha_1,\ldots,\alpha_k \in \P_n$ and contexts
$E_1,\ldots,E_k$, then we have in particular
$E_1[\alpha_1] \comp_i v_1 = E_1[\alpha_1] \comp_i v_2$ so that we can
conclude $v_1 = v_2$ by the previous case.
* Suppose that $u,v_1,v_2 \in \freecat\P_{i+1}$, $v_1,v_2 \in
\freecat\P_{i+1}$ with $n > 0$. By the unicity of normal forms, we can
uniquely write $u$ as $E^1[\alpha^1] \comp_i \cdots \comp_i E^k[\alpha^k]$
and $v_j$ as $E^1_j[\beta^1_j] \comp_i \cdots \comp_i
E^{l_j}_j[\beta^{l_j}_j]$ for $j \in \set{1,2}$ for some adequate $k,l_1,l_2
\in \N$, $i$$E^{\cdot}$, $E^{\cdot}_1$, $E^{\cdot}_2$ and
$(i{+}1)$$\alpha^{\cdot}$, $\beta^\cdot_1$ and
$\beta^\cdot_2$. By considering the induced normal forms on $u \comp_i
v_1$ and $u \comp_i v_2$ by concatenation, we deduce by unicity of normal
forms that $l_1 = l_2$ and $E^\cdot_1 = E^\cdot_2$ and $\beta^\cdot_1
= \beta^\cdot_2$, so that $v_1 = v_2$.
Note that such a property does not hold for polygraphs of strict categories.
Indeed, considering the $2$of strict precategories $\P$ defined by
\begin{align*}
\P_0 &= \set{x}
\P_1 &= \set{f \co x \to x}
\P_2 &= \set{\alpha \co \unit x \To f}
\end{align*}
we have $\alpha \comp_0 \unit f \neq \unit f \comp_0 \alpha$ while
\[
\alpha \comp_1(\alpha \comp_0 \unit f)
\alpha \comp_0 \alpha
\alpha \comp_1 (\unit f \comp_0 \alpha)
\]
in the free strict $2$$\freecat\P$. Graphically,
\[
\begin{tikzcd}
\ar[rr,bend left=50,equals]
\ar[rr,bend left=25,phantom,"\alpha\!\Downarrow"]
\ar[r,bend left,equals]\ar[r,bend right,"f"']\ar[r,phantom,"\alpha\!\Downarrow"]
\end{tikzcd}
\begin{tikzcd}
\ar[r,bend left,equals]
\ar[r,phantom,"\alpha\!\Downarrow"]
\ar[r,bend right,"f"']
\ar[r,bend left,equals]
\ar[r,phantom,"\alpha\!\Downarrow"]
\ar[r,bend right,"f"']
\end{tikzcd}
\begin{tikzcd}
\ar[rr,bend left=50,equals]
\ar[rr,bend left=25,phantom,"\alpha\!\Downarrow"]
\ar[r,"f"']
\ar[r,bend left,equals]\ar[r,bend right,"f"']\ar[r,phantom,"\alpha\!\Downarrow"]
\end{tikzcd}
\]
*Conduché functors.
We now introduce the notion of (strict) Conduché functor for precategories,
following the work of Guetta in the case of strict
categories <cit.>. Informally, these functors have a
co-functoriality property, in the sense that cells mapped to composites are
themselves composites. The notion of weak Conduché functor was introduced by
Guiraud in a seemingly unrelated context <cit.> as a necessary
and sufficient condition for a functor $F \co C \to D$ between strict
$n$to be exponentiable, for the pullback functor
$F^\leftarrow \co \Cat/D \to \Cat/C$ to have a right adjoint.
Let $n \in \Ninf$, $C,D \in \nPCat n$ and $F \co C \to D$ be an
$n$. We say that $F$ is Conduché functor$n$
when it satisfies that, for all $i,k_1,k_2,k\in \N^*_n$
with $i = \min(k_1,k_2) - 1$ and $k = \max(k_1,k_2)$, $u \in C_k$,
$i$ $v_1 \in D_{k_1}$ and $v_2 \in D_{k_2}$ such
\[
F(u) = v_1 \pcomp_i v_2,
\]
there exist unique $i$ $u_1 \in C_{k_1} $ and $u_2\in C_{k_2}$ such
\[
F(u_1) = v_1 \qqtand F(u_2) = v_2 \qqtand u_1 \pcomp_i u_2 = u.
\]
As in the case of strict categories, the Conduché property implies a unique
lifting of identities:
Given $n \in \Ninf$ and an $n$prefunctor $F\co C \to D \in \nPCat
n$, if
\[
F(u) = \unit v
\]
for some $k \in \Nlt{n}$, ${u \in C_{k+1}}$, and ${v \in D_k}$, then there
exists a unique $u' \in C_k$ such that
\[
F(u') = v
\qqtand
u = \unit {u'}
\zbox.
\]
Since $\unit v = \unit v \pcomp_k \unit v$, by the Conduché property, there
exist unique $u_1,u_2 \in C_{k+1}$ such that $F(u_1) = \unit v$,
$F(u_2) = \unit v$ and $u = u_1 \pcomp_k u_2$. Moreover, we have that
$F(\unit {\csrc_k (u)}) = v$ and $u = \unit {\csrc_k (u)} \pcomp_k u$ so that
$u_1 = \unit {\csrc_k(u)}$ and $u_2 = u$. Symmetrically, we have that
$u_1 = u$ and $u_2 = \unit{\ctgt_k(u)}$. Thus,
$u = \unit {\csrc_k(u)} \pcomp_k \unit {\ctgt_k(u)}$, so that
$\csrc_k(u) = \ctgt_k(u)$ and $u = \unit{u'}$ with $u' =
\csrc_k(u)$. Uniqueness is immediate.
Unlike for strict categories, we have the remarkable property that all free
functors of precategories are Conduché:
Given $m \in \Ninf$ and a morphism $F \co \P \to \Q \in\nPol m$, the
prefunctor $\freecat F\co\freecat\P\to\freecat\Q$ is Conduché.
For the sake of simplicity, we only handle the case $m = \omega$. Suppose
given $n \in \N$ and an $n$-cell $u\in\freecat\P_n$ such that $F(u)=\bar
u_1\comp_{i}\bar u_2$. We reason by case analysis on the relative dimensions
of $\bar u_1$ and $\bar u_2$.
* If $\bar u_1,\bar u_2\in\freecat\Q_n$ then $i=n-1$. By the unicity of
normal forms and its compatibility with $\pcomp_{n-1}$, there are unique
$u_1,u_2$ such that $\freecat F(u_k) = \bar u_k$ for $k \in \set{1,2}$ and
$u = u_1 \pcomp_{n-1} u_2$.
* Suppose $\bar u_1\in \freecat \Q_{i+1}$ and
$\bar u_2 \in \freecat\Q_{n}$. If there are $u_1,u_2$ such that
$\freecat F(u_k) = \bar u_k$ for $k \in \set{1,2}$ and
$u = u_1 \comp_i u_u$, then they are unique since, by the previous point, $u_1$
and $\csrc_{i+1}(u_2)$ are uniquely determined by
$\csrc_{i+1}(u) = u_1 \pcomp_{i} u^-_2$ and $\freecat F(u_1) = \bar u_1$ and
$\freecat F(u^-_2) = \csrc_{i+1}(u_2)$. Moreover, since
$u = u_1 \pcomp_i u_2$, we have that $u_2$ is unique by
<Ref>. So unicity holds. For existence, we reason by
induction on $n$ and $\bar u_2$.
* If $\bar u_2 = \bar E[\bar \alpha]$ for some $(i{+}1)$$\bar E = (\bar l,\bar E',\bar r)$, then, by unicity of normal forms,
$u = E[\alpha]$ for some $\alpha \in \P$, and $(i{+}1)$$E = (l, E',r)$, and we moreover have
$\freecat F(l) = \bar u_1 \pcomp_i \bar l$,
$\freecat F(E[\alpha]) = \bar E[\bar \alpha]$ and
$\freecat F(r) = \bar r$. By the first part, there are $u_1$ and
$\tilde l$ such that $l = u_1 \pcomp_i \tilde l$, so that
$\tilde E = (\tilde l,E',r)$ satisfies that
$u = u_1 \pcomp_{i} \tilde E[\alpha]$, $\freecat F(u_1) = \bar u_1$ and
$\freecat F(\tilde E[\alpha]) = \bar E[\bar \alpha]$.
* If $\bar u_2 = \bar E[\bar \alpha]$ for some $(j{+}1)$$\bar E = (\bar l,\bar E',\bar r)$ with $j > i$, then, by unicity of
normal forms, $u = E[\alpha]$ for some $\alpha \in \P$, and
$(j{+}1)$$E = (l, E',r)$, and we moreover have
$\freecat F(l) = \bar u_1 \pcomp_i \bar l$,
$\freecat F(E'[\alpha]) = \bar u_1 \pcomp_i \bar E'[\bar \alpha]$ and
$\freecat F(r) = \bar u_1 \pcomp_i \bar r$. By the other induction
hypothesis, there are $u^1_1,u^2_1,u^3_1$, $\tilde l, \tilde u', \tilde r$
such that $l = u^1_1 \pcomp_i \tilde l$,
$E'[\alpha] = u^2_1 \pcomp_i \tilde u'$ and $r = u^3_1 \pcomp_i \tilde
r$. Since
$u^1_1 \pcomp_i \ctgt_j(\tilde l) = \ctgt_{j}(l) = \csrc_j(\tilde u') =
u^2_1 \pcomp_i \csrc_j(\bar E'[\bar \alpha])$ and
$\freecat F(u^1_1) = \freecat F(u^2_1) = \bar u_1$ and
$\freecat F(\ctgt_j(\tilde l)) = \freecat F(\csrc_j(\tilde u')) =
\ctgt_j(\bar l)$, by unicity, we have $u^1_1 = u^2_1$ and
$\ctgt_j(\tilde l) = \csrc_j(\tilde u')$. Similarly, $u^2_1 = u^3_1$ and
$\ctgt_j(\tilde u') = \csrc_j(\tilde r)$. Thus, writing $u_1$ for $u^1_1$
and $u_2$ for $\tilde l \pcomp_j \tilde u' \pcomp_j \tilde r$, we have
that $u = u_1 \pcomp_i u_2$ is the wanted decomposition for $u$.
* If
$\bar u_2 = \bar E_1[\bar \alpha_1] \pcomp_{n-1} \cdots \pcomp_{n-1} \bar
E_k[\bar \alpha_k]$, then, by unicity of normal forms, we have that
$u = E_1[\alpha_1] \pcomp_{n-1} \cdots \pcomp_{n-1} E_k[\alpha_k]$ such
$\freecat F (E_l[\alpha_l]) = \bar u_1 \pcomp_i \bar E_l[\bar
\alpha_l]$. By induction hypothesis, we get $u^l_1$ and $u^l_2$ such that
$E_l[\alpha_l] = u^l_1 \pcomp_i u^l_2$, $\freecat F(u^l_1) = \bar u_1$ and
$\freecat F(u^l_2) = \bar E_l[\bar \alpha_l]$. Using the same argument as
earlier, we get that $u^1_1 = \cdots = u^k_1$ and $u^1_2,\ldots,u^k_2$ are
$(n{-}1)$so that, writing $u_1$ for $u^1_1$ and $u_2$ for
$u^1_2 \pcomp_{n-1} \cdots \pcomp_{n-1} u^k_2$, we have a decomposition
$u = u_1 \pcomp_i u_2$ satisfying the wanted properties.
* Suppose $\bar u_1\in \freecat \Q_{n}$ and
$\bar u_2 \in \freecat\Q_{i+1}$. This case is similar to the previous
As a counter-example for the above property in the context of strict
categories, consider the polygraphs $\P$ and $\Q$ defined by
\begin{align*}
\P_0 &= \set{x}
\P_1 &= \emptyset
\P_2 &= \set{\alpha \co \unit x \To \unit x, \beta\co \unit x \To \unit x}
\\
\Q_0 &= \set{y}
\Q_1 &= \emptyset
\Q_2 &= \set{\gamma \co \unit y \To \unit y}
\zbox.
\end{align*}
We then have a morphism $F \co \P \to \Q$ sending $\alpha$ and $\beta$ to
$\gamma$, and the associated prefunctor $\freecat F$ sends both $\alpha \comp_0 \beta$
and $\beta \comp_0 \alpha$ to $\gamma \comp_0 \gamma$.
A nice application of the above Conduché properties is the characterization of
monomorphisms of polygraphs. First, we briefly observe the equivalence between
monomorphisms of precategories and dimensionwise injections.
Given $n \in \Ninf$ and $F \co C \to D \in \nPCat n$, the following are equivalent:
* $F$ is a monomorphism,
* $F_k$ is a monomorphism for every $k\le n$.
The theory of $n$is sketchable and the functor
$(-)_k \co \nPCat n \to \Set$, which to a precategory associates its set of
$k$-cells is induced by a sketch morphism. It is thus a right
adjoint <cit.>. In particular, it preserves monomorphisms. Thus,
<ref> implies
<ref>. Moreover, since the functors $(-)_k$
for $k < n+1$ are jointly faithful, we have that
<ref> implies
We then have the following characterization property for monomorphisms
of polygraphs, which are in particular preserved by the functor $\freecat-$:
Given $n \in \Ninf$ and $F \co \P \to \Q \in \nPol n$, the following are equivalent:
* $F$ is a monomorphism,
* $F_k$ is a monomorphism for every $k\le n$,
* $\freecat F$ is a
monorphism in $\nPCat n$.
We show this property by induction on $n$.
<Ref> clearly implies
Conversely, assuming <Ref>, by induction
hypothesis, we have that $F_k$ and $\freecat F_k$ are monomorphisms for $k <
n$. Now, let $x,y \in \P_n$ such that $F_n(x) = F_n(y)$. In particular, we
have $\freecat F_{n-1}(\csrctgt\eps(x)) = \freecat F_{n-1}(\csrctgt\eps(y))$
for $\eps \in\set{-,+}$, so that $\csrctgt\eps(x) = \csrctgt\eps(y)$ for
$\eps\in\set{-,+}$, by injectivity of $\freecat F_{n-1}$. Consider the
$n$$\R$ such that $\poltruncf_{n-1} \R = \poltruncf_{n-1} \P$ and
$\R_n = {z}$ with $\gsrctgt\eps_{n-1}(z) =\csrctgt\eps(x)$ for $\eps \in
\set{-,+}$. Then, we have two canonical morphisms $G^x,G^y \co \R \to \P$,
verifying $G^x(z) = x$ and $G^y(z) = y$. We then have $F\circ G^x = F\circ
G^y$, so that $G^x = G^y$ since $F$ is a monomorphism. In particular, we have
$x = y$. Thus, $F_n$ is injective, so <Ref>
By <Ref>, the embedding
$\polinje^\P_k:\P_k\to\freecat \P_k$ ($\polinje^\Q_k:\Q_k\to\freecat \Q_k$) is a monomorphism. Thus,
<Ref> implies
<Ref>, since
$\polinje^\Q_{k} \circ F_k = \freecat F_k \circ \polinje^\P_{k}$ and the
right-hand side of the latter equation is a monomorphism by
Conversely, assume <Ref>. Let $u,v \in
\freecat\P_n$ such that $\freecat F(u) = \freecat F(v)$. We show that $u = v$
by induction on an expression defining $u$. If $u = \unit{u'}$ for some $u'
\in \freecat\P_{n-1}$, by <Ref>, there
exists $v' \in \freecat\P_{n-1}$ such that $v = \unit{v'}$. We thus have
$\freecat F(u') = \freecat F(v')$ and $u' = v'$ by induction hypothesis. If $u
= u_1 \comp_i u_2$ for some $i < n$ and $i$$u_1,u_2 \in
\freecat\P$, then by <Ref>, there exists $i$$v_1,v_2 \in \freecat\P$ such that $v = v_1\comp_i v_2$ and $\freecat F(u_k) =
\freecat F(v_k) $ for $k \in \set{1,2}$, so that $u_k = v_k$ by induction
hypothesis, and $u = v$. Finally, if $u = \polinje_{\P,n}(g)$ for some $g \in \P_n$
then $v = \polinje^\P_n(h)$ for some $h \in \P_n$ by <Ref>.
But then, we have
\[
\polinje^\Q_n(F_n(g))
\freecat F(\polinje^\P_n(g))
\freecat F(\polinje^\P_n(h))
\polinje^\Q_n(F_n(h))
\]
where $\polinje^\Q_n \circ F_n$ is a monomorphism by hypothesis and
<Ref>. Thus, $g = h$ and $u = v$. Hence,
<Ref> holds.
§ MAKKAI'S CRITERION FOR PRESHEAF CATEGORIES
We now recall the criterion given by Makkai <cit.> to detect
whether a category $\cC$ is a presheaf category in the expected way, relatively to a concretization functor $\cC \to \Set$. In the case of a presheaf
category, the objects of the base category are recognized as the suitably
initial elements of the concretization. Makkai used this criterion to show
that polygraphs for strict categories do not form a presheaf categories in the
expected way, where the concretization functor maps a polygraph to the set of
all generators. We will use this criterion in <Ref> to prove that,
in the case of precategories, we do get a presheaf category.
A concrete categoryconcrete category is a category $\mcal C$
endowed with a functor
\[
\concrf[\mcal C]- \co \mcal C \to \Set\zbox.
\]
The above concretization functorconcretization functor should be
understood as a candidate set-theoretic representation of $\mcal C$: for
$c$ an object of $C$, the set $\concrf c$ describes the candidate elements of the associated
presheaf. The following canonical example should provide a good illustration of
this intuition.
Let $C$ be a small category. $\hat C$ has a canonical structure of concrete
category, where $\concrf[\hat C]-$ is defined on preasheaves $P \in \hat C$ by
\[
\concrf[\hat C]{P} = \bigsqcup_{c \in C_0} P(c)
\]
and extended naturally to morphisms between presheaves.
In the following, we will be interested in the concretization functor
given by the following example:
The functor $\poltoset{-}\co \nPol\omega \to \Set$ which maps $\P \in \nPol\omega$ to
\[
\poltoset \P = \bigsqcup_{k \in \N} \P_k
\]
equips $\nPol\omega$ with a structure of concrete category.
Later, we will study the properties of $\oPol$ equipped with the above
concretization functor. Another concretization functor on $\oPol$ that will be
of interest for us is given by the example below:
There is a functor $\cattoset-\co \nCat\omega \to \Set$ which
maps $C\in\nCat\omega$ to
\[
\cattoset C=\bigsqcup_{k \in \N}C_k
\]
By precomposition with the functor $(-)^*:\nPol\omega\to\nCat\omega$, we
obtain a functor $\polconcrb-\co \nPol\omega \to \Set$ which
maps $\P \in \nPol\omega$ to
\[
\polconcrb\P = \bigsqcup_{k \in \N} \freecat\P_k
\]
and also equips $\nPol\omega$ with a structure of concrete category.
In order to distinguish between the two preceding structures of concrete
category on $\oPol$, we use the convention that we write $\oPol$ when
considering the concrete category structure on $\oPol$ given by $\poltoset-$
(Polostar)$\oPolb$the category $\oPol$ equipped with the
concretization functor $\polconcrb -$and $\oPolb$ when considering the
concrete category structure on $\oPol$ given by $\polconcrb -$.
An equivalence of concrete categoriesequivalence of concrete
categories between concrete categories $(\mcal C,\concrf[\mcal C]{-})$
and $(\mcal D,\concrf[\mcal D]{-})$ is the data of an equivalence of
categories $\mcal E\co \mcal C \to \mcal D$ and a natural isomorphism
\[
{\Phi\co \concrf[\mcal D]{-} \circ \mcal E \To \concrf[\mcal C]{-}}\zbox.
\]
When such an equivalence exists, $(\mcal C,\concrf[\mcal C]{-})$ and $(\mcal
D,\concrf[\mcal D]{-})$ are said concretely equivalent
categoriesconcretely equivalent. One might then consider the following
natural question:
When is some concrete category $(\mcal C,\concrf[\mcal C]{-})$ concretely equivalent
to a presheaf category $(\hat C,\concrf[\hat C]{-})$ for some small
category $C$?
When it is the case, we say that $(\mcal C,\concrf[\mcal C]{-})$ is a
concrete presheaf categoryconcrete presheaf category.
Given a concrete category $(\mcal C,\concrf[\mcal C]{-})$, the
category of elements $\Elt(\mcal C)$ of $\mcal C$ is the category
* whose objects are the pairs $(X,x)$ where ${X \in \mcal C_0}$ and ${x \in
\concrf[\mcal C]{X}}$, and
* whose morphisms from $(X,x)$ to $(Y,y)$ are the morphisms $f \co X \to
Y \in \mcal C$ such that $\concrf[\mcal C]{f}(x) = y$.
Given a morphism $f\co (X,x) \to (Y,y)$ as above, we say that $y$ is a
specializationspecialization of $x$.
An object $(X,x) \in \Elt(\mcal C)$ is principal!elementprincipal when, for every
morphism $f\co (Y,y) \to (X,x) \in \Elt(\mcal C)$ such that $f$ is a
monomorphism in $\mcal C$, we have that $f$ is an isomorphism; it is
primitive elementprimitive when it is principal and, for all $f\co (Y,y) \to (X,x)
\in\Elt(\mcal C)$ where $(Y,y)$ is principal, $f$ is an isomorphism.
Let $C$ be a small category and consider the canonical concrete category
structure on $\hat C$ given by <Ref>. Given
$P\in\hat C$ and $c\in C$, we write $\iota_c:P(c)\to\bigsqcup_{c\in C}P(c)$
for the canonical injection. The category $\Elt(\hat C)$ has
* as objects the pairs $(P,\iota_c(x))$ where $P \in \hat C$ and $x \in
P(c)$, and
* as morphisms from $(P,\iota_c(x))$ to $(Q,\iota_d(y))$ the natural
transformations $\alpha\co P \To Q$ such that $c = d$ and $\alpha_c(x) = y$.
Given $(P,\iota_c(x)) \in \Elt(\hat C)$, we have the following.
* $(P,\iota_c(x))$ is principal when $P$ is the smallest subpresheaf $P'$
of $P$ such that $x \in P'(c)$. In particular, for
all $c \in C$, $(C(-,c),\iota_c(\unit c))\in \Elt(\hat C)$ is principal.
* $(P,\iota_c(x))$ is primitive when the natural transformation $\theta\co
C(-,c) \to P$ which maps $\unit c$ to $x$ is an isomorphism.
The characterization of concrete presheaf categories given by Makkai
is the following <cit.>:
Let $(\mcal C,\concrf[\mcal C]{-})$ be a concrete category. $\mcal C$ is
concretely equivalent to a presheaf category if and only if the following
conditions are all satisfied:
* $\concrf[\mcal C]{-}$ reflects isomorphisms,
* $\mcal C$ is cocomplete
and $\concrf[\mcal C]{-}$ preserves all small colimits,
* the collection of isomorphism
classes of primitive elements of $\Elt(\mcal C)$ is small,
* for every element $(X,x) \in
\Elt(C)$, there is a morphism $(U,u) \to (X,x)$ for some primitive
element $(U,u)$,
* given two morphisms $f,g\co
(U,u) \to (X,x) \in \Elt(\mcal C)$ where $(U,u)$ is primitive, we have $f =
* given two morphisms $f\co (U,u)
\to (X,x)$ and $g\co (V,v) \to (X,x)$ of $\Elt(\mcal C)$ where both $(U,u)$
and $(V,v)$ are primitive, there is an isomorphism $\theta\co (U,u) \to
(V,v)$ such that $g \circ \theta = f$.
§ THE SUPPORT FUNCTION
It is often useful to consider the support of a cell in a precategory, which
informally consists in the set of generators occurring in this cell. In
particular, the support will allow us to retrieve some properties of a morphism
of polygraphs $F$ from the associated free functor $\freecat F$, which will turn
out to be useful when studying polyplexes. A support function for free strict
categories was already introduced by Makkai for his study of the word problem on
these categories <cit.>.
Given $n \in \Ninf$ and an $n$ $\P$, we define the
support functionsupport function
\[
\supp[\P]\co \cattoset{\freecat\P}
\to
% \finsets {\sqcup_{i < n+1} \P_i}
\pset{\poltoset\P}
\]
which to any cell in $\freecat\P$ associates a set of generators of $\P$, by
induction on $u \in \freecat\P$ as follows:
* if $u = g \in \P_0$, then $\supp(u) = \set g$,
* if $u = g \in \P_{k+1}$ for some $k < n$, then $\supp(u) = \set g \cup
\supp(\gsrc(g)) \cup \supp(\gtgt(g))$,
* if $u = \unit{u'}$ for some $k < n$ and $u' \in \freecat{\P}_{k}$, then $\supp(u) = \supp(u')$,
* if $u = u_1 \pcomp_i u_2$ for some $0 < k_1,k_2 < n+1$, $i =
\min(k_1,k_2)-1$ and $i$$u_1 \in \freecat{\P_{k_1}}$ and $u_2 \in
\freecat{\P_{k_2}}$, then $\supp(u) = \supp(u_1) \cup \supp(u_2)$.
One can easily verify that $\supp$ respects the axioms of precategories, so that:
The function $\supp$ is well-defined.
The function $\supp$ is moreover natural:
Let $n \in \Ninf$ and $F \co \P \to \Q \in \nPol n$. Then, we have that
$\supp[\Q] \circ \cattoset{\freecat F} = \poltoset F \circ \supp[\P]$.
By induction on $u \in \freecat\P$.
Given a polygraph $\P$ and a cell $u \in \freecat\P$, the support of $u$ is
always finite. By restricting $\P$ to the generators occurring in this support,
on can show the following:
Given $n \in \Ninf$, an $n$ $\P$ and $u \in \freecat\P$, there
exist a finite $n$ $\tilde{P}$, a monomorphism $F\co \tilde{P}
\to \P$ and $\tilde u \in \freecat{{\tilde{P}}}$ such that $\freecat F(\tilde
u) = u$ and $\supp(\tilde u) = \poltoset{\tilde\P}$.
Given $F \co \P \to \Q$ and $u \in \freecat\P$, we write $F/u \co \supp(u) \to
\supp(\freecat F (u))$ for the restriction of $F$ to the support and the image
of the support of $u$.
Given a pair of parallel morphisms
\[
\begin{tikzcd}
\P
\ar[r,shift left,"F"]
\ar[r,shift right,"G"']
\Q
\end{tikzcd}
\]
of $\oPol$ such that $\freecat F(u) = \freecat G(u)$ for some $u \in
\freecat \P$, we have $F/u = G/u$.
By induction on $n$ and a formula defining $u$.
* If $u = \alpha$ for some $\alpha \in \P$, then $F(\alpha) =
G(\alpha)$. We then also have that
$\freecat F(\csrctgt\eps(\alpha)) =\freecat G(\csrctgt\eps(\alpha))$ for
$\eps \in \set{-,+}$, so that $F/\csrctgt\eps(u) =G/\csrctgt\eps(u)$ by
induction. Thus, $F/\alpha = G/\alpha$.
* If $u = \unit{u'}$, then the property follows by induction hypothesis.
* If $u = u_1 \pcomp_i u_2$. Then, we have $\freecat F(u_1) \pcomp_i
\freecat F(u_2) = \freecat G(u_1) \pcomp_i \freecat G(u_2)$. Writing
$\toterm[\P]\co \P \to \polterm$ for the terminal morphism in $\oPol$,
we have $\freecat{\toterm[\Q]}(\freecat
F(u_j))=\freecat{\toterm[\P]}(u_j)=\freecat{\toterm[\Q]}(\freecat G(u_j))$
for $j \in \set{1,2}$. Since $\freecat{\toterm[\P]}$ is Conduché by
<Ref>, we have $\freecat F(u_j) = \freecat G(u_j)$ for $j \in
\set{1,2}$. Thus, $F/u_j = G/u_j$ for $j \in \set{1,2}$ so that $F / u = G / u$.
We have the following nice description of principal elements of
$\Elt(\oPol)$ and $\Elt(\oPolb)$ using support:
An element $(\P,u)$ of $\Elt(\oPol)$ ($\Elt(\oPolb)$) is principal if
and only if $\supp(u) = \poltoset{\P}$.
Assume that $(\P,u)$ is principal. Then, by <Ref>,
there exist an element $(\tilde\P,\tilde u)$ and $F \co(\tilde\P,\tilde u) \to
(\P,u)$ such that $F$ is a monomorphism of $\oPol$ and $\supp(\tilde u) =
\poltoset{\tilde\P}$. Since $(\P,u)$ is principal, we have that $F$ is an
isomorphism. Thus, by <Ref>, we have that $\supp(u) =
\poltoset\P$.
Conversely, assume that $\supp(u) = \poltoset\P$. Let $(\Q,v)$ be an element
and $F \co (\Q,v) \to (\P,u)$ be a morphism where $F$ is a monomorphism in
$\oPol$. By <Ref>, we have that $\poltoset{F(\supp(v))} =
\supp(u) = \poltoset\P$. Thus, $F_k$ is surjective for every $k \in \N$.
Moreover, $F_k$ is injective by <Ref>, so that $F_k$
is an isomorphism for every $k$. Since $\poltoset -$ reflects isomorphisms
(exercise to the reader), we have that $F$ is an isomorphism. Thus, $(\P,u)$
is principal.
Finally, as a consequence of <Ref>,
we have:
Given a pair of parallel morphisms
\[
\begin{tikzcd}
\ar[r,shift left,"F"]
\ar[r,shift right,"G"']
\end{tikzcd}
\]
of $\Elt(\oPolb)$ where $(\P,u)$ is principal, then $F = G$.
§ POLYPLEXES
We now introduce the construction of polyplexes for the cells of free
precategories. Those are polygraphs representing composition shapes such that
every such cell in a polygraph is the composite of a polyplex in a unique way.
Polyplexes are themselves composed of plexes (see next section) which are
polygraphs representing generators in a polygraph.
These notions are due to Burroni <cit.>, and were further
developed by Henry <cit.>.
Formally, a polyplex is an element $(\P,u)\in\Elt(\oPolb)$ which is
primitive (for the concrete structure introduced in <Ref>). Given
an element $(\Q,v)$ in $\Elt(\oPolb)$, a polyplex lifting is the data of
a polyplex $(\P,u)$ and a morphism of elements $F \co (\P,u) \to (\Q,v) \in
\Elt(\oPolb)$.
The construction of polyplexes will be carried out by induction on a formula
defining a cell. The inductive case of identities is handled by the following
Given an element $(\P,u) \in \Elt(\oPolb)$, $(\P,u)$ is a polyplex if and
only if $(\P,\unit u)$ is a polyplex.
By <Ref>, $(\P,u)$ is principal if and only if
$(\P,\unit u)$ is principal. So we can assume that both are principal.
Suppose that $(\P,u)$ is primitive. Let $F \co (\Q,v) \to (\P,\unit u)$ be a
morphism of elements where $\Q$ is principal. Then, by
<Ref>, we have that $v = \unit {v'}$ for some $v' \in
\freecat\Q$, and, by compatibility of $\freecat F$ with $\csrc$, we moreover
have $\freecat F(v') = u$. Since $\supp(v) = \supp(v')$, $(\Q,v')$ is still a
principal element. Thus, $F$ is an isomorphism since $(\P,u)$ is primitive.
Hence, $(\P,\unit u)$ is primitive. The converse is similar.
The lemmas and propositions until the end of this section, describing the
remaining cases characterizing polyplexes for composites and generators together
with global existence and unicity properties, are proved by mutual
induction on a formula defining the cell $u$ appearing in the
statements. First, the case of generators:
Let $(U,u) \in \Elt(\oPolb)$. Then, the following are equivalent:
* $(U,u)$ is a polyplex and there exist
$\alpha \in U$ such that $u = \alpha$,
* there exist polyplex
\[
G^\eps \co (U^\eps,u^\eps) \to (U,\csrctgt\eps(u))
\]
for $\eps \in \set{-,+}$, principal elements $(S,s)$ and $(T,t)$, and
\begin{align*}
F^{\eps-} \co (S,s)&\to (U^\eps,\csrc(u^\eps))
F^{\eps+} \co (T,t)&\to (U^\eps,\ctgt(u^\eps))
\end{align*}
for $\eps \in \set{-,+}$, such that, considering the pushout
\[
\begin{tikzcd}[column sep=20mm]
S \sqcup T
\ar[r,"{[F^{+-},F^{++}]}"]
\ar[d,"{[F^{--},F^{-+}]}"']
\ar[d,"\bar G^+",dotted]
\\
\ar[r,"\bar G^-"',dotted]
\partial U
\end{tikzcd}
\]
$(U,u)$ is isomorphic to $(\bar U,\bar \alpha)$, where $\bar U$ is
obtained from $\partial U$ by adding a generator
\[
\bar \alpha : \bar G^-(u^-) \to \bar G^+(u^+)
\zbox.
\]
Suppose that <Ref> holds. By the unicity of normal
forms (<Ref>), it is enough to show that
$(\bar U,\bar \alpha)$ is primitive. First, it is principal by
<Ref> since
\[
\supp(\bar \alpha) = \set{\alpha} \cup \supp(\bar G^-(u^-)) \cup \supp(\bar G^+(u^+)) = \poltoset{\bar U}
\zbox.
\]
Second, consider a morphism $H \co (V,v) \to (\bar U,\alpha)$ with $(V,v)$
principal. By induction hypothesis on <Ref>, we have
polyplex liftings
\[
H^\eps \co (\tilde U^\eps,\tilde u^\eps) \to (V,\csrctgt\eps(v))
\]
for $\eps \in \set{-,+}$. Since
$\freecat H(\csrctgt\eps(v)) = \csrctgt\eps(\bar u)$,
by <Ref>, we can assume that
$(\tilde U^\eps,\tilde u^\eps) = (U^\eps,u^\eps)$ for $\eps \in
\set{-,+}$. Since $(S,s)$ is principal, we have, by
\[
H^- \circ F^{--} = H^+ \circ F^{+-}
\qqtand
H^- \circ F^{-+} = H^+ \circ F^{++}
\zbox.
\]
Thus, we derive a morphism $\partial H' \co \partial U \to V$ from the
pushout. By unicity of normal forms, $v = \beta$ for some $\beta \in V$. Thus,
$\partial H'$ can be extended to $H' \co \bar U \to V$ by putting
$H'(\alpha) = \beta$. Using <Ref>, we can easily
verify that $H'$ is the inverse of $H$. Hence, $(\bar U,\bar u)$ is a
Now, assume that <Ref> holds. By induction hypothesis,
there are polyplex liftings
\[
G^\eps \co (U^\eps,u^\eps) \to (\P,\csrctgt\eps(u))
\]
for $\eps \in \set{-,+}$. By induction hypothesis
on <Ref>, there exists a polyplex lifting
$F^{--} \co (S,s) \to (U^-,\csrc(u^-))$. Similarly, there is a polyplex
lifting of $(U^+,\csrc(u^+))$ and, since
$\freecat {(G^-)}(\csrc(u^-)) = \freecat{(G^+)}(\csrc(u^+))$,
by <Ref>, it can be chosen to be of the form
\[
F^{+-} \co (S,s) \to (U^+,\csrc(u^+))
\zbox.
\]
Similarly, there are polyplex liftings
\[
F^{-+} \co (T,t) \to (U^-,\ctgt(u^-))
\qqtand
F^{++} \co (T,t) \to (U^+,\ctgt(u^+))
\zbox.
\]
Writing $F^\eps$ for $[F^{\eps-},F^{\eps+}]$ for $\eps \in \set{-,+}$,
consider the pushout
\[
\begin{tikzcd}
S \sqcup T
\ar[r,"F^+"]
\ar[d,"F^+"']
\ar[d,"\bar G^+",dotted]
\\
\ar[r,"\bar G^-"',dotted]
\partial U
\end{tikzcd}
\]
and write $\bar U$ for the obtained from $\partial U$ by adding a generator
$\bar \alpha \co \bar G^-(u^-) \to \bar G^+(u^+)$ (this is well-defined, since
the definition of $\partial U$ ensures that $\csrctgt\eps(\bar G^-(u^-)) =
\csrctgt\eps(\bar G^+(u^+))$ for $\eps \in \set{-,+}$). By the first part,
$(\bar U,\bar \alpha)$ is a polyplex, and we easily deduce a polyplex lifting $H
\co (\bar U,\bar \alpha) \to (U,\alpha)$ from the above pushout. Since
$(U,\alpha)$ is primitive, $H$ is an isomorphism.
Thus, <Ref> holds.
The next lemma deals with the case of composites of the polyplex construction:
Let $(U,u) \in \Elt(\oPolb)$, $u_1 \in \freecat U_k$, $u_2 \in \freecat U_l$
for some $k,l \in \N$, with $u_1$ and $u_2$ are $i$for $i =
\min(k,l)$. Then, the following are equivalent:
* $(U,u)$ is a polyplex and $u = u_1
\pcomp_i u_2$,
* there exist a principal element
$(U',u')$ and polyplexes $(U_1,u_1)$ and $(U_2,u_2)$, and morphisms $F_j \co
U'\to U_j \in \oPol$ and $G_j \co U_j \to U$ for $j \in \set{1,2}$, such
\[
\begin{tikzcd}
\ar[d,"F_1"']
\ar[r,"F_2"]
\ar[d,"G_2"]
\\
\ar[r,"G_1"']
\end{tikzcd}
\]
is a pushout diagram in $\oPol$, $\freecat F_1(u') = \ctgt_i(u_1)$, $\freecat F_2(u')
= \csrc_i(u_2)$ and $u = G_1(u_1) \pcomp_i G_2(u_2)$.
Suppose that <Ref> holds.
We have
\begin{align*}
\supp(\freecat G_1(u_1) \pcomp_i \freecat G_2(u_2))
&= \supp(\freecat G_1(u_1)) \cup
\supp(\freecat G_2(u_2)) \\
&= G_1(\supp(u_1)) \cup G_2(\supp(u_2)) \\
& = G_1(\poltoset{U_1}) \cup G_2(\poltoset{U_2})\\
&= \poltoset{U} % colimits computed as in Set
\end{align*}
thus $(U,u)$ is principal by <Ref>. Now, consider $H
\co (\R,w) \to (U,u) \in \Elt(\oPolb)$ with $(\R,w)$ principal. We have
\[
H(\poltoset{\R}) = H(\supp(w)) =
\supp(\freecat H(w)) = \supp(u) = \poltoset{U}
\]
so that the functions $H_j \co \R_j \to U_j$ are surjective for every $j$.
Thus, $H$ is an epimorphism. Since $\freecat H$ is Conduché
by <Ref> and $\freecat H(w) = \freecat G_1(u_1) \pcomp_i
\freecat G_2(u_2)$, there exist unique $w_1,w_2$ such that $\freecat H(w_j) =
\freecat G_j(u_j)$ for $j \in \set{1,2}$ and $w = w_1 \pcomp_i w_2$. By
induction hypothesis on <Ref>, there exist polyplex
liftings $H'_j \co (\tilde U_j,\tilde u_j) \to (\R,w_j)$ for $j \in
\set{1,2}$. By induction hypothesis on <Ref>,
since both $(\tilde U_j,\tilde u_j)$ and $(U_j,u_j)$ are polyplex liftings of
$(U,\freecat G_j(u_j))$, we may assume that $(\tilde U_j,\tilde u_j)
=(U_j,u_j)$ for $j \in \set{1,2}$. By <Ref>, we
have $H'_1 \circ F_1 = H'_2 \circ F_2$, so that we obtain $H' \co U \to \R$
from the pushout. We compute that
\[
H'(u) = H'(\freecat G_1(u_1) \pcomp_i \freecat G_2(u_2)) = H'_1(u_1)
\pcomp_i H'_2(u_2) = w_1 \pcomp_i w_2 = w\zbox.
\]
Thus, using <Ref>, we easily have that $H' \circ H
= \unit {\R}$ and $H \circ H' = \unit{U}$. Hence, $(U,u)$ is primitive.
Conversely, suppose that <Ref> holds. Then, by
induction hypothesis on <Ref>, there exist $G_k \co
(U_k,\bar u_k) \to (\P,u_k)$ with $(U_k,\bar u_k)$ primitive for $k \in
\set{1,2}$. By induction hypothesis on <Ref>, there
exist $\tilde F_k \co (\tilde U_k,\tilde u_k) \to
(U_k,\csrctgt{\eps(k)}_i(u_k))$ with $(\tilde U_k,\tilde u_k)$ primitive for
$i \in \set{1,2}$ and $\eps(1) = +$ and $\eps(2) = -$. In particular, $(\tilde
U_1,\tilde u_1)$ and $(\tilde U_2,\tilde u_2)$ are both polyplex liftings of
$(U,\ctgt_i(u_1))$. By induction hypothesis on
<Ref>, we can assume that $(\tilde U_1,\tilde
u_1) = (\tilde U_2,\tilde u_2)$ and write $(\tilde U,\tilde u)$ for this
element. By <Ref>, we have $G_1 \circ F_1 = G_2
\circ F_2$. Consider the pushout
\[
\begin{tikzcd}
\tilde U
\ar[d,"F_1"']
\ar[r,"F_2"]
\ar[d,"\bar G_2",dotted]
\\
\ar[r,"\bar G_1"',dotted]
\bar U
\end{tikzcd}
\]
By its universal property, we get a morphism $H \co \bar U \to U$ from $G_1$
and $G_2$. By the first implication, $(\bar U,\freecat G_1(\bar u_1) \pcomp_i
\freecat G_2(\bar u_2))$ is a primitive element. Moreover, $H$ induces a
\[
H \co (\bar U,\freecat G_1(\bar u_1) \comp_i \freecat G_2(\bar u_2)) \to (U,u_1\pcomp_i u_2)
\]
of $\Elt(\oPolb)$. Thus, since $(U,u_1\pcomp_i u_2)$ is primitive, $H$
is an isomorphism.
The previous lemmas lead to the following polyplex lifting existence property:
Given an element $(\P,u)\in\Elt(\oPolb)$, there exists a polyplex lifting
\[
F \co (U,\bar u) \to (\P,u)
\]
where $(U,\bar u)$ is primitive.
We reason by case analysis on a formula for $u$.
* If $u = \unit {u'}$, then, by <Ref>, the conclusion
follows from induction hypothesis.
* If $u = u_1 \pcomp_i u_2$, then, by induction hypothesis, there are
\[
G^k \co (U^k,\bar u_k) \to (\P,u)
\]
with $(U^k,\bar u_k)$ primitive for $k \in \set{1,2}$. By induction
hypothesis, there are polyplex liftings
\[
F^k \co (\tilde U^k,\tilde u_k) \to (U^k,\csrctgt{\eps(k)}_i(\bar u_k))
\]
with $\eps(1) = +$ and $\eps(2) = -$. By induction hypothesis on
<Ref>, we can assume that
$(\tilde U^1,\tilde u_1) = (\tilde U^2,\tilde u_2)$ and write
$(\tilde U,\tilde u)$ for this element. Since $(\tilde U,\tilde u)$ is
principal, we have $G^1 \circ F^1 = G^2 \circ F^2$. Consider the pushout
\[
\begin{tikzcd}
\tilde U
\ar[d,"F^1"']
\ar[r,"F^2"]
\ar[d,"\bar G^2",dotted]
\\
\ar[r,"\bar G^1"',dotted]
\bar U
\end{tikzcd}
\]
Then, by <Ref>,
$(\bar U,\bar G^1(\bar u_1) \pcomp_i \bar G^2(\bar u_2))$ is a polyplex, and
the universal property of pushouts gives a polyplex lifting
\[
H \co (\bar U,\bar G^1(\bar u_1) \pcomp_i \bar G^2(\bar u_2)) \to (\P,u)
\zbox.
\]
* If $u = \alpha$ for some generator $\alpha \in \P$, by induction, there
are polyplex liftings
\[
G^\eps \co (U^\eps,u^\eps) \to (\P,\csrctgt\eps(u))
\]
for $\eps \in \set{-,+}$. By induction on <Ref>,
there exists a polyplex lifting
\[
F^{--} \co (S,s) \to (U^-,\csrc(u^-))
\zbox.
\]
Similarly, there is a polyplex lifting of $(U^+,\csrc(u^+))$ and, since
\[
\freecat {(G^-)}(\csrc(u^-)) = \freecat{(G^+)}(\csrc(u^+))
\]
by <Ref>, it can be chosen to be of the form
\[
F^{+-} \co (S,s) \to (U^+,\csrc(u^+))
\zbox.
\]
Similarly, there are polyplex liftings
\[
F^{-+} \co (T,t) \to (U^-,\ctgt(u^-))
\qqtand
F^{++} \co (T,t) \to (U^+,\ctgt(u^+))
\zbox.
\]
Writing $F^\eps$ for $[F^{\eps-},F^{\eps+}]$ for $\eps \in \set{-,+}$,
consider the pushout
\[
\begin{tikzcd}
S \sqcup T
\ar[r,"F^+"]
\ar[d,"F^+"']
\ar[d,"\bar G^+",dotted]
\\
\ar[r,"\bar G^-"',dotted]
\partial U
\end{tikzcd}
\]
and write $U$ for the obtained from $\partial U$ by adding a
\[
\bar \alpha : \bar G^-(u^-) \to \bar G^+(u^+)
\]
(this is well-defined, since the definition of $\partial U$ ensures that
$\csrctgt\eps(\bar G^-(u^-)) = \csrctgt\eps(\bar G^+(u^+))$ for
$\eps \in \set{-,+}$). By <Ref>, $(U,\bar \alpha)$ is a
polyplex, and we easily deduce a polyplex lifting
$H \co (U,\bar \alpha) \to (\P,\alpha)$.
Finally, we have the following uniqueness property of polyplex liftings:
Given two morphisms $L^1\co (U^1,u_1) \to (\P,u)$ and $L^2\co (U^2,u_2) \to (\P,u)$
of $\Elt(\oPolb)$ where both $(U^1,u_1)$ and $(U^2,u_2)$ are primitive, there is an
isomorphism $\Theta\co (U^1,u_1) \to (U^2,u_2)$ such that $L^2 \circ \Theta = L^1$.
We reason by case analysis on a formula for $u$.
* If $u = \unit {u'}$, then the conclusion follows from induction hypothesis on $u$.
* If $u = \alpha$ for some $\alpha \in \P$, then,
by <Ref>, $U^1$ and $U^2$ are obtained by adding
respective top-level generators $\alpha^1$ and $\alpha^2$ to polygraphs
$\partial U^1$ and $\partial U^2$, the latter being expressed as pushouts
\[
\begin{tikzcd}[column sep=20mm]
S^i \sqcup T^i
\ar[r,"{[F^{i,+-},F^{i,++}]}"]
\ar[d,"{[F^{i,--},F^{i,-+}]}"']
\ar[d,"\bar G^{i,+}",dotted]
\\
\ar[r,"\bar G^{i,-}"',dotted]
\partial U^i
\end{tikzcd}
\]
for some principal $(S^i,s^i)$, $(T^i,t^i)$ and some primitive
$(U^{i,-},u^{i,-})$, $(U^{i,+},u^{i,+})$ for $i \in \set{1,2}$ as in the
statement of that lemma. In particular, $(U^{i,\eps},u^{i,\eps})$ are
polyplex liftings of $\csrctgt\eps(\alpha)$ for $i \in \set{1,2}$ and $\eps
\in \set{-,+}$. By induction hypothesis, for $\eps \in \set{-,+}$, there are
isomorphisms $\Theta^\eps \co (U^{1,\eps},u^{1,\eps}) \to
(U^{2,\eps},u^{2,\eps})$. Since $(S^1,s^1)$ and $(T^1,t^1)$ are principal,
we can easily verify with <Ref> that
\[
G^{2,-} \circ \Theta^- \circ [F^{1,--},F^{1,-+}] = G^{2,+} \circ \Theta^+
\circ [F^{1,+-},F^{1,++}]
\]
so that we get a morphism $\partial\Theta \co \partial U^1 \to \partial
U^2$, which extends to a morphism $\Theta \co U^1 \to U^2$ such that
$\Theta(\alpha^1) = \alpha^2$. Symmetrically, a morphism $\Theta' \co
(U^2,\alpha^2) \to (U^1,\alpha^1)$ can be built.
Using <Ref>, we easily verify that $\Theta$ and
$\Theta'$ are inverse of each other.
* If $u = u_1 \pcomp_i u_2$, we use the pushout description
from <Ref> and this case is then handled just like the
previous one.
A consequence of the existence and unicity properties above, together with
<Ref>, is that the functor
$\polconcrb-\co \nPol\omega \to \Set$ of <Ref> is
familially representable <cit.>, can be
expressed as a functor of the form
\[
\bigsqcup_{i \in I} \Hom(U^i,-) \co \oPol \to \Set\zbox.
\]
Here, $I$ is a set of representatives $(U^i,u^i)$ of all polyplexes
(considered up to isomorphism of elements in $\Elt(\oPolb)$) of any
dimension. Those can for instance be enumerated by constructing one polyplex
liftings for each cell of the free precategory on the terminal polygraph. A
similar description holds for the functor $\freecat -_k$, mapping a polygraph
to the set of $k$of the associated free precategory: the family $I$ is
now a set of representatives for the polyplexes of dimension $k$ up to
A consequence of the canonicity of a polyplex liftings given by the above
properties is that one can define a polyplex measure on the cells of free
precategories. Let $\P \in \oPol$, and write $\Z\P$ for the free $\Z$-module
on $\poltoset\P$. Given $u \in \freecat\P$, one can define $\meas[\P](u)$ as
follows. Consider a polyplex lifting $F \co (V,v) \to (\P,u)$ and define
$S_V \in \Z V$ by $S_V = \sum_{g\in V} g$. Then, one defines $\meas[\P](u)$ as
$\Z F(S_V)$. The definition of $\meas[\P](u)$ does not depend on the choice of
$(V,v)$ by <Ref>. The
question of the existence of a similar measure for free strict categories was
raised by Makkai in <cit.>. Later, using the standard
Eckmann–Hilton for strict categories, the non-existence of such a measure was
proven <cit.>.
§ POLYGRAPHS AS A PRESHEAF CATEGORY
We can now use the results of the previous section in order to conclude that
$\oPol$ is a (concrete) presheaf category on the base category (also called
shape category) of plexes, which are the elementary shapes polygraphs are
made of. In addition to the works of Burroni <cit.> and
Henry <cit.>, this notion was also studied by
Makkai <cit.> under the name “computopes”.
Formally, a plex is an element $(\P,u)\in\Elt(\oPol)$ which is primitive
(for the concrete structure introduced in <Ref>). Given an
element $(\Q,v)$ in $\Elt(\oPol)$, a plex lifting is the data of a plex
$(\P,u)$ and a morphism of elements $F \co (\P,u) \to (\Q,v)\in\Elt(\oPol)$.
In order to relate the properties of plexes to the ones of polyplexes proved in
the previous section, we first need to briefly discuss the link between
$\Elt(\oPol)$ and $\Elt(\oPolb)$. We write $\poltopolb\co \Elt(\oPol) \to
\Elt(\oPolb)$ for the canonical embedding. First note that, as a consequence of
<Ref><ref>, that
The functor $\mfunctorb U$ is fully faithful.
We then have the following.
Let $(\P,g) \in \Elt(\oPol)$. Then
* $(\P,g)$ is principal if and only
if $\mfunctorb U(\P,g)$ is principal,
* $(\P,g)$ is a plex if and only if $\mfunctorb U(\P,g)$ is a polyplex.
By <Ref>, <ref> holds.
Suppose now that both $(\P,g)$ and $\mfunctorb U(\P,g)$ are principal. By
<Ref><ref>, $\mfunctorb U$ is
fully faithful, so that it reflects isomorphisms. Thus, if $\mfunctorb
U(\P,g)$ is a polyplex, then $(\P,g)$ is a plex. For the converse, note that
if $f\co (\Q,v) \to \mfunctorb U(\P,g)$ is a morphism of $\Elt(\oPolb)$,
then, by <Ref><ref>, $v \in
\Q$, so that
\[
= \mfunctorb U(\Q,v)
\qtand
f = \mfunctorb U(f)\zbox.
\]
Hence, if $(\P,g)$ is a plex, then $\mfunctorb U(\P,g)$ is a polyplex.
The category $\oPol$ is a concrete presheaf category.
We verify that the various conditions of Makkai's criterion
(<ref>) are satisfied.
<ref> Clear from the definition of $\oPol$.
<ref> A consequence of general properties
satisfied by categories of polygraphs derived from a globular monad (see
Propositions 1.3.3.7 and 1.3.3.15 of <cit.>).
<ref> Since a primitive element $(\P,g)$ is
principal, the polygraph $\P$ is finite. Thus, up to isomorphism, the sets
$\P_i$ can be assumed to be subsets of $\N$. So
that <ref> holds.
<ref> Given an element
$(\P,g) \in \Elt(\oPol)$, by <Ref>, there exists a
polyplex lifting $F \co (U,u) \to (\P,g)$ of $\mfunctorb U (\P,g)$. By
<Ref><ref>, we have that
$u \in U$. Moreover, by
<Ref><ref>, we have
that $(U,u)$ is a plex, so <ref> holds.
<ref> Given
$f,g\co (U,u) \to (X,x) \in \Elt(\oPol)$ with $(U,u)$ a primitive plex, then
we have that
$\mfunctorb U (f),\mfunctorb U(g)\co (U,u) \to (X,x) \in \Elt(\oPolb)$, so
that $\mfunctorb U (f),\mfunctorb U(g)$ by <Ref>,
and $f = g$ by faithfulness.
<ref> Given two
morphisms $f\co (U,u) \to (X,x)$ and $g\co (V,v) \to (X,x)$ of $\Elt(\oPol)$
where both $(U,u)$ and $(V,v)$ are primitive, we have by
<Ref> that there is an isomorphism
\[
\theta\co \poltopolb (U,u) \to \poltopolb (V,v) \in \Elt(\oPolb)
\]
such that $\poltopolb (g) \circ \theta = \poltopolb (f)$. We conclude by the
full faithfulness of $\poltopolb$.
Following Makkai's proof of <cit.>, the base
category of the presheaf category given by the above theorem is a small full
subcategory of $\oPol$, whose objects are (the underlying polygraphs of)
plexes, and such that every (underlying polygraph of a) plex is isomorphic to
exactly one object of this subcategory. The objects of the latter can thus be
easily enumerated, since they are in correspondence with the generators of the
terminal polygraph $\polterm$, as plex liftings.
Like the familial representability observed in <Ref>,
the conditions
proved above entails a familial representability for the functor
$\poltoset{-}$ of <Ref>, which can be expressed as
\[
\bigsqcup_{i\in I} \Hom(U^i,-) \co \oPol \to \Set
\zbox.
\]
Here, $I$ is a set of representatives $(U^i,g^i)$ of all plexes (considered up
to isomorphism of $\Elt(\oPol)$). By taking $I$ to be a set of representatives
of all plexes of dimension $k$ for some $k \in \N$, one get a familial
representability of the functor $(-)_k \co \oPol \to \Set$.
In <cit.>, relies on
<cit.>, which gives sufficient conditions
for a category to be a presheaf category on a given full subcategory. The
difference with <cit.> is that the latter is
relative to a concrete presheaf structure, and is able to characterize the
shape category as a full subcategory of primitive elements.
§ PARAMETRIC ADJUNCTION AND GENERICIC FACTORIZATION
While <Ref> asserts that the cells of free precategories
on polygraphs are instances of universal shapes (polyplexes), a more
conceptual and general syntactical result can be given, which encompasses both
the existence of those universal shapes and the Conduché property of free
functors. This result relies on the existence of a parametric adjunction and an
associated generic factorization for the free functor $\freecat-$. Parametric
adjunctions and generic factorizations appear frequently in the context of
algebraic higher
category <cit.>: for example,
the free monad functor on globular sets is parametric right adjoint, and
has an associated generic factorization. While the classical parametric right
adjoints are monad functors on presheaf categories (for which characterization
criteria have been developed, for example <cit.>), the unusual fact here is that the parametric right
adjoint $\freecat-$ is a left adjoint, whose codomain is not a presheaf
category, but the category of $n$: for us, this fact reflects and
summarize the good syntactical properties of the theory of precategories.
While parametric adjunctions can easily be deduced from familial
representability properties (like <Ref>) in a presheaf
setting (see <cit.>), there is no direct
criterion in our setting, so that we have to show the parametric
representability by hand: we need to show that the functor
$\freecat-_\polterm \co \oPol \to \defslicecat$ is a right adjoint, where
$\defslicecat$ is the slice category of $\oPCat$ over the free precategory on
the terminal polygraph $\polterm$, and $\freecat-_\polterm$ the functor induced
by $\freecat-$. Since both $\oPol$ and $\defslicecat$ are locally presentable
categories, and that $\freecat-$ is a left adjoint, we are only required to show
that $\freecat-_\polterm$ preserves limits (see <cit.>). Since $\oPol$ has a terminal object and the
computation of limits in $\defslicecat$ amounts to the computation of a
connected limit in $\oPCat$, we simply need to show that connected limits are
preserved, recovering <cit.> in our context. In
the following, given $k,n \in \N$, we write $D^{k,n}$, or simply $D^k$ for the
free $n$with one non-identity $k$.
Given $n \in \Ninf$, the functor $\freecat - \co \nPol n \to \nPCat n$
preserves connected limits.
First note that the functor $\cattoset- \co \nPCat n \to \Set$ is
conservative; it is moreover familially representable by the $D^k$'s for $k <
n+1$ (a $k$of an $n$$C$ is the same thing as a functor $D^k
\to C$) and thus preserves connected limits by <cit.> and <cit.>. Since
$\nPCat n$ is complete, it is sufficient to show that the functor
$\polconcrb-\co \nPol n \to \Set$ preserves connected limits. But this functor
is familially representable by <Ref>, so that it
preserves connected limits by <cit.>.
By the argument exposed earlier, we can conclude that:
Given $n \in \Ninf$, the functor $\freecat-_\polterm \co \nPol n \to \defslicecat[n]$ is a right
adjoint. In other words, $\freecat-$ is a parametric right adjoint.
As a consequence, we have a generic factorization for the functor $\freecat-$.
We recall from <cit.> the notion of generic morphism in the
present case: given $C \in \nPCat n$ and $\P \in \nPol n$, a morphism $F \co C
\to \freecat\P$ is generic when, for any commutative square of the form
\[
\begin{tikzcd}
\ar[r,"G"]
\ar[d,"F"']
\freecat\Q
\ar[d,"\freecat H"]
\\
\freecat\P
\ar[r,"\freecat K"']
\ar[ur,dotted,"\freecat L"description]
\freecat\R
\end{tikzcd}
\]
for some $\Q,\R \in \nPol n$, $G \co C \to \freecat\Q$ in $\nPCat n$, $H \co \Q
\to \R$ and $K \co \P \to \R$ in $\nPol n$, there exists a unique $L \co \P \to
\Q$ such that $G = \freecat L \circ F$ and $K = H \circ L$. Now, given a
morphism $F \co C \to \freecat \P$, a generic factorization is a
decomposition of $F$ as $\freecat H \circ G$ for some $\Q \in \nPol n$, some
generic $G \co C \to \freecat\Q$ and $H \co \Q \to \P \in \nPol n$. By the
universal property of generic morphisms, such a decomposition is unique up to an
isomorphism $\Q \to \Q'$.
Given $n\in \Ninf$, $C \in \nPCat n$ and $\P \in \nPol n$, every $F \co C \to
\freecat\P \in \nPCat n$ admits a generic factorization.
By <cit.>, the existence of generic
factorizations follows from the fact that $\freecat-_\polterm$ is a parametric
right adjoint.
Some generic morphisms are easy to identify:
Given $n \in \N$ and $\P \in \oPol$, writing $u$ for the non-identity $n$of $D^n$, a functor $F \co D^n \to \freecat\P$ is generic if and only if
$(\P,F(u))$ is a polyplex.
We start with the first implication. Let $H \co (\Q,v) \to (\P,F(u))$ be a
polyplex lifting of $(\P,F(u))$. Then, writting $G \co D^n \to \freecat\Q$
sending $u$ to $v$, we have $\freecat H \circ G = \freecat{(\unit{\P})} \circ
F$. Thus, there exists a unique lifting $L \co \P \to \Q$ such that $\freecat
L \circ F = G$ and $H \circ L = \unit{\P}$. In particular, we have that
$\freecat L(F(u)) = v$ and $L$ is a monomorphism. Thus, since $(\Q,v)$ is
principal, $L \co (\P,F(u)) \to (\Q,v)$ is an isomorphism.
Conversely, let
\[
\begin{tikzcd}
\ar[r,"G"]
\ar[d,"F"']
\freecat\Q
\ar[d,"\freecat H"]
\\
\freecat\P
\ar[r,"\freecat K"']
\freecat\R
\end{tikzcd}
\]
be a commutative square where $(\P,F(u))$ is assumed to be a polyplex.
Consider a polyplex lifting $L \co (\bar \P,\bar u) \to (\Q,G(u))$. By
applying $\freecat H$, $(\bar \P,\bar u)$ is a polyplex lifting of $\freecat
H(G(u)) = \freecat K(F(u))$ and so is $(\P,F(u))$. By
<Ref>, we may assume $(\bar \P,\bar u) =
(\P,F(u))$ with $H \circ L = K$. Moreover, since $\freecat L(F(u)) = G(u)$, we
have $L \circ F = G$ by freeness of $D^n$. Finally, the unicity of the lifting
$L$ of the above square is a consequence of <Ref>.
In a related manner, given $n \in \N$ and $v \in \freecat\polterm_n$, the
image of $D^n \xto{v} \freecat\polterm$, seen as an object of $\defslicecat$,
by a left adjoint to $\freecat-_\polterm$ is the underlying polygraph of a
polyplex lifting of $v$.
The above generic factorization can be seen as a stronger version of
<Ref>. Indeed, given $k,l > 0$ and $i = \min(k,l) - 1$, there
exists a polygraph $\D^{k,l}$ such that $\freecat{(\D^{k,l})}$ is the free
with one $k$$u_1$ and one $l$$u_2$, such that $\ctgt_i(u_1)
= \csrc_i(u_2)$. The construction of $\D^{k,l}$ can be seen to induce a
polyplex $(\D^{k,l},u_1\comp_i u_2)$ by <Ref>. Writting $n$
for $\max(k,l)$ and $F^{k,l} \co D^n \to \freecat{(\D^{k,l})}$ for the functor
sending the non-trivial $n$of $D^n$ to $u_1\comp_i u_2$,
<Ref> amounts to observe that the $F^{k,l}$'s are generic by
§ TOWARD HOMOTOPICAL PROPERTIES OF PRECATEGORIES
In this section, we report on failed attempts to study homotopical properties of
categories, leaving open questions for future works.
A folk model structure on precategories?
In the setting of strict $n$-categories, the usefulness of polygraphs can be
explained by the facts that they are free objects such that every category
admits a description by such an object, and any two descriptions are suitably
equivalent. In more precise and modern terms, this was formalized by Lafont,
Métayer and Worytkiewicz <cit.>, who constructed a structure of
model category on the category $\nCat\omega$ of strict $\omega$-categories, in
which weak equivalences are the expected equivalences of $\omega$-categories and
cofibrant objects are $\omega$-categories freely generated by polygraphs.
One could expect that we could perform a similar construction on precategories,
and construct a model structure where weak equivalences are the expected ones
and cofibrant objects are polygraphs in the sense of this article. Whether this
is possible or not is left as an open question, but explain here that a direct
adaptation of the proof of <cit.> does not go through easily.
Let us first introduce some terminology. Given an $C$, we make the
following coinductive mutual definitions:
* two cells $x,y \in C$ of the same dimension are equivalent,
denoted $x\sim y$, when there exists an equivalence $u : x \to y$;
* a cell $u : x \to y$ is an equivalence when there exists $\bar
u : y \to x$ such that $u \comp \bar u \sim \unit x$ and $\bar u \comp u
\sim \unit y$.
We could then have hope for the following definition of weak equivalences. Given
an $F : C \to D$, $F$ is a weak equivalence when it is
“essentially surjective in every dimension”,
* for every $0$$y \in D_0$, there exists $x \in C_0$ such that
$Fx \sim y$,
* for every pair of parallel cells $u,u' \in C$ and cell
$\bar v : F(u) \to F(u')$, there exists $v \in C$ such that
$F(v) \sim \bar v$.
The above definitions directly generalize the ones for strict categories. The
construction of the folk model structure on strict then requires a
weak division property <cit.>, which the authors
present as being “crucial”. The direct generalization of it in the setting of
precategories is as follows:
[Weak division]
Given an $C$ and an equivalence $u : x \to y \in C_1$, for any
$1$$s,t : y \to z$ and for any $2$$w : u \comp_0 s \To u \comp_0 t$,
* there is a $2$$v : s \To t$ such that
$u \comp_0 v \sim w$,
\[
\begin{tikzcd}[row sep=1ex]
&y\ar[dr,bend left,"s"]\\
x\ar[ur,bend left,"u"]\ar[dr,bend right,"u"']\ar[rr,phantom,"w\Downarrow"]&&z\\
&y\ar[ur,bend right,"t"']
\end{tikzcd}
\qquad\sim\qquad
\begin{tikzcd}
x\ar[r,"u"]&y\ar[r,bend left=50,"s"]\ar[r,bend right=50,"t"']\ar[r,phantom,"v\Downarrow"]&z
\end{tikzcd}
\]
* for any $2$$v,v':s\To t$ such that
$u\comp_0v\sim w\sim u\comp_0 v'$ we have $v\sim v'$.
We would also need a generalization of the above property for $n$-cells, but we
will see that the proof of the stated property in dimension 1 already fails to
generalize from strict categories to precategories. Consider cells $u$ and $w$
as in the above property, with $u$ reversible, and let us try to define the
cell $v$. Writing $r:\bar u\comp_0 u\to\id x$ for the $2$-cell witnessing that
$u$ is reversible, following <cit.>, we are tempted to define $v$
\[
v=\ol{(r\comp_0 s)}\comp_1(\bar u\comp_0w)\comp_1(r\comp_0t)
\]
If we picture $r$ and $w$ as on the left, $v$ can be pictured as on the right:
\begin{align*}
\end{align*}
In particular, in the case where $w$ is of the form $w=u\comp_0 v'$ for some
$2$-cell $v':s\To t$, we should have $v\sim v'$ by <ref>. In the case of
strict categories, this holds thanks to the interchange law:
\[
\begin{array}{r@{\ =\ }c@{\ =\ }c@{\ \sim\ }c}
\ol{(r\comp_0 s)}\comp_1(\bar u\comp_0u\comp_0 v')\comp_1(r\comp_0t)
\ol{(r\comp_0 s)}\comp_1(r\comp_0s)\comp_1v'
\\
\satex{ex1a}
\satex{ex1b}
\satex{ex1c}
\satex{ex1d}
\end{array}
\]
However, in the case of precategories there is no reason why this should hold.
Of course, this does not directly imply that <ref> does not hold
or that there is no suitable model structure on precategories, but more work is
required than a mere adaptation of <cit.>. The above also suggests
that it could be interesting to investigate structures “in between” precategories
and strict categories, where the interchange law is only required to hold for
some morphisms (such as $r$ in the above example).
A cone construction?
Another homotopy-related question one might ask is whether the underlying shape
category of the presheaf category of polygraphs of precategories is able to
model homotopy types. A now standard approach to get a positive answer is to
show that this shape category is a weak test
category <cit.>, a
category $C$ whose presheaf category $\hat C$ can be equipped with a canonical
class of weak equivalences $\cW$, such that the induced localization $\hat
C[\finv\cW]$ is canonically isomorphic to the homotopy category $\Hot$, so that,
in particular, $\hat C$ models all homotopy types.
A common way to show that a category is a weak test category is to exhibit a
separating décalage <cit.> on this category.
Formally, a décalage on a catégorie $C$ is given by a functor $D : C
\to C$ together with natural transformations
\[
\begin{tikzcd}
\ar[r,"\alpha"]
\ar[l,"\beta"']
\top
\end{tikzcd}
\]
where $\top$ is an object of $C$ seen as a constant functor. Such a décalage is
separating when we moreover have that
* for every $c \in C_0$, the arrow $\alpha_c : c \to D(c)$ is a monomorphism,
* $\alpha$ is cartesian: for every morphism $f : c \to c' \in C$, the
\[
\begin{tikzcd}
\ar[r,"\alpha_c"]
\ar[d,"f"']
\ar[d,"D(f)"]
\\
\ar[r,"\alpha_{c'}"']
\end{tikzcd}
\]
is a pullback,
* for every $c \in C_0$, there is no commutative diagram of the form
\[
\begin{tikzcd}
\ar[r,"g"]
\ar[d,"f"']
\top
\ar[d,"\beta_c"]
\\
\ar[r,"\alpha_c"']
\end{tikzcd}
\]
for some $c' \in C_0$ and $f\co c' \to c$ and $g\co c' \to \top$ in $C_1$.
Following Henry's line of proof for the case of regular
plexes <cit.>, a promising choice of décalage in a polygraphic
setting is the one where $D$ is cone construction functor, also called
expansion functor: starting from a polygraph $\P$, this functor adds to
$\P$ a $0$$o$, a $1$$x \to o$ for each $x \in \P_0$, and
more generally an $(i{+}1)$for each $i$of $\P$, so that
$D\P$ appears as a combinatorial description of a cone over the space defined
by $\P$. Then, continuing the definition of a décalage, one can take $\alpha$ to
be the canonical embedding of a polygraph into the base of its cone, $\top$ to
be the polygraph with only one $0$ $o$, and $\beta$ to be the marking
of $o$ as the top of each constructed cone.
While Henry <cit.> used the join of strict
categories <cit.> to define the expansion functor on regular
plexes, a more direct description of this construction was used by
<cit.> in the case of strict
categories that we unsuccessfully tried to adapt to precategories. In the
following, we describe this attempt, hoping it can still benefit other settings.
Write $\oPCatp$ for the category of pointed , that is, the
category whose objects are the pairs $(C,o)$ where $o \in C_0$ and the morphisms
$(C,o_C) \to (D,o_D)$ are the functor $F\co C \to D$ such that $F(o_C) = o_D$.
We have an evident adjunction
\begin{equation}
\label{eq:cone-first-adj}
\begin{tikzcd}
\oPCat
\ar[r,bend left,"\pointedfreef-"]
\phar[r,"\bot"]
\oPCatp
\ar[l,bend left,"\pointedfgf"]
\end{tikzcd}
\end{equation}
where $\pointedfgf$ simply forgets the pointed $0$$o$. In order to define
an expansion functor on precategories, one wants to introduce a functor
\[
\conef \co \oPCatp \to \oPCatp
\]
such that $\conef (C,o)$ is the of $i$ on $(C,o)$ for $i \in
\N$: a $0$is some base $0$$\cb x \in C$ together with some
$1$$\cc x \co \cb x \to o$ of $C$, a $1$between $(\cb x,\cc x)$ and
$(\cb y,\cc y)$ is a base $1$$\cb f \co x \to y$ and $\cc f \co \cb f
\comp_0 \cc y \To \cc x$, and so on. There is then a natural embedding
$\gamma_{(C,o)} \co \conef (C,o) \to (C,o)$, mapping every $i$to its
base $i$. If such a functor exists, one could then define the category
of conic precategories $\oPCatc$ whose objects are the triples $(C,o,\sigma)$,
where $(C,o)$ is a pointed and $\sigma$ is a section of $\gamma_{(C,o)}$
satisfying adequate degeneracy conditions (see <cit.>), and whose morphisms are the ones of $\oPCatp$ which
adequately commute with the sections. In other words, an object of $\oPCatc$ is
a pointed with the data of a compatible $i$for every $i$,
satisfying degeneracy conditions. The forgetful functor $\conicfgf\co\oPCatc \to
\oPCatp$ should then admit a left adjoint, so that we get an adjunction
\begin{equation}
\label{eq:cone-second-adj}
\begin{tikzcd}
\oPCatp
\ar[r,bend left,"\conicfreef-",pos=0.54]
\phar[r,"\bot"]
\oPCatc
\ar[l,bend left,"\conicfgf"]
\end{tikzcd}
\zbox.
\end{equation}
The expansion functor is then the functor
\[
\tilde D = \pointedfgf \circ \conicfgf \circ \conicfreef- \circ \pointedfreefpar-
\co \oPCat \to \oPCat
\]
which is the underlying functor of the monad of the composition of the two
adjunctions (<ref>) and (<ref>). Then, one
could show that this functor restricts well to polygraphs (just like for the case
of strict categories <cit.>), so that we get $D \co \oPol \to
\oPol$, and then show that $D$ is the underlying functor of a separating
Sadly, the definition of $\conef$ does not go through for precategories. Given a
pointed $(C,o)$, even though one can follow the concrete definition of
<cit.> to get a globular set $\conef(C,o)$ equipped with
precategorical compositions operations, one can show that the latter do not
satisfy axiom <Ref> of precategories in general: the lack of
interchange law for precategories is to blame here.
While it is not formally excluded that the shape category of plexes is a weak
test category, the fact that it does not admit an expansion functor while the
one of regular plexes does is already a bad sign which suggests, in addition to
the difficulty to define a notion of weak equivalences with good properties (as
discussed at the beginning of this section), that bare precategories are
not an adequate tool for homotopical purposes (but that does not prevent them to
be used to define other adequate tools, like Gray
categories <cit.>). Maybe this could be linked to the fact
that the underlying operad of precategories is not contractile, and should be
better understood in future work.
The authors would like to thank Manuel Araújo for the discussions about their
shared interest on precategories and their applications. They also would like to
thank Léonard Guetta for his useful comments about this work, in particular on
the use of strict Conduché functors in a precategorical setting.
Missing 'biblatex' package
The bibliography requires the 'biblatex' package.
Cambridge University Press
titleLocally presentable and accessible categories
titleOrientals as free algebras
titleJoint et tranches pour les $\omega$-catégories strictes
titleCoherence for adjunctions in a $4$-category
titleComputads and string diagrams for $n$-sesquicategories
titleSimple string diagrams and $n$-sesquicategories
Cambridge university press
titleTerm rewriting and all that
titleGlobular: an online proof assistant for higher-dimensional rewriting
booktitleLogic in Computer Science (LICS), 32nd Annual Symposium on
titleData structures for quasistrict higher categories
titleToposes, triples, and theories
American Mathematical Society
journaltitleContemporary Mathematics
titleComputads for finitary monads on globular sets
University of Sydney, Australia
titleLimits in $2$-categories of locally-presented categories
journaltitleTheoretical computer science
titleHigher-dimensional word problems with applications to equational logic
titleAutomates et grammaires polygraphiques
Cambridge University Press
journaltitleMathematical Structures in Computer Science
titleConnected limits, familial representability and Artin glueing
Cambridge University Press
journaltitleMathematical Structures in Computer Science
titleCorrigenda for ‘Connected limits, familial representability and Artin glueing’
titleA direct proof that the category of 3-computads is not cartesian closed
titleComputads for weak $\omega$-categories as an inductive type
journaltitleMathematische Annalen
titleÜber unendliche diskontinuierliche Gruppen
Institut Polytechnique de Paris
titleComputational descriptions of higher categories
Higher categories ; Strict categories ; Gray categories ; Rewriting ; Word problem ; Pasting diagrams ; Catégories supérieures ; Catégories strictes ; Catégories de Gray ; Réécriture ; Problème du mot ; Diagrammes de recollement
titleAn extension of Batanin's approach to globular algebras
Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik
booktitle3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)
titleCoherence of Gray categories via rewriting
titleRewriting in Gray categories with applications to coherence
London Mathematical Society
journaltitleCompositio Mathematica
titleCoherent presentations of Artin monoids
Société mathématique de France
seriesMémoires de la Société Mathématique de France
titleMéthode de la descente
American Mathematical Society
titleCoherence for tricategories
titlePursuing stacks
journaltitleHigher Structures
titlePolygraphs and Discrete Conduché $\omega$-Functors
journaltitleMathematische Zeitschrift
titleConvergent presentations and polygraphic resolutions of associative algebras
journaltitleTheory and Applications of Categories
titleHigher-dimensional categories with finite derivation type
Cambridge University Press
journaltitleMathematical Structures in Computer Science
titleCoherence in monoidal track categories
titleRegular polygraphs and the Simpson conjecture
journaltitleHigher structures
titleNon-unital polygraphs form a presheaf category
titleWeak units and homotopy 3-types
journaltitleJournal of Pure and Applied Algebra
titleA new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier)
journaltitleJournal of Pure and Applied Algebra
titleTowards an algebraic theory of boolean circuits
journaltitleAdvances in Mathematics
titleA folk model structure on omega-cat
noteAvailable on the author's web page <http://www.math.mcgill.ca/makkai/>
titleThe word problem for computads
Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik
booktitleRTA-24th International Conference on Rewriting Techniques and Applications-2013
titleA homotopical completion procedure with applications to coherence of monoids
booktitle1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)
titleHomological computations for term rewriting systems
booktitleProceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation
titleCompletion in operads via essential syzygies
Société mathématique de France
titleLa théorie de l'homotopie de Grothendieck
International Press of Boston
journaltitleHomology, Homotopy and Applications
titleCofibrant objects among higher-dimensional categories
journaltitleLogical Methods in Computer Science
titleTowards 3-Dimensional Rewriting Theory
journaltitleJournal of Pure and Applied Algebra
titleWord problems and a homological finiteness condition for monoids
journaltitleTheoretical Computer Science
titleA finiteness condition for rewriting systems
journaltitleJournal of Pure and Applied Algebra
titleLimits indexed by category-valued 2-functors
journaltitleHandbook of algebra
titleCategorical structures
journaltitleJournal of Pure and Applied Algebra
titleThe petit topos of globular sets
Cambridge University Press
titleTerm rewriting systems
journaltitleChristiana Videnskabs-Selskabs Skrifter, I. Matematisk-naturvidenskabelig Klasse
titleProbleme über Veränderungen von Zeichenreihen nach gegebenen Regeln
journaltitleLogical Methods in Computer Science
titleGlobular: an online proof assistant for higher-dimensional rewriting
journaltitleTheory and Applications of Categories
titleGeneric morphisms, parametric representations and weakly Cartesian monads
journaltitleTheory and Applications of Categories
titleFamilial 2-functors and parametric right adjoints
|
# Patterns of gauge symmetry in the background field method
A. C. Aguilar University of Campinas - UNICAMP, Institute of Physics “Gleb
Wataghin”, 13083-859 Campinas, São Paulo, Brazil. M. N. Ferreira Department
of Theoretical Physics and IFIC,
University of Valencia and CSIC, E-46100, Valencia, Spain. D. Ibañez
University Centre EDEM, Muelle de la Aduana, La Marina de Valencia, 46024,
Valencia, Spain. B. M. Oliveira University of Campinas - UNICAMP, Institute
of Physics “Gleb Wataghin”, 13083-859 Campinas, São Paulo, Brazil. J.
Papavassiliou Department of Theoretical Physics and IFIC,
University of Valencia and CSIC, E-46100, Valencia, Spain.
###### Abstract
The correlation functions of Yang-Mills theories formulated in the background
field method satisfy linear Slavnov-Taylor identities, which are naive
generalizations of simple tree level relations, with no deformations
originating from the ghost-sector of the theory. In recent years, a stronger
version of these identities has been found to hold at the level of the
background gluon self-energy, whose transversality is enforced separately for
each special block of diagrams contributing to the gluon Schwinger-Dyson
equation. In the present work we demonstrate by means of explicit calculations
that the same distinct realization of the Slavnov-Taylor identity persists in
the case of the background three-gluon vertex. The analysis is carried out at
the level of the exact Schwinger-Dyson equation for this vertex, with no
truncations or simplifying assumptions. The demonstration entails the
contraction of individual vertex diagrams by the relevant momentum, which
activates Slavnov-Taylor identities of vertices and multi-particle kernels
nested inside these graphs; the final result emerges by virtue of a multitude
of extensive cancellations, without the need of performing explicit
integrations. In addition, we point out that background Ward identities amount
to replacing derivatives of propagators by zero-momentum background-gluon
insertions, in exact analogy to standard properties of Abelian gauge theories.
Finally, certain potential applications of these results are briefly
discussed.
## I Introduction
In recent years, the systematic exploration of Green’s (correlation) functions
has afforded important insights on the nonperturbative properties of non-
Abelian gauge theories, such as pure Yang-Mills theories and Quantum
Chromodynamics Roberts and Williams (1994); Alkofer and von Smekal (2001);
Fischer (2006); Roberts (2008); Binosi and Papavassiliou (2009); Binosi _et
al._ (2015); Cloet and Roberts (2014); Aguilar _et al._ (2016a); Binosi _et
al._ (2016, 2017); Huber (2020); Papavassiliou (2022). This ongoing scrutiny
relies on continuum studies based on nonperturbative functional methods Maris
and Roberts (1997, 2003); Braun _et al._ (2010); Eichmann _et al._ (2009);
Cloet _et al._ (2009); Boucaud _et al._ (2008); Eichmann _et al._ (2010);
Fischer _et al._ (2009); Boucaud _et al._ (2008); Dudal _et al._ (2008);
Rodriguez-Quintero (2011); Tissier and Wschebor (2010); Pennington and Wilson
(2011); Huber _et al._ (2012); Cloet and Roberts (2014); Fister and Pawlowski
(2013); Cyrol _et al._ (2015); Boucaud _et al._ (2008); Pawlowski _et al._
(2004); Pawlowski (2007); Cyrol _et al._ (2018a, b); Corell _et al._ (2018);
Gao _et al._ (2018); Blaizot _et al._ (2021); Roberts (2020, 2020); Horak
_et al._ (2021); Gao _et al._ (2021) carried out almost exclusively in the
linear covariant ($R_{\xi}$) gauges, where the Landau gauge is the preferred
choice, and on lattice simulations performed in the same gauge Cucchieri _et
al._ (2006); Cucchieri and Mendes (2007); Bogolubsky _et al._ (2007, 2009);
Cucchieri _et al._ (2008); Cucchieri and Mendes (2010); Oliveira and Silva
(2009); Oliveira and Bicudo (2011); Maas (2013); Ayala _et al._ (2012);
Oliveira and Silva (2012); Athenodorou _et al._ (2016); Duarte _et al._
(2016); Boucaud _et al._ (2017, 2018); Aguilar _et al._ (2021). However, the
background field method (BFM) DeWitt (1967); Honerkamp (1972); Kallosh (1974);
Kluberg-Stern and Zuber (1975); Arefeva _et al._ (1974); Abbott (1981);
Weinberg (1980); Abbott (1982); Shore (1981); Abbott _et al._ (1983) has also
been employed in several occasions, furnishing useful vantage points, and
exposing key properties of the theory that are normally distorted by standard
quantization procedures Aguilar and Papavassiliou (2006); Aguilar _et al._
(2008, 2016a); Papavassiliou (2022).
The BFM is a powerful framework that enables the implementation of the gauge-
fixing procedure necessary for quantizing gauge theories without losing
explicit gauge invariance, in contradistinction to the conventional
quantization schemes DeWitt (1967); Honerkamp (1972); Kallosh (1974); Kluberg-
Stern and Zuber (1975); Arefeva _et al._ (1974); Abbott (1981); Weinberg
(1980); Abbott (1982); Shore (1981); Abbott _et al._ (1983). The starting
point of the BFM is the splitting of the gauge field $A_{\mu}^{a}$ appearing
in the classical action of the theory according to
$A_{\mu}^{a}=B_{\mu}^{a}+Q_{\mu}^{a}$, where $B_{\mu}^{a}$ and $Q_{\mu}^{a}$
are the background and quantum (fluctuating) fields, respectively. The quantum
field is the variable of integration in the generating functional $Z(J)$, and
external sources are coupled only to it, as $J\cdot Q$. The background field
does not enter in loops; it couples externally to Feynman diagrams, connecting
them with the asymptotic states to form S-matrix elements. Then, by virtue of
a special gauge-fixing condition, the resulting action (with the corresponding
ghost terms included) is no longer invariant under transformations of the
quantum field, but retains its invariance intact with respect to the
background field Abbott (1982); Binosi and Papavassiliou (2009).
A key consequence of the background gauge invariance of the action is that
Green’s functions involving the $B_{\mu}^{a}$ field satisfy ghost-free
Slavnov-Taylor identities (STIs), akin to the Takahashi identity known from
QED: the STIs are straightforward generalization of tree level relations,
receiving no ghost-related contributions after the inclusion of quantum
corrections. Instead, in the standard STIs Taylor (1971); Slavnov (1972) of
the $R_{\xi}$ gauges Fujikawa _et al._ (1972), starting already at one-loop,
the ghost sector modifies these tree level relations non-trivially. To fix the
ideas with a simple example, the quark-gluon vertex with either a $B$ or a $Q$
gluon satisfies at tree level the simple identity (suppressing color)
$q^{\mu}\gamma_{\mu}=\not{q}=(-\not{p}-m)-(\not{r}-m)=S_{0}^{-1}(-\not{p})-S_{0}^{-1}(\not{r})$,
where $S_{0}$ is the tree level version of the quark propagator $S$. In the
case of a $B$ gluon, the all-order STI is obtained from the above tree level
identity by simply substituting $S_{0}\to S$, namely
$q^{\mu}\widehat{\Gamma}_{\mu}(q,r,p)=S^{-1}(-\not{p})-S^{-1}(\not{r})$.
Instead, in the case of a $Q$ gluon ($R_{\xi}$ gauges), the STI gets modified
by quantum corrections Taylor (1971); Slavnov (1972), which induce a
dependence on the ghost dressing function and the quark-ghost kernel Marciano
and Pagels (1978); Davydychev _et al._ (2001); Aguilar and Papavassiliou
(2011); Aguilar _et al._ (2017).
As was pointed in earlier studies, the STI satisfied by the background-gluon
self-energy, $\widehat{\Pi}_{\mu\nu}(q)$, namely the standard transversality
condition $q^{\mu}\widehat{\Pi}_{\mu\nu}(q)=0$, is implemented in a very
special way, which has been denominated “block-wise” Aguilar and Papavassiliou
(2006); Aguilar _et al._ (2008, 2016a). Specifically, the diagrammatic
representation of the SDE governing $\widehat{\Pi}_{\mu\nu}(q)$ is composed by
four distinct blocks, namely one- and two-loop diagrams containing only
gluons, and one- and two-loop diagrams containing ghost fields, as shown in
Fig. 1. The block-wise realization of the STI in this case is the simple
statement that the transversality of $\widehat{\Pi}_{\mu\nu}(q)$ is enforced
independently for each of the four blocks. This is in sharp contrast to what
happens in the $R_{\xi}$ gauges, where, already at the one-loop perturbative
level, it is only the sum of gluon and ghost diagrams that is transverse
Itzykson and Zuber (1980); Peskin and Schroeder (1995). The proof of this
property is particularly simple; it proceeds by contracting the various
diagrams by $q^{\mu}$ from the side of the fully dressed vertices, thus
triggering the corresponding naive STIs of the BFM. It is important to
emphasize that the proof holds for any value of the gauge-fixing parameter
$\xi_{{\scriptscriptstyle Q}}$ used to define the propagators of the $Q$-type
gluons entering in the various loops.
The basic question that arises naturally in this context is whether the
special block-wise realization of the STI described above is particular to the
two-point function, or if it is a common feature of all Green’s functions
containing only $B$ gluons. In the present work we take a first step in the
exploration of this issue, and demonstrate that the same pattern persists in
the STI of the BFM vertex with three incoming background gluons, to be denoted
by $\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$. This Abelian STI relates the
contraction $q^{\mu}\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$ to the difference
$\widehat{\Pi}_{\mu\nu}(r)-\widehat{\Pi}_{\mu\nu}(p)$ Cornwall and
Papavassiliou (1989); Aguilar and Papavassiliou (2006); Binosi and
Papavassiliou (2009). It turns out that the diagrams comprising the SDE of
$\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$ may also be classified into four
subsets in a way completely analogous to the case of
$\widehat{\Pi}_{\mu\nu}(q)$. Then, the contraction of each subset by $q^{\mu}$
generates the difference of the corresponding subsets of
$\widehat{\Pi}_{\mu\nu}$, confirming the block-wise realization of this STI.
We emphasize that, as in the case of the gluon self-energy, this property is
completely $\xi_{{\scriptscriptstyle Q}}$-independent.
An additional noteworthy aspect of the BFM Green’s functions is the Ward
identities (WIs) they satisfy, namely the relations that emerge when the
momentum that triggers the STIs is taken to vanish. For example, in the case
of the $\widehat{\Gamma}_{\mu}(q,r,p)$ mentioned above, a Taylor expansion of
the STI around $q=0$ and subsequent matching of terms linear in $q$ yields the
relation $\widehat{\Gamma}_{\mu}(0,-p,p)=\partial S^{-1}(\not{p})/\partial
p^{\mu}$, which is the precise equivalent of the text-book WI known from QED,
relating the photon-electron vertex with the electron propagator Itzykson and
Zuber (1980); Peskin and Schroeder (1995). In fact, exactly as happens in QED,
this WI admits a simple diagrammatic interpretation: the derivative of the
inverse quark propagator may be depicted as the insertion of a background
gluon carrying zero momentum. These observations may be straightforwardly
extended to the case of the three-gluon vertex
$\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$, allowing for a completely analogous
pictorial representation of the corresponding WI. In fact, the block-wise
realization of the STI leads to a corresponding pattern for the WIs that
emerges from it: the derivative acting on any of the blocks of
$\widehat{\Pi}_{\mu\nu}(q)$ is identical to the diagrams comprising the
associated block of $\widehat{\Gamma}_{\alpha\mu\nu}(q,r,p)$, when the
corresponding momentum is set to zero. To the best of our knowledge, the
notions described above appear for the first time in the literature.
The article is organized as follows. In Sec. II we review certain pivotal
properties of the BFM, and explain the notion of the block-wise transversality
at the level of the SDE that governs the background gluon propagator. Sec. III
contains the main result of this work, namely the demonstration of the block-
wise realization of the STI for the case of the background three-gluon vertex.
Then, in Sec. IV we focus on the WIs of the BFM, their graphical
representation in terms of zero-momentum gluon insertions, and demonstrate the
block-wise realization of the three-gluon WI, for the operationally simplest
subset of graphs. In Sec. V we summarize our findings and discuss future
directions. Finally, in four Appendices we present complementary material that
facilitates the perusal of the article.
## II General theoretical framework
In this section we highlight some of the significant features of the BFM
formalism that are relevant for the demonstrations that follow; for further
details the reader is referred to the extensive literature on the subject,
see, e.g., Abbott (1981); Abbott _et al._ (1983); Binosi and Papavassiliou
(2009).
(i) The initial decomposition of the gauge field into $B_{\mu}^{a}$ and
$Q_{\mu}^{a}$ components increases considerably the number of Green’s
functions that can be defined, which may be classified into three broad
subsets: those with $B_{\mu}^{a}$ fields only, those with $Q_{\mu}^{a}$ fields
only (corresponding to the standard Green’s functions of the $R_{\xi}$
gauges), and mixed ones, with both $B_{\mu}^{a}$ and $Q_{\mu}^{a}$ fields. We
will occasionally denote Green’s functions according to the type of incoming
fields, such as “BB” for the case of the propagator connecting two background
fields, or “BBB” for the case of the three-gluon vertex connecting three such
fields.
(ii) The gluon propagator QQ that enters in the quantum loops will be denoted
by $\Delta_{\mu\nu}^{ab}(q)=-i\delta^{ab}\Delta_{\mu\nu}(q)$, with
$\displaystyle\Delta_{\mu\nu}(q)=P_{\mu\nu}(q)\Delta(q)+\xi_{{\scriptscriptstyle
Q}}\frac{q_{\mu}q_{\nu}}{q^{4}}\,,\qquad
P_{\mu\nu}(q)=g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\,,$ (1)
whose inverse is
$\displaystyle\Delta_{\mu}^{\nu}(q)\Delta_{\nu\rho}^{-1}(q)=g_{\mu\rho}\,;\quad\quad\Delta_{\nu\rho}^{-1}(q)=\Delta^{-1}(q)P_{\nu\rho}(q)+\xi_{{\scriptscriptstyle
Q}}^{-1}q_{\nu}q_{\rho}\,.$ (2)
The scalar function $\Delta(q)$ is related to the gluon self-energy
$\Pi_{\mu\nu}(q)=P_{\mu\nu}(q)\Pi(q)$ through $\Delta^{-1}(q)=q^{2}+i\Pi(q)$,
and $\xi_{{\scriptscriptstyle Q}}$ is the quantum gauge-fixing parameter. Note
that $\xi_{{\scriptscriptstyle Q}}$ enters also in the tree level expressions
of the vertices BQQ and BBQQ, given in Table 1.
(iii) In what follows we will use extensively a number of three- and four-
particle vertices, which we list here. In particular, the relevant three-
particle vertices are
$\displaystyle\Gamma_{\bar{c}^{m}c^{n}Q_{\mu}^{a}}(r,p,q)=-gf^{mna}\Gamma_{\mu}(r,p,q)\,,\qquad$
$\displaystyle\Gamma_{\\!Q_{\alpha}^{a}Q_{\mu}^{b}Q_{\nu}^{c}}(q,r,p)=gf^{abc}\widetilde{\Gamma}_{\alpha\mu\nu}(q,r,p)\,,$
$\displaystyle\Gamma_{\bar{c}^{m}c^{n}B_{\mu}^{a}}(r,p,q)=-gf^{mna}\widetilde{\Gamma}_{\mu}(r,p,q)\,,\qquad$
$\displaystyle\Gamma_{\\!B_{\alpha}^{a}Q_{\mu}^{b}Q_{\nu}^{c}}(q,r,p)=gf^{abc}\widetilde{\Gamma}_{\alpha\mu\nu}(q,r,p)\,,$
(3)
while the four-particle vertices are
$\displaystyle\Gamma_{B_{\mu}^{a}B_{\nu}^{b}\bar{c}^{m}c^{n}}(q,r,p,t)=-ig^{2}\widehat{\Gamma}_{\mu\nu}^{abmn}(q,r,p,t),$
$\displaystyle\Gamma_{\\!B_{\alpha}^{a}Q_{\beta}^{b}Q_{\mu}^{c}Q_{\nu}^{d}}(q,r,p,t)=-ig^{2}\widetilde{\Gamma}_{\alpha\beta\mu\nu}^{abcd}(q,r,p,t),$
$\displaystyle\Gamma_{B_{\mu}^{a}Q_{\nu}^{b}\bar{c}^{m}c^{n}}(q,r,p,t)=-ig^{2}\widetilde{\Gamma}_{\mu\nu}^{abmn}(q,r,p,t),$
$\displaystyle\Gamma_{\\!B_{\alpha}^{a}B_{\beta}^{b}Q_{\mu}^{b}Q_{\nu}^{c}}(q,r,p,t)=-ig^{2}\widehat{\Gamma}_{\alpha\beta\mu\nu}^{abcd}(q,r,p,t),$
(4)
where $g$ denotes the gauge coupling constant, and $f^{abc}$ are the SU(3)
structure constants.
(iv) A central quantity in our analysis is the background self-energy,
$\widehat{\Pi}_{\mu\nu}(q)$, related to the inverse background gluon
propagator $\widehat{\Delta}^{-1}_{\mu\nu}(q)$ by 111The definition of the BB
propagator $\widehat{\Delta}_{\mu\nu}(q)$ requires the addition to the action
of a supplementary gauge-fixing term, which introduces the “classical” gauge-
fixing parameter, $\xi_{{\scriptscriptstyle C}}$ Abbott (1981); Abbott _et
al._ (1983); Binosi and Papavassiliou (2009). Note that this step is necessary
only when connecting the background gluon to external states in order to
construct S-matrix elements, and will be omitted here.
$\widehat{\Delta}^{-1}_{\mu\nu}(q)=q^{2}P_{\mu\nu}(q)+i\widehat{\Pi}_{\mu\nu}(q)\,.$
(5)
The gauge symmetry enforces the fundamental STI
$q^{\mu}\widehat{\Pi}_{\mu\nu}(q)=0\,,$ (6)
from which follows that
$\widehat{\Pi}_{\mu\nu}(q)=P_{\mu\nu}(q)\widehat{\Pi}(q)$, where
$\widehat{\Pi}(q)$ is a scalar function. Thus, Eq. (5) may be cast in the form
$\widehat{\Delta}^{-1}_{\mu\nu}(q)=P_{\mu\nu}(q)\left[q^{2}+i\widehat{\Pi}(q)\right]\,.$
(7)
The SDE that defines $\widehat{\Pi}_{\mu\nu}(q)$ is diagrammatically
represented in Fig. 1. Note the separation of the dressed Feynman diagrams
into the following four distinct groups:
(1) One-loop gluonic graphs, enclosed in the blue box; their total
contribution is denoted by $\widehat{\Pi}^{(1)}_{\mu\nu}(q)$.
(2) One-loop ghost graphs, enclosed in the orange box; their total
contribution is denoted by $\widehat{\Pi}^{(2)}_{\mu\nu}(q)$.
(3) Two-loop gluonic graphs, enclosed in the purple box; their total
contribution is denoted by $\widehat{\Pi}^{(3)}_{\mu\nu}(q)$.
(4) Two-loop ghost graphs, enclosed in the green box; their total contribution
is denoted by $\widehat{\Pi}^{(4)}_{\mu\nu}(q)$.
Figure 1: Diagrammatic representation of the self-energy
$\widehat{\Pi}_{\mu\nu}(q)$. The small gray circles at the end of gluon legs
indicate a background field. The orange and green circles represent
conventional fully dressed propagators and vertices, respectively, while the
blue circles represent fully dressed vertices with one background gluon.
One of the most exceptional properties of $\widehat{\Pi}_{\mu\nu}(q)$ is its
block-wise transversality Aguilar and Papavassiliou (2006); Binosi and
Papavassiliou (2008a, b). Specifically, the fundamental relation given in Eq.
(6) is realized in a very special way: each of the four subsets of diagrams in
Fig. 1 is individually transverse, i.e.,
$q^{\mu}\widehat{\Pi}^{(i)}_{\mu\nu}(q)=0\,,\qquad i=1,2,3,4\,.$ (8)
This particular result is a direct consequence of the Abelian STIs satisfied
by the fully dressed vertices entering in the diagrams comprising the
$\widehat{\Pi}^{(i)}_{\mu\nu}(q)$ Aguilar and Papavassiliou (2006); Binosi and
Papavassiliou (2008a, b), namely BQQ, ${\rm B\bar{c}c}$, BQQQ and ${\rm
BQ\bar{c}c}$, reported in Table 2 of the Appendix C.
(v) In order to elucidate with a simple example how this special
transversality is enforced at the diagrammatic level, we consider the case of
$\widehat{\Pi}^{(2)}_{\mu\nu}(q)$, whose diagrams are enclosed by the orange
box of Fig. 1.
The diagrams $(a_{{\scriptscriptstyle 3}})$ and $(a_{{\scriptscriptstyle 4}})$
are given by
$\displaystyle(a_{{\scriptscriptstyle 3}})_{\mu\nu}(q)$
$\displaystyle=\lambda\int_{k}(2k-q)_{\mu}D(k-q)D(k)\widetilde{\Gamma}_{\nu}(q-k,k,-q)\,,$
(9) $\displaystyle(a_{{\scriptscriptstyle 4}})_{\mu\nu}(q)$
$\displaystyle=2\lambda\,g_{\mu\nu}\int_{k}D(k)\,,$ (10)
where we have used the Feynman rules given in Eq. (95) and Eq. (99) of the
Appendix B, and factored out the trivial color structure $\delta^{ab}$ from
both expressions. In addition, we have defined
$\lambda:=g^{2}C_{\rm A}\,,$ (11)
where $C_{\rm A}$ is the Casimir eigenvalue of the adjoint representation [$N$
for SU($N$)]. Furthermore, we have introduced
$\int_{k}:=\frac{1}{(2\pi)^{4}}\\!\int_{-\infty}^{+\infty}\\!\\!\mathrm{d}^{4}k\,,$
(12)
where the use of a symmetry-preserving regularization scheme is implicitly
assumed.
We next contract graph $(a_{{\scriptscriptstyle 3}})_{\mu\nu}(q)$ by
$q^{\nu}$, thus triggering the STI satisfied by
$\widetilde{\Gamma}_{\nu}(q-k,k,-q)$, given in Eq. (101), to obtain
$\displaystyle q^{\nu}(a_{{\scriptscriptstyle 3}})_{\mu\nu}(q)$
$\displaystyle=$
$\displaystyle\lambda\int_{k}(2k-q)_{\mu}D(k-q)D(k)\left[D^{-1}(k-q)-D^{-1}(k)\right]$
(13) $\displaystyle=$
$\displaystyle\lambda\int_{k}(2k-q)_{\mu}\left[D(k)-D(k-q)\right]$
$\displaystyle=$ $\displaystyle-2\lambda\,q_{\mu}\int_{k}D(k)\,,$
which is exactly the negative of the contraction
$q^{\nu}(a_{{\scriptscriptstyle 4}})_{\mu\nu}(q)$. Hence,
$\displaystyle q^{\nu}\left[(a_{{\scriptscriptstyle
3}})_{\mu\nu}(q)+(a_{{\scriptscriptstyle
4}})_{\mu\nu}(q)\right]=q^{\nu}\widehat{\Pi}^{(2)}_{\mu\nu}(q)=0\,.$ (14)
We emphasize that the above strategy of contracting directly individual
diagrams and triggering the corresponding STIs will be followed unaltered in
the more complicated case of the three-gluon vertex treated in the next
section. Note finally that the entire demonstration leading to Eq. (8) is
carried out for a general value of the gauge-fixing parameter
$\xi_{{\scriptscriptstyle Q}}$ Aguilar and Papavassiliou (2006); Binosi and
Papavassiliou (2008a, b).
## III Block-wise STI of the three-gluon vertex
In this section we demonstrate the block-wise realization of the STI satisfied
by the BBB three-gluon vertex.
### III.1 General considerations
(i) The one-particle irreducible three-gluon vertex,
$\widehat{\bm{\Gamma}}_{\alpha\mu\nu}^{abc}(q,r,p)$, is defined from the
vacuum expectation value of the time ordered product of three background
gluons (in momentum space), as
$\langle
0|\,T\\!\left[{B}^{a}_{\alpha^{\prime}}(q)\,{B}^{b}_{\mu^{\prime}}(r)\,{B}^{c}_{\nu^{\prime}}(p)\right]\\!|0\rangle=g\,\widehat{\bm{\Gamma}}_{\alpha\mu\nu}^{abc}(q,r,p)\widehat{\Delta}^{\alpha}_{\alpha^{\prime}}(q)\widehat{\Delta}^{\mu}_{\mu^{\prime}}(r)\widehat{\Delta}^{\nu}_{\nu^{\prime}}(p)\,.$
(15)
The three-gluon vertex $\widehat{\bm{\Gamma}}_{\alpha\mu\nu}^{abc}(q,r,p)$ is
naturally cast in the form
$\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)=\widehat{\Gamma}^{(0)abc}_{\alpha\mu\nu}(q,r,p)+\widehat{\Gamma}^{abc}_{\alpha\mu\nu}(q,r,p)\,,$
(16)
where the tree level component
$\widehat{\Gamma}^{(0)abc}_{\alpha\mu\nu}(q,r,p)=f^{abc}\widehat{\Gamma}^{(0)}_{\alpha\mu\nu}(q,r,p)$
coincides with that of the conventional three-gluon vertex (QQQ), i.e.,
$\widehat{\Gamma}^{(0)}_{\alpha\mu\nu}(q,r,p)=\Gamma^{(0)}_{\alpha\mu\nu}(q,r,p)=(q-r)_{\nu}g_{\alpha\mu}+(r-p)_{\alpha}g_{\mu\nu}+(p-q)_{\mu}g_{\alpha\nu}\,,$
(17)
while $\widehat{\Gamma}^{abc}_{\alpha\mu\nu}(q,r,p)$ captures all quantum
corrections, both perturbative and nonperturbative.
Figure 2: The block-wise structure of the SDE which describes the three-gluon
vertex $\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)$ (BBB). The blue
circles represent full one-particle irreducible vertices, while the purple
ones the four- and five-point scattering kernels.
(ii) The SDE that defines $\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)$
is shown diagrammatically in Fig. 2, written with respect to the gluon that
carries momentum $q$; therefore, the corresponding vertices to which this leg
is attached are kept at tree level. The corresponding Feynman diagrams have
been classified in four blocks, applying the exact same criterion as in the
case of $\widehat{\Pi}_{\mu\nu}(q)$, and employing the same color code for the
individual boxes as in Fig. 1. Thus,
$\widehat{\Gamma}^{abc}_{\alpha\mu\nu}(q,r,p)=\sum_{i=1}^{4}\widehat{\Gamma}^{(i)abc}_{\alpha\mu\nu}(q,r,p)\,,$
(18)
where, as shown in Fig. 2, the four blocks are comprised by the diagrams
(suppressing indices)
$\widehat{\Gamma}^{(1)}=(b_{{\scriptscriptstyle 1}})+(b_{{\scriptscriptstyle
2}})+(b_{{\scriptscriptstyle
3}})\,,\qquad\widehat{\Gamma}^{(2)}=(c_{{\scriptscriptstyle
1}})+(c_{{\scriptscriptstyle 2}})+(c_{{\scriptscriptstyle 3}})\,,\qquad\
\widehat{\Gamma}^{(3)}=(d)\,,\qquad\widehat{\Gamma}^{(4)}=(e)\,.$ (19)
(iii) It is well-known that
$\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)$ satisfies the Abelian STI
Cornwall and Papavassiliou (1989); Aguilar and Papavassiliou (2006); Binosi
and Papavassiliou (2009)
$p^{\nu}\widehat{\bm{\Gamma}}^{abc}_{\alpha\mu\nu}(q,r,p)=f^{cae}\left[\widehat{\Delta}^{be}_{\alpha\mu}(r)\right]^{-1}-f^{bce}\left[\widehat{\Delta}^{ae}_{\alpha\mu}(q)\right]^{-1}\,,$
(20)
and cyclic permutations thereof. The STI of Eq. (20) may be obtained by means
of formal manipulations of the BFM generating functional, or simply from the
STI of the conventional QQQ vertex Marciano and Pagels (1978); Ball and Chiu
(1980); Davydychev _et al._ (1996), by setting all ghost-related
contributions to their tree level values.
From Eq. (17) it is elementary to show that
$\displaystyle
p^{\nu}\widehat{\Gamma}^{(0)}_{\alpha\mu\nu}(q,r,p)=r^{2}P_{\alpha\mu}(r)-q^{2}P_{\alpha\mu}(q)\,.$
(21)
Then, from Eqs. (5), (16) and (20) follows that
$p^{\nu}\widehat{\Gamma}^{abc}_{\alpha\mu\nu}(q,r,p)=if^{cae}\widehat{\Pi}^{be}_{\alpha\mu}(r)-if^{bce}\widehat{\Pi}^{ae}_{\alpha\mu}(q)\,.$
(22)
(iv) The central observation of the present study is that, as happens in the
case of Eq. (6), the STI in Eq. (22) admits a block-wise realization.
Specifically, as we will demonstrate in this section,
$p^{\nu}\widehat{\Gamma}^{(i)abc}_{\alpha\mu\nu}(q,r,p)=if^{cae}\widehat{\Pi}^{(i)be}_{\alpha\mu}(r)-if^{bce}\widehat{\Pi}^{(i)ae}_{\alpha\mu}(q)\,,\qquad
i=1,2,3,4\,.$ (23)
In diagrammatic terms, Eq. (23) states that the contraction by $p^{\nu}$ of
the diagrams within a given block (color) in Fig. 2 generates the difference
between diagrams within the corresponding block (color) of Fig. 1. In fact,
the validity of Eq. (23) will be demonstrated by acting with $p^{\nu}$ on
vertex diagrams, and exploiting the STIs triggered by this contraction in
order to cast the result in the form of self-energy contributions.
(v) Note that the diagrams shown in Fig. 2 have a factor $g$ removed from
them, which cancels against the $g$ appearing in the definition of the three-
gluon vertex in Eq. (15). In addition a factor of $i$ will be factored out,
which will cancel against the explicit $i$ appearing on the r.h.s. of Eq.
(23). Thus, on the r.h.s. of all intermediate formulas will appear directly
the self-energy diagrams $(a_{i})$ of Fig. 1. Furthermore, with the exception
of Sec. IV, we will suppress the argument $(q,r,p)$ in all vertex graphs.
(vi) We introduce the definitions
$\displaystyle h_{1}^{abmn}=f^{abe}f^{mne}\,,\qquad
h_{2}^{abcde}=f^{abm}f^{cmn}f^{dne}\,.$ (24)
Note that $h_{1}^{abmn}=h_{1}^{banm}$ and $h_{2}^{abcde}=h_{2}^{baced}$.
(vii) We introduce the short-hand notation
$\displaystyle
R^{\alpha\beta}_{\mu}(r,p)\equiv\Delta^{\alpha\sigma}(r)\Delta^{\beta\rho}(p)\Gamma_{\mu\sigma\rho}(-r-p,r,p),\,$
$\displaystyle R_{\mu}(r,p)\equiv D(r)D(p)\Gamma_{\mu}(r,p,-r-p),$
$\displaystyle\widetilde{R}^{\alpha\beta}_{\mu}(r,p)\equiv\Delta^{\alpha\sigma}(r)\Delta^{\beta\rho}(p)\widetilde{\Gamma}_{\mu\sigma\rho}(-r-p,r,p),\,$
$\displaystyle\widetilde{R}_{\mu}(r,p)\equiv
D(r)D(p)\widetilde{\Gamma}_{\mu}(r,p,-r-p).$ (25)
The special relations
$\displaystyle\int_{k}\widetilde{R}^{\sigma\beta}_{\mu}(k,r-k)=\int_{k}\widetilde{R}_{\mu}(k,r-k)=0\,,$
(26)
will be employed in the analysis that follows. Their validity may be
established by appealing to the Bose symmetry of the BQQ vertex with respect
to its two quantum legs or the ghost-antighost symmetry of the ${\rm
B\bar{c}c}$ vertex, and the change of integration variable $r-k\to k$. An
alternative demonstration proceeds by noting that
$\int_{k}\widetilde{R}_{\mu}(k,r-k)=\frac{r_{\mu}}{r^{2}}I(r^{2})\,,$ (27)
with
$\displaystyle I(r^{2})=$
$\displaystyle\int_{k}D(k)D(r-k)\,r^{\mu}\widetilde{\Gamma}_{\mu}(k,r-k,-r)$
$\displaystyle=$
$\displaystyle\int_{k}D(k)D(r-k)\left[D^{-1}(k)-D^{-1}(r-k)\right]$
$\displaystyle=$ $\displaystyle\int_{k}\left[D(r-k)-D(k)\right]=0\,,$ (28)
where the STI of Eq. (101) was used. The first relation in Eq. (26) may be
proved in the exact same way, employing the STI of Eq. (100).
(viii) Lastly, from now on we adopt the convention that 1PI vertices
containing at least one background gluon will be represented diagrammatically
by a blue circle (see, e.g., the BBQQ vertex in diagram
$(b_{{\scriptscriptstyle 3,2}})$ of Fig. 3).
### III.2 One-loop gluonic sector (first block)
We begin by considering the one-loop gluonic vertex graphs, namely the set
$\\{(b_{{\scriptscriptstyle 1}}),(b_{{\scriptscriptstyle
2}}),(b_{{\scriptscriptstyle 3}})\\}$, enclosed in the blue box of Fig. 2.
As a first step, we recognize that, by virtue of Eq. (26), diagrams
$(b_{{\scriptscriptstyle 1}})$ and $(b_{{\scriptscriptstyle 2}})$ vanish,
$\displaystyle(b_{{\scriptscriptstyle 1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{g^{2}}{2}f^{ced}\\!\\!\int_{k}\\!\widehat{\Gamma}_{\alpha\mu\beta\sigma}^{(0)abde}\widetilde{R}^{\sigma\beta}_{\nu}(-k,k-p)=0\,,$
$\displaystyle(b_{{\scriptscriptstyle 2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{g^{2}}{2}f^{bed}\\!\\!\int_{k}\\!\widehat{\Gamma}_{\alpha\nu\beta\sigma}^{(0)acde}\widetilde{R}^{\sigma\beta}_{\mu}(-k,k-r)=0\,,$
(29)
since $\widehat{\Gamma}_{\alpha\mu\beta\sigma}^{(0)abde}$ is momentum-
independent, and may be pulled out of the integral sign.
Figure 3: The two contributions which arise from diagram
$(b_{{\scriptscriptstyle 3}})$ of Fig. 2 after performing the skeleton
expansion of the four-gluon scattering kernel (purple blob).
Diagram $(b_{{\scriptscriptstyle 3}})$ has two contributions,
$\displaystyle(b_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=(b_{{\scriptscriptstyle
3,1}})^{abc}_{\alpha\mu\nu}+(b_{{\scriptscriptstyle
3,2}})^{abc}_{\alpha\mu\nu}\,,$ (30)
given in Fig. 3, with
$\displaystyle(b_{{\scriptscriptstyle 3,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{\lambda}{2}f^{abc}\\!\\!\int_{k}\\!\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(q,k-q,-k)\Delta^{\rho\sigma}(k)\widetilde{R}_{\mu}^{\lambda\beta}(k+p,q-k)\widetilde{\Gamma}_{\nu\rho\lambda}(p,k,-k-p)\,,$
$\displaystyle(b_{{\scriptscriptstyle 3,2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{g^{2}}{2}f^{ade}\\!\\!\int_{k}\\!\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(q,k-q,-k)\Delta^{\beta\rho}(k-q)\Delta^{\sigma\lambda}(k)\widehat{\Gamma}^{cbde}_{\nu\mu\rho\lambda}(p,r,q-k,k)\,.$
(31)
The contraction $p^{\nu}(b_{{\scriptscriptstyle 3}})^{abc}_{\alpha\mu\nu}$ may
be evaluated using the STIs in Eqs. (100) and (2) and a moderate amount of
algebra, yielding
$\displaystyle p^{\nu}(b_{{\scriptscriptstyle 3}})^{abc}_{\alpha\mu\nu}=$
$\displaystyle-\frac{\lambda}{2}f^{abc}\\!\\!\int_{k}\\!\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(q,k-q,-k)\left[\widetilde{R}^{\sigma\beta}_{\mu}(k,q-k)+\widetilde{R}^{\sigma\beta}_{\mu}(-k-p,k-q)\right]\,.$
(32)
The first term in Eq. (32) is exactly $-f^{bce}(a_{{\scriptscriptstyle
1}})^{ae}_{\alpha\mu}(q)$. The second term, after $k\rightarrow-k-p$,
generates $f^{cae}(a_{{\scriptscriptstyle 1}})^{be}_{\alpha\mu}(r)$; note, in
particular, that
$\displaystyle\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(q,k-q,-k)\rightarrow-\widetilde{\Gamma}_{\alpha\beta\sigma}^{(0)}(r,k-r,-k)+2p_{\beta}g_{\alpha\sigma}-p_{\alpha}g_{\beta\sigma}-(\xi_{{\scriptscriptstyle
Q}}^{-1}+1)p_{\sigma}g_{\alpha\beta}\,,$ (33)
where, due to Eq. (26), the last three terms give vanishing contributions.
Thus, one arrives at
$\displaystyle p^{\nu}(b_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=f^{cae}(a_{{\scriptscriptstyle
1}})^{be}_{\alpha\mu}(r)-f^{bce}(a_{{\scriptscriptstyle
1}})^{ae}_{\alpha\mu}(q)\,.$ (34)
At this point, we add and subtract on the r.h.s. of Eq. (34) the momentum-
independent seagull graph $(a_{{\scriptscriptstyle 2}})$, to obtain
$\displaystyle p^{\nu}(b_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=f^{cae}[(a_{{\scriptscriptstyle
1}})+(a_{{\scriptscriptstyle
2}})]_{\alpha\mu}^{be}(r)-f^{bce}[(a_{{\scriptscriptstyle
1}})+(a_{{\scriptscriptstyle 2}})]_{\alpha\mu}^{ae}(q)\,,$ (35)
which, in view of Eq. (29), is precisely Eq. (23) for $i=1$.
### III.3 One-loop ghost sector (second block)
Figure 4: The three contributions which emerge from the diagram
$(c_{{\scriptscriptstyle 3}})$ shown in the Fig. 2, after implementing the
skeleton expansion of the four-point scattering kernel (purple blob) formed by
two background gluons with a ghost-antighost pair.
We next focus on the one-loop ghost graphs, forming the set
$\\{(c_{{\scriptscriptstyle 1}}),(c_{{\scriptscriptstyle
2}}),(c_{{\scriptscriptstyle 3}})\\}$, enclosed in the orange box of Fig. 2.
The demonstration that follows is completely analogous to that of the previous
subsection.
Due to Eq. (26), diagrams $(c_{{\scriptscriptstyle 1}})$ and
$(c_{{\scriptscriptstyle 2}})$ vanish,
$\displaystyle(c_{{\scriptscriptstyle 1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=$ $\displaystyle
g^{2}f^{edc}\\!\\!\int_{k}\\!\widehat{\Gamma}_{\alpha\mu}^{(0)abde}\widetilde{R}_{\nu}(-k,k-p)=0\,,$
$\displaystyle(c_{{\scriptscriptstyle 2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=$ $\displaystyle
g^{2}f^{edb}\\!\\!\int_{k}\\!\widehat{\Gamma}_{\alpha\nu}^{(0)acde}\widetilde{R}_{\mu}(-k,k-r)=0\,,$
(36)
and we only need to consider the contraction of graph $(c_{{\scriptscriptstyle
3}})$. This diagram contains three contributions, depicted in Fig. 4,
$\displaystyle(c_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{3}(c_{{\scriptscriptstyle
3,j}})^{abc}_{\alpha\mu\nu}\,,$ (37)
with
$\displaystyle(c_{{\scriptscriptstyle 3,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{\lambda}{2}f^{abc}\\!\\!\int_{k}\\!(2k-q)_{\alpha}D(k-q)\widetilde{R}_{\mu}(k,-k-r)\widetilde{\Gamma}_{\nu}(k+r,q-k,p)\,,$
$\displaystyle(c_{{\scriptscriptstyle 3,2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{\lambda}{2}f^{abc}\\!\\!\int_{k}\\!(2k-q)_{\alpha}D(k-q)\widetilde{R}_{\mu}(-k-r,k)\widetilde{\Gamma}_{\nu}(q-k,k+r,p)\,,$
$\displaystyle(c_{{\scriptscriptstyle 3,3}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-g^{2}f^{eda}\\!\\!\int_{k}\\!(2k-q)_{\alpha}D(k)D(q-k)\widehat{\Gamma}_{\mu\nu}^{bcde}(r,p,q-k,k)\,.$
(38)
The contraction of diagram $(c_{{\scriptscriptstyle 3}})$ by the momentum
$p^{\nu}$ activates the STIs of Eqs. (101) and (2), and one obtains.
$\displaystyle p^{\nu}(c_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=-\lambda
f^{abc}\\!\\!\int_{k}\\!(2k-q)_{\alpha}\\!\left[\widetilde{R}_{\mu}(k-q,-p-k)+\widetilde{R}_{\mu}(q-k,k)\right]\,.$
(39)
The second term of Eq. (39) is simply $-f^{bce}(a_{{\scriptscriptstyle
3}})^{ae}_{\alpha\mu}(q)$, while the first term, after the shift
$k\rightarrow-k-p$ and use of Eq. (26), furnishes
$f^{cae}(a_{{\scriptscriptstyle 3}})^{be}_{\alpha\mu}(r)$. Thus, we conclude
that
$\displaystyle p^{\nu}(c_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=f^{cae}(a_{{\scriptscriptstyle
3}})^{be}_{\alpha\mu}(r)-f^{bce}(a_{{\scriptscriptstyle
3}})^{ae}_{\alpha\mu}(q)\,,$ (40)
which, after adding and subtracting the momentum-independent ($a_{4}$), is
tantamount to the validity of Eq. (23) for $i=2$.
### III.4 Two-loop gluonic sector (third block)
We turn to the two-loop dressed gluonic contributions, contained within the
diagram $(d)$, enclosed by the purple box in Fig. 2. To that end, in Fig. 5 we
show the individual diagrams that emerge upon implementing the skeleton
expansion of the five-gluon kernel (purple circle) inside $(d)$. Note that all
vertices appearing in these graphs are one-particle irreducible.
Figure 5: The seven contributions originating from diagram $(d)$ of the Fig.
2. This group splits into two subsets: the first composed by the diagrams
$\\{(d_{{\scriptscriptstyle 1,1}}),(d_{{\scriptscriptstyle 1,2}})\\}$ and the
second by $\\{(d_{{\scriptscriptstyle 2,1}}),(d_{{\scriptscriptstyle
2,2}}),(d_{{\scriptscriptstyle 2,3}}),(d_{{\scriptscriptstyle
2,4}}),(d_{{\scriptscriptstyle 2,5}})\\}$.
The diagrams in Fig. 5 are naturally organized into two subsets,
$\displaystyle(d)^{abc}_{\alpha\mu\nu}=(d_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}+(d_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}\,,$ (41)
with
$\displaystyle(d_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{2}(d_{{\scriptscriptstyle
1,j}})^{abc}_{\alpha\mu\nu}\,,\qquad(d_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{5}(d_{{\scriptscriptstyle
2,j}})^{abc}_{\alpha\mu\nu}\,,$ (42)
where
$\displaystyle(d_{{\scriptscriptstyle 1,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}f^{cen}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\widetilde{R}_{\nu}^{\rho\lambda}(l,-p-l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(r,l+p,s,k)\,,$
$\displaystyle(d_{{\scriptscriptstyle 1,2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{ig^{4}}{6}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\Delta^{\rho\lambda}(l)\widehat{\Gamma}_{\nu\mu\lambda\eta\tau}^{cbemd}(p,r,l,s,k)\,,$
(43)
and
$\displaystyle(d_{{\scriptscriptstyle 2,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}f^{ben}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\widetilde{R}_{\mu}^{\rho\lambda}(l,-r-l)\widetilde{\Gamma}_{\nu\lambda\eta\tau}^{cnmd}(p,l+r,s,k)\,,$
$\displaystyle(d_{{\scriptscriptstyle 2,2}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}h_{2}^{dmbec}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\rho\lambda}(l)R_{\tau}^{\sigma\beta}(s,k)\widetilde{R}_{\mu}^{\eta\tau}(l+p,q-l)\widetilde{\Gamma}_{\nu\lambda\eta}(p,l,-l-p)\,,$
$\displaystyle(d_{{\scriptscriptstyle 2,3}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}h_{2}^{dmceb}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)R_{\tau}^{\eta\sigma}(l-q,s)\widetilde{R}_{\mu}^{\rho\lambda}(l,-r-l)\widetilde{\Gamma}_{\nu\lambda\eta}(p,l+r,q-l)\,,$
$\displaystyle(d_{{\scriptscriptstyle 2,4}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{cdmbe}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\eta}(k)\widetilde{R}_{\mu}^{\rho\tau}(l,-r-l)R_{\tau}^{\sigma\lambda}(s,k+p)\widetilde{\Gamma}_{\nu\lambda\eta}(p,-k-p,k)\,,$
$\displaystyle(d_{{\scriptscriptstyle 2,5}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{ig^{4}}{2}f^{mdn}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\rho\lambda}(l)R_{\tau}^{\eta\sigma}(l-q,s)\widehat{\Gamma}^{cben}_{\nu\mu\lambda\eta}(p,r,l,q-l)\,.$
(44)
This particular separation is motivated by the observation that the
contraction by $p^{\nu}$ of each subset generates a concrete term of the STI,
namely
$\displaystyle p^{\nu}(d_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}=f^{cae}(a_{{\scriptscriptstyle
5}})^{be}_{\alpha\mu}(r)-f^{bce}(a_{{\scriptscriptstyle
5}})^{ae}_{\alpha\mu}(q)\,,$ $\displaystyle p^{\nu}(d_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}=f^{cae}(a_{{\scriptscriptstyle
6}})^{be}_{\alpha\mu}(r)-f^{bce}(a_{{\scriptscriptstyle
6}})^{ae}_{\alpha\mu}(q)\,,$ (45)
where the self-energy diagrams $(a_{{\scriptscriptstyle 5}})$ and
$(a_{{\scriptscriptstyle 6}})$ (purple box in Fig. 1) are given by
$\displaystyle(a_{{\scriptscriptstyle 5}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=-\frac{ig^{4}}{6}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\Delta^{\rho\lambda}(l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bemd}(-q,l,s,k)$
$\displaystyle=ig^{4}h_{1}^{adme}\int_{k}\int_{l}\Delta^{\beta\tau}(k)\Delta^{\eta}_{\beta}(s)\Delta^{\lambda}_{\alpha}(l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bemd}(-q,l,s,k)\,,$
$\displaystyle(a_{{\scriptscriptstyle 6}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=\frac{ig^{4}}{2}h_{1}^{bedm}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\widetilde{R}_{\mu}^{\rho\lambda}(l,q-l)R_{\lambda}^{\sigma\beta}(s,k)\,,$
(46)
with $s=q-k-l$. In passing from the first to the second expression for
$(a_{{\scriptscriptstyle 5}})$, the explicit form of
$\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}$, given in Eq. (96), has
been used.
The contraction of the momentum $p^{\nu}$ with the above diagrams will
activate a series of STIs, which will furnish the desired structures, together
with a considerable number of terms that will cancel exactly among each other.
In what follows, we briefly outline how this calculation may be best
organized.
We begin with the diagrams comprising $(d_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}$. The action of $p^{\nu}$ on
$(d_{{\scriptscriptstyle 1,1}})^{abc}_{\alpha\mu\nu}$ leads to the contraction
$p^{\nu}\widetilde{R}_{\nu}^{\rho\lambda}(l,-p-l)$, triggering the STI of Eq.
(100), and yielding
$\displaystyle
p^{\nu}\widetilde{R}_{\nu}^{\rho\lambda}(l,-p-l)=\Delta^{\rho\alpha}(l)\Delta^{\lambda\beta}(l+p)\left[\Delta^{-1}_{\alpha\beta}(l+p)-\Delta^{-1}_{\alpha\beta}(l)\right]=\Delta^{\rho\lambda}(l)-\Delta^{\rho\lambda}(l+p)\,.$
(47)
The second term in the above expression will give
$\displaystyle p^{\nu}(d_{{\scriptscriptstyle 1,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{ig^{4}}{2}f^{cen}\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\Delta^{\rho\lambda}(l+p)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(r,l+p,s,k)+\cdots,$
(48)
where the ellipsis denotes terms that do not contribute to the r.h.s. of Eq.
(45). We now change the integration variables as $k\rightarrow-k$ and
$l\rightarrow-l-p$, and exploit Lorentz invariance to replace
$\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(r,-l,-t,-k)\rightarrow\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(-r,l,t,k)$.
Then, substituting the Feynman rule of Eq. (96) and using the Bose symmetry of
the vertices, we arrive at
$\displaystyle p^{\nu}(d_{{\scriptscriptstyle 1,1}})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}f^{cae}h_{1}^{edmn}\int_{k}\int_{l}\Delta^{\beta\tau}(k)\Delta^{\eta}_{\beta}(t)\Delta^{\lambda}_{\alpha}(l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bnmd}(-r,l,t,k)+\cdots\,,$
(49)
which is precisely $f^{cae}(a_{{\scriptscriptstyle 5}})^{be}_{\mu\nu}(r)$, in
the form given in the second line of Eq. (46).
Similarly, the contraction of $(d_{{\scriptscriptstyle
1,2}})^{abc}_{\alpha\mu\nu}$ by $p^{\nu}$ activates the STI for the five-point
function $\widehat{\Gamma}_{\nu\mu\lambda\eta\tau}^{cbemd}(p,r,l,s,k)$, given
in Eq. (106), namely
$\displaystyle
p^{\nu}\widehat{\Gamma}^{cbemd}_{\nu\mu\lambda\eta\tau}(p,r,l,s,k)$
$\displaystyle=f^{bcx}\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{xemd}(r+p,l,s,k)+f^{ecx}\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bxmd}(r,l+p,s,k)$
$\displaystyle+f^{mcx}\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bexd}(r,p,s+p,k)+f^{dcx}\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{bemx}(r,p,s,k+p)\,.$
(50)
Note that only the first term on the r.h.s. of Eq. (50) contains the four-
point function with a $q$ entry, since $r+p=-q$. Thus, one gets
$\displaystyle p^{\nu}(d_{{\scriptscriptstyle
1,2}})^{abc}_{\alpha\mu\nu}=\frac{ig^{4}}{6}f^{bcx}\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}$
$\displaystyle\Delta^{\beta\tau}(k)\Delta^{\sigma\eta}(s)\Delta^{\rho\lambda}(l)\widetilde{\Gamma}_{\mu\lambda\eta\tau}^{xemd}(-q,l,s,k)+\cdots\,,$
(51)
which is precisely $f^{bce}(a_{{\scriptscriptstyle 5}})^{ae}(q)$.
After appropriate changes in the integration variables and judicious use of
Bose symmetry, one may show that all remaining terms, denoted by the ellipses
in Eqs. (49) and (51), cancel against each other. Thus, one is left with the
first equation in Eq. (45).
A similar line of reasoning reveals that the term
$f^{cae}(a_{{\scriptscriptstyle 6}})^{be}_{\alpha\mu}(r)$ originates from the
contraction of $p^{\nu}$ with the diagrams $(d_{{\scriptscriptstyle 2,2}})$
and $(d_{{\scriptscriptstyle 2,4}})$ of the second group. Specifically, one
triggers the STI of Eq. (100) to obtain
$\displaystyle p^{\nu}\left[(d_{{\scriptscriptstyle
2,2}})\\!+\\!(d_{{\scriptscriptstyle
2,4}})\right]^{abc}_{\alpha\mu\nu}\\!=\frac{ig^{4}}{2}\left[h_{2}^{dmbec}\\!+\\!2h_{2}^{cdmbe}\right]\\!\\!\int_{k}\\!\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\widetilde{R}_{\mu}^{\rho\lambda}(l,r-l)R_{\lambda}^{\sigma\beta}(t,k)+\cdots\,.$
(52)
Using the Feynman rule given by Eq. (96) for the tree level vertex one gets
that
$\displaystyle\frac{ig^{4}}{2}\left[h_{2}^{dmbec}\\!+\\!2h_{2}^{cdmbe}\right]\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}=\frac{ig^{4}}{2}f^{cax}h_{1}^{bedm}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)xdme}+\cdots\,;$
(53)
the substitution of the first term into Eq. (52) gives precisely
$f^{cae}(a_{{\scriptscriptstyle 6}})^{be}_{\alpha\mu}(r)$, while the ellipsis
contains the terms that will cancel.
Finally, the contraction of $(d_{{\scriptscriptstyle 2,5}})$ by $p^{\nu}$
activates the STI of Eq. (2), giving
$\displaystyle p^{\nu}(d_{{\scriptscriptstyle 2,5}})^{abc}_{\alpha\mu\nu}=$
$\displaystyle-\frac{ig^{4}}{2}f^{bcx}h_{1}^{xedm}\\!\\!\\!\int_{k}\int_{l}\widetilde{\Gamma}_{\alpha\beta\sigma\rho}^{(0)adme}\widetilde{R}_{\mu}^{\rho\lambda}(l,q-l)R_{\lambda}^{\sigma\beta}(s,k)+\cdots\,,$
(54)
which is exactly $-f^{bce}(a_{{\scriptscriptstyle 6}})^{ae}_{\alpha\mu}(q)$.
Once again, all the terms inside the ellipses in Eqs. (52) and (54), cancel
exactly against the terms coming from $p^{\nu}\left[(d_{\scriptscriptstyle
2,1})+(d_{\scriptscriptstyle 2,3})\right]_{\alpha\mu\nu}$, leading to the
second line in Eq. (45).
Thus, the above considerations demonstrate the validity of Eq. (23) for $i=3$.
### III.5 Two-loop ghost sector (fourth block)
Finally, the two-loop dressed ghost graphs, given by the diagram $(e)$,
enclosed by the green box in Fig. 2, have twenty two contributions, depicted
in Fig. 6, which have been further separated into three subgroups as
$\displaystyle(e)^{abc}_{\alpha\mu\nu}=\sum_{i=1}^{3}(e_{{\scriptscriptstyle
i}})^{abc}_{\alpha\mu\nu}\,,$ (55)
with
$\displaystyle(e_{{\scriptscriptstyle
1}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{4}(e_{{\scriptscriptstyle
1,j}})^{abc}_{\alpha\mu\nu}\,,\qquad(e_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{6}(e_{{\scriptscriptstyle
2,j}})^{abc}_{\alpha\mu\nu}\,,\qquad(e_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}=\sum_{j=1}^{12}(e_{{\scriptscriptstyle
3,j}})^{abc}_{\alpha\mu\nu}\,,$ (56)
such that
$\displaystyle p^{\nu}(e_{{\scriptscriptstyle 1}})^{abc}_{\alpha\mu\nu}\\!$
$\displaystyle=\\!f^{cae}(a_{{\scriptscriptstyle
7}})^{be}_{\alpha\mu}(r)\\!-\\!f^{bce}(a_{{\scriptscriptstyle
7}})^{ae}_{\alpha\mu}(q)\,,$ $\displaystyle p^{\nu}(e_{{\scriptscriptstyle
2}})^{abc}_{\alpha\mu\nu}\\!$ $\displaystyle=\\!f^{cae}(a_{{\scriptscriptstyle
8}})^{be}_{\alpha\mu}(r)\\!-\\!f^{bce}(a_{{\scriptscriptstyle
8}})^{ae}_{\alpha\mu}(q)\,,$ $\displaystyle p^{\nu}(e_{{\scriptscriptstyle
3}})^{abc}_{\alpha\mu\nu}\\!$
$\displaystyle=\\!f^{cae}\left[(a_{{\scriptscriptstyle
9}})\\!+\\!(a_{{\scriptscriptstyle
10}})\right]^{be}_{\alpha\mu}(r)\\!-\\!f^{bce}\left[(a_{{\scriptscriptstyle
9}})\\!+\\!(a_{{\scriptscriptstyle 10}})\right]^{ae}_{\alpha\mu}(q)\,.$ (57)
The expressions for all the diagrams in Fig. 6, together with the associated
self-energy graphs, are given in the Appendix D.
A close inspection of these expressions reveals that
$\displaystyle p^{\nu}(e_{{\scriptscriptstyle
1,4}})^{abc}_{\alpha\mu\nu}=-f^{bce}(a_{{\scriptscriptstyle
7}})^{ae}_{\alpha\mu}(q)+\cdots\,,\qquad\quad p^{\nu}(e_{{\scriptscriptstyle
3,6}})^{abc}_{\alpha\mu\nu}=-f^{bce}(a_{{\scriptscriptstyle
9}})^{ae}_{\alpha\mu}(q)+\cdots\,,$ $\displaystyle
p^{\nu}(e_{{\scriptscriptstyle
2,6}})^{abc}_{\alpha\mu\nu}=-f^{bce}(a_{{\scriptscriptstyle
8}})^{ae}_{\alpha\mu}(q)+\cdots\,,\qquad\quad p^{\nu}(e_{{\scriptscriptstyle
3,12}})^{abc}_{\alpha\mu\nu}=-f^{bce}(a_{{\scriptscriptstyle
10}})^{ae}_{\alpha\mu}(q)+\cdots\,,$ (58)
and
$\displaystyle p^{\nu}\left[(e_{{\scriptscriptstyle
1,1}})\\!+\\!(e_{{\scriptscriptstyle 1,2}})\\!+\\!(e_{{\scriptscriptstyle
1,3}})\right]^{abc}_{\alpha\mu\nu}=$ $\displaystyle
f^{cae}(a_{{\scriptscriptstyle 7}})^{be}_{\alpha\mu}(r)+\cdots\,,$
$\displaystyle p^{\nu}\left[(e_{{\scriptscriptstyle
2,2}})\\!+\\!(e_{{\scriptscriptstyle 2,4}})\right]^{abc}_{\alpha\mu\nu}=$
$\displaystyle f^{cae}(a_{{\scriptscriptstyle
8}})^{be}_{\alpha\mu}(r)+\cdots\,,$ $\displaystyle
p^{\nu}\left[(e_{{\scriptscriptstyle 3,3}})\\!+\\!(e_{{\scriptscriptstyle
3,4}})\\!+\\!(e_{{\scriptscriptstyle 3,9}})\\!+\\!(e_{{\scriptscriptstyle
3,10}})\right]^{abc}_{\alpha\mu\nu}=$ $\displaystyle
f^{cae}\left[(a_{{\scriptscriptstyle 9}})\\!+\\!(a_{{\scriptscriptstyle
10}})\right]^{be}_{\alpha\mu}(r)+\cdots\,.$ (59)
Figure 6: Contributions from the diagram $(e)$ in Fig. 2 after expanding the
five-point kernel represented by the purple blob. This group is organized in
the three subsets $(e_{{\scriptscriptstyle 1}})$, $(e_{{\scriptscriptstyle
2}})$, and $(e_{{\scriptscriptstyle 3}})$.
It is then a matter of straightforward algebra to demonstrate that all terms
contained in the ellipses of Eqs. (III.5) and (III.5) cancel against each
other and with the other diagrams, as
$\displaystyle p^{\nu}[(e_{{\scriptscriptstyle
2,1}})\\!+\\!(e_{{\scriptscriptstyle 2,3}})\\!+\\!(e_{{\scriptscriptstyle
2,5}})]^{abc}_{\alpha\mu\nu}+\cdots=0\,,$ $\displaystyle
p^{\nu}[(e_{{\scriptscriptstyle 3,1}})\\!+\\!(e_{{\scriptscriptstyle
3,2}})\\!+\\!(e_{{\scriptscriptstyle 3,3}})\\!+\\!(e_{{\scriptscriptstyle
3,5}})\\!+\\!(e_{{\scriptscriptstyle 3,7}})\\!+\\!(e_{{\scriptscriptstyle
3,8}})\\!+\\!(e_{{\scriptscriptstyle
3,11}})]^{abc}_{\alpha\mu\nu}+\cdots=0\,,$ (60)
leaving Eq. (57) as the final result. Thus, the validity of Eq. (23) for $i=4$
is confirmed.
The final conclusion drawn from the analysis presented in subsections III.2,
III.3, III.4, and III.5 is that the block-wise realization of the STI
announced in subsection III.1, holds. Notice, in fact, that the validity of
Eq. (23) has been demonstrated for an arbitrary value of the gauge-fixing
parameter $\xi_{{\scriptscriptstyle Q}}$.
## IV Abelian Ward identities with Background gluons
In this section we derive Abelian WIs from the STIs satisfied by the BFM
vertices, and apply to them the text-book diagrammatic representation for the
WIs known from QED Itzykson and Zuber (1980). In addition, we demonstrate the
block-wise realization of the WI that connects the vertex
$\widehat{\Gamma}_{\alpha\mu\nu}(0,r,-r)$ with the derivative of
${\widehat{\Delta}}(r)$.
Figure 7: Diagrammatic representation of the WI for the ${\rm B\bar{c}c}$
vertex, where the derivative of the ghost propagator can be identified as the
insertion of a zero-momentum background gluon leg. Here we define the notation
of a perforated circle as being the derivative acting on the Green’s function.
As is well-known in the context of Abelian gauge theories, such as spinor or
scalar QED, the implementation of the limit $q\to 0$ of the Takahashi identity
gives rise to the corresponding WI. In order to fix the ideas consider the
latter theory, describing the interaction of a photon with a complex scalar,
where the full photon-scalar vertex $\Gamma_{\mu}(r,p,q)$ satisfies the
Abelian STI (Takahashi identity)
$q^{\mu}\Gamma_{\mu}(r,p,q)={\cal D}^{-1}(p)-{\cal D}^{-1}(r),$ (61)
with ${\cal D}(p)$ denoting the fully dressed propagator of the scalar field.
Then, the standard WI is determined by expanding both sides of Eq. (61) around
$q=0$, and equating the linear terms. Specifically, this procedure yields
$\Gamma_{\mu}(r,-r,0)=\frac{\partial\,{\cal D}^{-1}(r)}{\partial r^{\mu}}\,,$
(62)
or, equivalently,
$\frac{\partial\,{\cal D}(r)}{\partial r^{\mu}}=-{\cal
D}(r)\Gamma_{\mu}(r,-r,0){\cal D}(r)\,.$ (63)
The version of the WI given in Eq. (63) admits the text-book diagrammatic
interpretation: the derivative of the propagator ${\cal D}(r)$ is equivalent
to the insertion of a zero-momentum photon in it Itzykson and Zuber (1980).
It turns out that the Abelian STIs satisfied by the BFM three-point functions
give rise to WIs completely analogous to that of Eq. (63), which admit the
same diagrammatic interpretation given above, but now in terms of zero-
momentum insertions of a background gluon.
The simplest case is that of the ghost-gluon vertex
$\widetilde{\Gamma}_{\mu}(r,p,q)$, whose WI is identical to that of Eqs. (62)
and (63), after the replacement $\Gamma_{\mu}\to\widetilde{\Gamma}_{\mu}$ and
${\cal D}\to D$, i.e.,
$\widetilde{\Gamma}_{\mu}(r,-r,0)=\frac{\partial\,{D}^{-1}(r)}{\partial
r^{\mu}}\,\Longrightarrow\,\frac{\partial\,{D}(r)}{\partial
r^{\mu}}=-{D}(r)\widetilde{\Gamma}_{\mu}(r,-r,0){D}(r)\,;$ (64)
the corresponding diagrammatic representation is shown in Fig. 7.
Turning to the case of the BQQ vertex
$\widetilde{\Gamma}_{\alpha\mu\nu}(q,r,p)$, it is rather straightforward to
deduce from the STI of Eq. (100) the corresponding WI, namely
$\widetilde{\Gamma}_{\alpha\mu\nu}(0,-p,p)=-\frac{\partial\Delta^{-1}_{\mu\nu}(p)}{\partial
p^{\alpha}}\,\Longrightarrow\,\frac{\partial\Delta^{\mu\nu}(p)}{\partial
p^{\alpha}}=\Delta^{\mu\rho}(p)\,\widetilde{\Gamma}_{\alpha\rho\sigma}(0,-p,p)\,\Delta^{\nu\sigma}(p)\,;$
(65)
the last relation is diagrammatically depicted in Fig. 8. Note that the above
WI, when applied at tree level, reproduces from Eq. (2) the expression for
$\widetilde{\Gamma}_{\alpha\mu\nu}^{(0)}(q,r,p)$ given in Eq. (1), capturing
correctly its dependence on the gauge-fixing parameter
$\xi_{{\scriptscriptstyle Q}}$.
Figure 8: Diagrammatic representation of the WI for the BQQ vertex, where the
derivative of the gluon propagator can be identified as the insertion of a
zero-momentum background gluon leg.
We next focus our attention on the WIs satisfied by BFM vertices with more
than three incoming fields. As a concrete example, consider the vertex ${\rm
BB\bar{c}c}$; when contracted with respect to the momentum carried by one of
the background gluons, it satisfies the STI given by Eq. (2). Expanding both
sides of Eq. (2) around $q=0$, and using the Jacobi identity to eliminate the
zeroth order term on the r.h.s., we obtain the WI
$\widetilde{\Gamma}^{abmn}_{\mu\nu}(0,-p-t,p,t)=\left(f^{amx}f^{nbx}\frac{\partial}{\partial
p^{\mu}}+f^{anx}f^{bmx}\frac{\partial}{\partial
t^{\mu}}\right)\widetilde{\Gamma}_{\nu}(p,t,-p-t)\,.$ (66)
Exactly analogous expressions may be deduced for higher point Green’s
functions; for a formal derivation of the STI satisfied by a general vertex of
the form ${\rm BQ}^{n}$, see Appendix A.
Now we want to explore the block-wise realization of the WI of the BBB vertex
for the case of the one-loop ghost group, which satisfies the STI of Eq. (23)
for $i=2$, or, equivalently, Eq. (40). In the soft-gluon limit, we obtain
simply
$\displaystyle\widehat{\Gamma}^{(2)}_{\alpha\mu\nu}(q,-q,0)=\frac{\partial\widehat{\Pi}^{(2)}_{\alpha\mu}(q)}{\partial
q^{\nu}}\,,$ (67)
or, in terms of diagrams
$\displaystyle(c_{{\scriptscriptstyle
3}})_{\alpha\mu\nu}(q,-q,0)=\frac{\partial(a_{{\scriptscriptstyle
3}})_{\alpha\mu}(q)}{\partial q^{\nu}}\,.$ (68)
In arriving at Eq. (68) we have used that $(a_{{\scriptscriptstyle 4}})$ is
$q$-independent, and that, in the soft-gluon limit, $(c_{{\scriptscriptstyle
1}})=(c_{{\scriptscriptstyle 2}})=0$ (see Sec. III.3).
To prove Eq. (68), we first symmetrize the process of differentiation of
$(a_{{\scriptscriptstyle 3}})_{\alpha\mu}$ by shifting the loop momentum
($k\rightarrow u-k$, with $u=q/2$), to get
$\displaystyle\frac{\partial(a_{{\scriptscriptstyle
3}})_{\alpha\mu}(q)}{\partial q^{\nu}}=(a^{\prime}_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q)+(a^{\prime}_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q)+(a^{\prime}_{{\scriptscriptstyle
3,3}})_{\alpha\mu\nu}(q)\,,$ (69)
with
$\displaystyle(a^{\prime}_{{\scriptscriptstyle 3,1}})_{\alpha\mu\nu}(q)$
$\displaystyle=$
$\displaystyle-2\lambda\int_{k}k_{\alpha}\left[\frac{\partial}{\partial
q^{\nu}}D(k-u)\right]D(k+u)\widetilde{\Gamma}_{\mu}(k+u,u-k,-q)\,,$
$\displaystyle(a^{\prime}_{{\scriptscriptstyle 3,2}})_{\alpha\mu\nu}(q)$
$\displaystyle=$
$\displaystyle-2\lambda\int_{k}k_{\alpha}D(k-u)\left[\frac{\partial}{\partial
q^{\nu}}D(k+u)\right]\widetilde{\Gamma}_{\mu}(k+u,u-k,-q)\,,$
$\displaystyle(a^{\prime}_{{\scriptscriptstyle 3,3}})_{\alpha\mu\nu}(q)$
$\displaystyle=$
$\displaystyle-2\lambda\int_{k}k_{\alpha}D(k-u)D(k+u)\left[\frac{\partial}{\partial
q^{\nu}}\widetilde{\Gamma}_{\mu}(k+u,u-k,-q)\right]\,;$ (70)
the last three contributions are depicted graphically in the first line of
Fig. 9.
Next, for the terms $(a^{\prime}_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q)$ and $(a^{\prime}_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q)$ we use Eq. (64) to write
$\displaystyle(a^{\prime}_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q)\\!=\\!-$
$\displaystyle\lambda\\!\\!\int_{k}k_{\alpha}\left[D(k\\!-\\!u)\widetilde{\Gamma}_{\nu}(k\\!-\\!u,u\\!-\\!k,0)D(k\\!-\\!u)\right]\\!D(k\\!+\\!u)\widetilde{\Gamma}_{\mu}(k\\!+\\!u,u\\!-\\!k,-q)\,,$
$\displaystyle(a^{\prime}_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q)\\!=\\!-$
$\displaystyle\lambda\\!\\!\int_{k}k_{\alpha}D(k\\!-\\!u)\left[D(k\\!+\\!u)\widetilde{\Gamma}_{\nu}(k\\!+\\!u,-u\\!-\\!k,0)D(k\\!+\\!u)\right]\widetilde{\Gamma}_{\mu}(k\\!+\\!u,u\\!-\\!k,-q)\,.$
(71)
A direct comparison of these last expressions with the contributions to
$(c_{{\scriptscriptstyle 3}})_{\alpha\mu\nu}(q,-q,0)$ in Eq. (37) [for
$(q,r,p)\to(q,-q,0)$] allows one to establish that
$\displaystyle(a^{\prime}_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q)=(c_{{\scriptscriptstyle
3,1}})_{\alpha\mu\nu}(q,-q,0)\,,\qquad(a^{\prime}_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q)=(c_{{\scriptscriptstyle
3,2}})_{\alpha\mu\nu}(q,-q,0)\,.$ (72)
Figure 9: The diagrammatic representation of the differentiation of the graph
$(a_{{\scriptscriptstyle 3}})$ with respect to $q^{\nu}$. The effect of
differentiating the ghost propagator (dressed background ghost-gluon vertex)
is the insertion of a zero-momentum background gluon leg in the propagator
(vertex).
Consider finally the $(c_{{\scriptscriptstyle 3,3}})^{abc}_{\alpha\mu\nu}$ in
Eq. (III.3); setting $(q,r,p)\to(q,-q,0)$, shifting $k\rightarrow u-k$, and
employing Eq. (66), we get
$\displaystyle(c_{{\scriptscriptstyle 3,3}})^{abc}_{\alpha\mu\nu}(q,-q,0)$
$\displaystyle=2g^{2}f^{eda}\int_{k}k_{\alpha}D(k-u)D(k+u)\widehat{\Gamma}_{\mu\nu}^{bcde}(-q,0,k+u,u-k)\,,$
$\displaystyle=-\frac{\lambda}{2}\\!\\!\int_{k}\\!\\!k_{\alpha}D(k\\!-\\!u)D(k\\!+\\!u)\\!\\!\left(\\!\frac{\partial}{\partial(k+u)^{\nu}}\\!+\\!\frac{\partial}{\partial(u-k)^{\nu}}\\!\\!\right)\\!\widetilde{\Gamma}_{\mu}(k\\!+\\!u,u\\!-\\!k,-q)\,,$
$\displaystyle=-2\lambda\int_{k}k_{\alpha}D(k-u)D(k+u)\frac{\partial}{\partial
q^{\nu}}\widetilde{\Gamma}_{\mu}(k+u,u-k,-q)\,,$ (73)
and therefore
$\displaystyle(a^{\prime}_{{\scriptscriptstyle
3,3}})_{\alpha\mu\nu}(q)=(c_{{\scriptscriptstyle
3,3}})_{\alpha\mu\nu}(q,-q,0)\,,$ (74)
which completes the proof of Eq. (68). The interpretation of the previous
steps in terms of background-gluon insertions is given in the second line of
Fig. 9.
## V Discussion and Conclusions
It has been known for some time Aguilar and Papavassiliou (2006) that the
transversality of the background self-energy is enforced in a special way,
namely independently for each one of the four subsets (blocks) of diagrams
comprising the corresponding SDE. In the present work we have shown that the
Abelian STI of the background three-gluon vertex is also realized according to
the exact same pattern, at the level of the corresponding SDE: the momentum
contraction of each subset of vertex diagrams generates the difference of the
corresponding self-energy subsets.
The demonstration of this property has been carried out at the level of the
fully dressed Feynman diagrams that comprise the relevant SDEs. In particular,
the contraction of all three-gluon vertex diagrams by the appropriate momentum
triggers STIs satisfied by the vertices and the kernels embedded in them,
giving rise to crucial rearrangements and cancellations, which are implemented
algebraically, with no need to resort to any integrations. Note that the
extensive reorganization of diagrams observed here has been first identified
in the context of the pinch technique Cornwall (1982); Papavassiliou (1990);
Pilaftsis (1997); Binosi and Papavassiliou (2002a, 2009), where the “gauge-
invariant” three-gluon vertex was first studied at the one-loop level Cornwall
and Papavassiliou (1989); Hashimoto _et al._ (1994); Binger and Brodsky
(2006). Evidently, it would be particularly interesting to explore the origin
of the block-wise STIs at a formal level, and establish its validity by means
of the Batalin-Vilkovisky functional machinery Batalin and Vilkovisky (1977,
1983); Binosi and Papavassiliou (2002b, 2009); Binosi and Quadri (2012, 2013)
.
It is natural to conjecture that the STI of the background four-gluon vertex,
B4, given by Papavassiliou (1993); Hashimoto _et al._ (1994)
$\displaystyle
q^{\mu}\widehat{\bm{\Gamma}}^{mnrs}_{\mu\alpha\beta\gamma}(q,r,p,t)$
$\displaystyle=$ $\displaystyle
f^{mse}f^{ern}\widehat{\bm{\Gamma}}_{\alpha\beta\gamma}(r,p,q+t)+f^{mne}f^{esr}\widehat{\bm{\Gamma}}_{\beta\gamma\alpha}(p,t,q+r)$
(75) $\displaystyle+$ $\displaystyle
f^{mre}f^{ens}\widehat{\bm{\Gamma}}_{\gamma\alpha\beta}(t,r,q+p)\,,$
is realized according to the same block-wise pattern described above. A
diagrammatic demonstration along the lines presented in this work appears to
be quite feasible, and would give further support to the notion that the STI
of any Bn-type of vertex is enforced in this characteristic manner.
Some of the results presented in Sec. IV may be used in order to explore the
numerical impact of certain truncations or approximations, in the spirit of
the recent study presented in Aguilar _et al._ (2022a). For example, the
equality shown in Fig. 9 will be distorted if the vertex BB${\rm\bar{c}}\rm c$
were to be replaced by its tree level value, given by Eq. (99). The amount of
discrepancy induced between the two sides of this equation is a quantitative
indicator of the veracity of such an approximation.
Throughout the present analysis we have assumed that the BBB vertex does not
contain irregularities in the form of massless poles. However, as has been
shown in detail in a series of studies, the emergence of a dynamical gluon
mass Cornwall (1982) through the operation of the Schwinger mechanism
Schwinger (1962a, b) hinges on the inclusion of longitudinally coupled
massless poles in the fundamental vertices of the theory Jackiw and Johnson
(1973); Eichten and Feinberg (1974); Aguilar _et al._ (2008, 2012); Ibañez
and Papavassiliou (2013); Aguilar _et al._ (2016b); Eichmann _et al._ (2021)
Quite importantly: (a) the STIs satisfied by the vertices are resolved with
the nontrivial participation of these poles, and (b) in the soft-gluon limit,
the associated WIs are displaced by an amount controlled by the corresponding
pole residues Aguilar _et al._ (2016b, 2022b, 2022c). In particular, ongoing
research reveals that the STIs impose stringent conditions on the pole content
of the three-gluon vertex, which must, at the same time, be dynamically
realized. The treatment of this problem within the BFM (i.e., at the level of
the BBB rather than the QQQ vertex) eliminates structures originating from the
ghost-sector of the theory, which tend to complicate and obscure the
underlying physical picture. We expect that the completion of this study will
shed light on the question of how symmetry-induced constraints are dynamically
enforced at the level the corresponding SDEs.
## VI Acknowledgments
The work of A. C. A. and B. M. O. are supported by the CNPq grants
307854/2019-1 and 141409/2021-5, respectively. A. C. A also acknowledges
financial support from project 464898/2014-5 (INCT-FNA). M. N. F. and J. P.
are supported by the Spanish MICINN grant PID2020-113334GB-I00. M. N. F.
acknowledges financial support from Generalitat Valenciana through contract
CIAPOS/2021/74. J. P. also acknowledges funding from the regional
Prometeo/2019/087 from the Generalitat Valenciana.
## Appendix A Derivation of Abelian STIs
In this Appendix we employ the Batalin-Vilkovisky formalism Batalin and
Vilkovisky (1977, 1983); Binosi and Papavassiliou (2002b, 2009); Binosi and
Quadri (2012, 2013) to derive the Abelian STI satisfied by the generic vertex
${\rm BQ}^{n}$ when contracted by the momentum carried by the gluon $\rm B$.
We start with the WI functional, given by Binosi and Papavassiliou (2009)
$\displaystyle
W\\!=\\!\\!\int\\!\\!d^{4}x\\!\left[\delta_{\vartheta}Q^{x,\mu}(x)\frac{\delta\Gamma}{\delta
Q_{\mu}^{x}(x)}\\!+\\!\delta_{\vartheta}B^{x,\mu}(x)\frac{\delta\Gamma}{\delta
B_{\mu}^{x}(x)}\\!+\\!\delta_{\vartheta}c^{x}(x)\frac{\delta\Gamma}{\delta
c^{x}(x)}\\!+\\!\delta_{\vartheta}\bar{c}^{x}(x)\frac{\delta\Gamma}{\delta\bar{c}^{x}(x)}\right]\\!\\!=0\,,$
(76)
where $\vartheta^{a}$ are the local infinitesimal parameters which correspond
to the SU(3) generators $t^{a}$, and play the role of the ghost field.
$\Gamma$ in Eq. (76) is the “reduced” effective action, defined as the full
effective action without the gauge-fixing term Binosi and Papavassiliou
(2002b, 2009). Consequently, the Green’s functions obtained from $\Gamma$ will
be missing the corresponding gauge-dependent contribution at tree level.
Finally, the gauge transformations of the fields are given by
$\displaystyle\delta_{\vartheta}Q_{\mu}^{x}=gf^{xdc}Q_{\mu}^{d}\vartheta^{c}\,,\qquad$
$\displaystyle\delta_{\vartheta}B_{\mu}^{x}=\partial_{\mu}\vartheta^{x}+gf^{xdc}B_{\mu}^{d}\vartheta^{c}\,,$
(77)
$\displaystyle\delta_{\vartheta}c^{x}=-gf^{xdc}c^{d}\vartheta^{c}\,,\qquad$
$\displaystyle\delta_{\vartheta}\bar{c}^{x}=-gf^{xdc}\bar{c}^{d}\vartheta^{c}\,.$
To obtain the background Abelian STIs the first step is to differentiate the
functional $W$ with respect to the parameter $\vartheta^{a}(x)$, furnishing
$\displaystyle\frac{\delta
W}{\delta\vartheta^{a}(x)}=gf^{eda}Q_{\mu}^{d}(x)\Gamma_{Q_{\mu}^{e}}(x)+\partial_{\mu}\Gamma_{B_{\mu}^{a}}(x)=0\,,$
(78)
where we have already set to zero the vacuum expectation values (VEVs) of the
ghost, antighost, and background gluon fields222In the end of the procedure
all of the VEVs are set to zero. Since the ${\rm BQ}^{n}$ vertex has only one
external $\rm B$ and no external ghost and antighost fields these VEVs can be
set to zero from the outset.. Moreover, we introduce the shorthand notation
for functional derivatives
$\Gamma_{\phi_{1}\phi_{2}\ldots\phi_{n}}(x_{1},x_{2},\ldots,x_{n}):=\frac{\delta^{n}\Gamma}{\delta\phi_{1}(x_{1})\delta\phi_{2}(x_{2})\ldots\delta\phi_{n}(x_{n})}\,,$
(79)
where $\phi_{i}(x_{i})$ denotes a generic field.
The STIs of interest are then obtained by differentiating Eq. (78) $n$ times
with respect to the quantum gluon. Note, in particular, that the functional
derivatives of the term $\partial_{\mu}\Gamma_{B_{\mu}^{a}}(x)$ in Eq. (78)
generate divergences such as
$\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{{\scriptscriptstyle
1}}}^{b_{{\scriptscriptstyle 1}}}\ldots Q_{\nu_{{\scriptscriptstyle
n}}}^{b_{{\scriptscriptstyle n}}}}(x,y_{1},\ldots,y_{n})$ which, after Fourier
transformation, result in the typical l.h.s. of the Abelian STIs, i.e., a
Green’s function contracted with a background gluon momentum.
To fix the ideas, let us consider as two special cases the STIs for the BQ and
BQQ functions.
Differentiating Eq. (78) with respect to $Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}(y_{1})$ we obtain
$\displaystyle\frac{\delta^{2}W}{\delta Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}(y_{1})\delta\vartheta^{a}(x)}\\!\\!=$ $\displaystyle
gf^{eb_{{\scriptscriptstyle
1}}a}\delta(x\\!-\\!y_{1})\Gamma_{\\!Q_{\nu_{1}}^{e}}\\!\\!(x)\\!+\\!gf^{eda}Q_{\mu}^{d}(x)\Gamma_{\\!Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\mu}^{e}}\\!(y_{1},x)\\!+\\!\partial_{\mu}\Gamma_{\\!B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}}\\!(x,y_{1})\\!=\\!0\,.$ (80)
Setting the gluon field $Q=0$, and the one-point function
$\Gamma_{\\!Q_{\nu_{1}}^{e}}\\!\\!(x)=0$, we obtain
$\displaystyle\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}}(x,y_{1})=0\,,$ (81)
where $\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle 1}}}}(x,y_{1})$
is the inverse BQ propagator, with its $1/\xi_{Q}$ term removed. In momentum
space notation, Eq. (81) becomes
$q^{\mu}\left[q^{2}P_{\mu\nu}(q)+i{{\widetilde{\Pi}}_{\mu\nu}(q)}\right]=0\quad\Longrightarrow\quad
q^{\mu}{{\widetilde{\Pi}}_{\mu\nu}(q)}=0\,,$ (82)
expressing the exact transversality of the BQ self-energy.
Then, an additional differentiation of Eq. (80) with respect to
$Q_{\nu_{2}}^{b_{2}}(y_{2})$ yields
$\displaystyle\frac{\delta^{3}W}{\delta Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}(y_{1})\delta Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}(y_{1})\delta\vartheta^{a}(x)}=$ $\displaystyle
gf^{eb_{{\scriptscriptstyle
1}}a}\delta(x-y_{1})\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}}(x,y_{2})+gf^{eb_{{\scriptscriptstyle
2}}a}\delta(x-y_{2})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}}(y_{1},x)$ $\displaystyle+gf^{eda}Q_{\mu}^{d}(x)$
$\displaystyle\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)+\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}}(x,y_{1},y_{2})=0\,.$ (83)
At this point, by setting all the fields to zero we obtain the Abelian STI for
the BQQ vertex in configuration space, namely
$\displaystyle\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}}(x,y_{1},y_{2})\\!=\\!-gf^{eab_{{\scriptscriptstyle
1}}}\delta(x\\!-\\!y_{1})\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}}(x,y_{2})\\!-\\!gf^{eab_{{\scriptscriptstyle
2}}}\delta(x\\!-\\!y_{2})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}}(y_{1},x)\,.$ (84)
Now, Fourier transforming the above equation to momentum space leads to
$\displaystyle q^{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}}(q,r,p)=gf^{ab_{{\scriptscriptstyle 1}}b_{{\scriptscriptstyle
2}}}\left[p^{2}P_{\mu\nu}(p)+{\Pi_{\mu\nu}(p)}\right]-gf^{ab_{{\scriptscriptstyle
1}}b_{{\scriptscriptstyle
2}}}\left[r^{2}P_{\mu\nu}(r)+{\Pi_{\mu\nu}(r)}\right]\,,$ (85)
which, with the definition of Eq. (3), becomes Eq. (100). The derivation of
the STI for the BBB vertex, given in Eq. (20), is completely analogous.
Next, we prove that the STI of the ${\rm BQ}^{n}$ vertex is given by
$\displaystyle-\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(x,y_{1},y_{2},\cdots,y_{n})=gf^{eab_{{\scriptscriptstyle
1}}}\delta(x-y_{1})\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(x,y_{2},y_{3},\cdots,y_{n})$ $\displaystyle\hskip
113.81102pt+gf^{eab_{{\scriptscriptstyle
2}}}\delta(x-y_{2})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(y_{1},x,y_{3},\cdots,y_{n})+\cdots$ $\displaystyle\hskip
113.81102pt+gf^{eab_{n}}\delta(x-y_{n})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n-1}}^{b_{n-1}}Q_{\nu_{n}}^{e}}(y_{1},y_{2},\cdots,y_{n-1},x)\,.$ (86)
To that end, we first differentiate Eq. (A) $n-2$ times. This procedure yields
$\displaystyle-\partial_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(x,y_{1},y_{2},\cdots,y_{n})=gf^{eab_{{\scriptscriptstyle
1}}}\delta(x-y_{1})\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(x,y_{2},y_{3},\cdots,y_{n})$ $\displaystyle\hskip
85.35826pt+gf^{eab_{{\scriptscriptstyle
2}}}\delta(x-y_{2})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(y_{1},x,y_{3},\cdots,y_{n})$ $\displaystyle\hskip
85.35826pt+gf^{eax}\left\\{\frac{\delta^{n-2}}{\delta
Q_{\nu_{n}}^{b_{n}}(y_{n})\cdots\delta Q_{\nu_{3}}^{b_{{\scriptscriptstyle
3}}}(y_{3})}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)\right]\right\\}_{Q\rightarrow 0}\\!\\!\\!\,.$
(87)
Clearly, to demonstrate Eq. (A) we need to prove that
$\displaystyle gf^{eax}$ $\displaystyle\left\\{\frac{\delta^{n-2}}{\delta
Q_{\nu_{n}}^{b_{n}}(y_{n})\cdots\delta Q_{\nu_{3}}^{b_{{\scriptscriptstyle
3}}}(y_{3})}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)\right]\right\\}_{Q\rightarrow 0}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad gf^{eab_{{\scriptscriptstyle
3}}}\delta(x-y_{3})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{e}Q_{\nu_{4}}^{b_{4}}\cdots
Q_{\nu_{n}}^{b_{n}}}(y_{1},y_{2},x,y_{4},\cdots,y_{n})+\cdots$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+gf^{eab_{n}}\delta(x-y_{n})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n-1}}^{b_{n-1}}Q_{\nu_{n}}^{e}}(y_{1},y_{2},\cdots,y_{n-1},x)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+gf^{eax}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}Q_{\mu}^{e}}(y_{1},y_{2},\cdots,y_{n},x)\right]_{Q\rightarrow
0}\,.$ (88)
The proof proceeds by induction. First, it is clear that Eq. (A) holds for
$n=3$. Indeed, in this case one has to take a single derivative
$\displaystyle gf^{eax}\\!\left\\{\frac{\delta}{\delta
Q_{\nu_{{\scriptscriptstyle 3}}}^{b_{{\scriptscriptstyle
3}}}(y_{3})}\\!\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)\right]\right\\}_{Q\rightarrow
0}\\!\\!\\!=gf^{eab_{{\scriptscriptstyle
3}}}\delta(x-y_{3})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}Q_{\nu_{3}}^{e}}(y_{1},y_{2},x)$
$\displaystyle\hskip
170.71652pt+gf^{eax}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{3}}Q_{\mu}^{e}}(y_{1},y_{2},y_{3},x)\right]_{Q\rightarrow
0}\,,$ (89)
which is Eq. (A) for $n=3$.
Then, assume that Eq. (A) is true for $n=k$. Differentiating the result once
more with respect to $Q_{\nu_{k+1}}^{b_{k+1}}$ we obtain
$\displaystyle gf^{eax}\left\\{\frac{\delta^{k-1}}{\delta
Q_{\nu_{k+1}}^{b_{k+1}}(y_{k+1})\delta Q_{\nu_{k}}^{b_{k}}(y_{k})\cdots\delta
Q_{\nu_{3}}^{b_{{\scriptscriptstyle
3}}}(y_{3})}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\mu}^{e}}(y_{1},y_{2},x)\right]\right\\}_{Q\rightarrow 0}=$
$\displaystyle\hskip 85.35826ptgf^{eab_{{\scriptscriptstyle
3}}}\delta(x-y_{3})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{e}Q_{\nu_{4}}^{b_{4}}\cdots
Q_{\nu_{k}}^{b_{k}}Q_{\nu_{k+1}}^{b_{k+1}}}(y_{1},y_{2},x,y_{4},\cdots,y_{k},y_{k+1})+\cdots$
$\displaystyle\hskip
85.35826pt+gf^{eab_{k}}\delta(x-y_{k})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{k-1}}^{b_{k-1}}Q_{\nu_{k}}^{e}Q_{\nu_{k+1}}^{b_{k+1}}}(y_{1},y_{2},\cdots,y_{k-1},x,y_{k+1})$
$\displaystyle\hskip
85.35826pt+gf^{eab_{k+1}}\delta(x-y_{k+1})\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{k}}^{b_{k}}Q_{\nu_{k+1}}^{e}}(y_{1},y_{2},\cdots,y_{k},x)\,.$
$\displaystyle\hskip
85.35826pt+gf^{eax}\left[Q_{\mu}^{x}(x)\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{k}}^{b_{k}}Q_{\nu_{k+1}}^{b_{k+1}}Q_{\mu}^{e}}(y_{1},y_{2},\cdots,y_{k},y_{k+1},x)\right]_{Q\rightarrow
0}\,,$ (90)
which is Eq. (A) for $n=k+1$. This completes the proof.
In momentum space, Eq. (A) is given by (suppressing a factor of $g$)
$\displaystyle iq_{\mu}\Gamma_{B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(q,p_{1},p_{2},\cdots,p_{n})=$ $\displaystyle
f^{eab_{{\scriptscriptstyle
1}}}\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(p_{1}+q,p_{2},p_{3},\cdots,p_{n})$ $\displaystyle+$
$\displaystyle f^{eab_{{\scriptscriptstyle
2}}}\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(p_{1},p_{2}+q,p_{3},\cdots,p_{n})+\cdots$
$\displaystyle+$ $\displaystyle
f^{eab_{n}}\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n-1}}^{b_{n-1}}Q_{\nu_{n}}^{e}}(p_{1},p_{2},p_{3},\cdots,p_{n}+q)\,.$
(91)
The corresponding WI is obtained by expanding Eq. (A) around $q=0$ and
collecting terms linear in $q$. Using $p_{n}=-\sum\limits_{i=1}^{n-1}p_{i}$,
we obtain
$\displaystyle i\Gamma_{\\!B_{\mu}^{a}Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(0,p_{1},p_{2},\cdots,p_{n})\\!=\\!\left(\\!f^{eab_{{\scriptscriptstyle
1}}}\frac{\partial}{\partial
p_{1}^{\nu_{1}}}+\\!\cdots\\!+f^{eab_{n-1}}\frac{\partial}{\partial
p_{n-1}^{\nu_{n-1}}}\\!\right)\Gamma_{\\!Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}\cdots Q_{\nu_{n}}^{e}}(p_{1},\cdots,p_{n})\,.$ (92)
Note that the absence of a zeroth order term on the l.h.s. of Eq. (A) implies
the relation
$\displaystyle 0=$ $\displaystyle f^{eab_{{\scriptscriptstyle
1}}}\Gamma_{Q_{\nu_{1}}^{e}Q_{\nu_{2}}^{b_{{\scriptscriptstyle
2}}}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(p_{1},p_{2},p_{3},\cdots,p_{n})+f^{eab_{{\scriptscriptstyle
2}}}\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{e}Q_{\nu_{3}}^{b_{{\scriptscriptstyle 3}}}\cdots
Q_{\nu_{n}}^{b_{n}}}(p_{1},p_{2},p_{3},\cdots,p_{n})+\cdots$
$\displaystyle\hskip
170.71652pt+f^{eab_{n}}\Gamma_{Q_{\nu_{1}}^{b_{{\scriptscriptstyle
1}}}Q_{\nu_{2}}^{b_{{\scriptscriptstyle 2}}}\cdots
Q_{\nu_{n-1}}^{b_{n-1}}Q_{\nu_{n}}^{e}}(p_{1},p_{2},p_{3},\cdots,p_{n})\,,$
(93)
whose validity we have checked explicitly for $n=3,4$.
## Appendix B Feynman rules for BFM vertices
In the Table 1 of this Appendix we list the Feynman rules for BFM vertices at
tree level.
Vertex | Feynman rule
---|---
|
$\displaystyle\widetilde{\Gamma}_{\alpha\mu\nu}^{(0)}(q,r,p)=(q-r)_{\nu}g_{\alpha\mu}$
$\displaystyle+$
$\displaystyle(r-p)_{\alpha}g_{\mu\nu}+(p-q)_{\mu}g_{\alpha\nu}$
$\displaystyle+$ $\displaystyle\xi_{{\scriptscriptstyle
Q}}^{-1}(g_{\alpha\nu}r_{\mu}-g_{\alpha\mu}p_{\nu})\,,$
| $\displaystyle\widetilde{\Gamma}^{(0)}_{\mu}(r,p,q)=(r-p)_{\mu}\,,$ (95)
|
$\displaystyle\widetilde{\Gamma}^{(0)abcd}_{\alpha\beta\mu\nu}=f^{adx}f^{cbx}\left(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\beta}g_{\mu\nu}\right)+f^{abx}f^{dcx}\left(g_{\alpha\nu}g_{\beta\mu}-g_{\alpha\mu}g_{\beta\nu}\right)$
(96)
$\displaystyle+f^{acx}f^{dbx}\left(g_{\alpha\nu}g_{\beta\mu}-g_{\alpha\beta}g_{\mu\nu}\right)\,,$
| $\displaystyle\widehat{\Gamma}^{(0)abcd}_{\alpha\beta\mu\nu}$
$\displaystyle=\Gamma_{\alpha\beta\mu\nu}^{(0)abcd}+\xi_{{\scriptscriptstyle
Q}}^{-1}(f^{adx}f^{bcx}g_{\alpha\nu}g_{\beta\mu}-f^{acx}f^{dbx}g_{\alpha\mu}g_{\beta\nu})\,,$
(97)
|
$\displaystyle\widetilde{\Gamma}^{(0)abmn}_{\mu\nu}=f^{max}f^{xbn}g_{\mu\nu}\,,$
(98)
|
$\displaystyle\widehat{\Gamma}^{(0)abmn}_{\mu\nu}=g_{\mu\nu}\left(f^{max}f^{bnx}+f^{mbx}f^{anx}\right)\,.$
(99)
Table 1: The diagrammatic representations of the new vertices appearing in the
BFM and their respective Feynman rules at tree level Binosi and Papavassiliou
(2009). Notice that for the three-point functions we have factored out the
coupling $g$ and their respective color structure, following the definitions
of Eq. (3), while for the four-point functions, we have factored out only
$-ig^{2}$ as shown in Eq. (4).
## Appendix C Abelian Slavnov-Taylor identities in the BFM
In the Table 2 we collect all the Abelian STI in the BFM necessary to
demonstrate the block-wise realization of the STI for the background three-
gluon vertex.
Vertex | Abelian STI
---|---
BQQ | $q^{\alpha}\,\widetilde{\Gamma}_{\alpha\mu\nu}(q,r,p)=\Delta^{-1}_{\mu\nu}(p)-\Delta^{-1}_{\mu\nu}(r)\,,\vspace{-0.9cm}$ (100)
${\rm B\bar{c}c}$ | $q^{\mu}\,\widetilde{\Gamma}_{\mu}(r,p,q)=D^{-1}(p)-D^{-1}(r)\,,\vspace{-0.9cm}$ (101)
BQQQ | $\displaystyle q^{\alpha}\widetilde{\Gamma}^{abcd}_{\alpha\beta\mu\nu}(q,r,p,t)=$ $\displaystyle f^{abx}f^{dcx}\Gamma_{\beta\mu\nu}(r+q,p,t)+f^{acx}f^{bdx}\Gamma_{\beta\mu\nu}(r,p+q,t)$ $\displaystyle+f^{adx}f^{cbx}\Gamma_{\beta\mu\nu}(r,p,t+q)\,,\vspace{-3cm}$
BBQQ | $\displaystyle q^{\alpha}\widehat{\Gamma}^{abcd}_{\alpha\beta\mu\nu}(q,r,p,t)=$ $\displaystyle f^{abx}f^{dcx}\widetilde{\Gamma}_{\beta\mu\nu}(r+q,p,t)+f^{acx}f^{bdx}\widetilde{\Gamma}_{\beta\mu\nu}(r,p+q,t)$ $\displaystyle+f^{adx}f^{cbx}\widetilde{\Gamma}_{\beta\mu\nu}(r,p,t+q)\,,\vspace{-7cm}$
${\rm BQ\bar{c}c}$ | $\displaystyle q^{\mu}\widetilde{\Gamma}^{abmn}_{\mu\nu}(q,r,p,t)=$ $\displaystyle f^{nax}f^{bmx}\Gamma_{\nu}(p,q+t,r)+f^{nbx}f^{max}\Gamma_{\nu}(q+p,t,r)$ $\displaystyle+f^{nmx}f^{abx}\Gamma_{\nu}(p,t,q+r)\,,\vspace{-5.5cm}$
${\rm BB\bar{c}c}$ | $\displaystyle q^{\mu}\widehat{\Gamma}_{\mu\nu}^{abmn}(q,r,p,t)=$ $\displaystyle f^{abx}f^{mnx}\widetilde{\Gamma}_{\nu}(p,t,q+r)+f^{amx}f^{nbx}\widetilde{\Gamma}_{\nu}(q+p,t,r)$ $\displaystyle+f^{anx}f^{bmx}\widetilde{\Gamma}_{\nu}(p,q+t,r)\,,\vspace{-5.5cm}$
BBQQQ | $\displaystyle q^{\alpha}\widehat{\Gamma}^{abcde}_{\alpha\beta\mu\nu\rho}(q,r,p,t,u)=f^{bax}\widetilde{\Gamma}_{\beta\mu\nu\rho}^{xcde}(r+q,p,t,u)+f^{cax}\widetilde{\Gamma}_{\beta\mu\nu\rho}^{bxde}(r,p+q,t,u)$ (106) $\displaystyle+f^{dax}\widetilde{\Gamma}_{\beta\mu\nu\rho}^{bcxe}(r,p,t+q,u)+f^{eax}\widetilde{\Gamma}_{\beta\mu\nu\rho}^{bcdx}(r,p,t,u+q)\,,$
${\rm BBQ\bar{c}c}$ | $\displaystyle q^{\alpha}\widehat{\Gamma}^{abcmn}_{\alpha\mu\nu}(q,r,p,t,u)=f^{bax}\widetilde{\Gamma}_{\mu\nu}^{xcmn}(r+q,p,t,u)+f^{cax}\widetilde{\Gamma}_{\mu\nu}^{bxmn}(r,p+q,t,u)$ (107) $\displaystyle+f^{max}\widetilde{\Gamma}_{\mu\nu}^{bcxn}(r,p,t+q,u)+f^{nax}\widetilde{\Gamma}_{\mu\nu}^{bcmx}(r,p,t,u+q)\,.$
Table 2: The Abelian STIs satisfied by the BQQ, ${\rm B\bar{c}c}$, BQQQ, BBQQ,
${\rm BQ\bar{c}c}$, ${\rm BB\bar{c}c}$, BBQQQ and ${\rm BBQ\bar{c}c}$
vertices.
## Appendix D Expressions for the two-loop ghost sector of the BBB SDE
The two-loop ghost sector of the SDE of the vertex BBB given by diagram $(e)$
in Fig. 2, whose expansion is given in Fig. 6, relate with the background
gluon self-energy by Eq. (57), where the expression for the diagrams
$(a_{{\scriptscriptstyle 7}})$, $(a_{{\scriptscriptstyle 8}})$,
$(a_{{\scriptscriptstyle 9}})$ and $(a_{{\scriptscriptstyle 10}})$, in Fig. 1
can be expressed as
$\displaystyle(a_{{\scriptscriptstyle 7}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=ig^{4}h_{1}^{aedm}\\!\\!\int_{k}\int_{l}D(l)D(s)\Delta^{\beta}_{\alpha}(k)\widetilde{\Gamma}_{\mu\beta}^{bmde}(-q,k,l,s)\,,$
(108) $\displaystyle(a_{{\scriptscriptstyle 8}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=\frac{1}{2}i\lambda^{2}\delta^{ab}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}R_{\sigma}(-l,l-k)\widetilde{R}_{\mu}^{\sigma\beta}(-k,q+k)\,,$
(109) $\displaystyle(a_{{\scriptscriptstyle 9}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=-i\lambda^{2}\delta^{ab}\\!\\!\int_{k}\int_{l}D(q+l)\Delta_{\alpha}^{\beta}(k)R_{\beta}(k-l,l)\widetilde{\Gamma}_{\mu}(-l,q+l,-q)\,,$
(110) $\displaystyle(a_{{\scriptscriptstyle 10}})^{ab}_{\alpha\mu}(q)$
$\displaystyle=-\frac{1}{2}i\lambda^{2}\delta^{ab}\\!\\!\int_{k}\int_{l}D(q+l)\Delta_{\alpha}^{\beta}(k)R_{\beta}(l,k-l)\widetilde{\Gamma}_{\mu}(q+l,-l,-q)\,.$
(111)
The decomposition of diagram $(e)$, given in Eqs. (55) and (56), can be
separated in three groups: $(e_{1})^{abc}_{\alpha\mu\nu}$, with
$\displaystyle(e_{1,1})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{amdce}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(k)D(s)\widetilde{R}_{\nu}^{\beta\sigma}(l,-p-l)\widetilde{\Gamma}^{bedm}_{\mu\sigma}(r,l+p,s,k)\,,$
$\displaystyle(e_{1,2})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{cmade}\\!\\!\int_{k}\int_{l}D(s)\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\nu}(-p-k,k)\widetilde{\Gamma}^{bedm}_{\mu\beta}(r,l,s,k+p)\,,$
$\displaystyle(e_{1,3})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{amecd}\\!\\!\int_{k}\int_{l}D(k)\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\nu}(k,-k-p)\widetilde{\Gamma}^{bedm}_{\mu\beta}(r,l,k+p,s)\,,$
$\displaystyle(e_{1,4})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{1}^{amed}\\!\\!\int_{k}\int_{l}D(k)D(s)\Delta_{\alpha}^{\beta}(l)\widehat{\Gamma}_{\mu\nu\beta}^{bcedm}(r,p,l,s,k)\,,$
(112)
$(e_{2})^{abc}_{\alpha\mu\nu}$, with
$\displaystyle(e_{2,1})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{amebd}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(k)D(s)\widetilde{R}_{\mu}^{\beta\sigma}(l,-r-l)\widetilde{\Gamma}_{\nu\sigma}^{cdem}(p,r+l,s,k)\,,$
$\displaystyle(e_{2,2})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{4}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta^{\beta}_{\alpha}(l)R_{\rho}(s,k)\widetilde{R}_{\mu}^{\sigma\rho}(p+l,q-l)\widetilde{\Gamma}_{\nu\beta\sigma}(p,l,-p-l)\,,$
$\displaystyle(e_{2,3})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{4}\lambda^{2}f^{abc}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}\Delta^{\lambda\rho}(q-l)R_{\lambda}(s,k)\widetilde{R}_{\mu}^{\beta\sigma}(l,-r-l)\widetilde{\Gamma}_{\nu\rho\sigma}(p,q-l,l+r)\,,$
$\displaystyle(e_{2,4})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{4}\lambda^{2}f^{abc}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(s)\widetilde{R}_{\nu}(-k-p,k)\widetilde{R}_{\mu}^{\beta\sigma}(l,-r-l)\Gamma_{\sigma}(s,k+p,l+r)\,,$
$\displaystyle(e_{2,5})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{1}^{xmbe}h_{2}^{amecx}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(s)\widetilde{R}_{\nu}(k,-k-p)\widetilde{R}_{\mu}^{\beta\sigma}(l,-r-l)\Gamma_{\sigma}(k+p,s,l+r)\,,$
$\displaystyle(e_{2,6})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{2}g^{2}\lambda
f^{aem}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)\Delta^{\rho\sigma}(q-l)R_{\rho}(s,k)\widehat{\Gamma}_{\mu\nu\beta\sigma}^{bcem}(r,p,l,q-l)\,,$
(113)
and finally $(e_{3})^{abc}_{\alpha\mu\nu}$, with
$\displaystyle(e_{3,1})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{bmade}\\!\\!\int_{k}\int_{l}D(s)\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\mu}(-k-r,k)\widetilde{\Gamma}^{cedm}_{\nu\beta}(p,l,s,k+r)\,,$
$\displaystyle(e_{3,2})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{1}^{xmce}h_{2}^{amebx}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(k)\widetilde{R}_{\nu}^{\beta\sigma}(l,-l-p)R_{\sigma}(s,k+r)\widetilde{\Gamma}_{\mu}(-k-r,k,r)\,,$
$\displaystyle(e_{3,3})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{4}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\nu}(k,-k-p)R_{\beta}(k-q,s)\widetilde{\Gamma}_{\mu}(k+p,q-k,r)\,,$
$\displaystyle(e_{3,4})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{4}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)\Gamma_{\beta}(s+p,k+r,l)\widetilde{R}_{\mu}(-k-r,k)\widetilde{R}_{\nu}(s,-s-p)\,,$
$\displaystyle(e_{3,5})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{2}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)R_{\beta}(s,k-q)\widetilde{R}_{\mu}(-k-r,k)\widetilde{\Gamma}_{\nu}(q-k,k+r,p)\,,$
$\displaystyle(e_{3,6})^{abc}_{\alpha\mu\nu}$
$\displaystyle=-\frac{i}{2}g^{2}\lambda
f^{ade}\\!\\!\int_{k}\int_{l}D(k)\Delta_{\alpha}^{\beta}(l)R_{\beta}(k-q,s)\widehat{\Gamma}^{cbde}_{\mu\nu}(r,p,k,q-k)\,,$
$\displaystyle(e_{3,7})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{2}^{amedb}\\!\\!\int_{k}\int_{l}D(s)\Delta_{\alpha}^{\beta}(l)\widetilde{R}_{\mu}(k,-k-r)\widetilde{\Gamma}^{cedm}_{\nu\beta}(p,l,k+r,s)\,,$
$\displaystyle(e_{3,8})^{abc}_{\alpha\mu\nu}$
$\displaystyle=ig^{4}h_{1}^{xmce}h_{2}^{amebx}g_{\alpha\beta}\\!\\!\int_{k}\int_{l}D(k)\widetilde{R}_{\nu}^{\beta\sigma}(l,-l-p)R_{\sigma}(k+r,s)\widetilde{\Gamma}_{\mu}(k,-k-r,r)\,,$
$\displaystyle(e_{3,9})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{2}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)R_{\beta}(s,k-q)\widetilde{\Gamma}_{\mu}(q-k,k+p,r)\widetilde{R}_{\nu}(-p-k,k)\,,$
$\displaystyle(e_{3,10})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{4}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)\Gamma_{\beta}(r-k,-t,l)\widetilde{R}_{\mu}(-k,k-r)\widetilde{R}_{\nu}(t,q+k+l)\,,$
$\displaystyle(e_{3,11})^{abc}_{\alpha\mu\nu}$
$\displaystyle=\frac{i}{2}\lambda^{2}f^{abc}\\!\\!\int_{k}\int_{l}\Delta_{\alpha}^{\beta}(l)R_{\beta}(k-q,s)\widetilde{R}_{\mu}(k,-k-r)\widetilde{\Gamma}_{\nu}(k+r,q-k,p)\,,$
$\displaystyle(e_{3,12})^{abc}_{\alpha\mu\nu}$ $\displaystyle=-ig^{2}\lambda
f^{ade}\\!\\!\int_{k}\int_{l}D(k)\Delta_{\alpha}^{\beta}(l)R_{\beta}(s,k-q)\widehat{\Gamma}^{bcde}_{\mu\nu}(r,p,q-k,k)\,.$
(114)
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|
# Text Representation Enrichment Utilizing Graph based Approaches: Stock
Market Technical Analysis Case Study
Sara Salamat , Nima Tavassoli , Behnam Sabeti and Reza Fahmi sara,
nima, behnam<EMAIL_ADDRESS>EveinceWALLSTRAßE 18, 10179BerlinGermany
###### Abstract.
Graph neural networks (GNNs) have been utilized for various natural language
processing (NLP) tasks lately. The ability to encode corpus-wide features in
graph representation made GNN models popular in various tasks such as document
classification. One major shortcoming of such models is that they mainly work
on homogeneous graphs, while representing text datasets as graphs requires
several node types which leads to a heterogeneous schema. In this paper, we
propose a transductive hybrid approach composed of an unsupervised node
representation learning model followed by a node classification/Edge
prediction model. The proposed model is capable of processing heterogeneous
graphs to produce unified node embeddings which are then utilized for node
classification or link prediction as the downstream task. The proposed model
is developed to classify stock market technical analysis reports, which to our
knowledge is the first work in this domain. Experiments, which are carried
away using a constructed dataset, demonstrate the ability of the model in
embedding extraction and the downstream tasks.
## 1\. Introduction
The way we understand and process natural language mainly depends on the way
it is represented. While representing language as a bag of tokens and sequence
of tokens are largely used in the NLP community, graph representation is
another approach for structuring texts, as well as utilizing relational
information between different text elements. Representing text as a graph has
a long history as several specialized text graphs, including dependency graphs
(Zhang et al., 2019), constituency graphs (Marcheggiani and Titov, 2019),
lexical networks (Radev and Mihalcea, 2008), knowledge graphs (Ye et al.,
2019), etc. has been created in this field. Moreover, it is also possible to
represent entities with different hierarchies such as document, passage,
sentence, and word in one graph. Because of this property, many NLP tasks
require heterogeneous graphs to model different entities and relations of the
problem. However, GNN models, as one of the most powerful frameworks of
analyzing graphs, such as GCN(Kipf and Welling, 2016), GAT(Veličković et al.,
2017), and GraphSAGE(Hamilton et al., 2017) are mostly designed for
homogeneous conditions, and therefore cannot fit in these problems directly.
We identify stock market technical analysis reports as a text dataset that
provides great analytical opportunities in financial markets. Hundreds of
these analyses in the form of text documents are published in
TradingView111Tradingview.com on a daily basis. Market analysts and investors
share their analysis, insights, and predictions within the platform. They can
also assign a tag to their posts from a predefined set (Long, Short,
Education) to indicate the overall status of that analysis. Table 1 shows a
sample of a post in our dataset.
As a novel work in this domain, we strive to automatically extract insights
and information from these reports with the help of graph representation
learning. In particular, this paper focuses on a specific version of that
vision that is the problem of classifying technical analysis reports in the
cryptocurrencies market. Classifying all documents into the Long, Short, and
Education classes helps us to perceive what position analysts and investors
generally take on a particular symbol in a given time window. In order to
classify properly, we use authors’ tags as the supervision of the
classification task and train a classifier based on those labels.
Table 1. Sample of the dataset Feature | Sample
---|---
author | EXCAVO
#comments | 7
content | ZECUSDT reached 50% Fibonacci and forming a bullish flag…
ID | L68rv5WO
#likes | 65
position | Long
signature | Channels: https://t.me/excavochannel …
symbol | BINANCE:ZECUSDT
time | Jan 21, 2021 @ 17:21:24.000
timeframe | 240
title | ZEC/USDT/BTC
To exploit richer relations among documents, we follow the graph
representation approach for the proposed document classification problem and
construct a graph based on the given input text. There are multiple methods
for static graph construction during preprocessing documents (Wu et al.,
2021). We take a hybrid approach, in which we capture both co-occurrence and
similarity relations between words and documents. We also cover the loss of
the text sequential information to some extent, by creating n-grams during the
preprocessing steps and using them in graph modelling.
Our base graph structure is solely constructed on the information contained in
the technical reports: documents and words, but formulating this
classification problem in a graph-based structure allows us to represent our
desired entities that are related to our specific problem as part of the graph
structure. We can take price patterns as an example, which are formations that
appear on stock price charts and are one of the key components of technical
analysis in financial markets. If we consider each price pattern as a separate
node type, it is possible to connect each one to its implication defined by
the tag node type, as well as the document containing that pattern. Here, we
have used an external knowledge base to enrich our document dataset. As a
result, we can incorporate different domain knowledge into the current
structure, and capture richer relationships among different elements.
In this work, we introduce a novel classification task of stock market
technical analysis documents and develop a hybrid transductive graph-based
solution. In the proposed approach, first, we store TradingView’s posts
published in the Ideas section and preprocess all documents. Then, we define
word and sentence representations by language models to construct our graph.
We also add price patterns, corpus topics, and post labels to the graph as
separate node types. After constructing the graph, node embeddings are trained
in an unsupervised manner, and we update the graph nodes with new
representations. This step can be considered optional. Finally, we train both
node classification on documents and link prediction on the edges between
documents and tags in order to estimate document labels. We conduct various
experiments to examine our proposed method and compare it with state-of-the-
art baselines in different settings. Experimental results show that our best
graph-based approach achieves 89% F1-score, while the top baseline method
(Fine-tuned BERT) has the F1-score 76%.
In short, we summarize the main contributions as follows:
* •
Proposing a novel heterogeneous graph construction method for representing
technical analysis documents.
* •
Proposing a graph neural network model for technical analysis reports
classification.
* •
Providing experimental results showing that our method outperforms several
state-of-the-art document classification baselines.
* •
Introducing the “Stock Market Technical Analysis Reports Classification” task,
which to our knowledge is a novel work in related domains.
* •
Presenting a dataset of technical analysis reports in the cryptocurrencies
market that enables further research in this field.
The remaining parts of this paper is structured as follows. Section 2 further
discusses the stock market technical analysis reports and price patterns in
financial markets. Following that, in Section 3 our graph construction
approach is described and our hybrid proposed model is explained in detail.
Section 4 introduces the baseline methods, our constructed dataset, and
experimental results. Finally, a categorization of graph-based approaches for
text classification is presented as related works in Section 5 and the paper
is concludeed in Section 6.
## 2\. Stock Market Technical Analysis Report
Predicting market behaviour from financial text data has been under extensive
research in recent years. Analyzing financial texts everyday and extracting
insights from them is a heavy task, therefore, researchers have been trying to
leverage machine learning methods to derive insights. Research in this area
has been mostly done on sentiment analysis of social media data to find out
whether people are talking about the market in a positive, negative or neutral
way (Araci, 2019; Chan and Chong, 2017; Chiong et al., 2018; Jangid et al.,
2018; Day and Lee, 2016; Daniel et al., 2017; Zhao et al., 2021b; Maia et al.,
2018). These approaches miss an important point. Most of the times a rise or
fall in market prices leads to social media posts and people’s reactions. It
is rare to get insights about market’s future trend from people’s emotions in
their posts. To address this matter, technical analysis reports are used in
this study. Technical analysts use price movements in price charts to decide
the future trends of the price (Kirkpatrick and Dahlquist, 2006). They use
different indicators and patterns for their analysis. Technical analysis
reports are the outcome of technical analysts’ investigation of the price
chart. They talk about the behaviour of the price chart and how they think the
behaviour affects the future trend of the chart. These reports carry much more
information about the movements of the market. Therefore, it is more likely
that technical reports are correlated with price movements because writers are
usually traders with expertise in market analysis.
### 2.1. Research Target
Market analysts usually talk about their analysis of the market behaviour and
how they think market will behave afterwards in their writings. In this study,
we aim to process their analysis and find out what they expect to see in the
chart. However, technical analysis can be prone to error as not every analysis
is correct and not everyone is expert enough to predict future behaviour
correctly. By looking at what the majority of the people are seeing in the
price chart and implications in the market, it is possible to derive insights
from their implications and finally predict the possible next move of the
market from the technical analysts’ point of view.
### 2.2. Price Pattern
Price patterns are configurations in price charts that are often used by
technical analysts for anticipating the direction of the market prices. Price
patterns have different shapes and are often identified by trendlines and
curves (Seo, 2017). Patterns can be categorized into two groups of reversal
and continuation patterns. Reversal patterns state a change in the direction
of the market and continuation patterns state that the trend remains the same.
In this work, most popular price patterns among technical analysts were used
as graph nodes. In stock trading, having a long position means that a rise is
expected in the price of the market as opposed to a short position which means
a fall is expected in the price. Price patterns usually signal a rise or fall
afterwards so their implications in the market is utilized in this research in
graph modeling. Table 2 shows some examples of patterns, their description and
implications in the market.
Table 2. Price patterns Name | Description | Implication
---|---|---
Head and shoulders | A baseline with three peaks where the middle is the highest. | Short
Double top | Two highs, looking like M | Short
Cup and handle | Two convex arcs which three top points touch a resistance level | Long
Bearish flag | Characterized by parallel trendlines over the consolidation area-lines have upward sloping | Short
## 3\. Proposed Approach
In this section, our approach for financial text analysis is explored. First,
the graph structure is explained and then graph processing approach is
introduced in details. Figure 1 illustrates the overall process.
DataBase Text Pre-processing Topic Modeling Text Embedding Graph Modeling Node
Classification Link Prediction Unsupervised Node Representation Learning Node
Classification Link Prediction
Figure 1. Overview of the proposed model
### 3.1. Graph Structure
In this section, details of our text graph construction approach are
explained. Every graph $\mathcal{G}(\mathcal{V},\mathcal{E})$ is characterised
by a set of nodes and edges. Our graph’s set of nodes includes five types of
nodes:
* •
Documents
* •
Words
* •
Topics
* •
Price patterns
* •
Document position (Link Prediction problem)
Document and word nodes are basics for representing a corpus as a graph and
other types of nodes are added to enrich the graph representation. Document
position nodes are labels of the documents that are only used in the link
prediction problem where the existence of a link between each document and
position nodes are predicted.
Table 3. Description of edges of the graph Source | Destination | Relation
---|---|---
Word | Word | PMI
Document | Word | TF-IDF
Word | Topic | $P(w|t)$
Document | Topic | $P(t|d)$
Document | Price Pattern | binary
Price Pattern | Topic | $P(t|pp)$
Document | Position | User-generated label (Link Prediction problem)
Price Pattern | Position | Price pattern implication (Link Prediction problem)
“icp formed a
double-bottom and a rise is expected” icp formed double-bottom topic 1 “… I
was bullish Bitcoin and expected it to rise towards 46k resistance…” topic 5
Long trend triple-top pattern rise expected double-bottom eth bitcoin
Figure 2. An illustration of sample graph representation
We connect words and documents based on word occurrence in documents so that
the weight of the edges are defined by term frequency-inverse document
frequency (TF-IDF) (Yao et al., 2019). TF-IDF evaluates how relevant a word is
to a document in the entire document set. Term frequency is the number of
times the word appears in the document and inverse document frequency is the
inverse of the number of documents that contain that word.
We use a fixed-size sliding window on all documents to calculate point-wise
mutual information (PMI) between every two words within the window. Then, we
connect words with positive PMI values together and assign the derived PMI as
the weight of that connection. The PMI value of a word pair $w_{1},w_{2}$ is
computed as
(1)
$\operatorname{PMI}(w_{1},w_{2})=\log\frac{p(w_{1},w_{2})}{p(w_{1})p(w_{2})}$
(2) $p(w_{1},w_{2})=\frac{\\#W(w_{1},w_{2})}{\\#W}$
(3) $p(w_{1})=\frac{\\#W(w_{1})}{\\#W}$
Where $\\#W(w_{1})$ is the number of sliding windows in a corpus that contain
word $w_{1}$, $\\#W(w_{1},w_{2})$ is the number of sliding windows that
contain both $w_{1}$ and $w_{2}$, and $\\#W$ is the total number of sliding
windows. In this approach, we have used both co-occurrence and similarity
relations between words and documents. Each of those methods only considers
one specific type of relation and has limitations in representing others.
To utilize high-level semantic information of documents, we connect topic node
$t$, resulted from the topic modeling, to document, word, and price pattern
nodes with the relations shown in the table 3. As aforementioned, we introduce
the price pattern node type that is connected to the documents containing that
pattern. We also connect patterns that have known implications in the
technical analysis literature to the position nodes. Figure 2 illustrates a
sample of nodes and their relations in our graph. Table 3 summarizes edges
between different node types.
### 3.2. Heterogeneous Graph Modeling
As we are dealing with a heterogeneous structure, we design a model that is
capable of processing these kinds of graphs. In this section, we describe the
components of our proposed model.
#### 3.2.1. HinSAGE
GraphSAGE (Hamilton et al., 2017) is a framework for inductive representation
learning on large graphs. GraphSAGE is used to generate low-dimensional vector
representations for nodes, and is capable of learning an embedding function
that generalizes to unseen nodes, without requiring a re-training procedure.
The key idea of this approach is training a set of aggregator functions that
learn to aggregate feature information from a node’s neighborhood.
We extend the GraphSAGE model to be able to aggregate feature vectors from
different node types in heterogeneous graphs. Our work is inspired by the
HinSAGE model introduced by the StellarGraph library (Data61, 2018). Algorithm
1 describes HinSAGE in the case where entire graph
$\mathcal{G}(\mathcal{V},\mathcal{E})$ and features for all nodes are provided
as input. First, all node embeddings are initialized to the node features.
Then, at each iteration (search depth), each node aggregates the
representations of the nodes in its one-hop neighborhood via edges of type
$r$, and concatenates its current representation $\mathbf{h}_{v}^{k-1}$ with
the aggregated neighborhood vector, $\mathbf{h}^{k}_{N_{r}(v)}$. As we can
observe in the algorithm, there are separate neighborhood weight matrices
$W_{\text{neigh}}$ for each relation between two node types. There are also
separate self-feature matrices $W_{\text{self}}$ for every node type. HinSAGE
finally passes the concatenated vector through a neural network layer with
non-linear activation function $\sigma$ to update the node embedding.
Algorithm 1 HinSAGE Algorithm
Input: Graph $\mathcal{G}(\mathcal{V},\mathcal{E})$; input features
$\left\\{\mathbf{x}_{v},\forall v\in\mathcal{V}\right\\}$; edge types
$r\in\\{1,\ldots,R_{e}\\}$; depth $K$; weight matrices $W^{k},\forall
k\in\\{1,\ldots,K\\}$; non-linearity $\sigma$; neighborhood function
$N:v\rightarrow 2^{\mathcal{V}}$
Output: Vector representations $\mathbf{z}_{v}$ for all $v\in\mathcal{V}$
1:$\mathbf{h}_{v}^{0}\leftarrow\mathbf{x}_{v},\forall v\in\mathcal{V};$
2:for $k=1\ldots K$ do
3: for $v\in\mathcal{V}$ do
4: for $r=1\ldots R_{e}$ do
5: ${\mathbf{h}^{k}}_{N_{r}(v)}\leftarrow\frac{1}{|N_{r}(v)|}\sum_{u\in
N_{r}(v)}D_{p}[{\mathbf{h}_{u}}^{k-1}]$
6: end for
7:
${\mathbf{h}_{v}}^{k}\leftarrow\sigma[\operatorname{CONCAT}[{W^{k}}_{t_{v},\text{self}}D_{p}[{\mathbf{h}_{v}}^{k-1}],{W^{k}}_{1,\text{neigh}}{\mathbf{h}^{k}}_{N_{1}(v)},$
$\ldots,{W^{k}}_{R_{e},\text{neigh}}{\mathbf{h}^{k}}_{N_{R_{e}}(v)}]+b^{k}]$
8: end for
9:
$\mathbf{h}_{v}^{k}\leftarrow\mathbf{h}_{v}^{k}/\left\|\mathbf{h}_{v}^{k}\right\|_{2},\forall
v\in\mathcal{V}$
10:end for
11:$\mathbf{z}_{v}\leftarrow\mathbf{h}_{v}^{K},\forall v\in\mathcal{V}$
In order to learn the weights of the aggregator functions, GraphSAGE applies a
differentiable, graph-based loss function:
(4)
$J_{\mathcal{G}}\left(\mathbf{z}_{u}\right)=-\log\left(\sigma\left(\mathbf{z}_{u}^{\top}\mathbf{z}_{v}\right)\right)-Q\cdot\mathbb{E}_{v_{n}\sim
P_{n}(v)}\log\left(\sigma\left(-\mathbf{z}_{u}^{\top}\mathbf{z}_{v_{n}}\right)\right),$
where node $v$ is a neighbor of node $u$, node $v_{n}$ is a distant node to
node $v$ and is sampled from a negative sampling distribution $P_{n}(v)$, and
$Q$ is the number of negative samples. This loss function enforces close nodes
to have similar representations and distant nodes to have dissimilar
representations.
#### 3.2.2. DGI
Deep Graph Infomax (DGI) (Velickovic et al., 2019) is an unsupervised node
representation learning approach for graph-structured data. DGI’s core idea is
learning to distinguish between the original graph and the corrupted one that
derives from a corruption procedure. Given the true input graph $G$, they
change it into the mutated version $H=C(G)$ by randomly shuffling the node
features among nodes. $H$ has the same edges as $G$, but the features
associated with each node differ.
The model consists of an encoder that takes an input graph and computes an
embedding vector for each node. It is typically based on well-known graph
machine learning models such as GCN or GraphSAGE. The weights of the encoder
are trained according to the distinguishing step. Thus, DGI learns to
discriminate between nodes that have sensible connections and nodes that have
unexpected connections. After training, the encoder can be used independently
to compute node embedding vectors directly. We use DGI to obtain node
embeddings for the downstream tasks as shown in the figure 1. Because of
heterogeneity, we separately run the same algorithm with the same parameters
for each node type.
## 4\. Experiments and Results
### 4.1. Dataset
Tradingview is a platform and social network for traders and investors where
they can use different charts and technical features to make trading
decisions. Traders can also share their opinions and analysis of the market in
a section called Ideas. Users can choose between three options (i.e., Long,
Short, and Education) to describe general purpose of their post. These user-
generated labels were utilized for text classification tasks and a dataset was
put together from people’s posts about the cryptocurrency market and their
decision inferred from their analysis.
In this work, posts from 163 most popular cryptocurrency symbols were
collected and stored in a database. In addition to assigning a label, market
analysts select a timeframe for their posts to define the time range where
their analysis or decisions are valid. Each post has a signature part in which
writers usually share their websites or Telegram channels. Table 1 shows all
the crawled information from Tradingview Ideas and a sample data point. Posts
containing no text data were discarded. Overall 24420 posts are available in
the dataset. Table 4 shows some statistics of our dataset.
Table 4. Statistics of the dataset Field | Amount
---|---
Total posts | 24420
Total labelled posts | 16590
Mean length of posts | 70.05 words
Median length of posts | 34 words
Total posts containing at least one price pattern | 793
Posts labelled as Long | 13379(80.6% of labels)
Posts labelled as Short | 2984(18% of labels)
Posts labelled as Education | 227(1.4% of labels)
### 4.2. Pre-process
Text data needs to be normalized and preprocessed for further use. In order to
clean the documents in the corpus, these steps were taken:
* •
Normalizing documents
* •
Removing stop-words and punctuation marks
* •
Removing email addresses, newline characters, URLs and emojis
* •
Unifying price pattern variations, e.g., changing head n shoulders, head &
shoulders, h & s to head-and-shoulders.
After cleaning the documents, they were broken down into their words to form a
dictionary of words from the corpus.
### 4.3. Topic Modeling
As stated in previous sections, topic nodes are part of our graph structure.
To extract topics and their distribution, Latent Dirichlet Allocation (LDA)
method (Blei et al., 2003) is utilized. To find an optimal number of topics
for the corpus, coherence score was calculated for different topic numbers.
Figure 3 shows the result. This measure evaluates a single topic by measuring
the degree of semantic similarity between high scoring words in the topic.
This kind of measurements help distinguish between topics that are
semantically interpretable topics and topics that are artifacts of statistical
inference. There are various ways to calculate coherence score. Here we have
used the Context Vector coherence score introduced by (Aletras and Stevenson,
2013) that uses words co-occurrence counts. Based on the results shown in
Table 3 the optimal number of topics was set to 6.
$4$$5$$6$$7$$8$$9$$10$$0.4$$0.41$$0.42$$0.43$$0.44$$0.45$$0.46$Number of
topicsCoherence scorescore
Figure 3. Selecting the suitable number of topics
The final results of the topic modeling is shown in Table 5 in the form of
each topic’s most frequent words. Each topic’s distribution was calculated for
each document. Then the resulting vectors were mapped into two-dimensional
space using t-SNE method (van der Maaten and Hinton, 2008). The visualization
of the data-points is shown in Figure 4. The distributions of words in topics
and topics in documents is later used in our modeling process. The
effectiveness of topic nodes in classification task is studied and the results
show that these nodes improve the accuracy of the prediction.
Table 5. Topics and their frequent words 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---
time | buy | binance | drop | bch | price
bitcoin | term | analysis | trendline | crypto | support
market | tp | coin | beginning | structure | btc
wave | trade | position | monthly | charts | bullish
zec | short | dont | find | startegy | long
For initial embedding of graph nodes, two models were used. FastText library
(Grave et al., 2018) was used to encode words in our dictionary in a 100
dimensional embedding space. To encode the documents, sentence-transformers
(Reimers and Gurevych, 2019) pre-trained MiniLM model(Wang et al., 2020) with
6 layers was used. The MiniLM model maps documents to a 384 dimensional
embedding space. Summary of all node embeddings and their dimensions are shown
in Table 6.
Table 6. Graph nodes’ initial embedding Nodes | Embedding | Dim
---|---|---
Word node | fasttext word embedding | 100
Document node | Pretrained LM document embedding | 384
Topic node | One-hot | 6
Price Pattern node | One-hot | 29
Position node | One-hot (used in the Link Prediction problem.) | 3
(a)
(b)
(c)
(d)
Figure 4. Embedding of documents in 4 different settings. (a): embeddings
derived from topic distributions, (b): embeddings that are learned by DGI,
(c): embeddings learned through HinSAGE node classification, and (d):
embeddings learned through HinSAGE node classification on DGI representations.
The blue, green, and red dots correspond to the documents with Long, Short,
and Education labels respectively.
### 4.4. Baselines and Results
The following models are used as baselines for comparison purposes:
* •
A classification head was added to the pre-trained uncased BERT base model
introduced in (Devlin et al., 2018) and the model was fine-tuned on our
dataset using Tensorflow Model Garden (Yu et al., 2020).
* •
Sentence-transformers (Reimers and Gurevych, 2019) pre-trained MiniLM model
(Wang et al., 2020) was used to obtain 384-dimensional embeddings for
documents and then SVM was used to classify the document embeddings.
* •
FinBERT model (Araci, 2019) which is trained on financial text data for
sentiment analysis was used to get 768-dimensional document embeddings and
then SVM was used to classify the document embeddings.
Table 7 compares the final results of all the models developed in this study
and our baselines. Link prediction task using DGI and Hinsage models have the
highest f1-score among the others and has outperformed the baseline models.
Table 7. Results Model | Task | Precision | Recall | F1-score
---|---|---|---|---
DGI+HinSAGE | Link prediction | 0.9 | 0.89 | 0.89
HinSAGE | Link prediction | 0.89 | 0.88 | 0.89
Node2Vec(Grover and Leskovec, 2016)+Hinsage | Link prediction | 0.86 | 0.85 | 0.85
Fine-tuned BERT | Document classification | 0.80 | 0.73 | 0.76
SBERT embeddings+SVM | Document classification | 0.82 | 0.67 | 0.72
DGI+HinSAGE | Node classification | 0.7 | 0.68 | 0.68
HinSAGE | Node classification | 0.71 | 0.66 | 0.68
finBERT+SVM | Document classification | 0.78 | 0.61 | 0.66
Link prediction methods, in general, outperform node classification by a large
margin. By modeling the classification problem in the form of edge probability
estimation, we are required to introduce a position node, and connect each
tagged document to its corresponding position (label). We also connect some of
the price patterns to their implications (long or short), which brings new
kinds of information to this setting. Both modifications result in a more
informative and richer graph structure that is essential for GNN models to
learn better representations. We illustrate the output embedding of documents
in 4 different settings in Figure 4.
To examine the effectiveness of each group of nodes in the graph, experiments
were done using different combinations of nodes. Table 8 summarizes the
results for these experiments. The highest scoring result belongs to the link
prediction task using HinSAGE model with word, document, topic, and position
nodes. Using DGI and HinSAGE in the link prediction achieves almost the same
result, however, it can be seen that DGI improves the performance of node
classification in many cases.
Table 8. Experiments with different node settings Model | Task | Word | Document | Topic | Pattern | Position | F1-score
---|---|---|---|---|---|---|---
DGI + HinSAGE | Link Prediction | | | | | | 0.885
DGI + HinSAGE | Link Prediction | | | | | | 0.89
DGI + HinSAGE | Link Prediction | | | | | | 0.885
DGI + HinSAGE | Link Prediction | | | | | | 0.888
DGI + HinSAGE | Node Classification | | | | | | 0.68
DGI + HinSAGE | Node Classification | | | | | | 0.688
DGI + HinSAGE | Node Classification | | | | | | 0.68
DGI + HinSAGE | Node Classification | | | | | | 0.512
HinSAGE | Link Prediction | | | | | | 0.894
HinSAGE | Link Prediction | | | | | | 0.896
HinSAGE | Link Prediction | | | | | | 0.888
HinSAGE | Link Prediction | | | | | | 0.888
HinSAGE | Node Classification | | | | | | 0.674
HinSAGE | Node Classification | | | | | | 0.652
HinSAGE | Node Classification | | | | | | 0.662
HinSAGE | Node Classification | | | | | | 0.681
## 5\. Related works
Graph-based approaches have recently gained attraction in natural language
processing tasks. Graphs can express much more information in their structure
than the bag-of-words approach. Therefore, representing text documents in form
of graphs could result in more accurate text classification. Text graphs can
have different kinds of structures. Osman and Barukab (Osman and Barukub,
2020) categorized graph-based representation approaches in natural language
processing into 5 groups. For example, word co-occurrence graphs, document-
word graphs, phrase as a graph, etc. Graph representation approaches can be
divided into two groups: single graphs for a corpus that contain all documents
(Yao et al., 2019; Lin et al., 2021) and individual graphs for each text in
the document (Shanavas et al., 2020; Jiang et al., 2010; Zhang et al., 2020;
Zhao et al., 2021a). Graphs can also be heterogeneous or non-heterogeneous.
Non-heterogeneous graphs contain only one type of nodes and node
representations which is more common due to simplicity of the processing steps
(Lin et al., 2021; Li et al., 2021). Heterogeneous graphs contain different
types of nodes and representations (Jiang et al., 2010; Linmei et al., 2019;
Ragesh et al., 2021). A group of works in this area utilize graph-based
approach for extracting embeddings that have captured similarities in the
graph structures (Shanavas et al., 2020; Wang and Liu, 2010; Chen et al.,
2020). However, graph structures can be used in the downstream tasks by
turning the classification problem into node classification or link prediction
problems (Yao et al., 2019). Shanavas et al. (Shanavas et al., 2020) used
graph representation for each text document and defined a graph kernel for
measuring similarity between graphs and then used SVM to classify documents.
Jiang et al. (Jiang et al., 2010) used a graph representation approach for
documents with information of words like part of speech and their semantic
features and then used graph mining approaches for feature vector extraction
and classified the resulting vectors for each document.
In recent years, several deep learning methods have been proposed for graph-
based text classification. For example, Yao et al. (Yao et al., 2019)
leveraged Graph convolutional networks (Kipf and Welling, 2016) for text
classification. They constructed a graph with relations between words and
documents and turned the text classification problem into a node
classification task. Zhang et al. (Zhang et al., 2020) proposed TextING for
inductive text classification task and trained a graph neural network that
learns from detailed word-word relations from individual graphs for each
document. Another method was proposed by Zhao and Huang et al. (Zhao et al.,
2021a) for text classification using graphs for each document in the corpus.
They used Bi-LSTMs to extract sequential features from graphs and then used
graph convolutional networks for classification. Liu and You et al. (Liu et
al., 2020) constructed a text graph tensor and developed a learning method to
harmonize information from multiple graphs of their graph tensor for text
classification. In summary, this works builds on previous works in machine
learning on graphs and explores the text classification task on a corpus
represented as a heterogeneous graph. This study also applies text
classification in finance for predicting investors’ opinions which is new and
different from previous sentiment analysis studies in this field.
## 6\. Conclusion
We introduced a novel graph-based representation of stock market technical
analysis reports for extracting financial insights and information. A
heterogeneous graph with document, word, topic, price pattern, and position
nodes was constructed from technical analysis documents such that each group
of nodes had their specific initial node embeddings. Then, a graph neural
network model was proposed to learn expressive representations for the node
classification and edge prediction downstream tasks. Experimental results show
that our model is capable of predicting labels accurately on our highly
imbalanced dataset and outperforms the baseline methods. For the future
direction, it is worthwhile to explore more types of nodes and relations for
constructing more effective graph structures.
###### Acknowledgements.
We would like to thank everyone at Eveince for their valuable insights and
support.
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|
University of Birmingham, UK and http://gianlucacurzi.com
<EMAIL_ADDRESS>of Birmingham, UK and https://anupamdas.com.
<EMAIL_ADDRESS>Gianluca Curzi and Anupam Das [500]Theory of computation
Complexity theory and logic [500]Theory of computation Proof theory Both
authors are supported by a UKRI Future Leaders Fellowship, _Structure vs.
Invariants in Proofs_ , project reference MR/S035540/1.
###### Acknowledgements.
We thank the anonymous reviewers for their helpful comments and
suggestions.Bartek Klin and Elaine Pimentel 2 31st EACSL Annual Conference on
Computer Science Logic (CSL 2023) CSL 2023 CSL 2023 February 13–16, 2023
Warsaw, Poland 252 25
# Non-uniform complexity via non-wellfounded proofs
Gianluca Curzi Anupam Das
###### Abstract
Cyclic and non-wellfounded proofs are now increasingly employed to establish
metalogical results in a variety of settings, in particular for type systems
with forms of (co)induction. Under the Curry-Howard correspondence, a cyclic
proof can be seen as a typing derivation ‘with loops’, closer to low-level
machine models, and so comprise a highly expressive computational model that
nonetheless enjoys excellent metalogical properties.
In recent work, we showed how the cyclic proof setting can be further employed
to model computational complexity, yielding characterisations of the
polynomial time and elementary computable functions. These characterisations
are ‘implicit’, inspired by Bellantoni and Cook’s famous algebra of safe
recursion, but exhibit greater expressivity thanks to the looping capacity of
cyclic proofs.
In this work we investigate the capacity for _non-wellfounded_ proofs, where
finite presentability is relaxed, to model non-uniformity in complexity
theory. In particular, we present a characterisation of the class
$\mathbf{FP}/\mathit{poly}$ of functions computed by polynomial-size circuits.
While relating non-wellfoundedness to non-uniformity is a natural idea, the
precise amount of irregularity, informally speaking, required to capture
$\mathbf{FP}/\mathit{poly}$ is given by proof-level conditions novel to cyclic
proof theory. Along the way, we formalise some (presumably) folklore
techniques for characterising non-uniform classes in relativised function
algebras with appropriate oracles.
###### keywords:
Cyclic proofs, non-wellfounded proof-theory, non-uniform complexity,
polynomial time, safe recursion, implicit complexity
###### category:
## 1 Introduction
_Non-wellfounded proof theory_ is the study of possibly infinite (but finitely
branching) proofs, where appropriate global correctness criteria guarantee
logical consistency. This area originates (in its modern guise) in the context
of the modal $\mu$-calculus [32, 17], serving as an alternative framework to
manipulate least and greatest fixed points, and hence to model inductive and
coinductive reasoning. Since then, non-wellfounded proofs have been widely
investigated in many respects, such as predicate logic [9, 6], algebras [15,
16], arithmetic [33, 5, 13], proofs-as-programs interpretations [2, 18, 12,
25, 14], and continuous cut-elimination [31, 19]. Special attention in these
works is drawn to _cyclic_ (or _regular_) proofs, i.e. non-wellfounded proofs
with only finitely many distinct subproofs, comprising a natural notion of
finite presentability in terms of (possibly cyclic) directed graphs.
The _Curry-Howard_ reading of non-wellfounded proofs has revealed a deep
connection between proof-theoretic properties and computational behaviours
[12, 25, 14]. On the one hand, the typical correctness conditions ensuring
consistency, called _progressing_ (or _validity_) criteria, correspond to
totality: functions computed by progressing proofs are always well-defined on
all arguments. On the other hand, regularity has a natural counterpart in the
notion of _uniformity_ : circular proofs can be properly regarded as programs,
i.e. as finite sets of machine instructions, thus having a ‘computable’
behaviour.
In a recent work [11], the authors extended these connections between non-
wellfounded proof theory and computation to the realm of _computational
complexity_. We introduced the proof systems $\mathsf{C}\mathsf{B}$ and
$\mathsf{C}\mathsf{NB}$ capturing, respectively, the class of functions
computable in polynomial time ($\mathbf{F}\mathbf{P}$) and the elementary
functions ($\mathbf{F}\mathbf{ELEMENTARY}$). These proof systems are defined
by identifying global conditions on circular progressing proofs motivated by
ideas from _Implicit Computational Complexity_ (ICC).
ICC, broadly construed, is the study of machine-free (and often bound-free)
characterisations of complexity classes. One of the seminal works in the area
is Bellantoni and Cook’s function algebra $\mathsf{B}$ for
$\mathbf{F}\mathbf{P}$ based on _safe recursion_ [4]. The prevailing idea
behind safe recursion (and its predecessor, _ramified recursion_ [27]) is to
partition function arguments into ‘safe’ and ‘normal’, namely writing
$f(x_{1},\dots,x_{m};y_{1},\dots,y_{n})$ when $f$ takes $m$ normal inputs
$\vec{x}$ and $n$ safe inputs $\vec{y}$. In functions of $\mathsf{B}$, the
recursive parameters are always normal arguments, while recursive calls can
only appear in safe position; hence, no recursive call can be used as
recursive parameters of other previously defined functions. Our system
$\mathsf{C}\mathsf{B}$ morally represents a cyclic proof theoretic formulation
of $\mathsf{B}$.
To establish the characterisation result for $\mathsf{C}\mathsf{B}$ we
developed a novel function algebra for $\mathbf{F}\mathbf{P}$, called
$\mathsf{B}^{\subset}$. Roughly, the latter extends $\mathsf{B}$ with a more
expressive recursion mechanism on a special well-founded preorder,
‘$\subset$’, based on permutation of prefixes of normal arguments, and whose
definition requires relativisation of the algebra to admit _oracles_. The
characterisation theorem is then obtained by a ‘sandwich’ technique, where the
function algebras $\mathsf{B}$ and $\mathsf{B}^{\subset}$ serve, respectively,
as lower and upper bounds for $\mathsf{C}\mathsf{B}$.
In this paper we investigate the computational interpretation of more general
_non-wellfounded_ proofs, where finite presentability is relaxed in order to
model _non-uniform complexity_. In particular we consider the class
$\mathbf{FP}/\mathit{poly}$ of functions computable in polynomial time by
Turing machines with access to _polynomial advice_. Equivalently,
$\mathbf{FP}/\mathit{poly}$ is the class of functions computed by families of
polynomial-size circuits. Note, in particular, that
$\mathbf{FP}/\mathit{poly}$ includes _undecidable problems_ , and so cannot be
characterised by purely cyclic proof systems or usual function algebras, which
typically have only computable functions.
We define the system $\mathsf{nu}\mathsf{B}$ (‘non-uniform $\mathsf{B}$’),
allowing a form of non-wellfoundedness somewhere between arbitrary non-
wellfounded proofs and full regularity, and show that $\mathsf{nu}\mathsf{B}$
duly characterises $\mathbf{FP}/\mathit{poly}$. The characterisation theorem
for $\mathsf{nu}\mathsf{B}$ relies on an adaption of the aforementioned
sandwich technique for $\mathsf{C}\mathsf{B}$ to the current setting. This
requires a relativisation of both $\mathsf{B}$ and $\mathsf{B}^{\subset}$ to a
set of oracles, which we call $\mathbb{R}$, deciding properties of string
length. As a byproduct of our proof method we also obtain new relativised
function algebras for $\mathbf{FP}/\mathit{poly}$ based on safe recursion,
$\mathsf{B}(\mathbb{R})$ and $\mathsf{B}^{\subset}(\mathbb{R})$; these are
folklore-style results that, as far as we know, have not yet appeared in the
literature.
The overall structure of our result relies on a ‘grand tour’ of inclusions,
summarised as:
$\mathbf{FP}/\mathit{poly}\overset{\text{\tiny P.\ref{prop:fppoly-
in-B(R0)}}}{\subseteq}\mathsf{B}(\mathbb{R}_{1;0})\overset{\text{\tiny
P.\ref{prop:b(f)-in-
cbc(f)}}}{\subseteq}\mathsf{C}\mathsf{B}(\mathbb{R}_{1;0})\overset{\text{\tiny
P.\ref{prop:cbc(R0)-to-
nuB}}}{\subseteq}\mathsf{nu}\mathsf{B}\overset{\text{\tiny T.\ref{lem:non-
wellfounded-
oracles}}}{\subseteq}\mathsf{C}\mathsf{B}(\mathbb{R}_{1;1})\overset{\text{\tiny
L.\ref{lem:rel-tran-
lem}}}{\subseteq}\mathsf{B}^{\subset}(\mathbb{R}_{1;1})\overset{\text{\tiny
P.\ref{prop:relativised-
characterisation}}}{\subseteq}\mathbf{F}\mathbf{P}(\mathbb{R})\overset{\text{\tiny
P.\ref{prop:fppoly=fptime(RR)}}}{\subseteq}\mathbf{FP}/\mathit{poly}$
While this may seem like a long route to take, the structure of our argument
is designed so that each of the above inclusions are relatively simple to
establish and, as we said, yields several intermediate characterisions of
$\mathbf{FP}/\mathit{poly}$ of self-contained interest.
Related work. Characterisations of non-uniform complexity classes in the style
of ICC have been considered in the context of the $\lambda$-calculus [28] and
variants of linear logic [30]. The former captures the class
$\mathbf{P}/\mathit{poly}$, i.e., the languages decided by families of
polynomial circuits, while the latter also captures
$\mathbf{L}/\mathit{poly}$, i.e., the languages decided by families of
polynomial size branching programs (i.e. decision trees with sharing). However
this is the first work (as far as we know) that attempts to relate non-
wellfoundedness in proof theory to non-uniformity in complexity theory.
The relativised proof systems and function algebras presented in this paper
only query ‘bits of real numbers’. Proof systems based on linear logic and
implementing polytime computation over actual binary streams have been
considered, e.g., in [21], which provide an ICC-like characterisation of Ko’s
class of polynomial time computable functions over real numbers [24].
Outline of the paper. This paper is structured as follows. In Section 2 we
recall some preliminaries on non-uniform and implicit complexity, in
particular a proof theoretic formulation of the algebra $\mathsf{B}$. In
Section 3 we recall the circular system $\mathsf{C}\mathsf{B}$ from [11], and
introduce our new system $\mathsf{nu}\mathsf{B}$. In Section 4 we take an
interlude to present some _relativised_ characterisations of
$\mathbf{FP}/\mathit{poly}$, both in the machine setting and the implicit
setting, that will later serve use in our grand tour of inclusions. In Section
5 we employ those characterisations to establish the lower bound for
$\mathsf{nu}\mathsf{B}$, and in Section 6 we recast $\mathsf{nu}\mathsf{B}$ as
a sort of relativised circular system. Finally in Section 7 we adapt results
from [11] translating circular proofs to an appropriate function algebra to
the relativised setting, thereby achieving the upper bound for
$\mathsf{nu}\mathsf{B}$.
## 2 Preliminaries on computational complexity and safe recursion
Throughout this work we only consider (partial) functions on _natural
numbers_. We write $|x|$ for the length of the binary representation of a
number $x$, and for lists of arguments $\vec{x}=x_{1},\dots,x_{n}$ we write
$|\vec{x}|$ for the list $|x_{1}|,\dots,|x_{n}|$.
### 2.1 Non-uniform complexity classes
$\mathbf{F}\mathbf{P}$ is the class of (total) functions computable in
polynomial time on a Turing machine. The ‘non-uniform’ class
$\mathbf{FP}/\mathit{poly}$ is an extension of $\mathbf{F}\mathbf{P}$ that
intuitively has access to a polynomial amount of ‘advice’, determined only by
the _length_ of the input. Formally:
###### Definition 1 (Non-uniform polynomial time).
$\mathbf{FP}/\mathit{poly}$ is the class of functions $f(\vec{x})$ for which
there are strings $\alpha_{\vec{n}}\in\\{0,1\\}^{*}$, of size polynomial in
$\vec{n}$, and some $f^{\prime}(x,\vec{x})\in\mathbf{F}\mathbf{P}$ with:
* •
$|\alpha_{\vec{n}}|$ is polynomial in $\vec{n}$.
* •
$f(\vec{x})=f^{\prime}(\alpha_{|\vec{x}|},\vec{x})$.
The strings $\\{\alpha_{\vec{n}}\\}_{\vec{n}}$ represent the _polynomial
advice_ given to a polynomial-time computation, here $f^{\prime}(x,\vec{x})$.
Note that $f(\vec{x})$ only ‘receives advice’ depending on the lengths of its
inputs, $\vec{x}$.
Note, in particular, that $\mathbf{FP}/\mathit{poly}$ admits undecidable
problems. E.g. the function $f(x)=1$ just if $|x|$ is the code of a halting
Turing machine (and $0$ otherwise) is in $\mathbf{FP}/\mathit{poly}$. Indeed,
the point of the class $\mathbf{FP}/\mathit{poly}$ is to rather characterise a
more non-uniform notion of computation. In particular, the following is well-
known (see, e.g., [1, Theorem 6.11]):
###### Proposition 2.
$f(\vec{x})\in\mathbf{FP}/\mathit{poly}$ iff there are polynomial-size
circuits computing $f(\vec{x})$.
### 2.2 The Bellantoni-Cook algebra
A _two-sorted_ function is a function $f(\vec{x};\vec{y})$ whose arguments
have been delimited into ‘normal’ ones ($\vec{x}$, left of ‘;’), and ‘safe’
ones ($\vec{y}$, right of ‘;’).
The two-sorted algebra $\mathsf{B}$ was introduced in [4] and is defined as
follows:
###### Definition 3 (Bellantoni-Cook).
$\mathsf{B}$ is the smallest class of two-sorted functions containing,
* •
$0(;):=0$
* •
$\mathsf{s}_{0}(;x):=2x$
* •
$\mathsf{s}_{1}(;x):=2x+1$
* •
$\pi^{m;n}_{j;}(x_{0},\dots,x_{m-1};y_{0},\dots,y_{n-1}):=x_{j}$, whenever
$j<m$.
* •
$\pi^{m;n}_{;j}(x_{0},\dots,x_{m-1};y_{0},\dots,y_{n-1}):=y_{j}$, whenever
$j<n$.
* •
$\mathsf{p}(;x)=\lfloor\frac{x}{2}\rfloor$
* •
$\mathsf{cond}(;w,x,y,z):=\begin{cases}x&w=0\\\ y&w=0\mod 2,w\neq 0\\\
z&w=1\mod 2\end{cases}$
and closed under:
* •
(Safe composition)
* –
if $g(\vec{x};)\in\mathsf{B}$ and $h(\vec{x},x;\vec{y})\in\mathsf{B}$ then
also $f(\vec{x};\vec{y})\in\mathsf{B}$ where
$f(\vec{x};\vec{y}):=h(\vec{x},g(\vec{x};);\vec{y})$.
* –
if $g(\vec{x};\vec{y})\in\mathsf{B}$ and $h(\vec{x};\vec{y},y)\in\mathsf{B}$
then also $f(\vec{x};\vec{y})\in\mathsf{B}$ where
$f(\vec{x};\vec{y}):=h(\vec{x};\vec{y},g(\vec{x};\vec{y}))$
* •
(Safe recursion on notation) if $g(\vec{x};\vec{y})\in\mathsf{B}$ and
$h_{0}(x,\vec{x};\vec{y},y),h_{1}(x,\vec{x};\vec{y},y)\in\mathsf{B}$ then also
$f(x,\vec{x};\vec{y})\Achange{\in}\mathsf{B}$ where:
$\begin{array}[]{r@{\ := \ }ll}f(0,\vec{x};\vec{y})&g(\vec{x};\vec{y})\\\
f(\mathsf{s}_{0}x,\vec{x};\vec{y})&h_{0}(x,\vec{x};\vec{y},f(x,\vec{x};\vec{y}))&x\neq
0\\\
f(\mathsf{s}_{1}x,\vec{x};\vec{y})&h_{1}(x,\vec{x};\vec{y},f(x,\vec{x};\vec{y}))\end{array}$
Safe composition ensures that safe arguments may never appear in a normal
position. Note that, in the recursion scheme, the recursion parameter is
always a normal argument, whereas recursive calls must appear in safe
position. Along with the constraints on safe composition, this ensures that
the position of a recursive call is never the recursion parameter of another
recursion. This seemingly modest constraint duly restricts computation to
polynomial time, yielding Bellantoni and Cook’s main result:
###### Theorem 4 ([4]).
$f(\vec{x})\in\mathbf{F}\mathbf{P}$ if and only if $f(\vec{x};)\in\mathsf{B}$.
### 2.3 A proof-theoretic presentation of Bellantoni-Cook
We shall work with a formulation of $\mathsf{B}$ as a $S4$-style type system
in sequent-calculus style, where modalities are used to distinguish the two
sorts (similarly to [22]).
We consider _types_ (or _formulas_) $N$ (‘safe’) and $\Box N$ (‘normal’) which
intuitively vary over the natural numbers. We write $A,B,$ etc. to vary over
types. A _sequent_ is an expression $\Gamma\Rightarrow A$, where $\Gamma$ is a
list of types (called the _context_ or _antecedent_) and $A$ is a type (called
the _succedent_). For a list of types
$\Gamma=\overset{k}{\overbrace{{N},{\ldots},{N}}}$, we write $\Box\Gamma$ for
$\overset{k}{\overbrace{{{\Box N}},{\ldots},{{\Box N}}}}$.
###### Definition 5.
A _$\mathsf{B}$ -derivation_ is a (finite) derivation built from the rules in
Figure 1.
$\small\begin{array}[]{c}{\vbox{\hbox{\kern
7.75pt\hbox{\vbox{\offinterlineskip\hbox{\kern
15.21248pt\hbox{\hbox{\hbox{$$}}}\kern 15.21248pt}\kern 1.43518pt\hbox{\hbox
to0.0pt{\hss\hbox{$\smash{\lower
0.0pt\hbox{$\scriptstyle\mathsf{id}\;$}}$}}\vbox{\vbox to0.4pt{\vfill\hbox
to30.42497pt{\hrulefill}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower
0.0pt\hbox{$$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{\hbox{$\kern
0.0pt\hbox{$N\Rightarrow N$}\kern 0.0pt$}}}}}\kern
0.0pt}}}\quad{{}{}\vbox{\hbox{\kern
14.53499pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma}\Rightarrow
N$}}\kern
10.00002pt}\hbox{\hbox{${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N}\Rightarrow
B$}}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower
0.0pt\hbox{$\scriptstyle\mathsf{cut}_{N}\;$}}$}}\vbox{\vbox
to0.4pt{\vfill\hbox to70.45317pt{\hrulefill}\vfill}}\hbox
to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}\hss}\kern 0.0pt}\kern
1.43518pt\hbox{\kern
23.025pt\hbox{\hbox{${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma}\Rightarrow
B$}}\kern 23.025pt}}}\kern 0.0pt}}}\quad{{}{}\vbox{\hbox{\kern
14.05pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma}\Rightarrow{\Box
N}$}}\kern
10.00002pt}\hbox{\hbox{${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma}\Rightarrow
B$}}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower
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1.43518pt\hbox{\kern
17.66876pt\hbox{\hbox{${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N}\Rightarrow
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23.75002pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma}\Rightarrow
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0.0pt\hbox{$\scriptstyle|\mathsf{cond}|_{\Box}\;$}}$}}\vbox{\vbox
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to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}\hss}\kern 0.0pt}\kern
1.43518pt\hbox{\kern
17.66876pt\hbox{\hbox{${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma}\Rightarrow
N$}}\kern 17.66876pt}}}\kern 0.0pt}}}\end{array}$ Figure 1: $\mathsf{B}$ as a
sequent-style type system.
The colouring of type occurrences in Figure 1 may be ignored for now, they
will become relevant in the next section. We may write
$\mathcal{D}=\mathsf{r}(\mathcal{D}_{1},\ldots,\mathcal{D}_{n})$ (for $n\leq
3$) if $\mathsf{r}$ is the bottom-most inference step of a derivation
$\mathcal{D}$ whose immediate subderivations are, respectively,
$\mathcal{D}_{1},\ldots,\mathcal{D}_{n}$. As done in [11], we shall assume
w.l.o.g. that sequents have shape ${{{\Box N}},\ldots,{{\Box
N}},{N},\ldots,{N}\Rightarrow A}$, i.e. in the left-hand side all ${\Box N}$
occurrences are placed before all $N$ occurrences.
We construe the system of $\mathsf{B}$-derivations as a class of two-sorted
functions by identifying each rule instance as an operation on two-sorted
functions as follows:
###### Definition 6 (Semantics of $\mathsf{B}$).
Given a $\mathsf{B}$-derivation $\mathcal{D}$ of
$\overset{m}{\overbrace{{{\Box N}},{\ldots},{{\Box
N}}}},\overset{n}{\overbrace{{N},{\ldots},{N}}}\Rightarrow A$ we define a two-
sorted function $f_{\mathcal{D}}(x_{1},\dots,x_{m};y_{1},\dots,y_{n})$ in
Figure 2 by induction on the structure of $\mathcal{D}$ (all rules as typeset
in Figure 1).
$\small\begin{array}[]{cc}\begin{array}[]{rcl}f_{\mathsf{id}}(;y)&:=&y\\\
f_{\mathsf{cut}_{N}(\mathcal{D}_{0},\mathcal{D}_{1})}(\vec{x};\vec{y})&:=&f_{\mathcal{D}_{1}}(\vec{x};\vec{y},f_{\mathcal{D}_{0}}(\vec{x};\vec{y}))\\\
f_{\mathsf{cut}_{\Box}(\mathcal{D}_{0},\mathcal{D}_{1})}(\vec{x};\vec{y})&:=&f_{\mathcal{D}_{1}}(f_{\mathcal{D}_{0}}(\vec{x};\vec{y}),\vec{x};\vec{y})\\\
f_{\mathsf{w}_{N}(\mathcal{D}_{0})}(\vec{x};\vec{y},y)&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y})\\\
f_{\mathsf{w}_{\Box}(\mathcal{D}_{0})}(x,\vec{x};\vec{y})&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y})\\\
f_{\mathsf{e}_{N}(\mathcal{D}_{0})}(\vec{x};\vec{y},y,y^{\prime},\vec{y}^{\prime})&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y},y^{\prime},y,\vec{y}^{\prime})\\\
f_{\mathsf{e}_{\Box}(\mathcal{D}_{0})}(\vec{x},x,x^{\prime},\vec{x}^{\prime};\vec{y})&:=&f_{\mathcal{D}_{0}}(\vec{x},x^{\prime},x,\vec{x}^{\prime};\vec{y})\\\
f_{\Box_{l}(\mathcal{D}_{0})}(x,\vec{x};\vec{y})&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y},x)\\\
f_{\Box_{r}(\mathcal{D}_{0})}(\vec{x};)&:=&f_{\mathcal{D}_{0}}(\vec{x};)\\\
f_{i}(;)&:=&i\\\
f_{\mathsf{s}_{i}(\mathcal{D}_{0})}(\vec{x};\vec{y})&:=&\mathsf{s}_{i}(;f_{\mathcal{D}_{0}}(\vec{x};\vec{y}))\end{array}&\begin{array}[]{rcl}f_{\mathsf{srec}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2})}(0,\vec{x};\vec{y})&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y})\\\
f_{\mathsf{srec}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2})}(\mathsf{s}_{i}x,\vec{x};\vec{y})&:=&f_{\mathcal{D}_{i+1}}(x,\vec{x};\vec{y},\\\
&&f_{\mathsf{srec}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2})}(x,\vec{x};\vec{y}))\\\
f_{\mathsf{cond}_{N}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2})}(\vec{x};\vec{y},0)&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y})\\\
f_{\mathsf{cond}_{N}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2})}(\vec{x};\vec{y},\mathsf{s}_{i}y)&:=&f_{\mathcal{D}_{i+1}}(\vec{x};\vec{y},y)\\\
f_{\mathsf{cond}_{\Box}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2})}(0,\vec{x};\vec{y})&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y})\\\
f_{\mathsf{cond}_{\Box}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2})}(\mathsf{s}_{i}x,\vec{x};\vec{y})&:=&f_{\mathcal{D}_{i+1}}(x,\vec{x};\vec{y})\\\
f_{|\mathsf{cond}|_{N}(\mathcal{D}_{0},\mathcal{D}_{1})}(\vec{x};\vec{y},0)&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y})\\\
f_{|\mathsf{cond}|_{N}(\mathcal{D}_{0},\mathcal{D}_{1})}(\vec{x};\vec{y},\mathsf{s}_{i}y)&:=&f_{\mathcal{D}_{1}}(\vec{x};\vec{y},y)\\\
f_{|\mathsf{cond}|_{\Box}(\mathcal{D}_{0},\mathcal{D}_{1})}(0,\vec{x};\vec{y})&:=&f_{\mathcal{D}_{0}}(\vec{x};\vec{y})\\\
f_{|\mathsf{cond}|_{\Box}(\mathcal{D}_{0},\mathcal{D}_{1})}(\mathsf{s}_{i}x,\vec{x};\vec{y})&:=&f_{\mathcal{D}_{1}}(x,\vec{x};\vec{y})\end{array}\end{array}$
Figure 2: Semantics of system $\mathsf{B}$, where $i\in\\{0,1\\}$ and
$\mathsf{s}_{i}x\neq 0$ and $\mathsf{s}_{i}y\neq 0$.
This formal semantics exposes how $\mathsf{B}$-derivations and functions in
the algebra $\mathsf{B}$ relate. The rule $\mathsf{srec}$ in Figure 1
corresponds to safe recursion, and safe composition along safe parameters is
expressed by $\mathsf{cut}_{N}$. Note, however, that the interpretation of
$\mathsf{cut}_{\Box}$ in Figure 2 apparently does not satifsfy the required
constraint on safe composition of a function $g$ along a normal parameter of a
function $h$, which forbids the presence of safe parameters in $g$. However,
this admission turns out to be harmless, and we are able to obtain the
following result that justifies the overloading of the notation
‘$\mathsf{B}$’:
###### Proposition 7 ([11]).
$f(\vec{x};\vec{y})\in\mathsf{B}$ iff there is a $\mathsf{B}$-derivation
$\mathcal{D}$ for which $f_{\mathcal{D}}(\vec{x};\vec{y})=f(\vec{x};\vec{y})$.
###### Remark 8 (Bootstrapping).
Note that the rules $1$, $|\mathsf{cond}|_{N}$ and $|\mathsf{cond}|_{\Box}$
are semantically redundant, being derivable from the others by:
$f_{1}=f_{\mathsf{s}_{1}(0)}$,
$f_{|\mathsf{cond}|_{N}(\mathcal{D}_{0},\mathcal{D}_{1})}=f_{\mathsf{cond}_{N}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{1})}$,
and
$f_{|\mathsf{cond}|_{\Box}(\mathcal{D}_{0},\mathcal{D}_{1})}=f_{\mathsf{cond}_{\Box}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{1})}$.
Indeed, our original presentation of the system in [11] did not include these
rules, but we have ‘bootstrapped’ our system here in order to facilitate the
definitions of our restricted ‘non-wellfounded’ systems later for
characterising $\mathbf{FP}/\mathit{poly}$, in particular in Section 3.
## 3 Non-wellfounded systems based on Bellantoni-Cook
In this section we recall a ‘coinductive’ version of $\mathsf{B}$ that was
recently introduced in our earlier work [11], and go on to introduce the new
system $\mathsf{nu}\mathsf{B}$ of this work. In particular we shall give
global criteria that control the computational strength of non-wellfounded
typing derivations. Throughout this section we shall work with the system
$\mathsf{B}^{-}:=\mathsf{B}-\\{\mathsf{srec}\\}$.
###### Definition 9 (Coderivations).
A ($\mathsf{B}^{-}$-)_coderivation_ $\mathcal{D}$ is a possibly infinite
rooted tree generated by the rules of $\mathsf{B}^{-}$. Formally, we identify
$\mathcal{D}$ with a (labelled) prefix-closed subset of $\\{0,1,2\\}^{*}$
(i.e. a ternary tree). Each node is labelled by an inference step from
$\mathsf{B}^{-}$ such that, whenever $\nu\in\mathcal{D}$ is labelled by a step
$S_{1}$$\cdots$$S_{n}$ $S$ , for $n\leq 3$, $\nu$ has $n$ children in
$\mathcal{D}$ labelled by steps with conclusions $S_{1},\dots,S_{n}$
respectively. Sub-coderivations of a coderivation $\mathcal{D}$ rooted at
position $\nu\in\\{0,1,2\\}^{*}$ are denoted $\mathcal{D}_{\nu}$, so that
$\mathcal{D}_{\varepsilon}=\mathcal{D}$.
$\scriptstyle\mathsf{id}\;$ $N\Rightarrow N$ $\scriptstyle\mathsf{s}_{1}\;$
$N\Rightarrow N$ $\scriptstyle\Box_{l}\;$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}}\Rightarrow N$ $\scriptstyle\Box_{r}\;$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}}\Rightarrow{\Box N}$ $\vdots$ $\scriptstyle\mathsf{cut}_{\Box}\;$
$\;\bullet$ ${\Box N}\Rightarrow N$ $\scriptstyle\mathsf{cut}_{\Box}\;$
$\;\bullet$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}}\Rightarrow N$ $\textstyle{\scriptstyle\mathcal{G}}$
$\Box\vec{N}\Rightarrow N$ $\vdots$ $\scriptstyle\mathsf{cond}_{\Box}\;$
$\;\bullet$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},\Box\vec{N}\Rightarrow{N}$ $\scriptstyle\Box_{r}\;$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},\Box\vec{N}\Rightarrow{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}}$ $\textstyle{\scriptstyle\mathcal{H}_{i}}$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},\Box\vec{N},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}}\Rightarrow N$ $\scriptstyle\mathsf{cut}_{\Box}\;$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},\Box\vec{N}\Rightarrow N$ $\scriptstyle\mathsf{cond}_{\Box}\;$
$\;\bullet\quad{\scriptstyle{i=0,1}}$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\underline{{\Box
N}}},\Box\vec{N}\Rightarrow N$
$\scriptstyle\mathsf{id}\;$ $N\Rightarrow N$ $\vdots$
$\scriptstyle\mathsf{cond}_{\Box}\;$ $\;\circ$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}},N\Rightarrow N$ $\scriptstyle\mathsf{s}_{i}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}},N\Rightarrow N$ $\scriptstyle\mathsf{cond}_{\Box}\;$
$\;\circ\quad{\scriptstyle{i=0,1}}$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{{\Box
N}}},N\Rightarrow N$ $\vdots$ $\scriptstyle\mathsf{cond}_{\Box}\;$ $\;\bullet$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}},N\Rightarrow N$ $\scriptstyle\mathsf{s}_{i}\;$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}},N\Rightarrow N$ $\scriptstyle\mathsf{cond}_{\Box}\;$
$\;\bullet\quad{\scriptstyle{i=0,1}}$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\underline{{\Box
N}}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}},N\Rightarrow N$
$\scriptstyle\mathsf{id}\;$
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}N}\Rightarrow
N$ $\scriptstyle\mathsf{s}_{0}\;$
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}N}\Rightarrow
N$ $\vdots$ $\scriptstyle\mathsf{cond}_{\Box}\;$ $\;\bullet$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}N}\Rightarrow{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}N}$
$\vdots$ $\scriptstyle\mathsf{cond}_{\Box}\;$ $\;\bullet$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}N}\Rightarrow
N$ $\scriptstyle\mathsf{cut}_{N}\;$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Box
N}},{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}N}\Rightarrow
N$ $\scriptstyle\mathsf{cond}_{\Box}\;$ $\;\bullet$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\underline{{\Box
N}}},{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}N}\Rightarrow
N$ $\textstyle{\scriptstyle f(0)}$ $\Rightarrow N$ $\textstyle{\scriptstyle
f(1)}$ $\Rightarrow N$ $\vdots$ $\scriptstyle|\mathsf{cond}|_{\Box}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{{\Box
N}}}\Rightarrow N$ $\scriptstyle|\mathsf{cond}|_{\Box}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{{\Box
N}}}\Rightarrow N$ $\scriptstyle|\mathsf{cond}|_{\Box}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{{\Box
N}}}\Rightarrow N$
$\scriptstyle 0\;$ $\Rightarrow N$ $\vdots$
$\scriptstyle|\mathsf{cond}|_{\Box}\;$ $\;\bullet$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}}\Rightarrow N$ $\scriptstyle\mathsf{s}_{0}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}}\Rightarrow{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N}$
$\vdots$ $\scriptstyle|\mathsf{cond}|_{\Box}\;$ $\;\bullet$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}}\Rightarrow N$ $\scriptstyle\mathsf{s}_{1}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}}\Rightarrow{\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}N}$
$\textstyle{\scriptstyle\mathcal{F}(r)}$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}}\Rightarrow{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}N}$
$\scriptstyle\mathsf{id}\;$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N}\Rightarrow
N$ $\scriptstyle\mathsf{id}\;$
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N}\Rightarrow
N$ $\scriptstyle\mathsf{id}\;$
${\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}N}\Rightarrow
N$ $\scriptstyle\mathsf{cond}_{N}\;$
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}N},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N},{\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}N}\Rightarrow
N$ $\scriptstyle\mathsf{cut}_{N}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N},{\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}N}\Rightarrow
N$ $\scriptstyle\mathsf{cut}_{N}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N}\Rightarrow
N$ $\scriptstyle\mathsf{cut}_{N}\;$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\Box
N}}\Rightarrow N$ $\scriptstyle|\mathsf{cond}|_{\Box}\;$ $\;\bullet$
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{{\Box
N}}}\Rightarrow N$
Figure 3: Examples of coderivations: $\mathcal{I}$ (top left), $\mathcal{R}$
(top right), $\mathcal{C}$ (second line), $\mathcal{E}$ (third line, left),
$\mathcal{F}(f)$ with $f:\mathbb{N}\to\mathbb{N}$ (third line, right),
$\mathcal{A}(r)$ with $r:\mathbb{N}\to\\{0,1\\}$ (bottom).
Examples of coderivations can be found in Figure 3 (some of them are from
[11]), whose computational meaning is discussed in 13, and employ the
following conventions:
###### Convention 10 (Representing coderivations).
Henceforth, we may mark steps by $\bullet$ (or similar) in a coderivation to
indicate roots of identical sub-coderivations. Moreover, to avoid ambiguities
and to ease parsing of (co)derivations, we shall often underline principal
formulas of a rule instance in a given coderivation and omit instances of
structural rules $\mathsf{e}_{N}$, $\mathsf{e}_{\Box}$, $\mathsf{w}_{N}$ and
$\mathsf{w}_{\Box}$, absorbing them into other steps (typically cuts) when it
causes no confusion. Finally, when the sub-coderivations $\mathcal{D}_{0}$ and
$\mathcal{D}_{1}$ above the second and the third premise of the conditional
rule (from left) are similar, we may compress them into a single
‘parametrised’ sub-coderivation $\mathcal{D}_{i}$ (with $i=0,1$).
As discussed in [14, 12, 25], coderivations can be identified with Kleene-
Herbrand-Gödel style equational programs, in general computing partial
recursive functionals (see, e.g., [23, §63] for further details). We shall
specialise this idea to our two-sorted setting.
###### Definition 11 (Semantics of coderivations).
To each $\mathsf{B}^{-}$-coderivation $\mathcal{D}$ we associate a two-sorted
Kleene-Herbrand-Gödel partial function $f_{\mathcal{D}}$ obtained by
construing the semantics of 6 as a (possibly infinite) equational program.
Given a two-sorted function $f(\vec{x};\vec{y})$, we say that $f$ is _defined_
by a $\mathsf{B}^{-}$-coderivation $\mathcal{D}$ if
$f_{\mathcal{D}}(\vec{x};\vec{y})=f(\vec{x};\vec{y})$.
###### Remark 12.
The notion of _computation_ for equational programs is given by (finitary)
reasoning in equational logic (see, e.g., [23, §63]): for numerals
$\vec{m},\vec{n}$, we have that $f_{\mathcal{D}}(\vec{m};\vec{n})$ is well-
defined and returns some numeral $k$ just if the equation
$f_{\mathcal{D}}(\vec{m};\vec{n})=k$ can be (finitely) derived in equational
logic (with basic numerical axioms) over the equational program for
$\mathcal{D}$. Implicit here is the fact that the semantics of
$\mathsf{B}^{-}$-coderivations yield _coherent_ equational programs: whenever
$f_{\mathcal{D}}(\vec{m};\vec{n})=k$ and
$f_{\mathcal{D}}(\vec{m};\vec{n})=k^{\prime}$ are derivable then
$k=k^{\prime}$ [14, 12].
###### Example 13.
By purely equational reasoning, we can simplify the Kleene-Gödel-Herbrand
style semantics in 11 of the coderivations in Figure 3 to get the equational
programs in Figure 4: $f_{\mathcal{I}}$ represents a function that is always
undefined, as its equational program keeps increasing the length of the input;
$f_{\mathcal{R}}$ is an instance of a _non-safe_ recursion scheme (on
notation), as the recursive call appears in normal position; $f_{\mathcal{C}}$
computes concatenation of the binary representation of three natural numbers;
$f_{\mathcal{E}}$ has exponential growth rate (as long as $y\neq 0$), since
$f_{\mathcal{E}}(x;y)=2^{2^{|{x}|}}\cdot|y|$; the (infinite) equational
program for $f_{\mathcal{F}(f)}$ computes $f(|x|)$ by simply exhausting the
values of $|x|$; finally, $f_{\mathcal{A}(r)}$ on input $x$ returns the binary
string $r(0)\cdot r(1)\cdot\cdots\cdot r(|x|-1)$ if $x>0$, and $0$ otherwise.
$\small\begin{array}[]{cc}\begin{array}[]{rcl}f_{\mathcal{I}}(x;)&=&f_{\mathcal{I}}(\mathsf{s}_{1}x;)\\\
f_{\mathcal{R}}(0,\vec{x};)&=&f_{\mathcal{G}}(\vec{x};)\\\
f_{\mathcal{R}}(\mathsf{s}_{i}x,\vec{x};)&=&f_{\mathcal{H}_{i}}(x,\vec{x},f_{\mathcal{R}}(x,\vec{x};);)\\\
f_{\mathcal{C}}(0,0;z)&=&z\\\
f_{\mathcal{C}}(0,\mathsf{s}_{i}y;z)&=&\mathsf{s}_{i}f_{\mathcal{C}}(0,y;z)\\\
f_{\mathcal{C}}(\mathsf{s}_{i}x,y;z)&=&\mathsf{s}_{i}f_{\mathcal{C}}(x,y;z)\end{array}&\begin{array}[]{rcl}f_{\mathcal{E}}(0;y)&=&\mathsf{s}_{0}(;y)\\\
f_{\mathcal{E}}(\mathsf{s}_{i}x;y)&=&f_{\mathcal{E}}(x;f_{\mathcal{E}}(x;y))\\\
\\{f_{\mathcal{F}(f)}(x;)&=&\Achange{f(|x|)\\}_{|x|\in\mathbb{N}}}\\\
f_{\mathcal{A}(r)}(0;)&=&0\\\
\Achange{f_{\mathcal{A}(r)}(\mathsf{s}_{i}x;)}&=&\begin{cases}\mathsf{s}_{0}f_{\mathcal{A}(\Achange
r)}(x;)&\text{if }\Achange{f_{\mathcal{F}(r)}(x;)=0}\\\
\mathsf{s}_{1}f_{\mathcal{A}(\Achange
r)}(x;)&\text{otherwise}\end{cases}\par\end{array}\end{array}$ Figure 4:
Equational programs derived from the coderivations in Figure 3, where
$i\in\\{0,1\\}$.
The above examples illustrate several recursion theoretic features of
$\mathsf{B}^{-}$-coderivations that we shall seek to control in the remainder
of this section:
1. (I)
_non-totality_ (e.g., the coderivation $\mathcal{I}$);
2. (II)
_non-computability_ (e.g., the coderivation $\mathcal{F}(f)$, with $f$ non-
computable);
3. (III)
_non-safety_ (e.g., the coderivation $\mathcal{R}$), despite the presence of
modalities implementing the normal/safe distinction of function arguments;
4. (IV)
_nested recursion_ (e.g., the coderivation $\mathcal{E}$).
To address (I) we shall adapt to our setting a well-known ‘totality criterion’
from non-wellfounded proof theory (similar to those in [14, 12, 25]). First we
need to recall some standard structural proof theoretic notions:
###### Definition 14 (Ancestry).
Fix a coderivation $\mathcal{D}$. We say that a type occurrence $A$ is an
_immediate ancestor_ of a type occurrence $B$ in $\mathcal{D}$ if they are
types in a premiss and conclusion (respectively) of an inference step and, as
typeset in Figure 1, have the same colour. If $A$ and $B$ are in some
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\Gamma}$
or
${\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\Gamma^{\prime}}$,
then furthermore they must be in the same position in the list.
For a definition of immediate ancestry avoiding colours, we point the reader
to standard proof theory references, e.g. [10, Sec. 1.2.3]. Being a binary
relation, immediate ancestry forms a directed graph upon which our totality
criterion is built:
###### Definition 15 (Progressing coderivations).
Fix a coderivation $\mathcal{D}$. A _thread_ is a maximal path in the graph of
immediate ancestry. We say that a (infinite) thread is _progressing_ if it is
eventually constant ${\Box N}$ and infinitely often principal for a
$\mathsf{cond}_{\Box}$ rule or a $|\mathsf{cond}|_{\Box}$ rule. A coderivation
is _progressing_ if each of its infinite branches has a progressing thread.
In [11] we showed that the progressing criterion is indeed sufficient (but
obviously not necessary) to guarantee that the partial function computed by a
coderivation is, in fact, total (see also [25, 12, 14]):
###### Proposition 16 (Progressing implies totality, [11]).
If $\mathcal{D}$ is progressing, then $f_{\mathcal{D}}$ is total.
The argument for this proposition is by contradiction: assuming non-totality,
construct an infinite ‘non-total branch’, whence a contradiction to well-
orderedness of $\mathbb{N}$ is implied by a progressing thread along it. We
shall use similar argument later in the proof of 36.
###### Example 17.
In Figure 3, $\mathcal{I}$ has precisely one infinite branch (that loops on
$\bullet$) which contains no instances of $\mathsf{cond}_{\Box}$ or
$|\mathsf{cond}|_{\Box}$ at all, so $\mathcal{I}$ is not progressing. On the
other hand, $\mathcal{C}$ has two simple loops, one on $\bullet$ and the other
one on $\circ$. For any infinite branch $B$ we have two cases: if $B$ crosses
the bottommost conditional infinitely many times, it contains a progressing
blue thread; otherwise, $B$ crosses the topmost conditional infinitely many
times, so that it contains a progressing red thread. Therefore, $\mathcal{C}$
is progressing. By applying the same reasoning, we conclude that
$\mathcal{E}$, $\mathcal{F}(f)$, $\mathcal{A}(r)$, and $\mathcal{R}$ are
progressing (if $\mathcal{G}$ and $\mathcal{H}_{i}$ are).
To address (III)-(IV) we recall the following properties of coderivations from
[11]:
###### Definition 18 (Safety, left-leaning).
We say that a coderivation $\mathcal{D}$ is _safe_ if each branch crosses only
finitely many $\mathsf{cut}_{\Box}$-steps, and _left-leaning_ if each branch
goes right at a $\mathsf{cut}_{N}$-step only finitely often.
###### Example 19.
In Figure 3, the only non-safe coderivations are $\mathcal{R}$ and
$\mathcal{I}$, as the branches looping on $\bullet$ contain infinitely many
$\mathsf{cut}_{\Box}$. $\mathcal{E}$ is the only non-left-leaning
coderivation, as it has a branch looping at $\bullet$ that crosses infinitely
many times the rightmost premise of a $\mathsf{cut}_{N}$.
Finally, concerning (II), recall that the aim of this work is to characterise
non-uniform classes, which may contain non-computable predicates and
functions. To this end we introduce a generalisation of the notion of
‘regularity’, typically corresponding to computability (e.g. in [12, 14, 25]),
that is commonplace in cyclic proof theory:
###### Definition 20 (Generalised regularity).
Let $\mathsf{R}\subseteq\mathsf{B}^{-}$. A $\mathsf{B}^{-}$-coderivation
$\mathcal{D}$ is _$\mathsf{R}$ -regular_ if it has only finitely many distinct
sub-coderivations containing rules among $\mathsf{R}$. If
$\mathsf{R}=\mathsf{B}^{-}$, i.e. it has only finitely many distinct sub-
coderivations, then we say that $\mathcal{D}$ is _regular_ (or _circular_).
Note that, while usual derivations may be naturally written as finite trees or
dags, regular coderivations may be naturally written as finite directed
(possibly cyclic) graphs. Also, from a regular coderivation $\mathcal{D}$ we
obtain a _finite_ equational program for $f_{\mathcal{D}}$. In particular,
while there are continuum many (non-wellfounded) coderivations, there are only
countably many regular ones.
###### Example 21.
In Figure 3, $\mathcal{F}(f)$ and $\mathcal{A}(r)$ are the only non-regular
coderivations (as long as $\mathcal{G}$, $\mathcal{H}_{i}$ are regular). Also,
$\mathcal{A}(r)$ is $\mathsf{R}$-regular for any
$\mathsf{R}\subseteq\mathsf{B}^{-}-\\{0,1,|\mathsf{cond}|_{\Box}\\}$, since
$r(i)$ is computed by just a $0$ or $1$ step when $r:\mathbb{N}\to\\{0,1\\}$.
We are now ready to present the non-wellfounded proof systems that will be
considered in this paper:
###### Definition 22 ($\mathsf{C}\mathsf{B}$ and $\mathsf{nu}\mathsf{B}$).
$\mathsf{C}\mathsf{B}$ is the class of regular progressing safe and left-
leaning $\mathsf{B}^{-}$-coderivations. $\mathsf{nu}\mathsf{B}$ is the class
of
$\\{\mathsf{cond}_{\Box},\mathsf{cond}_{N},\mathsf{s}_{0},\mathsf{s}_{1},\mathsf{id}\\}$-regular
progressing safe and left-leaning $\mathsf{B}^{-}$-coderivations. A two-sorted
function $f(\vec{x};\vec{y})$ is _$\mathsf{C}\mathsf{B}$ -definable_ (resp.
_$\mathsf{nu}\mathsf{B}$ -definable_) if there is a coderivation
$\mathcal{D}\in\mathsf{C}\mathsf{B}$ (resp.
$\mathcal{D}\in\mathsf{nu}\mathsf{B}$ ) such that
$f_{\mathcal{D}}(\vec{x};\vec{y})=f(\vec{x};\vec{y})$.
Recalling Examples 17, 19 and 21, $\mathcal{C}$ is the only coderivation in
$\mathsf{C}\mathsf{B}$, while $\mathcal{A}(r)$ is an example of coderivation
in $\mathsf{nu}\mathsf{B}$ for any $r:\mathbb{N}\to\\{0,1\\}$. The system
$\mathsf{C}\mathsf{B}$ was already introduced in [11], where we showed that
$\mathsf{C}\mathsf{B}=\mathbf{F}\mathbf{P}$ (among other results), whereas
$\mathsf{nu}\mathsf{B}$ (read ‘non-uniform $\mathsf{B}$’) is new. The main
result of this paper is to show that $\mathsf{nu}\mathsf{B}$ admits just the
right amount of non-wellfoundedness to duly characterise the analogous non-
uniform class:
###### Theorem 23.
$\mathsf{nu}\mathsf{B}=\mathbf{FP}/\mathit{poly}$
The rest of this work is devoted to the proof of this result. In particular,
the two directions of the equality are given by 34 and 43.
###### Remark 24 (On proof checking).
Let us point out that all conditions on coderivations we have considered so
far are _decidable_ on regular coderivations. In particular, progressiveness
may be decided by reduction to universality of Büchi automata. In the presence
of safety, however, it turns out that proof checking becomes easier: checking
whether a regular coderivation is in $\mathsf{C}\mathsf{B}$ is actually
decidable in $\mathbf{NL}$ [11, Cor. 32]. Of course, as
$\mathsf{nu}\mathsf{B}$ coderivations are not finitely presented (indeed like
$\mathbf{FP}/\mathit{poly}$ programs), such decidability issues are no longer
relevant.
## 4 On relativised characterisations of $\mathbf{FP}/\mathit{poly}$
In this section we consider recursion theoretic characterisations of
$\mathbf{FP}/\mathit{poly}$ via relativised function algebras. This will serve
not only as a ‘warm up’ to motivate our main characterisation, but will also
provide several of the intermediate results necessary to that end. The results
of this section are based on textbook techniques and are (presumably)
folklore.
### 4.1 Non-uniformity via resource-bounded oracle machines
A _relation_ is a function $r(\vec{x})$ such that we always have
$r(\vec{x})\in\\{0,1\\}$.
###### Definition 4.1 (Relativised complexity classes).
Let $R$ be a set of relations. The class $\mathbf{F}\mathbf{P}(R)$ consists of
just the functions computable in polynomial time by a Turing machine with
access to an oracle for each $r\in R$.
For instance, using this notion of relativised computation, we can define the
levels of the functional polynomial hierarchy $\mathbf{FPH}$ by
$\Box^{p}_{1}:=\mathbf{F}\mathbf{P}$,
$\Box^{p}_{2}:=\mathbf{F}\mathbf{P}(\mathbf{NP})$,
$\Box^{p}_{3}:=\mathbf{F}\mathbf{P}(\Sigma^{p}_{2})$, etc.
Let us write $\mathbb{R}:=\\{r:\mathbb{N}^{k}\to\Achange{\\{0,1\\}}\ |\
|\vec{x}|=|\vec{y}|\implies r(\vec{x})=r(\vec{y})\\}$. Note that the notation
$\mathbb{R}$ is suggestive here, since its elements are essentially maps from
lengths/positions to Booleans, and so may be identified with Boolean streams.
###### Proposition 25.
$\mathbf{FP}/\mathit{poly}=\mathbf{F}\mathbf{P}(\mathbb{R})$.
###### Proof 4.2 (Proof sketch).
11todo: 1clean this up a bit
For the left-right inclusion, let $p(n)$ be a polynomial and
$\mathbf{C}=(C_{n})_{n<\omega}$ be a circuit family with each $C_{n}$ taking
$n$ Boolean inputs and having size $<p(n)$. We need to show that the language
computed by $\mathbf{C}$ is also computed in
$\mathbf{F}\mathbf{P}(\mathbb{R})$. Let $c\in\mathbb{R}$ be the function that,
on inputs $x,y$ returns the $|y|$th bit of $C_{|x|}$. Using this oracle we can
compute $C_{|x|}$ by polynomially queries to $c$, and this may be evaluated as
usual using a polynomial-time evaluator in $\mathbf{F}\mathbf{P}$.
For the right-left inclusion, notice that a polynomial-time machine can only
make polynomially many calls to oracles with inputs of only polynomial size.
Thus, if $f\in\mathbf{F}\mathbf{P}(\mathbb{R})$ then there is some $p_{f}$
with $f\in\mathbf{F}\mathbf{P}(\mathbb{R}^{<p_{f}})$, where
$\mathbb{R}^{<p_{f}}$ is the restriction of each $r\in\mathbb{R}$ to only its
first $p_{f}(|\vec{x}|)$ many bits. Now, since $f$ can only call a fixed
number of oracles from $\mathbb{R}$, we can collect these finitely many
polynomial-length prefixes into a single advice string for computation in
$\mathbf{FP}/\mathit{poly}$.
### 4.2 A relativised Bellantoni-Cook characterisation of
$\mathbf{FP}/\mathit{poly}$
We shall employ the following writing conventions for the remainder of this
work. For a set of (single-sorted) functions $F$, let us write:
* •
$F_{1;0}$ for the set of two-sorted functions $f(\vec{x};)$ for each
$f(\vec{x})\in F$;
* •
$F_{1;1}$ for the set of two-sorted functions $f(\vec{x};\vec{y})$ for each
$f(\vec{x},\vec{y})\in F$.
Given a set $F$ of two-sorted functions, the algebra $\mathsf{B}(F)$ is
defined just like $\mathsf{B}$ but with additional initial (two-sorted)
functions $F$. Note that, since functions of $\mathsf{B}(F)$ are given by
finite programs, they can only depend on finitely many members of $F$.
###### Proposition 26.
$\mathbf{FP}/\mathit{poly}\subseteq\mathsf{B}(\mathbb{R}_{1;0})$
One natural way to prove this result would be to go via
$\mathbf{F}\mathbf{P}(\mathbb{R})$, in light of 25. Indeed Bellantoni
established foundational results relating $\mathbf{F}\mathbf{P}(R)$ and
versions of $\mathsf{B}(R)$, for $R$ a set of relations, in [3], but
unfortunately the sorting of the corresponding arguments is subtle and does
not immediately give the result we are after. For this reason we give a direct
proof, that nonetheless inlines some ideas from [3].
###### Proof 4.3 (Proof of 26).
Let $\mathbf{C}=(C_{n})_{n<\omega}$ be a circuit family with each $C_{n}$
taking $n$ inputs and having size $<p(n)$, for some (monotone) polynomial $p$.
We need to show that the language computed by $\mathbf{C}$ is also computed in
$\mathsf{B}(\mathbb{R}_{1;0})$.
First, let $\mathrm{Eval}(x,y)$ evaluate the circuit described by $x$ on the
input $y$. Since $\mathrm{Eval}\in\mathbf{F}\mathbf{P}$, we have as standard
(e.g. by [4, Lemma 3.2]) a function $\mathrm{Eval}(m;x,y)\in\mathsf{B}$ and a
monotone polynomial $q$ such that $|m|\geq
q(|x|,|y|)\implies\mathrm{Eval}(m;x,y)=\mathrm{Eval}(x,y)$. Now, in
particular, if $x$ is the description of some $C_{n}$ and $n=|y|$, then also
$|x|\leq p(|y|)$, and so $|m|\geq
q(p(|y|),|y|)\implies\mathrm{Eval}(m;x,y)=\mathrm{Eval}(x,y)$. Finally,
denoting
$\overset{n}{\overbrace{\mathsf{s}_{1}\ldots\mathsf{s}_{1}\phantom{|}}}0$ by
$1^{n}$, this means that we have
$\mathrm{Eval}(y;x):=\mathrm{Eval}(1^{q(p(|y|),|y|)};x,y)\in\mathsf{B}$, that
in particular evaluates, when $x$ describes $C_{|y|}$, the circuit $C_{|y|}$
on input $y$.
Now, let $c\in\mathbb{R}_{1;0}$ with $c(y,z;)=$ $|z|$th bit of
$C_{|y|}$.color=redcolor=redtodo: color=redA: read the description left-right,
with infinitely many trailing zeroes. We assume that circuits are coded in a
prefix-free way that the evaluator takes into consideration. We show that the
function $C(y,z;)=c(y,0;)\cdot c(y,1;)\cdot\cdots\cdot c(y,1^{|z|-1};)$ is in
$\mathsf{B}(c)$ by the following instance of safe recursion:
$\begin{array}[]{r@{\ = \ }l}C(y,0;)&0\\\
C(y,\mathsf{s}_{i}z;)&\mathsf{cond}(;c(y,z;),\mathsf{s}_{0}(;C(y,z;)),\mathsf{s}_{1}(;C(y,z;)))\end{array}$
So we have that $C(y;):=C(y,1^{p(|y|)};)$ computes the description of
$C_{|y|}$.color=redcolor=redtodo: color=redA: …up to some trailing 0s that the
evaluator ignores Now we can decide whether $y$ is accepted by $C_{|y|}$
simply by calling the function $\mathrm{Eval}(y;C(y;))\in\mathsf{B}(c)$.
It turns out that we also have the converse inclusion too. This will be
subsumed by our later results but we include it here for the sake of
completeness. The key is to establish a general form of Bellantoni and Cook’s
polymax bounding lemma to account for modulus of continuity as well as growth:
###### Lemma 27 (Relational bounding lemma for $\mathsf{B}$).
Let $R$ be a set of two-sorted relations, and suppose
$f(R)(\vec{x};\vec{y})\in\mathsf{B}(R)$. Then there is a polynomial $p_{f}$
such that, setting $m_{f}(\vec{m},\vec{n}):=p_{f}(\vec{m})+\max\vec{n}$, we
have:
* •
(Polynomial modulus of growth)
$|f(R)(\vec{x};\vec{y})|<m_{f}(|\vec{x}|,|\vec{y}|)$
* •
(Polynomial modulus of continuity)
$f(R)(\vec{x};\vec{y})=f(\lambda|\vec{u}|,|\vec{v}|<m_{f}(|\vec{x}|,|\vec{y}|).r(\vec{u};\vec{v}))_{r\in
R}(\vec{x};\vec{y})$
Using this we may establish:
###### Proposition 28 (E.g. see [3]).
Let $R$ be a set of relations.
$\mathsf{B}(R_{1;1})\subseteq\mathbf{F}\mathbf{P}(R)$.
We shall not actually need this result directly in this work, rather
recovering (a version of) it from a more refined grand tour of inclusions.
However this does lead to the first ‘implicit’ characterisation of
$\mathbf{FP}/\mathit{poly}$ of this work:
###### Corollary 29.
$\mathsf{B}(\mathbb{R}_{1;0})=\mathbf{FP}/\mathit{poly}$
## 5 $\mathbf{FP}/\mathit{poly}\subseteq\mathsf{nu}\mathsf{B}$ via
relativised circular systems
In this section we establish one direction of 23. In particular, by the end of
this section, we will have established the following inclusions,
$\mathbf{FP}/\mathit{poly}\subseteq\mathsf{B}(\mathbb{R}_{1;0})\subseteq\mathsf{C}\mathsf{B}(\mathbb{R}_{1;0})\subseteq\mathsf{nu}\mathsf{B}$
where $\mathsf{C}\mathsf{B}(F)$ is an extension of $\mathsf{C}\mathsf{B}$ by
new initial sequents for two-sorted functions in $F$.
### 5.1 Relativised simulation of $\mathsf{B}$ in $\mathsf{C}\mathsf{B}$
We shall consider ‘relativised’ versions of $\mathsf{C}\mathsf{B}$, that may
include new initial sequents. Formally:
###### Definition 30.
Let $F$ be a set of two-sorted functions. A $\mathsf{B}^{-}(F)$-coderivation
is just a usual $\mathsf{B}^{-}$-coderivation that may use initial sequents of
the form $\scriptstyle f\;$ ${\Box N}^{n_{i}},N^{m_{i}}\Rightarrow N$ , when
$f\in F$ takes $n_{i}$ normal and $m_{i}$ safe inputs. We write
$\mathsf{C}\mathsf{B}(F)$ for the set of $\mathsf{C}\mathsf{B}$-coderivations
allowing initial sequents from $F$. The semantics of such coderivations and
the notion of $\mathsf{C}\mathsf{B}(F)$-definability are as expected, with
$f_{\mathcal{D}(F)}$ denoting the induced interpretation of
$\mathcal{D}(F)\in\mathsf{C}\mathsf{B}(F)$.
Note, again, that since $\mathsf{C}\mathsf{B}(F)$ coderivations are regular,
they only depend on finitely many members of $F$. By a modular extension of
the result that $\mathsf{B}\subseteq\mathsf{C}\mathsf{B}$ from [11], we
obtain:
###### Proposition 31.
Let $F$ be a set of two-sorted functions.
$\mathsf{B}(F)\subseteq\mathsf{C}\mathsf{B}(F)$.
The proof is simply by structural induction on the definition of a
$\mathsf{B}(F)$ function, where the recursion cases are handled by circularity
as in [11]. In particular, if $f$ is defined by safe recursion on notation
from $g,h_{0},h_{1}$ then the corresponding
$\mathsf{C}\mathsf{B}$-coderivation is given by:
$\textstyle{\scriptstyle g}$ $\Gamma\Rightarrow N$ $\vdots$
$\scriptstyle\mathsf{cond}_{\Box}\;$ $\;\bullet$ ${\Box N},\Gamma\Rightarrow
N$ $\textstyle{\scriptstyle h_{i}}$ ${\Box N},\Gamma,N\Rightarrow N$
$\scriptstyle\mathsf{cut}_{N}\;$ ${\Box N},\Gamma\Rightarrow N$
$\scriptstyle\mathsf{cond}_{\Box}\;$
$\;\bullet\quad{\text{\scriptsize$i=0,1$}}$ ${\Box N},\Gamma\Rightarrow N$
The only new cases in the induction are for an initial function from $F$,
which is simply translated into the appropriate initial sequent.
### 5.2 Simulating $\mathbb{R}_{1;0}$ oracles in $\mathsf{nu}\mathsf{B}$
In this subsection we shall establish:
###### Proposition 32.
$\mathsf{C}\mathsf{B}(\mathbb{R}_{1;0})\subseteq\mathsf{nu}\mathsf{B}$.
By definition of $\mathsf{nu}\mathsf{B}$ and $\mathsf{C}\mathsf{B}$, it
suffices to only consider the new initial sequents from $\mathbb{R}_{1;0}$.
For this we simply appeal to the following lemma:
###### Lemma 33.
For each $r(\vec{x};)\in\mathbb{R}_{1;0}$, there is a
$\mathsf{nu}\mathsf{B}$-coderivation defining it, in particular using only the
rules $0,1,|\mathsf{cond}|_{\Box}$.
###### Proof 5.1.
We proceed by induction on the length of $\vec{x}$. When the list is empty,
then $r(;)$ is just a Boolean, in which case we can derive it with just the
$0$ or $1$ step. Now, for $r(x,\vec{x};)$ we have:
IH($r_{0}$) $\Box\vec{N}\Rightarrow N$ IH($r_{1}$) $\Box\vec{N}\Rightarrow N$
$\quad\vdots\quad$ $\scriptstyle|\mathsf{cond}|_{\Box}\;$ $\underline{{\Box
N}},\Box\vec{N}\Rightarrow N$ $\scriptstyle|\mathsf{cond}|_{\Box}\;$
$\underline{{\Box N}},\Box\vec{N}\Rightarrow N$
where $r_{i}(\vec{x})$ is the function $r(1^{i},\vec{x})$ and the
coderivations marked IH($r_{i}$) are obtained by the inductive hypothesis for
$r_{i}$.
Note that, by putting together 26, 31 (setting $F=\mathbb{R}_{1;0}$) and 32,
we now have one half of our main result:
###### Corollary 34.
$\mathbf{FP}/\mathit{poly}\subseteq\mathsf{nu}\mathsf{B}$
## 6 $\mathsf{nu}\mathsf{B}$ as relativised regular coderivations
To facilitate the other direction of 23, let us first address a form of
converse to 32 above, that duly embeds $\mathsf{nu}\mathsf{B}$ into a
relativised circular system, which we shall rely on in the next section:
###### Theorem 35.
$\mathsf{nu}\mathsf{B}\subseteq\mathsf{C}\mathsf{B}(\mathbb{R}_{1;1})$.
Before giving the proof, we need to first establish some intermediate results:
###### Lemma 36.
If $\mathcal{D}$ is progressing and
$\\{\mathsf{s}_{0},\mathsf{s}_{1},\mathsf{id}\\}$-free then $f_{\mathcal{D}}$
is a relation, i.e. $f_{\mathcal{D}}(\vec{x};\vec{y})\leq 1$.
###### Proof 6.1 (Proof sketch).
We proceed by contradiction, always assuming 16, that progressing
coderivations compute total functions.
If $f_{\mathcal{D}}(\vec{x};\vec{y})>1$ then we argue that there is an
immediate sub-coderivation $\mathcal{D}^{\prime}$ of $\mathcal{D}$ and
arguments $\vec{x}^{\prime},\vec{y}^{\prime}$ such that
$f_{\mathcal{D}^{\prime}}(\vec{x}^{\prime};\vec{y}^{\prime})>1$. Some of the
critical cases are:
* •
If $\mathcal{D}=\mathsf{cut}_{N}(\mathcal{D}_{0},\mathcal{D}_{1})$ then
$f_{\mathcal{D}}(\vec{x};\vec{y})=f_{\mathcal{D}_{1}}(\vec{x};\vec{y},f_{\mathcal{D}_{0}}(\vec{x};\vec{y}))$,
and we set $\mathcal{D}^{\prime}:=\mathcal{D}_{1}$,
$\vec{x}^{\prime}:=\vec{x}$ and
$\vec{y}^{\prime}:=\vec{y},f_{\mathcal{D}_{0}}(\vec{x};\vec{y})$ (since
$f_{\mathcal{D}_{0}}(\vec{x};\vec{y})$ is well-defined). The case for
$\mathsf{cut}_{\Box}$ is similar.
* •
If
$\mathcal{D}=\mathsf{cond}_{\Box}(\mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2})$
then $\vec{x}=x,\vec{z}$ with
$f_{\mathcal{D}}(0,\vec{z};\vec{y})=\mathcal{D}_{0}(\vec{z};\vec{y})$ and
$f_{\mathcal{D}}(\mathsf{s}_{i}x^{\prime},\vec{z};\vec{y})=f_{\mathcal{D}_{i+1}}(x^{\prime},\vec{z};\vec{y})$.
If $x=0$ we set $\mathcal{D}^{\prime}:=\mathcal{D}_{0}$,
$\vec{x}^{\prime}:=\vec{z}$, and $\vec{y}^{\prime}:=\vec{y}$. If
$x=\mathsf{s}_{i}x^{\prime}$ then we set
$\mathcal{D}^{\prime}:=\mathcal{D}_{i+1}$,
$\vec{x}^{\prime}:=x^{\prime},\vec{z}$, and $\vec{y}^{\prime}:=\vec{y}$. The
cases for $\mathsf{cond}_{N},|\mathsf{cond}|_{N}$, and
$|\mathsf{cond}|_{\Box}$ are similar.
* •
In all other cases $\mathcal{D}$ ends with a unary rule so that
$\mathcal{D}^{\prime}$ is the only immediate sub-coderivation, and
$\vec{x}^{\prime},\vec{y}^{\prime}$ are determined by the semantics of the
rule (cf. Figure 2).
Note that, in the absence of $\mathsf{s}_{0},\mathsf{s}_{1}$, we indeed have
that $f^{\prime}(\vec{x}^{\prime};\vec{y}^{\prime})=f(\vec{x};\vec{y})>1$ so
we may continually apply this process to build up a branch
$B=({\mathcal{D}=\mathcal{D}^{(0)}},\mathcal{D}^{(1)},\mathcal{D}^{(2)},\dots)$
and arguments
${(\vec{x};\vec{y})=(\vec{x}^{(0)};\vec{y}^{(0)})},(\vec{x}^{(1)};\vec{y}^{(1)}),(\vec{x}^{(2)};\vec{y}^{(2)}),\dots$
such that
$f_{\mathcal{D}^{(k)}}(\vec{x}^{(k)};\vec{y}^{(k)})=f_{\mathcal{D}}(\vec{x};\vec{y})>1$.
Observe that $B$ cannot end at an $\mathsf{id}$ step, by assumption that
$\mathcal{D}$ is $\mathsf{id}$-free. Also, if $B$ ends at a $0$ or $1$ step we
have by construction that $f_{\mathcal{D}}(\vec{x};\vec{y})\in\\{0,1\\}$, a
contradiction. Thus $B$ must be infinite. Since $\mathcal{D}$ is progressing
there is a progressing thread along $B$, say $({\Box N}^{i})_{i\geq k}$, where
each ${\Box N}^{i}$ is an occurrence of ${\Box N}$. Let us examine the values,
say $x^{i}$, assigned to each ${\Box N}^{i}$. Notice that:
* •
by inspection of the rules and their interpretations from 11, we have that
$x^{i+1}\leq x^{i}$; and,
* •
if ${\Box N}^{i}$ is principal for a $\mathsf{cond}_{\Box}$ or a
$|\mathsf{cond}|_{\Box}$ step then $x^{i+1}<x^{i}$.
If follows that $(x^{i})_{i\geq k}$ is a non-increasing sequence of natural
numbers that does not converge, contradicting the well-ordering property of
$\mathbb{N}$.
###### Lemma 37.
If $\mathcal{D}$ is
$\\{\mathsf{cond}_{\Box},\mathsf{cond}_{N},\mathsf{id}\\}$-free,
$|\vec{x}|=|\vec{x}\textquoteright|$ and $|\vec{y}|=|\vec{y}\textquoteright|$,
then
$f_{\mathcal{D}}(\vec{x};\vec{y})=f_{\mathcal{D}}(\vec{x}^{\prime};\vec{y}^{\prime})$,
whenever $f(\vec{x};\vec{y})$ is well-defined.
###### Proof 6.2 (Proof sketch).
Being given by an equational program, we have that
$f_{\mathcal{D}}(\vec{x};\vec{y})=m$ has a (finite) equational derivation for
some $m\in\mathbb{N}$, by assumption that it is well-defined (cf. 12).
Replacing $\vec{x},\vec{y}$ by $\vec{x}^{\prime},\vec{y}^{\prime}$ in this
derivation yields $f_{\mathcal{D}}(\vec{x}^{\prime};\vec{y}^{\prime})=m$ too.
The only critical cases are the steps $|\mathsf{cond}|_{N}$,
$|\mathsf{cond}|_{\Box}$, whose semantics only depend on the length of their
arguments.
Putting the two above Lemmata together we have:
###### Proposition 38.
If $\mathcal{D}$ is progressing and
$\\{\mathsf{cond}_{\Box},\mathsf{cond}_{N},\mathsf{s}_{0},\mathsf{s}_{1},\mathsf{id}\\}$-free,
then $f_{\mathcal{D}}\in\mathbb{R}_{1;1}$.
Now we can prove 35:
###### Proof 6.3 (Proof sketch).
Let $\mathcal{D}$ be a $\mathsf{nu}\mathsf{B}$-coderivation and let $V$ be the
set of minimal nodes $\nu$ such that $\mathcal{D}_{\nu}$ is
$\\{\mathsf{cond}_{\Box},\mathsf{cond}_{N},\mathsf{s}_{0},\mathsf{s}_{1},\mathsf{id}\\}$-free,
and so by 38 we have that each $f_{\mathcal{D}_{\nu}}\in\mathbb{R}_{1;1}$.
Now, let $\mathcal{D}^{V}$ be obtained from $\mathcal{D}$ by simply deleting
each sub-coderivation $\mathcal{D}_{\nu}$, for $\nu\in V$, and construing each
of their conclusions as new initial sequents. By definition of
$\mathsf{nu}\mathsf{B}$, note that $\mathcal{D}^{V}$ is now a coderivation in
$\mathsf{C}\mathsf{B}(f_{\mathcal{D}_{\nu}})_{\nu\in
V}\subseteq\mathsf{C}\mathsf{B}(\mathbb{R}_{1;1})$ and we are done.
## 7 $\mathsf{nu}\mathsf{B}\subseteq\mathbf{FP}/\mathit{poly}$: a relativised
algebra subsuming circular typing
The final part of our chain of inclusions requires us to translate
(relativised) circular coderivations into an appropriate function algebra. The
idea is that, in the presence of safety, one can reduce circularity to a form
of recursion on ‘permutations of prefixes’ that nonetheless remains feasible.
This was (one of) the main result(s) of [11] and, fortunately, we are able to
import those results accounting only for additional initial relations.
### 7.1 Safe recursion on permutations of prefixes
Let us write $\vec{x}\subseteq\vec{y}$ if $\vec{x}$ is a permutation of
prefixes of $\vec{y}$, i.e. $\vec{x}=x_{0},\dots,x_{n-1}$ and
$\vec{y}=y_{0},\dots,y_{n-1}$ and there is a permutation $\pi:[n]\to[n]$ s.t.
each $x_{i}$ is a prefix of $y_{\pi i}$. We shall write
$\vec{x}\subset\vec{y}$ if for at least one $i<n$ we have that $x_{i}$ is a
strict prefix of $y_{\pi i}$. Note in particular that $\subset$ is a well-
founded pre-order so admits an induction and recursion principles.
To formulate recursion over well-founded relations it is convenient to employ
(two-sorted) oracles as placeholders for recursive calls. Due to the necessary
constraints on composition, we shall formally distinguish these oracles
(metavariables, $a,b,$ etc.) from additional initial functions (metavariables
$f,g$ etc. until now).
###### Definition 39.
Let $F$ be a set of two-sorted functions. The algebra
$\mathsf{B}^{\subset}(F,\vec{a})$ is the smallest class of two-sorted
functions containing,
* •
all the initial functions of $\mathsf{B}$;
* •
each (two-sorted) function $a_{i}$ among $\vec{a}$;
* •
each (two-sorted) function $f\in F$;
and closed under:
* •
(Relativised safe composition)
* –
if $g(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(F,\vec{a})$ and
$h(\vec{x};\vec{y},y)\in\mathsf{B}^{\subset}(F,\vec{a})$ then
${f(\vec{x};\vec{y}):=h(\vec{x};\vec{y},g(\vec{x};\vec{y}))\in\mathsf{B}^{\subset}(F,\vec{a})}$;
* –
if $g(\vec{x};)\in\mathsf{B}^{\subset}(F)$ and
$h(\vec{x},x;\vec{y})\in\mathsf{B}^{\subset}(F,\vec{a})$ then
${f(\vec{x};\vec{y}):=h(\vec{x},g(\vec{x};);\vec{y})\in\mathsf{B}^{\subset}(F,\vec{a})}$;
* •
(Safe recursion on $\subset$)
if $h(a)(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(F,a,\vec{a})$ then
${f(\vec{x};\vec{y}):=h(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{u};\vec{v}))(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(F,\vec{a})}$.
To be clear, the ‘guarded’ abstraction notation above is formally defined as
$\left(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{u};\vec{v})\right)(\vec{u}^{\prime};\vec{v}^{\prime})\
:=\
\begin{cases}f(\vec{u}^{\prime};\vec{v}^{\prime})&\vec{u}^{\prime}\subset\vec{x},\vec{v}^{\prime}\subseteq\vec{y}\\\
0&\text{otherwise}\end{cases}$
$\mathsf{B}^{\subset}(\varnothing,\vec{a})$ is the same as the notion
$\mathsf{B}^{\subset}(\vec{a})$ from [11]. Note in particular the distinction
between $F$ and $\vec{a}$ in the safe composition scheme: when composing along
a normal parameter (second line), the function $g(\vec{x};)$ must not contain
any oracles among $\vec{a}$.
Adapting the _Bounding Lemma_ from [11, Lemma 38] to account for further
initial relations gives:
###### Lemma 40 (Relational Bounding lemma).
Let $R$ be a set of two-sorted relations and
$f(R)(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(R)$. There is a polynomial
$p_{f}(\vec{n})$ such that, writing
$m_{f}(\vec{x},\vec{y})=p_{f}(|\vec{x}|)+\max|\vec{y}|$, we have:
* •
$|f(R)(\vec{x};\vec{y})|<m_{f}(\vec{x},\vec{y})$
* •
$f(R)(\vec{x};\vec{y})=f(\lambda|\vec{u}_{r}|,|\vec{v}_{r}|<m_{f}(\vec{x},\vec{y}).r(\vec{u}_{r};\vec{v}_{r}))_{r\in
R}(\vec{x};\vec{y})$
The first point is common to implicit complexity, being essentially Bellantoni
and Cook’s ‘polymax bounding lemma’ from [4]. The second point expresses a
dual property: while the first bounds the modulus of _growth_ , the second
bounds the modulus of _continuity_.
Here it is important the the new initial functions are relations, or at least
that they have constant/limited growth rate. In fact, for the proof, cf. [11],
one needs a more complicated statement accounting for growth properties of the
intermediate oracles $\vec{a}$ used for recursion, even though we only
ultimately need the statement above for our purposes, once all such oracles
$\vec{a}$ are ‘discharged’.
Using the Bounding Lemma we have from [11] (again accounting for further
initial relations) the main characterisation result for
$\mathsf{B}^{\subset}$:
###### Proposition 41 (Relativised characterisation).
For a set $R$ of relations,
$\mathsf{B}^{\subset}(R_{1;1})\subseteq\mathbf{F}\mathbf{P}(R)$.
The main point for proving this result is that the graph of $\subseteq$ is
relatively small, in particular for each $\vec{y}$ there are only polynomially
many $\vec{x}\subseteq\vec{y}$.111Of course, this polynomial depends on the
length of $\vec{y}$, but for a given function of the algebra this is some
global constant. So we can calculate a function
$f(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(R)$ simply by polynomial-time
induction (at the meta level) on $\subset$, storing all previous values in a
lookup table. This table will have only polynomially many entries, by the
previous observation about the size of the graph of $\subseteq$, and each
entry will have only polynomial size by the Bounding Lemma.
### 7.2
$\mathsf{C}\mathsf{B}(\mathbb{R}_{1;1})\subseteq\mathsf{B}^{\subset}(\mathbb{R}_{1;1})$:
from circular proofs to recursive functions
The point of $\mathsf{B}^{\subset}$ in [11] was to play the role of a target
algebra to translate circular coderivations into. The _Translation Lemma_ from
that work [11, Lemma 47], accounting for further initial relations, gives:
###### Lemma 42 (Relativised translation).
Let $R$ be a set of relations.
$\mathsf{C}\mathsf{B}(R_{1;1})\subseteq\mathsf{B}^{\subset}(R_{1;1})$.
Note that we specialise the statement above only to sets of relations to avoid
size issues potentially caused be new initial funtions of arbitrary growth
rate. In fact, this proof requires closure of
$\mathsf{B}^{\subset}(F,\vec{a})$ under a _simultaneous_ version of its
recursion scheme, upon which a careful translation from circular coderivations
in ‘cycle normal form’ (see, e.g., [8, Definition 6.2.1]) to an equational
specification can be duly resolved in $\mathsf{B}^{\subset}$.
Now by setting $R=\mathbb{R}$, we have the following consequence of 41:
###### Corollary 43.
$\mathsf{nu}\mathsf{B}\subseteq\mathbf{FP}/\mathit{poly}$
###### Proof 7.1.
We have $\mathsf{nu}\mathsf{B}\subseteq\mathsf{C}\mathsf{B}(\mathbb{R}_{1;1})$
by 35,
$\mathsf{C}\mathsf{B}(\mathbb{R}_{1;1})\subseteq\mathsf{B}^{\subset}(\mathbb{R}_{1;1})$
by 42,
$\mathsf{B}^{\subset}(\mathbb{R}_{1;1})\subseteq\mathbf{F}\mathbf{P}(\mathbb{R})$
by 41, and finally
$\mathbf{F}\mathbf{P}(\mathbb{R})\subseteq\mathbf{FP}/\mathit{poly}$ by 25.
Along with 34, we have now established both directions of our main result 23
that $\mathsf{nu}\mathsf{B}=\mathbf{FP}/\mathit{poly}$, hence completing the
proof.
## 8 Conclusions
In this work we presented the two-sorted non-wellfounded proof system
$\mathsf{nu}\mathsf{B}$ and proved that it characterises the complexity class
$\mathbf{FP}/\mathit{poly}$. Our results build on previous work [11], where we
defined the cyclic proof systems $\mathsf{C}\mathsf{B}$ and
$\mathsf{C}\mathsf{NB}$ capturing, respectively, $\mathbf{F}\mathbf{P}$ and
$\mathbf{F}\mathbf{ELEMENTARY}$ [11]. The system $\mathsf{nu}\mathsf{B}$ is
obtained from $\mathsf{C}\mathsf{B}$ by associating non-uniformity in
computation to a form of non-wellfoundedness in proof theory. To establish the
characterisation theorems, we also formalised some (presumably) folklore
results on relativised function algebras for $\mathbf{FP}/\mathit{poly}$.
For future research, the first author is investigating non-wellfounded
approaches to $\mathbf{FP}/\mathit{poly}$ in the setting of linear logic [20].
In particular, we are studying a non-wellfounded version of Mazza’s
Parsimonious Logic [29], a variant of linear logic where the exponential
modality $\oc$ satisfies Milner’s law ($\oc A\simeq\oc A\otimes A)$. This
provides a natural computational interpretation of formulas $!A$ as types of
streams on $A$. Mazza showed in [30] that Parsimonious Logic can be used to
capture $\mathbf{P}/\mathit{poly}$ using _wellfounded_ proofs that are
essentially _infinitely branching_. We conjecture that a similar
characterisation can be obtained in a non-wellfounded (and finitely branching)
setting, using ideas from this work.
Another direction is to explore applications of the results of this paper to
probabilistic complexity. In particular, we aim to study fragments of
$\mathsf{nu}\mathsf{B}$ modelling the class $\mathbf{BPP}$ (bounded-error
probabilistic polynomial time), essentially by leveraging on well-known
derandomisation methods showing the inclusion of $\mathbf{BPP}$ in
$\mathbf{FP}/\mathit{poly}$, and hence in $\mathbf{F}\mathbf{P}(\mathbb{R})$
(see 25). A challenging aspect of this task is to obtain characterisation
results that are entirely in the style of ICC, since $\mathbf{BPP}$ is defined
by explicit (error) bounds, as observed in [26]. We suspect that
$\mathsf{nu}\mathsf{B}$ represents the right framework for investigating fully
implicit characterisations of this class, where additional proof-theoretic
conditions can be introduced to restrict computationally the access to oracles
and, consequently, to model bounded-error probabilistic computation.
As [11] also established a system $\mathsf{C}\mathsf{NB}$ for
$\mathbf{F}\mathbf{ELEMENTARY}$, it would be pertinent to ask whether ideas in
this work can be applied to $\mathsf{C}\mathsf{NB}$ to characterise
$\mathbf{F}\mathbf{ELEMENTARY}/\mathit{poly}$, i.e., the class of functions
computable in elementary time by a Turing machine with access to a polynomial
advice. Unfortunately, the modulus of continuity established for
$\mathsf{C}\mathsf{NB}$ in [11] is super-polynomial (indeed elementary),
meaning that the same technique, a priori, would not restrict computation to
only polynomial advice. Consideration of this issue is left to future
research.
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## Appendix A Some properties of progressing $\mathsf{B}^{-}$-coderivations
established in [11]
Proof of 16. We proceed by contradiction. If $f_{\mathcal{D}}$ is non-total
then, since each rule preserves totality top-down, we must have that
$f_{\mathcal{D}^{\prime}}$ is non-total for one of $\mathcal{D}$’s immediate
sub-coderivations $\mathcal{D}^{\prime}$. Continuing this reasoning we can
build an infinite leftmost ‘non-total’ branch
$B=(\mathcal{D}^{i})_{i<\omega}$. Let $({\Box N}^{i})_{i\geq k}$ be a
progressing thread along $B$, and assign to each ${\Box N}^{i}$ the least
natural number $n_{i}\in\mathbb{N}$ such that $f_{\mathcal{D}^{i}}$ is non-
total when $n_{i}$ is assigned to the type occurrence ${\Box N}^{i}$.
Now, notice that:
* •
$(n_{i})_{i\geq k}$ is monotone non-increasing, by inspection of the rules and
their interpretations from 6.
* •
$(n_{i})_{i\geq k}$ does not converge, since $({\Box N}^{i})_{i\geq k}$ is
progressing and so is infinitely often principal for $\mathsf{cond}_{\Box}$ or
$|\mathsf{cond}|_{\Box}$, where the value of $n_{i}$ must strictly decrease
(cf., again, 6).
This contradicts the well-ordering property of the natural numbers. ∎
###### Proposition 44 ([11]).
Given a progressing $\mathsf{B}^{-}$-coderivation
$\mathcal{D}:\Box\Gamma,\vec{N}\Rightarrow{\Box N}$, there is a progressing
$\mathsf{B}^{-}$-coderivation $\mathcal{D}^{*}:\Box\Gamma\Rightarrow{\Box N}$
such that:
$f_{\mathcal{D}}(\vec{x};\vec{y})=f_{\mathcal{D}^{*}}(\vec{x};).$
###### Proof A.1.
By progressiveness, any infinite branch contains a $\mathsf{cond}_{\Box}$-step
or a $|\mathsf{cond}|_{\Box}$-step, which have non-modal succedents. Thus
there is a set of $\mathsf{cond}_{\Box}$-occurrences and
$|\mathsf{cond}|_{\Box}$-occurrences that forms a bar across $\mathcal{D}$. By
König Lemma, the set of all nodes of $\mathcal{D}$ below this bar, say
$X_{\mathcal{D}}$, is finite. The proof now follows by induction on the
cardinality of $X_{\mathcal{D}}$. The case where the last rule of
$\mathcal{D}$ is an instance of
$\mathsf{id},0,1,\Box_{r},\mathsf{cond}_{N},\mathsf{cond}_{\Box},|\mathsf{cond}|_{N},|\mathsf{cond}|_{\Box}$,
or $\mathsf{srec}$ are trivial. If the last rule of $\mathcal{D}$ is an
instance of $\mathsf{e}_{N},\mathsf{e}_{\Box},\Box_{l},\mathsf{s}_{i}$, and
$\mathsf{w}_{\Box}$ then we apply the induction hypothesis. Let us now suppose
that $\mathcal{D}$ has been obtained from a derivation $\mathcal{D}_{0}$ by
applying an instance of $\mathsf{w}_{N}$. By induction hypothesis, there
exists a derivation $\mathcal{D}_{0}^{*}:\overset{n}{\overbrace{{{\Box
N}},{\ldots},{{\Box N}}}}\Rightarrow{\Box N}$ such that
$f_{\mathcal{D}_{0}}(\vec{x};\vec{y})=f_{\mathcal{D}_{0}^{*}}(\vec{x};)$.
Since
$f_{\mathcal{D}}(\vec{x};\vec{y},y)=f_{\mathcal{D}_{0}}(\vec{x};\vec{y})=f_{\mathcal{D}_{0}^{*}}(\vec{x};)$
we just set $\mathcal{D}^{*}=\mathcal{D}_{0}^{*}$. Suppose now that
$\mathcal{D}$ is obtained from two derivations $\mathcal{D}_{0}$ and
$\mathcal{D}_{1}$ by applying an instance of $\mathsf{cut}_{N}$. By induction
hypothesis, there exists $\mathcal{D}_{1}^{*}$ such that
$f_{\mathcal{D}_{1}}(\vec{x};\vec{y},y)=f_{\mathcal{D}_{1}^{*}}(\vec{x};)$.
Since
$f_{\mathcal{D}_{1}}(\vec{x};\vec{y},f_{\mathcal{D}_{0}}(\vec{x};\vec{y}))=f_{\mathcal{D}_{1}^{*}}(\vec{x};)$,
we set $\mathcal{D}^{*}=\mathcal{D}_{1}^{*}$. As for the case where the last
rule is $\mathsf{cut}_{\Box}$, by induction hypothesis, there exist
derivations $\mathcal{D}_{0}^{*}$ and $\mathcal{D}_{1}^{*}$ such that
$f_{\mathcal{D}_{0}}(\vec{x};\vec{y})=f_{\mathcal{D}_{0}^{*}}(\vec{x};)$ and
$f_{\mathcal{D}_{1}}(\vec{x},x;\vec{y})=f_{\mathcal{D}_{1}^{*}}(\vec{x},x;)$,
so that we define $\mathcal{D}^{*}$ as the derivation obtained from
$\mathcal{D}_{0}^{*}$ and $\mathcal{D}_{1}^{*}$ by applying the rule
$\mathsf{cut}_{\Box}$.
## Appendix B Simultaneous recursion schemes
For our main results, we will ultimately need that the function algebra
$\mathsf{B}^{\subset}(F)$ is closed under simultaneous versions of its
recursion scheme.
###### Definition 45 (Simultaneous schemes).
Let $F$ be a set of two-sorted functions. We define the scheme
$\mathsf{ssrec}_{\subset}$ as follows, for arbitrary
$\vec{a}=a_{1},\dots,a_{k}$:
* •
from $h_{i}(\vec{a})(\vec{x};\vec{y})$ over $\vec{a},\vec{b}$, $F$, for $1\leq
i\leq k$, define $f_{i}(\vec{x};\vec{y})$ over $\vec{b}$, $F$, for $1\leq
i\leq k$, by:
$f_{i}(\vec{x};\vec{y})=h_{i}((\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f_{j}(\vec{u};\vec{v}))_{1\leq
j\leq k})(\vec{x};\vec{y})$
###### Proposition 46.
If $\vec{f}(\vec{x};\vec{y})$ over $\vec{b}$,$F$ are obtained by applying
$\mathsf{ssrec}_{\subset}$ to
$\vec{h}(\vec{a})(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(\vec{a},\vec{b}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},F})$,
then also
$\vec{f}(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(\vec{b}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},F})$
###### Proof B.1.
In what follows we shall neglect relativisation to $F$, as these oracles play
no role in the proof. Let $f_{i}(\vec{x};\vec{y})$ and
$h_{i}(a_{1},\dots,a_{k})(\vec{x};\vec{y})$ be as given in 45, and temporarily
write $f_{j}^{\vec{x};\vec{y}}$ for
$\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f_{j}(\vec{u};\vec{v})$,
so we have:
$f_{i}(\vec{x};\vec{y})=h_{i}(f_{1}^{\vec{x};\vec{y}},\dots,f_{k}^{\vec{x};\vec{y}})(\vec{x};\vec{y})$
For $i\in\mathbb{N}$, let us temporarily write $\underline{i}$ for $i$ in
binary notation222In fact, any notation will do, but we pick one for
concreteness., and $\vec{i}$ for the list
$\underline{i},\underline{i+1},\dots,\underline{k},\underline{1},\underline{2},\dots,\underline{i-1}$
Note that, for all $i=1,\dots,k$, $\vec{i}$ is a permutation (in fact a
rotation) of $\underline{1},\dots,\underline{k}$.
Now, let $f(\vec{x};\vec{y},\vec{z})$ over oracles $\vec{b}$ be given as
follows:
$f(\vec{x};\vec{y},\vec{z}):=\begin{cases}h_{1}(f_{1}^{\vec{x};\vec{y}},\dots,f_{k}^{\vec{x};\vec{y}})(\vec{x};\vec{y})&\vec{z}=\vec{1}\\\
\vdots\\\
h_{k}(f_{1}^{\vec{x};\vec{y}},\dots,f_{k}^{\vec{x};\vec{y}})(\vec{x};\vec{y})&\vec{z}=\vec{k}\\\
0&\text{otherwise}\end{cases}$ (1)
Note that this really is a finite case distinction since each of the boundedly
many $\vec{i}$ has bounded size, both bounds depending only on $k$, and so is
computable in $\mathsf{B}^{-}$ over $\vec{h}$.
By definition, then, we have that
$f(\vec{x};\vec{y},\vec{i})=f_{i}(\vec{x};\vec{y})$. Moreover note that, for
each $j=1,\dots,k$, we have,
$\begin{array}[]{rcl}f_{j}^{\vec{x};\vec{y}}(\vec{u}^{\prime};\vec{v}^{\prime})&=&(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f_{j}(\vec{u};\vec{v}))(\vec{u}^{\prime};\vec{v}^{\prime})\\\
&=&(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{u};\vec{v},\vec{j}))(\vec{u}^{\prime};\vec{v}^{\prime})\\\
&=&(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y},\lambda\vec{w}\subseteq\vec{z}.f(\vec{u};\vec{v},\vec{w}))(\vec{u}^{\prime};\vec{v}^{\prime},\vec{j})\end{array}$
as long as $\vec{z}$ is some $\vec{i}$, so indeed (1) has the form,
$f(\vec{x};\vec{y},\vec{z})=h(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y},\lambda\vec{w}\subseteq\vec{z}.f(\vec{u};\vec{v},\vec{w}))(\vec{x};\vec{y})$
and $f(\vec{x};\vec{y},\vec{z})\in\mathsf{B}^{\subset}(\vec{b})$ by
$\mathsf{srec}_{\subset}$. Finally, since
$f_{i}(\vec{x};\vec{y})=f(\vec{x};\vec{y},\vec{i})$, we indeed have that each
$f_{i}(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(\vec{b})$.
## Appendix C Proof of 42
In this section we prove 42. In what follows, when clear from the context, we
shall simply write $\mathcal{D}$ in place of $\mathcal{D}(F)$ to facilitate
readability. We recall that oracles in a coderivation are initial sequents.
First, we observe that a regular coderivation (with oracles) can be naturally
seen as a finite tree with ‘backpointers’, a representation known as _cycle
normal form_ , cf. [7, 9].
###### Definition 47 (Cycle normal form).
Let $F$ be a set of two-sorted functions, and let $\mathcal{D}$ be a regular
$\mathsf{B}^{-}(F)$-coderivation. The _cycle normal form_ (or simply _cycle
nf_) of $\mathcal{D}$ is a pair $\langle\mathcal{D},{R}_{\mathcal{D}}\rangle$,
where ${R}_{\mathcal{D}}$ is a partial self-mapping on the nodes of
$\mathcal{D}$ whose domain of definition is denoted
$\mathit{Bud}(\mathcal{D})$ and:
1. (i)
every infinite branch of $\mathcal{D}$ contains some (unique)
$\nu\in\mathit{Bud}(\mathcal{D})$;
2. (ii)
if $\nu\in\mathit{Bud}(\mathcal{D})$ then both
${R}_{\mathcal{D}}(\nu)\sqsubset\nu$ and
$\mathcal{D}_{{R}_{\mathcal{D}}(\nu)}=\mathcal{D}_{\nu}$;
3. (iii)
for any two distinct nodes $\mu\sqsubset\nu$ strictly below
$\mathit{Bud}(\mathcal{D})$, $\mathcal{D}_{\mu}\neq\mathcal{D}_{\nu}$
We call any $\nu\in\mathit{Bud}(\mathcal{D})$ a _bud_ , and
${R}_{\mathcal{D}}(\nu)$ its _companion_. A _terminal_ node is either one of
the leaves of $\mathcal{D}$ (among which, the functions in $F$) or a bud. The
set of nodes of $\mathcal{D}$ bounded above by a terminal node is denoted
$T_{\mathcal{D}}$. Given a node $\nu\in T_{\mathcal{D}}$, we define
$\mathit{Bud}_{\nu}(\mathcal{D})$ as the restriction of buds to those above
$\nu$.
###### Remark 48.
The cycle normal form of a regular coderivation $\mathcal{D}$ always exists,
as by definition any infinite branch contains a node $\nu$ such that
$\mathcal{D}_{\nu}=\mathcal{D}_{\mu}$ for some node $\mu$ below $\nu$.
$\mathit{Bud}(\mathcal{D})$ is designed to consist of just the _least_ such
nodes, so that by construction the cycle normal form is unique. Note that
$\mathit{Bud}(\mathcal{D})$ must form an antichain: if
$\mu,\nu\in\mathit{Bud}(\mathcal{D})$ with $\mu\sqsubset\nu$, then
${R}_{\mathcal{D}}(\mu)\sqsubset\mu$ are below $\mathit{Bud}(\mathcal{D})$ but
we have $\mathcal{D}_{{R}_{\mathcal{D}}(\mu)}=\mathcal{D}_{\mu}$ by (ii)
above, contradicting the (iii).
Also, notice that any branch of $\mathcal{D}$ contains a leaf of
$T_{\mathcal{D}}$. Moreover, since $\mathit{Bud}(\mathcal{D})$ is an
antichain, the leaves of $T_{\mathcal{D}}$ defines a ‘bar’ across
$\mathcal{D}$, and so $T_{\mathcal{D}}$ is a finite tree.
The following proposition allows us to reformulate progressiveness, safety and
left-leaning conditions for cycle normal forms.
###### Proposition 49.
Let $F$ be two-sorted functions, and let $\mathcal{D}$ be a regular
$\mathsf{B}^{-}(F)$-coderivation with cycle nf
$\langle\mathcal{D},{R}_{\mathcal{D}}\rangle$. For any
$\nu\in\mathit{Bud}(\mathcal{D})$, the (finite) path $\pi$ from
${R}_{\mathcal{D}}(\nu)$ to $\nu$ satisfies:
1. 1.
if $\mathcal{D}$ is progressing, $\pi$ must contain the conclusion of an
instance of $\mathsf{cond}_{{\Box N}}$ or $|\mathsf{cond}|_{\Box}$;
2. 2.
if $\mathcal{D}$ is progressing and safe, $\pi$ cannot contain the conclusion
of $\mathsf{cut}_{{\Box N}}$, $\Box_{l}$, $\mathsf{w}_{\Box}$, and the
leftmost premise of $\mathsf{cond}_{\Box}$ and of $|\mathsf{cond}|_{\Box}$;
3. 3.
if $\mathcal{D}$ is a $\mathsf{C}\mathsf{B}(F)$-coderivation, $\pi$ cannot
contain the conclusion of $\mathsf{w}_{N}$, the leftmost premise of
$\mathsf{cond}_{N}$ and of $|\mathsf{cond}|_{N}$, and the rightmost premise of
$\mathsf{cut}_{N}$.
###### Proof C.1.
By definition of cycle nf, each path from ${R}_{\mathcal{D}}(\nu)$ to $\nu$ in
$\langle\mathcal{D},{R}_{\mathcal{D}}\rangle$ is contained in a branch of
$\mathcal{D}$ such that each rule instance in the former appears infinitely
many times in the latter. Hence:
1. (i)
if $\mathcal{D}$ is progressing, the path contains the conclusion of an
instance of $\mathsf{cond}_{{\Box N}}$ or $|\mathsf{cond}|_{\Box}$;
2. (ii)
if $\mathcal{D}$ is safe, the path cannot contain the conclusion of a
$\mathsf{cut}_{{\Box N}}$ rule;
3. (iii)
if $\mathcal{D}$ is left-leaning, the path cannot contain the rightmost
premise of a $\mathsf{cut}_{N}$ rule.
This shows point 1. Let us consider point 2. By point (ii), if $\mathcal{D}$
is safe then, going from a node $\mu$ of the path to each of its children
$\mu^{\prime}$, the number of modal formulas in the context of the
corresponding sequents cannot increase. Moreover, the only cases where this
number strictly decreases is when $\mu$ is the conclusion of $\Box_{l}$,
$\mathsf{w}_{{\Box N}}$, or when $\mu^{\prime}$ is the leftmost premise of
$\mathsf{cond}_{\Box}$ and of $|\mathsf{cond}|_{\Box}$. Since
${R}_{\mathcal{D}}(\nu)$ and $\nu$ must be labelled with the same sequent, all
such cases are impossible. As for point 3 we notice that, by point (iii) and
the above reasoning, if $\mathcal{D}$ is safe and left-leaning then, going
from a node $\mu$ of the path to each of its children $\mu^{\prime}$, the
number of non-modal formulas in the context of the corresponding sequents
cannot increase. Moreover, the only cases where this number strictly decreases
is when $\mu$ is the conclusion of $\mathsf{w}_{N}$, or when $\mu^{\prime}$ is
the leftmost premise of $\mathsf{cond}_{N}$ and of $|\mathsf{cond}|_{N}$.
Since ${R}_{\mathcal{D}}(\nu)$ and $\nu$ must be labelled with the same
sequent, all such cases are impossible.
In what follows we shall indicate circularities in cycle nfs explicitly by
extending $\mathsf{C}\mathsf{B}(F)$ with a new inference rule called
$\mathsf{dis}$:
$\Gamma\Rightarrow A$ $\scriptstyle\mathsf{dis}\;$ $\;X$ $\Gamma\Rightarrow A$
where $X$ is a finite set of nodes. In this presentation, we insist that each
companion $\nu$ of the cycle nf $\langle\mathcal{D},{R}_{\mathcal{D}}\rangle$
is always the conclusion of an instance of $\mathsf{dis}$, where $X$ denotes
the set of buds $\nu^{\prime}$ such that
${R}_{\mathcal{D}}(\nu^{\prime})=\nu$. This expedient will allow us to
formally distinguish cases when a node of $T_{\mathcal{D}}$ is a companion
from those where it is the conclusion of a standard rule of $\mathsf{B}^{-}$.
To facilitate the translation, we shall define two disjoint sets ${C}_{\nu}$
and ${O}_{\nu}$. Intuitively, given a cycle nf
$\langle\mathcal{D},{R}_{\mathcal{D}}\rangle$ and $\nu\in T_{\mathcal{D}}$,
${C}_{\nu}$ is the set of companions above $\nu$, while ${O}_{\nu}$ is the set
of buds whose companion is strictly below $\nu$.
###### Definition 50.
Let $F$ be a set of two-sorted functions, and let
$\langle\mathcal{D},{R}_{\mathcal{D}}\rangle$ be the cycle normal form of a
$\mathsf{B}^{-}(F)$-coderivation $\mathcal{D}$. We define the following two
sets for any $\nu\in T_{\mathcal{D}}$:
$\displaystyle{C}_{\nu}$
$\displaystyle:=\\{\mu\in{R}_{\mathcal{D}}(\mathit{Bud}_{\nu}(\mathcal{D}))\
|\ \nu\sqsubseteq\mu\\}$ $\displaystyle{O}_{\nu}$
$\displaystyle:=\\{\mu\in\mathit{Bud}_{\nu}(\mathcal{D}))\ |\
{R}_{\mathcal{D}}(\mu)\sqsubset\nu\\}$
We are now ready to prove 42, i.e.,
$\mathsf{C}\mathsf{B}(R_{1;1})\subseteq\mathsf{B}^{\subset}(R_{1;1})$ for $R$
a set of relations. Actually, we shall establish a stronger result, 51 below.
Given $\mathcal{D}\in\mathsf{C}\mathsf{B}(R_{1;1})$, the proof proceeds by
analysing each node $\nu_{0}\in T_{\mathcal{D}}$ and associates with it an
instance of the scheme $\mathsf{ssrec}_{\subset}$ (45) that simultaneously
defines the functions $\\{f_{\mathcal{D}_{\nu}}\ |\
\nu\in{C}_{\nu}\cup\\{\nu_{0}\\}\\}$, with the help of an additional set of
oracles $\\{f_{\mathcal{D}_{\mu}}\ |\ \mu\in{O}_{\nu_{0}}\\}$. When $\nu_{0}$
is the root of $\mathcal{D}$, note that ${O}_{\nu_{0}}=\varnothing$ and so the
function thus defined will be oracle-free. Thus we obtain an instance of
$\mathsf{ssrec}_{\subset}$ defining $f_{\mathcal{D}}$, and so
$f_{\mathcal{D}}\in\mathsf{B}^{\subset}(R_{1;1})$ by Proposition 46.
###### Lemma 51 (Translation Lemma, general version).
Let $R$ be a set of relations. Given
$\langle\mathcal{D},{R}_{\mathcal{D}}\rangle$ the cycle nf of a
$\mathsf{C}\mathsf{B}(R_{1;1})$-coderivation $\mathcal{D}$, and $\nu_{0}\in
T_{\mathcal{D}}$:
1. 1.
If ${O}_{\nu_{0}}=\varnothing$ then
$f_{\mathcal{D}_{\nu_{0}}}\in\mathsf{B}^{\subset}(R_{1;1})$.
2. 2.
If ${O}_{\nu_{0}}\neq\varnothing$ then
$\forall\nu\in{C}_{\nu_{0}}\cup\\{\nu_{0}\\}$:
$\displaystyle f_{\mathcal{D}_{\nu}}(\vec{x};\vec{y})$
$\displaystyle=h_{\nu}((\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{C}_{\nu_{0}}\cup{O}_{\nu_{0}}})(\vec{x};\vec{y})$
where:
1. (a)
$h_{\nu}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu_{0}}},(a_{\mu})_{\mu\in{C}_{\nu_{0}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$
and hence
$f_{\mathcal{D}_{\nu}}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu_{0}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$;
2. (b)
the order $\vec{u}\subseteq\vec{x}$ is strict if either
$\nu,\mu\in{C}_{\nu_{0}}$ or the path from $\nu$ to $\mu$ in
$\mathcal{D}_{\nu_{0}}$ contains the conclusion of an instance of
$\mathsf{cond}_{\Box}$ or an instance of $|\mathsf{cond}|_{\Box}$;
###### Proof C.2.
Points 1 and 2 are proven by simultaneous induction on the longest distance of
$\nu_{0}$ from a leaf of $T_{\mathcal{D}}$. Notice that in the following
situations only point 1 applies:
* •
$\nu$ is the conclusion of an initial sequent $r\in R$.
* •
$\nu$ is the conclusion of an instance of $\mathsf{id}$, $0$, or $1$;
* •
$\nu$ is the conclusion of an instance of $\mathsf{w}_{N}$,
$\mathsf{w}_{\Box}$, $\Box_{l}$ and $\mathsf{cut}_{{\Box N}}$;
In particular, the last two cases hold by Proposition 49.2-3, as it must be
that ${O}_{\nu^{\prime}}=\varnothing$ for any premise $\nu^{\prime}$ of
$\nu_{0}$. Let us discuss the case where $\nu_{0}$ is the conclusion of a
$\mathsf{cut}_{\Box}$ rule with premises $\nu_{1}$ and $\nu_{2}$. By induction
on point 1 we have
$f_{\mathcal{D}_{\nu_{1}}}(\vec{x};\vec{y}),f_{\mathcal{D}_{\nu_{2}}}(\vec{x},x;\vec{y})\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathsf{B}^{\subset}(R_{1;1})}$.
Since the conclusion of $\mathcal{D}_{\nu_{1}}$ has modal succedent, by
Proposition 44 there must be a coderivation $\mathcal{D}^{*}$ such that
$f_{\mathcal{D}^{*}}(\vec{x};)=f_{\mathcal{D}_{\nu_{1}}}(\vec{x};\vec{y})\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathsf{B}^{\subset}(R_{1;1})}$.
Hence, we define
$f_{\mathcal{D}_{\nu_{0}}}(\vec{x};\vec{y})=f_{\mathcal{D}_{\nu_{2}}}(\vec{x},f_{\mathcal{D}^{*}}(\vec{x};);\vec{y})\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathsf{B}^{\subset}(R_{1;1})}$.
Let us now consider point 2. If $\nu_{0}$ is the conclusion of a bud then
${O}_{\nu_{0}}=\\{\nu_{0}\\}$, ${C}_{\nu_{0}}=\varnothing$, and all points
hold trivially. The cases where $\nu_{0}$ is an instance of an initial sequent
$r\in R$, $\mathsf{w}_{N}$, $\mathsf{e}_{N}$, $\mathsf{e}_{\Box}$, $\Box_{r}$,
$\mathsf{s}_{0}$ or $\mathsf{s}_{1}$ are straightforward. Suppose that
$\nu_{0}$ is the conclusion of a $\mathsf{cond}_{\Box}$ step with premises
$\nu^{\prime}$, $\nu_{1}$, and $\nu_{2}$, and let us assume
${O}_{\nu_{1}}\neq\varnothing$, ${O}_{\nu_{2}}\neq\varnothing$. By Proposition
49.2 we have ${O}_{\nu^{\prime}}=\varnothing$, so that
$f_{\mathcal{D}_{\nu^{\prime}}}\in\mathsf{B}^{\subset}(R_{1;1})$ by induction
hypothesis on point 1. By definition,
${O}_{\nu_{0}}={O}_{\nu_{1}}\cup{O}_{\nu_{2}}$ and
${C}_{\nu_{0}}={C}_{\nu_{1}}\cup{C}_{\nu_{2}}$. Then, we set:
$\begin{array}[]{c}f_{\mathcal{D}_{\nu_{0}}}(x,\vec{x};\vec{y})=\mathsf{cond}(;x,f_{\mathcal{D}_{\nu^{\prime}}}(\vec{x};\vec{y}),f_{\mathcal{D}_{\nu_{1}}}(\mathsf{p}(x;),\vec{x};\vec{y}),f_{\mathcal{D}_{\nu_{2}}}(\mathsf{p}(x;),\vec{x};\vec{y}))\end{array}$
where $\mathsf{p}(x;)$ can be defined from $\mathsf{p}(;x)$ and projections.
By induction hypothesis on $\nu_{i}$:
$\begin{array}[]{rcl}f_{\mathcal{D}_{\nu_{i}}}(\mathsf{p}(x;),\vec{x};\vec{y})&=&h_{\nu_{i}}((\lambda
u,\vec{u}\subseteq\mathsf{p}(x;),\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(u,\vec{u};\vec{v}))_{\mu\in{C}_{\nu_{i}}\cup{O}_{\nu_{i}}})(\mathsf{p}(x;),\vec{x};\vec{y})\\\
&=&h_{\nu_{i}}((\lambda u,\vec{u}\subset
x,\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(u,\vec{u};\vec{v}))_{\mu\in{C}_{\nu_{i}}\cup{O}_{\nu_{i}}})(\mathsf{p}(x;),\vec{x};\vec{y})\end{array}$
hence
$\begin{array}[]{c}f_{\mathcal{D}_{\nu_{0}}}(x,\vec{x};\vec{y})=h_{\nu_{0}}((\lambda
u,\vec{u}\subset
x,\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(u,\vec{u};\vec{v}))_{\mu\in{C}_{\nu_{0}}\cup{O}_{\nu_{0}}})(x,\vec{x};\vec{y})\end{array}$
for some $h_{\nu_{0}}$. By IH,
$h_{\nu_{0}}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu_{0}}},(a_{\mu})_{\mu\in{C}_{\nu_{0}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$
and
$f_{\mathcal{D}_{\nu_{0}}}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu_{0}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$.
This shows point 2a. Point 2b is trivial.
Let us now consider the case where $\nu_{0}$ is an instance of
$\mathsf{cond}_{N}$, assuming ${O}_{\nu_{1}}\neq\varnothing$ and
${O}_{\nu_{2}}\neq\varnothing$. By Proposition 49.3 we have
${O}_{\nu^{\prime}}=\varnothing$, so that
$f_{\mathcal{D}_{\nu^{\prime}}}\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathsf{B}^{\subset}(R_{1;1})}$
by induction hypothesis on point 1. By definition,
${O}_{\nu_{0}}={O}_{\nu_{1}}\cup{O}_{\nu_{2}}$ and
${C}_{\nu_{0}}={C}_{\nu_{1}}\cup{C}_{\nu_{2}}$. Then, we set:
$\begin{array}[]{c}f_{\mathcal{D}_{\nu_{0}}}(\vec{x};y,\vec{y})=\mathsf{cond}(;y,f_{\mathcal{D}_{\nu^{\prime}}}(\vec{x};\vec{y}),f_{\mathcal{D}_{\nu_{1}}}(\vec{x};\mathsf{p}(;y),\vec{y}),f_{\mathcal{D}_{\nu_{2}}}(\vec{x};\mathsf{p}(;y),\vec{y}))\end{array}$
By induction hypothesis on $\nu_{i}$:
$\begin{array}[]{rcl}f_{\mathcal{D}_{\nu_{i}}}(\vec{x};\mathsf{p}(;y),\vec{y})&=&h_{\nu_{i}}((\lambda\vec{u}\subseteq
x,\vec{x},\lambda
v,\vec{v}\subseteq\vec{z}.f_{\mathcal{D}_{\mu}}(\vec{u};v,\vec{v}))_{\mu\in{C}_{\nu_{i}}\cup{O}_{\nu_{i}}})(\vec{x};\vec{z})\\\
&=&h_{\nu_{i}}((\lambda\vec{u}\subset\vec{x},\lambda v,\vec{v}\subset
y,\vec{y}.f_{\mathcal{D}_{\mu}}(\vec{u};v,\vec{v}))_{\mu\in{C}_{\nu_{i}}\cup{O}_{\nu_{i}}})(\vec{x};\vec{z})\end{array}$
where $\vec{z}=\mathsf{p}(;y),\vec{y}$, and hence
$\begin{array}[]{c}f_{\mathcal{D}_{\nu_{0}}}(\vec{x};y,\vec{y})=h_{\nu_{0}}((\lambda\vec{u}\subseteq\vec{x},\lambda
v,\vec{v}\subset
y,\vec{y}.f_{\mathcal{D}_{\mu}}(\vec{u};v,\vec{v}))_{\mu\in{C}_{\nu_{0}}\cup{O}_{\nu_{0}}})(\vec{x};y,\vec{y})\end{array}$
for some $h_{\nu_{0}}$. Point 2a and 2b are given by the induction hypothesis.
The cases for $|\mathsf{cond}|_{\Box}$ and $|\mathsf{cond}|_{N}$ are similar.
Let us now consider the case where $\nu_{0}$ is the conclusion of an instance
of $\mathsf{dis}$ with premise $\nu^{\prime}$, where $X$ is the set of nodes
labelling the rule. We have ${O}_{\nu_{0}}={O}_{\nu^{\prime}}-X$ and
${C}_{\nu_{0}}={C}_{\nu^{\prime}}\cup\\{\nu_{0}\\}$. We want to find
$(h_{\nu})_{\nu\in{C}_{\nu}\cup\\{\nu_{0}\\}}$ defining the equations for
$(f_{\mathcal{D}_{\nu}})_{\nu\in{C}_{\nu}\cup\\{\nu_{0}\\}}$ in such a way
that points 2a-2b hold. We shall start by defining $h_{\nu_{0}}$. First, note
that, by definition of cycle nf,
$f_{\mathcal{D}_{\nu_{0}}}(\vec{x};\vec{y})=f_{\mathcal{D}_{\nu^{\prime}}}(\vec{x};\vec{y})=f_{\mathcal{D}_{\mu}}(\vec{x};\vec{y})$
for all $\mu\in X$. By induction hypothesis on $\nu^{\prime}$ there exists a
family $(g_{\nu})_{\nu\in{C}_{\nu^{\prime}}\cup\\{\nu^{\prime}\\}}$ such that:
$\begin{array}[]{c}f_{\mathcal{D}_{\nu^{\prime}}}(\vec{x};\vec{y})=g_{\nu^{\prime}}((\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{C}_{\nu^{\prime}}\cup{O}_{\nu^{\prime}}})(\vec{x};\vec{y})\end{array}$
(2)
and, moreover, for all $\nu\in{C}_{\nu^{\prime}}$:
$\begin{array}[]{c}f_{\mathcal{D}_{\nu}}(\vec{x};\vec{y})=g_{\nu}((\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{C}_{\nu^{\prime}}\cup{O}_{\nu^{\prime}}})(\vec{x};\vec{y})\end{array}$
(3)
Since ${O}_{\nu^{\prime}}={O}_{\nu_{0}}\cup X$ and the path from
$\nu^{\prime}$ to any $\mu\in X$ must cross an instance of
$\mathsf{cond}_{\Box}$ by Proposition 49.1, the induction hypothesis on
$\nu^{\prime}$ (point 2b) allows us to rewrite (2) as follows:
$\begin{array}[]{rl}f_{\mathcal{D}_{\nu^{\prime}}}(\vec{x};\vec{y})=g_{\nu^{\prime}}(&(\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\nu}}(\vec{u};\vec{v}))_{\nu\in{C}_{\nu^{\prime}}},\\\
&(\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{O}_{\nu_{0}}},\\\
&(\lambda\vec{u}\subset\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in
X}\quad)(\vec{x};\vec{y})\end{array}$ (4)
On the other hand, for all $\nu\in\mathcal{C}_{\nu^{\prime}}$, for all
$\vec{u}\subseteq\vec{x}$ and $\vec{v}$, the equation in (3) can be rewritten
as:
$\begin{array}[]{rl}f_{\mathcal{D}_{\nu}}(\vec{u};\vec{v})=g_{\nu}(&(\lambda\vec{w}\subset\vec{u},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v^{\prime}}\subseteq\vec{v}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v^{\prime}}))_{\mu\in{C}_{\nu^{\prime}}},\\\
&(\lambda\vec{w}\subseteq\vec{u},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v^{\prime}}\subseteq\vec{v}}.f_{\mathcal{D}_{\mu}}(\vec{w};\vec{v^{\prime}}))_{\mu\in{O}_{\nu_{0}}},\\\
&(\lambda\vec{w}\subseteq\vec{u},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v^{\prime}}\subseteq\vec{v}}.f_{\mathcal{D}_{\mu}}(\vec{w};\vec{v^{\prime}}))_{\mu\in
X}\quad)(\vec{u};\vec{v})\end{array}$ (5)
and so, for all $\nu\in{C}_{\nu^{\prime}}$:
$\begin{array}[]{rl}\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda
v\subseteq\vec{y}}.f_{\mathcal{D}_{\nu}}(\vec{u};\vec{v})=\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda
v\subseteq\vec{y}}.g_{\nu}(&(\lambda\vec{w}\subset\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v^{\prime}}\subseteq\vec{v}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v^{\prime}}))_{\mu\in{C}_{\nu^{\prime}}},\\\
&(\lambda\vec{w}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v^{\prime}}\subseteq\vec{v}}.f_{\mathcal{D}_{\mu}}(\vec{w};\vec{v^{\prime}}))_{\mu\in{O}_{\nu_{0}}},\\\
&(\lambda\vec{w}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v^{\prime}}\subseteq\vec{v}}.f_{\mathcal{D}_{\mu}}(\vec{w};\vec{v^{\prime}}))_{\mu\in
X}\quad)(\vec{u};\vec{v})\end{array}$ (6)
Now, since the paths from $\nu^{\prime}$ to any $\mu\in X$ in $\mathcal{D}$
must contain an instance of $\mathsf{cond}_{\Box}$ or an instance of
$|\mathsf{cond}|_{\Box}$, for all $\nu\in{C}_{\nu^{\prime}}$ and all $\mu\in
X$, we have that either the path from $\nu^{\prime}$ to $\nu$ contains an
instance of $\mathsf{cond}_{\Box}$ or an instance of $|\mathsf{cond}|_{\Box}$
or the path from $\nu$ to $\mu$ does. By applying the induction hypothesis on
$\nu^{\prime}$ (point 2b), given $\nu\in{C}_{\nu^{\prime}}$ and $\mu\in X$,
either
$\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\nu}}(\vec{u};\vec{v})$
in (4) is such that $\vec{u}\subset\vec{x}$, or
$\lambda\vec{w}\subseteq\vec{u},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v^{\prime}}\subseteq\vec{v}}.f_{\mathcal{D}_{\mu}}(\vec{w};\vec{v^{\prime}})$
in (5) is such that $\vec{w}\subset\vec{u}$. This means that, for any $\mu\in
X$,
$\lambda\vec{w}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v^{\prime}}\subseteq\vec{v}}.f_{\mathcal{D}_{\mu}}(\vec{w};\vec{v^{\prime}})$
in (6) is such that $\vec{w}\subset\vec{x}$. For each
$\nu\in{C}_{\nu^{\prime}}$, by rewriting
$\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\nu}}(\vec{u};\vec{v})$
in (4) according to the equation in (6) we obtain:
$\begin{array}[]{c}f_{\mathcal{D}_{\nu_{0}}}(\vec{x};\vec{y})=t_{\nu_{0}}((\lambda\vec{u}\subset\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{C}_{\nu^{\prime}}\cup
X},(\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{O}_{\nu_{0}}})(\vec{x};\vec{y})\end{array}$
for some $t_{\nu_{0}}$. Since
$f_{\mathcal{D}_{\mu}}=f_{\mathcal{D}_{\nu_{0}}}$ for all $\mu\in X$, and
since ${C}_{\nu_{0}}={C}_{\nu^{\prime}}\cup\\{\nu_{0}\\}$, by setting
$h_{\nu_{0}}:=t_{\nu_{0}}$ the above equation gives us the following:
$\begin{array}[]{c}f_{\mathcal{D}_{\nu_{0}}}(\vec{x};\vec{y})=h_{\nu_{0}}((\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{C}_{\nu_{0}}\cup{O}_{\nu_{0}}})(\vec{x};\vec{y})\end{array}$
(7)
which satisfies point 2b. From (7) we are able to find the functions
$(h_{\nu})_{\nu\in{C}_{\nu}}$ defining the equations for
$(f_{\mathcal{D}_{\nu}})_{\nu\in{C}_{\nu}}$. Indeed, the induction hypothesis
on $\nu^{\prime}$ gives us (3) for any $\nu\in{C}_{\nu^{\prime}}$. We rewrite
in each such equation any
$\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq
y}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v})$ such that $\mu\in X$ according to
equation (7), as $f_{\mathcal{D}_{\mu}}=f_{\mathcal{D}_{\nu_{0}}}$ for any
$\mu\in X$. We obtain the following equation for any
$\nu\in{C}_{\nu^{\prime}}$:
$\begin{array}[]{c}f_{\mathcal{D}_{\nu}}(\vec{x};\vec{y})=t_{\nu}((\lambda\vec{u}\subset\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{C}_{\nu^{\prime}}\cup\\{\nu_{0}\\}},(\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{O}_{\nu_{0}}})(\vec{x};\vec{y})\end{array}$
for some $t_{\nu}$. Since ${C}_{\nu_{0}}={C}_{\nu^{\prime}}\cup\\{\nu_{0}\\}$
and the above equation satisfies point 2b, we set $h_{\nu}:=t_{\nu}$ and we
obtain, for all $\nu\in{C}_{\nu_{0}}$:
$f_{\mathcal{D}_{\nu}}(\vec{x};\vec{y})=h_{\nu}((\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{C}_{\nu_{0}}\cup{O}_{\nu_{0}}})(\vec{x};\vec{y})$
(8)
It remains to show that (7) and (8) satisfy point 2a. On the one hand for all
$\nu\in{C}_{\nu_{0}}$ we have
$h_{\nu}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu_{0}}},(a_{\mu})_{\mu\in{C}_{\nu_{0}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$,
with $(a_{\mu})_{\mu\in{C}_{\nu_{0}}}$ oracle functions. On the other hand, by
applying the induction hypothesis, we have
$f_{\mathcal{D}_{\nu_{0}}}=f_{\mathcal{D}_{\nu^{\prime}}}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu^{\prime}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$
and
$f_{\mathcal{D}_{\nu}}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu^{\prime}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$,
for all $\nu\in{C}_{\nu_{0}}$. Since ${O}_{\nu^{\prime}}={O}_{\nu_{0}}\cup X$
and $f_{\mathcal{D}_{\nu_{0}}}=f_{\mathcal{D}_{\mu}}$ for all $\mu\in X$, we
have both
$f_{\mathcal{D}_{\nu_{0}}}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu_{0}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$
and
$f_{\mathcal{D}_{\nu}}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu_{0}}},f_{\mathcal{D}_{\nu_{0}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$
for all $\nu\in{C}_{\nu_{0}}$, and hence
$f_{\mathcal{D}_{\nu}}\in\mathsf{B}^{\subset}((f_{\mathcal{D}_{\mu}})_{\mu\in{O}_{\nu_{0}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$,
for all $\nu\in{C}_{\nu_{0}}$.
Last, suppose that $\nu_{0}$ is the conclusion of an instance of
$\mathsf{cut}_{N}$ with premises $\nu_{1}$ and $\nu_{2}$ then:
$f_{\mathcal{D}_{\nu_{0}}}(\vec{x};\vec{y})=f_{\mathcal{D}_{\nu_{2}}}(\vec{x};f_{\mathcal{D}_{\nu_{1}}}(\vec{x};\vec{y}),\vec{y})$
By Proposition 49.3 we have ${O}_{\nu_{2}}=\varnothing$, so that
$f_{\mathcal{D}_{\nu_{2}}}(\vec{x};y,\vec{y})\in\mathsf{B}^{\subset}(R_{1;1})$
by applying the induction hypothesis on $\nu_{1}$. We shall only consider the
case where ${O}_{\nu_{1}}\neq\varnothing$. Then, ${O}_{\nu_{0}}={O}_{\nu_{1}}$
and ${C}_{\nu_{0}}={C}_{\nu_{1}}$. Moreover, induction hypothesis on
$\nu_{1}$:
$\begin{array}[]{c}f_{\mathcal{D}_{\nu_{1}}}(\vec{x};\vec{y})=h_{\nu_{1}}((\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};v,\vec{v}))_{\mu\in{C}_{\nu_{1}}\cup{O}_{\nu_{1}}})(\vec{x};\vec{y})\end{array}$
So that:
$f_{\mathcal{D}_{\nu_{0}}}(\vec{x};\vec{y})=h_{\nu_{0}}((\lambda\vec{u}\subseteq\vec{x},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\lambda\vec{v}\subseteq\vec{y}}.f_{\mathcal{D}_{\mu}}(\vec{u};\vec{v}))_{\mu\in{C}_{\nu_{0}}\cup{O}_{\nu_{0}}})(\vec{x};\vec{y})$
for some $h_{\nu_{0}}$. Points 2a and 2b hold by applying the induction
hypothesis.
## Appendix D Proof of 40
Let us employ the notation $||\vec{x}||:=\sum|\vec{x}|$ throughout this
section. We will prove the following stronger statement.
###### Lemma 52 (Bounding lemma, general version).
Let $f(\vec{a})(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(\vec{a},R)$, with
$\vec{a}=a_{1},\dots,a_{k}$ oracles and $R$ a set of relations. There is a
function $m_{f}^{\vec{c}}(\vec{x},\vec{y})$ with,
$m_{f}^{\vec{c}}(\vec{x},\vec{y})=p_{f}(||\vec{x}||)+\sum\vec{c}+\max|\vec{y}|$
for a monotone polynomial $p_{f}(n)$ such that whenever there are constants
$\vec{c}=c_{1},\dots,c_{k}$ such that,
$|a_{i}(\vec{x}_{i};\vec{y}_{i})|\leq c_{i}+\sum_{j\neq
i}c_{j}+\max|\vec{y}_{i}|$ (9)
for $1\leq i\leq k$, we have:333To be clear, here we write $|\vec{y}_{i}|\leq
m_{f}^{\vec{c}}(\vec{x},\vec{y})$ here as an abbreviation for $\\{|y_{ij}|\leq
m_{f}^{\vec{c}}(\vec{x},\vec{y})\\}_{j}$.
$|f(\vec{a})(\vec{x};\vec{y})|\leq m_{f}^{\vec{c}}(\vec{x},\vec{y})$ (10)
$\begin{array}[]{rcl}f(\vec{a})(\vec{x};\vec{y})&=&f\left(\lambda\vec{x}_{i}.\lambda|\vec{y}_{i}|\leq
m_{f}^{\vec{c}}(\vec{x},\vec{y}).a_{i}(\vec{x}_{i};\vec{y}_{i})\right)_{i}(\vec{x};\vec{y})\\\
f(\vec{a})(\vec{x};\vec{y})&=&f\left(\lambda\vec{x}_{r}.\lambda|\vec{y}_{r}|\leq
m_{f}^{\vec{c}}(\vec{x},\vec{y}).r(\vec{x}_{r};\vec{y}_{r})\right)_{r\in
R}(\vec{x};\vec{y})\end{array}$ (11)
###### Remark 53.
Notice that 40 is obtained by setting $\vec{a}=\varnothing$. Notice also that,
in the case when $f(\vec{x};\vec{y})$ is just, say, $a_{i}(\vec{x};\vec{y})$,
we may choose to set $\vec{a}=a_{i}$ or $\vec{a}=a_{1},\dots,a_{n}$ in the
above lemma, yielding different bounds in each case. We shall exploit this in
inductive hypotheses in the proof that follows (typically when we write
‘WLoG’).
###### Proof D.1.
We prove Equation 10 and Equation 11 by induction on the definition of
$f(\vec{x};\vec{y})$, always assuming that we have $\vec{c}$ satisfying
Equation 9.
Let us start with Equation 10. If $f(\vec{x};\vec{y})$ is an initial function
or $f\in R$ then it suffices to set $p_{f}(n):=1+n$.
If $f(\vec{x};\vec{y})=a_{i}(\vec{x};\vec{y})$ then it suffices to set
$p_{f}(n):=0$.
If $f(\vec{x};\vec{y})=h(\vec{x};\vec{y},g(\vec{x};\vec{y}))$, let $p_{h}$ and
$p_{g}$ be obtained from the inductive hypothesis. Notice that either $g$ or
$h$ does not have oracles in $\vec{a}$ by 39, so that:
$\begin{array}[]{rcl}|f(\vec{x};\vec{y})|&=&|h(\vec{x};\vec{y},g(\vec{x};\vec{y}))|\\\
&\leq&p_{h}(||\vec{x}||)+\sum\vec{c}+\max(|\vec{y}|,|g(\vec{x};\vec{y})|)\\\
&\leq&p_{h}(||\vec{x}||)+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{h}}\sum\vec{c}+p_{g}(||\vec{x}||)+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\delta_{g}}\sum\vec{c}+\max|\vec{y}|\end{array}$
where $\delta_{g}+\delta_{h}\in\\{0,1\\}$ such that $\delta_{g}+\delta_{h}=1$,
so we may set $p_{f}(n):=p_{h}(n)+p_{g}(n)$.
If $f(\vec{x};\vec{y})=h(\vec{x},g(\vec{x};);\vec{y})$, let $p_{h}$ and
$p_{g}$ be obtained from the inductive hypothesis. Note that, by definition of
Safe Composition along a normal parameter, we must have that $g$ has no
oracles in $\vec{a}$ by 39, and so in fact $|g(\vec{x};)|\leq
p_{g}(||\vec{x}||)$. We thus have,
$\begin{array}[]{rcl}|f(\vec{x};\vec{y})|&=&|h(\vec{x},g(\vec{x};);\vec{y})|\\\
&\leq&p_{h}(||\vec{x}||,|g(\vec{x};)|)+\sum\vec{c}+\max|\vec{y}|\\\
&\leq&p_{h}(||\vec{x}||,p_{g}(||\vec{x}||))+\sum\vec{c}+\max|\vec{y}|\end{array}$
so we may set $p_{f}(n):=p_{h}(n,p_{g}(n))$.
Finally, if
$f(\vec{x};\vec{y})=h(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{u};\vec{v}))(\vec{x};\vec{y})$,
let $p_{h}$ be obtained from the inductive hypothesis. We claim that it
suffices to set $p_{f}(n):=np_{h}(n)$. Notice that, by monotonicity:
$p_{f}(n)\geq p_{h}(n)+p_{f}(n-1)$ (12)
for any $n>0$.
Now, to show Equation 10 we proceed by a sub-induction on $||\vec{x}||$. For
the base case, when $||\vec{x}||=0$ (and so, indeed, $\vec{x}=\vec{0}$), note
simply that $\lambda\vec{u}\subset\vec{x},\lambda\vec{v}.f(\vec{u};\vec{v})$
is the constant function $0$, and so we may appeal to the main inductive
hypothesis for $h(a)$ setting the corresponding constant $c$ for $a$ to be $0$
to obtain,
$\begin{array}[]{rcl}|f(\vec{0};\vec{y})|&=&|h(0)(\vec{0};\vec{y})|\\\
&\leq&p_{h}(0)+\sum\vec{c}+\max|\vec{y}|\\\
&\leq&p_{f}(0)+\sum\vec{c}+\max|\vec{y}|\end{array}$
as required. For the sub-inductive step, let $||\vec{x}||>0$. Note that,
whenever $\vec{u}\subset\vec{x}$ we have $||\vec{u}||<||\vec{x}||$ and so, by
the sub-inductive hypothesis and monotonicity of $p_{f}$ we have:
$|f(\vec{u};\vec{v})|\leq p_{f}(||\vec{x}||-1)+\sum\vec{c}+\max|\vec{v}|$
Now we may again appeal to the main inductive hypothesis for $h(a)$ by setting
$c=p_{f}(||\vec{x}||-1)$ to be the corresponding constant for
$a=\lambda\vec{u}\subset\vec{x},\lambda\vec{v}.f(\vec{u};\vec{v})$. We thus
obtain:
$\begin{array}[]{rclll}|f(\vec{x};\vec{y})|&=&|h(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{u};\vec{v}))(\vec{x};\vec{y})|\\\
&\leq&p_{h}(||\vec{x}||)+c+\sum\vec{c}+\max|\vec{y}|&&\text{main IH}\\\
&\leq&(p_{h}(||\vec{x}||)+p_{f}(||\vec{x}||-1))+\sum\vec{c}+\max|\vec{y}|&&\text{def.\leavevmode\nobreak\
of $c$}\\\ &\leq&p_{f}(||\vec{x}||)+\sum\vec{c}+\max|\vec{y}|&&\eqref{eq:bnd-
lem-rec-fn-inv}\end{array}$
Let us now prove Equation 11, and let $p_{f}$ be constructed by induction on
$f$ as above. We proceed again by induction on the definition of
$f(\vec{a})(\vec{x};\vec{y})$, always making explicit the oracles of a
function.
The initial functions and oracle calls are immediate, due to the
‘$\max|\vec{y}|$’ term in Equation 11.
If
$f(\vec{a})(\vec{x};\vec{y})=h(\vec{a})(\vec{x};\vec{y},g(\vec{a})(\vec{x};\vec{y}))$
then, by the inductive hypothesis for $h(\vec{a})$, any oracle call from
$h(\vec{a})$ only takes safe inputs of lengths:
$\begin{array}[]{rll}\leq&p_{h}(||\vec{x}||)+\sum\vec{c}+\max(|\vec{y}|,|g(\vec{a})(\vec{x};\vec{y})|)\\\
\leq&p_{h}(||\vec{x}||)+\sum\vec{c}+p_{g}(||\vec{x}||)+\sum\vec{c}+\max|\vec{y}|&\text{\eqref{eq:elem-
const-max-bound}}\\\
\leq&(p_{h}(||\vec{x}||+p_{g}(||\vec{x}||))+\sum\vec{c}+\max|\vec{y}|\\\
\leq&p_{f}(||\vec{x}||)+\sum\vec{c}+\max|\vec{y}|\end{array}$
Note that any oracle call from $g(\vec{a})$ will still only take safe inputs
of lengths $\leq p_{g}(||\vec{x}||)+\sum\vec{c}+\max|\vec{y}|$, by the
inductive hypothesis, and $p_{g}$ is bounded above by $p_{f}$.
If
$f(\vec{a})(\vec{x};\vec{y})=h(\vec{a})(\vec{x},g(\varnothing)(\vec{x};);\vec{y})$
then, by the inductive hypothesis, any oracle call will only take safe inputs
of lengths:
$\begin{array}[]{rlll}\leq&p_{h}(||\vec{x}||+|g(\varnothing)(\vec{x};)|)+\sum\vec{c}+\max|\vec{y}|\\\
\leq&p_{h}(||\vec{x}||+p_{g}(||\vec{x}||))+\sum\vec{c}+\max|\vec{y}|&&\eqref{eq:elem-
const-max-bound}\\\
\leq&p_{f}(||\vec{x}||)+\sum\vec{c}+\max|\vec{y}|\end{array}$
Last, suppose
$f(\vec{a})(\vec{x};\vec{y})=h(\vec{a},\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{a})(\vec{u};\vec{v}))(\vec{x};\vec{y})$.
We proceed by a sub-induction on $||\vec{x}||$. Note that, since
$\vec{u}\subset\vec{x}\implies||\vec{u}||<||\vec{x}||$, we immediately inherit
from the inductive hypothesis the appropriate bound on safe inputs for oracle
calls from
$\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{a})(\vec{u};\vec{v})$.
Now, recall from the proof of (10) that whenever $\vec{u}\subset\vec{x}$ (and
so $||\vec{u}||<||\vec{x}||$), we have $|f(\vec{u};\vec{v})|\leq
p_{f}(||\vec{x}||-1)+\sum\vec{c}+\max|\vec{v}|$. So by setting
$c=p_{f}(||\vec{x}||-1)$ in the inductive hypothesis for $h(\vec{a},a)$, with
$a=\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{a})(\vec{u};\vec{v})$,
any oracle call from $h(\vec{a},a)$ will only take safe inputs of lengths:
$\begin{array}[]{rlll}\leq&p_{h}(||\vec{x}||)+p_{f}(||\vec{x}||-1)+\sum\vec{c}+\max|\vec{y}|\\\
\leq&p_{f}(||\vec{x}||)+\sum\vec{c}+\max|\vec{y}|&&\text{\eqref{eq:bnd-lem-
rec-fn-inv}}\end{array}$
This completes the proof.
## Appendix E Proof of 41
In this section we prove 41, i.e., that
$\mathsf{B}^{\subset}(R_{1;1})\subseteq\mathbf{F}\mathbf{P}(R)$, for $R$ a set
of relations. We proceed by induction on the definition of
$f(\vec{x};\vec{y})$.
Each initial function is polynomial-time computable, and each (relativised)
complexity class considered is under composition, so it suffices to only
consider the respective recursion schemes. Suppose we have
$h(a)(\vec{x};\vec{y})\in\mathsf{B}^{\subset}(a,\vec{a}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},R_{1;1}})$
and let:
$f(\vec{x};\vec{y})=h(\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{u};\vec{v}))(\vec{x};\vec{y})$
We start by making some observations:
1. 1.
First, note that $|f(\vec{x};\vec{y})|\leq
p_{f}(|\vec{x}|)+\sum\vec{c}+\max|\vec{y}|$, by 40, and so
$|f(\vec{x};\vec{y})|$ is polynomial in $|\vec{x},\vec{y}|$.
2. 2.
Second, note that the set $[\vec{x};\vec{y}]:=\\{(\vec{u},\vec{v})\ |\
\vec{u}\subset\vec{x},\vec{v}\subseteq\vec{y}\\}$ has size polynomial in
$|\vec{x},\vec{y}|$:
* •
write $\vec{x}=x_{1},\dots,x_{m}$ and $\vec{y}=y_{1},\dots,y_{n}$.
* •
Each $x_{i}$ and $y_{j}$ have only linearly many prefixes, and so there are at
most
$|x_{1}|\cdot\cdots\cdot|x_{m}||y_{1}|\cdot\cdots\cdot|y_{n}|\leq||\vec{x},\vec{y}||^{m+n}$
many choices of prefixes for all the arguments $\vec{x},\vec{y}$. (This is a
polynomial since $m$ and $n$ are global constants).
* •
Additionally, there are $m!$ permutations of the arguments $\vec{x}$ and $n!$
permutations of the arguments $\vec{y}$. Again, since $m$ and $n$ are global
constants, we indeed have $|[\vec{x};\vec{y}]|=O(||\vec{x},\vec{y}||^{m+n})$,
which is polynomial in $|\vec{x},\vec{y}|$.
We describe a polynomial-time algorithm for computing $f(\vec{x};\vec{y})$
(over oracles $\vec{a}$, $R_{1;1}$) by a sort of ‘course-of-values’ recursion
on the order $\subset\times\subseteq$ on $[\vec{x};\vec{y}]$.
First, for convenience, temporarily extend $\subset\times\subseteq$ to a total
well-order on $[\vec{x};\vec{y}]$, and write $S$ for the associated successor
function. Note that $S$ can be computed in polynomial-time from
$[\vec{x};\vec{y}]$.
Define $F(\vec{x},\vec{y}):=\langle
f(\vec{u};\vec{v})\rangle_{\vec{u}\subset\vec{x},\vec{v}\subseteq\vec{y}}$,
i.e. it is the graph of
$\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{u};\vec{v})$
that we shall use as a ‘lookup table’. Note that $|F(\vec{x},\vec{y})|$ is
polynomial in $|\vec{x},\vec{y}|$ by Item 1 and Item 2 above. Now, we can
write:444Here, as abuse of notation, we are now simply identifying
$F(\vec{x};\vec{y})$ with
$\lambda\vec{u}\subset\vec{x},\lambda\vec{v}\subseteq\vec{y}.f(\vec{x};\vec{y})$.
$\begin{array}[]{c}F(S(\vec{x},\vec{y}))=\langle
f(S(\vec{x},\vec{y})),F(\vec{x},\vec{y})\rangle=\langle
h(F(\vec{x},\vec{y}))(\vec{x};\vec{y}),F(\vec{x},\vec{y})\rangle\end{array}$
Again by Item 2 (and since $F$ is polynomially bounded), this recursion
terminates in polynomial-time. We may now simply calculate
$f(\vec{x};\vec{y})$ as $h(F(\vec{x},\vec{y}))(\vec{x};\vec{y})$.
|
# Toward Accurate Modeling of Galaxy Clustering on Small Scales: Halo Model
Extensions and Lingering Tension
Gillian D. Beltz-Mohrmann Department of Physics and Astronomy, Vanderbilt
University, 2201 West End Ave, Nashville, TN 37235, USA High Energy Physics
Division, Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL
60439, USA Adam O. Szewciw Department of Physics and Astronomy, Vanderbilt
University, 2201 West End Ave, Nashville, TN 37235, USA Andreas A. Berlind
Department of Physics and Astronomy, Vanderbilt University, 2201 West End Ave,
Nashville, TN 37235, USA National Science Foundation, Division of
Astronomical Sciences, Alexandria, VA 22314, USA Manodeep Sinha Department of
Physics and Astronomy, Vanderbilt University, 2201 West End Ave, Nashville, TN
37235, USA SA 118, Center for Astrophysics & Supercomputing, Swinburne
University of Technology, 1 Alfred St., Hawthorn, VIC 3122, Australia ARC
Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
Australia
(Received January 3, 2023; Revised March 15, 2023; Accepted March 16, 2023)
###### Abstract
This paper represents an effort to provide robust constraints on the galaxy-
halo connection and simultaneously test the Planck $\Lambda\mathrm{CDM}$
cosmology using a fully numerical model of small-scale galaxy clustering. We
explore two extensions to the standard Halo Occupation Distribution model:
assembly bias, whereby halo occupation depends on both halo mass and the
larger environment, and velocity bias, whereby galaxy velocities do not
perfectly trace the velocity of the dark matter within the halo. Moreover, we
incorporate halo mass corrections to account for the impact of baryonic
physics on the halo population. We identify an optimal set of clustering
measurements to constrain this “decorated” HOD model for both low- and high-
luminosity galaxies in SDSS DR7. We find that, for low-luminosity galaxies, a
model with both assembly bias and velocity bias provides the best fit to the
clustering measurements, with no tension remaining in the fit. In this model
we find evidence for both central and satellite galaxy assembly bias at the
99% and 95% confidence levels, respectively. In addition, we find evidence for
satellite galaxy velocity bias at the 99.9% confidence level. For high
luminosity galaxies, we find no evidence for either assembly bias or velocity
bias, but our model exhibits significant tension with SDSS measurements. We
find that all of these conclusions still stand when we include the effects of
baryonic physics on the halo mass function, suggesting that the tension we
find for high luminosity galaxies may be due to a problem with our assumed
cosmological model.
Large-scale structure of the universe (902) — Galaxy dark matter halos (1880)
— Galaxy groups (597) — Clustering (1908) — Redshift surveys (1378)
††journal: ApJ
## 1 Introduction
Small-scale galaxy clustering contains a wealth of cosmological information.
However, harnessing the constraining power of small scales requires highly
accurate models of both dark matter structure formation and the galaxy-halo
connection. Halo models are motivated by our understanding that galaxies form
and reside in gravitationally bound, virialized regions of dark matter known
as halos (e.g., Neyman & Scott, 1952; Peebles, 1974; McClelland & Silk, 1977;
Scherrer & Bertschinger, 1991; Kauffmann et al., 1997; Jing et al., 1998;
Baugh et al., 1999; Kauffmann et al., 1999; Benson et al., 2000; Ma & Fry,
2000; Peacock & Smith, 2000; Seljak, 2000; Scoccimarro et al., 2001; Sheth et
al., 2001; White et al., 2001; Cooray & Sheth, 2002). These models assume that
the clustering of galaxies can be fully described by (i) the clustering of
their host halos and (ii) the way in which galaxies occupy these halos.
A key ingredient of the halo model is the Halo Occupation Distribution (HOD),
which specifies via a few parameters the probability that a halo of mass $M$
contains $N$ galaxies (above some luminosity threshold) as well as how the
galaxies are distributed within their halo (Berlind & Weinberg, 2002; Berlind
et al., 2003). The standard form of the HOD (Zheng et al., 2005) contains at
most five free parameters that specify the mean occupation number of central
and satellite galaxies. This HOD formulation assumes that central galaxies
reside at the center of their halo and move at the mean halo velocity, while
satellite galaxies trace the spatial and velocity distribution of dark matter
inside halos. Constraining these parameters when fitting to observational data
provides a useful empirical measurement against which we can test competing
theories of galaxy formation and evolution. Moreover, this framework provides
a useful avenue for testing cosmology on small scales, assuming that the HOD
model is sufficiently flexible to marginalize over the uncertainty of galaxy
formation. Compared to other methods like subhalo abundance matching, the HOD
model is more flexible and does not rely on having well-resolved subhalos from
simulations, making it a more robust framework for probing cosmology.
Many works have used the standard HOD to model the clustering of galaxies in
recent redshift surveys (e.g., Zehavi et al., 2011; Guo et al., 2016). Several
of these studies yield fits which would rule out the $\Lambda\mathrm{CDM}$ \+
HOD model if taken at face value. However, these studies typically rely on
analytic approximations for calculating clustering statistics, which can
introduce unknown systematic uncertainties. Additionally, the errors and
covariances used in these studies are typically derived via the jackknife
method, which has been shown to produce biased results (Norberg et al., 2009).
Sinha et al. (2018) (S18 hereafter) developed a fully numerical mock-based
forward modeling procedure, whereby the standard HOD model is applied to halo
catalogs from cosmological N-body simulations and observational survey
systematics are added to create realistic mock catalogs. These mocks are then
used both for model parameter exploration and for calculating covariance
matrices. This significantly improved the accuracy of the HOD modeling
framework and allowed for the use of arbitrary clustering statistics that
could be directly measured on mocks (as opposed to calculated analytically).
Using the projected correlation function, group multiplicity function, and
galaxy number density, S18 compared the clustering of galaxies in the Sloan
Digital Sky Survey (SDSS, York et al., 2000) to a $\Lambda\mathrm{CDM}$
cosmology (Planck Collaboration et al., 2014) \+ standard HOD model. Carefully
controlling for systematic errors allowed them to reliably interpret the
goodness of fit of their model. They found that their best-fit HOD model was
unable to jointly fit the clustering statistics, revealing moderate tension
with SDSS. Because this tension did not exist when they considered only
measurements of the projected correlation function (as is done in many
studies), S18 demonstrated the value of adding additional statistics in small-
scale clustering analyses.
Szewciw et al. (2022) (S22 hereafter) enhanced the procedure used in S18 in
order to maximize the return from spectroscopic surveys. They included the
same clustering statistics used in S18 (galaxy number density, projected
correlation function, and group multiplicity function) as well as four
additional clustering statistics: redshift-space correlation function, group
velocity dispersion, mark correlation function, and counts-in-cells
statistics. Additionally, they developed an algorithm to identify an optimal
set of clustering measurements at a variety of different scales in order to
maximize constraining power and minimize noise. With these optimal
observables, as well as several other improvements to the modeling procedure,
they were able to significantly tighten their HOD parameter constraints, as
well as dramatically increase the tension found in S18.
The tension found in S22 may be indicative of an issue with the cosmological
model used, but it also may be the case that the standard HOD model is simply
not flexible enough to accurately encompass the galaxy-halo connection. For
example, the standard HOD model assigns galaxies to halos based solely on the
halo’s mass, but it is possible that halo occupation depends on additional
(secondary) features of the halo (e.g., concentration) that correlate with the
halo’s larger scale environment, a phenomenon known as assembly bias (e.g.,
Sheth & Tormen, 2004; Gao et al., 2005; Wechsler et al., 2006; Gao & White,
2007; Croton et al., 2007; Salcedo et al., 2018; Wechsler & Tinker, 2018).
Evidence for galaxy assembly bias has been found in multiple hydrodynamic
simulations (e.g., Artale et al., 2018; Bose et al., 2019; Beltz-Mohrmann et
al., 2020; Xu & Zheng, 2020; Hadzhiyska et al., 2020, 2021a, 2021c, 2021b;
Contreras et al., 2021), indicating that it is an expected feature in a
$\Lambda\mathrm{CDM}$ universe. Additionally, the standard HOD model assumes
that galaxies trace the positions and velocities of the dark matter
distribution within their host halo, but, as has been seen in hydrodynamic
simulations, this may not be the case (e.g., Berlind et al., 2003; Beltz-
Mohrmann et al., 2020). Finally, the typical halo modeling framework relies on
dark matter only simulations for creating halo catalogs. These simulations
lack baryonic physics, which has been shown to have a significant impact on
the halo distribution itself (e.g., Beltz-Mohrmann & Berlind, 2021). It is
possible that failing to account for the impact of baryonic physics on the
halo population is contributing to the tension between the halo model and the
clustering of SDSS galaxies.
Several works have examined the potential for the presence of assembly bias to
affect constraints on the galaxy-halo connection and cosmology. For example,
Zentner et al. (2014) examined the potential for assembly bias to induce
systematic errors in inferred halo occupation statistics. They built mock
galaxy catalogs that exhibited assembly bias as well as companion mock
catalogs with identical HODs but no assembly bias. They fit HOD models to the
galaxy clustering in each catalog, and found that the inferred HODs described
the true HODs well in the mocks without assembly bias, but in the mocks with
assembly bias the inferred HODs exhibited significant systematic errors.
In a later study, McCarthy et al. (2019) used a mock galaxy catalog with
assembly bias to study how assembly bias might affect cosmological
constraints. Specifically, they used the large-scale redshift-space
distortions to probe $f\sigma_{8}$. They found that on small scales (a few to
tens of $h^{-1}\mathrm{Mpc}$) galaxy assembly bias can introduce systematic
uncertainties in cosmological constraints if unaccounted for. They concluded
that galaxy assembly bias can only be ignored when modeling scales above 8
$h^{-1}\mathrm{Mpc}$, where clustering is determined purely by the large scale
bias. Similarly, Lange et al. (2019) explored how galaxy assembly bias affects
cosmological inference and found a degeneracy between assembly bias and
$f\sigma_{8}$. Ultimately, they found that not including galaxy assembly bias
in the model leads to a small shift in the posterior of $f\sigma_{8}$,
indicating that it is important to account for galaxy assembly bias to obtain
unbiased cosmological constraints.
Several recent works have attempted to constrain the galaxy-halo connection
and/or cosmology in observational surveys using an extended HOD model that
includes assembly bias (e.g., Zentner et al., 2019; Vakili & Hahn, 2019;
Salcedo et al., 2022; Wang et al., 2022). Many of these works use the Hearin
et al. (2016) “decorated” HOD model, which adds two free parameters to the
standard HOD model to control the strength of central and satellite occupation
on a secondary property. Other works have extended the HOD model to include
galaxy velocity bias (Guo et al., 2015a, b), while a few recent works have
utilized an extended HOD that includes both assembly bias and velocity bias to
constrain the galaxy-halo connection and/or cosmology (e.g., McCarthy et al.,
2022; Lange et al., 2022; Zhai et al., 2022).
Table 1: SDSS Volume-limited Sample Parameters
$M_{r}^{\mathrm{lim}}$ | $z_{\mathrm{min}}$ | $z_{\mathrm{max}}$ | $z_{\mathrm{median}}$ | $V_{\mathrm{eff}}$ | $n_{g}$
---|---|---|---|---|---
| | | | $(h^{-3}\mathrm{Mpc}^{3})$ | $(h^{3}\mathrm{Mpc}^{-3})$
$-19$ | 0.02 | 0.07 | 0.0562 | 6,087,119 | 0.01453
$-21$ | 0.02 | 0.158 | 0.1285 | 67,174,396 | 0.00123
Note. — The columns list (from left to right): the absolute magnitude
threshold of each sample at $z=0.1$; the minimum, maximum, and median
redshifts; the effective volume; and the galaxy number density of each sample.
The volumes and number densities of the samples are corrected for survey
incompleteness.
In this work, we build on the procedure established in Sinha et al. (2018) and
Szewciw et al. (2022), extending the HOD model to include both assembly bias
and velocity bias. We explore two different halo properties for implementing
assembly bias, and identify an optimal set of clustering measurements to
constrain our model. We also implement corrections to our halo masses to
account for the impact of baryonic physics on the halo mass function. We use
this framework to model the small-scale clustering of both low- and high-
luminosity galaxies in SDSS. By using an optimal set of statistics, adding
flexibility to our HOD model and accounting for the potential impact of
baryonic physics, our goal is to make the most robust test to-date of our
assumed $\Lambda\mathrm{CDM}$ cosmological model using small-scale galaxy
clustering.
In Section 2 we describe our data, and in Section 3 we describe our
simulations and halo catalogs. In Section 4 we describe our halo model, and in
Section 5 we describe our full modeling procedure (including our mock galaxy
catalogs, covariance matrices, clustering measurements, and MCMC framework).
In Section 6 we describe our selection of optimal observables for constraining
our HOD model, and in Section 7 we describe our results. We summarize our
findings in Section 8.
Table 2: Simulation Parameters
Use | Sample | Simulation | Seeds | $L_{\mathrm{box}}$ | $N_{\mathrm{part}}$ | $m_{\mathrm{part}}$ | $\epsilon$ | Number
---|---|---|---|---|---|---|---|---
| | | | ($h^{-1}\mathrm{Mpc}$) | | ($h^{-1}\mathrm{M}_{\odot}$) | ($h^{-1}\mathrm{kpc}$) |
Covariance matrix | $-19$ | Consuelo | 4001 - 4100 | 420 | $1400^{3}$ | $2.26\times 10^{9}$ | 8 | 100
Covariance matrix | $-21$ | Carmen | 2001 - 2100 | 1000 | $1120^{3}$ | $5.97\times 10^{10}$ | 25 | 100
MCMC | $-19$ | ConsueloHD | 4002, 4022 | 420 | $2240^{3}$ | $5.53\times 10^{8}$ | 5 | 2
MCMC | $-21$ | CarmenHD | 2007, 2023 | 1000 | $2240^{3}$ | $7.46\times 10^{9}$ | 12 | 2
Note. — The columns list (from left to right): what each simulation is used
for, the absolute magnitude threshold of the corresponding SDSS sample, the
name of the simulation, the seeds used, the (comoving) boxsize, number of
particles, mass resolution, (comoving) force softening, and the number of
simulations.
## 2 Observational Data
In this work, we use the same observational dataset as that used in S22. We
utilize the large scale structure samples from the NYU Value Added Galaxy
Catalog (NYU-VAGC; Blanton et al., 2005) from the seventh data release (DR7;
Abazajian et al., 2009) of the Sloan Digital Sky Survey (SDSS; York et al.,
2000). The absolute magnitudes of the galaxies in this sample have been
k-corrected to rest-frame magnitudes at redshift $z=0.1$ but have not been
corrected for passive luminosity evolution.
From this sample, we construct two volume-limited subsamples, each complete
down to a specified r-band absolute magnitude threshold ($M_{r}<-19$ and
$M_{r}<-21$). We refer to these samples as the $-19$ and $-21$ samples
throughout this paper. The luminosity thresholds, redshift limits, median
redshifts, effective volumes, and number densities of our samples are listed
in Table 1. The co-moving distances of the SDSS galaxies in our samples are
determined using a flat $\Lambda\mathrm{CDM}$ cosmological model with
$\Omega_{\mathrm{m}}$ = 0.302 and $h=1$. Our distances are in units of
$h^{-1}\mathrm{Mpc}$, and our absolute magnitudes are actually
$M_{r}+5\mathrm{log}h$111Throughout this paper, $\mathrm{log}$ refers to
$\mathrm{log}_{10}$..
Fiber collisions are handled in the same way as in S22. Briefly, we first
adopt the nearest neighbor correction and then, informed by SDSS plate overlap
regions, we apply additional corrections to our galaxy clustering measurements
in order to account for errors in the nearest neighbor correction. This method
was applied in S22, and was recently further validated using the Uchuu-SDSS
galaxy lightcones (Dong-Páez et al., 2022). For more details on our
observational data and our treatment of fiber collisions, see S22.
## 3 Simulations and Halo Catalogs
In our modeling procedure, we make use of the same dark matter only
cosmological N-body simulations as those used in S22. These simulations are
from the Large Suite of Dark Matter Simulations project (LasDamas; McBride et
al., 2009) and were run on the Texas Advanced Computing Center’s Stampede
supercomputer using the public code gadget-2 (Springel, 2005). Power spectra
were generated with cmbfast (Seljak & Zaldarriaga, 1996; Zaldarriaga et al.,
1998; Zaldarriaga & Seljak, 2000), and initial conditions were generated with
2lptic (Scoccimarro, 1998; Crocce et al., 2006, 2012). All simulations were
run with the following cosmological parameters, based on results from the
Planck experiment (Planck Collaboration et al., 2014):
$\Omega_{\mathrm{m}}=0.302$, $\Omega_{\Lambda}=0.698$,
$\Omega_{\mathrm{b}}=0.048$, $h=0.681$, $\sigma_{8}=0.828$, and $n_{s}=0.96$.
For each observational sample of interest (i.e., -19 and -21), we run two sets
of simulations: 100 low resolution simulations to build a covariance matrix,
and 2 high resolution simulations for model parameter exploration. The details
of these simulations are given in Table 2. We identify halos with a spherical
over-density algorithm (SO; Lacey & Cole, 1994) using the rockstar phase-space
temporal halo finder (Behroozi et al., 2013). We adopt a virial mass
definition with a density threshold given by (Bryan & Norman, 1998). Finally,
for computational purposes, we randomly downsample to keep only 5% of the dark
matter particles in each halo, with no loss of accuracy (see S22).
## 4 Halo Model
### 4.1 The Standard HOD Model
The Halo Occupation Distribution framework governs the number, positions, and
velocities of galaxies within dark matter halos. The standard HOD model
assigns galaxies to halos based on five free parameters, which depend only on
the halo’s mass (Zheng et al., 2007). Galaxies are split into centrals and
satellites within their halos (Kravtsov et al., 2004; Zheng et al., 2005). In
this model, the mean number of central galaxies in a halo of mass $M$ is
described by
$\langle
N_{\mathrm{cen}}\rangle=\frac{1}{2}\bigg{[}1+\mathrm{erf}\bigg{(}\frac{\mathrm{log}M-\mathrm{log}M_{\mathrm{min}}}{\sigma_{\mathrm{log}M}}\bigg{)}\bigg{]},$
(1)
where $M_{\mathrm{min}}$ is the mass at which half of halos host a central
galaxy, $\sigma_{\mathrm{log}M}$ is the scatter around this halo mass, and
$\mathrm{erf}(x)$ is the error function,
$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\mathrm{exp}(-y^{2})dy$. For
a specific halo of mass $M$, we draw a random number $R$ from a uniform
distribution on the interval $[0,1)$. If $R<\langle N_{\mathrm{cen}}\rangle$,
then a central galaxy is assigned to the halo. The central galaxy is always
placed at the center of the halo and given the mean velocity of the halo.
The number of satellite galaxies in a given halo is drawn from a Poisson
distribution with mean
$\langle N_{\mathrm{sat}}\rangle=\langle
N_{\mathrm{cen}}\rangle\times\bigg{(}\frac{M-M_{0}}{M_{1}}\bigg{)}^{\alpha},$
(2)
where $M_{0}$ is the halo mass below which there are no satellite galaxies,
$M_{1}$ is the mass where halos contain one satellite galaxy on average, and
$\alpha$ is the slope of the power-law occupation function at high masses.
Each satellite galaxy is given the position and velocity of a randomly
selected dark matter particle within the halo.
### 4.2 Assembly Bias
One way in which we can extend the standard HOD model is to relax the
assumption that halo occupation depends solely on the halo’s mass. In other
words, we can allow for halo occupation to depend on both halo mass and a
secondary halo property, a phenomenon known as assembly bias (Gao et al.,
2005; Croton et al., 2007). To implement assembly bias, we use the decorated
HOD (dHOD) model of Hearin et al. (2016). In order to apply this decorated HOD
model, we first split halos by mass into bins of width 0.05 dex. Then, within
each mass bin, we split halos into two groups based on the median value of the
secondary property $s$ in each bin. We then assign galaxies to halos based on
the following conditional relations
$\langle N_{\mathrm{cen}}|M,s_{\mathrm{high}}\rangle=\langle
N_{\mathrm{cen}}|M\rangle+\delta N_{\mathrm{cen}},$ (3) $\langle
N_{\mathrm{cen}}|M,s_{\mathrm{low}}\rangle=\langle
N_{\mathrm{cen}}|M\rangle-\delta N_{\mathrm{cen}},$ (4) $\langle
N_{\mathrm{sat}}|M,s_{\mathrm{high}}\rangle=\langle
N_{\mathrm{sat}}|M\rangle+\delta N_{\mathrm{sat}},$ (5) $\langle
N_{\mathrm{sat}}|M,s_{\mathrm{low}}\rangle=\langle
N_{\mathrm{sat}}|M\rangle-\delta N_{\mathrm{sat}},$ (6)
where
$\delta N_{\mathrm{cen}}=A_{\mathrm{cen}}\mathrm{MIN}[\langle
N_{\mathrm{cen}}|M\rangle,1-\langle N_{\mathrm{cen}}|M\rangle]$ (7)
for central galaxies and
$\delta N_{\mathrm{sat}}=A_{\mathrm{sat}}\langle N_{\mathrm{sat}}|M\rangle$
(8)
for satellite galaxies. $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$ have values
between $-1$ and $1$; values of $0$ indicate no assembly bias. A key feature
of this dHOD model is that, regardless of the strength of the assembly bias,
$\langle N_{\mathrm{cen}}\rangle$ and $\langle N_{\mathrm{sat}}\rangle$ are
preserved for a given halo mass. In other words, at fixed mass, for the same
5-parameter standard HOD model, the decorated HOD has the same halo occupation
distribution when averaged over all halos.
Several works have explored the variety of different halo properties that can
be used to model assembly bias. Salcedo et al. (2018) explored halo assembly
bias in the LasDamas simulations and found that a clustering bias exists if
halos are binned by mass or by any other halo property, indicating that no
single halo property encompasses all the spatial clustering information of the
halo population. They also found that the mean values of some halo properties
depend on their halo’s distance to a more massive neighbor and concluded that
this “neighbor bias” largely accounts for the secondary bias seen in halos
binned by mass and split by concentration or age. However, they also found
that halos binned by other mass-like properties still show a secondary bias
even when the neighbor bias is removed.
Meanwhile, Mao et al. (2018) presented a summary of secondary halo biases of
high-mass halos due to various halo properties (e.g., concentration, spin,
several proxies of assembly history, and subhalo properties) in the MultiDark
Planck 2 simulation. They found that, while concentration, spin, and the
abundance and radial distribution of subhalos exhibit significant secondary
biases, properties that directly quantify halo assembly history do not.
Finally, Behroozi et al. (2022) examined the correlation of different
properties of dark matter halos (e.g., growth rate, spin, concentration) with
environment in the Small MultiDark Planck simulation and demonstrated that
these halo properties imprint distinct signatures in the galaxy two-point
correlation function and in the distribution of distances to galaxies’ $k$th
nearest neighbors. They demonstrated that the agreement with observed
clustering can be improved with a simple empirical model in which galaxy size
correlates with halo growth.
In this work, the first secondary halo property that we use to model assembly
bias is halo concentration, $c$, defined as the ratio of the virial radius
$R_{\mathrm{vir}}$ of the halo to the scale radius $R_{s}$ (Navarro et al.,
1997). The dependence of the galaxy-halo connenction on concentration or
circular velocity has been explored in a number of previous works (e.g.,
Lehmann et al., 2017; Xu & Zheng, 2020). For a given halo, concentration can
be found using the relationship between virial mass, maximum circular
velocity, and concentration at $z=0$:
$v_{\mathrm{circ}}(M_{\mathrm{vir}})=\frac{6.72\times
10^{-3}M_{\mathrm{vir}}^{1/3}\sqrt{c}}{\sqrt{ln(1+c)-c/(1+c)}}$ (9)
where $M_{\mathrm{vir}}$ is the virial mass of the halo in units of
$h^{-1}M_{\odot}$ and $v_{\mathrm{circ}}$ is the maximum circular velocity of
the halo in units of km/s (Klypin et al., 2011). In our case, when
implementing halo concentration as our secondary bias property, we determined
that the normalization is irrelevant and it is only the halo ranking that
matters; thus, we use $v_{\mathrm{circ}}/M_{\mathrm{vir}}^{1/3}$ as a proxy
for concentration. We refer to this assembly bias model using concentration as
“ABcon.”
Another halo property that can be used to model assembly bias is its larger
scale environment. The reason that conditioning the galaxy occupation on
concentration has an impact on clustering statistics is that concentration is
correlated with a halo’s larger scale environment at fixed mass. Since we do
not know a priori what secondary halo property to use in modeling assembly
bias, it makes sense to skip this intermediate step and condition galaxy
occupation directly on environment. Several works have explored the dependence
of the galaxy-halo connection on environment (e.g., Pujol et al., 2017; Shi &
Sheth, 2018; Hadzhiyska et al., 2020; Yuan et al., 2021). Motivated by these
works, we choose to also explore the effects of using local halo environment
to model assembly bias. We define local environment as the total mass of halos
within a 5 $h^{-1}\mathrm{Mpc}$ spheres centered on the halo of interest
(excluding the mass of the halo of interest itself). We do not impose any
lower mass limit on the halos included in this sum. We refer to this assembly
bias model using environment as “ABenv.”
### 4.3 Velocity Bias
Another way in which we can extend the HOD model is to relax the assumption
that satellite galaxies trace the velocities of the dark matter particles
within their host halo. In other words, we can introduce satellite velocity
bias (“VB”) into our model. We do this by introducing a new parameter,
$B_{\mathrm{vel}}$, to the model. $B_{\mathrm{vel}}$ is defined as the ratio
between the velocities of satellite galaxies and dark matter in the halo frame
of reference:
$B_{\mathrm{vel}}=\frac{v_{g}-v_{h}}{v_{m}-v_{h}}$ (10)
where $v_{g}$ is the velocity of the satellite galaxy, $v_{h}$ is the velocity
of the halo, and $v_{m}$ is the velocity of the randomly chosen dark matter
particle on which the satellite galaxy is placed. A value of
$B_{\mathrm{vel}}$ less than $1$ indicates that satellite galaxies are moving
with slower velocities than the dark matter, while a value of
$B_{\mathrm{vel}}$ greater than $1$ indicates that satellite galaxies are
moving faster than the dark matter, and a value of $B_{\mathrm{vel}}$ equal to
$1$ indicates no velocity bias.
In this study we only model satellite velocity bias and not central velocity
bias. In other words, we stick with the standard assumption that central
galaxies inherit the same velocity as their host halo. In principle, central
galaxies can move relative to their halo as predicted by some hydrodynamic
simulations (Berlind et al., 2003) and suggested for SDSS galaxies (e.g., Guo
et al., 2015a, b). However, when comparing HOD modeling to hydrodynamic
simulations Beltz-Mohrmann et al. (2020) found that the presence of central
velocity bias is likely to have a negligible effect on the galaxy clustering
statistics that we use in our analysis, unlike satellite velocity bias which
is likely important for low luminosity galaxies.
### 4.4 Accounting for Baryonic Effects
While not strictly part of the HOD model, another way in which we can extend
the standard halo modeling framework is to account for the effect of baryonic
physics on the halo mass function. The HOD model is typically applied to a
halo catalog generated from a dark matter only (DMO) simulation, which does
not account for the impact of baryonic physics on halo mass. Beltz-Mohrmann &
Berlind (2021) investigated the differences in halo mass functions between
matched DMO and hydrodynamic simulations in EAGLE, Illustris, and
IllustrisTNG, and found that, for halos at $z=0$, stellar feedback generally
reduces the masses of low mass halos ($\lesssim 10^{11}$
$h^{-1}\mathrm{M}_{\odot}$), while AGN feedback generally reduces the masses
of high mass halos (between $10^{12}$ and $10^{13}$
$h^{-1}\mathrm{M}_{\odot}$) compared to their DMO counterparts. However, the
exact effect that feedback has on the halo masses differs dramatically from
one hydrodynamic simulation to the next. By matching halos according to mass
between dark matter and hydrodynamic simulations, Beltz-Mohrmann & Berlind
(2021) produced formulae which can be used to adjust the halo masses in a DMO
simulation in order to reproduce the halo mass function from a given
hydrodynamic simulation. Additionally, they produced fits based on matching
halos between dark matter and hydrodynamic simulations based on both mass and
local halo environment. By taking halo environment into account, these fits
can be used to adjust halo masses in a DMO simulation to not only reproduce
the global halo mass function from a hydrodynamic simulation, but also to
reproduce the conditional mass function, which then also reproduces the halo
correlation function. In Section 7.4 we apply several of the halo mass
corrections from Beltz-Mohrmann & Berlind (2021) to our halo catalogs, in
order to assess the robustness of our results to changes in the mass function
due to baryonic physics. It should be noted that these mass corrections do not
modify the halo profile, nor do they alter the velocity dispersion of dark
matter within the halo, they adjust only the mass of each halo.
### 4.5 Summary
In this work, we extend the standard HOD model in several ways. We first
explore the effects of extending the standard HOD model to include
concentration-based assembly bias. We then explore the effects of instead
using environment-based assembly bias. Next we extend the model to include
both assembly bias and satellite velocity bias. Finally, we implement halo
mass corrections to account for the impact of baryonic physics, and we
investigate the effects this has on the results from our most complete halo
model (i.e., the model with both assembly bias and velocity bias).
## 5 Modeling Procedure
### 5.1 Building mock galaxy catalogs
We build mock galaxy catalogs to use as our model by populating the two high-
resolution simulations for each sample (ConsueloHD and CarmenHD) with
galaxies. Once we populate our dark matter halos with galaxies, we build
realistic mock galaxy catalogs that resemble our SDSS samples of interest. To
do this, we transpose the mock galaxies from Cartesian to spherical
coordinates by positioning a virtual observer at the center of the box and
converting the positions of the galaxies into RA, DEC, and comoving distances.
We then carve out four independent mock galaxy catalogs from each simulation
box and incorporate the same systematic effects that plague our observational
dataset, such as redshift-space distortions, sample geometry, and
incompleteness. For more information on the forward modeling details, see S22.
### 5.2 Covariance Matrices
If we wish to take advantage of the information present at small scales to
constrain the galaxy-halo connection, it is essential that we construct
accurate covariance matrices for our clustering measurements. To do this, we
run 100 low-resolution simulations for each sample (Consuelo and Carmen) which
differ in the phases of the density modes of the power spectrum, which is
controlled by a seed supplied to 2lptic. We populate these low-resolution
simulations with galaxies using the same HOD parameters222The covariance
matrices are built using a model that does not include assembly bias or
velocity bias. as were used to build the matrices in S22. These parameters are
listed in Table 3.
We then build 400 independent mock galaxy catalogs for each sample, from which
we can construct a covariance matrix to represent cosmic variance. The
elements of the covariance matrix are given by
$C_{ij}=\frac{1}{N-1}\sum_{1}^{N}(y_{i}-\overline{y_{i}})(y_{j}-\overline{y_{j}})$
(11)
where the sum is taken over the $N=400$ mocks. The values $y_{i}$ and $y_{j}$
are the $i$th and $j$th observables measured on each mock, while
$\overline{y_{i}}$ and $\overline{y_{j}}$ are the mean values of the $i$th and
$j$th observables, respectively. Each diagonal element, $C_{ii}$, of the
matrix is the variance across 400 mocks for observable $i$, and
$\sqrt{C_{ii}}$ is the cosmic variance uncertainty of observable $i$. For an
arbitrary observable, we refer to this uncertainty as $\sigma_{\mathrm{obs}}$.
S22 showed that the noise from using a finite number (400) of mock catalogs
does not significantly affect our clustering analysis. Additionally, S22
examined the impact of resolution on the accuracy of our covariance matrices,
and determined that the lower resolution of the Carmen and Consuelo
simulations causes us to overestimate the error on the smallest scales of the
correlation function by 10-20%. However, not only is this a small effect, but
larger cosmic variance uncertainties result in lower chi-square measurements
and in general make it more difficult to rule out incorrect models, and we
would rather have slightly broader constraints than artificially tight
constraints.
Table 3: Fiducial HOD parameters for covariance matrices
$M_{r}^{\mathrm{lim}}$ | $\mathrm{log}{M_{\mathrm{min}}}$ | $\sigma_{\mathrm{log}M}$ | $\mathrm{log}{M_{0}}$ | $\mathrm{log}{M_{1}}$ | $\alpha$
---|---|---|---|---|---
$-19$ | 11.54 | 0.22 | 12.01 | 12.74 | 0.92
$-21$ | 12.72 | 0.46 | 7.87 | 13.95 | 1.17
Note. — The HOD parameters used to construct the covariance matrices in our
analysis. Note that the matrices were constructed assuming zero assembly bias
and velocity bias.
### 5.3 Clustering Statistics
Several works have demonstrated the power of using a variety of different
clustering statistics to constrain both the galaxy-halo connection as well as
cosmology (Berlind & Weinberg, 2002; Sinha et al., 2018; Hadzhiyska et al.,
2021a; Szewciw et al., 2022; Storey-Fisher et al., 2022). In our analysis, we
employ the following clustering statistics: the projected correlation function
$w_{\mathrm{p}}(r_{\mathrm{p}})$ (e.g., Zehavi et al., 2002, 2004; Zheng,
2004; Zehavi et al., 2005; Zheng et al., 2007; Zehavi et al., 2011; Leauthaud
et al., 2012; Zentner et al., 2014; Coupon et al., 2015), the redshift-space
correlation function $\xi(s)$ (e.g., Tinker et al., 2006b; Parejko et al.,
2013; Guo et al., 2015b; Padilla et al., 2019; Beltz-Mohrmann et al., 2020;
Tonegawa et al., 2020), the group multiplicity function $n(N)$ (e.g., Berlind
et al., 2006; Zheng & Weinberg, 2007; Sinha et al., 2018; Beltz-Mohrmann et
al., 2020), the average group velocity dispersion $\sigma_{v}(N)$, the mark
correlation function $\mathrm{mcf}(s)$ (e.g., Zu & Mandelbaum, 2018; Storey-
Fisher et al., 2022), and two special cases of counts-in-cells $P_{N}(R)$: the
void probability function $P_{0}$ ($\mathrm{VPF}(R)$) and the singular
probability function $P_{0}$ ($\mathrm{SPF}(R)$) (e.g., Tinker et al., 2006a,
2008; McCullagh et al., 2017; Walsh & Tinker, 2019; Wang et al., 2019; Beltz-
Mohrmann et al., 2020; Perez et al., 2021). A detailed description of each of
these clustering statistics is given in S22. To calculate
$w_{\mathrm{p}}(r_{\mathrm{p}})$, $\xi(s)$, $\mathrm{mcf}(s)$,
$\mathrm{VPF}(R)$, and $\mathrm{SPF}(R)$, we make use of the publicly
available code corrfunc (Sinha & Garrison, 2019, 2020). In our modeling
procedure, we measure each clustering statistic in the same way (i.e., either
on the full box/es or on the mock galaxy catalogs) as was done in S22. It is
important to note that our clustering statistics range in scale from about
$0.1$ to $20$ $h^{-1}\mathrm{Mpc}$ for both samples; thus, our analysis
extends from the highly-nonlinear regime all way out to the “quasi-linear”
regime of clustering.
One of the main motivations for including so many higher-order statistics in
our analysis is to ultimately obtain constraining power for both our model of
the galaxy-halo connection and our cosmological model. For example, redshift-
space correlation function contains information about galaxy peculiar
velocities due to redshift-space distortions that change the apparent
positions of galaxies along the line of sight, which can help us constrain
cosmic structure growth. Additionally, Storey-Fisher et al. (2022) found that
statistics beyond the standard galaxy clustering statistics (e.g.
$w_{\mathrm{p}}(r_{\mathrm{p}})$) significantly increase the constraining
power on cosmological parameters of interest. Specifically, they found that
including counts in cells statistics and the mark correlation function
improves the precision of constraints on $\sigma_{8}$ by 33%, $\Omega_{m}$ by
28%, and the growth of structure parameter, $f\sigma_{8}$, by 18% compared to
standard statistics. While we do not vary our cosmological model in this work,
and thus cannot comment on the specific ability of each clustering measurement
to constrain cosmological parameters, we include such a wide variety of
clustering statistics in this work with the goal of ultimately constraining
both HOD and cosmological parameters.
### 5.4 MCMC
We explore the HOD parameter space with a Markov Chain Monte Carlo (MCMC)
algorithm, using a privately developed C-implementation of the popular affine-
invariant sampler emcee (Foreman-Mackey et al., 2013), which we call
emcee_in_c.333https://github.com/aszewciw/emcee_in_c We impose flat priors on
the same parameter ranges given in S18, as well as flat priors of [-1.0,1.0]
on $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$, and a flat prior of [0.5,1.5] on
$B_{\mathrm{vel}}$.
At each point in the chain, we evaluate the likelihood that a particular HOD
model could have generated a dataset with the same clustering as SDSS. This
likelihood is given by
$\mathcal{L}(\mathbf{D}|\mathbf{M})=\frac{\exp(-\frac{1}{2}{(\mathbf{D}-\mathbf{M})\mathbf{C}^{-1}(\mathbf{D}-\mathbf{M})^{T}})}{\sqrt{(2\pi)^{K}\mathrm{det}(\mathbf{C})}},$
(12)
where D is the K-dimensional vector of observables measured on the SDSS
dataset, M is the corresponding vector of observables measured on the HOD
model, and C is the K-dimensional covariance matrix of these observables
representing cosmic variance (see Equation 11). (The the term within the
exponential is essentially $\chi^{2}$, multiplied by a factor of $-1/2$.)
It is worth noting that our likelihood calculation assumes that all of our
observables are Gaussian. However, Hahn et al. (2019) found that assuming a
Gaussian likelihood in a group multiplicity function analysis slightly
underestimates the uncertainties and biases HOD parameter constraints by $\sim
0.5\sigma$. We have examined all of our observables and determined that the
vast majority of them (including the group multiplicity function) appear to
follow Gaussian distributions and pass typical tests of Gaussianity, so we
have proceeded with assuming a Gaussian likelihood.
In the HOD framework, the process of populating halos with galaxies is
stochastic, and is controlled with a “population seed.” For a fixed HOD model,
changes in this population seed can lead to significant differences in
clustering statistics. To minimize the noise in our results due to this random
variation, at each point in the chain we populate halos four times, using four
fixed population seeds. Thus the clustering measurements for a given point in
HOD parameter space are the average measurements over these four population
seeds. (See S22 for details.)
Table 4: Optimal Observable Order
Index | -19 sHOD | -19 dHOD | -21 sHOD | -21 dHOD
---|---|---|---|---
1 | $n_{\mathrm{gal}}$ | $n_{\mathrm{gal}}$ | $n_{\mathrm{gal}}$ | $n_{\mathrm{gal}}$
2 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 2 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 2 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 2 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 2
3 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 4 | $\sigma_{v}(N)$ 3 | $\xi(s)$ 8 | $\sigma_{v}(N)$ 1
4 | $\mathrm{VPF}(R)$ 3 | $\xi(s)$ 8 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 4 | $\xi(s)$ 9
5 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 8 | $n(N)$ 3 | $\mathrm{mcf}(s)$ 9 | $\xi(s)$ 3
6 | $\xi(s)$ 1 | $\mathrm{SPF}(R)$ 1 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 1 | $\mathrm{mcf}(s)$ 10
7 | $n(N)$ 3 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 3 | $\xi(s)$ 9 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 5
8 | $\xi(s)$ 5 | $n(N)$ 2 | $\mathrm{mcf}(s)$ 7 | $n(N)$ 1
9 | $n(N)$ 2 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 8 | $\xi(s)$ 4 | $\sigma_{v}(N)$ 3
10 | $n(N)$ 4 | $\xi(s)$ 1 | $\xi(s)$ 7 | $\mathrm{mcf}(s)$ 3
11 | $n(N)$ 1 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 4 | $\mathrm{mcf}(s)$ 10 | $\xi(s)$ 1
12 | $\mathrm{SPF}(R)$ 4 | $\mathrm{VPF}(R)$ 2 | $\xi(s)$ 1 | $\xi(s)$ 8
13 | $\xi(s)$ 13 | $\mathrm{mcf}(s)$ 1 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 14 | $\xi(s)$ 5
14 | $\mathrm{mcf}(s)$ 14 | $\xi(s)$ 10 | $n(N)$ 1 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 1
15 | $\xi(s)$ 6 | $\mathrm{SPF}(R)$ 2 | $\mathrm{SPF}(R)$ 4 | $n(N)$ 2
16 | $n(N)$ 5 | $\xi(s)$ 4 | $\mathrm{mcf}(s)$ 3 | $\mathrm{SPF}(R)$ 4
17 | $\xi(s)$ 2 | $n(N)$ 1 | $\xi(s)$ 6 | $\mathrm{mcf}(s)$ 5
18 | $\mathrm{SPF}(R)$ 2 | $n(N)$ 5 | $\sigma_{v}(N)$ 4 | $\sigma_{v}(N)$ 4
19 | $\xi(s)$ 10 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 1 | $\xi(s)$ 5 | $\mathrm{mcf}(s)$ 14
20 | $\mathrm{mcf}(s)$ 2 | $\mathrm{SPF}(R)$ 4 | $\xi(s)$ 3 | $\mathrm{SPF}(R)$ 3
21 | $\mathrm{mcf}(s)$ 3 | $\mathrm{mcf}(s)$ 7 | $n(N)$ 4 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 3
22 | $\sigma_{v}(N)$ 1 | $\mathrm{mcf}(s)$ 11 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 7 | $\sigma_{v}(N)$ 5
23 | $\sigma_{v}(N)$ 3 | $\sigma_{v}(N)$ 5 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 3 | $\sigma_{v}(N)$ 2
24 | $\xi(s)$ 9 | $\mathrm{SPF}(R)$ 3 | $\mathrm{mcf}(s)$ 8 | $\xi(s)$ 7
25 | $\sigma_{v}(N)$ 4 | $\xi(s)$ 3 | $\mathrm{VPF}(R)$ 3 | $n(N)$ 3
26 | $\mathrm{mcf}(s)$ 1 | $n(N)$ 4 | $\xi(s)$ 2 | $n(N)$ 4
27 | $\sigma_{v}(N)$ 2 | $\mathrm{mcf}(s)$ 2 | $n(N)$ 5 | $\xi(s)$ 4
28 | $n(N)$ 6 | $\sigma_{v}(N)$ 2 | $n(N)$ 2 | $\xi(s)$ 2
29 | $\mathrm{VPF}(R)$ 1 | $\mathrm{VPF}(R)$ 4 | $\xi(s)$ 11 | $n(N)$ 5
30 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 1 | $\mathrm{mcf}(s)$ 8 | $\sigma_{v}(N)$ 3 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 8
31 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 6 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 6 | $\sigma_{v}(N)$ 2 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 4
32 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 5 | $\xi(s)$ 9 | $\mathrm{SPF}(R)$ 2 | $\xi(s)$ 6
33 | $\sigma_{v}(N)$ 5 | $n(N)$ 6 | $\mathrm{mcf}(s)$ 5 | $\mathrm{mcf}(s)$ 8
34 | $w_{\mathrm{p}}(r_{\mathrm{p}})$ 3 | $\mathrm{mcf}(s)$ 14 | $\mathrm{SPF}(R)$ 3 | $\xi(s)$ 10
35 | $\sigma_{v}(N)$ 7 | $\mathrm{VPF}(R)$ 5 | $\mathrm{mcf}(s)$ 4 | $\xi(s)$ 11
36 | $n(N)$ 7 | $\mathrm{mcf}(s)$ 12 | $\mathrm{SPF}(R)$ 1 | $\mathrm{mcf}(s)$ 7
37 | – | – | – | $\mathrm{mcf}(s)$ 12
38 | – | – | – | $\xi(s)$ 14
39 | – | – | – | $\mathrm{VPF}(R)$ 5
40 | – | – | – | $\mathrm{VPF}(R)$ 2
41 | – | – | – | $\mathrm{mcf}(s)$ 1
Note. — The type of clustering statistic and the bin number (1-indexing) for
the observables chosen (in order) for each sample. “sHOD” refers to the
observables chosen for each sample using the standard HOD model in S22. “dHOD”
refers to the observables chosen in this work using the decorated HOD model.
The observables chosen in this work that were not chosen in S22 are shown in
bold.
## 6 Choosing Optimal Observables
In order to constrain the dHOD when fit to SDSS, we must first choose a set of
observables to use in our MCMC. We cannot arbitrarily continue to increase the
number of observables we use, because doing so increases the noise in our
modeling and degrades our final constraints. Noise is introduced into the
covariance matrix due to the fact that we are constructing it from only 400
mocks. This noise propagates into the likelihood function and ultimately into
our posterior results. Thus, we need to choose our observables wisely. We seek
a subset of observables that produce the tightest constraints on our HOD
parameters, at the cost of little noise.
To choose an “optimal” set of high-information, low-noise observables, we
employ the importance sampling algorithm described in S22. In this algorithm,
we first create four mock SDSS catalogs for which we will determine optimal
statistics. We use four mocks instead of the actual SDSS data in order to
minimize the impact of cosmic variance on the selection of optimal statistics.
We build these four mocks using a fiducial dHOD (with concentration) model
with parameters that we obtain by fitting this model to the S22 set of
clustering statistics (listed in Table 4 under “sHOD”) measured on the SDSS.
We run an initial MCMC on each of the four mock galaxy catalogs, fitting the
dHOD (with concentration) model to only two observables: $n_{\mathrm{gal}}$
and $w_{\mathrm{p}}(r_{\mathrm{p}}\sim 0.3\ h^{-1}\mathrm{Mpc})$. This results
in a fairly broad MCMC non-uniform grid of points in parameter space for each
mock. We then use importance sampling on these grids to explore the
constraining power of different combinations of clustering statistics. The
algorithm chooses observables one by one, each time selecting the observable
that, when combined with all previously chosen observables, produces the
tightest projected constraints on all HOD parameters of interest. When
choosing an observable, we consider how it performs on across all four grids,
minimizing any bias due to cosmic variance. Thus, at the end of running this
algorithm, we have a list of observables (ordered in terms of cumulative
constraining power) and a corresponding list of cumulative projected
constraints for each sample. (We refer the reader to S22 for a more complete
description of this procedure.)
Figure 1: Constraints on each HOD parameter as we increase the number of
observables, for the $-19$ sample (blue) and the $-21$ sample (red). The solid
line in each panel shows the average mock constraint (1-$\sigma$) across four
mocks, and the shaded region is an estimate of the uncertainty (inner 68%) in
these constraints due to the noise present in the covariance matrix. The dot
indicates the optimal number of observables for each sample, and the dashed
line indicates the corresponding constraining power for each parameter.
There are two key differences in our implementation of this algorithm compared
to S22. First, when choosing the third observable for each sample, we only
attempt to constrain $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$. This is
because these parameters are entirely unconstrained when using only
$n_{\mathrm{gal}}$ and $w_{\mathrm{p}}(r_{\mathrm{p}}\sim 0.3\
h^{-1}\mathrm{Mpc})$, which causes the MCMC to explore unrealistic HOD models;
thus, it is essential to choose an observable early on that provides
constraining power for these parameters. After the third observable is chosen,
we make all successive choices by attempting to jointly constrain all HOD
parameters (excluding $\mathrm{log}{M_{0}}$ for the $-21$ sample). Second, in
the S22 algorithm, new grids are created (by running new MCMCs using the
already chosen observables) whenever the old grids become insufficiently dense
for importance sampling. S22 creates these new grids after choosing five
observables for each sample, and again for the $-19$ sample after choosing
eight observables. In our case, because we are trying to constrain two
additional parameters, our grids become insufficiently dense more quickly, and
so we ultimately build denser grids after choosing three, five, ten, and
twenty observables for each sample.
In Figure 1, we show our estimated constraint for each HOD parameter
(excluding $\mathrm{log}{M_{0}}$) as we choose successive observables. The
results for the $-19$ sample are shown in blue, and the results for the $-21$
sample are shown in red. The solid lines show the average constraint across
the four mocks used in the algorithm described above. In Table 4, we list the
observables chosen (in order) that we use for each sample (labeled “dHOD”). We
also list the observables chosen in S22 using a standard HOD model (i.e., no
assembly bias, labeled “sHOD”). The observables chosen in this work that were
not chosen in S22 are shown in bold.
Figure 2: Projected constraints (1-$\sigma$) of each clustering statistic
(combined with $n_{\mathrm{gal}}$) for each HOD parameter. The constraints for
the $-19$ and $-21$ mocks are shown in blue and red, respectively. The height
of each smaller vertical bar shows the projected constraints on one mock,
while the larger open bar shows the average constraint across four mocks.
After ordering the observables from greatest to least constraining power, we
need to choose the total number of observables to use in our analysis to
maximize our constraining power and minimize the noise in our procedure due to
building our covariance matrix from a finite number of mocks (i.e., 400). To
do this, we employ the same procedure as S22. Briefly, we estimate an
uncertainty associated with each projected constraint (for a given number of
observables $K$) by resampling the covariance matrix 100 times and then
importance sampling the chain with each of these resampled matrices. Doing so
lets us approximate the uncertainty in our constraints due to noise in the
covariance matrix for each combination of observables that we consider. The
shaded regions in each panel of Figure 1 show this uncertainty for each HOD
parameter as we increase $K$. We choose the lowest value of $K$ such that the
constraint at this value is within one standard error of the constraint at all
higher values of $K$. We require that this condition is met for all HOD
parameters (except $\mathrm{log}{M_{0}}$ for the $-21$ sample). The optimal
number of observables for each sample is indicated with a dot in each panel,
and the corresponding constraining power is shown with a dashed line. For the
$-19$ sample, the optimal number of observables is 36. For the $-21$ sample,
the optimal number of observables is 41. Thus, the size of our data vector for
each sample is 36 and 41, respectively.
Using these observables, we confirm that we can recover the truth when running
chains on mocks created with different HOD parameters (i.e., different amounts
of assembly bias) for each sample. In all of our validation tests with mocks,
all parameters for the -21 sample are always recovered within $1\sigma$, and
in the -19 sample all parameters are always recovered within or just outside
the $1\sigma$ region. Additionally, for both samples, the best-fit result is
always a good fit (i.e., it always has a p-value greater than 0.85).
Looking at the optimal observables in Table 4, it is noteworthy that for both
the $-19$ and $-21$ samples, the third observable (chosen to constrain only
$A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$) is a small bin of the average group
velocity dispersion function ($\sigma_{v}(N)$ 3 for $-19$ and $\sigma_{v}(N)$
1 for $-21$). It is also noteworthy that for both samples, the majority of the
first twenty observables chosen in this analysis (16/20 or 17/20) were also
chosen in S22 to constrain an HOD model without assembly bias. Meanwhile,
about half of the observables chosen beyond the initial twenty (8/16 or 9/21)
in this analysis are unique to the model with assembly bias (i.e., they were
not chosen in S22). This possibly indicates that the initial observables are
chosen for their ability to constrain the standard HOD parameters, while the
later observables are selected for their ability to constrain the assembly
bias parameters. This may also indicate that it is difficult to constrain
assembly bias until the standard HOD parameters are constrained, or that
assembly bias is a smaller signal on top of the global clustering signal.
For the $-19$ sample, the unique observables chosen for this analysis include
a large and small scale of $\xi(s)$, five scales of $P_{N}(R)$, and four large
scales of $\mathrm{mcf}(s)$. For the $-21$ sample, the unique observables
chosen for this analysis include two bins of $\sigma_{v}(N)$, two intermediate
scales of $w_{\mathrm{p}}(r_{\mathrm{p}})$, one small scale and two large
scales of $\mathrm{mcf}(s)$, one intermediate bin of $n(N)$, two large scales
of $\xi(s)$, and two bins of $\mathrm{VPF}(R)$. It is worth mentioning that
for the $-19$ sample, it is difficult to accurately constrain the decorated
HOD model until the parameter $\mathrm{log}{M_{0}}$ is constrained. This
occurs by about 15 observables, particularly after $\xi(s)$ 1 and
$w_{\mathrm{p}}(r_{\mathrm{p}})$ 4 are included. In the $-21$ sample, the
parameter $\mathrm{log}{M_{0}}$ remains unconstrained. This is consistent with
the results of S22, which found that constraining $\mathrm{log}{M_{0}}$ is
important for obtaining accurate results in the $-19$ sample, but not in the
$-21$ sample.
Figure 3: Parameter constraints for the SDSS $-19$ sample, using concentration
as the secondary halo property and the “dHOD” optimal observables (listed in
Table 4). The crosshairs in the third panel indicate
$A_{\mathrm{cen}}=A_{\mathrm{sat}}=0$ (i.e., no assembly bias).
Figure 4: Parameter constraints for the SDSS $-21$ sample, using concentration
as the secondary halo property and the 41 “dHOD” optimal observables (listed
in Table 4). The crosshairs in the third panel indicate
$A_{\mathrm{cen}}=A_{\mathrm{sat}}=0$ (i.e., no assembly bias).
Given the MCMC grids that we obtained from the first three observables in our
optimal selection algorithm, we can use importance sampling to estimate the
constraining power we would achieve for each HOD parameter had we run a chain
using only one clustering statistic (e.g., $w_{\mathrm{p}}(r_{\mathrm{p}})$)
plus $n_{\mathrm{gal}}$. We display the results of this exercise in Figure 2.
In each panel, the y-axis shows the projected constraint (1-$\sigma$) for a
particular HOD parameter as we use different clustering statistics. The
constraints for the $-19$ and $-21$ mocks are shown in blue and red,
respectively. The height of each smaller vertical bar shows the projected
constraints on one mock, while the larger open bar shows the average
constraint across four mocks.
For the central and satellite parameters, our results are similar (though not
identical) to the results from S22. For the assembly bias parameters, it is
interesting to note that for both samples, no single clustering statistic
provides significant constraining power for either $A_{\mathrm{cen}}$ or
$A_{\mathrm{sat}}$. $\xi(s)$ seems to have the most constraining power for
both $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$ in both samples, but it
performs only slightly better than the other clustering statistics.
Due to the nature of importance sampling, these results should be interpreted
as estimates, purely for visual purposes. However, this figure illustrates
that while no single clustering statistic provides significant constraining
power for assembly bias, the combination of different scales of each
clustering statistic is able to produce tighter constraints on the assembly
bias parameters than any one statistic.
Figure 5: Residuals between the best-fit model and the SDSS measurements for
the $-19$ (top) and $-21$ (bottom) samples. For each sample, the model
includes concentration-based assembly bias. We show residuals for all
observables, but the model was constrained using the “dHOD” optimal
observables for each sample (listed in Table 4), which are displayed with
larger points.
## 7 Results
### 7.1 Concentration-based Assembly Bias
Here we present the results from using the optimal observables identified in
the previous section to constrain the galaxy-halo connection in SDSS using a
decorated HOD model with concentration-based assembly bias. The results for
the $-19$ sample are shown in Figure 3, while the results for the $-21$ sample
are shown in Figure 4. Dark and light regions depict the 1- and 2-$\sigma$
regions, respectively. The best-fit parameters are listed in Table 5, along
with their corresponding p-values, as well as the results from the previous
S22 analysis for comparison. The constraints for each parameter are listed in
Table 6.
For the $-19$ sample, our best-fit results suggest positive central galaxy
assembly bias ($A_{\mathrm{cen}}$ = 0.793) and negative satellite galaxy
assembly bias ($A_{\mathrm{sat}}$ = -0.368). In other words, central galaxies
preferentially reside in halos with higher concentrations, while satellite
galaxies preferentially reside in halos with lower concentrations, at fixed
mass. This is consistent with previous results (e.g., Lange et al., 2022; Wang
et al., 2022) which also found positive central galaxy assembly bias and
negative satellite galaxy assembly bias when using concentration as the
secondary halo property. Additionally, this best-fit model yields a
significant decrease in tension compared to the results of S22 ($2.0\sigma$
compared to $4.5\sigma$). Unfortunately, even for our optimal combination of
observables, it is difficult to tightly constrain central galaxy assembly bias
for this sample (see the third panel in Figure 3). Despite the lack of tight
constraints on $A_{\mathrm{cen}}$, we are able to rule out a model with zero
assembly bias (i.e., the standard HOD model).
Figure 6: Parameter constraints for the SDSS $-19$ sample, from a model with
assembly bias (with concentration as the secondary halo property) and
satellite velocity bias, using the “dHOD” optimal observables (listed in Table
4). The crosshairs in the third and fourth panels indicate no assembly bias
and no velocity bias ($A_{\mathrm{cen}}=A_{\mathrm{sat}}=B_{\mathrm{vel}}=0$).
Figure 7: Parameter constraints for the SDSS $-21$ sample, from a model with
assembly bias (with concentration as the secondary halo property) and
satellite velocity bias, using the “dHOD” optimal observables (listed in Table
4). The crosshairs in the third and fourth panels indicate no assembly bias
and no velocity bias ($A_{\mathrm{cen}}=A_{\mathrm{sat}}=B_{\mathrm{vel}}=0$).
For the $-21$ sample, we obtain slightly tighter constraints on
$A_{\mathrm{cen}}$ than we are able to achieve in the $-19$ sample. However,
our best-fit results are consistent with zero assembly bias. Additionally,
this model does not result in any decrease in tension compared to the results
from S22.444In fact, the tension actually increased slightly compared to the
previous analysis, from $4.1\sigma$ up to $4.7\sigma$. This slight increase in
tension can be attributed to the change in observables between this work and
the previous work. This finding is consistent with the results of Beltz-
Mohrmann et al. (2020), which found assembly bias to be present in
hydrodynamic simulations for lower luminosity galaxies but not significant for
higher luminosity galaxies. It is thus to be expected that for the $-21$
sample, the addition of assembly bias parameters to the model did not result
in any relief of tension. Furthermore, the constraints on the standard HOD
parameters in the $-21$ sample do not change considerably compared to what
they were in S22, indicating that the addition of assembly bias has very
little affect on the outcome of the model.
In Figure 5 we show the deviation between each observable as measured on SDSS
(D) and on our best-fit model (M) for each sample. This deviation is shown as
a factor of the cosmic variance uncertainty, $\sigma_{\mathrm{obs}}$, for each
observable. This quantity is shown for all observables, where each point is a
different scale or bin of a given clustering statistic. Each clustering
statistic is plotted in a different color and is labeled on the x-axis. The
specific observables actually used in our analysis are plotted as larger bold
points. The results for the $-19$ sample are shown in the top panel, and the
results for the $-21$ panel are shown in the bottom panel. In the $-19$
sample, much of the remaining tension seems to be coming from several scales
of $\xi(s)$ and, to a lesser extent, $n_{\mathrm{gal}}$, $n(N)$,
$\mathrm{VPF}(R)$, and $\mathrm{SPF}(R)$. In the $-21$ sample, most of the
clustering statistics exhibit a high degree of tension at various scales.
Overall, the $-19$ sample exhibits a greater improvement in the observable
residuals compared to the S22 results than the $-21$ sample does, which
explains the greater overall reduction in tension seen in this sample.
The remaining tension found for both the $-19$ and $-21$ samples could
indicate that the HOD model needs to be made even more flexible with the
inclusion of spatial and velocity bias parameters (e.g., Beltz-Mohrmann et
al., 2020). Additionally, it is possible that a different secondary halo
property other than concentration could be more strongly correlated with
galaxy occupation and is thus a more appropriate choice for our assembly bias
model. It is also possible that accounting for the impact of baryonic physics
on the halo mass function could relieve some of the remaining tension.
Finally, these results are for a fixed cosmology sample; it is possible that a
slight change in cosmological parameter values could also result in a further
relief of this tension. We explore some of these possibilities in the
remaining sections.
Figure 8: Residuals between the best-fit model and the SDSS measurements for
the $-19$ (top) and $-21$ (bottom) samples. For each sample, the model
includes assembly bias (with concentration as the secondary halo property) as
well as velocity bias. We show residuals for all observables, but the model
was constrained using the “dHOD” optimal observables for each sample (listed
in Table 4), which are displayed with larger points.
### 7.2 Satellite Velocity Bias
Here we investigate whether adding satellite velocity bias to our HOD model
results in better agreement with SDSS. Using the same set of optimal
observables for each sample listed in Table 4, we run chains on our SDSS
samples using an HOD model with both concentration-based central and satellite
galaxy assembly bias (i.e., $A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$) and
additionally with satellite galaxy velocity bias ($B_{\mathrm{vel}}$). The
results for the $-19$ sample are shown in Figure 6, while the results for the
$-21$ sample are shown in Figure 7. The best-fit parameter values and
constraints are listed in Tables 5 and 6. Additionally, in Figure 8 we show
the deviation between each observable as measured on SDSS (D) and on our best-
fit model (M) for each sample, with the same layout as in Figure 5.
We do not identify a new set of optimal observables for constraining this new
model, but rather run chains for each of our SDSS samples using the same
optimal observables listed in Table 4 (“dHOD”). While choosing a new set of
optimal observables could potentially lead to tighter constraints on
$B_{\mathrm{vel}}$, we emphasize that our goal is not to optimally constrain
satellite velocity bias, but rather to allow $B_{\mathrm{vel}}$ to vary with
the hope of alleviating the lingering tension with SDSS. Additionally, we note
that our optimal set of observables already includes many measurements that
are sensitive to galaxy velocities (e.g., $\xi(s)$, $n(N)$, $\sigma_{v}(N)$)
and so should contain constraining power for $B_{\mathrm{vel}}$.
For the $-19$ sample, our best-fit results for this model indicate moderate
satellite galaxy velocity bias, with satellite galaxies moving slightly slower
than the dark matter distribution ($B_{\mathrm{vel}}$ = 0.898). However, when
velocity bias is included in the model, the strength of the assembly bias
signal is significantly reduced. This is in part due to the anti-correlation
between $B_{\mathrm{vel}}$ and (concentration-based) $A_{\mathrm{sat}}$, which
can be seen in the fourth panel of Figure 6: when $B_{\mathrm{vel}}=1$, lower
values of $A_{\mathrm{sat}}$ are preferred by the model, but when
$B_{\mathrm{vel}}$ is allowed to be less than 1, $A_{\mathrm{sat}}$ increases.
While the best-fit parameter values still suggest positive central assembly
bias and negative satellite assembly bias ($A_{\mathrm{cen}}$ = 0.825 and
$A_{\mathrm{sat}}$ = -0.251), the constraints on $A_{\mathrm{cen}}$ and
$A_{\mathrm{sat}}$ are much weaker, and we can no longer rule out a model with
zero assembly bias (though we can rule out a model with zero velocity bias).
Figure 9: Parameter constraints for the SDSS $-19$ sample, from a model with
assembly bias (with environment as the secondary halo property) and satellite
velocity bias, using the “dHOD” optimal observables (listed in Table 4). The
crosshairs in the third and fourth panels indicate no assembly bias and no
velocity bias ($A_{\mathrm{cen}}=A_{\mathrm{sat}}=B_{\mathrm{vel}}=0$).
Figure 10: Parameter constraints for the SDSS $-21$ sample, from a model with
assembly bias (with environment as the secondary halo property) and satellite
velocity bias, using the “dHOD” optimal observables (listed in Table 4). The
crosshairs in the third and fourth panels indicate no assembly bias and no
velocity bias ($A_{\mathrm{cen}}=A_{\mathrm{sat}}=B_{\mathrm{vel}}=0$).
The fact that adding velocity bias to our model reduces the strength of the
assembly bias signature is strong evidence in favor of having a sufficiently
flexible HOD model, without which we cannot claim to have made a robust
detection of assembly bias, nor can we hope to reliably constrain cosmology.
It is likely that our analysis is sensitive to $B_{\mathrm{vel}}$ because of
our use of clustering measurements that are particularly affected by galaxy
velocities (e.g., $\xi(s)$). It is possible that an analysis that does not
include these statistics would have less constraining power for
$B_{\mathrm{vel}}$ and therefore might not affirm the presence of velocity
bias.
After including $B_{\mathrm{vel}}$ as well as $A_{\mathrm{cen}}$ and
$A_{\mathrm{sat}}$ in our HOD model, the best-fit result for the $-19$ sample
exhibits a substantial decrease in tension, down from $2.0\sigma$ to
$1.4\sigma$. The $\chi^{2}/\mathrm{d.o.f}$ also decreased, from 1.53 to 1.27.
Thus, when we include both assembly bias and velocity bias in our model, our
clustering results are in agreement with SDSS. Looking at the top panel of
Figure 8 and comparing it to Figure 5, we can see that this relief in tension
comes from the improvement in observables across the board, but particularly
in $n_{\mathrm{gal}}$, $w_{\mathrm{p}}(r_{\mathrm{p}})$ 555We note that the
improvement in $n_{\mathrm{gal}}$ and $w_{\mathrm{p}}(r_{\mathrm{p}})$ comes
not from their relationship with satellite velocity, but rather from the
changes in other HOD parameters as a result of including velocity bias in the
model., $\xi(s)$, and $n(N)$.
For the $-21$ sample, our best-fit results for this model yield minimal
satellite galaxy velocity bias ($B_{\mathrm{vel}}$ = 0.976), as well as
minimal central and satellite galaxy assembly bias ($A_{\mathrm{cen}}$ = 0.144
and $A_{\mathrm{sat}}$ = -0.198). For this sample, the constraints on
$A_{\mathrm{cen}}$ and $A_{\mathrm{sat}}$ do not significantly degrade after
adding velocity bias to the model, and the constraints on $B_{\mathrm{vel}}$
comparable to what they are in the $-19$ sample. In spite of this, we cannot
claim a significant detection of assembly or satellite galaxy velocity bias in
the SDSS $-21$ sample. Once again, the constraints on the standard HOD
parameters remain roughly the same, suggesting that the further addition of
velocity bias to the model has negligible impact. Additionally, we are still
unable to relieve the tension present in the $-21$ sample even after adding
this new flexibility to the HOD model.
Figure 11: Residuals between the best-fit model and the SDSS measurements for
the $-19$ (top) and $-21$ (bottom) samples. For each sample, the model
includes assembly bias (with environment as the secondary halo property) as
well as velocity bias. We show residuals for all observables, but the model
was constrained using the “dHOD” optimal observables for each sample (listed
in Table 4), which are displayed with larger points.
Looking at the lower panel of Figure 8, we see very little improvement in our
residuals compared to Figure 5, which illustrates why the addition of velocity
bias to the model results in no relief in tension for this sample. It is
particularly noteworthy that even the statistics that contain velocity
information (like $\xi(s)$ and $\sigma_{v}(N)$) do not show any substantial
improvement after adding velocity bias to the model. While it is possible that
a different velocity bias prescription could lead to more improvement, it is
also possible that these dynamical clustering measurements are sensitive to an
issue that exists not within our HOD model but within our cosmological model.
### 7.3 Environment-based Assembly Bias
The previous section shows that central galaxy occupation is, at most, only
loosely tied to the concentration of the host halo. We next investigate
whether modeling assembly bias with a different halo property can improve the
goodness of fit of our model. Given that we do not know which halo property
other than mass most strongly affects the presence of a central galaxy in a
given halo, we choose to use local halo environment as our new assembly bias
property. This allows us to investigate the general assumption that central
galaxy occupation is tied to some halo property that is correlated with the
local environment. For this reason, environment is a useful property for
probing assembly bias and gives our model the flexibility that it needs to
model galaxy clustering without knowing the “true” halo property that leads to
assembly bias. We define local environment as the total mass in halos within a
5 $h^{-1}\mathrm{Mpc}$ radius.
Once again, we do not identify a new set of optimal observables for
constraining this environment-based assembly bias model, but rather run chains
for each of our SDSS samples using the same optimal observables listed in
Table 4 (“dHOD”). The results for the $-19$ sample are shown in Figure 9,
while the results for the $-21$ sample are shown in Figure 10. The best-fit
parameter values and constraints are listed in Tables 5 and 6.
For the $-19$ sample, our best-fit results once again indicate positive
central galaxy assembly bias ($A_{\mathrm{cen}}$ = 0.533), and negative
satellite galaxy assembly bias ($A_{\mathrm{sat}}$ = -0.224). In other words,
low-luminosity central galaxies preferentially reside in halos with denser
environments, while satellite galaxies preferentially reside in halos with
less dense environments, at fixed mass. We can rule out a model with no
central galaxy assembly bias at the 99% confidence level and a model with no
satellite galaxy assembly bias at the 95% confidence level. We again find that
satellite galaxies have velocities that are slightly slower than the dark
matter distribution ($B_{\mathrm{vel}}$ = 0.826). We can rule out a model with
no satellite velocity bias at the 99.9% confidence level. It is noteworthy
that when environment is used to model assembly bias instead of concentration,
$B_{\mathrm{vel}}$ and $A_{\mathrm{sat}}$ are correlated rather than anti-
correlated.
Additionally, the constraints on $A_{\mathrm{cen}}$ are much improved compared
to the constraints when using concentration, and the constraints on
$A_{\mathrm{sat}}$ are slightly improved. This improvement in constraints is
seen in spite of the fact that the observables used were chosen to optimally
constrain a concentration-based assembly bias model. We attribute this
improvement to the fact that environment is more directly associated with
clustering than concentration.
Table 5: SDSS best-fit results for different halo models
$M_{r}^{\mathrm{lim}}$ | Model | Obs. | $\mathrm{log}{M_{\mathrm{min}}}$ | $\sigma_{\mathrm{log}M}$ | $\mathrm{log}{M_{0}}$ | $\mathrm{log}{M_{1}}$ | $\alpha$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$ | $B_{\mathrm{vel}}$ | p-value | AIC
---|---|---|---|---|---|---|---|---|---|---|---|---
$-19$ | Standard HOD | S22 | 11.445 | 0.099 | 11.651 | 12.703 | 0.958 | – | – | – | $6.8\cdot 10^{-6}$ | 87.77
| ABcon | BM22 | 11.455 | 0.141 | 11.757 | 12.685 | 0.925 | 0.793 | -0.368 | – | 0.047 | 56.83
| ABcon + VB | BM22 | 11.474 | 0.132 | 11.877 | 12.715 | 0.950 | 0.825 | -0.251 | 0.898 | 0.155 | 51.54
| ABenv + VB | BM22 | 11.490 | 0.125 | 11.855 | 12.783 | 0.985 | 0.533 | -0.224 | 0.826 | 0.364 | 45.99
$-21$ | Standard HOD | S22 | 12.728 | 0.467 | 9.015 | 13.929 | 1.112 | – | – | – | $3.5\cdot 10^{-5}$ |
| ABcon | BM22 | 12.774 | 0.554 | 9.447 | 13.926 | 1.067 | -0.090 | -0.240 | – | $2.6\cdot 10^{-6}$ | 99.38
| ABcon + VB | BM22 | 12.756 | 0.525 | 9.804 | 13.915 | 1.108 | 0.144 | -0.198 | 0.976 | $1.8\cdot 10^{-6}$ | 100.96
| ABenv + VB | BM22 | 12.740 | 0.495 | 9.984 | 13.917 | 1.079 | -0.025 | 0.165 | 1.011 | $4.1\cdot 10^{-6}$ | 98.43
Note. — Best-fit HOD parameters for each SDSS sample using four different
models: the standard 5-parameter model, a model with concentration-based
assembly bias (“ABcon”), a model with concentration-based assembly bias plus
satellite velocity bias (“ABcon + VB”), and a model with environment-based
assembly bias plus satellite velocity bias (“ABenv + VB”). The Standard HOD
results are taken from S22 and thus use the S22 observables, while the chains
using extended HOD models use the optimal observables identified in this work
(listed in Table 4). We indicate the goodness-of-fit of each parameter
combination with a p-value, as well as assess the success of the model using
the AIC.
Ultimately, this model is an even better fit to the data than the model with
concentration-based assembly bias plus velocity bias ($0.9\sigma$ compared to
$2.0\sigma$), and we can rule out a model with zero assembly bias and zero
velocity bias. Looking at the top panel of Figure 11 and comparing it to
Figure 8, we can see that switching the assembly bias property from
concentration to environment leads to improvement in almost every observable
that we measure, which explains the reduction in tension. In particular,
$n_{\mathrm{gal}}$, $\xi(s)$, $\mathrm{mcf}(s)$, $\mathrm{VPF}(R)$, and
$\mathrm{SPF}(R)$ see sizeable improvement.
For the $-21$ sample, our best-fit results indicate negligible central galaxy
assembly bias ($A_{\mathrm{cen}}$ = -0.025) and minimal satellite galaxy
assembly bias ($A_{\mathrm{sat}}$ = 0.165), as well as negligible velocity
bias ($B_{\mathrm{vel}}$ = 1.011). A model with with no assembly bias and no
velocity bias is entirely consistent with the data. This means that for high-
luminosity galaxies, neither central nor satellite galaxies show any
meaningful preference toward local halo environment, and satellite galaxies
move with velocities similar to the dark matter within the halo.
Like in the $-19$ sample, the constraints on $A_{\mathrm{cen}}$ and
$A_{\mathrm{sat}}$ are substantially improved for the $-21$ sample when using
environment as the secondary halo property as opposed to concentration.
However, the constraints on $M_{\mathrm{min}}$ and $\sigma_{\mathrm{log}M}$
are actually degraded when environment-based assembly bias is used. Once again
we see no improvement in tension between our model and SDSS ($4.6\sigma$).
Looking at the lower panel of Figure 11, we see little to no improvement in
our residuals compared to Figure 8, which illustrates why switching the
assembly bias parameter from concentration to environment fails to reduce the
tension for this sample.
These results demonstrate that low-luminosity galaxies exhibit an assembly
bias signature that is present in some capacity regardless of the secondary
halo property used, although the exact strength of the central and satellite
assembly bias may differ for different secondary properties. Meanwhile, high-
luminosity galaxies do not display any assembly bias for either secondary halo
property. Moreover, the tension found between our model and SDSS is not easily
alleviated with a change in secondary halo property. Investigating many
different secondary halo properties for modeling assembly bias is beyond the
scope of this paper; however, given our results using concentration and
environment, we do not anticipate that some other secondary halo property
would alleviate all of the tension that we find in the $-21$ sample.
Table 6: SDSS Constraints
$M_{r}^{\mathrm{lim}}$ | Model | $\mathrm{log}{M_{\mathrm{min}}}$ | $\sigma_{\mathrm{log}M}$ | $\mathrm{log}{M_{0}}$ | $\mathrm{log}{M_{1}}$ | $\alpha$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$ | $B_{\mathrm{vel}}$
---|---|---|---|---|---|---|---|---|---
$-19$ | sHOD | $11.442^{+0.016}_{-0.015}$ | $0.106^{+0.074}_{-0.065}$ | $11.674^{+0.089}_{-0.094}$ | $12.691^{+0.028}_{-0.029}$ | $0.954^{+0.019}_{-0.019}$ | – | – | –
| ABcon | $11.469^{+0.019}_{-0.017}$ | $0.159^{+0.074}_{-0.077}$ | $11.750^{+0.093}_{-0.095}$ | $12.685^{+0.029}_{-0.031}$ | $0.930^{+0.025}_{-0.028}$ | $0.673^{+0.245}_{-0.529}$ | $-0.361^{+0.107}_{-0.103}$ | –
| ABcon+VB | $11.484^{+0.022}_{-0.021}$ | $0.121^{+0.072}_{-0.067}$ | $11.864^{+0.100}_{-0.099}$ | $12.724^{+0.037}_{-0.037}$ | $0.947^{+0.024}_{-0.028}$ | $0.338^{+0.472}_{-0.720}$ | $-0.194^{+0.145}_{-0.134}$ | $0.876^{+0.042}_{-0.041}$
| ABenv+VB | $11.481^{+0.021}_{-0.021}$ | $0.132^{+0.073}_{-0.052}$ | $11.786^{+0.119}_{-0.140}$ | $12.776^{+0.041}_{-0.040}$ | $0.986^{+0.027}_{-0.027}$ | $0.444^{+0.265}_{-0.190}$ | $-0.173^{+0.108}_{-0.115}$ | $0.857^{+0.041}_{-0.042}$
$-21$ | sHOD | $12.748^{+0.015}_{-0.015}$ | $0.517^{+0.029}_{-0.029}$ | $9.015^{+2.017}_{-2.036}$ | $13.919^{+0.014}_{-0.014}$ | $1.088^{+0.031}_{-0.033}$ | – | – | –
| ABcon | $12.737^{+0.019}_{-0.020}$ | $0.494^{+0.038}_{-0.040}$ | $8.980^{+2.019}_{-2.013}$ | $13.914^{+0.015}_{-0.015}$ | $1.110^{+0.035}_{-0.039}$ | $-0.236^{+0.290}_{-0.297}$ | $-0.148^{+0.181}_{-0.156}$ | –
| ABcon+VB | $12.747^{+0.020}_{-0.020}$ | $0.505^{+0.038}_{-0.040}$ | $9.525^{+1.676}_{-1.879}$ | $13.923^{+0.018}_{-0.017}$ | $1.111^{+0.043}_{-0.050}$ | $0.154^{+0.305}_{-0.283}$ | $-0.101^{+0.215}_{-0.208}$ | $0.971^{+0.047}_{-0.041}$
| ABenv+VB | $12.773^{+0.039}_{-0.043}$ | $0.546^{+0.067}_{-0.081}$ | $9.984^{+1.161}_{-1.456}$ | $13.925^{+0.022}_{-0.022}$ | $1.059^{+0.052}_{-0.062}$ | $0.014^{+0.038}_{-0.045}$ | $0.115^{+0.108}_{-0.118}$ | $1.000^{+0.050}_{-0.044}$
Note. — Marginalized constraints on SDSS for both samples using four different
models: the standard HOD model from S22 (using the optimal observables from
S22), an HOD model with concentration-based assembly bias, a model with
concentration-based assembly bias plus satellite velocity bias, and a model
with environment-based assembly bias plus satellite velocity bias, using the
optimal observables identified in this work. We present the median parameter
values along with upper and lower limits corresponding to the 84 and 16
percentiles respectively. All of these chains were run using the optimal
observables identified in this work (listed in Table 4).
### 7.4 Baryonic Effects
In this section, we present the results from applying the halo mass
corrections from Beltz-Mohrmann & Berlind (2021) to our halo catalogs and then
repeating our analysis using an HOD model with both assembly bias and velocity
bias. Specifically, we utilize the mass corrections for $M_{\mathrm{vir}}$
halos at $z=0$ according to the IllustrisTNG and EAGLE simulations, as well as
the environment-dependent mass correction from IllustrisTNG. (The EAGLE mass
correction shows very little environmental dependence, so we do not employ it
here.) The best-fit model parameters for these analyses are listed in Table 7.
The constraints on the model parameters remain similar in size after the
different mass corrections, and thus are not listed separately; we refer the
reader to Table 6 for the constraints on the assembly bias + velocity bias
(ABe+VB) model.
In Figure 12, we show the results of applying these halo mass corrections to
the $-19$ halo catalogs and performing our analysis. The panels show the model
parameters, using the same layout as in Figure 6. The original results (i.e.,
with no mass correction) are depicted in blue. The results from the EAGLE mass
correction are shown in yellow, the results from the TNG mass correction are
shown in green, and the results from the environment-dependent TNG mass
correction (“TNG,env”) are shown in purple. In Figure 13, we show the results
of applying these halo mass corrections to the $-21$ sample. The original
results (i.e., with no mass correction) are depicted in red, and the mass-
corrected results are shown with the same colors as in Figure 12.
For both samples, the mass corrections produce minimal changes to our HOD
parameter constraints (with the exception of $\mathrm{log}{M_{\mathrm{min}}}$,
which does experience significant shifts in each sample). Additionally, our
conclusions about the presence of assembly bias and velocity bias remain the
same for both samples after the mass corrections: the $-19$ samples exhibits
significant postive central assembly bias and negative satellite assembly
bias, as well as significant velocity bias, while the $-21$ sample exhibits no
such biases. Furthermore, the goodness-of-fit of the model remains roughly the
same after each of the mass corrections: an HOD model with environment-based
assembly bias and satellite velocity bias produces good agreement with the
clustering of low-luminosity SDSS galaxies, while the same model yields
significant tension with the clustering of high-luminosity SDSS galaxies. None
of the mass corrections are able to alleviate this tension.
It is unsurprising that the mass corrections mainly affect the best-fit value
of $\mathrm{log}{M_{\mathrm{min}}}$ in each sample, have a slight impact on
the other standard HOD parameters, and have a negligible affect on the
assembly bias and velocity bias parameters. This is because the mass
corrections shift the masses of our halos (albeit in a non-trivial way), and
so the parameter that governs the minimum halo mass that can host a galaxy
shifts to compensate. To a lesser extent, the parameter that governs the
scatter in this minimum halo mass ($\sigma_{\mathrm{log}M}$), and the
parameters that determine the number of satellite galaxies in a halo of a
given mass ($\mathrm{log}{M_{1}}$ and $\alpha$) also shift to compensate for
the halo mass correction. Meanwhile, the parameters that govern the dependence
of halo occupation on a halo property other than mass ($A_{\mathrm{cen}}$ and
$A_{\mathrm{sat}}$) and the parameter that governs the relative velocities of
the satellite galaxies to the dark matter ($B_{\mathrm{vel}}$) are unaffected
by changes to the halo mass function.
Figure 12: HOD parameter constraints for the SDSS $-19$ sample, from a model
with assembly bias (with environment as the secondary halo property) and
velocity bias, after applying three different mass corrections. The crosshairs
in the third and fourth panels indicate no assembly bias and no velocity bias
($A_{\mathrm{cen}}=A_{\mathrm{sat}}=B_{\mathrm{vel}}=0$).
Figure 13: HOD parameter constraints for the SDSS $-21$ sample, from a model
with assembly bias (with environment as the secondary halo property) and
velocity bias, after applying three different mass corrections. The crosshairs
in the third and fourth panels indicate no assembly bias and no velocity bias
($A_{\mathrm{cen}}=A_{\mathrm{sat}}=B_{\mathrm{vel}}=0$).
Overall, it is difficult to distinguish between any of the mass corrections
based on their agreement with the clustering of SDSS, nor would we rule out
any of these models of baryonic physics based on our results. It is possible
that with different clustering statistics, we could tighten some of our
constraints and thus differentiate between the results of the different mass
corrections. It is also possible that with a better fitting model for the
$-21$ sample, the effect of different mass corrections on the overall tension
could be seen. Ideally, we would be able to vary HOD parameters, cosmology
parameters, and mass correction prescriptions simultaneously and use our
results to rule out certain baryonic physics models; however, this is a
challenge left for future work.
Table 7: SDSS best-fit results with different mass corrections
$M_{r}^{\mathrm{lim}}$ | Model | Mass Correction | $\mathrm{log}{M_{\mathrm{min}}}$ | $\sigma_{\mathrm{log}M}$ | $\mathrm{log}{M_{0}}$ | $\mathrm{log}{M_{1}}$ | $\alpha$ | $A_{\mathrm{cen}}$ | $A_{\mathrm{sat}}$ | $B_{\mathrm{vel}}$ | p-value
---|---|---|---|---|---|---|---|---|---|---|---
$-19$ | ABenv + VB | – | 11.490 | 0.125 | 11.855 | 12.783 | 0.985 | 0.533 | -0.224 | 0.826 | 0.364
| ABenv + VB | TNG | 11.478 | 0.140 | 11.780 | 12.760 | 0.963 | 0.606 | -0.286 | 0.844 | 0.250
| ABenv + VB | EAGLE | 11.412 | 0.146 | 11.758 | 12.766 | 0.963 | 0.575 | -0.273 | 0.825 | 0.311
| ABenv + VB | TNG, env. | 11.481 | 0.122 | 11.777 | 12.777 | 0.971 | 0.506 | -0.228 | 0.816 | 0.301
$-21$ | ABenv + VB | – | 12.740 | 0.495 | 9.984 | 13.917 | 1.079 | -0.025 | 0.165 | 1.011 | $4.1\cdot 10^{-6}$
| ABenv + VB | TNG | 12.671 | 0.455 | 9.258 | 13.924 | 1.122 | -0.035 | 0.145 | 1.001 | $1.1\cdot 10^{-6}$
| ABenv + VB | EAGLE | 12.720 | 0.539 | 11.028 | 13.925 | 1.059 | -0.026 | 0.177 | 0.970 | $1.5\cdot 10^{-6}$
| ABenv + VB | TNG, env. | 12.745 | 0.575 | 8.594 | 13.914 | 1.053 | 0.023 | 0.074 | 0.998 | $8.5\cdot 10^{-7}$
Note. — Best-fit HOD parameters for each SDSS sample using different HOD
models as well as different mass corrections from Beltz-Mohrmann & Berlind
(2021). The model includes concentration-based assembly bias plus velocity
bias. All of these chains were run using the optimal observables identified in
this work (listed in Table 4). We list the best-fit values of each parameter
and indicate the goodness of fit of each parameter combination with a p-value.
## 8 Conclusions
In this work we have explored several extensions to the standard HOD model and
employed an optimal set of galaxy clustering measurements to constrain this
model for both high- and low-luminosity galaxies in SDSS. We first extended
the standard HOD model to include parameters for central and satellite galaxy
assembly bias, using halo concentration as the secondary halo property for
implementing this assembly bias. We identified a set of observables to best
constrain this model, using the algorithm laid out in Szewciw et al. (2022).
We then further extended our model to include an additional parameter for
satellite galaxy velocity bias and repeated our analysis for both SDSS samples
using this new model with both concentration-based assembly bias and satellite
velocity bias. We then repeated this analysis using local halo environment as
the assembly bias property instead of concentration. Lastly, we applied three
different halo mass corrections to our dark matter halos to account for the
impact of baryonic physics on the halo mass function; we repeated our analysis
by applying our extended halo model (with environment-based assembly bias and
satellite velocity bias) to these corrected halo masses. This is the first
time that an extended HOD modeling framework, with assembly bias and velocity
bias, a prescription to account for the impact of baryonic physics on the halo
mass function, and a variety of galaxy clustering statistics measured on a
wide range of scales, has been used to constrain the galaxy-halo connection in
the SDSS $-19$ and $-21$ samples. Our conclusions are listed below:
* •
Low-luminosity galaxies in SDSS exhibit both central and satellite galaxy
assembly bias when fit with an HOD model that includes concentration-based
assembly bias, with satellite galaxies displaying a negative dependence of
occupation on concentration and central galaxies displaying a positive
dependence on concentration (although central galaxy assembly bias is
difficult to constrain). With this model, we find evidence for satellite
assembly bias at the 99.8% confidence level. Additionally, this model is a
substantially better fit to the clustering of low-luminosity galaxies than the
standard HOD model (i.e., a model with no assembly bias).
* •
When fitting the clustering of low-luminosity galaxies with an HOD model that
includes both concentration-based assembly bias and satellite galaxy velocity
bias, we find evidence for satellite velocity bias at the 99.8% confidence
level, with satellite galaxies moving $\sim 10-15\%$ slower than the dark
matter. The assembly bias is quite unconstrained, making it difficult to rule
out a model with zero assembly bias. However, this model does further reduce
the tension with SDSS.
* •
When fit with an HOD model that instead uses environment-based assembly bias,
low-luminosity galaxies exhibit significant negative satellite assembly bias
and significant positive central assembly bias. Using environment also helps
to tighten the constraints on the assembly bias parameters. This model also
results in significant satellite velocity bias. We find evidence for satellite
assembly bias, central assembly bias, and satellite velocity bias at the 95%,
99%, and 99.9% confidence levels, respectively. This model ultimately results
in the tightest constraints on assembly bias and velocity bias, as well as the
best agreement with SDSS, with essentially no remaining tension.
* $\blacklozenge$
High-luminosity galaxies exhibit negligible assembly bias when using either
concentration or local environment as the assembly bias property (although the
constraints are once again tighter using environment.) They also exhibit
negligible satellite velocity bias when fit with a model that includes both
assembly bias and velocity bias. Additionally, none of these models yield good
agreement with SDSS ($>4\sigma$ tension).
* $\bigstar$
While each different treatment of baryonic physics leads to a slight change in
best-fit HOD parameters, none of them significantly change our conclusions
about the presence of assembly bias and velocity bias in each sample, nor do
they change the goodness-of-fit of the HOD model used. Thus, we cannot draw
any conclusions on the accuracy of our baryonic physics models based on this
analysis, nor can we use baryonic physics to explain the tension we find in
the $-21$ sample.
For low-luminosity galaxies, our results using either concentration or
environment are consistent with recent results from semi-analytic models and
hydrodynamic simulations (e.g., Artale et al., 2018; Zehavi et al., 2018; Bose
et al., 2019). Additionally, the presence of assembly bias and velocity bias
among low-luminosity galaxies but not among high-luminosity galaxies is
consistent with recent findings from semi-analytic models and hydrodynamic
simulations (e.g., Contreras et al., 2019, 2021, 2023; Beltz-Mohrmann et al.,
2020).
Our findings are also consistent with several recent observational studies.
For example, Zentner et al. (2019) used concentration to model assembly bias
in SDSS and found evidence for satellite assembly bias among faint galaxies
($M_{r}<-19$) but found no evidence for assembly bias in the $M_{r}<-21$
sample. Similarly, Vakili & Hahn (2019) used concentration to model assembly
bias in SDSS and detected moderate central assembly bias among faint galaxies
($M_{r}<-20.5,-20,-19.5$) but did not detect central galaxy assembly bias
among bright galaxies ($M_{r}<-21.5,-21$). Meanwhile, Salcedo et al. (2022)
instead used environment to model assembly bias in SDSS and similarly found no
evidence for assembly bias among bright galaxies. Wang et al. (2022) also used
concentration to model assembly bias in SDSS, and detected positive central
assembly bias for faint galaxies ($M_{r}<-20.5,-20,-19.5,-19$), and marginal
negative satellite galaxy assembly bias in the $M_{r}<-20$ and $M_{r}<-19$
samples, but did not detect assembly bias in the $M_{r}<-21$ sample.
The assembly bias signature among low-luminosity galaxies can be understood as
follows: Early forming halos ultimately contain fewer satellites, because they
acquired their satellites earlier, and thus these satellites were subject to
the destructive processes of the host halo (i.e., merging) for a longer period
of time (Zentner et al., 2005). Thus, it is reasonable that satellite galaxies
would preferentially reside in late-forming halos. Formation time is strongly
correlated with halo concentration, with early forming halos having higher
concentrations (e.g., Wechsler et al., 2002; Zhao et al., 2003), and thus
fewer satellite galaxies. This also explains why satellite galaxies
preferentially reside in halos with low-density environments: satellites
residing in high-density environments are more vulnerable to mergers, and thus
host halos in high-density environments will ultimately contain fewer
satellites.
Meanwhile, among Milky Way-sized halos, galaxies residing in higher
concentration halos tend to be more luminous (Zentner et al., 2019). This is
because at fixed mass, halos with higher concentrations have deeper potential
wells, allowing gravity to more strongly bind the stellar and gas contents of
these halos, possibly leading to more rapid star formation (or less
vulnerability to processes that suppress star formation). Thus, we can
understand why central galaxies residing in Milky Way-sized halos seem to
preferentially reside in high concentration halos: the deep potential well of
the host halo ultimately leads to a more luminous central galaxy, which is
more likely to pass our luminosity cut than a central galaxy living in a
shallow potential well. The same logic can be used to understand why central
galaxies prefer to reside in halos with high-density environments - such
environments are more conducive to merging, leading to a more luminous central
galaxy in the end.
The lack of assembly bias signature among high-luminosity galaxies has a few
possible explanations. It has been found that among Milky Way-sized halos, the
number of subhalos in a given host halo depends heavily on host halo
environment, but among cluster-mass halos, the abundance of subhalos exhibits
no such environmental-dependence (e.g., Zentner et al., 2019). This explains
why when using halo environment as our secondary halo property, we detect
satellite assembly bias among low-luminosity galaxies but not among high-
luminosity galaxies. Another possible explanation is that satellite galaxies
in the -21 sample are all recent additions to the halo, and so they have not
had enough time to be destroyed via mergers; thus, whether the host halo is
high- or low-concentration (or lives in a high- or low-density environment)
makes no difference for the presence of satellite galaxies in this sample.
This could also explain why we detect satellite galaxy velocity bias among
low-luminosity galaxies but not among high-luminosity galaxies: satellite
galaxies in the -19 sample have likely been slowed down via dynamical friction
over time, whereas satellite galaxies in the -21 sample are all more recent
acquisitions to the halo and thus have not had time to be significantly
affected by dynamical friction.
While several physical explanations are reasonable, the fact remains that
among low-luminosity galaxies the model that includes both assembly bias and
satellite velocity bias exhibits minimal tension; in other words, the model is
in good agreement with the clustering of $-19$ SDSS galaxies. This is
consistent with our expectation for low-luminosity galaxies, and with the
minimal tension found in previous studies of this nature for low-luminosity
galaxies. By contrast, the model is not in good agreement with the clustering
of $-21$ SDSS galaxies. This high degree of tension for high-luminosity
galaxies is in contrast with these previous studies, which did not find any
significant tension with SDSS. The larger constraining power of our results is
likely, in part, due to the large set of optimal clustering statistics that we
use.
This tension that we find among the higher luminosity sample could be
indicative of several things. For example, it is possible that central
galaxies do indeed exhibit significant velocity bias (e.g., Guo et al., 2015a,
b), and including this in the model would lead to better agreement with SDSS.
However, we have examined the impact on our observables after adding central
velocity bias to our best-fit model at the level found in previous works, and
found that this has a negligible effect on our clustering measurements. Thus,
if central velocity bias is indeed present in SDSS, it is not currently
detectable with our clustering measurements given our uncertainty due to
cosmic variance. It is also possible that galaxies do not trace the spatial
distribution of dark matter within halos (i.e., there is spatial bias Watson
et al., 2012; Piscionere et al., 2015). Additionally, the standard HOD model
assumes that the number of satellite galaxies in each halo is governed by a
Poisson distribution, but recent results indicate that this is probably not
the case (Boylan-Kolchin et al., 2010; Mao et al., 2015; Jiménez et al.,
2019).
It is also possible that a change in halo definition or halo finder could
alter our results. S18 repeated their analysis twice, once using $M_{200b}$
halos and again using $M_{vir}$ halos, and found similar results, with
$M_{vir}$ halos producing slightly tighter constraints. For this reason we
have proceeded using only $M_{vir}$ halos. Because a small change in halo
definition simply leads to a small change in mass for all halos, we think it
likely that the HOD parameters can compensate for any small change in halo
definition; in this case, our best-fit parameter values would change slightly,
but our overall conclusions about the presence of assembly bias or the
goodness-of-fit of our model would not change. However, a significant change
in halo definition or halo finder could potentially lead to changes in our
conclusions. For example, several works have found that the proper treatment
of splashback halos could lead to a reduction in the assembly bias signature
for low-luminosity galaxies (Villarreal et al., 2017; Mansfield & Kravtsov,
2020).
In future work, it is worth investigating whether accounting for these
possibilities leads to improved agreement between our model and the observed
clustering of high-luminosity galaxies. However, we think it unlikely that
accounting for these affects would be enough to explain the amount of tension
we are finding. In fact, in hydrodynamic simulations, the standard HOD model
proved to be a good fit to the clustering of high-luminosity galaxies,
provided that the model was applied to a DMO simulation with the same
cosmological model as the hydrodynamic simulation in question (Beltz-Mohrmann
et al., 2020). Thus, the fact that we find such significant tension in our
analysis between our best-fit HOD model and the clustering of high-luminosity
galaxies leads us to believe that there may be an issue with our cosmological
model. It is possible that our clustering statistics are able to detect such
an issue for our high-luminosity sample, but are not sensitive enough to pick
up on a cosmological discrepancy among low-luminosity galaxies. Such a result
would be consistent with the findings of several other recent analyses (e.g.,
Chapman et al., 2022; Lange et al., 2022; Wibking et al., 2020; Zhai et al.,
2022). For example, Lange et al. (2022) used an HOD model with both assembly
bias and velocity bias parameters to obtain cosmological constraints from the
BOSS LOWZ sample. Using $V_{\mathrm{max}}$ as their assembly bias property,
they did not find significant evidence of either central or satellite galaxy
assembly bias, and found only minimal evidence for central velocity bias and
no evidence of satellite velocity bias. However, they found that their best
cosmological constraints were slightly inconsistent with the Planck
observations. Similarly, Zhai et al. (2022) used the Aemulus suite of
cosmological N-body simulations to model the clustering of BOSS galaxies,
using an HOD model with both assembly bias (based on environment) and velocity
bias. They found some evidence for positive galaxy assembly bias but no
evidence for satellite galaxy velocity bias. Additionally, they found that
their cosmological constraints exhibited some tension with the Planck
observations.
In future work, we intend to explore whether a change in cosmological
parameters could be the key to alleviating this tension that we are finding.
It is worth noting that the Dark Energy Spectroscopic Instrument (DESI, DESI
Collaboration et al., 2016) will have better precision than the SDSS due to
its larger volume, allowing it to potentially detect even smaller differences
in clustering measurements. Applying our model to upcoming DESI data could
allow us to gain better constraints on our halo model parameters,
differentiate between different baryonic feedback implementations, and
ultimately constrain cosmology using small-scale galaxy clustering.
We would like to sincerely thank our anonymous referee for helpful comments
which improved the quality of this paper. This project has been supported by
the National Science Foundation (NSF) through Award (AST-1909631), and has
made use of NASA’s Astrophysics Data System; matplotlib, a Python library for
publication quality graphics (Hunter, 2007); scipy (Virtanen et al., 2020);
the ipython package (Pérez & Granger, 2007); astropy, a community-developed
core Python package for Astronomy (Astropy Collaboration et al., 2018, 2013);
numpy (Harris et al., 2020); pandas (McKinney, 2010, 2011), and chainconsumer
(Hinton, 2016). Funding for the SDSS and SDSS-II has been provided by the
Alfred P. Sloan Foundation, the Participating Institutions, the National
Science Foundation, the U.S. Department of Energy, the National Aeronautics
and Space Administration, the Japanese Monbukagakusho, the Max Planck Society,
and the Higher Education Funding Council for England. The SDSS Web Site is
http://www.sdss.org/. The SDSS is managed by the Astrophysical Research
Consortium for the Participating Institutions. The Participating Institutions
are the American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, University of Cambridge, Case Western Reserve University,
University of Chicago, Drexel University, Fermilab, the Institute for Advanced
Study, the Japan Participation Group, Johns Hopkins University, the Joint
Institute for Nuclear Astrophysics, the Kavli Institute for Particle
Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of
Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute
for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New
Mexico State University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States Naval
Observatory, and the University of Washington. The mock catalogues used in
this paper were produced by the LasDamas project
(http://lss.phy.vanderbilt.edu/lasdamas/); we thank NSF XSEDE for providing
the computational resources for LasDamas. The MCMCs in this work were run on
the Texas Advanced Computing Center’s Stampede2 supercomputer. Some of the
computational facilities used in this project were provided by the Vanderbilt
Advanced Computing Center for Research and Education (ACCRE). Parts of this
research were conducted by the Australian Research Council Centre of
Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through
project number CE170100013. These acknowledgements were compiled using the
Astronomy Acknowledgement Generator
(http://astrofrog.github.io/acknowledgment-generator/).
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|
# Vector meson photoproduction in UPCs with FoCal
A. Bylinkin1, J. Nystrand1 and D. Tapia Takaki2 1 Department of Physics and
Technology, University of Bergen, Bergen, Norway 2 Department of Physics and
Astronomy, University of Kansas, Lawrence, KS, USA<EMAIL_ADDRESS>
###### Abstract
We discuss the physics prospects of photon-induced measurements using the
high-granularity FoCal detector to be installed at the ALICE experiment,
covering the pseudorapidity interval $3.4\leq\eta\leq 5.8$. This new detector,
scheduled to be in operation from Run 4, will explore the small Bjorken-$x$
physics region in an unprecedented way. In this region the gluon saturation
phenomenon is expected to be dominant. Combined with the rest of the ALICE
subdetectors, including the zero degree calorimenters, FoCal will serve to
reconstruct in a model-independent way the measured photoproduction cross
sections for vectors mesons in a wide range of photon-target energies, down to
$x$ values of about $7\times 10^{-6}$ and $2\times 10^{-6}$ in ultra-
peripheral photon–proton and photon–lead collisions, respectively.
Keywords: ultra-peripheral collisions, UPC, gluon saturation, LHC Run 4
upgrades
## 1 Introduction
The FoCal detector [1] of ALICE (A Large Ion Collider Experiment) [2] at
CERN’s Large Hadron Collider will address fundamental physics questions about
the dynamics of partons inside nucleons and nuclei, exploring a unique
kinematic region at high energies where the growth of the gluon density is
expected to recede. This is a phenomenon known as gluon saturation that is
predicted by unitary principles in the theory of Quantum Chromodynamics (QCD),
the most compelling theory for describing the strong interactions of quarks
and gluons. Gluon saturated matter implies reaching a dynamical equilibrium
between the splitting and recombination of gluons. Such a regime can be
modelled with the introduction of non-linear QCD evolution equations [3, 4,
5]. It is of fundamental interest to experimentally determine the saturation
scale where this phenomenon appears, for different configurations in nucleons
and nuclei.
It is also of great interest to explore whether the existence of gluon
saturated protons implies the formation of a new state of matter with
universal features present in hadronic matter at high energies [6]. While
measurements at the HERA ep collider at the Deutsches Elektronen-Synchrotron
(DESY) have provided evidence for a significant rise of the gluon density in
protons as a function of energy or Bjorken-$x$ [7], the explored energies were
not sufficient to reach a consensus opinion on the formation of gluonic
saturated matter.
Early measurements from fixed target experiments, followed by studies at the
Relativistic Heavy-Ion Collider (RHIC) in Brookhaven National Laboratory (BNL)
[8, 9, 10, 11] indicate a reduction of the density of low momentum gluons in
heavy nuclei, as compared to the density in individual protons at high
energies. This experimental observation has been explained in terms of nuclear
gluon shadowing, a multi-re-scattering process present in nuclei that can be
linked to gluon saturation in nucleons [12]. The dynamics of the “initial
state” of nuclei before they collide impacts many measurements in nucleus-
nucleus collisions and is important for assessing the properties of the Quark
Gluon Plasma (QGP), a phase of QCD matter where quarks and gluons are
deconfined. For recent reviews see Refs. [13, 14].
The future Electron Ion Collider (EIC) [15] planned to be built at BNL will
focus on addressing several fundamental questions in nuclear physics [16],
including the study of gluon saturated matter. Being a dedicated QCD
accelerator, it promises detailed studies in a wide variety of reactions,
colliding species and energies. At the same time, several unique measurements
in small Bjorken-$x$ physics can only be carried out using the high energy
beams provided by the LHC. Since the end of the HERA collider, the highest
photon energies are being explored using ultra-peripheral heavy ion collisions
(UPC) at the LHC. The protons and heavy ions accelerated at the LHC carry
themselves electromagnetic fields which can be treated as a flux of quasi-real
photons. Using UPCs, the LHC can explore $\gamma\mathrm{p}$ center of mass
energies ($W_{\gamma\mathrm{p}}$) up to several TeV. The center of mass
energies for $\gamma\rm{A}$ and two-photon interactions go up to
$\sim\leavevmode\nobreak\ $700 GeV/nucleon and up to $\sim\leavevmode\nobreak\
$150 GeV, respectively. Altogether, such high energy collisions open the
possibility to study QCD physics and carry out searches for physics beyond the
standard model in an unexplored kinematic region [17, 18, 19, 20, 21, 22, 23].
The ALICE collaboration has recently completed a major detector and system
upgrade. The new data taking period has started this year with Run 3
(2022–2025) and will be followed by Run 4 (2029–2032). The current ALICE plans
are to collect data from Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.36$
TeV, pp collisions at various collision energies, p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=8$ TeV, and O–O interactions at
$\sqrt{s_{\mathrm{NN}}}=6.5$ TeV [24]. We anticipate an integrated luminosity
of about $\mathcal{L}$ = 13 nb-1 in Pb–Pb collisions in Runs 3 & 4, an order
of magnitude increase with respect to Runs 1 & 2\. The increase in the number
of reconstructed events for ALICE UPC measurements will be much more
significant because of the continuous detector read out implemented from Run
3. Despite the excellent performance and high-level productivity of the ALICE
UPC program, the use of dedicated triggers in Runs 1 & 2 has introduced
additional systematic uncertainties to the measurements, and some analyses
were not possible to be done due to trigger and bandwidth limitations. The
prospects for the UPC measurements at the LHC during Runs 3 & 4 are discussed
in Ref. [24]. Some future measurements have also been highlighted in Refs.
[25, 26].
In this context, and to fully exploit the potential of the ALICE experiment in
the study of the small-$x$ parton structure of nucleons and nuclei, a high-
granularity calorimeter (FoCal) has been proposed [1]. It is expected to be in
operation from Run 4. FoCal is a high-granularity, compact silicon-tungsten
(Si+W) sampling electromagnetic calorimeter with longitudinal segmentation
backed by a conventional high granularity metal and scintillating hadronic
calorimeter. It will be located at a distance of $7$ m from the interaction
point, outside the solenoid magnet. The Technical Design Report of FoCal is in
preparation and expected to be completed by summer 2023. FoCal has been
designed for measuring direct photons at forward rapidity with high precision,
as well as jets and $\gamma$-jet and jet-jet events, in pp and p–Pb collisions
[1].
The study of UPCs is also an integral part of the FoCal physics program. The
study of ultra-peripheral heavy-ion collisions with FoCal will contribute to a
more systematic and global study of nucleons and nuclei at high energies in
$\gamma\rm{p}$ and $\gamma\rm{Pb}$ interactions. Although the center of mass
energies in these interactions will be lower than in pp and PbPb, photon-
induced interactions have a clear advantage in probing the structure of the
target. In this document, we first present a brief review of recent UPC vector
meson measurements at the LHC. We then provide the expected yields of VM
photoproduction in the FoCal acceptance, and discuss some high-profile
measurements where FoCal will have a competitive advantage with respect to any
other LHC measurement.
## 2 Brief review of UPC vector meson photoproduction at the LHC
The ALICE, ATLAS, CMS and LHCb collaborations have carried out UPC
measurements in $\gamma\rm{p}$, $\gamma\rm{A}$, and two-photon interactions.
The cross sections for these processes at the LHC are very high. The studies
of photoproduced vector mesons have been of particular interest. One important
goal of these studies is to determine the nuclear gluon distributions. To
leading order perturbative QCD, there is a direct proportionality between the
photoproduction cross section of heavy vector mesons and the square of the
gluon distribution function [27]. Various attempts have been made in order to
translate the photonuclear production cross section to a gluon distribution,
including work towards next-to-leading order calculations of parton
distribution functions and generalized parton distributions [28, 29, 30].
Exclusive photoproduction of $\rm{J/\psi}$ off proton targets has been studied
at the LHC by ALICE in p–Pb collisions [31, 32] and by LHCb in pp collisions
[33, 34, 35]. For a $\rm{J/\psi}$ produced at rapidity $y$ it follows from the
kinematics that the photon-proton center of mass energy squared is
$\rm{W_{\gamma p}^{2}=2E_{p}M_{J/\psi}e^{\pm y}}.$ (1)
Here, $E_{p}$ is the proton energy in the laboratory frame. The $\pm$ in the
exponent corresponds to the cases when the photon emitting beam moves in the
positive and negative direction, respectively. In pp collisions, it is equally
probable that either beam particle emits the photon, whereas in p–Pb
collisions the Pb nucleus is the dominant photon source. This two-fold
ambiguity in determining $W_{\gamma p}$ is the main reason why p–Pb is the
preferred configuration to study photoproduction off a proton target. The
formula also shows that by going to higher rapidities higher values of
$W_{\gamma p}$ are obtained. From the $\gamma\rm{p}$ center of mass energy the
corresponding Bjorken-$x$ is determined, $x=\rm{(M_{J/\psi}/W_{\gamma
p})^{2}}$.
Exclusive $\rm{J/\psi}$ production on proton target has been studied at HERA
up to center of mass energies of 300 GeV, corresponding to $x\sim 10^{-4}$.
The ALICE measurement in p–Pb collisions has extended the energy range to
$W_{\gamma p}=706$ GeV.
Results from HERA showed that the photoproduction cross section,
$\sigma(\gamma p\rightarrow\rm{J/\psi}p)$, increases with center of mass
energy following a power law. The result from ALICE, extending down to $x\sim
10^{-5}$, showed no deviation from this behaviour, although the statistical
error was large.
While the LHCb collaboration has done studies for exclusive $\rm{J/\psi}$ in
pp collisions at $\sqrt{s}=7$ TeV [33, 34] and $\sqrt{s}=13$ TeV [35], their
analyses of $\sigma(\gamma\mathrm{p})$ are strongly model dependent because of
the ambiguity in the photon direction mentioned above. The photoproduction
cross section in a pp collision can be expressed as
$\frac{d\sigma}{dy}=n(+y)\sigma(\rm\gamma\rm{p,+y})+\rm{n(-y})\sigma(\gamma\rm{p,-y})\,,$
(2)
where $\rm{n(+y)}$ and $\rm{n(-y)}$ are the photon fluxes for positive and
negative rapidity, respectively. For each of the measured $d\sigma/dy$ points,
they report two “solutions” for the photon-proton cross section
$\sigma(\rm\gamma p)$: one for the photon emitter and one for the photon
target, corresponding to two different $W_{\gamma\mathrm{p}}$. It is well
known that the corresponding photon fluxes decrease rapidly as the rapidity
increases, thus most of their reported solutions are effectively from low
$W_{\gamma\mathrm{p}}$. In their 7 TeV measurements they have assumed that the
proton dissociation background is not $W_{\gamma\mathrm{p}}$ dependent, in
contradiction to the measurements performed by H1 and ALICE [36, 31]. The 13
TeV pp results from LHCb incorporated the use of a forward detector called
HeRSCheL to mitigate the background from proton dissociation, while the photon
direction ambiguity remains [35].
For these reasons, only measurements in UPC p–Pb collisions can provide a
model independent way for studying the energy dependence of
$\sigma({\gamma\mathrm{p}})$. While LHCb and ATLAS have not reported
measurements in UPC p–Pb collisions, CMS has reported results on exclusive
$\rho^{0}$ [37] and $\Upsilon$ [38] photoproduction. FoCal serves this program
by extending the exclusive $\rm{J/\psi}$ measurements further forward in
rapidity than what it is possible using the ALICE muon spectrometer and LHCb,
reaching down to $x\sim 7\times 10^{-6}$.
The measurements of coherent $\rm{J/\psi}$ photoproduction cross section in
UPC Pb–Pb collisions by the ALICE, CMS and LHCb collaborations have been found
to be compatible with moderate nuclear gluon shadowing, and in good agreement
with the central values of the EPS09 nuclear parton parameterization [39, 40,
41, 42]. Theoretical effort is underway for incorporating the data in parton
distributions [30], albeit some theoretical uncertainties remain [43].
Contrary to pp collisions, it might be possible to determine the photon
direction of the photon emitter and the photon target in Pb–Pb collisions via
the study of the neutron dependence using zero degree calorimeters (ZDCs).
Only ALICE, ATLAS and CMS have ZDCs, although no separation of the photon
sources has been reported at present. FoCal is special since can reach down to
$x\sim 2\times 10^{-6}$ for coherent $\rm{J/\psi}$ photoproduction by doing a
neutron-dependent analysis.
## 3 Physics measurements in p–Pb UPCs with FoCal
The cross sections and expected yields for exclusive production of heavy VMs,
$\rm{p+Pb\rightarrow p+Pb+VM}$, in the dielectron decay channel have been
calculated using STARlight[44], assuming the geometrical acceptance of FoCal
(both electrons within $3.4\leq\eta\leq 5.8$). The yields have been calculated
for both the p-Pb and Pb-p configurations, assuming an integrated luminosity
of $150\,\rm{nb^{-1}}$ in both cases. The results of these calculations are
shown in Table 1.
Hereafter, for the FoCal projections shown in all figures, we have considered
a 60% detector efficiency for the quarkonia measurements as discussed in the
Letter of Intent [1]. A 7% systematic uncertainty is added for the projected
points for exclusive $\rm{J/\psi}$ photoproduction, and 15% for $\rm{J/\psi}$
mesons with proton dissociation following the Run 2 measurements performed by
ALICE. These numbers include the uncertainty on the luminosity determination.
To obtain the uncertainty of the projected numbers, the statistical
uncertainties calculated from the expected yields have been added in
quadrature with the systematical uncertainties reported in ALICE Run 2
analyses. In addition, we only show data point projections when the yield,
before applying efficiency corrections, is at least 80 events per bin in
rapidity. The projected points are also smeared randomly according to the
Gaussian distribution with the width corresponding to the expected statistics
of each measurement. Finally, the FoCal energy reach corresponding to its
acceptance is shown as a box on the figures.
Table 1: Cross sections and yields for VMs calculated from STARlight. The top three lines correspond to the case when the proton is moving towards FoCAL, corresponding to the low energy regime ($11\leq W_{\gamma\rm{p}}\leq 36$ GeV for the $\rm{J/\psi}$). The lower three lines corresponds to the opposite configuration when the Pb-nucleus moves towards FoCAL, corresponding to the high energy regime ($1100\leq W_{\gamma\rm{p}}\leq 3600$ GeV for the $\rm{J/\psi}$). VM | $\sigma(\rm p+Pb\rightarrow p+Pb+VM)$ | $\sigma(3.4\leq\eta_{1,2}\leq 5.8)$ | Yield
---|---|---|---
| | p $\rightarrow$ FoCal | p $\rightarrow$ FoCal
$\rho^{0}$ | 35 mb | 140 nb | 21,000
$\phi$ | 1.7 mb | 51 nb | 7,700
$\rm{J/\psi}$ | 98 $\mu$b | 400 nb | 60,000
$\psi(2S)$ | 16 $\mu$b | 8.9 nb | 1,300
$\Upsilon(1S)$ | 220 nb | 0.38 nb | 60
| | Pb $\rightarrow$ FoCal | Pb $\rightarrow$ FoCal
$\rho^{0}$ | 35 mb | 17 nb | 2,600
$\phi$ | 1.7 mb | 5.3 nb | 800
$\rm{J/\psi}$ | 98 $\mu$b | 36 nb | 5,400
$\psi(2S)$ | 16 $\mu$b | 0.53 nb | 80
$\Upsilon(1S)$ | 220 nb | 0.67 pb | $\sim$ 0
Figure 1: ALICE data [45] (full red squares) on exclusive photoproduction of
$\rm{J/\psi}$ off protons as a function of $W_{\gamma\mathrm{p}}$, obtained in
p–Pb UPCs at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, compared to a power-law fit
(green band), to the data from HERA [46, 36] (full violet triangles), to the
STARlight projection (full black circles) in the FoCal acceptance, and to
theoretical models (see text). The LHCb solutions (open circles and squares)
for pp collisions at $\sqrt{s}=7$ TeV and $13$ TeV are also shown [33, 34,
35]. The uncertainties of the ALICE data are the quadratic sum of the
statistical and systematic uncertainties. The uncertainties of the STARlight
prediction are the quadratic sum of the statistical uncertainty of the
expected number of events (Table 1) multiplied by the reconstruction
efficiency and the systematic uncertainty taken at the level of ALICE Run 2
measurements [45]. The FoCal energy reach corresponding to its acceptance is
illustrated with a box.
Figure 1 shows the energy dependence of exclusive $\rm{J/\psi}$
photoproduction measured by ALICE, LHCb, H1 and ZEUS, and the projected data
expected by FoCal. The ALICE measurements have been obtained using ultra-
peripheral p–Pb collisions where there is no ambiguity in the photon
direction, which is not the case for symmetric systems such as pp collisions.
As mentioned above, LHCb has made several assumptions for extracting the
energy dependence of exclusive $\rm{J/\psi}$ in pp. Observing a deviation of a
power-law trend would require the most model-independent calculation possible.
For this figure, the projected FoCal data are shown for the case when the Pb
travels in the direction of the FoCal detector. The STARlight cross section is
a parameterization based on the HERA data, which follows a power-law trend. At
high energies, there are three sets of non-linear QCD models, namely, the Hot
Spot model (CCT) [47], the NLO BFKL [48] and the CGC-based calculations [49].
The data are fitted to the power-law function $N(W_{\gamma p}/W_{0})^{\delta}$
with $W_{0}=90$ GeV following the HERA measurements and $N$ and $\delta$ being
free parameters. The green band shows the experimental uncertainty obtained
from the fit to the existing ALICE data alone. The blue band corresponds to
the uncertainty obtained from the power-law fit to the ALICE data combined
with the STARlight points projected in the FoCal acceptance. Figure 2 shows
the ratio of the existing ALICE data, the STARlight projection and theoretical
models to the power-law fit to the existing ALICE data on exclusive
$\rm{J/\psi}$ photoproduction cross sections. The incorporation of the FoCal
data leads to a significant reduction in the measured uncertainty at high
energies, even in the case of no saturation scenario (STARlight projection).
Figure 2: Ratio of the ALICE data, STARlight projection and theoretical
models shown in Fig. 1 to the power-law fit through the ALICE data points. The
green band shows the uncertainty of the power-law fit to the existing ALICE
data while the blue band shows the reduction of the fit uncertainty at high
energy when the ALICE data is combined with the STARlight projected points.
The FoCal energy reach corresponding to its acceptance is illustrated with a
box.
The FoCal detector will provide access to an unexplored kinematic regime at
small-$x$ where a different trend in the growth of the cross section might
occur. For this reason the projection of the $\rm{J/\psi}$ photoproduction
cross section in the FoCal acceptance was also obtained using the NLO BFKL
model [48], as it is shown in Figure 3. Since saturation models are expected
to deviate from the power-law dependence, the following function is used to
fit the existing ALICE data together with the projected data points:
$\sigma(\gamma p)\approx\frac{N}{\frac{1}{W_{\gamma p}^{\delta}}+A},$ (3)
which has three free parameters: $N$ that gives the overall normalization,
$\delta$ that describes the power-law rise of the cross section at lower
energies, and $A$ that determines the saturation behavior (broken power-law)
at higher $W_{\gamma p}$ values. As one can see, such functional shape
provides a good description of both the existing ALICE data and the NLO BFKL
projection.
Figure 3: ALICE data [45] (full red squares) on exclusive photoproduction of
$\rm{J/\psi}$ off protons as a function of $W_{\gamma\mathrm{p}}$, obtained in
p–Pb UPCs at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, compared to a power-law fit
(green band), to the data from HERA [46, 36] (full purple triangles), to the
NLO BFKL projection (full black stars) in the FoCal acceptance, and to
theoretical models (see text). LHCb solutions (open circles and squares) for
pp collisions at $\sqrt{s}=7$ TeV and $13$ TeV are also shown [33, 34, 35].
The uncertainties of the ALICE data are the quadratic sum of the statistical
and systematic uncertainties. The uncertainties of the NLO BFKL prediction are
the quadratic sum of the statistical uncertainty of the expected number of
events (Table 1) multiplied by the reconstruction efficiency and the
systematic uncertainty taken at the level of ALICE Run 2 measurements [45].
The FoCal energy reach corresponding to its acceptance is illustrated with a
box.
To illustrate it in a more quantitative manner, Figure 4 shows the ratio of
the NLO BFKL projection to the same power-law fit to the ALICE data (shown in
Figure 3). Thus, one can conclude that if saturation occurs as predicted by
these models the expected precision of the FoCal measurement on exclusive
$\rm{J/\psi}$ photoproduction in UPC p–Pb collisions would be sufficient to
observe the deviation from the power-law trend at high energies.
Figure 4: Ratio of the ALICE data, NLO BFKL projection and theoretical models
shown in Fig. 3 to the power-law fit through the ALICE data points. The green
band shows the uncertainty of the power-law fit to the existing ALICE data
while the blue band shows the reduction of the fit uncertainty at high energy
when the ALICE data is combined with the NLO BFKL projected points and fitted
to the formula shown in Eq. 3. The FoCal energy reach corresponding to its
acceptance is illustrated with a box.
Recent theoretical calculations suggest that the ratio of the cross sections
between $\psi(2S)$ and $\rm{J/\psi}$ has a larger sensitivity to gluon
saturation than the individual cross section themselves [26]. This is partly
due to the different wave functions, and the radius dependence of the color
dipole for the two mesons. Figure 5 shows this measured ratio from H1 and ZEUS
data [50, 51], and the projected STARlight events within the FoCal acceptance.
The LHCb collaboration has not reported the measurement for this ratio with
systematic uncertainties [33, 34, 35], so no LHCb points are shown in Fig. 5.
The HERA points are compared to two different types of model calculations: the
calculations based on the color dipole model (GBW) and on the BGK model,
respectively [26]. For these two types of calculations, the linear and the
non-linear (saturation) predictions are shown. There is a very visible
separation between the linear and the non-linear calculation, making this
ratio a very promising measurement to observe gluon saturation at the LHC. The
HERA data at lower energies does not provide any conclusive determination. The
projected FoCal points obtained from STARlight agree with the linear-based
calculations, which is as expected since it does not include any saturation
effects. While this figure does not show the theoretical uncertainty bands
displayed on Figure 2, the comparison between these two different sets of
calculations, the GBW and BGK, provide some confidence that this effect can be
seen regardless of which of the two commonly used QCD models is considered.
The figure also shows the kinematic region that can be explored utilizing the
ALICE muon spectrometer during Run 3. Although there is a sensitivity to
observe saturation effects with the Run 3 data alone, a comprehensive science
program would require exploring the highest energy points which is only
possible with the use of FoCal. Also, there is an added value of having
independent measurements using two different detectors at different rapidity
intervals in Runs 3&4.
Figure 5: Energy dependence of the ratio between $\psi(2S)$ and $\rm{J/\psi}$
photoproduction cross sections. The points correspond to the HERA data (full
triangles) measured by the H1 and ZEUS collaborations, respectively [50, 51],
and to the STARlight projections (full circles). For the STARlight projected
points only the statistical uncertainty is shown based on the number of
expected events multiplied by the reconstruction efficiency. Theoretical
calculations based on BKG and GBW saturation models and their corresponding
linearized versions [26], are also shown. The two boxes illustrate the
kinematic regions that can be explored utilizing the ALICE muon detectors
during Run 3 and the ALICE FoCal detector during Run 4, respectively.
Besides the study of exclusive photoproduction of vector mesons, the
dissociative process has received recent interest. According to the Good-
Walker formalism [52, 53], dissociative diffraction probes the variance of the
different target configurations, contrary to exclusive processes that only
probes the average of such substructures. It has been shown that the momentum
transfer distribution of dissociative $\rm{J/\psi}$ in proton targets measured
by H1 [36] can be described by shape fluctuations (the BM model) [54, 55]. In
addition, the CCT model [47] considers that the proton target configurations
measured using the dissociative $\rm{J/\psi}$ process can be described by
gluonic hot spots, which as the energy increases, their number increases and
they start overlapping. When the hot spot overlap is significant, the
different configurations look similar and the variance should significantly
decrease in the gluon saturation regime. Figure 6 shows the ratio of
dissociative-to-exclusive $\rm{J/\psi}$ photoproduction as a function of
$W_{\gamma\rm{p}}$, including the H1 data [36] and ALICE preliminary data, and
comparisons to the BM and CCT predictions. The FoCal projected points shown
uses the STARlight yield for the exclusive process, and the BM model
prediction for the dissociative process. Note that STARlight does not make
predictions for dissociative photoproduction. Other more recent predictions
for the energy dependence of the proton geometry fluctuations have also been
reported [56]. The FoCal data are uniquely positioned to probe the
fluctuations of the proton target configurations for gluonic saturated matter.
The observation of a significant reduction of the measured cross section as
energy increases will provide a clear signature of gluon saturation at high
energies.
Figure 6: Energy dependence of the ratio of dissociative to exclusive
$\rm{J/\psi}$ photoproduction cross sections. The points correspond to the H1
measurement (full triangles), the preliminary ALICE data (full squares) and to
the projected data (open and full circles) for the measurements with the ALICE
muon detectors during Run 3 and the ALICE FoCal detector during Run 4,
respectively. The two boxes correspond to the kinematic regions that can be
explored in the FoCal measurements. Theoretical calculations based on the CCT
and BM saturation models are also shown.
Using the FoCal detector for the case of ultra-peripheral Pb–p collisions in
which the proton travels in its direction might also be of interest. At low
energy $W_{\gamma\rm{p}}$, there are several theoretical motivations related
to a better description of nucleon mass, quarkonia production and others [57].
Figure 7 (top) shows the projected $\rm{J/\psi}$ photoproduction cross
sections predicted by STARlight, together with the previous LHC, HERA and
fixed-target measurements. With FoCal the energies that can be achieved are
much lower than those from the current collider experiments, probing as low as
$W_{\gamma\rm{p}}$ = 12 GeV, where an onset of near-threshold effects in the
$\rm{J/\psi}$ photoproduction can already be seen. This measurement will also
provide a cross check, with unprecedented experimental precision, of the data
reported by the E401 experiment which is found to be significantly below the
current theoretical predictions in this kinematic region. Recently, the GlueX
experiment at the Thomas Jefferson National Accelerator Facility (JLab) has
performed precise measurements of near threshold $\rm{J/\psi}$ photoproduction
[58]. Their data are also shown in Fig. 7 as a function of $W_{\gamma\rm{p}}$.
There is a significant gap between these low-energy points and the existing
LHC data. Before the start of the future EIC, this kinematic region can be
explored by having a special Pb–p run at the LHC. We consider a potential
special run such that the proton and lead beams would be colliding at
$\sqrt{s_{\mathrm{NN}}}=1.26$ TeV. It is interesting to point out that with
only two days of machine operation, corresponding to a collected luminosity
$\mathcal{L}$ = 20 nb-1, the number of recorded events should be sufficient to
precisely map out this kinematic region, see Figure 7 (bottom). In this center
of mass energy range, the photon energy is low leading to very large fluxes
from the lead ion. This explains the large yields even for modest integrated
luminosities in this configuration.
Figure 7: ALICE data [45] (full red squares) on exclusive photoproduction of
$\rm{J/\psi}$ off protons as a function of $W_{\gamma\mathrm{p}}$, obtained in
Pb–p UPCs at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, compared to a power-law fit
(green band), to the data from HERA [46] (full blue triangles), to the
STARlight projection (full black circles) in the FoCal acceptance, and to
theoretical models (see text). LHCb solutions (open squares) for pp collisions
at $\sqrt{s}=7$ TeV and $13$ TeV are also shown together with various
measurements performed by fixed target experiments. The lower panel shows in
addition the STARlight projection for the Pb–p UPCs at
$\sqrt{s_{\mathrm{NN}}}=1.26$ TeV (full stars).
## 4 Physics measurements in Pb–Pb UPCs with FoCal
Exclusive photonuclear production of VMs has been studied by several
experiments at the LHC [39, 59, 60, 61]. The cross sections and expected
yields for exclusive production of heavy VMs in heavy-ion collisions,
$\rm{Pb+Pb\rightarrow Pb+Pb+V}$, in the dielectron decay channel have been
calculated from STARlight[44], assuming the geometrical acceptance of FoCal
(both electrons within $3.4\leq\eta\leq 5.8$). The yields have been calculated
assuming an integrated luminosity $\mathcal{L}$ = $7.0\,{\rm nb^{-1}}$ for LHC
Run 4. The results are shown in Table 2. The expected statistics for the
$\rm{J/\psi}$ is very large, and it should thus be possible to map out the
detailed shape of the rapidity distribution with high precision over the range
$3.4<y<5.8$.
A recent study, based on a next-to-leading order calculation, has found that
there is a strong contribution to the cross section not just from gluons but
also from quarks [30]. Figure 8 shows the prediction from this model (NLO)
compared with the existing ALICE and LHCb data. The figure also includes the
STARlight projection into the FoCal acceptance, scaled by a factor 0.42 as
documented in the recent CERN Yellow Report [24]. The latter is included since
STARlight, which does not include gluon shadowing, is known to overpredict the
measured cross section at midrapidity. As can be seen in the figure, the
interference between the quark and gluon contributions is largest in the
region covered by the FoCal acceptance.
Both STARlight and the NLO calculations show a bump in the cross section at
very forward rapidities ($y>5.3$), see Figure 8 (bottom). This bump comes from
the second diffractive peak in the nuclear form factor, which cuts off the
cross section for very low photon energies. This means that in this region the
$\rm{J/\psi}$ transverse momentum spectrum will be modified, leading to a
larger $<p_{T}>$ than for the bulk of the events at lower rapidities.
Table 2: Cross sections and yields for VMs calculated from STARlight assuming an integrated luminosity $\mathcal{L}$ = 7.0 nb-1. VM | $\sigma(\rm Pb+Pb\rightarrow Pb+Pb+VM)$ | $\sigma(3.4\leq\eta_{1,2}\leq 5.8)$ | Yield
---|---|---|---
$\rho^{0}$ | 5.0 b | 20 $\mu$b | 140,000
$\phi$ | 440 mb | 10 $\mu$b | 70,000
$\rm{J/\psi}$ | 39 mb | 53 $\mu$b | 370,000
$\psi(2S)$ | 7.5 mb | 1.1 $\mu$b | 7,500
$\Upsilon(1S)$ | 94 $\mu$b | 5.0 nb | 35
Figure 8: ALICE data [45] (full red squares) for the rapidity dependence of
the measured cross section for coherent $\rm{J/\psi}$ photoproduction in
ultra-peripheral Pb–Pb collisions compared to the recent theoretical
calculation (see text) [30], to the LHCb data (open circles) and to the
STARlight projection (full black circles) in the FoCal acceptance. The box
corresponds to the acceptance of the FoCal detector. The bottom figure shows
the same data in the logarithmic scale, where the contribution from the form
factor is more visible in the most forward region.
Contrary to analyses based on pp data at forward rapidity, it has been
suggested that in ultra-peripheral Pb–Pb collisions it is possible to
distinguish the photon direction by studying the emission of forward neutrons
that accompany the vector meson [24, 62]. The method to identify which of the
two Pb nucleus is the photon emitter or the photon target relies on the
measurement of $\rm{J/\psi}$ events in terms of the following nuclear break up
configurations: 0n0n (no neutron on both sides of the interaction point), Xn0n
or 0nXn (at least one neutron on one side of the interaction point and no
neutrons on the opposite side), and XnXn (at least one neutron on both sides
of interaction point). The measured cross section can be expressed as
$\frac{d\sigma}{dy}=\frac{d\sigma\rm(0n0n)}{dy}+2\frac{d\sigma\rm(0nXn)}{dy}+\frac{d\sigma\rm(XnXn)}{dy}\,,$
(4)
where for coherent $\rm{J/\psi}$ is reasonable to consider that the cross
sections are the same for the Xn0n and 0nXn configurations. The relative
fractions of the neutron break up modes can be computed using STARlight [44]
or nOOn [63]. Table 3 presents the predicted STARlight cross sections and the
expected yields.
Table 3: Cross sections and yields for coherent $\rm{J/\psi}$ photoproduction as a function of the different break up scenarios, calculated from STARlight assuming an integrated luminosity $\mathcal{L}$ = 7.0 nb-1. The small difference in the $\rm{J/\psi}$ yield compared to Table 2 is due to rounding errors in the calculation. Neutron configuration | $\sigma(\rm Pb+Pb\rightarrow\rm{J/\psi}+Pb+Pb)$ | $\sigma(3.4\leq\eta_{1,2}\leq 5.8)$ | Yield
---|---|---|---
0n0n | 28.8 mb | 47 $\mu$b | 329,000
0nXn + Xn0n | 7.3 mb | 5.0 $\mu$b | 35,000
XnXn | 3.0 mb | 2.0 $\mu$b | 14,000
For each of three differential equations shown on the left hand side of Eq. 4,
one can express them in terms of the total cross section
$\rm\sigma(\gamma+Pb)$ and the corresponding photon flux for both rapidity
directions using Eq. 2, thus resulting in three coupled equations. The
solution of this linear system results in the determination of
$\sigma(\rm\gamma\rm{Pb,+y})$ and $\sigma(\rm\gamma\rm{Pb,-y})$. In practice,
there are several ways how to solve the system of coupled equations. As
described in [24, 62], at least the 0n0n and 0nXn are needed since the XnXn
component only contributes to less than 4% of the yield. Also, the XnXn sample
might suffer from a large background from non-UPC events. It is also possible
to extract the cross sections $\sigma(\rm\gamma\rm{Pb,+y})$ and
$\sigma(\rm\gamma\rm{Pb,-y})$ by considering all the measurements together
using a $\chi^{2}$ fit or employing the singular value decomposition (SVD)
method [64].
For this study, we have considered the SVD method and that the experimental
uncertainties of the measurement are about 5% for 0n0n, about 6% for 0nXn (or
Xn0n) and about 14% for XnXn. As done above, a 60% efficiency correction is
applied beyond the acceptance effect. We have also required that there should
be about 80 $\rm{J/\psi}$ events in each of the neutron configurations. This
imposes a careful selection of the number of rapidity intervals, thus the
$\rm\sigma(\gamma+Pb)$ ranges, that can be studied. This results in different
rapidity bin sizes, specially for the lowest and highest energy points. We
found that in order to optimize the extraction of the high energy points (a
small fraction of the total sample since the photon flux decreases rapidly
with rapidity), only five rapidity intervals can be considered, resulting in
10 $\rm\sigma(\gamma+Pb)$ points. Figure 9 shows the $\sigma(\gamma\rm{Pb})$
cross section for coherent $\rm{J/\psi}$ photoproduction in ultra-peripheral
Pb–Pb collisions as function of $W_{\gamma\rm{Pb}}$, using the SVD method to
reconstruct the photon direction. The uncertainty of the measurement results
from propagating the experimental uncertainties of the $\rm d{\sigma}/{dy}$s
in the various neutron configurations. Clearly a different analysis of the
data could also be done so to optimize the low energy points, while here we
preferred to illustrate the projected statistics and uncertainties by
optimizing the extraction of the high energy points where the effects of non-
linear QCD should be dominant. Figure 9 also shows the predictions of the
impulse approximation, and the CCT and b-BK-A models [47, 65, 66]. The latter
models include gluon saturation effects, while STARlight consider Glauber-like
corrections not present in the impulse approximation. At present, no
experimental data exist for this measurement. Note that LHCb does not have
ZDCs installed at present or for Run 4, making ALICE, thanks to the FoCal
detector, the only experiment capable of carrying out this analysis, reaching
down to $x$ values of about $2\times 10^{-6}$.
Figure 9: STARlight projection (full black circles) for the cross section of
coherent $\rm{J/\psi}$ photoproduction in ultra-peripheral Pb–Pb collisions
shown as a function of the photon-lead center-of-mass energy
$W_{\gamma\rm{Pb}}$. To separate the two photon directions, this analysis will
use both the FoCal and ZDC detectors. Predictions from the impulse
approximation, CCT and b-BK models are also shown. The two boxes indicate the
FoCal detector acceptance for the two explored kinematic regions.
## 5 Other possible UPC measurements with FoCal
There are several additional UPC measurements where FoCal can contribute. Such
measurements include the study of low-mass vector mesons such as $\rho^{0}$
and $\phi(1020)$ [67], dielectron production from two-photon interactions,
excited $\rho$ states [68], open charm [69], inclusive photonuclear dijets
[70] and diffractive dijets [71]. The light-by-light scattering process could
also be explored [72] in future performance studies. UPC processes have also
been suggested as a way to probe the Einstein-Podolsky-Rosen relationship
[73]. It would be interested to utilize new techniques such as quantum
tomography [74] to analyze the UPC data in new ways. Finally, photon-induced
interactions might be of future interest in the search for physics beyond the
standard model.
## 6 Summary
The high-granularity FoCal calorimeter at ALICE will provide access to an
unexplored kinematic region where gluon saturation phenomena should be very
visible. We have provided the expected yield of vector meson photoproduction
in the dielectron decay channel. The expected statistics will be very large in
both ultra-peripheral p–Pb and Pb–Pb collisions. The future measurements of
the energy dependence of $\sigma(\gamma\rm{p})$ and $\sigma(\gamma\rm{Pb})$
for the photoproduction of $\rm{J/\psi}$ mesons should provide a clear
signature of gluon saturation if it is there. Moreover, FoCal will enable
measuring the energy dependence of the ratio of exclusive
$\psi(2S)$–to-$\rm{J/\psi}$, which is a measurement with high sensitivity. The
study of the dissociative production of exclusive $\rm{J/\psi}$ is sensitive
to quantum fluctuations of the proton constituents, thus also sensitive of
gluon saturation. Finally, we have discussed the prospects of studying the low
energy points of $\rm{J/\psi}$ photoproduction in UPC p–Pb collisions, and the
impact of having a short special run in p–Pb to explore the photoproduction of
$\rm{J/\psi}$ near threshold. These prospects have been discussed in the
context of existing and future measurements at LHC, concluding that FoCal
provides a unique physics program for these processes, probing the high-energy
limit of QCD in an unprecedented way.
## Acknowledgments
We thank colleagues from the ALICE UPC and FoCal groups for interesting
discussions, and Martin Hentschinski, Heikki Mäntysaari and the authors of
[43] for providing their theory predictions. We acknowledge support by the
Norwegian Research Council and the U.S. Department of Energy, Office of
Science, Nuclear Physics.
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|
A group action on higher Chow cycles on Kummer surfaces]A group action on higher Chow cycles on a family of Kummer surfaces
Ken Sato
Graduated School of Mathematical Sciences, The University of Tokyo
We construct a family of Kummer surfaces $\X^\circ \rightarrow T^\circ$ from the Legendre family of elliptic curves. Then we construct a family of higher Chow cycles on $\X^\circ \rightarrow T^\circ$ and calculate their values under the transcendental regulator map. For the calculation, we use a finite group action on $\X^\circ\rightarrow T^\circ$. We show that the rank of the space of the indecomposable cycles of $\X_t$ is greater than or equal to $18$ for very general $t\in T^\circ(\C)$. To show the linearly independence of indecomposable higher Chow cycles, we consider the image of normal functions associated to higher Chow cycles under a Picard-Fuchs differential operator of $\X^\circ\rightarrow T^\circ$.
§ INTRODUCTION
§.§ Contents of this paper
In the celebrated paper [4], Bloch defined higher Chow groups $\CH^p(X,q)$ for a variety $X$ over a field $k$. Higher Chow groups are a natural generalization of Chow groups. For a closed subvariety $Z\subset X$ of codimension $c$, the localization exact sequence of Chow groups
\begin{equation*}
\CH^{p-c}(Z)\rightarrow \CH^p(X)\rightarrow \CH^p(X-Z)\rightarrow 0
\end{equation*}
fits into the localization exact sequence of higher Chow groups
\begin{equation}
\cdots \rightarrow \CH^{p}(X,1)\rightarrow \CH^{p}(X-Z,1)\rightarrow \CH^{p-c}(Z)\rightarrow \CH^p(X)\rightarrow \CH^p(X-Z)\rightarrow 0.
\end{equation}
Thus higher Chow groups are an analogue of the singular cohomologies for algebraic varieties. Furthermore, there exists a canonical isomorphism
\begin{equation}
\CH^p(X,q)\otimes_{\Z} \Q\simeq H^{2p-q}_{\mathcal M}(X,\Q(p))
\end{equation}
where $H^{2p-q}_{\mathcal M}(X,\Q(p))$ is the motivic cohomology of $X$. Motivic cohomologies and higher Chow groups appear in many aspects of algebraic geometry and number theory. However, its structure is still mysterious for many varieties.In this paper, we study higher Chow cycles in $\CH^2(X,1)$ for a certain type of $K3$ surfaces, which are regarded as 2-dimensional analogues of elliptic curves. Higher Chow groups of general $K3$ surfaces are studied in [6]. We treat a special type of Kummer surfaces and study their higher Chow groups in detail.We consider the following map induced by the intersection product.
\begin{equation}\label{introintersect}
\CH^1(X,1)\otimes_\Z \CH^1(X) \lra \CH^2(X,1)
\end{equation}
Since $\CH^1(X)\simeq \mathrm{Pic}(X)$ and $\CH^1(X,1)\simeq \Gamma(X,\O_X^\times)$, the image of (<ref>) can be described by the known invariants. Hence we are interested in the cokernel $\CH^2(X,1)_\ind$ of (<ref>), which is called the indecomposable part of $\CH^2(X,1)$. In this paper, we give an estimate for the rank of $\CH^2(X,1)_\ind$. For the estimation, we construct elements in $\CH^2(X,1)$ explicitly, and consider their images under the following regulator map defined by Beilinson.
\begin{equation}\label{introreg}
\begin{tikzcd}
H^3_{\mathcal M}(X,\Q(2)) \arrow[r] & H^3_{\mathcal D}(X,\Q(2))
\end{tikzcd}
\end{equation}
Here the target $ H^3_{\mathcal D}(X,\Q(2))$ is the Deligne cohomology of $X$. In the articles [9],[13], [6], [8],[7] and [3], they consider families of varieties $\{X_t\}_{t\in T}$ and construct families of higher Chow cycles $\{\xi_t\}_{t\in T}$. Then they show that $\xi_t$ does not vanish for very general[We use the word “very general" for the meaning that “outside of a countable union of proper(= not the whole space) analytic subsets".] $t\in T$ by studying the behavior of the images of these cycles under the regulator map as a function of $t$. We follow this strategy.In this paper, we consider a family of Kummer surfaces $\X^\circ \rightarrow T^\circ$, which is constructed in Section 3. We construct a family of higher Chow subgroups $\Xi = \{\Xi_t \subset \CH^2(\X_t,1)\}_{t\in T^\circ}$ and compute their images under the following transcendental regulator maps $r$ at fibers.
\begin{equation}
\begin{tikzcd}
r: \CH^2(\X_t,1) \arrow[r] \arrow[d]& H^3_{\D}(\X_t^\an,\Z(2)) \arrow[r] & (H^{2,0}(\X_t^\an))^\vee/H_2(\X_t^\an,\Z) \\
\CH^2(\X_t,1)_\ind \arrow[urr,dashed] &
\end{tikzcd}
\end{equation}
Here the upper left map is the regulator map. The transcendental regulator map factors through $\CH^2(\X_t,1)_\ind$. Thus we can use the transcendental regulator map for the rank estimate for indecomposable parts. The main theorem of this paper is as follows.
$(\mathrm{Theorem}$ $\ref{mainthmpart3})$
For a very general $t\in T^\circ(\C)$,
\begin{equation}
\rank r(\Xi_t) = 18.
\end{equation}
Especially, $\rank \CH^2(\X_t,1)_\ind \ge 18$.
Since $\X^\circ\rightarrow T^\circ$ is a certain base change of the Kummer surface family treated in Section 6 of [6], $\CH^2(\X_t,1)_\ind \neq 0$ was already known for very general $t$. Theorem <ref> improves the estimate for the rank of $\CH^2(\X_t,1)_\ind$. While the construction of a higher Chow cycle in [6] is based on a certain elliptic fibration structure of $\X_t$, our construction of $\Xi_t$ is based on the fact that $\X_t$ is the minimal desingularization of a double covering of $\P^1_k\times_k \P^1_k$. Thus we give a new way of construction of higher Chow cycles on such type of Kummer surfaces in this paper. The merit of our construction is that the values of the transcendental regulator maps can be represented by relatively simple integrals. e.g. (<ref>)For the computation of the image of the transcendental regulator map, we construct topological chains on $\X_t^\an$ explicitly (Section 8) and use the formula obtained by Levine ([11]). By Levine's formula, the following multivalued holomorphic function appears in the image of an element of $\Xi$ under the transcendental regulator map (Proposition <ref>).
\begin{equation}\label{ldayo}
\Lc (a,b) = \int_{\triangle}\frac{dxdy}{\sqrt{x(1-x)(1-ax)}\sqrt{y(1-y)(1-by)}}
\end{equation}
Here $\triangle =\{(x,y)\in \R^2\::\: 0<y<x<1\}$. (<ref>) is similar to the integral representation of Appell's hypergeometric functions. A difference is that the boundary of the domain of integral is not necessarily contained in the branching locus of the integrand. In other words, (<ref>) is a kind of incomplete integrals.The Beilinson conjecture predicts that if $X$ is defined over a number field, the values (in a suitable sense) of the regulator map (<ref>) are related to the special values of $L$-functions of motives of $X$. Hence it is an interesting problem what kinds of functions appear in the image of the regulator map.Recently, in [2], Asakura and Otsubo gives examples of special varieties (which have hypergeometric fibrations) whose values of the regulator maps are represented by the value at $z=1$ of a generalized hypergeometric function ${}_3 F_2$. Furthermore, by deforming such varieties, they give a 1-dimensional family of varieties such that the value of the regulator map of members of such family is represented by generalized hypergeometric function ${}_3 F_2$ ([1]). Hence some relations between the value of the regulator map and hypergeometric functions were known. The object we treat in this paper can be regarded as a certain $\Q^{\oplus 18}$-extension of the exterior tensor product of two Gauss hypergeometric differential equations ${}_2F_1$.To compute the value of transcendental regulator for each element in $\Xi$, we use automorphisms of the Kummer surface family. We consider the following type of automorphisms of a family of algebraic varieties.
Let $X\rightarrow S$ be a family of algebraic varieties over a field $k$. The automorphism group $\Aut_k(X\rightarrow S)$ of $X\rightarrow S$ consists of a pair $(g,\underline{g})$ with $g\in \Aut_k(X)$ and $\underline{g}\in \Aut_k(S)$ such that the following diagram commutes.
\begin{equation}\label{autoasfamilydiagram}
\begin{tikzcd}
X \arrow[r,"g"]\arrow[d] & X \arrow[d] \\
S \arrow[r,"\underline{g}"] & S
\end{tikzcd}
\end{equation}
In this paper, we construct the following finite group action explicitly on the Kummer surface family $\X^\circ \rightarrow T^\circ$. Let $V$ be the Klein four-group and $\pi$ be the natural projection $\mathfrak S_4 \rightarrow {\mathfrak S}_4/V = {\mathfrak S}_3$. We set $G = (\{(h_1,h_2)\in \mathfrak{S}_4\times \mathfrak{S}_4: \pi(h_1) = \pi(h_2)\})^2$. We define a $\Z/2\Z$-extension $\widetilde{G}$ of $G$ (Definition <ref>).
$(\mathrm{Proposition}$ $\ref{mainprop1})$
The group $\widetilde{G}$ acts faithfully on the family $\X^\circ\rightarrow T^\circ$.
Then we construct a subgroup $\Xi^\can\subset \CH^2(\X^\circ,1)$ and define $\Xi$ as the sum of $\widetilde{\rho}_*\Xi^\can\:(\widetilde{\rho}\in \widetilde{G})$. The author is informed of the constructions of several elements in $\Xi$ by Terasoma in seminars. We generalize his idea of the constructions of higher Chow cycles so that we can use automorphisms of $\X^\circ\rightarrow T^\circ$.We compute the image of $\Xi$ under the regulator map by using $\widetilde{G}$-action as follows: since $\Xi$ is constructed as a family over $T^\circ$, we can define a “relative transcendental regulator map" $R_\omega$ (Definition <ref>)
\begin{equation}\label{reltransregin}
\begin{tikzcd}
R_\omega : \Xi \arrow[r] & \Qc_\omega(T^\circ)
\end{tikzcd}
\end{equation}
where $\Qc_\omega$ is a sheaf on $(T^\circ)^\an$ such that restriction of $\Qc_\omega$ at $t\in T^\circ(\C)$ is isomorphic to $(H^{2,0}(\X_t^\an))^\vee/H_2(\X_t^\an,\Q)$. The reason why $R_\omega$ is called “relative transcendental regulator" is that the restriction of $R_\omega(\Xi)$ at $t\in T^\circ(\C)$ coincides with $r(\Xi_t)$ modulo torsion part. This relative transcendental regulator map associates families of higher Chow cycles to (a generalization of) normal functions. Though this kind of maps can be defined in more general setting (cf.[15] and [6]), we employ an ad hoc definition since we need only the explicit description for special cases.We define a $\widetilde{G}$-action on $\Qc_\omega$ so that $R_\omega$ is equivariant under this action. Thus we reduce the computation of $r(\Xi_t)$ to that of $R_\omega(\Xi)$ and the $\widetilde{G}$-action on $\Qc_\omega(T^\circ)$. In Section 6, we construct two subgroups $\widetilde{I}\simeq (\mathfrak S_4\times_{\mathfrak S_3} \mathfrak S_4)\times (\Z/2\Z)$ and $\widetilde{G}_\fib\simeq (\Z/2\Z)^5$ of $\widetilde{G}$ which stabilize $R_\omega(\Xi^\can)\subset \Qc_\omega(T^\circ)$. Since $\Xi$ is defined as the sum of $\widetilde{\rho}_*\Xi^\can$, we can show that the rank of $R_\omega(\Xi)$ is at most 18 by examining the size of the stabilizer of $R_\omega(\Xi^\can)$. To show that the rank of $R_\omega(\Xi)$ is exactly 18, we consider the image of $R_\omega(\Xi)\subset \Qc_\omega(T^\circ)$ under a Picard-Fuchs differential operator
\begin{equation}
\begin{tikzcd}
\Dc: \Qc_\omega(T^\circ) \arrow[r] & \O(T^\circ)^{\oplus 2}.
\end{tikzcd}
\end{equation}
Similar methods are used in [13], [8] and [6]. We define a $\widetilde{G}$-action on $\O(T^\circ)^{\oplus 2}$ so that $\Dc$ is $\widetilde{G}$-equivariant. To prove the equivariance, we show the transformation formulae of Picard-Fuchs differential operators by $\widetilde{G}$-action (Proposition <ref>). This result is interesting by itself from the point of view of differential equations. Using a simple description of $\Dc\circ R_\omega(\Xi)$, we show that $\Dc\circ R_\omega(\Xi)$ has 18 $\Q$-linearly independent elements (Table <ref>). Thus we can show Theorem <ref>.
§.§ Outline of this paper
This paper is divided into 3 parts. Part 1 consists of Section 2, Section 3 and Section 4. The purpose of Part 1 is to fix the notation and to prove Proposition <ref>. In Section 2, we introduce a category $\Schgpk$, which is used to consider multiple finite group actions on multiple schemes simultaneously. In Section 3, we construct the Kummer surface family $\X\rightarrow T$. In Section 4, we prove Proposition <ref>.Part 2 consists of Section 5 and Section 6. The purpose of Part 2 is to explain the construction of $\Xi \subset \CH^2(\X^\circ,1)$ and consider the $\widetilde{G}$-action on $\Xi$. In Section 5, we first construct a subgroup of the higher Chow group $\Xi^\can\subset \CH^2(\X^\circ,1)$ and define $\Xi$ as the sum of its images under $\widetilde{G}$-action. In Section 6, we construct two subgroups $\widetilde{I}$ and $\widetilde{G}_\fib$ which stabilize the image of $\Xi^\can$ under the transcendental regulator map.The purpose of Part 3 is to prove Theorem <ref>. Part 3 consists of Section 7, Section 8 and Section 9. In Section 7, we fix relative differential forms $\omega$ on $\X^\circ\rightarrow T^\circ$ and examine $\widetilde{G}$-action on $\omega$. Furthermore, we find a Picard-Fuchs differential operator $\Dc$ which annihilates period functions of $\X^\circ\rightarrow T^\circ$. In Section 8, we calculate the image of an element of $\Xi^\can_t$ under the transcendental regulator map. In Section 9, we define the relative transcendental regulator map $R_\omega$ in (<ref>) and prove $\widetilde{G}$-equivariance of $\Dc$ and $R_\omega$. Finally, we prove Theorem <ref>.In Appendix A, we recall the definition of decomposable cycles in higher Chow groups and how decomposable cycles are represented by elements of the homology group of the Gersten complex (cf. Proposition <ref>).
§.§ Acknowledgement
The author expresses his sincere gratitudes to his supervisor Professor Tomohide Terasoma, who gave the author the idea of the construction of higher Chow cycles in Section 5 and also the idea of the construction of the topological 2-chains in Section 8 and let the author know a technique of checking the non-triviality of higher Chow cycles as in [13]. Furthermore, he gave the author many valuable comments which simplifies the arguments in this paper. He thanks Professor Shuji Saito sincerely, who gave the author many helpful comments on this paper and encouraged him during the doctoral course. He is also very grateful to Professor Takeshi Saito, who simplified the proof of Proposition <ref> and improved sentences in the draft. The author is supported by the FMSP program by the University of Tokyo.
§.§ Conventions
§.§.§ Conventions for algebraic geometry
* For a field $k$, a variety over $k$ is an integral separated scheme of finite type over $k$. For a variety $X$, its function field of $X$ is denoted by $R(X)$.
* For a morphism $X\rightarrow S$ and $s\in S$, we usually denote the fiber over $s$ by $X_s$.
For $\varphi \in \Hom_S(Y,X)$, $\varphi^\sharp$ denotes the morphism of sheaves of rings $\varphi^\sharp : \O_X\rightarrow \varphi_*\O_Y$.
* For $S$-schemes $Y$ and $X$, $\Hom_S(Y,X)$ denotes the set of $S$-morphisms. If $Y=\Spec R$, elements in $\Hom_S(Y,X)$ are called $R$-rational points and we also use the notation $X(R)$ for $\Hom_S(Y,X)$. The group of $S$-automorphisms of $X$ is denoted by $\Aut_S(X)$. For any morphism $S' \rightarrow S$, we have a natural map $\Hom_S(Y,X)\rightarrow \Hom_{S'}(Y\times_S S', X\times_{S} S')$. For a subset $\Sigma$ of $\Hom_S(Y,X)$, the image of $\Sigma$ under this map is called the base change of $\Sigma$ by $S'\rightarrow S$.
* For closed subschemes $Y_1$ and $Y_2$ of $X$ which satisfy $Y_1\cap Y_2 = \emptyset$, $Y_1\sqcup Y_2\subset X$ denotes the closed subscheme corresponding to the ideal sheaf $\I_{Y_1}\cap \I_{Y_2}$ where $\I_{Y_i}$ is the ideal sheaf corresponding to $Y_i$.
§.§.§ Conventions for group theory
* In this paper, we always consider left group actions. For a group $G$, the opposite $G$-action is a (left) action of the opposite group $G^{\mathrm{op}}$. Let $G$ be a group and $M$ be an abelian group with a $G$-action. For a subgroup $N\subset M$, the $G$-action of $M$ stabilizes $N$ if and only if for any $g\in G$ and $n\in N$, we have $g\cdot n\in N$.
* For a set $\Sigma$, $\mathfrak{S}(\Sigma)$ denotes the symmetric group of $\Sigma$. For $n\in \Z_{\ge 1}$, $\mathfrak{S}_n$ denotes the symmetric group of the set $\{0,1,\dots,n-1\}$. For $\sigma \in \mathfrak{S}(\Sigma)$, $\sgn(\sigma)\in \{\pm 1\}$ denotes its image under the sign character of $\mathfrak{S}(\Sigma)$.
* For a set $A$ and an abelian group $M$, the set of maps from $A$ to $M$ is denoted by $M^A$. The set $M^A$ has a natural structure of an abelian group.
§.§.§ Others
* For a set $\Sigma$, $|\Sigma|$ denotes the cardinality of $\Sigma$.
* For a ring $A$, the multiplicative group of $A$ is denoted by $A^\times$. If $A$ is an integral domain, its fraction field is denoted by $\Frac(A)$.
* For $n\in \Z_{>1}$ and a field $k$, $\mu_n(k)$ denotes the subgroup of $k^\times$ consisting of $n$-th roots of unity.
* We use the symbol $\lrcorner$ for the fiber product as follows.
\begin{equation}
\begin{tikzcd}
X\times_S Y \arrow[d,"pr_1"']\arrow[r,"pr_2"]\arrow[dr, phantom, "\lrcorner",very near start] & Y \arrow[d] \\
X \arrow[r] & S
\end{tikzcd}
\end{equation}
§ GENERALITIES OF DISCRETE GROUP ACTIONS ON SCHEMES
In this section, we introduce a category $\Schgpk$ of schemes with group actions and prove some properties which we use in Section 4 to construct group actions on a family of Kummer surfaces.All results in this section are more or less formal and proofs are often straightforward. Hence we omit proofs or give only sketches. Throughout in this section, we fix a field $k$ and assume all schemes and morphisms are over $k$.
§.§ Schemes with group actions
(The definition of $\Schgpk$)
* A scheme with a group action $(S,H,\varphi)$ is a triplet consisting of a $k$-scheme $S$, a group $H$ and a group homomorphism $\varphi: H\rightarrow \Aut_k(S)$. We usually omit $\varphi$ from the notation and write $(S,H)$. In that case, we use the same symbol for $h\in H$ and its image in $\Aut_k(S)$.
* A pair $(f,\psi)$ of a morphism of $k$-schemes $f:T\rightarrow S$ and a group homomorphism $\psi: G\rightarrow H$ is called a morphism of schemes with group actions from $(T,G)$ to $(S,H)$ if the following diagram commutes for any $g\in G$.
\begin{equation}
\begin{tikzcd}
T \arrow[r,"f"]\arrow[d,"g"] & S \arrow[d,"\psi(g)"] \\
T \arrow[r,"f"] & S
\end{tikzcd}
\end{equation}
Then we have a category $\Schgpk$ of schemes with group actions by the natural composition of morphisms.
* Let $(S,H)\in \Schgpk$. For a $S$-scheme $X$, we define $\Aut(X;S,H)$ as the following group.
\begin{equation}\label{defnautoxsh}
\Aut(X;S,H) = \left\{(\mu, \nu) \in \Aut_k(X)\times H :
\begin{tikzcd}
X \arrow[r] \arrow[d,"\mu"] & S \arrow[d,"\nu"] \\
X \arrow[r] & S
\end{tikzcd}
\quad\text{commutes.}
\right\}
\end{equation}
By the natural projection $\Aut(X;S,H)\rightarrow \Aut_k(X)$ and $\Aut(X;S,H)\rightarrow H$, we have the following object and morphism in $\Schgpk$.
\begin{equation}
\begin{tikzcd}
(X,\Aut(X;S,H))\arrow[r] & (S,H)
\end{tikzcd}
\end{equation}
* For a morphism $(f,\varphi): (X,G) \rightarrow (S,H)$ be a morphism in $\Schgpk$, we have a group homomorphism[See Definition <ref> for the notation $\Aut_k(X\rightarrow S)$.]
\begin{equation}
G\lra \Aut_k(X\rightarrow S); g \mapsto (g, \varphi(g)).
\end{equation}
If the $G$-action on $X$ is faithful, this group homomorphism is injective.
In this paper, we often use the following fiber product construction in $\Schgpk$.
Consider the following diagram in $\Schgpk$.
\begin{equation}
\begin{tikzcd}
& (S_1,H_1) \arrow[d,"(f_1\mathrm{,}\varphi_1)"] \\
(S_2,H_2) \arrow[r,"(f_2\mathrm{,}\varphi_2)"] & (S_3,H_3)
\end{tikzcd}
\end{equation}
Then the fiber product $(S_1,H_1)\times_{(S_3,H_3)}(S_2,H_2)$ exists and isomorphic to $(S_1\times_{S_3} S_2 ,H_1\times_{H_3} H_2)$. Here $H_1\times_{H_3} H_2$ is the fiber product of groups. i.e.
\begin{equation}
H_1\times_{H_3} H_2 = \{(h_1,h_2)\in H_1\times H_2: \varphi_1(h_1)=\varphi_2(h_2)\}.
\end{equation}
* Let $(X,G)\rightarrow (S,H)$ be a morphism in $\Schgpk$. For $g\in G$, $\underline{g}$ denotes its image in $H$. A subset $\Sigma$ of $\Hom_S(S,X)$ is compatible with $(X,G)\rightarrow (S,H)$ if and only if for any $\sigma\in \Sigma$ and $g\in G$, $g\circ\sigma\circ\underline{g}^{-1}\in \Sigma$.
* If $\Sigma$ is compatible with $(X,G)\rightarrow (S,H)$, we have a $G$-action on $\Sigma$ defined by
\begin{equation}\label{settheoreticaction}
G\times \Sigma \lra \Sigma;\:\:(g, \sigma) \mapsto g\circ \sigma\circ\underline{g}^{-1}.
\end{equation}
We can keep track this group action on $\Sigma$ after fiber product operations.
* Let $(X_i,G_i)\rightarrow (S_i,H_i)$ be a morphism in $\Schgpk$ for $i=1,2$. Put $(S,H) =(S_1,H_1)\times (S_2,H_2)$ and $(X,G) = (X_1,G_1)\times (X_2,G_2)$. Then we have the following morphism.
\begin{equation}
\begin{tikzcd}
(X_1,G_1) \arrow[d] &(X,G) \arrow[d,dashed] \arrow[l,"pr_1"']\arrow[r,"pr_2"] & (X_2,G_2)\arrow[d] \\
(S_1,H_1)& (S,H) \arrow[l,"pr_1"']\arrow[r,"pr_2"] & (S_2,H_2)
\end{tikzcd}
\end{equation}
Suppose $\Sigma_i\subset \Hom_{S_i}(S_i,X_i)$ is compatible with $(X_i,G_i)\rightarrow (S_i,H_i)$ for $i=1,2$. Then
\begin{equation}
\Sigma = \{\sigma_1\times \sigma_2: \sigma_1\in \Sigma_1,\sigma_2\in \Sigma_2\}\subset \Hom_S(S,X)
\end{equation}
is compatible with $(X,G)\rightarrow (S,H)$. The $G$-action on $\Sigma$ is given by
\begin{equation}
G\times \Sigma \lra \Sigma; \:\:((g_1,g_2),(\sigma_1,\sigma_2))\mapsto (g_1\cdot\sigma_1,g_2\cdot\sigma_2).
\end{equation}
* Consider the following fiber product diagram in $\Schgpk$.
\begin{equation}
\begin{tikzcd}
(X',G')\arrow[dr,phantom,"\lrcorner",very near start] \arrow[r] \arrow[d] & (S',H') \arrow[d]\\
(X,G) \arrow[r] & (S,H)
\end{tikzcd}
\end{equation}
Suppose $\Sigma\subset \Hom_S(S,X)$ is compatible with $(X,G)\rightarrow (S,H)$. Then its base change $\Sigma' \subset \Hom_{S'}(S',X')$ is compatible with $(X',G')\rightarrow (S',H')$. Furthermore, the natural map $\Sigma\rightarrow \Sigma'$ is $G'$-equivariant.
§.§ Linearizations of $\O_X$-modules
We recall the definition of $G$-linearizations of $\O_X$-modules. In some references, $\O_X$-module with a $G$-linearization is called $G$-equivalent sheaf.
Let $(X,G)\in \Schgpk$ and $\L$ be an $\O_X$-module. A $G$-linearization of $\L$ is a collection of $\O_X$-module isomorphisms $\{\Phi_{g}: g^*\L\xrightarrow{\sim} \L\}_{g\in G}$ such that for any $g,h\in G$, the following diagram commutes.
\begin{equation}\label{cocyclecond}
\begin{tikzcd}
(g\circ h)^* \L \arrow[d,"\Phi_{gh}"] & h^*g^*\L \arrow[d,"h^*(\Phi_g)"]\arrow[l,"\sim"'] \\
\L & h^* \L \arrow[l, "\Phi_h"]
\end{tikzcd}
\end{equation}
The commutativity of (<ref>) is called the cocycle condition.
Sheaves of relative differentials are fundamental examples of linearized sheaves.
Let $(f,\varphi): (X,G)\rightarrow (S,H)$ be a morphism in $\Schgpk$. We have a canonical $G$-linearization $\{\Phi_g\}_{g\in G}$ of the sheaf of differentials $\Omega_{X/S}^1$.
For $g\in G$, we have the following diagram.
\begin{equation}
\begin{tikzcd}\label{universalitydifferential}
X \arrow[r,"g"]\arrow[d,"f"] & X \arrow[d,"f"] \\
S \arrow[r,"\varphi(g)"] & S
\end{tikzcd}
\end{equation}
By the universality of the sheaf of differentials, we have an $\O_{X}$-module homomorphism $g^*\Omega_{X/S}^1\rightarrow \Omega_{X/S}^1$. By the universality, this satisfies the cocycle condition.
We list constructions of new linearized sheaves from other linearized sheaves.
Let $(X,G) \in \Schgpk$ and $\L$ be an $\O_X$-module with a $G$-linearization $\{\Phi_g\}_{g\in G}$.
* Let $(f,\varphi): (Y,H)\rightarrow (X,G)$ be a morphism in $\Schgpk$. For $h\in H$, put
\begin{equation}
f^*\Phi_{\varphi(h)} : h^*(f^*\L) \simeq (f\circ h)^*\L = (\varphi(h)\circ f)^*\L \simeq f^*\varphi(h)^*\L\xrightarrow{f^*\Phi_{\varphi(h)}} f^*\L.
\end{equation}
Then $\{f^*\Phi_{\varphi(h)}\}_{h\in G}$ is a $H$-linearization of $f^*\L$.
* Let $\mathscr M$ be a $\O_X$-modules with $G$-linearization $\{\Psi_g\}_{g\in G}$. For $g\in G$, put
\begin{equation}
\Phi_g\otimes \Psi_g : g^*(\L\otimes_{\O_X} \mathscr M)\simeq g^*\L\otimes_{\O_X} g^*\mathscr M\xrightarrow{\Phi_g\otimes \Psi_g} \L\otimes_{\O_X} \mathscr M
\end{equation}
Then $\{\Phi_g\otimes \Psi_g\}_{g\in G}$ is a $G$-linearization on $\L\otimes_{\O_X}\mathscr M$.
* Assume that $\L$ is invertible sheaf. For $g\in G$, put
\begin{equation}
\Phi^{\otimes(-1)}_g : g^*\mathcal{H}om_{\O_X}(\L,\O_X) \simeq \mathcal{H}om_{\O_X}(g^*\L,\O_X) \xrightarrow{(\Phi_g^{-1})^\vee } \mathcal{H}om_{\O_X}(\L,\O_X)
\end{equation}
Then $\{\Phi_g^{\otimes(-1)}\}_{g\in G}$ is a $G$-linearization of $\L^{\otimes(-1)}$.
The group cocycles have close relations with sheaves with linearizations. In this paper, explicit cocycle calculations play an important role for the main result.
Assume an abelian group $M$ has an opposite $G$-action. An opposite 1-cocycle on $M$ is a 1-cocycle of $G^\op$ on $M$. In other words, an opposite 1-cocycle is a map $\chi: G\rightarrow M$ which satisfies the following condition: For any $g,h\in G$,
\begin{equation}\label{oppositecocycleequation}
\chi(gh) = \chi(h) + h\cdot (\chi(g)).
\end{equation}
Let $(X,G)\in \Schgpk$. We have a natural opposite $G$-action on the $k$-algebra $\Gamma(X,\O_X)$ defined by
\begin{equation}
G\times \Gamma(X,\O_X) \rightarrow \Gamma(X,\O_X); (g, a)\mapsto g^\sharp(a).
\end{equation}
We also have an opposite $G$-action on the abelian group $\Gamma(X,\O_X^\times)$. If $X$ is an integral scheme, by the similar method, we have an opposite $G$-action on $R(X)^\times$.We can get opposite 1-cocycles from linearizations of invertible sheaves and rational sections of them
Let $(X,G)\in \Schgpk$ where $X$ is an integral scheme. Let $\L$ be an invertible sheaf, $\{\Phi_g\}_{g\in G}$ be a $G$-linearization on $\L$ and $\eta$ be a non-zero rational section. For $g\in G$, we define $\phi(g)\in R(X)^\times$ by
\begin{equation}\label{phigdefn}
\Phi_g(g^*(\eta)) = \phi(g)^{-1}\cdot \eta
\end{equation}
Then $\phi: G\rightarrow R(X)^\times$ is an opposite $G$-cocycle, which is called the opposite 1-cocycle associated with $(\L,\{\Phi_g\}_{g\in G}, \eta)$. Furthermore, if we take another rational section $\eta' =f\eta\: (f\in R(X)^\times)$, opposite 1-cocycle $\phi$ changes by the coboundary 1-cocycle associated with $f$.
§.§ Lifting of group actions by cyclic coverings and blowing-ups
Finally, we prove the liftability of group actions by a cyclic covering and a blowing-up. We recall the construction of cyclic coverings.
Let $X$ be a scheme and $m\in \Z_{>1}$. Let $\L$ be an invertible sheaf on $X$ and $h \in \Gamma(X,\L^{\otimes(-m)})$. We define a commutative $\O_X$-algebra structure on $\bigoplus_{i=0}^{m-1}\L^{\otimes i}$ by the following rule: For an open subset $U\subset X$, $x\in \L^{\otimes i}(U)$ and $y \in \L^{\otimes j}(U)$ where $i,j\in \{0,1,\dots, m-1\}$, we define
\begin{equation}\label{multiplicativerule}
x\cdot y = \left\{
\begin{aligned}
&x\otimes y \in \L^{\otimes (i+j)}(U) &(i+j < m)\\
&x\otimes y\otimes h|_U \in \L^{\otimes (i+j-m)}(U) &(i+j\ge m)
\end{aligned}
\right.
\end{equation}
We extend this multiplication rule $\O_X$-bilinearly. Note that commutativity and associativity follows from that $\L$ is an invertible sheaf. Then $m$-uple covering associated with $(\L, h)$ is defined by
\begin{equation}
\begin{tikzcd}
\Specu \bigoplus_{i=0}^{m-1}\L^{\otimes i} \arrow[r] & X.
\end{tikzcd}
\end{equation}
Here $\Specu$ denotes the relative spectrum of $\O_X$-algebras.
Let $(X,G) \in \Schgpk$. Let $\L$ be an invertible sheaf with $G$-linearization $\{\Phi_g\}_{g\in G}$. Let $\eta \in \Gamma(X,\L^{\otimes(-m)})$ be a global section and $\pi: Y\rightarrow X$ be a $m$-uple covering associated with $(\L,\eta)$. Suppose that
\begin{equation}\label{linearizationinvariantcondition}
\Phi_g^{\otimes(-m)}(g^*(\eta)) = \eta.
\end{equation}
Then we have a $G$-action on $Y$ such that $(\pi,\id_G):(Y,G)\rightarrow (X,G)$ is a morphism in $\Schgpk$.
For $g\in G$, we define an automorphism $\tilde{g}: Y\rightarrow Y$ as follows.
* Let $Y_1$ be the $m$-uple covering associated with $(g^*\L, g^*(\eta))$. Then $Y_1$ is a fiber product of $Y\rightarrow X$ and $X\xrightarrow{\:g\:} X$. Since $g$ is an isomorphism, $Y_1\rightarrow Y$ is so.
* By the isomorphism $\Phi_g$, $(g^*\L, g^*(\eta))$ is isomorphic to $(\L,\eta)$. Hence we have an isomorphism $Y\xrightarrow{\sim} Y_1$ over $X$.
By composing these isomorphism, we get an automorphism $\tilde{g}\in \Aut_k(X)$.
\begin{equation}
\begin{tikzcd}
Y \arrow[r,"\mathrm{(2)}","\sim"']\arrow[d,"\pi"] & Y_1 \arrow[d,"\pi"] \arrow[r,"\mathrm{(1)}","\sim"'] \arrow[dr,phantom,"\lrcorner",very near start] &Y \arrow[d,"\pi"] \\
X\arrow[r,equal]& X \arrow[r,"g","\sim"']& X
\end{tikzcd}
\end{equation}
We can show that $G\rightarrow \Aut_k(Y);g\mapsto \tilde{g}$ is a group homomorphism by the cocycle condition. Hence we can construct $G$-action on $Y$ and by construction, $(\pi, \id_G): (Y,G)\rightarrow (X,G)$ becomes a morphism in $\Schgpk$
Finally, we prove liftability of group actions by blowing-ups. This follows from the universal property of the blowing-up.
Let $(X,G)\in \Schgpk$ and $Y$ be a closed subscheme of $X$ which is stable under the $G$-action. Let $b: \Bl_YX\rightarrow X$ be a blowing up of $X$ along $Y$. Then we have a $G$-action on $\Bl_Y X$ such that $b$ is equivariant to $G$-actions.
§ CONSTRUCTION OF A FAMILY OF KUMMER SURFACES
Hereafter we fix a field $k$ whose characteristic is not $2$. In this section, we explicitly construct the family of Kummer surfaces $\X\rightarrow T$.
§.§ Construction of the Legendre family of elliptic curves
* We set $A=k\left[c,\frac{1}{c(1-c)}\right]$, which is a localization of the polynomial ring of one variable $k[c]$ and $S = \Spec A$. Let $\P^1_{S} = \Proj A[Z_0,Z_1]$ be the projective line over $S$.
* We use the notations $U_0 = D_+(Z_{0})\subset \P^1_S$ and $U_1 = D_+(Z_1)\subset \P^1_S$. We define the local coordinate $z= Z_1/Z_0$ on $U_0$.
* We define $h(z) = z(1-z)(1-cz)\in A[z]$ and $\widetilde{h} = h(z)dz^{\otimes(-2)}\in \Gamma(\P^1_{S},(\Omega_{\P^1_{S}/S}^1)^{\otimes(-2)})$.
We construct the Legendre family $\E\rightarrow S$ of elliptic curves as a double covering of $\P^1_S$.
Let $\mathcal E \rightarrow \P^1_{S}$ be the double covering associated with[See Definition <ref> for this notation.] $(\Omega_{\P^1_{S}/S}^1,\tilde{h})$. On the open subset $U_0\subset \P^1_S$, $\mathcal E\rightarrow \P^1_{S}$ can be described as the following morphism.
\begin{equation}\label{ellipticmorphism}
E_0 = \Spec A[u,z]/(u^2-h(z)) \lra \Spec A[z] = U_0
\end{equation}
(Definition of $\Sigma$)
* We define a set of $A$-rational points $\Sigma$ on $\P^1_S$ by
\begin{equation}
\Sigma = \{0,1,1/c,\infty\} \subset \Hom_S(S,\P^1_S).
\end{equation}
Here $0,1,1/c,\infty$ denotes $A$-rational points corresponding to $z = 0,1,1/c,\infty$.
* Similarly, we use the same symbol $\Sigma$ for a set of $A$-rational points on $\E$ corresponding to $z = 0,1,1/c,\infty$ and $u=0$.
* For a morphism of schemes $Z\rightarrow S$, we use the same symbol $\Sigma$ for its base change by $Z\rightarrow S$.
* If we would like to indicate the variety which points in $\Sigma$ are on, we use the notation like $\Sigma(\P^1_S)$ or $\Sigma(\E)$.
We have the description of the involution $\iota$ on $\E$ associated with the structure of elliptic curves as follows.
Let $\iota$ be an automorphism of $\mathcal E$ defined by the following $A$-algebra homomorphism.
\begin{equation}
\begin{tikzcd}[row sep=tiny,ampersand replacement = \&]
A[u,z]/(u^2-h(z)) \arrow[r] \&A[u,z]/(u^2-h(z)) \\
u, z \arrow[r,mapsto] \& -u,z
\end{tikzcd}
\end{equation}
Then $\iota$ is the involution with respect to the elliptic curve structure $(\E,O)$ over $S$ where $O\in \Sigma$.
Since $E_0$ is written in Weierstrass form, if $O = \infty$, we have the result. If $O=0,1,1/c$, we use the following lemma. The proof is standard.
Let $E$ be a smooth projective curve of genus 1 over a field $K$. Let $O$ and $O'$ be $K$-rational points of $E$. Morphisms $\iota$ and $\iota'$ are involutions on $E$ of taking inverses associated with the elliptic curve structure $(E,O)$ and $(E,O')$. Suppose $O'$ is a 2-torsion point for the elliptic curve $(E,O)$. Then $\iota = \iota'$.
§.§ A family of Kummer surfaces associated with products of Elliptic curves
We use the following notations.
* Let $B$ denote a $k$-algebra $A\otimes_{k} A$. We set $a = c\otimes 1,b = 1\otimes c \in B$ and $T = \Spec B$.
* Let $\Y=\P^1_{S}\times_{k} \P^1_{S}$. We regard $\Y$ as a scheme over $T = S\times_k S$. For $i,j\in \{0,1\}$, $Y_{i,j} = U_i\times_{k} U_j$ are open subschemes of $\Y$.
* Let $x,y$ denote local coordinates on $Y_{0,0}$ corresponding to $z \otimes 1$ and $1\otimes z$ in $A[z]\otimes_{k} A[z]$, respectively. Using $x$ and $y$, we can write $Y_{0,0} = \Spec B[x,y]$.
* We define the following polynomial with coefficients in $B$.
\begin{equation}
\begin{aligned}\label{fg}
f(x) & = x(1-x)(1-ax) \\
g(y) &= y(1-y)(1-by)
\end{aligned}
\end{equation}
* Let $\L$ be an invertible sheaf on $\Y$ corresponding to $pr_{1}^*\Omega^1_{\P^1_S/S}\otimes_{\O_{\Y}} pr_2^*\Omega^1_{\P^1_S/S}$ where $pr_i: \Y \rightarrow \P^1_{S}$ denotes the $i$-th projection. Furthermore, we define a global section $\eta$ by $\eta = pr_1^*(\tilde{h}) \otimes pr_2^*(\tilde{h}) \in \Gamma(\Y,\L^{\otimes(-2)})$.
We define $\widetilde{\Y}\lra \Y$ as the double covering associated with[See Definition <ref> for this notation.] $(\L, \eta)$. On $Y_{0,0}\subset \Y$, $\widetilde{\Y}\rightarrow \Y$ is described as follows.
\begin{equation}\label{ymorphism}
\widetilde{Y}_{0,0} = \Spec B[u,x,y]/(u^2-f(x)g(y)) \rightarrow \Spec B[x,y] = Y_{0,0} \subset \Y
\end{equation}
We define an open subscheme $\widetilde{Y}_{0,0}\subset \widetilde{\Y}$ as above.
The double covering $\widetilde{\Y}$ and $\mathcal{E}\times_k\mathcal{E}$ are related as follows. Note that the coordinate ring of $E_0\times_k E_0\subset \E\times_k \E$ is described as follows.
\begin{equation}\label{ellproddescription}
\begin{tikzcd}[row sep = tiny, ampersand replacement = \&]
A[u,z]/(u^2-h(z))\otimes_k A[u,z]/(u^2-h(z)) \arrow[r,"\sim"] \&[-10pt] B[u_1,u_2,x,y]/(u_1^2-f(x),u_2^2-g(y)) \\
u\otimes 1, 1\otimes u, z\otimes 1, 1\otimes z \arrow[r,mapsto] \& u_1,u_2,x,y
\end{tikzcd}
\end{equation}
We have a morphism $\E\times_k \E\rightarrow \widetilde{\Y}$ over $T$ described as the following $B$-algebra homomorphism.
\begin{equation}\label{etimeseytilde}
\begin{tikzcd}[row sep = tiny,ampersand replacement = \&]
B[u,x,y]/(u^2-f(x)g(y)) \arrow[r] \& B[u_1,u_2,x,y]/(u_1^2-f(x),u_2^2-g(y)) \\
u,x,y \arrow[r,mapsto] \& u_1u_2,x,y
\end{tikzcd}
\end{equation}
Then $\E\times_k \E\rightarrow \widetilde{\Y}$ corresponds to the universal categorical quotient of $\mathcal{E}\times_k\mathcal{E}$ under the $\Z/2\Z$-action induced by $\iota\times \iota$ .
By the description of $\iota$ in Proposition <ref>, $\iota\times \iota$ acts on $E_0\times_k E_0$ as
\begin{equation}
\begin{tikzcd}[row sep = tiny,ampersand replacement=\&]
B[u_1,u_2,x,y]/(u_1^2-f(x),u_2^2-g(y))\arrow[r] \& B[u_1,u_2,x,y]/(u_1^2-f(x),u_2^2-g(y)) \\
u_1,u_2 \arrow[r,mapsto] \& -u_1,-u_2
\end{tikzcd}
\end{equation}
Hence the image of (<ref>) generates the ring of invariants under the involution. Since the map (<ref>) is injective, we have the result.
(Definition of $\Sigma^2$)
* We define a set $\Sigma^2$ of $B$-rational points on $\Y$ by
\begin{equation}
\Sigma^2 = \{\sigma_1\times \sigma_2 : \sigma_1,\sigma_2\in \Sigma\}
\end{equation}
where $\sigma_1\times \sigma_2: T= S\times_k S\rightarrow \P^1_S\times_k \P^1_S = \Y$ is the direct product of $\sigma_1$ and $\sigma_2$.
* Similarly, we define a set $\Sigma^2$ of $B$-rational points on $\E\times_k \E$ by $ \{\sigma_1\times \sigma_2 : \sigma_1,\sigma_2\in \Sigma\}$. We also use the same symbol $\Sigma^2$ for its image under the map $\Hom_T(T,\E\times_k \E)\rightarrow \Hom_T(T,\widetilde{\Y})$ induced by the morphism $\E\times_k\E\rightarrow \widetilde{\Y}$ in (<ref>).
* More specifically, $\Sigma^2$ is the set of $B$-rational points whose $x$-coordinate and $y$-coordinate are in $\{0,1,1/a,\infty\}$ and $\{0,1,1/b,\infty\}$ respectively. We often identify
\begin{equation}
\Sigma^2 =\{0,1,1/a,\infty\}\times \{0,1,1/b,\infty\}
\end{equation}
and elements in $\Sigma^2$ is written like $(0,0),(1,1)$ and $(1/a,1/b)$. Each $\sigma\in \Sigma^2$ can be regarded as a closed subscheme. We use the same symbol $\Sigma^2$ for the closed subscheme which is the disjoint union of each $\sigma \in \Sigma^2$.
* For a morphism of schemes $Z\rightarrow T$, we use the same symbol $\Sigma^2$ for its base change by $Z\rightarrow T$.
* If we would like to indicate the variety which points in $\Sigma^2$ are on, we use the notation like $\Sigma^2(\Y)$ or $\Sigma^2(\widetilde{\Y}')$.
We define $\X\rightarrow \widetilde{\Y}$ as the blowing up of $\widetilde{\Y}$ along $\Sigma^2$. Then $\X$ is described locally on $\widetilde{Y}_{0,0}$ as follows.
\begin{equation}\label{xdescriptioneqn}
\begin{tikzcd}[row sep = tiny]
V_{0,0}=\Spec B[v,x,y]/(v^2f(x)-g(y)) \arrow[dr] &[-40pt]&[-20pt] \\
& \widetilde{Y}_{0,0} \arrow[r,phantom,"="]& \Spec B[u,x,y]/(u^2-f(x)g(y))\\
W_{0,0}=\Spec B[w,x,y]/(w^2g(y)-f(x)) \arrow[ur] &&
\end{tikzcd}
\end{equation}
These morphisms are defined by $u \mapsto vf(x)$ and $u\mapsto wg(y)$. The local coordinates $v$ and $w$ are glued by the relation $v = \frac{1}{w}$. We define open subschemes $V_{0,0}$ and $W_{0,0}$ of $\X$ as above.
For $\sigma \in \Sigma^2$, we define $Q_\sigma\subset \X$ by the following fiber product.
\begin{equation}
\begin{tikzcd}
Q_\sigma \arrow[r,hook]\arrow[d]\arrow[dr,phantom,"\lrcorner",very near start] & \X \arrow[d] \\
T \arrow[r,"\sigma"] & \widetilde{\Y}
\end{tikzcd}
\end{equation}
See Figure <ref> for the configurations of $Q_\sigma$ on $\X$.
The exceptional divisors $Q_\sigma$ on $\X$
We constructed the following $T$-schemes.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
&[50pt]\mathcal{E}\times_k\mathcal{E}\arrow[d,"\mathrm{quotient\:by\;}\iota\times\iota"] &[20pt]\\[13pt]
\X \arrow[r,"\mathrm{blowing-up\:along}","\Sigma^2"'] & \widetilde{\Y} \arrow[r,"\mathrm{double\; cover}","\mathrm{by\:}(\L\mathrm{,}\eta)"'] & \Y = \P^1_S \times_k \P^1_S \\
V_{0,0}\arrow[u,"\cup",phantom] \arrow[dr] &[30pt]&\\[-5pt]
&\widetilde{Y}_{0,0} \arrow[uu,"\cup",phantom] \arrow[r] & Y_{0,0}= U_0\times_k U_0 \arrow[uu,"\cup",phantom]&\\[-5pt]
W_{0,0}\arrow[ur] &&\\[5pt]
\end{tikzcd}
\end{equation}
We can check that these constructions are all stable under any base change of $T$.
Let $Z$ be any scheme over $T$. Let $\X_Z,\widetilde{\Y}_Z$, $(\mathcal{E}\times_k\mathcal{E})_Z$ and $\Y_Z$ denote the base changes of $\X,\widetilde{\Y}$, $\mathcal{E}\times_k\mathcal{E}$ and $\Y$ by $Z\rightarrow T$. Then we have the following.
* $\widetilde{\Y}_Z \rightarrow \Y_Z$ is the double cover associated with $(\L,\eta)$. Here we use the same symbol $(\L, \eta)$ for its pull back by $\Y_Z \rightarrow \Y$.
* $(\mathcal{E}\times_k \mathcal{E})_{Z} \rightarrow \widetilde{\Y}_Z$ is the quotient by $(\iota\times\iota)_Z$. Here $(\iota\times\iota)_Z$ is the base change of $\iota\times\iota$.
* $\X_Z\rightarrow \widetilde{\Y}_Z$ is the blowing up along $\Sigma^2$.
(3) is not so obvious since the blowing-up is not stable under the base change. But in this case the result follows from the fact that $\O_{\widetilde{\Y}}/\I^n$ is flat over $T$ for any $n>0$ where $\I$ is the ideal sheaf corresponding to $\Sigma^2$.By the properties of the Legendre family $\E\rightarrow S$, we have the following.
Let $t\in T$ and $O \in \Sigma^2$. Then the abelian surface $(\E\times_k\E)_{t}$ whose identity element is $O$ has the following properties.
* $\Sigma^2$ is the set of 2-torsion points of this abelian surface structure.
* $(\iota\times \iota)_{t}$ is the involution of taking inverse.
* Let $a(t), b(t)\in \kappa(t)$ be the images of elements $a,b\in \O_T(T)$ at the residue field of $t$. Then $(\E\times_k \E)_{t}$ is isomorphic to the direct product of the elliptic curves $y^2 = x(1-x)(1-a(t)x)$ and $y^2 = x(1-x)(1-b(t)x)$ over $\kappa(t)$.
Finally, we prove that $\X\rightarrow T$ is a family of Kummer surfaces.
For $t \in T$, the fiber $\X_{t}$ is isomorphic to the Kummer surface associated with the abelian surface $\left((\E \times_k \E)_{t}, O\right)$ where $O \in \Sigma^2$.
By Proposition <ref>, $(\iota\times \iota)_{t}$ is the involution of taking inverses on the abelian surface $(\E \times_k \E)_{t}$. By Proposition <ref> (2), $(\E \times_k\E)_{t} \rightarrow \widetilde{\Y}_{t}$ corresponds to the quotient by $(\iota\times\iota)_{t}$. Since $\Sigma^2\subset (\E\times_k \E)_t(\kappa(t))$ is the set of 2-torsion points on $(\E\times_k \E)_t$, its image $\Sigma^2\subset \widetilde{\Y}_t(\kappa(t))$ corresponds to the set of 16 singular points on $\widetilde{\Y}_{t}$. By Proposition <ref> (3), $\X_{t} \rightarrow \widetilde{\Y}_{t}$ is the blowing-up of $\widetilde{\Y}_{t}$ along these singular points. Hence $\X_{t}$ is isomorphic to the Kummer surface associated with $(\mathcal{E}\times_k \mathcal{E})_{t}$.
§.§ Construction of other smooth families of varieties over $T$
In this subsection, we define other smooth families of varieties $(\E\times_k\E)\wph$ and $\overline{\X}$ over $T$ and explain their relations with $\X$. These families of varieties are used for relating periods of $\X$ with those of elliptic curves in the Legendre family (Section 7) and for a construction of topological 2-chains on fibers $\X_t$ (Section 8).
Let $(\E\times_k\E)\wph$ (resp. $\overline{\X}$) be the blowing-up of $\E\times_k\E$ (resp. $\Y$) along $\Sigma^2$. By the universal property of the blowing-up, we have unique morphisms $(\E\times_k\E)\wph\rightarrow \X$ and $\X\rightarrow \overline{\X}$ such that the following diagram commutes.
\begin{equation}
\begin{tikzcd}
(\E\times_k \E)\widetilde{\phantom{h}} \arrow[d,"\mathrm{blowing-up}"'{yshift = 4pt},"\mathrm{along\:}\Sigma^2"'{yshift = -6pt}] \arrow[r,dashed] &[60pt] \X \arrow[d,"\mathrm{blowing-up}"'{yshift = 4pt},"\mathrm{along\:}\Sigma^2"'{yshift = -6pt}] \arrow[r,dashed]
&[60pt] \overline{\X} \arrow[d,"\mathrm{blowing-up}"{yshift = 4pt},"\mathrm{along\:}\Sigma^2"{yshift = -6pt}] \\
\E\times_k \E \arrow[r,"\mathrm{quotient\:by\:}\iota\times \iota"']
& \widetilde{\Y} \arrow[r,"\mathrm{double}\:\mathrm{cover}"'] & \Y
\end{tikzcd}
\end{equation}
The morphism $\X\rightarrow \overline{\X}$ is described by the following $B$-algebra homomorphisms.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
B[\overline{v},x,y]/(\overline{v}f(x)-g(y)) \arrow[r]& B[v,x,y]/(v^2f(x)-g(y)); &[-30pt] \overline{v}\arrow[r,mapsto] &v^2 \\
B[\overline{w},x,y]/(\overline{w}g(y)-f(x)) \arrow[r] & B[w,x,y]/(w^2g(y)-f(x)); & \overline{w}\arrow[r,mapsto] &w^2
\end{tikzcd}
\end{equation}
Finally, we name exceptional divisors on $\overline{\X}$. We use this notation in Section 8.
For $\sigma\in \Sigma^2$, we define the exceptional divisor $\overline{Q}_\sigma\subset \overline{\X}$ by the following fiber product.
\begin{equation}
\begin{tikzcd}
\overline{Q}_\sigma\arrow[d]\arrow[dr,phantom,"\lrcorner",very near start] \arrow[r,hook] & \overline{\X} \arrow[d] \\
T \arrow[r,"\sigma"] & \Y
\end{tikzcd}
\end{equation}
The morphism $\X\rightarrow \overline{\X}$ induces the $2:1$ map $Q_\sigma\rightarrow \overline{Q}_\sigma$.
§ CONSTRUCTION OF AUTOMORPHISMS OF THE FAMILY OF KUMMER SURFACES
As in Section 3, we fix a field $k$ whose characteristic is not 2. Moreover, we assume $k$ contains $\sqrt{-1}$. Until subsection 7.1, we assume these conditions on $k$.In this section, we will construct a group $\widetilde{G}$ and its action to a scheme $\X'$, which is a base change of $\X$ in Definition <ref>. To construct $\widetilde{G}$-action on $\X'$, we construct following objects in $\Schgpk$.
\begin{equation}
\begin{tikzcd}[row sep = small, column sep = small]
&&&&&[-10pt](T',\underline{G})\arrow[dd,"pr_i"]\arrow[rr]&&[-13pt] (T,\underline{G}_0) \arrow[dd,"pr_i"]\\
(\X',\widetilde{G}) \arrow[rr] &&(\widetilde{\Y}',\widetilde{G})\arrow[rr]&& (\Y',G) \arrow[rr,crossing over] \arrow[ur]\arrow[dd,"pr_i"]&& (\Y,G_0)\arrow[ur]& \\[-10pt]
&&&&&(S',\underline{H})\arrow[rr]&& (S, \underline{H}_0) \\
&&&& (\P^1_{S'},H) \arrow[rr] \arrow[ur]&& (\P^1_S, H_0)\arrow[ur]\arrow[from = uu,crossing over,"pr_i"]&
\end{tikzcd}
\end{equation}
We will construct them in the following order.
* We start with $\underline{H}_0 = \Aut_k(S)=\mathfrak{S}(\{0,1,\infty\})$ (Definition <ref>). We define a $H_0\simeq \mathfrak S_4$-action on $\P^1_S$ so that $\Sigma \subset \Hom_S(S,\P^1_S)$ is compatible[See Definition <ref> for the definition of compatible sets.] with $(\P^1_S,H_0)\rightarrow (S,\underline{H}_0)$. Then we consider their base changes by a finte étale extension $\Spec A' = S'\rightarrow S$ (Definition <ref>) and get $(S',\underline{H})$ and $(\P^1_{S'},H)$. The group $\underline{H}$ is isomorphic to $\mathfrak S_4$ (Remark <ref>).
* We define the following objects in $\Schgpk$ (Definition <ref>)
\begin{equation}
\begin{aligned}
&(T,\underline{G}_0) = (S\times_k S, \underline{H}_0\times \underline{H}_0),&& (T',\underline{G}) = (S'\times_k S',\underline{H}\times \underline{H}) \\
&(\Y,G_0) = (\P^1_S\times_k \P^1_S, H_0\times H_0), && (\Y',G) = (\P^1_{S'}\times_k \P^1_{S'}, H\times H)
\end{aligned}
\end{equation}
* To lift the $G$-action on $\Y'$ by the double covering $\widetilde{\Y}' = \widetilde{\Y}\times_T T'\rightarrow \Y'$, we use Proposition <ref>. Since $\widetilde{\Y}$ is constructed from $(\L,\eta)$ in Definition <ref>, we will construct linearization on $\L$ satisfying the liftability condition (<ref>). For this purpose, we consider a group $\widetilde{G}$ which is a $\Z/2\Z$-extension of $G$ (Definition <ref>).
* Since the $\widetilde{G}$-action on $\widetilde{\Y}'$ stabilizes the blowing-up locus of $\X'\rightarrow \widetilde{\Y}'$, we can lift $\widetilde{G}$-action on $\widetilde{\Y}'$ to $\X' = \X\times_T T'$ (Proposition <ref>).
We calculate some opposite 1-cocycles in Subsection 4.4. They are important for the description of the group action on the higher Chow subgroup $\Xi^\can$ (Section 6), on the 2-form $\omega \in \Gamma(X,\Omega_{\X'/T'}^2)$ (Section 7) and on the sheaf $\Qc_\omega$ (Section 9).
§.§ Automorphisms on $S$ and $\P^1_S$
In this subsection, we construct objects and a morphism $(\P^1_S,H_0)\rightarrow (S,\underline{H}_0)$ in $\Schgpk$.
We define $\underline{H}_0 = \Aut_k(S)$. If we regard $S = \P^1_k-\{0,1,\infty\}$, every $\underline{\tau}_0\in \underline{H}_0$ extends to an automorphism on $\P^1_k$ which stabilizes the $k$-rational point set $\{0,1,\infty\}$. Hence we have the following group isomorphism.
\begin{equation}\label{autosg}
\begin{tikzcd}[row sep = tiny]
\Aut_k(S)\arrow[r,equal] &[-15pt] \underline{H}_0 \arrow[r,"\sim"] & \mathfrak{S}(\{0,1,\infty\}) \\
&\underline{\tau}_0 \arrow[r,mapsto] \aru& \left(\bullet \mapsto \underline{\tau}_0(\bullet)\right)\aru
\end{tikzcd}
\end{equation}
We often identify $\underline{H}_0$ with $\mathfrak{S}(\{0,1,\infty\})$. The correspondence of $\underline{H}_0$ and $\mathfrak{S}(\{0,1,\infty\})$ is given in Table <ref>. Note that the composition on $ \mathfrak{S}(\{0,1,\infty\})$ is defined as the usual order. For example, $(0\:1)(0\:\infty) = (0\:\infty\:1)$. Thus $\underline{H}_0$ induces an opposite action on the ring $A$.
The correspondence of $\underline{H}_0\simeq \mathfrak{S}(\{0,1,\infty\})$
$\underline{\tau}_0$ $\underline{\tau}_0^\sharp(c) $ $\underline{\tau}_0$ $\underline{\tau}_0^\sharp(c)$
$\id $ $c$ $ (0\;1) $ $1-c$
$(1\;\infty)$ $\frac{c}{c-1}$ $ (0\;1\;\infty)$ $ \frac{1}{1-c} $
$(0\;\infty)$ $\frac{1}{c}$ $(0\;\infty\;1) $ $ \frac{c-1}{c} $
Next, we define a subgroup $H_0$ of the automorphism group of $\P^1_S$. Using the notation in Definition <ref>, we have the following group.
\begin{equation}
\Aut(\P^1_S;S,\underline{H}_0) = \left\{(\tau_0,\underline{\tau}_0)\in \Aut_k(\P^1_S)\times \underline{H}_0:
\begin{tikzcd}
\P^1_S \arrow[r]\arrow[d,"\tau_0"] & S \arrow[d,"\underline{\tau}_0"] \\
\P^1_S \arrow[r] & S
\end{tikzcd}
\quad\text{commutes.}
\right\}
\end{equation}
Since the natural projection $\Aut(\P^1_S;S,\underline{H}_0) \rightarrow \Aut_k(\P^1_S)$ is injective, we identify $\Aut(\P^1_S;S,\underline{H}_0)$ as a subgroup of $\Aut_k(\P^1_S)$. We often denote an element in $\Aut(\P^1_S;S,\underline{H}_0) $ by $\tau_0$. For $\tau_0\in \Aut(\P^1_S;S,\underline{H}_0)$, the image of $\tau_0$ under the natural projection $\Aut(\P^1_S;S,\underline{H}_0) \rightarrow \underline{H}_0$ is denoted by $\underline{\tau}_0$.
We define $H_0$ as the following subgroup of $\Aut(\P^1_S;S,\underline{H}_0)$.
\begin{equation}
H_0 =\left\{\tau_0 \in \Aut(\P^1_S;S,\underline{H}_0): \text{For any }\sigma\in \Sigma\text{, } \tau_0\circ \sigma\circ\underline{\tau}_0^{-1}\in \Sigma.
\right\}
\end{equation}
Then we have a natural morphism $\alpha: (\P^1_S,H_0)\rightarrow (S,\underline{H}_0)$ in $\Schgpk$. By the construction, $\Sigma$ is compatible with $\alpha: (\P^1_S,H_0)\rightarrow (S,\underline{H}_0)$. By Definition <ref>, $H_0$ has the following natural set-theoretic action on $\Sigma$.
\begin{equation}\label{grouphom}
\begin{tikzcd}[row sep = tiny]
H_0 \arrow[r] & \mathfrak{S}(\Sigma) \arrow[r,equal] &[-15pt] \mathfrak{S}(\{0,1,1/c,\infty\}) \\
\tau_0 \aru \arrow[rr,mapsto] & & \left(\sigma\mapsto \tau_0\circ \sigma \circ \underline{\tau}_0^{-1}\right)\aru
\end{tikzcd}
\end{equation}
The group homomorphism $(\ref{grouphom})$ is an isomorphism.
Let $\tau_0\in H_0$. We have the following diagram.
\begin{equation}
\begin{tikzcd}
\P^1_S \arrow[r,"\tau_0"]\arrow[d] & \P^1_S \arrow[d] \arrow[r,"\underline{\tau}_0^{-1}",dashed] & \P^1_S \arrow[d] \\
S\arrow[r,"\underline{\tau}_0"] & S \arrow[r,"\underline{\tau}_0^{-1}"] & S
\end{tikzcd}
\end{equation}
where $\underline{\tau}_0^{-1}: \P^1_S\rightarrow \P^1_S$ is the morphism $\id_{\P^1_k}\times \underline{\tau}_0^{-1} : \P^1_S = \P^1_k\times_k S \rightarrow \P^1_k\times_k S = \P^1_S$. Then $(\underline{\tau}_0^{-1})^\sharp(z) = z$ where $z$ is the inhomogeneous coordinate on $\P^1_S$ in Definition <ref>. Since $\underline{\tau}_0^{-1}\circ \tau_0: \P^1_S\rightarrow \P^1_S$ is a morphism over $S$, we can write $(\underline{\tau}_0^{-1}\circ \tau_0)^\sharp(z) = \frac{pz+q}{rz+s}$ where $p,q,r,s\in A$. Hence we can write
\begin{equation}\label{fracrepn}
\tau_0^\sharp(z) = \frac{pz+q}{rz+s}\quad(p,q,r,s\in A).
\end{equation}
First, we check (<ref>) is injective. Suppose $\tau_0\in H_0$ lies in the kernel of (<ref>). Since $\tau_0$ acts trivially on $\Sigma$, $\tau_0(0) = 0$, $\tau_0(1) = 1$, $\tau_0(1/c) = 1/c$ and $\tau_0(\infty) = \infty$. Especially we have
\begin{equation}
\begin{aligned}
&\frac{p\cdot 0 +q}{r\cdot 0 + s} = 0, \quad&& \frac{p\cdot 1+q}{r\cdot 1 + s} = 1, \\
& \frac{p\cdot \frac{1}{c} + q}{ r\cdot \frac{1}{c} + s} = \frac{1}{\underline{\tau}_0^\sharp(c)},\quad && \frac{p\cdot \infty + q}{r\cdot \infty+ s} = \infty.
\end{aligned}
\end{equation}
Hence we see that $\tau_0^\sharp(z) = z$ and $\underline{\tau}_0^\sharp(c) = c$. i.e. $\tau_0 = \id_{H_0}$.Next, we check that (<ref>) is surjective. It is enough to find elements in $H_0$ corresponding to $(0\:1),(0\:1\:1/c\:\infty)\in \mathfrak{S}(\Sigma)$ since they are generators of $\mathfrak{S}(\Sigma)$. We use the presentation in (<ref>) again. For example, to find $\tau_0\in H_0$ corresponding to $(0\:1\:1/c\:\infty)$, it is enough to find $p,q,r,s\in A$ such that
\begin{equation}\label{cond01cinfty}
\begin{aligned}
&\frac{p\cdot 0 +q}{r\cdot 0 + s} = 1, \quad&& \frac{p\cdot 1+q}{r\cdot 1 + s} = \frac{1}{\underline{\tau}_0^\sharp(c)}, \\
& \frac{p\cdot \frac{1}{c} + q}{ r\cdot \frac{1}{c} + s} = \infty,\quad && \frac{p\cdot \infty + q}{r\cdot \infty+ s} = 0.
\end{aligned}
\end{equation}
From these conditions, we can find a pair of automorphisms $(\tau_0,\underline{\tau}_0)\in \Aut_k(\P^1_S)\times \underline{H}_0$ such that $\tau_0^\sharp(z) = \frac{1}{1-cz}$ and $\underline{\tau}_0^\sharp(c) = 1-c$, which is in $H_0$ and its image under the map (<ref>) is $(0\:1\:1/c\:\infty)\in \mathfrak{S}(\Sigma)$. Similarly, we can find the element of $H_0$ such that its image under the map (<ref>) is $(0\:1)\in \mathfrak{S}(\Sigma)$.
By Proposition <ref>, we often identify $H_0$ with $\mathfrak{S}(\Sigma)$. The explicit correspondence of $H_0\simeq \mathfrak{S}(\Sigma)$ is given in Table <ref>. We can find these correspondence by the same method we use in the proof of Proposition <ref>. In the table, for each $\tau_0\in H_0$, the image of $c$ under $\underline{\tau}_0^\sharp : A\rightarrow A$ and the image of the local coordinate $z$ under $\tau_0^\sharp:\O_{\P^1_S} \rightarrow (\tau_0)_*\O_{\P^1_S}$ are given.
The correspondence of $H_0\simeq \mathfrak{S}(\{0,1,1/c,\infty\})$
$\tau_0$ $\underline{\tau}_0^\sharp(c)$ $\tau_0^\sharp(z)$
$\id$ $c $ $z$
$\tau_0$ $\underline{\tau}_0^\sharp(c) $ $\tau_0^\sharp(z) $ $\tau_0$ $\underline{\tau}_0^\sharp(c) $ $\tau_0^\sharp(z) $ $\tau_0$ $\underline{\tau}_0^\sharp (c) $ $\tau_0^\sharp(z)$
$(0\; 1) $ $\frac{c}{c-1}$ $1-z$ $(0\;1/c) $ $1-c$ $\frac{1-cz}{1-c}$ $(0\;\infty) $ $\frac{1}{c}$ $\frac{1}{z}$
$(1/c\;\infty)$ $\frac{c}{c-1} $ $\frac{(1-c)z}{1-cz}$ $(1\;\infty) $ $ 1-c $ $\frac{z}{z-1}$ $(1\; 1/c) $ $\frac{1}{c}$ $cz$
$\tau_0$ $\underline{\tau}_0^\sharp(c) $ $\tau_0^\sharp(z) $ $\tau_0$ $\underline{\tau}_0^\sharp(c) $ $\tau_0^\sharp(z)$ $\tau_0$ $ \underline{\tau}_0^\sharp(c) $ $\tau_0^\sharp(z) $
$(0\;1)(1/c\;\infty)$ $c$ $\frac{1-z}{1-cz}$ $(0\;1/c)(1\;\infty)$ $ c$ $\frac{1-cz}{c(1-z)} $ $(0\;\infty)(1\;1/c) $ $ c$ $\frac{1}{cz}$
$\tau_0$ $\underline{\tau}_0^\sharp(c) $ $\tau_0^\sharp(z)$ $\tau_0$ $\underline{\tau}_0^\sharp(c)$ $ \tau_0^\sharp(z)$
$(0\;1\;1/c) $ $\frac{1}{1-c}$ $1-cz$ $(0\;1/c\;1) $ $\frac{c-1}{c}$ $\frac{c(1-z)}{c-1}$
$(0\;\infty\;1)$ $\frac{1}{1-c} $ $\frac{z-1}{z}$ $(0\;1\;\infty)$ $\frac{c-1}{c}$ $\frac{1}{1-z}$
$(0\;1/c\;\infty)$ $ \frac{1}{1-c}$ $\frac{1-c}{1-cz}$ $(0\;\infty\;1/c)$ $\frac{c}{c-1}$ $ \frac{1-cz}{(1-c)z}$
$(1\;\infty\;1/c)$ $\frac{1}{1-c}$ $\frac{(c-1)z}{1-z)}$ $(1\;1/c\;\infty)$ $\frac{c-1}{c}$ $\frac{cz}{cz-1}$
$\tau_0$ $\underline{\tau}_0^\sharp(c)$ $\tau_0^\sharp(z)$ $\tau_0$ $\underline{\tau}_0^\sharp(c)$ $\tau_0^\sharp(z) $ $\tau_0$ $\underline{\tau}_0^\sharp (c)$ $ \tau_0^\sharp(z)$
$(0\;1/c\;1\;\infty)$ $\frac{c}{c-1} $ $\frac{c-1}{c(1-z)}$ $(0\;1\;1/c\;\infty) $ $ 1-c$ $\frac{1}{1-cz}$ $ (0\;1\;\infty\;1/c)$ $\frac{1}{c} $ $\frac{1-cz}{1-z} $
$(0\;\infty\;1\;1/c)$ $\frac{c}{c-1}$ $\frac{cz-1}{cz}$ $(0\;\infty\;1/c\;1)$ $1-c$ $\frac{1-z}{(c-1)z}$ $(0\;1/c\;\infty\;1)$ $\frac{1}{c}$ $ \frac{c(1-z)}{1-cz}$
We have a bijection
\begin{equation}
\{\{\{0,1\},\{\infty,1/c\}\}, \{\{0,\infty\},\{1,1/c\}\},\{\{0,1/c\},\{1,\infty\}\}\} \simeq \{0,1,\infty\}
\end{equation}
defined by $\{\{0,1\},\{\infty,1/c\}\}\mapsto 0, \{\{0,\infty\},\{1,1/c\}\}\mapsto 1, \{\{0,1/c\},\{1,\infty\}\}\}\mapsto \infty$. Since $\mathfrak{S}(\Sigma)$ acts on the set on the left hand side, we have a group homomorphism
\begin{equation}\label{s4s3ver2}
\mathfrak S(\Sigma)\lra \mathfrak S(\{0,1,\infty\})
\end{equation}
The group homomorphism $H_0\rightarrow \underline{H}_0$ is identified with the group homomorphism (<ref>) under the identifications $H_0 = \mathfrak S(\Sigma)$ and $\underline{H}_0 = \mathfrak S(\{0,1,\infty\})$.
§.§ A finite étale covering $S'\rightarrow S$ and lifts of group actions
To get enough automorphisms of the family of Kummer surfaces, we have to enlarge the base scheme $S$. As we will see later in Section 5, this base change is also necessary for the construction of higher Chow cycles in $\Xi^\can$.
We define an $A$-algebra $A'$ as $A' = A[\sqrt{c},\sqrt{1-c}]$ and $S'=\Spec A'$. We have a natural morphism $S'\rightarrow S$ induced by $A\hookrightarrow A'$. Furthermore, we define $\underline{H} = \Aut(S';S,\underline{H}_0)$. i.e.
\begin{equation}
\underline{H} = \left\{(\underline{\tau},\underline{\tau}_0)\in \Aut_k(S')\times \underline{H}_0 : \begin{tikzcd}
S' \arrow[r] \arrow[d,"\underline{\tau}"] & S \arrow[d,"\underline{\tau}_0"] \\
S' \arrow[r] & S
\end{tikzcd}
\quad\text{commutes.}\right\}
\end{equation}
Then we have a natural morphism $\beta: (S',\underline{H})\rightarrow (S,\underline{H}_0)$ in $\Schgpk$. Since the natural projection $\underline{H}\rightarrow \Aut_k(S')$ is injective, we regard $\underline{H}$ as a subgroup of $\Aut_k(S')$. We often denote an element in $\underline{H}$ by $\underline{\tau}$. For $\underline{\tau}\in \underline{H}$, the image of $\underline{\tau}$ under the natural projection $\underline{H}\rightarrow \underline{H}_0$ is denoted by $\underline{\tau}_0\in \underline{H}_0$.
We have the following properties about $(S',\underline{H})$.
* $S'\rightarrow S$ is a finite étale morphism.
* We have the following isomorphism between $k$-algebras. Especially, $A'$ is an integral domain.
\begin{equation}\label{S'isom}
\begin{tikzcd}[ampersand replacement = \&]
A' \arrow[r,"\sim"] \& k\left[\gamma, \frac{1}{\gamma(\gamma^4-1)}\right]; \&[-20pt]
\sqrt{c}, \sqrt{1-c} \arrow[r, mapsto]\&
\frac{\gamma+\frac{1}{\gamma}}{2}, \frac{\gamma-\frac{1}{\gamma}}{2\sqrt{-1}}
\end{tikzcd}
\end{equation}
* The group homomorphism $\underline{H}\rightarrow \underline{H}_0$ is surjective.
* The kernel of $\underline{H}\rightarrow \underline{H}_0$ is isomorphic to $\mu_2(k)\times \mu_2(k)$.
Especially, $\underline{H}$ fits into the following exact sequence.
\begin{equation}
1 \lra \mu_2(k)\times \mu_2(k) \lra \underline{H} \lra \underline{H}_0\lra 1
\end{equation}
We can show (1), (2) and (4) by the ring theoretic calculation. To prove (3), we construct the lifts of $\underline{\tau}_0 \in \underline{H}_0$ explicitly. The result is summarized in Table <ref>. In the table, we give the image of $\gamma\in k\left[\gamma, \frac{1}{\gamma(\gamma^4-1)}\right]$ under the ring homomorphisms $\underline{\tau}^\sharp: k\left[\gamma, \frac{1}{\gamma(\gamma^4-1)}\right]\simeq A' \rightarrow A' \simeq k\left[\gamma, \frac{1}{\gamma(\gamma^4-1)}\right]$ corresponding to the lifts of each $\underline{\tau}_0^\sharp: A\rightarrow A$.
The lifts of $\tau_0\in \underline{H}_0$ to $\underline{H}$
$\underline{\tau}_0^\sharp(c) $ $\underline{\tau}^\sharp(\gamma) $ $\underline{\tau}_0^\sharp(c) $ $\underline{\tau}^\sharp(\gamma) $
$c $ $\pm \gamma, \pm \frac{1}{\gamma} $ $1-c $ $\pm\sqrt{-1}\gamma, \pm\sqrt{-1} \frac{1}{\gamma}$
$\frac{c}{c-1}$ $\pm\frac{\gamma+1}{\gamma-1}, \pm\frac{\gamma-1}{\gamma+1} $ $\frac{1}{1-c} $ $ \pm\sqrt{-1}\frac{\gamma+1}{\gamma-1}, \pm\sqrt{-1}\frac{\gamma-1}{\gamma+1} $
$\frac{c-1}{c} $ $ \pm \frac{\gamma+\sqrt{-1}}{\gamma-\sqrt{-1}},\pm\frac{\gamma-\sqrt{-1}}{\gamma+\sqrt{-1}}$ $\frac{1}{c}$ $\pm\sqrt{-1}\frac{\gamma+\sqrt{-1}}{\gamma-\sqrt{-1}},\pm\sqrt{-1}\frac{\gamma-\sqrt{-1}}{\gamma+\sqrt{-1}}$
More strongly, we can show that $\underline{H}$ is isomorphic to $\mathfrak{S}_4$ as follows. By the isomorphism (<ref>) in Proposition <ref>, we can regard $S' = \P^1_k-\{\pm 1,\pm\sqrt{-1},0,\infty\}$. Let $N = \{\pm 1,\pm\sqrt{-1},0,\infty\}\subset \P^1_k(k)$. If we plot points of $N$ on the Riemann sphere $\P^1(\C)$, $N$ forms an octahedron. We can check that $\underline{H}$ acts on this octahedron and $\underline{H}$ is naturally isomorphic to the octahedral group, which is isomorphic to $\mathfrak{S}_4$.
We define $(\P^1_{S'}, H)\in \Schgpk$ as a fiber product of $(\P^1_S, H_0)$ and $(S',\underline{H})$ over $(S,\underline{H}_0)$ in $\Schgpk$.
\begin{equation}\label{Hschemewithgroup}
\begin{tikzcd}
(\P^1_{S'}, H) \arrow[r]\arrow[d]\arrow[dr,"\lrcorner",phantom,very near start] & (S',\underline{H}) \arrow[d,"\beta"] \\
(\P^1_S, H_0) \arrow[r,"\alpha"] & (S,\underline{H}_0)
\end{tikzcd}
\end{equation}
By Proposition <ref>, $H$ is equal to the fiber product
\begin{equation}
H_0\times_{\underline{H}_0} \underline{H} =
\left\{\tau = (\tau_0,\underline{\tau})\in H_0\times\underline{H}: \alpha(\tau_0) = \beta(\underline{\tau})\right\}
\end{equation}
where $\alpha$ and $\beta$ are group homomorphisms corresponding to $\alpha: (\P^1_S,H_0)\rightarrow (S,\underline{H}_0)$ and $\beta: (S',\underline{H})\rightarrow (S,\underline{H}_0)$. Since $H_0\simeq \underline{H} \simeq \mathfrak{S}_4$ (Proposition <ref> and Remark <ref>) and $\underline{H}_0\simeq \mathfrak{S}_3$ (Definition <ref>), we have $H \simeq \mathfrak{S}_4\times_{\mathfrak{S}_3}\mathfrak{S}_4$.By definition, we have the following natural group homomorphisms $H\rightarrow \underline{H}$ and $H \rightarrow H_0$. By Remark <ref> and Proposition <ref>, they are surjective.
\begin{equation}\label{Hfamily}
\begin{tikzcd}
H \arrow[r,twoheadrightarrow]\arrow[d,twoheadrightarrow]\arrow[dr,"\lrcorner",very near start,phantom] & \underline{H}\arrow[d,twoheadrightarrow] & \tau \arrow[r,mapsto]\arrow[d,mapsto] & \underline{\tau}\arrow[d,mapsto] \\
H_0 \arrow[r,twoheadrightarrow] & \underline{H}_0 & \tau_0 \arrow[r,mapsto] & \underline{\tau}_0
\end{tikzcd}
\end{equation}
The images of $\tau \in H$ in $H_0$ and $\underline{H}$ are denoted by $\tau_0\in H_0$ and $\underline{\tau}\in \underline{H}$, respectively.
We define the following objects in $\Schgpk$.
\begin{equation}\label{Gdefeqn}
\begin{aligned}
& (T,\underline{G}_0) = (S,\underline{H}_0)\times (S,\underline{H}_0), && (\Y,G_0) = (\P^1_S, H_0)\times (\P^1_S, H_0) \\
&(T',\underline{G}) = (S',\underline{H})\times (S',\underline{H}), && (\Y',G) = (\P^1_{S'}, H)\times (\P^1_{S'}, H)
\end{aligned}
\end{equation}
By Proposition <ref>, $\underline{G}_0,G_0,\underline{G}$ and $G$ coincide with $\underline{H}_0\times \underline{H}_0, H_0\times H_0, \underline{H}\times \underline{H}$ and $H\times H$. By considering the direct products of morphisms in (<ref>), we have the following morphisms in $\Schgpk$.
\begin{equation}\label{directproddiagram}
\begin{tikzcd}
(\Y',G) \arrow[r]\arrow[d] & (T',\underline{G}) \arrow[d] & G &[-15pt] \rho \arrow[l,"\ni",phantom] \arrow[r,mapsto]\arrow[d, mapsto] & \underline{\rho} \arrow[d,mapsto]\arrow[r,"\in",phantom] &[-15pt] \underline{G} \\
(\Y,G_0) \arrow[r] & (T,\underline{G}_0) & G_0 & \rho_0 \arrow[l,"\ni",phantom] \arrow[r,mapsto] & \underline{\rho}_0 \arrow[r,"\in",phantom] & \underline{G}_0
\end{tikzcd}
\end{equation}
By checking the universality, we see that the left diagram in (<ref>) is the fiber product. Especially, $\Y'$ is the base change of $\Y$ by $T'\rightarrow T$. We denote the images of $\rho \in G$ under the group homomorphisms in (<ref>) by $\underline{\rho} \in \underline{G}$, $\rho_0\in G_0$ and $\underline{\rho}_0\in \underline{G}_0$ respectively. Furthermore, for $\rho \in G$, its first and second components are denoted by $\rho^{(1)}$ and $\rho^{(2)}$ respectively. i.e.
\begin{equation}
G = \{\rho = (\rho^{(1)},\rho^{(2)}): \rho^{(1)},\rho^{(2)}\in H\}
\end{equation}
We define $\underline{\rho}_0^{(i)}, \rho_0^{(i)}, \underline{\rho}^{(i)}$ for $i = 1,2$ similarly.
We define $B' = A'\otimes_k A'$. By definition, $T' = \Spec B'$. For any scheme $Z$ over $T$, $Z'$ denotes the base change of $Z$ by $T'\rightarrow T$. For example, $\widetilde{\Y}' = \widetilde{\Y}\times_T T'$, $\X' = \X\times_T T'$ and $Q'_\sigma = Q_\sigma\times_T T'$. This notation is compatible with $\Y' = \Y\times_T T'$.
The $B'$-rational point set $\Sigma^2(\Y')$ is compatible[See Definition <ref> for the definition of the $B'$-rational point set $\Sigma^2(\Y')$.] with $(\Y',G)\rightarrow (T',\underline{G})$. Especially, $G$ has a natural action on $\Sigma^2$.
By Definition <ref>, $\Sigma(\P^1_S)$ is compatible with $(\P^1_S,H_0)\rightarrow (S,\underline{H}_0)$. Then by Proposition <ref>, $\Sigma(\P^1_{S'})$ is compatible with $(\P^1_{S'},H)\rightarrow (S',\underline{H})$. Since $(\Y',G)\rightarrow (T',\underline{G})$ is the direct product of $(\P^1_{S'},H)\rightarrow (S',\underline{H})$, $\Sigma^2(\Y')$ is compatible with $(\Y',G)\rightarrow (T',\underline{G})$ by Proposition <ref> again.
§.§ Linearizations on $\L$ and cocycles $\phi,\chi$
In this subsection, we define a linearization of $\L$ which give rise to a lifting of the $G$-action on $\Y'$ to $\widetilde{\Y}'$. Since $\L = pr_1^*\Omega^1_{\P^1_{S'}/S'}\otimes_{\O_{\Y'}}pr_2^*\Omega^1_{\P^1_{S'}/S'}$, we have a $G$-linearization $\{\Psi_\rho\}_{\rho\in G}$ on $\L$. However, by this natural $G$-linearization, $\Psi_\rho^{\otimes(-2)}(\rho^*(\eta))$ and $\eta$ differs by
\begin{equation}
\Psi_\rho^{\otimes(-2)}(\rho^*(\eta)) = \chi_0(\rho)^{-1}\cdot \eta.
\end{equation}
where $\chi_0$ is an opposite 1-cocycle. The first aim of this subsection is to get the explicit description of this $\chi_0$. Then we will find an opposite 1-cocycle $\widetilde{\chi}$ such that $\widetilde{\chi}^2 = \chi_0$. For this purpose, we introduce opposite coboundary 1-cocycles $\chi$, $\chi^{(1)}$ and $\chi^{(2)}$ and take a $\Z/2\Z$-extension $\widetilde{G}$ of $G$. Finally, using $\widetilde{\chi}$, we modify the linearization $\{\Psi_\rho\}_{\rho\in G}$ on $\L$ and get a new $\widetilde{G}$-linearization $\{X_{\widetilde{\rho}}\}_{\widetilde{\rho}\in \widetilde{G}}$ on $\L$ which satisfies the liftability condition (<ref>) in Proposition <ref>.
We define $H$-linearization $\{\Phi_{\tau}\}_{\tau\in H}$ of $\Omega^1_{\P^1_{S'}/S'}$ as the canonical one induced by Proposition <ref>. By definition, $\{\Phi_{\tau}\}_{\tau\in H}$ satisfies
\begin{equation}
\Phi_{\tau}\left(\tau^*(dz)\right) = \frac{\partial}{\partial z}(\tau^\sharp(z))\cdot dz.
\end{equation}
We define an opposite 1-cocycle $\phi_0:H\rightarrow R(\P^1_{S'})^\times$ as the opposite 1-cocycle associated[See Proposition <ref> for the definition of associated 1-cocycles.] with $\left((\Omega^1_{\P^1_{S'}/S'})^{\otimes (-2)}, \{\Phi_{\tau}^{\otimes(-2)}\}_{\tau\in H}, \widetilde{h}\right)$, where $\widetilde{h}$ is the section defined in Definition <ref>. By definition, $\phi_0(\tau)$ can be computed as follows.
\begin{equation}\label{phiformula}
\phi_0(\tau) = \left(\frac{\partial}{\partial z}(\tau^\sharp(z))\right)^2 \frac{h(z)}{\tau^\sharp(h(z))}
\end{equation}
By the computation of $\phi_0(\tau)$ for each $\tau \in H$, we have the following properties.
* $\phi_0(\tau)$ is determined by the image of $\tau$ under $H\rightarrow \underline{H}_0$.
* The explicit description of $\phi_0(\tau)$ is given by the following table.
The opposite 1-cocycle $\phi_0$
$\underline{\tau}_0$ $\underline{\tau}_0^\sharp(c) $ $\phi_0(\underline{\tau}_0) $ $\underline{\tau}_0$ $\underline{\tau}_0^\sharp(c) $ $\phi_0(\underline{\tau}_0) $
$\id$ $c $ $ 1 $ $(0\:1)$ $1-c $ $-1 $
$(1\:\infty)$ $\frac{c}{c-1}$ $ 1-c$ $(0\:1\:\infty)$ $ \frac{1}{1-c}$ $ c-1$
$(0\:\infty)$ $\frac{1}{c}$ $c$ $(0\:\infty\:1)$ $\frac{c-1}{c}$ $ -c $
Especially, $\phi_0(\tau)\in A^\times$.
From these properties, we regard $\phi_0$ as the opposite 1-cocycle $\underline{H}_0 \rightarrow A^\times$.
(Definition of $\chi_0$) For $i=1,2$, we have an $G$-linearization $\{pr_i^*\Phi_{\rho^{(i)}}\}_{\rho\in G}$ of $pr_i^*\Omega^1_{\P^1_{S'}/S'}$ by pulling back (cf. Proposition <ref>) the $H$-linearization of $\Omega^1_{\P^1_{S'}/S'}$ in Definition <ref> by $pr_i: \Y'\rightarrow \P^1_{S'}$. Then we define a $G$-linearization $\{\Psi_\rho\}_{\rho\in G}$ of $\L = pr_1^*\Omega^1_{\P^1_{S'}/S'}\otimes_{\O_{\Y'}}pr_2^*\Omega^1_{\P^1_{S'}/S'}$ by
\begin{equation}
\Psi_\rho = pr_1^*\Phi_{\rho^{(1)}}\otimes pr_2^*\Phi_{\rho^{(2)}}.
\end{equation}
Since[See Definition <ref> for the definition of the polynomials $f(x),g(y)$.] $pr_1^\sharp(h(z)) = f(x)$ and $pr_2^\sharp(h(z)) = g(y)$, we have
\begin{equation}\label{transformfg}
\begin{aligned}
&pr_1^\sharp(\phi_0(\rho^{(1)})) = \left(\frac{\partial}{\partial x}(\rho^\sharp(x))\right)^2 \frac{f(x)}{\rho^\sharp(f(x))} \\
&pr_2^\sharp(\phi_0(\rho^{(2)}))= \left(\frac{\partial}{\partial y}(\rho^\sharp(y))\right)^2 \frac{g(y)}{\rho^\sharp(g(y))}.
\end{aligned}
\end{equation}
We define $\chi_0$ as the opposite 1-cocycle associated with $(\L, \{\Psi_\rho^{\otimes(-2)}\}_{\rho\in G},\eta)$. By definition, we have the following equations.
\begin{equation}
\Psi_{\rho}^{\otimes(-2)}(\rho^*(\eta)) = \chi_0(\rho)^{-1}\cdot \eta
\end{equation}
\begin{equation}\label{prchirel}
\chi_0(\rho) = pr_1^\sharp(\phi_0(\rho^{(1)})) pr_2^\sharp(\phi_0(\rho^{(2)}))\in B^\times
\end{equation}
By (<ref>), we can calculate $\chi_0$ from Table <ref>.
We will find an opposite 1-cocycle $\widetilde{\chi}$ such that $\widetilde{\chi}^2= \chi_0$. First, we will find an opposite coboundary 1-cocycle $\phi$ of $\underline{H}$ whose square coincides with $\phi_0$ up to sign.
$($Definition of $\phi$$)$
For $\tau \in \underline{H}$, we define
\begin{equation}
\phi(\underline{\tau}) = \underline{\tau}^\sharp\left(\frac{\sqrt{c}\sqrt{1-c}}{c^2-c+1}\right)\cdot \left(\frac{\sqrt{c}\sqrt{1-c}}{c^2-c+1}\right)^{-1}.
\end{equation}
The explicit description of $\phi$ is given in Table <ref> in Section 9. The opposite 1-cocycle $\phi$ of $\underline{H}$ has the following properties.
* For $\tau\in H$, we have
\begin{equation}\label{phiphi0}
\phi_0(\tau) = \sgn(\underline{\tau}_0)\cdot \phi(\underline{\tau})^2.
\end{equation}
where $\sgn: \underline{H}_0\simeq \mathfrak{S}(\{0,1,\infty\})\rightarrow\{\pm 1\}$ is the sign map.
* For $\underline{\tau}\in \underline{H}$, $\phi(\underline{\tau})\in (A')^\times$.
To prove (1), it is enough to calculate
\begin{equation}
\phi(\underline{\tau})^2 = \underline{\tau}^\sharp\left(\frac{c(1-c)}{(c^2-c+1)^2}\right)\cdot \left(\frac{c(1-c)}{(c^2-c+1)^2}\right)^{-1}
\end{equation}
Since the right hand side of the above equation depends only on the image $\underline{\tau}_0\in \underline{H}_0$ of $\underline{\tau}\in \underline{H}$ under $\underline{H}\rightarrow \underline{H}_0$ and $\phi_0(\tau)$ also depends only on $\underline{\tau}_0$ by Proposition <ref>, it is enough to check (<ref>) for each $\underline{\tau}_0\in \underline{H}_0$ by using Table <ref> and Table <ref>. (2) follows from (1).
We get an opposite 1-cocycle $\chi$ of $\underline{G}$ whose square coincides with $\chi_0$ up to sign.
(Definition of $\chi^{(1)},\chi^{(2)}$ and $\chi$) For $\underline{\rho} \in \underline{G}$, we define
\begin{equation}\label{chidef}
\begin{aligned}
&\chi^{(i)}(\underline{\rho}) = pr_i^\sharp(\phi(\underline{\rho}^{(i)})) && \in (B')^\times \quad \text{for }i = 1,2 \\
&\chi(\underline{\rho}) = \chi^{(1)}(\underline{\rho})\cdot \chi^{(2)}(\underline{\rho}) &&\in (B')^\times.
\end{aligned}
\end{equation}
By Proposition <ref>, $\chi$ satisfies the following equation[See Definition <ref> for the notation $\underline{\rho}_0^{(1)},\underline{\rho}_0^{(2)}$.] for $\rho\in G$.
\begin{equation}\label{chi0chi}
\chi_0(\rho) = \sgn(\underline{\rho}_0^{(1)})\sgn(\underline{\rho}_0^{(2)})\cdot \chi(\underline{\rho})^2
\end{equation}
By Definition <ref>, to find an opposite 1-cocycle $\widetilde{\chi}$ such that $\widetilde{\chi}^2 = \chi_0$, it is enough to find a square root of the group homomorphism $\underline{G}\rightarrow \{\pm 1\}; \rho\mapsto \sgn(\underline{\rho}_0^{(1)})\sgn(\underline{\rho}_0^{(2)})$. Hence we enlarge $G$ as follows.
(Definition of $\widetilde{G}$) Let $\widetilde{G}$ be the following fiber product of groups.
\begin{equation}\label{tildeGfiberproduct}
\begin{tikzcd}
\widetilde{G} \arrow[r,"\widetilde{\sgn}"]\arrow[d]\arrow[dr, phantom,"\lrcorner",very near start] & \mu_4(k) \arrow[d] &[-40pt] \zeta\arrow[l,"\ni",phantom] \arrow[d,mapsto] \\
G \arrow[r] & \mu_2(k) & \zeta^2 \arrow[l,"\ni",phantom]\\[-13pt]
\rho\arrow[u,phantom,"\rotatebox{90}{$\in$}"] \arrow[r,mapsto] & \sgn(\underline{\rho}_0^{(1)})\sgn(\underline{\rho}_0^{(2)})\arrow[u,phantom,"\rotatebox{90}{$\in$}"] &
\end{tikzcd}
\end{equation}
Then $\widetilde{G}$ can be written as follows.
\begin{equation}
\widetilde{G} = \{(\rho,\zeta)\in G\times \mu_4(k): \sgn(\underline{\rho}_0^{(1)})\sgn(\underline{\rho}_0^{(2)}) = \zeta^2\}
\end{equation}
We denote an element in $\widetilde{G}$ by $\widetilde{\rho}$ or $(\rho,\zeta)$. We define $\widetilde{\sgn}: \widetilde{G}\rightarrow \mu_4(k)$ as above. Since $\sqrt{-1}\in k$, $\mu_4(k)\rightarrow \mu_2(k); \zeta\mapsto \zeta^2$ is surjective and the kernel of this group homomorphism is $\mu_2(k)\subset \mu_4(k)$. Especially, we have the following exact sequence.
\begin{equation}\label{Gtildeexact}
\begin{tikzcd}[row sep = tiny]
1 \arrow[r]& \mu_2(k) \arrow[r] & \widetilde{G} \arrow[r] & G \arrow[r] & 1 \\
& & (\rho,\zeta) \arrow[r,mapsto]\aru & \rho\aru
\end{tikzcd}
\end{equation}
Finally, we get the desired cocycle $\widetilde{\chi}$.
For $\widetilde{\rho} = (\rho,\zeta)\in \widetilde{G}$, we define
\begin{equation}\label{defofchitilde}
\widetilde{\chi}(\widetilde{\rho}) = \widetilde{\sgn}(\widetilde{\rho})\cdot \chi(\underline{\rho}) = \zeta\cdot \chi(\underline{\rho}) \quad \in (B')^\times.
\end{equation}
where $\underline{\rho}\in \underline{G}$ is the image of $\rho\in G$ under $G\rightarrow \underline{G}$. Then $\widetilde{\chi}$ defines an opposite 1-cocycle of $\widetilde{G}$ on $(B')^\times$. Here $\widetilde{G}$ acts oppositely on $(B')^\times$ through $\widetilde{G}\rightarrow G\rightarrow \underline{G}$. Furthermore, $\widetilde{\chi}$ satisfies the following equation for any $\widetilde{\rho}= (\rho,\zeta)\in \widetilde{G}$.
\begin{equation}\label{chitilderelation}
\widetilde{\chi}(\widetilde{\rho})^2 = \chi_0(\rho)
\end{equation}
Since $\widetilde{\sgn}$ is the group homomorphism and $\underline{G}$ acts on $\mu_4(k)\subset B'$ trivially, $\widetilde{\sgn}$ is an opposite 1-cocycle of $\widetilde{G}$. Thus $\widetilde{\chi}$ is the product of opposite 1-cocycles and $\widetilde{\chi}$ is also an opposite 1-cocycle. Since $\widetilde{\sgn}$ satisfies $\widetilde{\sgn}(\rho)^2 = \sgn(\underline{\rho}_0^{(1)})\sgn(\underline{\rho}_0^{(2)})$, the equation (<ref>) follows from (<ref>) in Definition <ref>.
§.§ A $\widetilde{G}$-action on the family of Kummer surfaces $\X'$
Recall that $\widetilde{\Y}'$ is the base change of $\widetilde{\Y}$ by $T'\rightarrow T$ (Definition <ref>). Using the opposite 1-cocycle $\widetilde{\chi}$ in Definition <ref>, we can lift $G$-action on $\Y'$ to $\widetilde{G}$-action on $\widetilde{\Y}'$.
We have a $\widetilde{G}$-action on $\widetilde{\Y}'$ such that $(\widetilde{\Y}',\widetilde{G})\rightarrow (\Y',G)$ is a morphism in $\Schgpk$. For $\widetilde{\rho} = (\rho,\zeta) \in \widetilde{G}$, $\widetilde{\rho}^\sharp: \O_{\widetilde{\Y}'}\rightarrow \rho_*\O_{\widetilde{\Y}'}$ is described as follows.
\begin{equation}\label{gtildelocaldescriptioneqn}
x\mapsto \rho^\sharp(x),\quad y \mapsto \rho^\sharp(y), \quad u\mapsto \widetilde{\chi}(\widetilde{\rho})^{-1}\frac{\del}{\del x}(\rho^\sharp(x))\frac{\del}{\del y}(\rho^\sharp(y)) u
\end{equation}
where we use the notation in Proposition <ref>.
For $\widetilde{\rho}\in \widetilde{G}$, consider the following $\O_{\Y'}$-module isomorphism.
\begin{equation}\label{defofX}
X_{\widetilde{\rho}}: \rho^*\L \xrightarrow{\:\:\Psi_{\rho}\:\:} \L \xrightarrow{\:\:\widetilde{\chi}(\widetilde{\rho})^{-1}\:\:} \L.
\end{equation}
where $\widetilde{\chi}(\widetilde{\rho})^{-1}$ denotes the $\O_{\Y'}$-module isomorphism induced by the multiplication of $\widetilde{\chi}(\widetilde{\rho})^{-1}\in (B')^\times = \Gamma(\Y',\O_{\Y'}^\times)$. By the cocycle condition of $\Psi_{\rho}$ and the property of the opposite 1-cocycle, $\{X_{\widetilde{\rho}}\}_{\widetilde{\rho}\in \widetilde{G}}$ satisfies the cocycle condition. Hence we have the new $\widetilde{G}$-linearization $\{X_{\widetilde{\rho}}\}_{\widetilde{\rho}\in \widetilde{G}}$ of $\L$. Then by Definition <ref> and Proposition <ref>, we have
\begin{equation}
X_{\widetilde{\rho}}^{\otimes(-2)}(\rho^*(\eta)) = \widetilde{\chi}(\widetilde{\rho})^2\cdot \Psi_{\rho}^{\otimes(-2)}(\rho^*(\eta)) = \widetilde{\chi}(\widetilde{\rho})^2\cdot \chi_0(\rho)^{-1}\cdot\eta = \eta.
\end{equation}
Especially $\{X_\rho\}_{\rho\in \widetilde{G}}$ satisfies the condition (<ref>) in Proposition <ref>. Since $\widetilde{\Y}'$ is the double covering associated with $(\L, \eta)$ by Proposition <ref>, we have a $\widetilde{G}$-action on $\widetilde{\Y}'$ such that $(\widetilde{\Y}', \widetilde{G})\rightarrow (\Y',G)$ is a morphism in $\Schgpk$ by Proposition <ref>.We can calculate the local description of $\widetilde{G}$-action directly from the construction in Proposition <ref>. We can confirm that this action preserves the local equation $u^2-f(x)g(y)=0$ of $\widetilde{\Y}'$ as follows. For $\widetilde{\rho} = (\rho,\zeta) \in \widetilde{G}$, we have
\begin{equation}
\begin{aligned}
&\widetilde{\rho}^\sharp(u^2-f(x)g(y)) = \widetilde{\rho}^\sharp(u)^2-\rho^\sharp(f(x))\rho^\sharp(g(y)) \\
& = \widetilde{\chi}(\widetilde{\rho})^{-2}\left(\frac{\del}{\del x}(\rho^\sharp(x))\frac{\del}{\del y}(\rho^\sharp(y))\right)^2u^2 -\rho^\sharp(f(x))\rho^\sharp(g(y)) \\
& \underset{(\ref{transformfg}), (\ref{prchirel})}{=} \widetilde{\chi}(\widetilde{\rho})^{-2}\left(\frac{\del}{\del x}(\rho^\sharp(x))\frac{\del}{\del y}(\rho^\sharp(y))\right)^{2}u^2 - \chi_0(\rho)^{-1} \left(\frac{\del}{\del x}(\rho^\sharp(x))\frac{\del}{\del y}(\rho^\sharp(y))\right)^2 f(x)g(y) \\
& \underset{(\ref{chitilderelation})}{=} \chi_0(\rho)^{-1}\left(\frac{\del}{\del x}(\rho^\sharp(x))\frac{\del}{\del y}(\rho^\sharp(y))\right)^2(u^2-f(x)g(y)) =0.
\end{aligned}
\end{equation}
Recall that $\X'$ is the base change of $\X$ by $T'\rightarrow T$ (Definition <ref>). We lift the $\widetilde{G}$-action on $\widetilde{\Y}'$ to $\X'$. Since $\X'\rightarrow \widetilde{\Y}'$ is blowing-up, it is enough to show that the blowing-up locus is stable under $\widetilde{G}$-action.
We have the following.
* The set $\Sigma^2(\widetilde{\Y}')$ of $B'$-rational points is compatible with $(\widetilde{\Y}',\widetilde{G})\rightarrow(T',\underline{G})$.
* There exists a $\widetilde{G}$-action on $\X'$ such that $(\X',\widetilde{G})\rightarrow (\widetilde{\Y}',\widetilde{G})$ is a morphism in $\Schgpk$.
* For $\widetilde{\rho} = (\rho,\zeta) \in \widetilde{G}$, $\widetilde{\rho}^\sharp: \O_{\X'}\rightarrow \rho_*\O_{\X'}$ can be described locally as follows.
\begin{equation}
x\mapsto \rho^\sharp(x),\quad y\mapsto \rho^\sharp(y),\quad v\mapsto \frac{\sgn(\underline{\rho}_0^{(1)})}{\zeta}\frac{\chi^{(1)}(\underline{\rho})}{\chi^{(2)}(\underline{\rho})}\frac{\frac{\del}{\del y}(\rho^\sharp(y))}{\frac{\del}{\del x}(\rho^\sharp(x))}v
\end{equation}
Here we use the notation in Proposition <ref>.
By Proposition <ref>, $\Sigma^2(\Y')$ is compatible with $(\Y',G)\rightarrow (T',\underline{G})$. Since the $\widetilde{G}$-action on $\widetilde{\Y}'$ is a lift of $G$-action on $\Y'$ and each $\sigma \in \Sigma^2$ is contained in the branching locus of $\widetilde{\Y}'\rightarrow \Y'$, we can check that $\Sigma^2(\widetilde{\Y}')$ is compatible with $(\widetilde{\Y}',\widetilde{G})\rightarrow(T',\underline{G})$. Hence we show (1).By Proposition <ref>, $\X'\rightarrow \widetilde{\Y}'$ is the blowing-up along $\Sigma^2\subset \widetilde{\Y}'$. By (1), $\Sigma^2 \subset \widetilde{\Y}'$ is stable under the $\widetilde{G}$-action. Hence by applying Proposition <ref>, we have (2). (3) follows from the local description in Proposition <ref> and the definition of $\widetilde{\chi}$.
Recall that for $\sigma\in \Sigma^2$, $Q_\sigma \subset \X$ denotes the exceptional divisor over $\sigma\subset \widetilde{\Y}$ (Definition <ref>) and $Q'_{\sigma}$ denote the base change of $Q_{\sigma}$ by $T'\rightarrow T$ (Definition <ref>). The closed subscheme $Q'_\sigma \subset \X'$ is the same as the inverse image of $\sigma\subset \widetilde{\Y}'$ by $\X'\rightarrow \widetilde{\Y}'$. Hence we have the following.
For $\widetilde{\rho} = (\rho,\zeta) \in \widetilde{G}$ and $\sigma\in \Sigma^2$, the following holds.
\begin{equation}
\widetilde{\rho}(Q'_{\sigma}) = Q'_{\rho\cdot \sigma}.
\end{equation}
where $\rho\cdot \sigma$ is the image of $(\rho,\sigma)\in G\times \Sigma$ under the map $G\times \Sigma\rightarrow \Sigma$ induced by the $G$-action on $\Sigma$ in Proposition <ref>.
Finally, we can prove Proposition <ref> as follows.
The automorphism group of $\X'\rightarrow T'$ has a finite subgroup $\widetilde{G}$ which is isomorphic to a $\Z/2\Z$ extension of $(\mathfrak{S}_4\times_{\mathfrak{S}_3}\mathfrak{S}_4)^2$.
It is enough to show the following.
* We have an injective group homomorphism $\widetilde{G}\rightarrow \Aut_k(\X\rightarrow T)$.
* The group $\widetilde{G}$ is isomorphic to a $\Z/2\Z$ extension of $(\mathfrak{S}_4\times_{\mathfrak{S}_3}\mathfrak{S}_4)^2$.
By Definition <ref>, Proposition <ref> and Proposition <ref>, we have following morphisms in $\Schgpk$.
\begin{equation}\label{schgpkmatome}
(\X',\widetilde{G})\rightarrow (\widetilde{\Y}',\widetilde{G})\rightarrow (\Y',G) \rightarrow (T',\underline{G}).
\end{equation}
By the explicit description in Proposition <ref>, $\widetilde{G}$-action on $\X'$ is faithful. By Definition <ref>, we have (1). By the exact sequence (<ref>) in Definition <ref>, $\widetilde{G}$ is $\mu_2(k)\simeq \Z/2\Z$-extension of $G$. Furthermore, $G= H\times H$ (Definition <ref>) and $H\simeq \mathfrak{S}_4\times_{\mathfrak{S}_3}\mathfrak{S}_4$ (Definition <ref>). Hence we have (2).
For later use, we name $\widetilde{G}$-actions on fibers of $\X'\rightarrow T'$.
For a $k$-rational point $t\in T'(k)$, let $\X_t$ denote the fiber of $\X'\rightarrow T'$ over $t$. We denote the natural inclusion $\X_t\hookrightarrow \X'$ by $i_t$. For $\widetilde{\rho} = (\rho,\zeta) \in \widetilde{G}$, let $\underline{\rho}(t) \in T'(k)$ denote the $k$-rational point $\underline{\rho}\circ t$. We define $\rho_t: \X_t\rightarrow \X_{\underline{\rho}(t)}$ as a unique isomorphism over $k$ which makes the following diagram commute.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
\X' \arrow[rr]\arrow[dd,"\widetilde{\rho}"] && T' \arrow[dd]\arrow[dd,"\underline{\rho}"{yshift = 8pt}]& \\
&\X_t \arrow[rr,crossing over] \arrow[ul,"i_t"'] && \Spec k \arrow[ul,"t"'] \\
\X' \arrow[rr] && T' & \\
&\X_{\underline{\rho}(t)} \arrow[ul,"i_{\underline{\rho}(t)}"] \arrow[from = uu,crossing over,dashed]\arrow[rr] && \Spec k \arrow[from = uu, crossing over, equal]\arrow[ul,"\underline{\rho}(t)"']
\end{tikzcd}
\end{equation}
§ CONSTRUCTION OF A SUBGROUP $\XI$ OF THE HIGHER CHOW GROUP
In this section, we explain the construction of a higher Chow subgroup $\Xi\subset \CH^2(\X^\circ,1)$ where $\X^\circ$ is an open subset of $\X'$. First, we construct $\Xi^\can \subset \CH^2(\X^\circ,1)$ and we define $\Xi$ as the sum of $\widetilde{\rho}_*\Xi^\can$ where $\widetilde{\rho} \in \widetilde{G}$. For the construction of higher Chow cycles, we use the following results (Corollary 5.3 in [14]).
Let $X$ be a variety over $k$. The higher Chow group $\CH^2(X,1)$ of $X$ is canonically isomorphic to the homology group of the following sequence.
\begin{equation}
\begin{tikzcd}
K_2^{\mathrm{M}} (R(X)) \arrow[r,"T"] & \displaystyle\bigoplus_{Z\in X^{(1)}}R(Z)^\times \arrow[r,"\div"]&
\displaystyle\bigoplus_{p\in X^{(2)}}\Z\cdot p
\end{tikzcd}
\end{equation}
Here $X^{(1)}, X^{(2)}$ are the sets of integral closed subschemes of $X$ codimension $1$ and $2$, the map $\div$ is the sum of the divisor map $\div_Z$ for each $Z\in X^{(1)}$ and $T$ is the tame symbol map from the Milnor $K$-group $K_2^{\mathrm{M}} (R(X))$ of $R(X)$.
Hence to construct higher Chow cycles, it is enough to find a collection of rational functions which lies in the kernel of $\div$.
§.§ A familiy of curves $\Cc$ on $\X^\circ$
We construct a family of curves $\Cc$, which is the key for our construction of higher Chow cycles. First, we define an open subset $T^\circ\subset T'$. Hereafter we consider all things on this open subset.
Under the $\underline{G}$-action on $B'$, the orbit of $a-b\in B'$ consists of the following six elements up to multiplications of elements in $(B')^\times$.
\begin{equation}
a-b, a+b-1, a-\frac{b}{b-1}, a-\frac{1}{1-b}, a-\frac{1}{b}, a-\frac{b-1}{b}
\end{equation}
We define a $k$-algebra $B^\circ$ as the localization of $B'$ by these six elements. We define $T^\circ = \Spec B^\circ$, which is an open subscheme of $T'$. For a scheme $Z$ over $T'$, $Z^\circ$ denotes its base change by $T^\circ\hookrightarrow T'$. For example, $\Y^\circ = \Y' \times_{T'} T^\circ$, $\widetilde{\Y}^\circ = \widetilde{\Y}' \times_{T'} T^\circ$ and $\X^\circ = \X' \times_{T'} T^\circ$.
By the construction, $T^\circ\subset T'$ is stable under $\underline{G}$-action. Hence we have the following diagram in $\Schgpk$ whose vertical morphisms are open immersions.
\begin{equation}
\begin{tikzcd}
(\X^\circ, \widetilde{G}) \arrow[r]\arrow[d,hook'] & (\widetilde{\Y}^\circ, \widetilde{G}) \arrow[d,hook'] \arrow[r] & (\Y^\circ,G) \arrow[d,hook'] \arrow[r] & (T^\circ, \underline{G}) \arrow[d,hook'] \\
(\X',\widetilde{G}) \arrow[r] & (\widetilde{\Y}',\widetilde{G}) \arrow[r] & (\Y',G) \arrow[r] & (T',\underline{G})
\end{tikzcd}
\end{equation}
We define a closed subscheme $\D\subset \Y^\circ$ by the local equation $x=y$. Furthermore, we define a closed subscheme $\widetilde{\D}\hookrightarrow\widetilde{\Y}^\circ$ as the following fiber product.
\begin{equation}
\begin{tikzcd}
\widetilde{\D} \arrow[r,hook]\arrow[d]\arrow[dr,"\lrcorner",very near start, phantom] & \widetilde{\Y}^\circ \arrow[d] \\
\D \arrow[r,hook] & \Y^\circ
\end{tikzcd}
\end{equation}
The closed immersion $\widetilde{\D}\hookrightarrow \widetilde{\Y}^\circ$ is described locally on $\widetilde{Y}_{0,0}^\circ\subset \Y^\circ$ as follows.
\begin{equation}
\begin{tikzcd}
\Spec B^\circ[u,z]/(u^2-f(z)g(z)) \arrow[r] &[-15pt] \Spec B^\circ[u,x,y]/(u^2-f(x)g(y)) = \widetilde{Y}_{0,0}^\circ
\end{tikzcd}
\end{equation}
where $f(z),g(z)$ are polynomials in $(\ref{fg})$ in Definition <ref> and the morphism is induced by $x\mapsto z$ and $y\mapsto z$.
The figure of $\D$ on $\Y^\circ$
We define a closed subscheme $\Cc\subset \X^\circ$ as the strict transformation of $\widetilde{\D}\hookrightarrow \widetilde{\Y}^\circ$ by the blowing-up $\X^\circ\rightarrow \widetilde{\Y}^\circ$. The closed immersion $\Cc\hookrightarrow \X^\circ$ is described locally on $V_{0,0}^\circ, W_{0,0}^\circ\subset \X^\circ$ as follows.
\begin{equation}
\begin{tikzcd}[column sep = tiny, row sep = tiny]
\Spec B^\circ[v,z]/(v^2(1-az)-(1-bz)) \arrow[r]& \Spec B^\circ[v,x,y]/(v^2f(x)-g(y)) = V^\circ_{0,0} \\
\Spec B^\circ[w,z]/(w^2(1-bz)-(1-az)) \arrow[r] &\Spec B^\circ[w,x,y]/(w^2g(y)-f(x)) =W^\circ_{0,0}
\end{tikzcd}
\end{equation}
These morphisms are induced by $x\mapsto z$ and $y\mapsto z$.
By the description in Proposition <ref> and the fact $a-b$ is invertible on $T^\circ$, we see that $\Cc$ is a conic bundle on $T^\circ$ with a section (e.g. $(x,y,v) = (0,0,1)$). Hence we have the following corollary.
The $T^\circ$-scheme $\Cc$ is isomorphic to $\P^1_{T^\circ}$.
In this subsection, we constructed the following closed subschemes.
\begin{equation}
\begin{tikzcd}[row sep = small]
\X^\circ \arrow[r] &[40pt] \widetilde{\Y}^\circ \arrow[r] &[40pt] \Y^\circ \\[3pt]
\Cc \arrow[u,hook]\arrow[r,"\mathrm{strict\;transform}"'] & \widetilde{\D} \arrow[u,hook]\arrow [r,"\text{pull-back}"']\arrow[ur,"\urcorner",very near start,phantom] & \D \arrow[u,hook]
\end{tikzcd}
\end{equation}
§.§ Construction of a subgroup $\Xi^\can$ of the higher Chow group
In this section, we will construct a subgroup $\Xi^\can$ of the higher Chow group $\CH^2(\X^\circ,1)$. For the construction, we consider the closed subscheme $\Cc$ in the previous subsection and exceptional divisors $Q^\circ_{(0,0)},Q^\circ_{(1,1)}$ and $Q^\circ_{(\infty,\infty)}$ in Definition <ref>.To define rational functions on them, we use the following local descriptions of $Q^\circ_{(0,0)}, Q^\circ_{(1,1)}$ and $Q^\circ_{(\infty,\infty)}$. Since $Q^\circ_{(0,0)}$ and $Q^\circ_{(1,1)}$ are contained in $V_{0,0}$ and defined by the equation $x=y=0$ and $x= y=1$, we have the following description.
\begin{equation}\label{exdescription}
\begin{aligned}
V^\circ_{0,0}\cap Q^\circ_{(0,0)} &= \Spec B^\circ[v,x,y]/(v^2f(x)-g(y),x,y) \simeq \Spec B^\circ[v] \\
V^\circ_{0,0}\cap Q^\circ_{(1,1)} &= \Spec B^\circ[v,x,y]/(v^2f(x)-g(y),x-1,y-1) \simeq \Spec B^\circ[v] \\
\end{aligned}
\end{equation}
To get the local description of $Q^\circ_{(\infty,\infty)}$, we consider the following affine open subscheme $V^\circ_{1,1}$ of $\X^\circ$.
\begin{equation}\label{exdescription2}
V_{1,1}^\circ=\Spec B^\circ[v',\xi,\eta]/((v')^2\xi(\xi-1)(\xi-a)-\eta(\eta-1)(\eta-b))
\end{equation}
Here $\xi = \frac{1}{x}$,$\eta = \frac{1}{y}$ and $v' = \frac{y^2}{x^2}v$. Since $Q_{(\infty,\infty)}^\circ$ is defined by the equation $\xi = \eta = 0$, we have the following description.
\begin{equation}
V^\circ_{1,1} \cap Q^\circ_{(\infty,\infty)} = \Spec B^\circ[v',\xi,\eta]/((v')^2\xi(\xi-1)(\xi-a)-\eta(\eta-1)(\eta-b),\xi,\eta) \simeq \Spec B^\circ[v']
\end{equation}
We define six $B^\circ$-rational points $p_\bullet^\delta\: (\bullet\in \{0,1,\infty\},\delta\in\{+,-\})$ on $\X^\circ$ as follows.
* $p_0^+$ and $p_0^-$ correspond to $B^\circ$-rational points on $V_{0,0}^\circ\subset\X^\circ$ such that $(v,x,y) = \left(1,0,0\right)$ and $(v,x,y) = \left(-1,0,0\right)$ respectively.
* $p_1^+$ and $p_1^-$ correspond to $B^\circ$-rational points on $V_{0,0}^\circ\subset \X^\circ$ such that $(v,x,y) = \left(\frac{\sqrt{1-b}}{\sqrt{1-a}},1,1\right)$ and $(v,x,y) = \left(-\frac{\sqrt{1-b}}{\sqrt{1-a}},1,1\right)$ respectively.
* $p_{\infty}^+$ and $p_{\infty}^-$ correspond to $B^\circ$-rational points on $V_{1,1}^\circ\subset \X^\circ$ such that $(v',\xi,\eta) = \left(\frac{\sqrt{b}}{\sqrt{a}},0,0\right)$ and $(v',\xi,\eta) = \left(-\frac{\sqrt{b}}{\sqrt{a}},0,0\right)$ respectively.
By the local description, we have the following relations.
\begin{equation}\label{intersectionrelation}
\Cc \cap Q^\circ_{(0,0)} = p_0^+\sqcup p_0^-,\quad \Cc \cap Q^\circ_{(1,1)} = p_1^+\sqcup p_1^-,\quad \Cc \cap Q^\circ_{(\infty,\infty)} = p_\infty^+\sqcup p_\infty^-
\end{equation}
The relation between $p_\bullet^\delta$ and $\Cc, Q^\circ_{(\bullet, \bullet)}$
We define the following non-zero rational functions on $\Cc$, $Q^\circ_{(0,0)}$, $Q^\circ_{(1,1)}$ and $Q_{(\infty,\infty)}^\circ$ using the local description in Proposition <ref> and equation (<ref>),(<ref>).
* $\psi_0 = (v+1)\cdot(v-1)^{-1}$, $\psi_1 = \left(v + \frac{\sqrt{1-b}}{\sqrt{1-a}}\right)\cdot\left(v - \frac{\sqrt{1-b}}{\sqrt{1-a}}\right)^{-1}$,
$\psi_\infty = \left(v + \frac{\sqrt{b}}{\sqrt{a}}\right)\cdot\left(v - \frac{\sqrt{b}}{\sqrt{a}}\right)^{-1} \in R(\Cc)^\times$
* $\varphi_0 = (v-1)\cdot(v + 1)^{-1} \in R\left(Q^\circ_{(0,0)}\right)^\times$
* $\varphi_1 = \left(v-\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)\cdot\left(v+\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)^{-1} \in R\left(Q^\circ_{(1,1)}\right)^\times$
* $\varphi_\infty =\left(v'- \frac{\sqrt{b}}{\sqrt{a}}\right)\cdot\left(v'+ \frac{\sqrt{b}}{\sqrt{a}}\right)^{-1} \in R\left(Q^\circ_{(\infty,\infty)}\right)^\times$
Then the rational functions $\varphi_\bullet, \psi_\bullet$ satisfy the following relations.
* $\div_{\Cc}(\psi_0) = p_{0}^- - p_{0}^+ = - \div_{Q^\circ_{(0,0)}}(\varphi_0)$
* $\div_{\Cc}(\psi_1) = p_{1}^- - p_{1}^+ = - \div_{Q^\circ_{(1,1)}}(\varphi_1)$
* $\div_{\Cc}(\psi_\infty) = p_\infty^- - p_\infty^+ = - \div_{Q^\circ_{(\infty,\infty)}}(\varphi_\infty)$
Then we can construct a subgroup $\Xi^\can$ of $\CH^2(\X^\circ, 1)$ at most rank 3 as follows.
(Definition of $\Xi^\can$)
Consider the following elements of $\bigoplus_{Z\in (\X^\circ)^{(1)}}R(Z)^\times$.
\begin{equation}
\begin{aligned}
&\xi_0 = (\Cc, \psi_0) + (Q^\circ_{(0,0)},\varphi_0) \\
&\xi_1= (\Cc, \psi_1) + (Q^\circ_{(1,1)},\varphi_1) \\
&\xi_\infty = (\Cc, \psi_\infty) + (Q^\circ_{(\infty,\infty)},\varphi_\infty)
\end{aligned}
\end{equation}
By Proposition <ref>, they are in $\Ker\left(\bigoplus_{Z\in (\X^\circ)^{(1)}}R(Z)^\times\xrightarrow{\div} \bigoplus_{p\in (\X^\circ)^{(2)}}\Z\cdot p\right)$. Hence these elements define elements in $\CH^2(\X^\circ,1)$ which are denoted by the same symbols $\xi_0,\xi_1,\xi_\infty$ respectively. We define $\Xi^\can \subset \CH^2(\X^\circ,1)$ to be the subgroup generated by $\xi_0,\xi_1$ and $\xi_\infty$. For $\epsilon \in \Z^{\{0,1,\infty\}}$, we set
\begin{equation}
\xi(\epsilon) = \epsilon(0)\xi_0 + \epsilon(1)\xi_1+\epsilon(\infty)\xi_\infty \in \Xi^\can.
\end{equation}
By the following pull-back map, we can regard an element $\xi \in \CH^2(\X^\circ,1)$ as a family of higher Chow cycles $\{\xi_t\}_{t\in T^\circ}$. The existence of the following pull-back map is given in [12], Part I, Chapter II, 2.1.6.
For a $k$-rational point $t\in T^\circ(k)$, $i_t: \X_t\rightarrow \X^\circ$ in Definition <ref> is a $k$-morphism between smooth varieties. Hence we have a pull-back map
\begin{equation}
i_t^* : \CH^2(\X^\circ, 1) \lra \CH^2(\X_t, 1).
\end{equation}
For each $\xi \in \CH^2(\X^\circ, 1)$, $i_t^*\xi$ is denoted by $\xi_t$.
For a $k$-rational point $t\in T^\circ(k)$ and $\epsilon \in \Z^{\{0,1,\infty\}}$, $\xi(\epsilon)_t\in \CH^2(\X_t,1)$ is represented by the following element in $\bigoplus_{Z\in \X_t^{(1)}}R(Z)^\times$.
\begin{equation}\label{fiberxipresentation}
\begin{aligned}
&\left(\Cc_t, (\psi_0)_t^{\epsilon(0)}(\psi_1)_t^{\epsilon(1)}(\psi_\infty)_t^{\epsilon(\infty)}\right) \\
& + (Q_{(0,0)t}, (\varphi_0)^{\epsilon(0)}_t) + (Q_{(1,1)t}, (\varphi_1)^{\epsilon(1)}_t) + (Q_{(\infty,\infty)t}, (\varphi_\infty)^{\epsilon(\infty)}_t)
\end{aligned}
\end{equation}
Here $\Cc_t, Q_{(\bullet,\bullet)t}$ are the fibers of $\Cc$ and $Q_{(\bullet,\bullet)}^\circ$ at $t$ and $(\psi_\bullet)_t, (\varphi_\bullet)_t$ are the pull back of the rational function $\psi_\bullet,\varphi_\bullet$ by $\Cc_t \hookrightarrow \Cc$ and $Q_{(\bullet,\bullet)t} \hookrightarrow Q_{(\bullet,\bullet)}^\circ$.
Recall that we regard elements in $\bigoplus_{Z\in (\X^\circ)^{(1)}}R(Z)^\times$ as elements in $Z^2(\X^\circ,1)$ $\subset Z^2(\X^\circ\times_k \Delta^1)$ ($\Delta^1 = \Spec k[T_0,T_1]/(T_0+T_1-1)$) by considering their graphs of rational functions. For example, $(\Cc,\psi_1)$ represents a codimension 1 integral closed subscheme of $\Cc\times_k \Delta^1$ defined by the local equation
\begin{equation}
\left(v-\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)T_0 + \left(v+\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)T_1 = 0.
\end{equation}
Here we use the local coordinates of $\Cc$ in Proposition <ref>. This closed subscheme intersect properly with $\X_t\times_k \Delta^1, \X_t\times_k \{T_0 = 0\}, \X_t\times_k \{T_1 = 0\}$. Hence the pull-back of the cycle corresponding to $(\Cc,\psi_1)$ by $\X_t\hookrightarrow \X^\circ$ is defined. The pull-back coincides with the intersection of this closed subscheme with $\X_t\times_k \Delta^1$ and the intersection is the graph of $(\psi_1)_t$. By considering pull-backs of $(\Cc,\psi_\bullet)$ and $(Q^\circ_{(\bullet,\bullet)},\varphi_\bullet)$ for $\bullet = 0,1,\infty$ similarly, we can show that
\begin{equation}
\begin{aligned}
& (\Cc_t, (\psi_0)_t) + (Q_{(0,0),t},(\varphi_0)_t) \\
& (\Cc_t, (\psi_1)_t) + (Q_{(1,1),t},(\varphi_1)_t) \\
& (\Cc_t, (\psi_\infty)_t) + (Q_{(\infty,\infty),t},(\varphi_\infty)_t)
\end{aligned}
\end{equation}
represents $\xi_{0,t}, \xi_{1,t}$ and $\xi_{\infty,t} \in \CH^2(\X_t,1)$. Hence we have the result.
§.§ Definition of a subgroup $\Xi$ of the higher Chow group
In this section, we define $\Xi \subset \CH^2(\X^\circ,1)$ and give representatives in $\bigoplus_{Z\in (\X^\circ)^{(1)}}R(Z)^\times$ for cycles in $\Xi$. In Section 6, we use these expressions to show that a subgroup $\widetilde{I}$ of $\widetilde{G}$ stabilize $\Xi^\can\subset \CH^2(\X^\circ,1)$.
We define a subgroup $\Xi$ of $\CH^2(\X^\circ,1)$ as
\begin{equation}
\Xi = \sum_{\widetilde{\rho}\in \widetilde{G}}\widetilde{\rho}_*\Xi^\can
\end{equation}
where $\Xi^\can \subset \CH^2(\X^\circ,1)$ is the subgroup of higher Chow group defined in Definition <ref> and $\widetilde{\rho}_* : \CH^2(\X^\circ,1)\rightarrow \CH^2(\X^\circ,1)$ is the push-forward map induced by an automorphism $\widetilde{\rho}:\X^\circ\rightarrow \X^\circ$. For a $k$-rational point $t\in T^\circ(k)$, we define $\Xi_t\subset \CH^2(\X_t,1)$ as the image of $\Xi$ under $i_t^*$ in Definition <ref>.
For $\rho\in G$, we define a closed subscheme $\D_\rho\subset \Y^\circ$ by the schematic image $\rho(\D)$. Note that $\D_\rho$ is determined by the image of $\rho \in G$ under $G\rightarrow G_0$. The local equation of $\D_\rho$ is given by $(\rho^{-1})^\sharp(x-y) = 0$. We define $\widetilde{\D}_\rho\hookrightarrow \widetilde{\Y}^\circ$ as the pull-back of $\D_\rho\hookrightarrow \Y^\circ$ by $\widetilde{\Y}^\circ \rightarrow \Y^\circ$. Furthermore, we define $\Cc_\rho\hookrightarrow \X^\circ$ as the strict transformation of $\widetilde{\D}_\rho$ by $\X^\circ\rightarrow \widetilde{\Y}^\circ$.
\begin{equation}
\begin{tikzcd}[row sep = small]
\X^\circ \arrow[r] &[40pt] \widetilde{\Y}^\circ \arrow[r] &[40pt] \Y^\circ \\[3pt]
\Cc_\rho \arrow[u,hook]\arrow[r,"\mathrm{strict\;transform}"'] & \widetilde{\D}_\rho \arrow[u,hook]\arrow [r,"\text{pull-back}"']\arrow[ur,"\urcorner",very near start,phantom] & \D_\rho \arrow[u,hook]
\end{tikzcd}
\end{equation}
Since $\rho(\D) = \D_\rho$, for $\widetilde{\rho}\in \widetilde{G}$, we have $\widetilde{\rho}(\widetilde{\D}) = \widetilde{\D}_\rho$ and $\widetilde{\rho}(\Cc) = \Cc_\rho$.
The following proposition follows from Proposition <ref> and Definition <ref>.
By an automorphism $\widetilde{\rho} = (\rho,\zeta)\in \widetilde{G}$ on $\X^\circ$, we have
\begin{equation}\label{rhotransfersubsch}
\widetilde{\rho}(\Cc) = \Cc_\rho,\quad \widetilde{\rho}(Q^\circ_{(0,0)}) = Q^\circ_{\rho\cdot(0,0)},\quad \widetilde{\rho}(Q^\circ_{(1,1)}) = Q^\circ_{\rho\cdot(1,1)},\quad \widetilde{\rho}(Q^\circ_{(\infty,\infty)}) = Q^\circ_{\rho\cdot(\infty,\infty)}.
\end{equation}
Let $\epsilon \in \Z^{\{0,1,\infty\}}$. Then $\rho_*\xi(\epsilon)\in \CH^2(\X^\circ,1)$ is represented by the following elements in $\bigoplus_{Z\in (\X^\circ)^{(1)}}R(Z)^\times$.
\begin{equation}
\begin{aligned}
&\left(\Cc_\rho, (\widetilde{\rho}^{-1})^\sharp(\psi_0^{\epsilon(0)}\psi_1^{\epsilon(1)}\psi_\infty^{\epsilon(\infty)})\right) + (Q_{\rho\cdot(0,0)}^\circ, (\widetilde{\rho}^{-1})^\sharp(\varphi_0^{\epsilon(0)})) \\
& + (Q_{\rho\cdot(1,1)}^\circ, (\widetilde{\rho}^{-1})^\sharp(\varphi_1^{\epsilon(1)})) + (Q_{\rho\cdot(\infty,\infty)}^\circ, (\widetilde{\rho}^{-1})^\sharp(\varphi_\infty^{\epsilon(\infty)}))
\end{aligned}
\end{equation}
where $(\widetilde{\rho}^{-1})^\sharp$ are the field isomorphisms $R(\Cc)\rightarrow R(\Cc_\rho)$ and $R(Q_{(\bullet,\bullet)}^\circ)\rightarrow R(Q_{\rho\cdot(\bullet,\bullet)}^\circ)$ induced by $\widetilde{\rho}$.
As we stated in the introduction, several elements in $\widetilde{\rho}_*\Xi^\can$ are at first constructed geometrically after T. Terasoma's idea. The keys for the geometric construction are the following.
* There exists the isomorphism $\Cc_\rho\simeq\P^1_{T^\circ}$ over $T^\circ$.
* For $\bullet = 0,1,\infty$, $\Cc_\rho\cap Q_{\rho\cdot(\bullet,\bullet)}^\circ$ decompose into the disjoint union of two $B^\circ$-rational points.
From these facts, we can construct higher Chow cycles in $\Xi$ directly by the similar method in subsection 5.2.
§ SUBGROUPS $\WIDETILDE{I}$ AND $\WIDETILDE{G}_\FIB$ OF $\WIDETILDE{G}$
In this section, we construct two subgroups $\widetilde{I}$ and $\widetilde{G}_\fib$ of $\widetilde{G}$. As we will see later (Proposition <ref>), these subgroups stabilize the image of $\Xi\subset \CH^2(\X^\circ,1)$ under the transcendental regulator maps at fibers $\X_t$. The subgroup $\widetilde{I}$ consists of automorphisms in $\widetilde{G}$ which stabilize a subgroup of symbols in $\bigoplus_{Z\in (\X^\circ)^{(1)}}R(Z)^\times$ which represents cycles in $\Xi^\can$. Hence $\widetilde{I}$ stabilize $\Xi^\can$ (Proposition <ref>). We describe the explicit $\widetilde{I}$-action on $\Xi^\can$.The subgroup $\widetilde{G}_\fib$ consists of automorphisms in $\widetilde{G}$ over $T^\circ$. Hence elements of $\widetilde{G}_\fib$ induce automorphisms of each fiber $\X_t$. Since $\widetilde{G}_\fib$ acts on a relative 2-form $\omega\in \Gamma(\X^\circ,\Omega^2_{\X^\circ/T^\circ})$ by the multiplication $\pm 1$, $\widetilde{G}_\fib$ stabilize the image of the transcendental regulator map (Proposition <ref>).
§.§ Definition of $\widetilde{I}$ and stability of $\Xi^\can$ under the $\widetilde{I}$-action
By Proposition <ref>, we identify $H_0=\mathfrak{S}(\{0,1,1/c,\infty\})$. We define a subgroup $I_0\subset G_0$ by the image of the stabilizer of $1/c\in\{0,1,1/c,\infty\}$ under the following diagonal embedding.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
H_0 \arrow[r,"\Delta"] & H_0\times H_0 \arrow[r,equal]&[-15pt] G_0; &[-30pt] \tau_0\arrow[r,mapsto] &(\tau_0,\tau_0)
\end{tikzcd}
\end{equation}
Consider the following diagram.
\begin{equation}\label{stabdiag}
\begin{tikzcd}
\mathfrak{S}(\{0,1,1/c,\infty\}) \arrow[r,equal] \arrow[d,"(\ref{s4s3ver2})",twoheadrightarrow] &[-15pt] H_0\arrow[r,"\Delta"]& H_0\times H_0\arrow[r,equal] &[-15pt] G_0 \arrow[d]\\
\mathfrak{S}(\{0,1,\infty\})\arrow[r,equal] & \underline{H}_0 \arrow[r,"\Delta"]& \underline{H}_0\times \underline{H}_0 \arrow[r,equal] & \underline{G}_0
\end{tikzcd}
\end{equation}
By the description of $H_0\rightarrow \underline{H}_0$ in Remark <ref> and the commutativity of diagram (<ref>), $I_0\hookrightarrow G_0 \twoheadrightarrow \underline{G}_0$ is injective and its image coincides with the image of the diagonal embedding of $\underline{H}_0$. We denote the image of $I_0$ in $\underline{G}_0$ by $\underline{I}_0$. By the argument above, $I_0\simeq \underline{I}_0$.
An element of the stabilizer of $1/c$ induces a permutation on $\{0,1,\infty\}\subset \{0,1,1/c,\infty\}$. Hence we often identify[This isomorphism is different from $I_0\xrightarrow{\:\sim\:} \underline{I}_0 \xrightarrow{\:\sim\:} H_0 = \mathfrak{S}(\{0,1,\infty\})$ where the second isomorphism is induced by the diagonal embedding.] $I_0$ with $\mathfrak{S}(\{0,1,\infty\})$. For each $\rho_0\in I_0 = \mathfrak{S}(\{0,1,\infty\})$, the action of $\rho_0$ on $\Y$ is given in the following Table <ref>.
The action of $I_0 = \mathfrak{S}(\{0,1,\infty\})$ on $\Y$
$\rho_0$ $( \underline{\rho}_0^\sharp(a), \underline{\rho}_0^\sharp(b) ) $ $(\rho_0^\sharp(x), \rho_0^\sharp(y)) $ $\rho_0$ $( \underline{\rho}_0^\sharp(a), \underline{\rho}_0^\sharp(b) ) $ $ (\rho_0^\sharp(x), \rho_0^\sharp(y)) $
$\id $ $(a,b) $ $(x,y)$ $ (0\;1) $ $ (\frac{a}{a-1},\frac{b}{b-1})$ $(1-x,1-y)$
$(1\;\infty)$ $ (1-a,1-b)$ $(\frac{x}{x-1},\frac{y}{y-1})$ $ (0\;1\;\infty)$ $ (\frac{a-1}{a},\frac{b-1}{b}) $ $(\frac{1}{1-x},\frac{1}{1-y})$
$(0\;\infty) $ $ (\frac{1}{a},\frac{1}{b})$ $ (\frac{1}{x},\frac{1}{y})$ $ (0\;\infty\;1)$ $(\frac{1}{1-a},\frac{1}{1-b})$ $(\frac{x-1}{x},\frac{y-1}{y})$
We define subgroups $\underline{I}\subset \underline{G}$, $I\subset G$ and $\widetilde{I}\subset \widetilde{G}$ as follows.
\begin{equation}\label{Isettheoretic}
\begin{aligned}
&\underline{I} = \{\underline{\rho}\in \underline{G}: \underline{\rho}_0\in \underline{I}_0\} \\
&I =\{\rho\in G: \rho_0\in I_0\} \\
&\widetilde{I} = \{(\rho,\zeta)\in \widetilde{G}: \rho\in I\}
\end{aligned}
\end{equation}
Then $I$ is isomorphic to $I_0\times_{\underline{I}_0} \underline{I}$. Since $I_0\rightarrow \underline{I}_0$ is an isomorphism by Definition <ref>, $I\rightarrow \underline{I}$ is also an isomorphism.
Since $\underline{I}_0\subset \underline{G}_0$ is the image of diagonal embedding (Definition <ref>), we have
\begin{equation}
\underline{I} = \{(\underline{\rho}^{(1)},\underline{\rho}^{(2)})\in \underline{H}\times \underline{H}: \underline{\rho}_0^{(1)}=\underline{\rho}_0^{(2)}\} = \underline{H}\times_{\underline{H}_0}\underline{H}
\end{equation}
Since $\underline{H}\simeq \mathfrak{S}_4$ by Remark <ref> and $\underline{H}_0\simeq \mathfrak{S}_3$ by Definition <ref>, $\underline{I}$ is isomorphic to $\mathfrak{S}_4\times_{\mathfrak{S}_3} \mathfrak{S}_4$. Since $I\simeq \underline{I}$, $I$ is also isomorphic to $\mathfrak{S}_4\times_{\mathfrak{S}_3} \mathfrak{S}_4$. Furthermore, since $\sgn(\underline{\rho}_0^{(1)})\sgn(\underline{\rho}_0^{(2)}) = 1$ for $\rho\in I$, we have a splitting of $\widetilde{I}\rightarrow I$ defined by $I\rightarrow \widetilde{I}; \rho\mapsto (\rho,1)$. By this splitting, we have an isomorphism $\widetilde{I}\simeq I \times \Z/2\Z$.
We will show that the $\widetilde{I}$-action stabilizes $\Xi^\can \subset\CH^2(\X^\circ,1)$. Hereafter in this subsection, we assume $\widetilde{\rho} = (\rho,\zeta) \in \widetilde{I}$. To prove $\widetilde{\rho}_*\Xi^\can\subset \Xi^\can$, we show that the symbol in Proposition <ref> which represents $\widetilde{\rho}_*\xi(\epsilon)$ coincides with the symbol which represents an element in $\Xi^\can$.
* Let $\Cc_\rho$ be the closed subscheme defined in Definition <ref>. Then we have $\Cc_\rho = \Cc$.
* Let $\rho_0$ be the image of $\rho$ by $I \rightarrow I_0\xrightarrow{\sim} \mathfrak{S}(\{0,1,\infty\})$ where the last isomorphism is the one in Remark <ref>. Then we have the following.
\begin{equation}
Q^\circ_{\rho\cdot (0,0)} = Q^\circ_{(\rho_0(0),\rho_0(0))},\quad Q^\circ_{\rho\cdot (1,1)} = Q^\circ_{(\rho_0(1),\rho_0(1))},\quad Q^\circ_{\rho\cdot (\infty,\infty)} = Q^\circ_{(\rho_0(\infty),\rho_0(\infty))}
\end{equation}
By the description of $I_0$-action in Table <ref>, the $I$-action on $\Y^\circ$ stabilizes the local equation $x=y$ of $\D$. Hence $\D_\rho = \D$ and by Definition <ref>, we have (1). (2) follows from the way of the identification $I_0 = \mathfrak{S}(\{0,1,\infty\})$ in Remark <ref>.
We will prove that the sets of rational functions $\{\varphi_\bullet^{\pm 1}: \bullet = 0,1,\infty\}$ and $\{\psi_\bullet^{\pm 1}: \bullet = 0,1,\infty\}$ are stable under the $\widetilde{I}$-action.
By Proposition <ref>, we have
\begin{equation}\label{intersectionpoints}
\widetilde{\rho}(p^+_\bullet)\sqcup \widetilde{\rho}(p^-_\bullet) = \widetilde{\rho}(\Cc\cap Q^\circ_{(\bullet,\bullet)}) = \Cc\cap Q^\circ_{(\rho_0(\bullet),\rho_0(\bullet))} = p^+_{\rho_0(\bullet)}\sqcup p^-_{\rho_0(\bullet)}
\end{equation}
for $\bullet = 0,1,\infty$ where $p^+_\bullet,p^-_\bullet$ are $B^\circ$-rational points in Definition <ref>. Then by comparing connected components in $(\ref{intersectionpoints})$, we have either
\begin{equation}
\mathrm{(A)}
\left\{
\begin{aligned}
&\widetilde{\rho}(p^+_\bullet) = p^+_{\rho_0(\bullet)} \\
&\widetilde{\rho}(p^-_\bullet) = p^-_{\rho_0(\bullet)}
\end{aligned}
\right.
\quad
\text{or}
\quad
\mathrm{(B)}
\left\{
\begin{aligned}
&\widetilde{\rho}(p^+_\bullet) = p^-_{\rho_0(\bullet)} \\
&\widetilde{\rho}(p^-_\bullet) = p^+_{\rho_0(\bullet)}
\end{aligned}
\right.
\end{equation}
for $\bullet = 0,1,\infty$. We define $\delta(\widetilde{\rho})\in \{\pm 1\}^{\{0,1,\infty\}}$ as follows.
* If the case $\mathrm{(A)}$ occurs for $\bullet = 0$, $\delta(\widetilde{\rho})(\rho_0(0)) = 1$, else $\delta(\widetilde{\rho})(\rho_0(0)) = -1$.
* If the case $\mathrm{(A)}$ occurs for $\bullet = 1$, $\delta(\widetilde{\rho})(\rho_0(1)) = 1$, else $\delta(\widetilde{\rho})(\rho_0(1)) = -1$.
* If the case $\mathrm{(A)}$ occurs for $\bullet = \infty$, $\delta(\widetilde{\rho})(\rho_0(\infty)) = 1$, else $\delta(\widetilde{\rho})(\rho_0(\infty)) = -1$.
Then we have the following.
* For $\bullet = 0,1,\infty$, we have the following.
\begin{equation}\label{equalitydelta}
(\widetilde{\rho}^{-1})^\sharp(\psi_\bullet) = \psi_{\rho_0(\bullet)}^{\delta(\widetilde{\rho})(\rho_0(\bullet))},\quad(\widetilde{\rho}^{-1})^\sharp(\varphi_\bullet) = \varphi_{\rho_0(\bullet)}^{\delta(\widetilde{\rho})(\rho_0(\bullet))}
\end{equation}
* We define an $\widetilde{I}$-action on $\{\pm 1\}^{\{0,1,\infty\}}$ by
\begin{equation}\label{Iactionpm1}
\begin{tikzcd}[row sep = tiny]
\widetilde{I}\times \{\pm 1\}^{\{0,1,\infty\}}\arrow[r] & \{\pm 1\}^{\{0,1,\infty\}}; &[-30pt] ((\rho,\zeta), \epsilon) \arrow[r,mapsto] & \epsilon\circ \rho_0^{-1}
\end{tikzcd}
\end{equation}
Then the map $\delta: \widetilde{I}\rightarrow \{\pm 1\}^{\{0,1,\infty\}};\widetilde{\rho}\mapsto \delta(\widetilde{\rho})$ defines a 1-cocycle with respect to this $\widetilde{I}$-action.
To prove this proposition, we use the following lemma.
Let $\varphi_1,\varphi_2\in R(\P^1_{T^\circ})^\times$. Assume $\varphi_1\not \in \Frac(B^\circ)$. Suppose that $\div(\varphi_1) = \div(\varphi_2)$ and $\div(\varphi_1+1)=\div(\varphi_2+1)$. Then we have $\varphi_1=\varphi_2$.
Since $\P^1_{T^\circ}$ is normal, $\div(\varphi_1) = \div(\varphi_2)$ and $\div(\varphi_1+1)=\div(\varphi_2+1)$ imply that there exist $p,q\in \Gamma(\P^1_{T^\circ},\O_{\P^1_{T^\circ}}^\times) = (B^\circ)^\times$ such that $\varphi_1 = p\varphi_2$ and $1+\varphi_1 = q(1+\varphi_2)$. Thus $(1-q)+(1-qp^{-1})\varphi_1 =0$. Since $\varphi_1\not \in \Frac(B^\circ)$, we have $p=q=1$. i.e. $\varphi_1 = \varphi_2$.
(Proposition <ref>) Note that $\Cc$ and $Q^\circ_{(\bullet,\bullet)}$ are isomorphic to $\P^1_{T^\circ}$ (Corollary <ref>). By the explicit presentations for $\varphi_\bullet,\psi_\bullet$ in Definition <ref>, we see that $\varphi_\bullet^{\pm 1},\psi_\bullet^{\pm 1}\not \in \Frac(B^\circ)$. Hence we can use Lemma <ref>. By the definition of $\delta$, we have the following relations for $\bullet = 0,1,\infty$.
\begin{equation}\label{psicond1}
\begin{aligned}
& \div_{\Cc}((\widetilde{\rho}^{-1})^\sharp(\psi_\bullet)) = \div_{\Cc} (\psi_{\rho_0(\bullet)}^{\delta(\widetilde{\rho})(\rho_0(\bullet))}) \\
& \div_{Q^\circ_{(\rho_0(\bullet),\rho_0(\bullet))}}((\widetilde{\rho}^{-1})^\sharp(\varphi_\bullet)) = \div_{Q^\circ_{(\rho_0(\bullet),\rho_0(\bullet))}}(\varphi_{\rho_0(\bullet)}^{\delta(\widetilde{\rho})(\rho_0(\bullet))})
\end{aligned}
\end{equation}
Here we use the relations in Proposition <ref>. Next, we see the divisors associated with $1+\varphi_\bullet$ and $1+\psi_\bullet$. We will consider a closed subscheme $\Zc\subset \X^\circ$ defined by the local equation $v = 0$. Then we have $B^\circ$-rational points $q_c, q_0,q_1,q_\infty$ on $\X^\circ$ such that
\begin{equation}
q_c = \Zc\cap \Cc,\quad q_\bullet = \Zc\cap Q_{(\bullet,\bullet)}^\circ \quad(\bullet = 0,1,\infty).
\end{equation}
Using these $B^\circ$-rational points, we can describe the divisors of $1+\psi_\bullet^{\pm 1}$ and $1+\varphi_\bullet^{\pm 1}$ as follows.
\begin{equation}
\left\{
\begin{aligned}
&\div_{\Cc}(1+\psi_\bullet) = q_c - p_\bullet^+ \\
&\div_{\Cc}(1+\psi_\bullet^{-1}) = q_c-p_\bullet^- \\
\end{aligned}
\right.
\quad
\left\{
\begin{aligned}
&\div_{Q^\circ_{(\bullet,\bullet)}}(1+\varphi_\bullet) = q_\bullet- p_\bullet^- \\
&\div_{Q^\circ_{(\bullet,\bullet)}}(1+\varphi_\bullet^{-1}) = q_\bullet- p_\bullet^+ \\
\end{aligned}
\right.
\end{equation}
where $\bullet = 0,1,\infty$. This follows from the explicit presentations of Definition <ref>. By the explicit description of $\widetilde{G}$-action in Proposition <ref>, we see that the closed subscheme $\Zc\subset \X^\circ$ is stable under the $\widetilde{I}$-action. Then we have
\begin{equation}
\begin{aligned}
&\widetilde{\rho}(q_c) = \widetilde{\rho}(\Zc\cap \Cc) = \Zc\cap \Cc = q_c \\
&\widetilde{\rho}(q_\bullet) = \widetilde{\rho}(\Zc\cap Q^\circ_{(\bullet,\bullet)}) = \Zc\cap Q^\circ_{(\rho_0(\bullet),\rho_0(\bullet))} = q_{\rho_0(\bullet)}
\end{aligned}
\end{equation}
By the definition of $\delta(\rho)(\bullet)$, we have the following relations for $\bullet = 0,1,\infty$.
\begin{equation}\label{psicond2}
\begin{aligned}
&\div_{\Cc}(1+(\widetilde{\rho}^{-1})^\sharp(\psi_\bullet)) = q_c -\widetilde{\rho}(p_\bullet^+) = \div_{\Cc}(1+\psi_{\rho_0(\bullet)}^{\delta(\widetilde{\rho})(\rho_0(\bullet))}) \\
& \div_{Q^\circ_{(\rho_0(\bullet),\rho_0(\bullet))}}(1+(\widetilde{\rho}^{-1})^\sharp(\varphi_\bullet)) = q_{\rho_0(\bullet)} - \widetilde{\rho}(p^-_\bullet) = \div_{Q^\circ_{(\rho_0(\bullet),\rho_0(\bullet))}}(1+\varphi_{\rho_0(\bullet)}^{\delta(\widetilde{\rho})(\rho_0(\bullet))})
\end{aligned}
\end{equation}
By (<ref>) and (<ref>), we have (1). (2) follows from (1).
We have $\widetilde{\rho}_*(\Xi^\can) = \Xi^\can$. The $\widetilde{I}$-action on $\Xi^\can$ is given as follows:
\begin{equation}\label{Iactionformula}
\begin{tikzcd}
\widetilde{\rho}_*: &[-25pt]\Xi^\can \arrow[r] & \Xi^\can \\[-10pt]
& \xi(\epsilon) \arrow[r,mapsto]\arrow[u,"\rotatebox{90}{$\in$}",phantom] & \xi\left(\delta(\widetilde{\rho})\cdot (\epsilon\circ \rho_0^{-1})\right) \arrow[u,"\rotatebox{90}{$\in$}",phantom]
\end{tikzcd}
\end{equation}
where $\delta(\widetilde{\rho})\cdot (\epsilon\circ \rho_0^{-1})$ denotes the product of functions $\delta(\widetilde{\rho})\in \{\pm 1\}^{\{0,1,\infty\}}\subset \Z^{\{0,1,\infty\}}$ and $\epsilon\circ \rho_0^{-1}\in \Z^{\{0,1,\infty\}}$.
By Proposition <ref>, $\left(\Cc_\rho, \prod_{\bullet = 0,1,\infty}(\widetilde{\rho}^{-1})^\sharp(\psi_\bullet)^{\epsilon(\bullet)}\right) = \left(\Cc,\prod_{\bullet = 0,1,\infty}\psi_\bullet^{\delta(\widetilde{\rho})(\bullet)\cdot \epsilon(\rho_0^{-1}(\bullet))}\right)$
and $\left(Q^\circ_{\rho\cdot (\bullet,\bullet)},
(\widetilde{\rho}^{-1})^\sharp\left(\varphi_\bullet\right)^{\epsilon(\bullet)}\right) = \left(Q^\circ_{(\rho_0(\bullet),\rho_0(\bullet))},
\varphi_{\bullet}^{\delta(\widetilde{\rho})(\rho_0(\bullet))\cdot\epsilon(\bullet)}\right)$ for $\bullet = 0,1,\infty$. Therefore, we have $\widetilde{\rho}_*\xi(\epsilon) = \xi\left(\delta(\widetilde{\rho})\cdot (\epsilon\circ \rho_0^{-1})\right)$ by Proposition <ref>. Hence we have the result.
We calculate $\delta, \underline{\chi}^{(i)}$ for some elements in $\widetilde{I}$. The result will be used in Section 9. For the calculation, we use the local description of $\widetilde{G}$-action on $\X^\circ$ in Proposition <ref>. Since $I\rightarrow \underline{I}$ is an isomorphism (Defintion <ref>), to specify elements in $I$, it is enough to give an automorphism on $B^\circ$ which belongs to $\underline{I}$.
* Let $\widetilde{\rho}^a = (\rho^a,1)\in \widetilde{I}$ be the element satisfying that
\begin{equation}
(\underline{\rho}^a)^\sharp: B^\circ\rightarrow B^\circ; \quad \sqrt{a},\sqrt{1-a},\sqrt{b},\sqrt{1-b} \mapsto \sqrt{a},-\sqrt{1-a},\sqrt{b},\sqrt{1-b}.
\end{equation}
Then we have $\rho_0^a= \id \in I = \mathfrak{S}(\{0,1,\infty\})$, $\chi^{(1)}(\underline{\rho}^a) = -1$ and $\chi^{(2)}(\underline{\rho}^a) = 1$. Furthermore, $\delta(\widetilde{\rho}^a)$ can be computed as
\begin{equation}
\delta(\widetilde{\rho}^a)(0) = -1,\quad \delta(\widetilde{\rho}^a)(1) = 1,\quad \delta(\widetilde{\rho}^a)(\infty) = -1.
\end{equation}
* Let $\widetilde{\rho}^b = (\rho^b,1)\in \widetilde{I}$ be the element satisfying that
\begin{equation}
(\underline{\rho}^a)^\sharp: B^\circ\rightarrow B^\circ; \quad \sqrt{a},\sqrt{1-a},\sqrt{b},\sqrt{1-b} \mapsto \sqrt{1-a},\sqrt{a},\sqrt{1-b},\sqrt{b}.
\end{equation}
Then we have $\rho_0^b = (1\:\infty) \in I_0 = \mathfrak{S}(\{0,1,\infty\})$, $\chi^{(1)}(\underline{\rho}^b) = 1$ and $\chi^{(2)}(\underline{\rho}^b)=1$. Furthermore, $\delta(\widetilde{\rho}^a)$ can be computed as
\begin{equation}
\delta(\widetilde{\rho}^a)(0) = -1,\quad \delta(\widetilde{\rho}^a)(1) = -1,\quad \delta(\widetilde{\rho}^a)(\infty) = -1.
\end{equation}
§.§ A fiber-preserving subgroup $\widetilde{G}_\fib$ of $\widetilde{G}$
In this section, we define another subgroup $\widetilde{G}_\fib$ of $\widetilde{G}$.
We define a normal subgroup $\widetilde{G}_\fib\subset \widetilde{G}$ as
\begin{equation}
\widetilde{G}_\fib = \Ker(\widetilde{G}\rightarrow \underline{G}).
\end{equation}
In other words, $\widetilde{G}_\fib$ consists of elements in $\widetilde{G}\subset \Aut_k(\X^\circ)$ which are automorphisms over $T^\circ$. Then we have $\widetilde{G}_\fib \simeq (\Z/2\Z)^5$.
First, we show $\Ker(H\rightarrow \underline{H})\simeq \Ker(H_0\rightarrow \underline{H}_0)\simeq (\Z/2\Z)^2$. We have the first isomorphism by the fact that a fiber product preserves kernels and the second isomorphism follows from Table <ref>. Hence we have $\Ker(G\rightarrow \underline{G}) \simeq (\Z/2\Z)^4$. Since $\underline{\rho}^{(1)}_0 = \underline{\rho}^{(2)}_0 = \id_{\underline{H}_0}$ for $\rho = (\rho^{(1)},\rho^{(2)})\in \Ker(G\rightarrow \underline{G})$, we have a splitting of $\Ker(\widetilde{G}\rightarrow \underline{G})\rightarrow \Ker(G\rightarrow \underline{G})$ defined by $\rho\mapsto (\rho,1)$. Hence $\widetilde{G}_\fib$ is isomorphic to the direct product of $\Ker(G\rightarrow \underline{G}) \simeq (\Z/2\Z)^4$ and $\Z/2\Z$.
$\widetilde{G}_\fib \cap \widetilde{I} = \{(\id_{G}, \pm 1)\}$.
Let $(\rho,\zeta)\in \widetilde{G}_\fib \cap \widetilde{I}$. By Definition <ref>, we have $\underline{\rho} = \id_{\underline{G}}$. Since $I\rightarrow \underline{I}$ is an isomorphism, we have $\rho = \id_G$. Hence $\zeta = \pm 1$. The other direction of the inclusion is clear.
Since $\widetilde{I}$ stabilize $\Xi^\can$ (Proposition <ref>) and $\widetilde{G}_\fib$ stabilize the image of $\Xi^\can(\subset \Xi)$ under the transcendental regulator (Proposition <ref>), the subgroup $\widetilde{G}_\fib\widetilde{I}\subset \widetilde{G}$ stabilize the image of $\Xi^\can(\subset \Xi)$ under the transcendental regulator map. Hence $\widetilde{\rho}_*\Xi^\can$ and $\widetilde{\rho}'_*\Xi^\can$ have the same image under the transcendental regulator map if $\widetilde{\rho}, \widetilde{\rho}'\in \widetilde{G}$ are in the same left coset by $\widetilde{G}_\fib\widetilde{I}$. The following proposition is useful to determine whether $\widetilde{\rho},\widetilde{\rho}'\in \widetilde{G}$ are in the same left coset or not.
The group homomorphism $\widetilde{G}\rightarrow \underline{G}_0$ induces the following bijection of sets.
\begin{equation}\label{bijectioncoset}
\begin{tikzcd}
\widetilde{G}/\widetilde{G}_\fib \widetilde{I} \arrow[r,"\sim"] & \underline{G}_0/\underline{I}_0
\end{tikzcd}
\end{equation}
Especially, we have $|\widetilde{G}/\widetilde{G}_\fib \widetilde{I}| = |\underline{G}_0/\underline{I}_0|=6$.
By the group homomorphism $\widetilde{G}\rightarrow \underline{G}_0$, $\widetilde{G}_\fib = \Ker(\widetilde{G}\rightarrow \underline{G})$ maps to $\{\id_{\underline{G}_0}\}$ and $\widetilde{I}$ maps to $\underline{I}_0$. Hence we see that the surjective map $\widetilde{G}\rightarrow \underline{G}_0$ induces a surjection (<ref>). We will see this is bijective. It is enough to compare the cardinality of $\widetilde{G}/\widetilde{G}_\fib I $ with that of $\underline{G}_0/\underline{I}_0$. By Definition <ref>, $|\underline{G}_0/\underline{I}_0| =|H_0|= 6$. On the other hand, by Definition <ref> and Remark <ref>, $|\widetilde{I}| = 192$. Hence by Proposition <ref> and Corollary <ref>, we have
\begin{equation}
|\widetilde{G}_\fib \widetilde{I} | =\frac{|\widetilde{G}_\fib|\cdot |\widetilde{I}|}{|\widetilde{G}_\fib\cap \widetilde{I}|} = 3072 = 2^{10}\cdot 3
\end{equation}
By Proposition <ref>, we have $|\widetilde{G}| = 18432 = 2^{11}\cdot 3^2$. Hence $|\widetilde{G}/\widetilde{G}_\fib I| = 6$ and we confirm that (<ref>) is bijective.
§ A DIFFERENTIAL FORM ON $\X$ AND A PICARD-FUCHS DIFFERENTIAL OPERATORS
Since $\X'\rightarrow T'$ is a family of $K3$ surfaces, we have the unique non-zero relative 2-form up to multiplication of elements in $(B')^\times$. We specify such a relative 2-form $\omega\in \Gamma(\X',\Omega^2_{\X'/T'})$ and observe the group action on $\omega$. Then we compute periods of each fiber $\X_t$ and find a Picard-Fuchs differential operator with respect to $\{\omega_t\}_{t\in T'}$. In other words, we find a differential operator on $(T')^\an$ which annihilate period functions associated with the relative 2-form $\omega\in \Gamma(\X',\Omega^2_{\X'/T'})$.
§.§ The definition of the relative 2-form $\omega$ and $\widetilde{G}$-action on $\omega$
We define a relative 2-form $\omega$ on $\X$ using a relative 2-form on $\E\times_k\E$. By Definition <ref>, we have the following morphisms over $T$.
\begin{equation}\label{XEEtildeEE}
\begin{tikzcd}
\X & (\E\times_k\E)\wph \arrow[l] \arrow[r] & \E\times_k \E
\end{tikzcd}
\end{equation}
We define $\theta \in \Gamma(\E,\Omega_{\E/S}^1)$ by $\theta = \frac{dz}{u}$ where we use the local coordinates in Proposition <ref>. Then we have the following 2-form on $\E\times_k\E$.
\begin{equation}
pr_1^*(\theta)\wedge pr_2^*(\theta) = \frac{dx\wedge dy}{u_1u_2} \in \Gamma(\E\times_k\E, \Omega_{\E\times_k \E/T}^2)
\end{equation}
where $pr_i: \E\times_k \E \rightarrow \E$ is the $i$-th projection and we use the local description of $\E\times_k\E$ in (<ref>). Furthermore, we define the 2-form $\widetilde{\omega}\in \Gamma((\E\times_k\E)\wph, \Omega_{(\E\times_k \E)\wph/T}^2)$ by the pull-back of $pr_1^*(\theta)\wedge pr_2^*(\theta)$ by $(\E\times_k \E)\wph\rightarrow \E\times_k \E$.Finally, since $\widetilde{\omega}$ is stable under the $\Aut_{\X}((\E\times_k\E)\wph)$-action, we have a unique element $\omega \in \Gamma(\X,\Omega_{\X/T}^2)$ such that the pull-back of $\omega$ to $(\E\times_k \E)^\wph$ coincides with $\widetilde{\omega}$. The 2-form $\omega$ is represented locally on $V_{0,0}$ as
\begin{equation}
\omega = \frac{dx \wedge dy}{vf(x)}.
\end{equation}
We use the same symbol $\omega$ for its base change by $\X'\rightarrow \X$. For a $k$-rational point $t'\in T'(k)$, We define $\omega_t \in \Gamma\left(\X_t, \Omega^2_{\X_t/k}\right)$ as the pull-back of $\omega$ by $i_t:\X_t\hookrightarrow \X'$.
Let $\widetilde{\rho} = (\rho,\zeta)\in \widetilde{G}$ and $t'\in T'(k)$ be a $k$-rational point. Recall the opposite 1-cocycle $\widetilde{\chi}(\widetilde{\rho})$ in Definition <ref>.
* Let $\omega$ be the relative 2-form defined in Definition <ref>. Then we have
\begin{equation}\label{actionformula}
\widetilde{\rho}^*\omega = \widetilde{\chi}(\widetilde{\rho})\cdot\omega.
\end{equation}
* Let $\widetilde{\chi}(\widetilde{\rho})(t)\in k$ be the image of $\widetilde{\chi}(\widetilde{\rho}) \in B'$ under $t^\sharp: B'\rightarrow k$. Then we have
\begin{equation}
\widetilde{\rho}_t^*\omega_{\underline{\rho}(t)} = \widetilde{\chi}(\widetilde{\rho})(t)\cdot \omega_t
\end{equation}
Since $\X'\rightarrow T'$ is smooth, $\Omega^2_{\X'/T'}$ is locally free. Hence it is enough to show that the formula (<ref>) on some non-empty open subset of $\X'$. We can show that
\begin{equation}
\rho^*\left(\frac{dx\wedge dy}{vf(x)}\right) = \frac{d\rho^\sharp(x)\wedge d\rho^\sharp(y)}{\rho^\sharp(u)} = \frac{\partial}{\partial x}\left(\rho^\sharp(x)\right)\frac{\partial}{\partial y}\left(\rho^\sharp(y)\right)
\frac{dx\wedge dy}{\rho^\sharp(u)} = \widetilde{\chi}(\widetilde{\rho})\cdot \frac{dx\wedge dy}{vf(x)}.
\end{equation}
Here we use Proposition <ref> and the relation $u=vf(x)$. Hence we have (1). (2) is the restriction of (1) at fibers.
§.§ Calculation of periods of $\X_t$
Hereafter we assume $k=\C$. In this subsection, we calculate periods of $\X_t$ at $t\in T'(\C)$ with respect to the 2-form $\omega_t$ in Definition <ref>.
Let $X$ be a smooth projective surface over $\C$ and $\eta \in \Gamma(X,\Omega^2_{X/\C})$ be an algebraic 2-form on $X$. We regard $\eta$ as a holomorphic 2-form on $X^\an$. We define a subgroup $\Pc(X,\eta)$ of $\C$ by
\begin{equation}
\Pc(X,\eta) = \left\{\displaystyle \int_{\Gamma}\eta \in \C : \Gamma \in Z_2(X^\an)\right\}.
\end{equation}
where $Z_2(X^\an)$ denotes the group of topological closed 2-cycles on $X^\an$. $\Pc(X,\eta)$ is a subgroup of periods of $X$ with respect to $\eta$.
Since $\X_t$ is a Kummer surface associated with a direct product of elliptic curves, $\Pc(\X_t,\omega_t)$ relates with periods of elliptic curves. We first compute periods of the member of the Legendre family of elliptic curves with respect to the relative 1-form $\theta\in \Gamma(\E,\Omega^1_{\E/S})$.
Let $s\in S(\C)$ be a $\C$-rational point on $S$ and $\E_s$ be the fiber of $\E\rightarrow S$ over $s$. We have the double covering $\E_s\rightarrow \P^1_\C$ by Proposition <ref>. Let $\gamma,\delta$ be $C^\infty$ paths on $(\P^1_\C)^\an$ such that the following conditions holds.
* $\gamma$ is a path from $0$ to $1$ and $\delta$ is a path from $1$ to $\infty$.
* $\gamma,\delta$ do not pass through $0,1,1/c,\infty$ unless edge points where $c\in \C$ is the image of $c\in A$ by $s^\sharp: A\rightarrow \C$.
* Let $\gamma_+,\gamma_-$ (resp. $\delta_+,\delta_-$) be lifts of $\gamma$ (resp. $\delta$) by $\E_s^\an \rightarrow ( \P^1_\C)^\an$. Then $[\gamma_+]-[\gamma_-]$ and $[\delta_+]-[\delta_-]$ are generators of $H_1(\E_s,\Z)$.
If $c \not\in \R_{\ge 0}$, the closed intervals in real axis $\gamma = [0,1]$ and $\delta = [1,\infty]$ satisfy the conditions for $\gamma$ and $\delta$.If $\gamma,\delta$ satisfy the conditions (1) to (3) at $s\in S(\C) = S^\an$, $\gamma,\delta$ satisfy the conditions for any $s'$ which is sufficiently close to $s\in S^\an$ in the classical topology. Hence we can define local holomorphic functions $P_1,P_2$ on $S^\an$ by the following integral representation. Note that $c$ is the coordinate of $S^\an$.
\begin{equation}
\begin{aligned}
P_1(c) &= \int_{\gamma_+}\theta_s = \int_\gamma \frac{dx}{\sqrt{x(1-x)(1-cx)}} \\
P_2(c) &= \int_{\delta_+}\theta_s = \int_\delta \frac{dx}{\sqrt{x(1-x)(1-cx)}}.
\end{aligned}
\end{equation}
where $\theta_s\in \Gamma(\E_s,\Omega^1_{\E_s/\C})$ is the pull-back of $\theta$ in Definition <ref> by $\E_s\hookrightarrow \E$. We define a differential operator $L: \O_{S^\an}\rightarrow \O_{S^\an}$ of order 2 by
\begin{equation}
L = c(1-c)\frac{d^2}{dc^2} +(1-2c)\frac{d}{dc} -\frac{1}{4}.
\end{equation}
Then we can check that $L(P_1) = L(P_2) = 0$ by the integral representation.
Let $t\in T(\C)$ and $pr_1(t), pr_2(t)\in S(\C)$ be its images by $pr_1,pr_2: T\rightarrow S$. By Proposition <ref>, $(\E\times_\C \E)_t$ is isomorphic to $\E_{pr_1(t)}\times_\C \E_{pr_2(t)}$. Using $P_1,P_2$, we can describe $\Pc(\E_{pr_1(t)} \times_\C \E_{pr_2(t)}, pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)}))$ as follows.
Let $t\in T(\C)$. Then $\Pc(\E_{pr_1(t)} \times_\C \E_{pr_2(t)}, pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)}))$ is generated by $4P_1(a)P_1(b)$, $4P_1(a)P_2(b)$, $4P_2(a)P_1(b)$ and $4P_2(a)P_2(b) \in \C$ where $a,b\in \C$ are images of $a,b\in B$ by $t^\sharp: B\rightarrow \C$.
By the condition (3) in Definition <ref>, the periods of the elliptic curve $\E_{pr_1(t)}$ with respect to $\theta_{pr_1(t)}$ is generated by $2P_1(a)$ and $2P_2(a)$. Similarly, the periods of the elliptic curve $\E_{pr_2(t)}$ with respect to $\theta_{pr_2(t)}$ is generated by $2P_1(b)$ and $2P_2(b)$. Then by the Künneth formula, we have the result.
Next, we see the relation between $\Pc(\E_{pr_1(t)} \times_\C \E_{pr_2(t)}, pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)}))$ and $\Pc(\X_t,\omega_t)$. By restricting the morphism (<ref>) to fibers at $t\in T(\C)$, we have the following diagram.
\begin{equation}\label{fiberdiagram}
\begin{tikzcd}
\X_t & (\E\times_\C \E)\wph_t \arrow[r]\arrow[l] & (\E\times_\C \E)_t \arrow[r,"\sim"] & \E_{pr_1(t)} \times_\C \E_{pr_2(t)}
\end{tikzcd}
\end{equation}
Let $p:(\E\times_\C \E)\wph_t \rightarrow \E_{pr_1(t)} \times_\C \E_{pr_2(t)}$ be the composition of the right arrows in (<ref>) and $\pi: (\E\times_\C \E)\wph_t\rightarrow \X_t$ be the left arrow in (<ref>). We have the following morphism $\phi$ of $\Z$-Hodge structures.
\begin{equation}\label{phiHodge}
\begin{tikzcd}
\phi:&[-28pt] H^2(\E_{pr_1(t)}^\an \times \E_{pr_2(t)}^\an) \arrow[r,"p^*"] & H^2(\left((\E\times_\C \E)\wph_t\right)^\an) \arrow[r,"\pi_!"] &H^2(\X_t^\an)
\end{tikzcd}
\end{equation}
where $p^*$ is the pull-back by $p$ and $\pi_!$ is the Gysin morphism ([16] p.178) induced by $\pi$. In other words, $\pi_!$ is the map
\begin{equation}\label{gysinmap}
\begin{tikzcd}
H^2(((\E\times_\C \E)\wph_t)^\an) \arrow[r,"\sim"] & H_2(((\E\times_\C \E)\wph_t)^\an) \arrow[r,"\pi_*"] & H_2(\X_t^\an) & H^2(\X_t^\an) \arrow[l,"\sim"']
\end{tikzcd}
\end{equation}
where $\pi_*$ is the push-forward map induced on the homology group and the first and the last isomorphisms are Poincaré duality.
The following relation holds in $H^2(\X_t^\an,\C)$.
\begin{equation}
\phi([pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})]) = 2[\omega_t]
\end{equation}
Under the isomorphism $(\E\times_\C \E)_t \simeq \E_{pr_1(t)} \times_\C \E_{pr_2(t)}$, the 2-form $pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})$ coincides with the pull-back of $pr_1^*(\theta)\wedge pr_2^*(\theta)$ in Definition <ref> at $t$. Let $\widetilde{\omega}_t \in \Gamma((\E\times_\C \E)\wph_t, \Omega_{(\E\times_\C \E)\wph_t/\C}^2)$ be the pull-back of $\widetilde{\omega}$ in Definition <ref> at $t$. Then we have
\begin{equation}
\begin{aligned}
& p^*\left(pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})\right) = \widetilde{\omega}_t \\
& \pi^*\omega_t = \widetilde{\omega}_t.
\end{aligned}
\end{equation}
Since $\pi: (\E\times_\C \E)\wph_t\rightarrow \X_t$ is the quotient by the involution (Proposition <ref>), $\pi$ is a generically $2:1$ map. Hence the mapping degree of $\pi: ((\E\times_\C \E)\wph_t)^\an\rightarrow \X_t^\an$ is $2$. By the definition of Gysin map, $\pi_!\circ \pi^*: H^2(\X_t^\an)\rightarrow H^2(\X_t^\an)$ equals to multiplication by $2$ (cf. [16], Remark 7.29). Then we have
\begin{equation}
\phi ([pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})]) = \pi_!p^*[pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})] = \pi_!\pi^*[\omega_t] = 2[\omega_t].
\end{equation}
For $i,j\in \{1,2\}$, we define a local holomorphic function $P_{ij}$ on $T^\an$ by
\begin{equation}
P_{ij}(t) = 2P_i(a)P_j(b)\quad(t\in T^\an)
\end{equation}
where $a,b\in \C$ are images of $a,b\in B$ by $t^\sharp: B\rightarrow \C$ and $P_1,P_2$ are local holomorphic functions defined in Definition <ref>. Note that $a,b$ are coordinates on $T^\an$. By pulling-back $P_{ij}$ by $(T')^\an\rightarrow T^\an$, we can regard $P_{ij}$ as a local holomorphic function on $(T')^\an$ for $i,j\in \{1,2\}$.Then for each $t'\in T'(\C)$, the subgroup $\Pc(\X_{t'},\omega_{t'})\subset \C$ is generated by $P_{11}(t')$, $P_{12}(t')$, $P_{21}(t')$ and $P_{22}(t')\in \C$.
For $t'\in T'(\C)$, let $t\in T(\C)$ be the image of $t'$ by $T'\rightarrow T$. Then we have $\Pc(\X_t,\omega_t) = \Pc(\X_{t'},\omega_{t'})$. Hence by Proposition <ref>, it is enough to show
\begin{equation}\label{periodrelation}
\Pc(\X_t,\omega_t) = \frac{1}{2}\Pc(\E_{pr_1(t)} \times_\C \E_{pr_2(t)}, pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})).
\end{equation}
Since $\X_t^\an$ is a $K3$ surface and $\E_{pr_1(t)}^\an \times \E_{pr_2(t)}^\an$ is an abelian surface, their singular cohomology groups with coefficients in $\Z$ are free of finite rank ([5], Chapter VIII, Proposition 3.2). Hence $H_2(\X_t^\an)$ and $H_2(\E_{pr_1(t)}^\an \times \E_{pr_2(t)}^\an)$ are duals of $H^2(\X_t^\an)$ and $H^2(\E_{pr_1(t)}^\an \times \E_{pr_2(t)}^\an)$ and the following morphism is the dual of $\phi$.
\begin{equation}
\begin{tikzcd}
\phi^\vee: &[-28pt] H_2(\X_t^\an) \arrow[r,"\pi^!"]& H_2(((\E\times_\C \E)\wph_t)^\an) \arrow[r,"p_*"] & H_2(\E_{pr_1(t)}^\an \times_\C \E_{pr_2(t)}^\an)
\end{tikzcd}
\end{equation}
where $\pi^!$ is the following morphism.
\begin{equation}
\begin{tikzcd}
H_2(\X_t^\an) & H^2(\X_t^\an) \arrow[l,"\sim"'] \arrow[r,"\pi^*"] & H^2(((\E\times_\C \E)\wph_t)^\an)\arrow[r,"\sim"] & H_2(((\E\times_\C \E)\wph_t)^\an)
\end{tikzcd}
\end{equation}
For any $\Gamma\in Z_2(\X_t^\an)$, we have
\begin{equation}
\begin{aligned}
&\int_\Gamma \omega_t = \langle[\omega_t], [\Gamma] \rangle = \frac{1}{2}\langle\phi([pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})]), [\Gamma] \rangle \\
&= \frac{1}{2}\langle[pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})], \phi^\vee([\Gamma]) \rangle = \frac{1}{2}\int_{\Gamma'}pr_1^*(\theta_{pr_1(t)})\wedge pr_2^*(\theta_{pr_2(t)})
\end{aligned}
\end{equation}
where $\langle \phantom{h},\phantom{h}\rangle$ is the canonical pairing of cohomology and homology and $\Gamma'\in Z_2(\E_{pr_1(t)}^\an \times \E_{pr_2(t)}^\an)$ is a representative of $\phi^\vee([\Gamma])\in H_2(\E_{pr_1(t)}^\an\times \E_{pr_2(t)}^\an)$. This equation proves the inclusion $(\subset)$ in (<ref>). To prove the other direction of the inclusion, it is enough to show that any element in $H_2(\E_{pr_1(t)}^\an\times \E_{pr_2(t)}^\an,\Z)$ can be written as $\phi^\vee([\Gamma])$ for some $[\Gamma]\in H_2(\X^\an_t)$. By [5], Chapter VIII, Proposition 5.1 and Corollary 5.6, $\phi: H^2(\E_{pr_1(t)}^\an\times \E_{pr_2(t)}^\an,\Z)\rightarrow H^2(\X_t^\an,\Z)$ is injective and its cokernel has no torsion. Hence its dual $\phi^\vee$ is surjective and we have the result.
Finally we can find a Picard-Fuchs differential operator $\Dc$, which annihilate the period functions $P_{ij}$.
We define differential operators $\Dc_1, \Dc_2 : \O_{(T')^\an}\rightarrow \O_{(T')^\an}$ by
\begin{equation}
\begin{aligned}
&\Dc_1 = a(1-a) \frac{\del^2}{\del a^2} +(1-2a)\frac{\del}{\del a} -\frac{1}{4} \\
&\Dc_2 = b(1-b) \frac{\del^2}{\del b^2} +(1-2b)\frac{\del}{\del b} -\frac{1}{4}
\end{aligned}
\end{equation}
Using these operators, we define a Picard-Fuchs differential operator $\Dc$ by
\begin{equation}
\Dc = \begin{pmatrix}
\Dc_1 \\
\Dc_2
\end{pmatrix}: \O_{(T')^\an} \rightarrow \O_{(T')^\an}^{\oplus 2}.
\end{equation}
These are $\C$-linear morphisms of sheaves. By Definition <ref> and Definition <ref>, the local holomorphic functions $P_{ij}$ are annihilated by the differential operator $\Dc$.
§ BASIC CALCULATION OF THE REGULATOR MAP
In this section, we calculate the image of the higher Chow cycle $\xi_{1,t}-\xi_{0,t} \in \CH^2(\X_t,1)$ in Definition <ref> under the transcendental regulator map by using Levine's formula. For this purpose, we construct topological 2-chains $K_+$ and $K_-$ on $\X_t^\an$ explicitly (Proposition <ref>) and express the value of $\xi_{1,t}-\xi_{0,t}$ under the transcendental regulator map using the local holomorphic function $\Lc$ (Definition <ref>). Hereafter we use the following notations.
* For a smooth variety $X$ over $\C$, its analytification is denoted by $X^\an$. As a set, we have $X^\an = X(\C)$.
* For a complex manifold $X^\an$, $S_n(X^\an)$ denote the free abelian group generated by $C^\infty$-singular chains on $X^\an$ of dimension $n$. The boundary operator is denoted by $\partial: S_\bullet(X^\an) \rightarrow S_{\bullet-1}(X^\an)$. We set $B_\bullet(X^\an) = \mathrm{Im}(S_{\bullet+1}(X^\an) \xrightarrow{\partial} S_\bullet(X^\an))$ and $Z_\bullet(X^\an) = \mathrm{Ker}(S_{\bullet}(X^\an) \xrightarrow{\partial} S_{\bullet-1}(X^\an))$.
* For a smooth variety $X$ over $\C$, we identify algebraic cycles on $X$ of dimension 0 with elements in $S_0(X^\an)$. Furthermore, we regard a $C^\infty$-path $\gamma: [0,1] \rightarrow X^\an$ as an element of $S_1(X^\an)$ such that $\del \gamma = \gamma(1)-\gamma(0)$.
§.§ Levine's formula for the regulator map
In this section, let $X$ be a smooth projective surface over $\C$ such that $H_1(X^\an, \Z) = 0$. We have the following canonical isomorphism for the Deligne cohomology of $X^\an$.
\begin{equation}\label{Delignefunctional}
H^3_\D(X^\an, \Z(2)) \simeq \frac{H^{2}(X^\an,\C)}{H^{2}(X^\an,\Z(2)) + F^2H^{2}(X^\an,\C)} \simeq \frac{(F^1H^2(X^\an,\C))^\vee}{H_2(X^\an,\Z)}.
\end{equation}
where we denote the dual of a $\C$-vector space $V$ by $V^\vee$. The last isomorphism is induced by the Poincaré duality. We regard $H_2(X^\an,\Z)$ as a subgroup of $(F^1H^2(X^\an,\C))^\vee$ by the integration. By this identification, we regard the Deligne cohomology as a quotient of the space of functionals of $F^1H^2(X^\an,\C)$.We will recall the formula for the regulator map in [11]. Let $\xi$ be an element of $\CH^2(X,1)$. By the Proposition <ref>, $\xi$ is represented by
\begin{equation}
\sum_{j}(C_j, f_j) \in \Ker\left(\bigoplus_{Z\in X^{(1)}}R(Z)^\times\xrightarrow{\div} \bigoplus_{p\in X^{(2)}}\Z\cdot p\right)
\end{equation}
where $C_j$ is a closed curve on $X$ and $f_j$ is a non-zero rational function on $C_j$. Let $D_j$ be the normalization of $C_j$. Hence $D_j$ is a smooth projective curve. $\mu_j: D_j\lra X$ denotes the composition of $D_j\lra C_j$ and $C_j\lra X$.First, we will define $\gamma_j \in S_1(D_j^\an)$. If $f_j\in \C^\times$, we set $\gamma_j = 0$. If $f_j$ is not a constant function, we regard $f_j$ as a finite morphism from $D_j$ to $\P^1_\C$ (because $D_j$ is smooth). Let $[\infty,0]\in S_1((\P^1_\C)^\an)$ be a path from $\infty$ to $0$ along the positive real axis. Since $D_j^\an\lra (\P^1_\C)^\an$ is a finite covering, we can define $\gamma_j$ as the pullback of $[\infty,0]$ by $D_j^\an\lra (\P^1_\C)^\an$. Then we have
\begin{equation}
\partial \gamma_j = \div_{D_j}(f_j)\quad \in S_0(D_j^\an).
\end{equation}
Next, we will define a 2-chain $\Gamma \in S_2(X^\an)$. Let $\gamma\in S_1(X^\an)$ be $\sum_j (\mu_j)_*\gamma_j$ where $(\mu_j)_*\gamma_j$ denotes the push-forward of $\gamma_j$ by $\mu_j: D_j^\an \rightarrow X^\an$. Since $\sum_j(C_j,f_j) \in \mathrm{Ker}(\div)$, we have $\gamma \in Z_1(X^\an)$. By the assumption $H_1(X^\an, \Z) = 0$, we can find a $\Gamma \in S_2(X^\an)$ such that $\partial\Gamma = \gamma$. We name these $\gamma$ and $\Gamma$ as follows.
In this paper, $\gamma \in S_1(X^\an)$ is called the 1-cycle associated with $\xi$ and $\Gamma \in S_2(X^\an)$ is called a 2-chain associated with $\xi$. Note that $\Gamma$ is determined only up to elements in $Z_2(X^\an)$.By [11], p.458–459, the following map is well-defined.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
\CH^2(X,1) \arrow[r] &[-18pt]\dfrac{F^1H^2(X^\an,\C)^\vee}{H_2(X^\an,\Z)} &[-40pt]\\[-10pt]
\left[\sum_j(C_j,f_j)\right] \arrow[r,mapsto] &\left([\omega] \mapsto \displaystyle\int_\Gamma\omega + \sum_j\dfrac{1}{2\pi\sqrt{-1}}\displaystyle\int_{D_j-\gamma_j}\log (f_j)\mu_j^*\omega\right)& \mod H_2(X^\an,\Z)
\end{tikzcd}
\end{equation}
Here $\log(f_j)$ is the pull-back of the principal branch of the holomorphic function $\log z$ on $(\P^1_\C)^\an-[\infty,0]$ by $f_j$. By the isomorphism $(\ref{Delignefunctional})$, this map is regarded as a map to $H^3_\D(X^\an,\Z(2))$. This map is called the regulator map[This definition of the regulator map is different from the map defined in [11] by the multiplication of $2\pi\sqrt{-1}$. The difference comes from the definition of the Poincaré duality.].
In this paper, we do not treat the whole Deligne cohomology group. We consider a certain quotient of the Deligne cohomology.
The transcendental regulator map is the composite of the following maps.
\begin{equation}
\begin{tikzcd}
r: \CH^2(X,1) \arrow[r] &\dfrac{F^1H^2(X^\an,\C)^\vee}{H_2(X^\an,\Z)} \arrow[r] &\dfrac{H^{2,0}(X^\an)^\vee}{H_2(X^\an,\Z)}
\end{tikzcd}
\end{equation}
where the first map is the regulator map in Definition <ref> and the second map is the projection induced by $H^{2,0}(X^\an)\hookrightarrow F^1H^2(X^\an,\C)$. We denote this map by $r$. The transcendental regulator map has the following properties.
* Let $\xi\in \CH^2(X,1)$ and $\Gamma$ be a 2-chain associated with $\xi$. For an algebraic 2-form $\eta$ on $X$, we have
\begin{equation}\label{transreg}
r(\xi)([\eta]) = \int_{\Gamma}\eta \mod \Pc(X,\eta).
\end{equation}
where $\Pc(X,\eta)\subset \C$ is the subgroup defined in Definition <ref>.
* For a decomposable cycle $\xi \in \CH^2(X,1)_{\dec}$, we have $r(\xi) = 0$. Especially, the transcendental regulator map factors through $\CH^2(X,1)_\ind$.
Since we regard $H_2(X^\an,\Z)$ as a subgroup of $F^1H^2(X^\an,\C)^\vee$ by integration, the evaluation by $[\eta]\in H^{2,0}(X^\an)$ induces the following map.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
H^{2,0}(X^\an)^\vee/H_2(X^\an,\Z) \arrow[r] & \C/\Pc(X,\eta);&[-30pt] \varphi \arrow[r,mapsto] & \varphi([\eta])
\end{tikzcd}
\end{equation}
Hence $r(\xi)([\eta])$ should be an element of $\C/\Pc(X,\eta)$. Since $\eta$ is a holomorphic 2-form and $D_j^\an$ is a complex manifold of dimension 1, we have $\mu_j^*\eta = 0$. Thus $\int_{D_j-\gamma_j}\log (f_j)\mu_j^*\eta = 0$ for all $j$. Hence (<ref>) follows from the formula in Definition <ref>. To prove (2), we use the fact that a decomposable cycle is represented by a sum of $(C,a)$ where $a\in \Gamma(X,\O_X^\times) = \C^\times$ by Proposition <ref>. In this case, $\gamma = 0$ and we can take $\Gamma = 0$. Thus (2) follows from (1).
When we compute the value of transcendental regulator map, it is sometimes convenient to replace a 1-cycle/2-chain associated with $\xi$ (Definition <ref>) by another 1-cycle/2-chain. Thus we define as follows.
Let $\xi$ be an element of $\CH^2(X,1)$ and $\gamma$ be the 1-cycle associated with $\xi$. In this paper, $\gamma' \in Z_1(X^\an)$ is called a 1-cycle associated with $\xi$ in a weak sense if there exists a family of smooth curves $\{D_\lambda\}_\lambda$ on $X$ such that $\gamma - \gamma' \in \sum_\lambda B_1(D_\lambda^\an)$. Here we regard $B_1(D_\lambda^\an)$ as a subgroup of $Z_1(X^\an)$ by the natural inclusions.Let $\Gamma \in S_2(X^\an)$ be a 2-chain associated with $\xi$. A 2-chain $\Gamma' \in S_2(X^\an)$ is called a 2-chain associated with $\xi$ in a weak sense if there exists a family of smooth curves $\{D_\lambda\}_\lambda$ on $X$ such that $\Gamma - \Gamma' \in Z_2(X^\an) + \sum_\lambda S_2(D_\lambda^\an)$.
The following proposition justifies this definition.
Let $\xi \in \CH^2(X,1)$.
* If $\gamma'$ is a 1-cycle associated with $\xi$ in a weak sense and $\Gamma' \in S_2(X^\an)$ satisfies $\partial \Gamma' = \gamma'$, then $\Gamma'$ is a 2-chain associated with $\xi$ in a weak sense.
* If $\Gamma'$ is a 2-chain associated with $\xi$ in a weak sense, we have
\begin{equation}
r(\xi)([\eta]) = \int_{\Gamma'} \eta \mod \Pc(X,\eta).
\end{equation}
(1) follows from the definition. (2) follows from the fact that the restriction of a holomorphic 2-form $\eta$ to each curve $D^\an_\lambda$ is $0$ since $D^\an_\lambda$ are 1-dimensional complex manifolds.
§.§ Construction of a 2-chain associated with $\xi_{1,t}-\xi_{0,t}$ in a weak sense
In this section, we fix a $\C$-rational point $t\in T^\circ(\C)$. By restricting the morphisms in Definition <ref>to fibers at $t$, we have the following morphisms.
\begin{equation}
\begin{tikzcd}
\X_t \arrow[r] & \overline{\X}_t \arrow[r] & \Y_t
\end{tikzcd}
\end{equation}
We will construct a topological 2-chain $K_+-K_-\in S_2(\X_t^\an)$ associated with $\xi_{1,t}-\xi_{0,t}$ in a weak sense from the following 2-chains on $\Y_t^\an$ and $\overline{\X}_t^\an$.
\begin{equation}
\begin{tikzcd}[row sep = small]
\X_t^\an \arrow[r] &[40pt] \overline{\X}_t^\an \arrow[r] &[80pt] \Y_t^\an \\
K_+\cup K_- \arrow[r,"\text{inverse}\:\text{image}"] \arrow[u,"\rotatebox{90}{$\subset$}",phantom]
& K \arrow[r,"\text{``strict transformation"}"] \arrow[u,"\rotatebox{90}{$\subset$}",phantom] &\overline{\triangle} \arrow[u,"\rotatebox{90}{$\subset$}",phantom]
\end{tikzcd}
\end{equation}
(Definition of $\overline{\triangle}$ and $K$) We use the same symbols $a,b,\sqrt{1-a},\sqrt{1-b}$ for their image by $t^\sharp: B^\circ \rightarrow \C$. We take a $C^\infty$-path $\gamma:[0,1]\rightarrow (\P^1_\C)^\an$ satisfying the following conditions.
* $\gamma(0) = 0$ and $\gamma(1) = 1$.
* $\gamma(s)\neq 0,1,\frac{1}{a},\frac{1}{b},\infty$ except $s = 0,1$.
* We can fix the branch of the functions $\sqrt{1-az}, \sqrt{1-bz}$ along $\gamma$ so that $\sqrt{1-a\gamma(0)} = \sqrt{1-b\gamma(0)} = 1$ and $\sqrt{1-a\gamma(1)} = \sqrt{1-a}, \sqrt{1-b\gamma(1)} = \sqrt{1-b}$.
* On a neighborhood of $0$, we have $\gamma(s) = s^2$. Furthermore, we can fix the branch of the function $\sqrt{z}$ along $\gamma$ so that $\sqrt{\gamma(1)} = 1$ and $\sqrt{\gamma(s)} = s$ on a neighborhood of $0$.
* On a neighborhood of $1$, we have $\gamma(s) = 1- (1-s)^2$. Furthermore, we can fix the branch of the function $\sqrt{1-z}$ along $\gamma$ so that $\sqrt{1-\gamma(0)} = 1$ and $\sqrt{1-\gamma(s)} = 1-s$ on a neighborhood of $1$.
The conditions (4) and (5) are necessary for $K_+$ and $K_-$ to be $C^\infty$-chains. If $\sqrt{1-a},\sqrt{1-b}\in \R_{>1}$, the closed interval $[0,1]$ along real axis (with suitable reparametrization) satisfies the conditions above. By the condition (3)(4)(5), we fix the branch of the local holomorphic functions $\sqrt{z(1-z)(1-az)}$ and $\sqrt{z(1-z)(1-bz)}$ along $\gamma$. We define $\triangle \subset \Y_t^\an$ as the image of the following map.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
\{(p,q)\in \R^2: 0 < q < p < 1\} \arrow[r] & \Y_t^\an \\
(p,q)\aru \arrow[r,mapsto] & (x,y) = (\gamma(p),\gamma(q)) \aru
\end{tikzcd}
\end{equation}
We define $\overline{\triangle}$ as the closure (in the sense of classical topology) of $\triangle$ in $\Y_t^\an$.Since $\triangle \subset \Y_t^\an$ does not intersect with the blowing-up locus of $\overline{\X}_t\rightarrow \Y_t$, the inverse image of $\triangle$ by $\overline{\X}_t\rightarrow \Y_t$ is homeomorphic to $\triangle$. We also denote the inverse image of $\triangle$ by $\triangle$. We define $K\subset \overline{\X}_t^\an$ as the closure (in the sense of classical topology) of $\triangle \subset \overline{\X}_t^\an$.
\begin{equation}
\begin{tikzcd}
\overline{\X}_t^\an \arrow[d] & \triangle \arrow[l,"\supset",phantom] \arrow[d,"\simeq"] \arrow[r,"\subset",phantom]& K\arrow[d] \\
\Y_t^\an &\triangle \arrow[l,"\supset",phantom] \arrow[r,"\subset",phantom] & \overline{\triangle}
\end{tikzcd}
\end{equation}
We define paths $\gamma_c,\gamma_{11},\gamma_{y},\gamma_{10}, \gamma_x$ and $\gamma_{00}$ on $\overline{\X}^\an_t$ appearing in the boundary $\del K$ as in Figure <ref>. We use the local coordinates $x,y,\overline{v}$ and $x,y,\overline{w}$ on $\overline{\X}_t$ in Definition <ref>. They satisfy the following properties.
* The path $\gamma_c$ is on the strict transformation of $\D_t^\an \subset \Y_t^\an $ by $\overline{\X}^\an_t\rightarrow \Y_t^\an$.
* The path $\gamma_y$ (resp. $\gamma_x$) is on a curve in $\overline{\X}_t^\an$ defined by $x -1 = \overline{w} = 0$ (resp. $y = \overline{v} = 0$).
* The paths $\gamma_{00},\gamma_{10}$ and $\gamma_{11}$ are on the exceptional curves $\overline{Q}^\an_{(0,0),t}$, $\overline{Q}^\an_{(1,0),t}$ and $\overline{Q}^\an_{(1,1),t}$ respectively. Here $\overline{Q}_{(0,0),t}$, $\overline{Q}_{(1,0),t}$ and $\overline{Q}_{(1,1),t}$ are fibers of $\overline{Q}_{(0,0)}$, $\overline{Q}_{(1,0)}$ and $\overline{Q}_{(1,1)}$ in Definition <ref> at $t$.
The figure of $K$ and paths on its boundary
Since $\triangle \subset \overline{\X}_t^\an$ does not intersect with the branching locus of the double covering $\X_t^\an \rightarrow \overline{\X}_t^\an$, the inverse image of $\triangle$ by $\X_t^\an \rightarrow \overline{\X}_t^\an$ decomposes into the disjoint union of $\triangle_+$ and $\triangle_-$, which are both homeomorphic to $\triangle\subset \overline{\X}_t$ (Note that $\triangle$ is simply connected). We define $K_+$ and $K_-$ as the closure of $\triangle_+$ and $\triangle_-$. We choose $K_+$ and $K_-$ so that $K_+$ contains $(x,y,v) = (0,0,1)$ and $K_-$ contains $(x,y,v) = (0,0,-1)$.
\begin{equation}
\begin{tikzcd}
\X_t^\an \arrow[d] & \triangle_+\sqcup \triangle_- \arrow[l,"\supset",phantom] \arrow[d,"\text{double cover}"{yshift = -2pt},"\text{\'etale}"{yshift = 6pt}] \arrow[r,"\subset",phantom]& K_+\cup K_- \arrow[d] \\
\overline{\X}_t^\an &\triangle \arrow[l,"\supset",phantom] \arrow[r,"\subset",phantom] & K
\end{tikzcd}
\end{equation}
By the condition (4)(5) in Definition <ref>, we can confirm that $K_{\pm}$ are $C^\infty$ manifolds with corners. Since $K_\pm$ are compact and have the natural orientation induced by $\triangle_{\pm}$, we can regard them as 2-chains on $\X_t^\an$.We define paths $\gamma_{c,\pm}, \gamma_{11,\pm}, \gamma_y,\gamma_{10,\pm},\gamma_x$ and $\gamma_{00,\pm}$ on $\X^\an_t$ appearing in the boundaries $\del K_+$ and $\del K_-$ as in Figure <ref>. They satisfy the following properties.
* The path $\gamma_{c,+}$ (resp. $\gamma_{c,-}$) is the lift of $\gamma_c$ to $K_+$ (resp. $K_-$) and it is on the curve $\Cc_t^\an\subset \X_t^\an$. Note that by the condition (3) in Definition <ref>, its terminal point is $(x,y,v) = (0,0,1)$ (resp. $(x,y,v) = (0,0,-1)$) and its initial point is $(x,y,v) = \left(1,1,\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)$ (resp. $(x,y,v) = \left(1,1,-\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)$).
* The paths $\gamma_{00,+},\gamma_{10,+}$ and $\gamma_{10,+}$ (resp. $\gamma_{00,-},\gamma_{10,-}$ and $\gamma_{10,-}$) are the lift of $\gamma_{00},\gamma_{10}$ and $\gamma_{11}$ to $K_+$ (resp. $K_-$) and they are on the exceptional curves $Q_{(0,0),t}^\an, Q_{(1,0),t}^\an$ and $Q_{(1,1),t}^\an$ respectively.
* Since $\gamma_x$ and $\gamma_y$ on $\overline{\X}_t^\an$ are contained in the branching locus of $\X^\an_t\rightarrow \overline{\X}^\an_t$, there exist unique lifts of them to $\X_t$. We denote their lifts by the same symbol $\gamma_x$ and $\gamma_y$.
The figure of $K_+,K_-$ and paths on their boundaries
The 2-chain $K_+-K_-\in S_2(\X_t^\an)$ is a 2-chain associated with $\xi_{1,t}-\xi_{0,t}$ in a weak sense.
We use the following lemma. The proof is immediate since $H_1((\P^1_\C)^\an) = 0$.
If $\gamma,\gamma'\in S_1((\P^1_\C)^\an)$ satisfy $\del \gamma = \del \gamma'$, then $\gamma-\gamma'\in B_1((\P^1_\C)^\an)$.
(Proposition <ref>)
By Proposition <ref>, $\xi_{1,t} -\xi_{0,t}$ is represented by the following element in $\bigoplus_{Z\in \X_t^{(1)}} R(Z)^\times$.
\begin{equation}
\left(\Cc_t, \frac{(v-1)\left(v+\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)}{(v+1)\left(v-\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)}\right) + \left(Q_{(0,0),t}, \frac{v+1}{v-1}\right) + \left(Q_{(1,1),t},\frac{v-\frac{\sqrt{1-b}}{\sqrt{1-a}}}{v+\frac{\sqrt{1-b}}{\sqrt{1-a}}}\right)
\end{equation}
By Figure <ref>, we see that that
\begin{equation}
\begin{aligned}
& \del(\gamma_{c,+}-\gamma_{c,-}) = \div_{\Cc_t}\left(\frac{(v-1)\left(v+\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)}{(v+1)\left(v-\frac{\sqrt{1-b}}{\sqrt{1-a}}\right)}\right),\quad \del(\gamma_{10,+} -\gamma_{10,-}) = 0 \\
&\del(\gamma_{00,+}-\gamma_{00,-}) = \div_{Q_{(0,0),t}}\left(\frac{v+1}{v-1}\right), \quad \del(\gamma_{11,+}-\gamma_{11,-}) = \div_{Q_{(1,1),t}}\left(\frac{v-\frac{\sqrt{1-b}}{\sqrt{1-a}}}{v+\frac{\sqrt{1-b}}{\sqrt{1-a}}}\right)
\end{aligned}
\end{equation}
Since $\Cc_t, Q_{(0,0),t}, Q_{(1,1),t}$ and $Q_{(1,0),t}$ are isomorphic to $\P^1_\C$, by Lemma <ref>, $(\gamma_{c,+}-\gamma_{c,-}) + (\gamma_{11,+}-\gamma_{11,-}) + (\gamma_{10,+} -\gamma_{10,-}) + (\gamma_{00,+}-\gamma_{00,-})$ is a 1-cycle associated with $\xi_{1,t} -\xi_{0,t}$ in a weak sense. Since we have
\begin{equation}
\del (K_+ -K_-) = (\gamma_{c,+}-\gamma_{c,-}) + (\gamma_{11,+}-\gamma_{11,-}) + (\gamma_{10,+} -\gamma_{10,-}) + (\gamma_{00,+}-\gamma_{00,-}),
\end{equation}
the result follows from Proposition <ref>.
§.§ Calculation of the transcendental regulator map at $t\in T^\circ(\C)$
Since we have constructed a 2-chain associated with $\xi_{1,t}-\xi_{0,t}$, we can compute the image of $\xi_{1,t}-\xi_{0,t}$ under the transcendental regulator map by Proposition <ref>.
Since $\X_t$ is a $K3$ surface and the holomorphic 2-form $\omega_t$ in Definition <ref> is non-zero, the following map is an isomorphism between abelian groups.
\begin{equation}\label{evt}
\begin{tikzcd}[row sep = tiny]
\ev_{t}: &[-30pt] H^{2,0}(\X_t^\an)^\vee/H_2(\X_t^\an, \Z) \arrow[r] & \C/\Pc(\X_t,\omega_t) \\
& \varphi \arrow[r,mapsto]\aru & \varphi([\omega_t])\aru
\end{tikzcd}
\end{equation}
We denote this map by $\ev_t$. Hereafter we concern periods of Kummer surfaces $\X_t$ for $t\in T^\circ(\C)$, we simply write $\Pc(\X_t,\omega_t) $ as $\Pc_{\omega_t}$. Furthermore, the image of $x\in \C$ under the natural projection $\C\rightarrow \C/\Pc_{\omega_t}$ is denoted by $[x]\in \C/\Pc_{\omega_t}$.
Let $t\in T^\circ(\C)$. Choose a path $\gamma$ satisfying the conditions in Definition <ref> at $t$. We can take an open neighborhood $U$ of $t$ in $(T^\circ)^\an$ in the classical topology such that $\gamma$ satisfies the conditions in Definition <ref> at every point on $U$. Then we have the following.
* The following integral converges and defines a holomorphic function $\Lc$ on $U$.
\begin{equation}\label{integralrepnL}
\Lc(t) = \int_{\triangle}\frac{\gamma'(p)\gamma'(q)dpdq}{\sqrt{\gamma(p)(1-\gamma(p))(1-a\gamma(p))}\cdot\sqrt{\gamma(q)(1-\gamma(q))(1-b\gamma(q))}}\quad (t\in U)
\end{equation}
Note that since the branch of $\sqrt{z(1-z)(1-az)}$ and $\sqrt{z(1-z)(1-bz)}$ along $\gamma$ is fixed by Definition <ref>, the branch of the integrand on $\triangle$ is also fixed.
* The image of $\xi_{1,t}-\xi_{0,t}$ under the transcendental regulator map $r$ is as follows.
\begin{equation}
\ev_{t}(r(\xi_{1,t}-\xi_{0,t})) = 2[\Lc(t)]\quad \in \C/\Pc_{\omega_t}
\end{equation}
* If we choose a different path $\gamma$, we get another local holomorphic function $\Lc'$. However, the difference $\Lc(t)-\Lc'(t)\in \C$ should lie in $\frac{1}{2}\Pc_{\omega_t}$.
By the construction of $\Delta_+\subset \X_t^\an$, we see that $\Delta_+$ coincides with the image of the following map.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
\{(p,q)\in \R^2: 0<q<p<1\} \arrow[r] &\X_t^\an \\
(p,q) \arrow[r,mapsto] \aru &(x,y,v) = \left(\gamma(p),\gamma(q),\frac{\sqrt{\gamma(q)(1-\gamma(q))(1-b\gamma(q))}}{\sqrt{\gamma(p)(1-\gamma(p))(1-a\gamma(p))}}\right) \aru
\end{tikzcd}
\end{equation}
Hence the right hand side of (<ref>) coincides with $\int_{\Delta_+} \omega_t$. Since the integrand is $C^\infty$ on the boundary of $\Delta_+$, we have $\int_{\Delta_+} \omega_t = \int_{K_+}\omega_t$. Thus the right hand side of (<ref>) can be regarded as integration of a $C^\infty$-function on a compact $C^\infty$-manifold with corners. Furthermore, the integrand is holomorphic with respect to $a,b$, which are local coordinates of $(T^\circ)^\an$. Hence we have (1). By Proposition <ref> and Proposition <ref>, we have
\begin{equation}
\ev_{t}(r(\xi_{1,t}-\xi_{0,t})) = \int_{K_+}\omega_t-\int_{K_-}\omega_t\in \C/\Pc_{\omega_t}
\end{equation}
Since $\int_{K_+}\omega_t = -\int_{K_-}\omega_t = \Lc(t)$ by definition, we have (2). Then by (2), $2[\Lc(t)]$ is determined up to elements in $\Pc_{\omega_t}$. Thus we have (3).
At last, we calculate the image of $\Lc$ under the Picard-Fuchs operator $\Dc$ in Definition <ref>. This calculation is used in the rank estimation of the image of $\Xi_t$ under the transcendental regulator map in Section 9. This theorem also gives a system of differential equations which $\Lc$ satisfies.
Let $\Lc$ be the local holomorphic function defined in Definition <ref>. Then we have
\begin{equation}
\Dc(\Lc) =
\frac{1}{a-b}\cdot
\begin{pmatrix}
\frac{\sqrt{1-b}}{\sqrt{1-a}}-1\\[1.5ex]
\end{pmatrix}
\end{equation}
A local holomorphic function $H(c,z) = -\frac{\sqrt{z(1-z)}}{2(1-c z)^{\frac{3}{2}}}$ satisfies
\begin{equation}
L_c\left(\frac{1}{\sqrt{z(1-z)(1-cz)}}\right) = \frac{\del H(c,z)}{\del z}.
\end{equation}
where $L_c$ is the differential operator defined in Definition <ref>. Then we have the following equation on 2-forms on $\X_t$.
\begin{equation}
\Dc_1\left(\frac{1}{\sqrt{x(1-x)(1-ax)}\cdot\sqrt{y(1-y)(1-by)}}\right)dx\wedge dy = d\left(
\frac{H(a,x)dy}{\sqrt{y(1-y)(1-by)}}\right)
\end{equation}
This equation holds on an open neighborhood of $K_+$. By the definition of $\Lc$ and Stokes theorem on $\X_t$, we have the following.
\begin{equation}
\begin{aligned}
&\Dc_1(\Lc)= \Dc_1\left( \int_{K_+}\frac{\gamma'(p)\gamma'(q)dpdq}{\sqrt{\gamma(p)(1-\gamma(p))(1-a\gamma(p))}\cdot\sqrt{\gamma(q)(1-\gamma(q))(1-b\gamma(q))}}\right) \\
& = \int_{K_+} d\left(
\frac{H(a,\gamma(p))\gamma'(q)dq}{\sqrt{\gamma(q)(1-\gamma(q))(1-b\gamma(q))}}
\right) = \int_{\del K_+} \frac{H(a,\gamma(p))\gamma'(q)dq}{\sqrt{\gamma(q)(1-\gamma(q))(1-b\gamma(q))}}
\end{aligned}
\end{equation}
Since the 1-form $\frac{H(a,x)dy}{\sqrt{y(1-y)(1-by)}}$ vanishes at $\{y=0\}$ and $\{x=1\}$, we have
\begin{equation}
\Dc_1(\Lc)= \frac{1}{2}\int_0^1 \frac{dz}{(1-bz)^{\frac{1}{2}}(1-az)^{\frac{3}{2}}} =\frac{1}{a-b} \int_1^{\frac{\sqrt{1-b}}{\sqrt{1-a}}} du = \frac{1}{a-b}\cdot\left(\frac{\sqrt{1-b}}{\sqrt{1-a}}-1\right).
\end{equation}
Here we use the coordinate transform $u= \frac{\sqrt{1-bz}}{\sqrt{1-az}}$. We can compute $\Dc_2(\Lc)$ similarly.
§ ESTIMATION OF THE RANK OF THE IMAGE OF $\XI$ UNDER THE TRANSCENDENTAL REGULATOR MAPS
In this section, we prove Theorem <ref>. The outline of the proof is as follows.
* We construct a $\Q$-linear sheaf $\Qc_\omega$ on $(T')^\an$ as a quotient of the sheaf of holomorphic functions $\O_{(T')^\an}$ by a locally constant subsheaf $\Pc_\omega$ generated by period functions $P_{ij}$. For each $t\in (T')^\an$, we have a “evaluation" map $\Qc_\omega(T')\rightarrow \C/\Q\Pc_{\omega_t}\simeq H^{2,0}(\X_t^\an)^\vee/H_2(\X_t,\Q)$. We see that the Picard-Fuchs differential operator $\Dc$ factors through the sheaf $\Qc_\omega$ (Definition <ref>).
* The $\Q$-linear space $\Qc_\omega(T^\circ)$ is the target of a “relative transcendental regulator map" $R_\omega: \Xi\rightarrow \Qc_\omega(T^\circ)$ (Definition <ref>). The “value" of $R_\omega(\xi)$ at $t\in T^{\circ}(\C)$ coincides with $r(\xi_t)$ modulo torsion.
* By the formula of the $\widetilde{G}$-action on $\omega_t$ in Proposition <ref>, we have the following commutative diagram (Proposition <ref>).
\begin{equation}
\begin{tikzcd}
\CH^2(\X_{t},1) \arrow[r,"r"]\arrow[d,"(\widetilde{\rho}_t)_*"] & H^{2,0} (\X_t^\an)^\vee/H_2(\X_t,\Z) \arrow[d,"(\widetilde{\rho}_t^*)^\vee"]\arrow[r,"\ev_{t}"] & \C/\Pc_{\omega_t} \arrow[d,"\widetilde{\chi}(\widetilde{\rho})(t)"] \\
\CH^2(\X_{\underline{\rho}(t)},1) \arrow[r,"r"] & H^{2,0}(\X_{\underline{\rho}(t)}^\an)^\vee/H_2(\X_t,\Z) \arrow[r,"\ev_{\underline{\rho}(t)}"]& \C/\Pc_{\omega_{\underline{\rho}(t)}}
\end{tikzcd}
\end{equation}
* By the diagram above, we can define a $\widetilde{G}$-action $\{\Upsilon_{\widetilde{\rho}}\}_{\widetilde{\rho}\in \widetilde{G}}$ on $\Qc_\omega$ (Definition <ref>) so that the relative transcendental regulator map $R_\omega$ is equivariant to $\widetilde{G}$-actions (Proposition <ref>). Furthermore, we can also define a $\widetilde{G}$-action $\{\Theta_{\widetilde{\rho}}\}_{\widetilde{\rho}\in\widetilde{G}}$ on $\O_{(T')^\an}^{\oplus 2}$ (Definition <ref>) so that the Picard-Fuchs differential operator $\Dc$ is equivariant to $\widetilde{G}$-actions (Proposition <ref>).
* In conclusion, we have the following diagram for $\widetilde{\rho}\in \widetilde{G}$.
\begin{equation}
\begin{tikzcd}
\Xi \arrow[d,"\widetilde{\rho}_*"]\arrow[r,"R_\omega"] & \Qc_\omega(T^\circ) \arrow[r,"\Dc"]\arrow[d,"\Upsilon_{\widetilde{\rho}}"] & \O_{(T')^\an}(T^{\circ})^{\oplus 2} \arrow[d,"\Theta_{\widetilde{\rho}}"] \\
\Xi \arrow[r,"R_\omega"] & \Qc_\omega(T^\circ) \arrow[r,"\Dc"] & \O_{(T')^\an}(T^{\circ})^{\oplus 2}
\end{tikzcd}
\end{equation}
By this diagram, we can compute the image $\Dc\circ R_\omega(\Xi)$ (Table <ref>) and get the desired rank estimate (Theorem <ref>).
§.§ The definition of the sheaves $\Pc_\omega$ and $\Qc_\omega$
In this section, we define the sheaves $\Pc_{\omega}$ and $\Qc_\omega$ and prove their properties.
We regard the sheaf $\O_{(T')^\an}$ of holomorphic functions on $(T')^\an$ as a $\Q$-linear sheaf. We define a subsheaf $\Pc_{\omega}\subset \O_{(T')^\an}$ as the unique sheaf satisfying the following property:
\begin{equation}\label{PisgeneratedbyPij}
\begin{aligned}
&&&\text{For any open set }U\subset (T')^\an\text{ in the classical topology such that }P_{ij}\text{ are defined, } \\
&&&\Pc_{\omega}|_U\text{ is the subsheaf generated (as a $\Q$-linear sheaf) by }P_{ij} \text{ for }i,j\in\{1,2\}.
\end{aligned}
\end{equation}
where $P_{ij}$ are the local holomorphic functions defined in Definition <ref>. Then we define a sheaf $\Qc_\omega$ as the quotient sheaf $\O_{(T')^\an}/\Pc_{\omega}$. For a local section $f$ of $\O_{(T')^\an}$, $[f]$ denotes the image of $f$ under the quotient map $\O_{(T')^\an}\rightarrow \Qc_{\omega}$.
The existence of $\Pc_\omega$ can be confirmed by the following remark.
Let $\pi: \X'\rightarrow T'$ be the structure morphism. We define the following sheaves $\Pc,\Qc$ on $(T')^\an$.
\begin{equation}
\begin{aligned}
&\Pc = \mathrm{Im}(R^2\pi_*\underline{\Q}_ {(\X')^\an}\rightarrow \mathcal{H}om_{\O_{(T')^\an}}(\pi_*\Omega^2_{\X'/T'}, \O_{(T')^\an})) \\
&\Qc =\mathrm{Coker}(R^2\pi_*\underline{\Q}_ {(\X')^\an}\rightarrow \mathcal{H}om_{\O_{(T')^\an}}(\pi_*\Omega^2_{\X'/T'}, \O_{(T')^\an}))
\end{aligned}
\end{equation}
where $\underline{\Q}_{(\X')^\an}$ is the constant sheaf with coefficients in $\Q$ on $(\X')^\an$ and the morphism $R^2\pi_*\underline{\Q}_ {(\X')^\an}\rightarrow \mathcal{H}om_{\O_{(T')^\an}}(\pi_*\Omega^2_{\X'/T'}, \O_{(T')^\an})$ is induced by the fiber integration. Since $\X'$ is a family of $K3$ surface, $\pi_*\Omega^2_{\X'/T'}$ is a locally free $\O_{(T')^\an}$-module of rank 1.
Then we have an isomorphism $\O_{(T')^\an} \simeq \mathcal{H}om_{\O_{(T')^\an}}(\pi_*\Omega^2_{\X'/T'}, \O_{(T')^\an})$ induced by $\varphi\mapsto \varphi\cdot \omega$ where $\omega$ is the 2-form in Definition <ref>. Under this isomorphism, we have $\Pc\simeq \Pc_\omega$ and $\Qc\simeq \Qc_\omega$. Since $\pi:\X'\rightarrow T'$ is a topologically locally trivial fibration, for a sufficiently small open neighborhood in the classical topology, we have a $\Q$-basis in $\Pc|_U$. The holomorphic functions $P_{ij} \:(i,j\in \{1,2\})$ are the images of such a basis under $\Pc|_U\simeq \Pc_\omega|_U$.
For each $t\in T'(\C)$, $\O_{(T')^\an,t}$ denotes the stalk of $\O_{(T')^\an}$ at $t$. We define the evaluation map $m_t$ by
\begin{equation}
\begin{tikzcd}[row sep = tiny]
m_t:&[-30pt] \O_{(T')^\an,t} \arrow[r]& \C; &[-30pt] \varphi \arrow[r,mapsto] & \varphi(t).
\end{tikzcd}
\end{equation}
For an open neighborhood $U$ of $t$ in the classical topology, composition of $m_t$ and a restriction map $\O_{(T')^\an}(U)\rightarrow \O_{(T')^\an,t}$ is also denoted by $m_t$. Furthermore, since $\Pc_{\omega_t}\subset \C$ is generated by the values of $P_{ij}$ at $t$ by Definition <ref>, $m_t: \O_{(T')^\an,t} \rightarrow \C$ induces the following map $\Qc_{\omega,t}\rightarrow \C/\Q\Pc_{\omega_t}$.
\begin{equation}
\begin{tikzcd}
\O_{(T')^\an,t} \arrow[r,"m_t"]\arrow[d,twoheadrightarrow] & \C \arrow[d,twoheadrightarrow]\\
\Qc_{\omega,t} \arrow[r,dashed,"m_t"] & \C/\Q\Pc_{\omega_t}
\end{tikzcd}
\end{equation}
where $\Q\Pc_{\omega_t}\subset \C$ is a $\Q$-linear subspace of $\C$ generated by $\Pc_{\omega_t}$. We also denote this map by $m_t$. Furthermore, the composite of $m_t$ and the restriction map of $\Qc_\omega$ is also denoted by $m_t$.
Let $U$ be an open subset of $(T')^\an$ in the classical topology and $\varphi\in \O_{(T')^\an}(U)$. Then $\varphi(t)\not \in \Q\Pc_{\omega_t}$ for very general[We use the word “very general" for the meaning of “outside of a countable union of proper ($=$ not the whole space) analytic subsets".] $t\in U$ if and only if $\varphi\not \in \Pc_\omega(U)$. Especially, if $\varphi\in \O_{(T')^\an}(U)$ satisfies that $\varphi(t)\in \Q\Pc_{\omega_t}$ holds for every $t\in U$, then $\varphi$ is a section of $\Pc_\omega(U)$.
We will prove the former part of the proposition. We may assume $U$ is so small that $P_{ij}$ are defined on $U$. For each quadruple $\underline{c}=(c_{ij})\in \Q^{\oplus 4}$, we define a holomorphic function $F_{\underline{c}}$ by
\begin{equation}
F_{\underline{c}} = \varphi- \sum_{i,j}c_{ij}P_{ij}.
\end{equation}
Consider the countable family $\{F_{\underline{c}}\}_{\underline{c}\in \Q^4}$ of holomorphic functions on $U$. If $\varphi\not \in \Pc_\omega(U)$, they are non-zero holomorphic functions. Especially, for very general $t\in U$, $F_{\underline{c}}(t)\neq 0$ holds for all $\underline{c}\in \Q^4$. Since $\Pc_{\omega_t}$ is generated (as a $\Q$-linear subspace of $\C$) by $P_{ij}(t)$, we see that $F_{\underline{c}}(t)\neq 0$ holds for all $\underline{c}\in \Q^4$ is equivalent to $\varphi(t) \not \in \Q\Pc_{\omega_t}$. Converse is clear. The latter part follows from the former part.
The sheaf $\Qc_\omega$ has the following property. This lemma enables us to reduce the computation of $\Qc_\omega$ to that of its restriction at each point on $U$.
For each open subset $U$ of $(T')^\an$ in the classical topology and non-zero section $x\in \Qc_\omega(U)$, the restriction $m_t(x)$ is non-zero for very general $t\in U$. Especially, the following map is injective.
\begin{equation}\label{MtPointwise}
\begin{tikzcd}[row sep = tiny]
\Qc_\omega(U) \arrow[r] & \displaystyle\prod_{t\in U}\C/\Q\Pc_{\omega_t}; &[-30pt] x \arrow[r,mapsto] & \left(m_t(x)\right)_{t}
\end{tikzcd}
\end{equation}
We can shrink $U$ so small that $x$ is of the form $x = [\varphi]$ for $\varphi\in \O_{(T')^\an}(U)$. Then the results follows from Proposition <ref>.
§.§ A $\widetilde{G}$-action on $\Qc_\omega$
First, we see that how $\widetilde{G}$ acts on $\C/\Pc_{\omega_t}$.
Let $t\in T^{\circ}(\C)$ and $\widetilde{\rho} = (\rho,\zeta)\in \widetilde{G}$. Let $\widetilde{\rho}_t: \X_t\xrightarrow{\:\sim\:} \X_{\underline{\rho}(t)}$ be the automorphism defined in Definition <ref>.
* We have $\Pc_{\omega_{\underline{\rho}(t)}} = \widetilde{\chi}(\widetilde{\rho})(t)\cdot\Pc_{\omega_{t}}$ as a subgroup of $\C$. Here $\widetilde{\chi}(\widetilde{\rho})(t)\in \C$ is the value of $\widetilde{\chi}(\widetilde{\rho})\in B'$ in Defintion <ref> at $t$.
* From (1), the following map is well-defined.
\begin{equation}\label{chimul}
\begin{tikzcd}[row sep = tiny]
\widetilde{\chi}(\widetilde{\rho})(t):&[-30pt] \C/\Pc_{\omega_t} \arrow[r] & \C/\Pc_{\omega_{\underline{\rho}(t)}}; &[-30pt] \lb x \rb \arrow[r,mapsto] & \lb\widetilde{\chi}(\widetilde{\rho})(t)\cdot x\rb
\end{tikzcd}
\end{equation}
* We have the following commutative diagram.
\begin{equation}\label{funddiag}
\begin{tikzcd}
\CH^2(\X_{t},1) \arrow[r,"r"]\arrow[d,"(\widetilde{\rho}_t)_*"] & H^{2,0} (\X_t^\an)^\vee/H_2(\X_t,\Z) \arrow[d,"(\widetilde{\rho}_t^*)^\vee"]\arrow[r,"\ev_{t}"] & \C/\Pc_{\omega_t} \arrow[d,"\widetilde{\chi}(\widetilde{\rho})(t)"] \\
\CH^2(\X_{\underline{\rho}(t)},1) \arrow[r,"r"] & H^{2,0}(\X_{\underline{\rho}(t)}^\an)^\vee/H_2(\X_t,\Z) \arrow[r,"\ev_{\underline{\rho}(t)}"]& \C/\Pc_{\omega_{\underline{\rho}(t)}}
\end{tikzcd}
\end{equation}
where the right vertical map is $(\ref{chimul})$ above.
Note that the following equation holds for every 2-chain $\Gamma\in S_2(\X_{t}^\an)$.
\begin{equation}\label{sekibuntransf}
\int_{(\widetilde{\rho}_t)_*\Gamma}\omega_{\underline{\rho}(t)} = \int_\Gamma (\widetilde{\rho}_t)^*\omega_{\underline{\rho}(t)} = \widetilde{\chi}(\widetilde{\rho})(t)\cdot \int_\Gamma\omega_t
\end{equation}
For the second equality, we use Proposition <ref>. By the equations (<ref>) for $\Gamma \in Z_2(\X_t^\an)$, we can show (1). If $\Gamma$ is a 2-chain associated with $\xi\in \CH^2(\X_t,1)$, then $(\widetilde{\rho}_t)_*\Gamma$ is a 2-chain associated with $(\widetilde{\rho}_t)_*\xi\in \CH^2(\X_{\underline{\rho}(t)},1)$. Hence by the equation (<ref>) for a 2-chain $\Gamma$ associated with $\xi$, we see that the whole rectangle in (<ref>) commutes. Since $\ev_t,\ev_{\underline{\rho}(t)}$ are isomorphisms by Definition <ref>, all squares in (<ref>) commute.
Then we will define a $\widetilde{G}$-linearization on $\O_{(T')^\an}$.
Let $\widetilde{\rho}=(\rho,\zeta)\in \widetilde{G}$. We define a morphism $\Upsilon_{\widetilde{\rho}}: \O_{(T')^\an}\rightarrow (\underline{\rho}^{-1})_*\O_{(T')^\an}$ as follows. Let $U$ be an open subset of $(T')^\an$ in the classical topology.
\begin{equation}\label{upsilondefn}
\begin{tikzcd}[row sep = tiny]
\Upsilon_{\rho}:&[-30pt] \O_{(T')^\an}(U) \arrow[r] & \O_{(T')^\an}(\underline{\rho}(U)) \arrow[r,equal]&[-20pt](\underline{\rho}^{-1})_*\O_{(T')^\an}(U) \\
& \varphi \aru \arrow[r,mapsto] & (\underline{\rho}^{-1})^\sharp\left(\widetilde{\chi}(\widetilde{\rho})\cdot \varphi\right)\aru
\end{tikzcd}
\end{equation}
Then $\{\Upsilon_{\widetilde{\rho}}\}_{\widetilde{\rho}\in \widetilde{G}}$ satisfies the cocycle condition. In other words, the following diagram commutes for $\widetilde{\rho},\widetilde{\rho}'\in \widetilde{G}$.
\begin{equation}\label{Upsiloncocycle}
\begin{tikzcd}
\O_{(T')^\an} \arrow[d,"\Upsilon_{\widetilde{\rho}'\widetilde{\rho}}"] \arrow[r,"\Upsilon_{\widetilde{\rho}}"] & (\underline{\rho}^{-1})_*\O_{(T')^\an}\arrow[d,"(\underline{\rho}^{-1})_*\Upsilon_{\widetilde{\rho}'}"] \\
((\underline{\rho}'\underline{\rho})^{-1})_*\O_{(T')^\an} \arrow[r,equal] & (\underline{\rho}^{-1})_*(\underline{\rho}')^{-1}_*\O_{(T')^\an}
\end{tikzcd}
\end{equation}
For $\widetilde{\rho}\in \widetilde{G}$ and an open subset $U\subset (T')^\an$ in the classical topology, we have
\begin{equation}
\Upsilon_{\widetilde{\rho}} (\Pc_\omega(U)) = \Pc_\omega(\underline{\rho}(U)).
\end{equation}
It is enough to show only $(\subset)$ by the cocycle condition. Let $\varphi\in \Pc_\omega(U)$. Then for $\underline{\rho}(t) \in \underline{\rho}(T)$, we have
\begin{equation}
\begin{aligned}
m_{\underline{\rho}(t)}( \Upsilon_{\widetilde{\rho}}(\varphi)) &= m_{\underline{\rho}(t)}((\underline{\rho}^{-1})^\sharp(\widetilde{\chi}(\widetilde{\rho})\cdot \varphi)) = m_t(\widetilde{\chi}(\widetilde{\rho})\cdot \varphi) \\
&= \widetilde{\chi}(\widetilde{\rho})(t)\cdot \varphi(t) \in \widetilde{\chi}(\widetilde{\rho})(t)\cdot \Q\Pc_{\omega_{t}} = \Q\Pc_{\omega_{\underline{\rho}(t)}}.
\end{aligned}
\end{equation}
The last equality follows from Proposition <ref>. By Proposition <ref>, $\Upsilon_{\widetilde{\rho}}(\varphi) \in \Pc_\omega(\underline{\rho}(U))$.
By the proposition above, the $\widetilde{G}$-linearization on $\O_{(T')^\an}$ induces a $\widetilde{G}$-linearization on $\Qc_\omega$.
By Proposition <ref>, $\Upsilon_{\widetilde{\rho}}:\O_{(T')^\an}\rightarrow (\underline{\rho}^{-1})_*\O_{(T')^\an}$ induces a morphism $\Qc_\omega \rightarrow (\underline{\rho}^{-1})_*\Qc_\omega $. Since $\underline{\rho}(T^\circ)=T^\circ$, we have the following $\Q$-linear map.
\begin{equation}\label{tildeGactionQ}
\begin{tikzcd}
\Upsilon_{\widetilde{\rho}}: \Qc_\omega(T^{\circ}) \arrow[r] & \Qc_\omega(T^{\circ})
\end{tikzcd}
\end{equation}
By the cocycle condition (<ref>), $\Upsilon_{\widetilde{\rho}}$ defines a $\widetilde{G}$-action on the $\Q$-linear space $\Qc_\omega(T^{\circ})$. By Definition <ref>, the following diagram commutes for $t\in (T^{\circ})^\an$.
\begin{equation}\label{equivariancemt}
\begin{tikzcd}
\Qc_\omega(T^{\circ}) \arrow[d,"\Upsilon_\rho"]\arrow[r,"m_t"] & \C/\Q\Pc_{\omega_t} \arrow[d,"\widetilde{\chi}(\rho)(t)"] \\
\Qc_\omega(T^{\circ}) \arrow[r,"m_{\underline{\rho}(t)}"] & \C/\Q\Pc_{\omega_{\underline{\rho}}(t)}
\end{tikzcd}
\end{equation}
where the right vertical map is induced by (2) of Proposition <ref>.
§.§ Construction of the relative transcendental regulator map $R_\omega$
In this section, we construct the relative transcendental regulator map and show the $\widetilde{G}$-equivariance of $R_\omega$. First, we construct an element in $\Qc_\omega(T^\circ)$ corresponding to a half of the image of $\xi_1-\xi_0$ under the relative transcendental regulator map.
There exists a unique element $[\Lc]\in \Qc_\omega(T^\circ)$ such that for $t\in T^{\circ}(\C)$,
\begin{equation}
m_t([\Lc]) = [\Lc(t)]
\end{equation}
where $\Lc(t)\in \C$ denotes the value of the local holomorphic function $\Lc$ in Definition <ref>.
The uniqueness follows from Lemma <ref>. We show the existence. We take an open cover $\{U_i\}_{i\in I}$ of $(T^\circ)^\an$ such that $\Lc$ is defined on each $U_i$. Let $\Lc_i\in \O_{(T')^\an}(U_i)$ denote a holomorphic function $\Lc$ on $U_i$. It is enough to glue $[\Lc_i]\in \Qc_\omega(U_i)$. By Proposition <ref>, for each $t\in U_i\cap U_j$, $\Lc_i(t)-\Lc_j(t) \in \Q\Pc_{\omega_t}$. Then we have $\Lc_i-\Lc_j\in \Pc_{\omega}(U_i\cap U_j)$ by Proposition <ref>. Hence we have $[\Lc_i]|_{U_i\cap U_j} = [\Lc_j]|_{U_i\cap U_j}$ in $\Qc_{\omega}(U_i\cap U_j)$ and we can check the gluing condition.
$($Definition of $R_\omega)$
There exists a unique group homomorphism
\begin{equation}
\begin{tikzcd}
R_\omega :\Xi \arrow[r] & \Qc_\omega(T^{\circ})
\end{tikzcd}
\end{equation}
which satisfies the following properties. The map $R_\omega$ is called the relative transcendental regulator map.
* For $t\in T^{\circ}(\C)$, the following diagram commutes.
\begin{equation}\label{fiberpointwise}
\begin{tikzcd}
\Xi \arrow[r,"R_\omega"]\arrow[d,"i_t^*"] &\Qc_\omega(T^{\circ})\arrow[dr,"m_t"] & \\
\CH^2(\X_t,1) \arrow[r,"\ev_t\circ r"] & \C/\Pc_{\omega_t}\arrow[r,twoheadrightarrow] & \C/\Q\Pc_{\omega_t}
\end{tikzcd}
\end{equation}
where $i_t^*$ is the pull-back map in Definition <ref>, $r$ is the transcendental regulator map in Definition <ref>, $\ev_t$ is the map defined in Definition <ref>, $m_t$ is the map defined in Definition <ref> and $\C/\Pc_{\omega_t} \rightarrow \C/\Q\Pc_{\omega_t}$ is the natural projection.
* For $\widetilde{\rho}\in \widetilde{G}$, the following diagram commutes.
\begin{equation}\label{GequivXi}
\begin{tikzcd}
\Xi \arrow[d,"\widetilde{\rho}_*"] \arrow[r,"R_\omega"] & \Qc_\omega(T^{\circ}) \arrow[d,"\Upsilon_{\widetilde{\rho}}"] \\
\Xi \arrow[r,"R_\omega"] & \Qc_\omega(T^{\circ})
\end{tikzcd}
\end{equation}
We will prove that there exists a unique map $R_\omega$ satisfying the condition (1) and $R_\omega$ satisfies (2).The uniqueness follows from Lemma <ref>. If we define $R_\omega(\xi_1-\xi_0) = 2[\Lc]$ where $[\Lc]$ is the element defined in Proposition <ref>, we can check the commutativity of (<ref>) for $\xi_1-\xi_0\in \Xi$ by Proposition <ref>. We can also define $R_\omega(\xi)$ for each $\xi\in \Xi$ so as to make the diagram (<ref>) commute as follows: By Proposition <ref>, $\xi$ is represented by a product of $(\widetilde{\rho}^{-1})^\sharp(\psi_\bullet)$ and $(\widetilde{\rho}^{-1})^\sharp(\varphi_\bullet)$. They are on smooth families of curves over $T^\circ$ and their zeros and poles are also smooth over $T^\circ$. Hence by the similar method in Section 8, we see that $\ev_t(r(\xi))$ is represented by the value of the local holomorphic function as in Proposition <ref>. Hence by the similar argument in Proposition <ref>, we can define $R_\omega(\xi)\in \Qc_\omega(T^{\circ})$. Hence we can check the existence of the map $R_\omega$ satisfying the condition (1).Next, we will prove that $R_\omega$ satisfies (2). Consider the following diagram.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
\Xi\arrow[rr,"R_\omega"{xshift = -20pt}] \arrow[dd,"i_t^*"]\arrow[dr,"\rho_*"'] && \Qc_\omega(T^{\circ})\arrow[dr,"\Upsilon_\rho"]\arrow[dd,"m_t"{yshift = 8pt}] & \\
& \Xi \arrow[rr,"R_\omega"{xshift = -20pt},crossing over]&& \Qc_\omega(T^{\circ})\arrow[dd,"m_{\underline{\rho}(t)}"]\\
\CH^2(\X_t,1)\arrow[dr,"(\rho_t)_*"'] \arrow[rr,"\ev_{t}\circ r"{xshift = 30pt}] && \C/\Q\Pc_{\omega_t}\arrow[dr,"\widetilde{\chi}(\rho)(t)"] &\\
& \CH^2(\X_{\underline{\rho}(t)},1)\arrow[from = uu,crossing over,"i_{\underline{\rho}(t)}^*"'{yshift = 8pt}]\arrow[rr,"\ev_{\underline{\rho}(t)}\circ r"'] && \C/\Q\Pc_{\omega_{\underline{\rho}(t)}}
\end{tikzcd}
\end{equation}
The left side face commutes by the associativity of pull-back maps on higher Chow groups[Note that since $\widetilde{\rho}\in \widetilde{G}$ is an isomorphism, $\widetilde{\rho}_* = (\widetilde{\rho}^{-1})^*$.] ([12] PartI, Chapter II, 2.1.6). The bottom face commutes by Proposition <ref> and the right side face commutes by (<ref>) in Definition <ref>. Since the front and back faces commute by (1), by Lemma <ref>, we see that the upper face commutes.
By $\widetilde{G}$-equivariance of $R_\omega$, we have a $\widetilde{G}$-action on $R_\omega(\Xi)$. Then we have the upper estimate for $\rank R_\omega(\Xi)$. The proof below is simplified by advice from T. Saito.
We have the following.
* For $\widetilde{\rho}\in \widetilde{G}_\fib$, we have $R_\omega(\widetilde{\rho}_*\Xi^\can) = R_\omega(\Xi^\can)$.
* We have $\rank R_\omega(\Xi) \le 18$.
* For each $t\in T^\circ(\C)$, $\rank r(\Xi_t)\le 18$.
By $\widetilde{G}$-equivariance of $R_\omega$, we have $R_\omega(\widetilde{\rho}_*\Xi^\can) = \Upsilon_{\widetilde{\rho}}(R_\omega(\Xi^\can))$. For $\widetilde{\rho}\in \widetilde{G}_\fib$, we have $\Upsilon_{\widetilde{\rho}} = \pm 1$ by definition of $\Upsilon_{\widetilde{\rho}}$. Hence we have (1).By (1) and Proposition <ref>, $R_\omega(\Xi^\can)\subset R_\omega(\Xi)$ is stabilized under the $\widetilde{G}_\fib\widetilde{I}$-action. Especially, we have a $\widetilde{G}_\fib\widetilde{I}$-representation on $R_\omega(\Xi^\can)$. Then the following $\widetilde{G}$-equivariant map is induced.
\begin{equation}\label{indrepn}
\mathrm{Ind}_{\widetilde{G}_\fib\widetilde{I}}^{\widetilde{G}} R_\omega(\Xi^\can) \lra R_\omega(\Xi)
\end{equation}
where $\mathrm{Ind}_{\widetilde{G}_\fib\widetilde{I}}^{\widetilde{G}} R_\omega(\Xi^\can)$ denotes the induced representation. Since $R_\omega(\Xi)$ is the sum of $R_\omega(\widetilde{\rho}_*\Xi^\can)$ for $\widetilde{\rho}\in \widetilde{G}$, the map (<ref>) is surjective. Then we have
\begin{equation}
\rank R_\omega(\Xi) \le \rank \mathrm{Ind}_{\widetilde{G}_\fib\widetilde{I}}^{\widetilde{G}} R_\omega(\Xi^\can) = |\widetilde{G}/\widetilde{G}_\fib\widetilde{I}|\cdot \rank R_\omega(\Xi^\can) \le 6\cdot 3.
\end{equation}
Here we use $|\widetilde{G}/\widetilde{G}_\fib\widetilde{I}| = 6$ by Proposition <ref>. Hence we have (2). By (2) and the commutative diagram (<ref>), we have (3).
§.§ The differential operator $\Dc$ and $\widetilde{G}$-actions
In this subsection, we define a $\widetilde{G}$-action on $\O_{(T')^\an}^{\oplus 2}$ so that $\Dc$ is $\widetilde{G}$-equivariant. For this purpose, we prove transformation formulae of $\Dc$.
Since the local holomorphic functions $P_{ij}$ are annihilated by the differential operator $\Dc: \O_{(T')^\an}\rightarrow \O_{(T')^\an}^{\oplus 2}$ in Definition <ref>, $\Pc_\omega \hookrightarrow \O_{(T')^\an} \xrightarrow{\Dc} \O_{(T')^\an}^{\oplus 2}$ is the 0-map. Hence the following morphism is induced. This morphism is also denoted by $\Dc$.
\begin{equation}
\begin{tikzcd}
\O_{(T')^\an} \arrow[r,"\Dc"] \arrow[d] &\O_{(T')^\an}^{\oplus2} \\
\Qc_\omega \arrow[ur,dashed,"\Dc"'] &
\end{tikzcd}
\end{equation}
Let $\widetilde{\rho} = (\rho,\zeta)\in \widetilde{G}$. Let $U$ be an open subset of $(T')^\an$ in the classical topology. We define a morphism $\Theta_{\widetilde{\rho}}: \O_{(T'^\an)}^{\oplus 2}\rightarrow (\underline{\rho}^{-1})_*\O_{(T'^\an)}^{\oplus 2}$ as the following map.
\begin{equation}
\begin{tikzcd}[row sep = tiny]
\Theta_{\widetilde{\rho}}: &[-30pt] \O_{(T'^\an)}^{\oplus 2}(U) \arrow[r] & \O_{(T'^\an)}^{\oplus 2}(\underline{\rho}(U))\arrow[r,equal] &[-50pt] (\underline{\rho}^{-1})_*\O_{(T'^\an)}^{\oplus 2}(U)\\
& \begin{pmatrix} \varphi_1 \\[5pt] \varphi_2\end{pmatrix} \aru \arrow[r,mapsto] &
\begin{pmatrix}
(\underline{\rho}^{-1})^\sharp\left( \widetilde{\chi}(\widetilde{\rho})\chi^{(1)}(\underline{\rho})^2 \cdot \varphi_1\right) \\[5pt]
(\underline{\rho}^{-1})^\sharp\left( \widetilde{\chi}(\widetilde{\rho})\chi^{(2)}(\underline{\rho})^2 \cdot \varphi_2\right)
\end{pmatrix} \aru &
\end{tikzcd}
\end{equation}
Here $\chi^{(1)}$ and $\chi^{(2)}$ are the opposite 1-cocycles defined in Definition <ref>. Then $\{\Upsilon_{\widetilde{\rho}}\}_{\rho\in \widetilde{G}}$ satisfies the following cocycle condition for $\widetilde{\rho},\widetilde{\rho}'\in \widetilde{G}$.
\begin{equation}\label{Picocycle}
\begin{tikzcd}
\O_{(T')^\an}^{\oplus 2} \arrow[d,"\Theta_{\widetilde{\rho}'\widetilde{\rho}}"] \arrow[r,"\Theta_{\widetilde{\rho}}"] & (\underline{\rho}^{-1})_*\O_{(T')^\an}^{\oplus 2}\arrow[d,"(\underline{\rho}^{-1})_*\Theta_{\widetilde{\rho}'}"] \\
((\underline{\rho}'\underline{\rho})^{-1})_*\O_{(T')^\an}^{\oplus 2} \arrow[r,equal] & (\underline{\rho}^{-1})_*(\underline{\rho}')^{-1}_*\O_{(T')^\an}^{\oplus 2}
\end{tikzcd}
\end{equation}
By the cocycle condition, $\Theta_{\widetilde{\rho}}: \O_{(T')^\an}(T^{\circ})^{\oplus 2}\rightarrow \O_{(T')^\an}(T^{\circ})^{\oplus 2}$ defines a $\widetilde{G}$-action on $\O_{(T')^\an}(T^{\circ})^{\oplus 2}$.
The main purpose of this subsection is to prove the following.
For $\rho\in \widetilde{G}$, the following diagram commutes.
\begin{equation}
\begin{tikzcd}
\Qc_\omega(T^{\circ})\arrow[r,"\Dc"]\arrow[d,"\Upsilon_\rho"] & \O_{(T')^\an}(T^{\circ})^{\oplus 2}\arrow[d,"\Theta_\rho"] \\
\Qc_\omega(T^{\circ}) \arrow[r,"\Dc"] & \O_{(T')^\an}(T^{\circ})^{\oplus 2}
\end{tikzcd}
\end{equation}
We need some preparation for proving Proposition <ref>. First, we define some differential operators twisted by $\underline{G}$-action.
For $\underline{\rho}\in \underline{G}$, we define differential operators $\Dc_i^{\underline{\rho}}$ for $i=1,2$ as follows.
\begin{equation}
\begin{aligned}
&\Dc^{\underline{\rho}}_1 = a'(1-a') \frac{\del^2}{(\del a')^2} +(1-2a')\frac{\del}{\del a'} -\frac{1}{4} \\
&\Dc^{\underline{\rho}}_2 = b'(1-b') \frac{\del^2}{(\del b')^2} +(1-2b')\frac{\del}{\del b'} -\frac{1}{4}
\end{aligned}
\end{equation}
where $a' = \underline{\rho}^\sharp(a)$ and $b' = \underline{\rho}^\sharp(b)$. Furthermore, we define $\Dc^{\underline{\rho}} = {}^t\left(\Dc_1^{\underline{\rho}}\:\: \Dc_2^{\underline{\rho}}\right): \O_{(T')^\an}\rightarrow \O_{(T')^\an}^{\oplus 2}$. By definition, for $\underline{\rho}\in \underline{G}$ and a local section $\varphi$ of $\O_{(T')^\an}$, we have $\Dc_i^{\underline{\rho}}(\underline{\rho}^\sharp(\varphi)) = \underline{\rho}^\sharp(\Dc_i\varphi)$ for $i= 1,2$. Hence the following commutes.
\begin{equation}\label{Drhotransdiag}
\begin{tikzcd}
\O_{(T')^\an} \arrow[r,"\Dc^{\underline{\rho}}_i"]\arrow[d,"(\underline{\rho}^{-1})^\sharp"] &[20pt] \O_{(T')^\an}\arrow[d,"(\underline{\rho}^{-1})^\sharp"] \\
(\underline{\rho}^{-1})_*\O_{(T')^\an} \arrow[r,"(\underline{\rho}^{-1})_*\Dc_i"'] & (\underline{\rho}^{-1})_*\O_{(T')^\an}
\end{tikzcd}
\end{equation}
We prove transformation formulae for $\Dc$. Since $\Dc_i$ is the “pull-back" of the hypergeometric differential operator $L$ in Definition <ref>, the following proposition is a key for the proof of the transformation formulae.
For $\underline{\tau}\in \underline{H}$, we define $L^{\underline{\tau}}:\O_{(S')^\an}\rightarrow \O_{(S')^\an}$ as follows.
\begin{equation}\label{twistL}
L^{\underline{\tau}} = c'(1-c')\frac{d^2}{(dc')^2} + (1-2c')\frac{d}{dc'} -\frac{1}{4}
\end{equation}
where $c' = \underline{\tau}^\sharp(c)$. Then we have the following relation in the ring of differential operators on $(S')^\an$.
\begin{equation}\label{Ltrans}
L^{\underline{\tau}} \cdot \phi(\underline{\tau}) = \phi(\underline{\tau})^3\cdot L
\end{equation}
Here we regard $\phi(\underline{\tau})\in A'$ as a differential operator by multiplication.
It is enough to prove the following.
\begin{equation}\label{enoughtoshowLtrans}
L^{\underline{\tau}} = \phi(\underline{\tau})^3\cdot L \cdot \phi(\underline{\tau})^{-1}
\end{equation}
To compute the right hand side of (<ref>), we need the explicit description of $\phi(\underline{\tau})$. By the relation $\phi_0 = \sgn\cdot \phi^2$ in Proposition <ref>, we can compute $\phi(\underline{\tau})$ up to $\pm 1$. The result is given by the following Table <ref>.
The opposite 1-cocycle $\phi$
$\underline{\tau}_0$ $\underline{\tau}^\sharp(c) $ $\phi(\underline{\tau}) $ $\underline{\tau}_0$ $\underline{\tau}^\sharp(c) $ $\phi(\underline{\tau}) $
$\id$ $c $ $ \pm 1 $ $(0\:1)$ $1-c $ $\pm 1 $
$(1\:\infty)$ $\frac{c}{c-1}$ $ \pm\sqrt{-1} \sqrt{1-c}$ $(0\:1\:\infty)$ $ \frac{1}{1-c}$ $ \pm\sqrt{-1} \sqrt{1-c}$
$(0\:\infty)$ $\frac{1}{c}$ $\pm\sqrt{-1} \sqrt{c}$ $(0\:\infty\:1)$ $\frac{c-1}{c}$ $ \pm\sqrt{-1} \sqrt{c}$
Thus we will compute $L\cdot \frac{1}{\sqrt{c}}$ and $L\cdot \frac{1}{\sqrt{1-c}}$. Using $\frac{d}{dc}\cdot c^\alpha = \alpha c^{\alpha-1} + c^\alpha \cdot \frac{d}{dc}$, we have
\begin{equation}
\begin{aligned}
&-(\sqrt{c})^3 L\cdot \frac{1}{\sqrt{c}} = -c^2(1-c)\frac{d^2}{dc^2}+c^2\frac{d}{dc}-\frac{1}{4} \\
&-(\sqrt{1-c})^3L\cdot \frac{1}{\sqrt{1-c}} = -c(1-c)^2\frac{d^2}{dc^2}-(1-c)^2\frac{d}{dc} -\frac{1}{4}
\end{aligned}
\end{equation}
We will compute the left hand side of (<ref>). Note that $L^{\underline{\tau}}$ is determined by the image of $\underline{\tau}$ in $\underline{H}_0$ since $\underline{\tau}^\sharp(c)$ depends only on the image of $\underline{\tau}$ in $\underline{H}_0$. Hence it is enough to check (<ref>) for six elements in $\underline{H}_0$. For example, we will check $\underline{\tau}_0 = (1\:\:\infty)$ case. In this case, $c' =\frac{c}{c-1}$, hence we have
\begin{equation}
\begin{aligned}
&\frac{d}{dc'} = \frac{dc}{dc'}\cdot \frac{d}{dc} = -\frac{1}{(c'-1)^2}\cdot \frac{d}{dc} = -(c-1)^2\cdot \frac{d}{dc} \\
&\frac{d^2}{(dc')^2}= \left(-(c-1)^2\cdot \frac{d}{dc}\right)^2 = (c-1)^4\frac{d^2}{dc^2}+2(c-1)^3\frac{d}{dc}.
\end{aligned}
\end{equation}
By substituting $c',\frac{d}{dc'},\frac{d^2}{(dc')^2}$ in (<ref>) by the above differential operators, we get
\begin{equation}
L^{\underline{\tau}} = -c(1-c)^2\frac{d}{dc^2}-(1-c)^2\frac{d}{dc}-\frac{1}{4}
\end{equation}
By the similar calculation, we get Table <ref> and confirm (<ref>) holds.
The differential operator $L^{\underline{\tau}}$
$\underline{\tau}_0$ $ L^{\underline{\tau}}$
$\id$ 2*$c(1-c)\dfrac{d^2}{dc^2} + (1-2c)\dfrac{d}{dc} -\dfrac{1}{4}$
$(1\:\infty)$ 2*$-c(1-c)^2\dfrac{d^2}{dc^2} - (1-c)^2\dfrac{d}{dc} -\dfrac{1}{4}$
$(0\:\infty)$ 2*$-c^2(1-c)\dfrac{d^2}{dc^2} +c^2\dfrac{d}{dc} -\dfrac{1}{4}$
Then we get the transformation formulae for $\Dc_i$.
For $\widetilde{\rho} = (\rho,\zeta) \in \widetilde{G}$, we have the following relations in the ring of differential operators on $(T')^\an$.
\begin{equation}
\begin{aligned}
& \Dc_1^{\underline{\rho}}\cdot \widetilde{\chi}(\widetilde{\rho}) = \widetilde{\chi}(\widetilde{\rho})\chi^{(1)}(\underline{\rho})^2\cdot \Dc_1 \\
& \Dc_2^{\underline{\rho}}\cdot \widetilde{\chi}(\widetilde{\rho}) = \widetilde{\chi}(\widetilde{\rho})\chi^{(2)}(\underline{\rho})^2 \cdot \Dc_2
\end{aligned}
\end{equation}
where we regard $\widetilde{\chi}(\widetilde{\rho}),\chi^{(1)}(\underline{\rho}),\chi^{(2)}(\underline{\rho})$ as differential operators by multiplication.
By Definition <ref> and Definition <ref>, we have
\begin{equation}
\widetilde{\chi}(\widetilde{\rho}) = \widetilde{\sgn}(\widetilde{\rho})\cdot \chi^{(1)}(\underline{\rho})\cdot \chi^{(2)}(\underline{\rho}) = \zeta \cdot pr_1^\sharp(\phi(\underline{\rho}^{(1)}))\cdot pr_2^\sharp(\phi(\underline{\rho}^{(2)})).
\end{equation}
For any section $\varphi\in \O_{(S')^\an}$, $\frac{\del}{\del a} (pr_2^\sharp(\varphi)) = 0$ by definition. Hence $\zeta\cdot pr_2^\sharp(\phi(\underline{\rho}^{(2)}))$ commutes with $\Dc_1^{\underline{\rho}}$. Furthermore, by Proposition <ref>, we have the following relation in the ring of differential operators.
\begin{equation}
\Dc_1^{\underline{\rho}}\cdot pr_1^\sharp(\phi(\underline{\rho}^{(1)})) = pr_1^\sharp(\phi(\underline{\rho}^{(1)})^3) \cdot \Dc_1
\end{equation}
Since $\chi^{(1)} = pr_1^\sharp(\phi)$, we have $\Dc_1^{\underline{\rho}}\cdot \widetilde{\chi}(\widetilde{\rho}) = \widetilde{\chi}(\widetilde{\rho})\chi^{(1)}(\underline{\rho})^2 \cdot \Dc_1$. We can prove $\Dc_2$ case similarly.
Finally, we can prove the $\widetilde{G}$-equivariance of $\Dc$.
(Proposition <ref>) For $\widetilde{\rho} = (\rho,\zeta)\in \widetilde{G}$ and $i=1,2$, the following diagram commutes by Proposition <ref> and (<ref>) in Definition <ref>.
\begin{equation}
\begin{tikzcd}
\O_{(T')^\an} \arrow[r,"\widetilde{\chi}(\widetilde{\rho})"] \arrow[d,"\Dc_i"']&[30pt] \O_{(T')^\an} \arrow[r,"(\underline{\rho}^{-1})^\sharp"]\arrow[d,"\Dc^{\underline{\rho}}_i"] &[30pt] (\underline{\rho}^{-1})_*\O_{(T')^\an} \arrow[d,"(\underline{\rho}^{-1})_*\Dc_i"] \\
\O_{(T')^\an} \arrow[r,"\widetilde{\chi}(\widetilde{\rho})\chi^{(i)}(\underline{\rho})^2"] & \O_{(T')^\an} \arrow[r,"(\underline{\rho}^{-1})^\sharp"] & (\underline{\rho}^{-1})_*\O_{(T')^\an}
\end{tikzcd}
\end{equation}
Hence we see that the whole rectangle of the following diagram commutes.
\begin{equation}
\begin{tikzcd}
\O_{(T')^\an}\arrow[d,"\Upsilon_{\widetilde{\rho}}"]\arrow[r,twoheadrightarrow] & \Qc_\omega \arrow[r,"\Dc"] \arrow[d,"\Upsilon_{\widetilde{\rho}}"]&[20pt] \O_{(T')^\an}^{\oplus 2} \arrow[d,"\Theta_{\widetilde{\rho}}"]\\
(\underline{\rho}^{-1})_*\O_{(T')^\an} \arrow[r,twoheadrightarrow] & (\underline{\rho}^{-1})_*\Qc_\omega \arrow[r,"(\underline{\rho}^{-1})_*\Dc"] & (\underline{\rho}^{-1})_*\O_{(T')^\an}^{\oplus 2}
\end{tikzcd}
\end{equation}
Since $\O_{(T')^\an}\rightarrow \Qc_\omega$ is an epimorphism and the left square commutes by definition, the right square commutes. By taking global section at $T^\circ$, we have the result.
§.§ The proof of the main theorem
Finally, we prove the main theorem by describing the image of $\Dc\circ R_\omega(\Xi)$ explicitly.
Let $R_\omega : \Xi^\can \rightarrow \Qc_\omega(T^{\circ})$ be the relative transcendental regulator map in Definition <ref>. For $\xi_0,\xi_1,\xi_\infty \in \Xi^\can$, we have
\begin{equation}\label{generalregvaleqn}
\Dc\circ R_\omega(\xi_0) = \frac{2}{a-b}
\begin{pmatrix}
1 \\[1.5ex]
\end{pmatrix}
\Dc\circ R_\omega(\xi_1) = \frac{2}{a-b}
\begin{pmatrix}
\frac{\sqrt{1-b}}{\sqrt{1-a}}\\[1.5ex]
\end{pmatrix}
\Dc\circ R_\omega(\xi_\infty) = \frac{2}{a-b}
\begin{pmatrix}
\frac{\sqrt{b}}{\sqrt{a}} \\[1.5ex]
\end{pmatrix}
\end{equation}
By Proposition <ref>, we have
\begin{equation}\label{keisanrei1}
\Dc\circ R_\omega(\xi_1-\xi_0) = \Dc(2[\Lc]) = \frac{2}{a-b}
\begin{pmatrix}
\frac{\sqrt{1-b}}{\sqrt{1-a}} - 1 \\[1.5ex]
1- \frac{\sqrt{1-a}}{\sqrt{1-b}}
\end{pmatrix}
\end{equation}
where $[\Lc]\in \Qc_\omega(T^{\circ})$ is the element defined in Proposition <ref>. Let $\widetilde{\rho}^a,\widetilde{\rho}^b \in \widetilde{I}$ be elements defined in Example <ref>. By Proposition <ref> and Proposition <ref>, $\Dc\circ R_\omega$ is equivariant to $\widetilde{G}$-actions. By the cocycle computation in Example <ref>, we have
\begin{equation}\label{keisanrei2}
\begin{aligned}
&\Dc\circ R_\omega(\xi_{0}+\xi_{1}) = \Dc \circ R_\omega(\widetilde{\rho}^a_*(\xi_{1}-\xi_0)) = \frac{2}{a-b}
\begin{pmatrix}
1+ \frac{\sqrt{1-b}}{\sqrt{1-a}} \\[1.5ex]
\end{pmatrix}
\\
&\Dc\circ R_\omega(\xi_0 - \xi_\infty) = \Dc\circ R_\omega(\widetilde{\rho}^b_*(\xi_{1}-\xi_0)) = \frac{2}{(1-a)-(1-b)}
\begin{pmatrix}
\frac{\sqrt{b}}{\sqrt{a}}-1 \\[1.5ex]
\end{pmatrix}
\end{aligned}
\end{equation}
From (<ref>) and (<ref>), we can deduce the result.
Finally, we can prove the main result. The proof of Theorem <ref> below is simplified by advice from T. Terasoma.
Let $\Xi\subset \CH^2(\X^{\circ},1)$ be the higher Chow subgroup defined in Definition <ref> and $\Xi_t\subset \CH^2(\X_t,1)$ be the restriction of $\Xi$ at the fiber over $t\in T^\circ(\C)$.
* Let $R_\omega: \Xi\rightarrow \Qc_\omega(T^{\circ})$ be the relative transcendental regulator map defined in Definition <ref>. Then we have
\begin{equation}
\rank R_\omega(\Xi) = 18.
\end{equation}
* Let $r:\CH^2(\X_t,1)\rightarrow H^{2,0}(\X_t)^\vee/H_2(\X_t,\Z)$ be the transcendental regulator map. Then we have
\begin{equation}
\rank r(\Xi_t) = 18
\end{equation}
for very general $t\in T^{\circ}(\C)$. Especially, we have the following inequality for very general $t\in T^{\circ}(\C)$.
\begin{equation}
\rank \CH^2(\X_t,1)_\ind \ge 18
\end{equation}
(1) Since $\Dc: \Qc_\omega(T^\circ)\rightarrow \O_{(T')^\an}(T^\circ)^{\oplus 2}$ is $\Q$-linear, it is enough to show $\rank \Dc\circ R_\omega(\Xi)\ge 18$ because we already know $\rank \Dc\circ R_\omega(\Xi)\le 18$ by Proposition <ref>. Since $\Xi$ is the sum of $\widetilde{\rho}_*\Xi^\can$, $\Dc\circ R_\omega(\Xi)$ is generated by
\begin{equation}\label{upsilonimage2}
\Dc\circ R_\omega(\widetilde{\rho}_*\Xi^\can) = \Theta_{\widetilde{\rho}} \left(\Dc\circ R_\omega(\Xi^\can)\right)\quad (\widetilde{\rho}\in \widetilde{G}).
\end{equation}
Here we use $\widetilde{G}$-equivariance of $\Dc\circ R_\omega$. Since $\Xi^\can$ is generated by $\xi_0,\xi_1$ and $\xi_\infty$,
\begin{equation}\label{upsilonimage}
\Theta_{\widetilde{\rho}} \left(\Dc\circ R_\omega(\xi_0)\right),\quad \Theta_{\widetilde{\rho}} \left(\Dc\circ R_\omega(\xi_1)\right),\quad \Theta_{\widetilde{\rho}} \left(\Dc\circ R_\omega(\xi_\infty)\right)
\end{equation}
are generators of (<ref>). By the definition of $\Theta_{\widetilde{\rho}}$ and Proposition <ref>, we can calculate (<ref>) for each $\widetilde{\rho}\in \widetilde{G}$. Since $\widetilde{G}_\fib\widetilde{I}$ stabilize $R_\omega(\Xi^\can)$, it is enough to calculate (<ref>) for six representatives of $\widetilde{G}/\widetilde{G}_\fib\widetilde{I}$. By Proposition <ref>, if we take lifts of $(\id,\id)$, $(\id,(0\:1))$, $(\id,(1\:\infty))$, $(\id,(0\:1\:\infty))$, $(\id,(0\:\infty))$ and $(\id,(0\:\infty\:1)) \in \underline{G}_0$ by $\widetilde{G}\rightarrow \underline{G}_0$, they become a complete system of representatives for $\widetilde{G}/\widetilde{G}_\fib\widetilde{I}$. Then we calculate (<ref>) for these lifts, we get the following Table <ref>.
The generators of the image of $\Xi$ under $\Dc\circ R_\omega$
The image in $\underline{G}_0$ generators of $\Theta_{\widetilde{\rho}} \left(\Dc\circ R_\omega(\Xi^\can)\right)$
\begin{pmatrix}
1 \\[1.5ex]
\end{pmatrix}
\begin{pmatrix}
\dfrac{\sqrt{1-b}}{\sqrt{1-a}} \\[1.5ex]
\end{pmatrix}$
\begin{pmatrix}
\dfrac{\sqrt{b}}{\sqrt{a}} \\[1.5ex]
\end{pmatrix}$
\begin{pmatrix}
1 \\[1.5ex]
\end{pmatrix}
\begin{pmatrix}
\dfrac{\sqrt{b}}{\sqrt{1-a}} \\[1.5ex]
\end{pmatrix}$,
\begin{pmatrix}
\dfrac{\sqrt{1-b}}{\sqrt{a}} \\[1.5ex]
\end{pmatrix}$
\begin{pmatrix}
\sqrt{1-b} \\[1.5ex]
\dfrac{1}{\sqrt{1-b}}
\end{pmatrix}
\begin{pmatrix}
\dfrac{1}{\sqrt{1-a}} \\[1.5ex]
\sqrt{1-a}
\end{pmatrix}$
\begin{pmatrix}
\dfrac{\sqrt{b}}{\sqrt{a}} \\[1.5ex]
\end{pmatrix}$
\begin{pmatrix}
\sqrt{b} \\[1.5ex]
\dfrac{1}{\sqrt{b}}
\end{pmatrix}
\begin{pmatrix}
\dfrac{1}{\sqrt{1-a}} \\[1.5ex]
\sqrt{1-a}
\end{pmatrix}$,
\begin{pmatrix}
\dfrac{\sqrt{1-b}}{\sqrt{a}} \\[1.5ex]
\end{pmatrix}$
\begin{pmatrix}
\sqrt{b} \\[1.5ex]
\dfrac{1}{\sqrt{b}}
\end{pmatrix}
\begin{pmatrix}
\dfrac{\sqrt{1-b}}{\sqrt{1-a}} \\[1.5ex]
\end{pmatrix}$
\begin{pmatrix}
\dfrac{1}{\sqrt{a}} \\[1.5ex]
\sqrt{a}
\end{pmatrix}$
\begin{pmatrix}
\sqrt{1-b} \\[1.5ex]
\dfrac{1}{\sqrt{1-b}}
\end{pmatrix}
\begin{pmatrix}
\dfrac{\sqrt{b}}{\sqrt{1-a}} \\[1.5ex]
\end{pmatrix}$
\begin{pmatrix}
\dfrac{1}{\sqrt{a}} \\[1.5ex]
\sqrt{a}
\end{pmatrix}$
It is enough to show that the vectors in Table <ref> are linearly independent over $\Q$. It is enough to show that the first component of these vectors are linearly independent over $\C$. Note that the first component of these vectors are written in the form of
\begin{equation}
c\cdot F_1\cdot F_2
\end{equation}
where $c\in \{\pm2, \pm2\sqrt{-1}\}$, $F_1$ is either
\begin{equation}\label{F1}
\dfrac{1}{a-b},\:\:\dfrac{1}{a+b-1},\:\:\dfrac{1}{ab-a-b},\:\:\dfrac{1}{ab-b+1},\:\:\dfrac{1}{ab-1}\text{ or }\dfrac{1}{a-ab-1} \in \Frac(B)
\end{equation}
and $F_2$ is either
\begin{equation}\label{F2}
1,\:\dfrac{\sqrt{b}}{\sqrt{a}},\:\dfrac{\sqrt{1-b}}{\sqrt{1-a}},\:\dfrac{\sqrt{1-b}}{\sqrt{a}},\:\dfrac{\sqrt{b}}{\sqrt{1-a}},\:\dfrac{1}{\sqrt{1-a}},\: \sqrt{1-b},\:\dfrac{1}{\sqrt{a}}\text{ or }\sqrt{b} \in \Frac(B').
\end{equation}
Since elements in (<ref>) are linearly independent over $\C$ and elements in (<ref>) are linearly independent over $\Frac(B)$, their products are linearly independent over $\C$. Hence we have the result.(2) By Lemma <ref>, we have $\rank m_t(R_\omega(\Xi)) = 18$ for very general $t\in T^\circ(\C)$. By the definition of relative transcendental regulator map, we see that $\rank \ev_t\circ r(\Xi_t) = 18$ in this case. Since $\ev_t$ is an isomorphism, we have $\rank r(\Xi_t) = 18$ for very general $t\in T^\circ(\C)$. The statement about indecomposable part follows from Proposition <ref>.
§ DECOMPOSABLE CYCLES IN HIGHER CHOW GROUP
In this section, we assume $X$ is a smooth variety over a field $k$. We define a subgroup $\CH^p(X,q)_\dec\subset \CH^p(X,q)$ called decomposable part.
For $p,p',q,q'\ge 0$, there exists a bilinear map
\begin{equation}
\CH^p(X,q)\times \CH^{p'}(X,q')\lra \CH^{p+p'}(X,q+q')
\end{equation}
called the intersection product. The intersection product is the composition of the external product $\CH^p(X,q)\times \CH^{p'}(X,q')\rightarrow \CH^{p+p'}(X\times_k X,q+q')$ and the pull-back by the diagonal embedding $X\rightarrow X\times_k X$.For $p,q\ge 0$, we define a subgroup $\CH^p(X,q)_{\dec}\subset\CH^p(X,q)$ by
\begin{equation}
\CH^p(X,q)_{\dec} = \sum_{\substack{s,t}} \mathrm{Im}\left(\CH^s(X ,t)\otimes_{\Z} \CH^{p-s}(X, q-t)\rightarrow \CH^p(X,q) \right)
\end{equation}
where $(s,t)$ runs over $0\le s \le p$, $0\le t \le q$ except $(s,t) = (0,0), (p,q)$ and $\CH^s(X ,t)\otimes_{\Z} \CH^{p-s}(X, q-t)\rightarrow \CH^p(X,q) $ is the map induced by the intersection product. Elements in $\CH^p(X,q)_{\dec}$ are called decomposable cycles. We define
\begin{equation}
\CH^p(X,q)_\ind = \CH^p(X,q)/\CH^p(X,q)_\dec.
\end{equation}
We describe the decomposable part of $\CH^2(X,1)$. Recall that an element of $\CH^2(X,1)$ is represented by an element in $\Ker\left(\bigoplus_{Z\in X^{(1)}}R(Z)^\times\xrightarrow{\div} \bigoplus_{p\in X^{(2)}}\Z\cdot p\right)$ as in Proposition <ref>.
An element of $\CH^2(X,1)_\dec$ can be represented by $\sum_\lambda(Y_\lambda,c_\lambda)\in \bigoplus_{Z\in X^{(1)}}R(Z)^\times$ such that $c_\lambda \in \Gamma(X,\O_X^\times)$.
Since $\CH^0(X,1) = 0$, $\CH^2(X,1)_\dec$ is the image of the map
\begin{equation}
\CH^1(X,1)\otimes_\Z \CH^1(X,0) \lra \CH^2(X,1)
\end{equation}
By [10] Section 8, the external product $\CH^1(X,1)\times \CH^1(X)\rightarrow \CH^2(X\times_k X,1)$ is induced by the following map.
\begin{equation}\label{external}
\begin{tikzcd}[row sep = tiny]
Z^1(X,1)\times Z^1(X)\arrow[r] & Z^2(X\times_k X,1); &[-30pt]([V],[W]) \arrow[r,mapsto]& \mathrm{[}V\times_k W\mathrm{]}
\end{tikzcd}
\end{equation}
where $V\subset X\times_k \Delta^1\:(\Delta^1 = \Spec k[T_0,T_1]/(T_0+T_1-1)), W\subset X$ are integral closed subschemes of codimension 1 and $[V], [W], [V\times_k W]$ denote the cycles corresponding to $V,W,V\times_k W$. Recall that we regard elements in $\Gamma(X,\O_X^\times)$ as cycles in $Z^1(X,1)$ by considering their graphs. Hence we can check that the external product of the graph of $c \in \Gamma(X,\O_X^\times)$ and an integral codimension 1-cycle $V$ intersects properly with the image of the diagonal embedding in $Z^2(X\times_k X,1)$. Moreover their intersection is the graph of $c$ on $V$. Hence we have the result.
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|
# Invariance of $\phi^{4}$ measure under nonlinear wave and Schrödinger
equations on the plane
Nikolay Barashkov, Petri<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematics and Statistics, University
of Helsinki
###### Abstract
We show probabilistic existence and uniqueness for the Wick-ordered cubic
nonlinear wave equation in a weighted Besov space over $\mathbb{R}^{2}$. To
achieve this, we show that a weak limit of $\phi^{4}$ measures on increasing
tori is invariant under the equation. We review and slightly simplify the
periodic theory and the construction of the weak limit measure, and then use
finite speed of propagation to reduce the infinite-volume case to the previous
setup. Our argument also gives a weak (Albeverio–Cruzeiro) invariance result
on the nonlinear Schrödinger equation in the same setting.
MSC (2020): 35L71, 60H30 (Primary); 35Q55, 60H15, 81T08
Keywords: nonlinear wave equation, nonlinear Schrödinger equation, invariant
measure, stochastic quantization, $\phi^{4}$ measure, weighted Besov space
## 1 Introduction
Since Jean Bourgain’s work in the 1990s, invariant measures have been an
important tool in probabilistic solution theory of dispersive PDEs. Bourgain
initially considered the nonlinear Schrödinger equation
$i\partial_{t}u(x,t)+\Delta u(x,t)=\pm\lambda\,{:\\!u^{3}\\!:}$ (NLS)
on one-dimensional torus $\mathbb{T}$ [8]. He proved almost sure well-
posedness in the Sobolev space $H^{1/2-\varepsilon}$ when the initial data is
distributed according to a so-called $\phi^{4}$ measure. Later on in [9], he
extended the result to $\mathbb{T}^{2}$. In two or more dimensions the
$\phi^{4}$ measure is supported on distributions, and it then becomes
necessary to renormalize the nonlinearity $u^{3}$ by Wick ordering, denoted by
${:\\!u^{3}\\!:}$.
Our main subject is the defocusing massive nonlinear wave equation
$\partial_{tt}u(x,t)+(m^{2}-\Delta)u(x,t)=-\lambda\,{:\\!u^{3}\\!:}$ (NLW)
on two-dimensional spatial domain $\mathbb{R}^{2}$. This equation was
previously solved on periodic domain by Oh and Thomann [46]. The main result
of this article can be stated as follows:
###### Theorem 1.1 (Global existence and uniqueness).
Let $\mu$ be the product of infinite-volume $\phi^{4}$ and white-noise
measures, and $\varepsilon>0$. For $\mu$-almost all initial data and any
$T>0$, the nonlinear wave equation (NLW) has a unique global solution in the
weighted space $L^{2}([0,T];\;H^{-2\varepsilon}(\rho))$.
Our approach is to first construct solutions in periodic domains
$[{-L},{L}]^{2}$, and then approximate infinite-volume solutions with them.
The high-level proof strategy in a periodic domain goes back to Bourgain:
1. 1.
Define a probability distribution on the initial data.
2. 2.
Prove deterministic well-posedness for time interval $[0,\tau]$ when the
initial data belongs to some set $A$ of large probability. The small time
$\tau$ depends on the size of $A$.
3. 3.
Prove that the probability measure is invariant in time under the equation.
4. 4.
Intersect the sets of initial and final values, which have same probability by
invariance. By iteration, the probability of blow-up by time $T=n\tau$ is
bounded by $n(1-\mathbb{P}(A))$.
5. 5.
Use stochastic estimates to show that an increase of $\mathbb{P}(A)$ cancels
the corresponding increase of iterations $n$; thus the probability of blow-up
can be made arbitrarily small.
This argument reduces the global solution theory into understanding the
invariance and large deviations of a suitable probability measure. To show
invariance, we use finite-dimensional approximation. Liouville’s theorem
states that the Gibbs measure associated to the Hamiltonian formulation of
(NLW) is invariant. These approximate measures converge in total variation to
the untruncated, periodic-domain measures.
The extension to infinite volume relies on two insights. The first is the
existence of uniform bounds for the measures in a polynomially weighted space
[38]. This yields a convergent subsequence of measures as $L\to\infty$. The
second step is to use the finite speed of propagation of (NLW) to reduce all
statements about measurable events to the periodic case.
### 1.1 The $\phi^{4}$ measure
What is the natural candidate for the invariant measure? As mentioned above,
Fourier-truncated versions of these equations converve the Hamiltonian $H$,
with which we can define the Gibbs measure proportional to $\exp(-\beta H)$.
The parameter $\beta>0$ is called the inverse temperature. For truncated and
periodic (NLW), the Gibbs measure has density222In the following, we set
$\beta=\lambda=1$ as they are not too relevant for our present topic.
$\exp\left(-\beta\int_{\mathbb{T}^{d}}\frac{\lambda\,{:\\!u^{4}\\!:}}{4}+\frac{m^{2}{\left|u\right|}^{2}+{\left|\nabla
u\right|}^{2}+{\left|\partial_{t}u\right|}^{2}}{2}\,\mathrm{d}x\right)$ (1.1)
with respect to Lebesgue measure on the Fourier coefficients. The second term
yields a Gaussian factor that can be utilized in removing the truncation.
The continuum versions of these Gibbs measures are studied in constructive
quantum field theory [23]. Stochastic quantization (see e.g. [50]) is a
rigorous PDE approach for their study. In this approach the $\phi^{4}_{d}$
measure is regarded as an invariant measure for a nonlinear heat equation with
white noise forcing. These equations are singular and cannot be solved
classically.
The periodic $\phi^{4}_{2}$ equation was solved by Da Prato and Debussche
[19]. The limit measure is absolutely continuous with respect to a Gaussian
measure. Existence of infinite-volume solutions for the 2D equation was later
shown by Mourrat and Weber in a polynomially weighted space [38]; see also
[37]. We will rely heavily on these ideas in Section 3.
The local well-posedness theory for the more singular 3D case came in three
approaches in mid-2010s: Hairer’s regularity structures [29]; Gubinelli,
Imkeller and Perkowski’s [25] paracontrolled distributions; and Kupiainen’s
renormalization group approach [33]. The bounds of Mourrat and Weber were then
exploited by Albeverio and Kusuoka [3] and Gubinelli and Hofmanová [24] to
give a self-contained construction of the $\phi^{4}_{3}$ measure.
In dimensions $d\geq 4$, the $\phi^{4}_{d}$ measures collapse to trivial
Gaussian measures. The final case $d=4$ was proved recently by Aizenman and
Duminil-Copin; see their article [1] for discussion.
The $\phi^{4}$ measure is expected to be invariant under three PDEs that share
essentially the same Hamiltonian: (NLS), (NLW), and the cubic stochastic
nonlinear heat equation. As shown in [13, Figure 1], the periodic-domain
invariance theory is almost done, with only the three-dimensional Schrödinger
missing.
This theory, and hence the global well-posedness of the equations, is much
less developed in the infinite volume. For wave and Schrödinger equations the
previous results are limited to one dimension [10] or radial setting [56].
The largest complication is that the infinite-volume $\phi^{4}$ measures are
only defined as weak limits of approximating sequences, and in particular they
are no longer absolutely continuous with respect to Gaussian measure. This
means that total variation convergence is no longer available and we have to
prove local well-posedness for non-Gaussian initial data. Depending on the
coupling constant $\lambda$, the accumulation point of the approximating
sequence might not be unique.
However, we are able to show that our invariant distribution can still be
coupled to a Gaussian and the perturbation term enjoys better analytic bounds.
A similar fact was exploited by Bringmann and collaborators in [11, 12, 13],
in situations where the singularity of the measure arises in finite volume due
to short scale divergencies.
For the nonlinear Schrödinger equation the situation is even more complicated,
as there is no finite speed of propagation. This means that we cannot reduce
the problem to the periodic setup. However, by giving up some
differentiability and thus uniqueness, we can still prove a weaker form of
invariance. This sense of invariance was initially developed for Euler and
Navier–Stokes equations by Albeverio and Cruzeiro [2], and was explored in the
case of periodic 2D NLS in [45].
As this manuscript was being prepared, Oh, Tolomeo, Wang, and Zheng published
their preprint [47] where similar ideas appear. They prove Theorem 1.1 for a
more challenging equation, (NLW) with additive stochastic forcing. Their
approach is based on an optimal transport argument developed in [39], whereas
our globalization argument depends more heavily on finite speed of
propagation. There are also slight differences in the use of stochastic
quantization, and we also consider the Schrödinger equation.
### 1.2 Previous literature and extensions
Let us take a moment here to review the history of this question. As mentioned
above, the general strategy to prove probabilistic well-posedness was
developed by Bourgain [8] in context of one-dimensional periodic (NLS). This
was in response to earlier work of Lebowitz, Rose, and Speer [34] in late
1980s.
Invariant measures for the one-dimensional wave equation were considered by
Zhidkov [57] and McKean and Vaninsky [36]. Radially symmetric (NLW) on a
three-dimensional ball was considered by Burq and Tzvetkov [17], and extended
by Xu to infinite volume [56]. Recently progress has been made in three
dimensions, culminating in the proof of invariance of periodic $\Phi^{4}_{3}$
under the wave equation [11, 12, 13].
NLW has also been considered with random data not sampled from the invariant
measure [30, 32]. Related to the invariance of Gibbs measures is the program
for showing quasi-invariance of Gaussian measures under Hamiltonian PDEs [55];
in this notion the distribution of solutions at any given time remains
absolutely continuous with respect to the initial measure. For the wave
equation this was carried out in [28, 49].
Another related development is the solution theory for (NLW) with additive
white noise forcing. This was achieved for the 2D cubic wave equation in [26]
and extended to global well-posedness in [27, 54]. The preprint [47] of Oh,
Tolomeo, Wang, and Zheng considers this case. If the equation also includes
dispersion, the invariant measure is moreover ergodic [53].
The nonlinearity can be replaced by a general polynomial or exponential term
as in [40, 47]. It is also possible to let the solution take values in a
manifold instead of $\mathbb{R}$. The invariant measures for these wave maps
equations [14, 16] are known as nonlinear sigma models in the physics
literature. In one dimension they can be interpreted as Brownian paths on a
manifold.
For (NLS) in one dimension it is possible to consider both focusing and
defocusing nonlinearities, due to the presence of an $L^{2}$ conservation law.
Restricting to a ball in $L^{2}$ leads to normalizable measure if the
nonlinearity is subquintic. In the quintic case the measure is normalizable
only if and only if the coupling is suffiently weak; remarkably, this
threshold is known exactly [43].
In two dimensions the defocusing case can still be investigated, as was done
by Bourgain [9] for the cubic case and later for general polynomial
nonlinearities by Deng, Nahmod, and Yue [20]. For the focusing NLS the $L^{2}$
cutoff does not lead to normalizable measure anymore [15]. Quasi-invariance
has also been investigated for the NLS [41, 44, 48].
In [42, 51] invariant measures of the Zakharov–Yukawa system have been
studied. This is a system of coupled wave and Schrödinger equations with
nonpositive Hamiltonian and an $L^{2}$ conservation law. Due to these
properties it behaves similarly to the defocusing NLS.
The activity described above has mostly taken place on the torus. In infinite
volume we mention the early result of Bourgain on one-dimensional NLS [10], as
well as the work of Cacciafesta and Suzzoni on the NLS and other Hamiltonian
equations [18]. These are in addition to the aforementioned papers [47, 56] on
two- and three-dimensional NLW.
Let us conclude this review with a comment on possible extensions of our work
and open problems. Our method extends in a straightforward way to more general
polynomial nonlinearities and to vector-valued models.
###### Example 1.2.
The mass term $m^{2}>0$ in (NLW) and (NLS) is used to avoid problems with the
zero Fourier mode. Still, negative mass is interesting to consider. One can
modify a negative-mass equation
$\partial_{tt}u(x,t)-(m^{2}+\Delta)u(x,t)=-\lambda\,{:\\!u^{3}\\!:}$
by adding $2m^{2}\,u(x,t)$ to both sides of the equation. Then the
nonlinearity will be of form $-\lambda\,{:\\!u^{3}\\!:}+2m^{2}u$, which is
still dominated by the cubic term.
For the weak invariance we also expect the extension to long-range models
(with fractional Laplacian) to be straightforward, provided the resulting
measures are not too singular. If one can find a suitable replacement for
finite speed of propagation in the wave case, we also expect strong invariance
to follow in a straighforward fashion.
It would be interesting to consider the strong invariance of $P(\phi)_{1}$
theories under 1-dimensional (NLS). The $\phi^{4}$ case has been solved by
Bourgain [10], but his argument does not apply to higher-order polynomial
nonlinearities. Of course the corresponding 2D problem in the full space is
also very interesting, as well as the case of cosine and exponential
nonlinearities.
Given the recent preprint [13] on invariance of three-dimensional periodic
(NLW), it is intriguing to ask about the extension to $\mathbb{R}^{3}$. While
the measure-theoretic part of our argument is dimension-independent, the
analytic estimates would require significant changes to account for the more
singular behaviour.
### 1.3 Outline of the article
Our argument consists of mainly putting together existing pieces, hence the
majority of this article presents the techniques at a more pedestrian pace.
All the estimates happen in Besov spaces, a generalization of Sobolev spaces.
Multiplicative estimates take a particularly nice form in these spaces, and
there are many (sometimes compact) embeddings between the spaces. We collect
the main results in Section 2.
We then collect the stochastic estimates for the measures in Section 3. There
we outline the proof of existence and bounds for the $\phi^{4}$ measure over
polynomially weighted $\mathbb{R}^{2}$ space. In particular, we show the
tightness of the sequence on increasing tori. Some of the calculations are
deferred to appendices.
We solve (NLW) on a periodic domain in Section 4. This argument is originally
due to Oh and Thomann [46], but we slightly simplify the argument by using
only Besov spaces. We also present explicitly the Bourgain-style global
existence argument omitted in the cited article.
The main result in this article is presented in Section 5. We use a measure-
theoretic argument to reduce the full flow to the periodic case, and thus
prove invariance of the infinite-volume $\phi^{4}$ measure.
In Section 6, we finally consider (NLS) on $\mathbb{R}^{2}$. We prove
invariance in Albeverio–Cruzeiro sense with some weaker estimates on the
solutions.
### 1.4 Notation
The weighted Besov spaces $B^{s}_{p,r}(\rho)$ are defined in Section 2 below.
We abbreviate $H^{s}(\rho)\coloneqq B^{s}_{2,2}(\rho)$ and
$\mathcal{C}^{s}(\rho)\coloneqq B^{s}_{\infty,\infty}(\rho)$. Here $\rho$ is a
polynomially decaying weight, the parameter of which may change from section
to section.
We also denote by $B^{s}_{p,r}(A)$ the flat weight on a set
$A\subset\mathbb{R}^{2}$, and abuse notation by writing
$B^{s}_{p,r}(\Lambda_{L})$ for the space of periodic functions. The periodic
domain is denoted by $\Lambda_{L}\coloneqq{[{-L},{L}]}^{2}$.
We use $\mu$ for either the $\phi^{4}_{2}$ measure, or the product of
$\phi^{4}_{2}$ and white noise measures; we use $\mu_{L}$ and $\mu_{L,N}$ for
the bounded-domain and bounded-domain Fourier-truncated versions respectively.
The product measure form is used in Sections 4 and 5. The $\phi^{4}_{2}$
measure is decomposed into a Gaussian free field $Z$ coupled with a more
regular part $\phi$. The full product measure is supported on a space
${\mathcal{H}}^{s}(\rho)\coloneqq H^{s}(\rho)\times H^{s-1}(\rho)$.
Capital $\Phi_{t}$ is reserved for the flow of (NLW). Subscripts similar to
those above are used for bounded and Fourier-truncated equations. We define
linear solution operators ${\mathcal{C}}_{t}$ and ${\mathcal{S}}_{t}$ for
(NLW) and ${\mathcal{T}}_{t}$ for (NLS) in the corresponding sections.
The Littlewood–Paley blocks $\Delta_{k}$ defined below are supported on dyadic
sets. We define the projection $P_{N}$ as a sharp Fourier cutoff to
$B(0,2^{N})$.
We use the general notation of $A\lesssim B$ if $A\leq cB$ for some
independent constant $c$, and $A\simeq B$ for $A\lesssim B\lesssim A$.
Positive constants $c,C$ may vary from line to line. The small constant
$\varepsilon>0$ appears mainly in regularity of the spaces and may change
between sections (but only finitely many times). We use $\delta$ to signify
other small parameters.
### 1.5 Acknowledgements
NB was supported by the ERC Advanced Grant 741487 “Quantum Fields and
Probability”. PL was supported by the Academy of Finland project 339982
“Quantum Fields and Probability”. PL would like to thank Kalle Koskinen,
Jaakko Sinko, and Aleksis Vuoksenmaa for useful conversations. Both authors
would like to thank Leonardo Tolomeo for helpful comments on the preprint.
## 2 Besov spaces
Besov spaces are a generalization of Sobolev spaces that support some useful
multiplication estimates and embeddings. We collect in this section the
necessary resuls, but largely omit the proofs. An excellent introduction to
the topic is in the article of Mourrat and Weber [38]. Some results are also
collected in the appendix of [24]. The textbook of Bahouri, Chemin, and
Danchin [4] treats the unweighted case.
###### Remark 2.1.
There are two conventions of weighted $L^{p}$ spaces in common use. [38] and
[24] respectively define
${\left\|f\right\|}_{L^{p}_{w}}^{p}\coloneqq\int
f(x)^{p}w(x)\,\mathrm{d}x\quad\text{and}\quad{\left\|f\right\|}_{L^{p}(w)}^{p}\coloneqq\int
f(x)^{p}w(x)^{p}\,\mathrm{d}x.$
We use the latter convention since it lets us apply a weight also when
$p=\infty$. For $p<\infty$ the conventions are interchangeable, and the
statements and their proofs require only minor changes.
###### Definition 2.2 (Littlewood–Paley blocks).
We fix $\Delta_{k}$ to be Fourier multipliers that restrict the support of
$\hat{u}$ to a partition of unity. More precisely, for $k\geq 0$ they are
smoothed indicators of the annuli $B(0,2^{k}\,8/3)\setminus B(0,2^{k}\,3/4)$,
and for $k=-1$ of the ball $B(0,3/4)$. The precise choice is irrelevant.
###### Definition 2.3 (Weighted Besov space).
We define the space $B^{s}_{p,r}(w)$ as the completion of
$C_{c}^{\infty}(\mathbb{R}^{d})$ with respect to the norm
${\left\|f\right\|}_{B^{s}_{p,r}(w)}\coloneqq{\left\|2^{ks}\,{\left\|w(x)[\Delta_{k}f](x)\right\|}_{L^{p}}\right\|}_{\ell^{r}}$
where the $L^{p}$ norm is taken over $x\in\mathbb{R}^{d}$ and the $\ell^{r}$
norm over $k\geq-1$. We abbreviate
$H^{s}(w)\coloneqq
B^{s}_{2,2}(w)\quad\text{and}\quad\mathcal{C}^{s}(w)\coloneqq
B^{s}_{\infty,\infty}(w).$
In particular, the space $H^{s}$ coincides with the usual (fractional-order)
Sobolev space, where the norm is defined as the $L^{2}$ norm of function
multiplied by ${\left\langle{\nabla}\right\rangle}^{s}$. One can also replace
the ball $B(0,3/4)$ with annuli for all $k\in-\mathbb{N}$ to define
_homogeneous_ Besov spaces: some of the following results translate to
homogeneous spaces, but the zero Fourier mode is then ignored and the
necessary embeddings are not available.
We will use throughout the article a nonhomogeneous polynomial weight
$\rho(x)\coloneqq{\left\langle{x}\right\rangle}^{-\alpha}\coloneqq(1+{\left|x\right|}^{2})^{-\alpha/2}$
for $\alpha\geq 0$ sufficiently large. What “sufficiently large” means may
vary from section to section, but the final choice is finite. In some sections
we also use the unweighted space ($\alpha=0$); this is indicated by omitting
$\rho$.
The following multiplicative inequality shows that products of distributions
and smooth enough functions are well-defined distributions. A recurring
‘trick’ in the following sections is to decompose stochastic objects into
distributional and more regular parts. There are also analogues of the usual
$L^{p}$ duality and interpolation.
###### Theorem 2.4 (Multiplicative inequality).
Let $s_{1}<s_{2}$ be non-zero such that $s_{1}+s_{2}>0$, and let
$1/p=1/p_{1}+1/p_{2}$. Then
${\left\|fg\right\|}_{B^{s_{1}}_{p,r}(\rho_{1}\rho_{2})}\lesssim{\left\|f\right\|}_{B^{s_{1}}_{p_{1},r}(\rho_{1})}{\left\|g\right\|}_{B^{s_{2}}_{p_{2},r}(\rho_{2})}.$
###### Theorem 2.5 (Duality).
Let $(p,p^{\prime})$ and $(r,r^{\prime})$ be Hölder conjugate pairs, and
$\rho_{1}$ and $\rho_{2}$ polynomial weights. Then
${\left\|fg\right\|}_{L^{1}(\rho_{1}\rho_{2})}\lesssim{\left\|f\right\|}_{B^{s}_{p,r}(\rho_{1})}{\left\|f\right\|}_{B^{-s}_{p^{\prime},r^{\prime}}(\rho_{2})}$
###### Theorem 2.6 (Interpolation).
Fix $\theta\in(0,1)$, $s=\theta s_{1}+(1-\theta)s_{2}$, and
$\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{2}},\quad\frac{1}{r}=\frac{\theta}{r_{1}}+\frac{1-\theta}{r_{2}},\quad\alpha=\theta\beta+(1-\theta)\gamma.$
Then
${\left\|f\right\|}_{B^{s}_{p,r}(\rho^{\alpha})}\leq{\left\|f\right\|}_{B^{s_{1}}_{p_{1},r_{1}}(\rho^{\beta})}^{\theta}{\left\|f\right\|}_{B^{s_{2}}_{p_{2},r_{2}}(\rho^{\gamma})}^{1-\theta}.$
We shall use the following three embedding results. The first lets us trade
smoothness for $L^{p}$ and $\ell^{r}$ regularity. The second plays a crucial
role in the weak convergence argument by letting us pass to a convergent
subsequence in a compact space.
###### Theorem 2.7 (Besov embeddings).
Let $s\in\mathbb{R}$, $1\leq q\leq p\leq\infty$, and
$s^{\prime}\geq s+d\left(\frac{1}{q}-\frac{1}{p}\right).$
Then
${\left\|f\right\|}_{B^{s}_{p,r}(\rho)}\lesssim{\left\|f\right\|}_{B^{s^{\prime}}_{q,r}(\rho)}.$
The usual chain $\ell^{\infty}\subset\ell^{r}\subset\ell^{1}$ applies. For any
$\varepsilon>0$ we also have
${\left\|f\right\|}_{B^{s}_{p,r}(\rho)}\lesssim{\left\|f\right\|}_{B^{s+\varepsilon}_{p,\infty}(\rho)}.$
Finally,
${\left\|f\right\|}_{B^{k}_{p,\infty}(\rho)}\lesssim{\left\|f\right\|}_{W^{k,p}(\rho)}\lesssim{\left\|f\right\|}_{B^{k}_{p,1}(\rho)}$
for all $k\in\mathbb{N}$; in particular, for $k=0$ that corresponds to
$L^{p}$.
###### Theorem 2.8 (Compact embedding).
Let $\rho_{2}$ and $\rho_{1}$ be polynomial weights with respective parameters
$\alpha_{2}>\alpha_{1}>d$, and $s_{2}<s_{1}$. The space
$B^{s_{1}}_{p,r}(\rho_{1})$ then embeds compactly into the less regular space
$B^{s_{2}}_{p,r}(\rho_{2})$.
For the finite-volume results, we also need periodic Besov spaces. The same
multiplicative inequalities work also in this case, and furthermore the
following lemma shows that we can move between periodic and polynomial-weight
spaces easily. We use the Mourrat–Weber [38] definition of these spaces.
###### Definition 2.9 (Periodic Besov space).
Given the set $\Lambda_{L}\coloneqq{[{-L},{L}]}^{d}$, we define the space
$B^{s}_{p,r}(\Lambda_{L})$ as the completion of $2L$-periodic
$C^{\infty}(\mathbb{R}^{d})$ functions with respect to the Besov norm. The
inner $L^{p}$ norm is unweighted but over $\Lambda_{L}$.
###### Lemma 2.10 (Embedding into polynomial-weight space).
Let $\rho$ be a polynomial weight with parameter $\alpha$, and let $f\in
C^{\infty}(\mathbb{R}^{d})$ be $2L$-periodic. Then
${\left\|\rho\right\|}_{L^{1}}{\left\|f\right\|}_{B^{s}_{p,r}(\rho)}\lesssim{\left\|f\right\|}_{B^{s}_{p,r}(\Lambda_{L})}\lesssim
L^{\alpha}{\left\|f\right\|}_{B^{s}_{p,r}(\rho)}.$
###### Proof.
The second inequality follows from
${\left\|f\right\|}_{B^{s}_{p,r}(\Lambda_{L})}\leq\left(\sup_{x\in\Lambda_{L}}\rho(x)^{-1}\right){\left\|f\right\|}_{B^{s}_{p,r}(\rho\mathbf{1}_{\Lambda_{L}})}\lesssim_{\rho}{\left\|f\right\|}_{B^{s}_{p,r}(\rho)}.$
Similarly, if we denote by $\Lambda_{L}^{j}$ the translates $\Lambda_{L}+j2L$,
the first inequality is
${\left\|f\right\|}_{B^{s}_{p,r}(\rho)}\leq\sum_{j\in\mathbb{Z}}{\left\|f\right\|}_{B^{s}_{p,r}(\rho\mathbf{1}_{\Lambda_{L}^{j}})}\leq{\left\|f\right\|}_{B^{s}_{p,r}(\Lambda_{L})}\sum_{j\in\mathbb{Z}}\sup_{x\in\Lambda_{L}^{j}}\rho(x).$
We must still justify that $f$ belongs to the polynomial-weight space.
Let $F$ be the restriction of $f$ to $\Lambda_{L}$, and let $F_{n}$ consist of
$2n+1$ repeats of $F$ along each axis, thus supported on
$[{-(2n+1)L},{(2n+1)L}]^{d}$. Then $F_{n}$ and $f$ coincide in any compact set
of $\mathbb{R}^{d}$ for large enough $n$.
However, $F_{n}$ might not belong to $B^{s}_{p,r}(\rho)$ because we did not
assume $F$ to vanish at the boundary. To fix that issue, we need to multiply
$F_{n}$ by the smoothed indicator of some slightly smaller set; call the
result $\tilde{F}_{n}$. Then the difference $\tilde{F}_{n+1}-\tilde{F}_{n}$
consists of two disjoint $C_{c}^{\infty}$ parts, and these parts are only
translated as $n$ increases.
With any $g\in C_{c}^{\infty}(\mathbb{R}^{d})$, we can approximate
${\left\|g\right\|}_{B^{s}_{p,r}(\rho)}\lesssim{\left\|g\right\|}_{L^{p}(\rho)}+{\left\|D^{\lceil
s\rceil}g\right\|}_{L^{p}(\rho)}.$
These norms are clearly local, so it is easier to see that
${\|\tilde{F}_{n+1}-\tilde{F}_{n}\|}_{B^{s}_{p,r}(\rho)}$ is an $f$-dependent
constant multiplied by $\sup_{x\in\Lambda_{L}^{n}}\rho(x)$. We can then use
triangle inequality to estimate
${\left\|\tilde{F}_{n+m}-\tilde{F}_{n}\right\|}_{B^{s}_{p,r}(\rho)}\lesssim
C_{f}\sum_{j=n}^{\infty}\sup_{x\in\Lambda_{L}^{j}}\rho(x),$
which proves that the approximating sequence $(\tilde{F}_{n})$ is Cauchy in
$B^{s}_{p,r}(\rho)$. ∎
## 3 Stochastic quantization
In this section we construct the $\phi^{4}$ measure in the infinite domain
$\mathbb{R}^{2}$ equipped with a suitable weight. This construction is well-
known in the literature of stochastic quantization, and we only outline the
results we will need.
We define the stochastic objects both on the periodic space
$\Lambda_{L}\coloneqq{[{-L},{L}]}^{2}$ and the full space $\mathbb{R}^{2}$.
The basic building block, Gaussian free field, is straightforwardly defined in
both cases, whereas for the $\phi^{4}_{2}$ we need to take a weak limit as
$L\to\infty$.
###### Remark 3.1.
In this section we denote by $\mu$ the $\phi^{4}$ measure, whereas in Sections
4 and 5 we need to consider the product of $\phi^{4}$ and white noise
measures. We go with this abuse of notation since the $\phi^{4}$ part is
always the “interesting” measure. However, it is important to keep in mind
that the NLW solution will also depend on the white noise.
### 3.1 Gaussian free field
###### Definition 3.2 (Gaussian free field).
The massive GFF $\nu_{L}$ is the Gaussian measure on
$\mathcal{S}^{\prime}(\Lambda_{L})$ with covariance
$\int\langle f,\phi\rangle\langle
g,\phi\rangle\,\mathrm{d}\nu_{L}(\phi)=\langle
f,(m^{2}-\Delta)^{-1}g\rangle_{L^{2}(\Lambda_{L})}.$
Similarly we can introduce the infinite-volume GFF $\nu$ supported on
$\mathcal{S}^{\prime}(\mathbb{R}^{2})$ and with covariance
$\int\langle f,\phi\rangle\langle
g,\phi\rangle\,\mathrm{d}\nu_{L}(\phi)=\langle
f,(m^{2}-\Delta)^{-1}g\rangle_{L^{2}(\mathbb{R}^{2})}.$
Note that we can view $\nu_{L}$ as a measure on
$\mathcal{S}^{\prime}(\mathbb{R}^{2})$ by periodic extension. The following
proposition is proved in [38, Theorem 5.1].
###### Theorem 3.3 (Uniform bounds for GFF).
$\nu_{L}$ and $\nu$ have samples almost surely in
$\mathcal{C}^{-\varepsilon}(\rho)$, and for all $p<\infty$ the expectations
are bounded (uniformly in $L$):
$\sup_{L}\int{\left\|\psi\right\|}_{\mathcal{C}^{-\varepsilon}(\rho)}^{p}\,\mathrm{d}\nu_{L}(\psi)<\infty,\quad\int{\left\|\psi\right\|}_{\mathcal{C}^{-\varepsilon}(\rho)}^{p}\,\mathrm{d}\nu(\psi)<\infty.$
We will denote random variables sampled from $\nu_{L},\nu$ by $Z_{L},Z$. With
some abuse of notation we will write the projections $Z_{L,N}=P_{N}Z_{L}$ and
$Z_{N}=P_{N}Z$. We can sample from the GFF by realizing it as
$Z_{L}=\frac{1}{L}\sum_{n\in
L^{-1}\mathbb{Z}^{2}}\frac{g_{n}e_{n}}{(m^{2}+|n|^{2})^{1/2}},$ (3.1)
where $g_{n}$ are complex Gaussians with variance $1$ and
$e_{n}(x)\coloneqq\exp(2\pi nx)$. We require $g_{-n}=\overline{g_{n}}$ to make
the field real, but otherwise $g_{n}$ are independent. For the full-space case
we can write
$Z=\int_{\mathbb{R}^{2}}\frac{\xi(y)e_{y}}{(m^{2}+|y|^{2})^{1/2}}\,\mathrm{d}y$
(3.2)
where $\xi$ is a white noise as defined below.
###### Definition 3.4 (White noise).
The white noise $\xi$ is a Gaussian process on
$\mathcal{S}^{\prime}(\mathbb{R}^{2})$ with covariance
$\mathbb{E}\,[{\left\langle\xi,\psi\right\rangle}_{L^{2}(\mathbb{R}^{2})}{\left\langle\xi,\phi\right\rangle}_{L^{2}(\mathbb{R}^{2})}]={\left\langle\psi,\phi\right\rangle}_{L^{2}(\mathbb{R}^{2})}.$
The measure $\nu_{L}$ does not have samples of positive regularity. This means
that taking powers of distributions sampled from $\nu_{L}$ does not make
sense. Yet the Gaussian structure of the randomness allows us to still define
powers of the field by so-called Wick ordering.
###### Definition 3.5 (Wick ordering, periodic space).
Let $a_{N,L}=\mathbb{E}\,[Z_{L,N}(0)^{2}]$. We then define
$\displaystyle{:\\!Z^{4}_{L,N}\\!:}_{L}$
$\displaystyle=Z^{4}_{L,N}-6a_{N,L}Z^{2}_{L,N}+3a^{2}_{N,L},$
$\displaystyle{:\\!Z^{3}_{L,N}\\!:}_{L}$
$\displaystyle=Z^{3}_{L,N}-3a_{N,L}Z_{L,N},$
$\displaystyle{:\\!Z^{2}_{L,N}\\!:}_{L}$ $\displaystyle=Z^{2}_{L,N}-a_{N,L}.$
As $N\to\infty$, the constants $a_{N,L}$ diverge logarithmically, and the
counterterms cancel the divergence of $Z^{k}_{L,N}$. Further Wick powers can
be defined using Hermite polynomials. Wick-ordered polynomials are defined by
Wick-ordering each monomial term separately. We remark that
$\mathbb{E}\,[Z_{L,N}(x)^{2}]$ does not depend on the choice of $x$ since the
Gaussian free field is translation-invariant.
It will be useful to define the Wick powers with a renormalization constant
that is independent of $L$. For this purpose we will use the expectation of
the full-space GFF.
###### Definition 3.6 (Wick ordering, full space).
We denote by $a_{N}=\mathbb{E}\,[Z_{N}(0)^{2}]$, and define
$\displaystyle{:\\!Z^{4}_{L,N}\\!:}$
$\displaystyle=Z^{4}_{L,N}-6a_{N}Z^{2}_{N,L}+3a^{2}_{N},$
$\displaystyle{:\\!Z^{3}_{L,N}\\!:}$
$\displaystyle=Z^{3}_{L,N}-3a_{N}Z_{L,N},$
$\displaystyle{:\\!Z^{2}_{L,N}\\!:}$ $\displaystyle=Z^{2}_{L,N}-a_{N}.$
The difference between these two renormalizations is a polynomial of strictly
lower degree; for the fourth Wick powers it is
${:\\!Z^{4}_{L,N}\\!:}_{L}-{:\\!Z^{4}_{L,N}\\!:}=-6(a_{N,L}-a_{N})Z^{2}_{L,N}+3(a_{N,L}^{2}-a_{N}^{2}).$
(3.3)
The next lemma asserts that the difference of renormalization constants goes
to zero as $N,L\to\infty$. This lets us always take Wick ordering with respect
to the full-space GFF.
###### Lemma 3.7.
For $a_{N,L}=\mathbb{E}\,[Z_{L,N}(0)^{2}]$ and
$a_{N}=\mathbb{E}\,[Z_{N}(0)^{2}]$ we have
$|a_{N,L}-a_{N}|\lesssim\frac{1}{N}+\frac{1}{L}.$
For every $L<\infty$ we can thus define
$r_{L}\coloneqq\lim_{N\to\infty}(a_{N,L}-a_{N})\lesssim\frac{1}{L}$.
###### Proof.
The first renormalization constant can be written as
$\mathbb{E}\,{\left|Z_{L,N}(0)^{2}\right|}=\frac{1}{L^{2}}\sum_{\begin{subarray}{c}n\in
L^{-1}\mathbb{Z}^{2}\\\ {\left|n\right|}\leq
N\end{subarray}}\frac{1}{m^{2}+{\left|n\right|}^{2}}=\sum_{\begin{subarray}{c}n\in
L^{-1}\mathbb{Z}^{2}\\\ {\left|n\right|}\leq
N\end{subarray}}\int_{P(n)}\frac{1}{m^{2}+{\left|n\right|}^{2}}\,\mathrm{d}x,$
where $P(n)$ is the rectangle $n+[{0},{1/L}]^{2}$. By covariance of the
continuum white noise, the second renormalization constant is
$\mathbb{E}\,{\left|Z_{L}(0)^{2}\right|}=\int_{{\left|x\right|}\leq
N}\frac{1}{m^{2}+{\left|x\right|}^{2}}\,\mathrm{d}x.$
The difference of these integrals is estimated in two parts. Some of the
rectangles $P(n)$ extend outside the ball $B(0,N)$; this introduces an error
of order
$\sum_{\begin{subarray}{c}n\in L^{-1}\mathbb{Z}^{2}\\\ P(n)\not\subset
B(0,N)\end{subarray}}\frac{1}{m^{2}+{\left|n\right|}^{2}}\lesssim
N\cdot\frac{1}{N^{2}}.$
Meanwhile for $x\in P(n)\cap B(0,N)$ we can estimate
$\displaystyle{\left|\frac{1}{m^{2}+{\left|n\right|}^{2}}-\frac{1}{m^{2}+{\left|x\right|}^{2}}\right|}$
$\displaystyle={\left|\frac{{\left|x\right|}^{2}-{\left|n\right|}^{2}}{(m^{2}+{\left|n\right|}^{2})(m^{2}+{\left|x\right|}^{2})}\right|}$
$\displaystyle\leq\frac{1/L}{{(m^{2}+{\left|n\right|}^{2})}^{2}}.$
This implies that the difference of integrals inside $B(0,N)$ is bounded by
$\frac{1}{L^{3}}\sum_{\begin{subarray}{c}n\in L^{-1}\mathbb{Z}^{2}\\\
{\left|n\right|}\leq
N\end{subarray}}\frac{1}{{(m^{2}+{\left|n\right|}^{2})}^{2}}\lesssim\frac{1}{L}.\qed$
We can now state that all Wick powers of the Gaussian free field are well-
defined. This result also translates into regular enough perturbations of the
GFF, in particular the $\phi^{4}$ measure defined below.
###### Lemma 3.8 (Moments of GFF powers).
Let $Z_{L}$ be sampled from $\nu_{L}$ and let $Z_{L,N}=P_{N}Z_{L}$. Then for
any $p<\infty$ and $i=1,2,\ldots$ we have
$\sup_{N}\mathbb{E}\left[{\left\|{:\\!Z^{i}_{L,N}\\!:}\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\Lambda_{L})}\right]<\infty\quad$
Furthermore ${:\\!Z^{i}_{L,N}\\!:}$ converges almost surely in
$\mathcal{C}^{-\varepsilon}(\mathbb{T}^{2})$ and in any
$L^{p}(\nu_{L},C^{-\varepsilon}(\rho))$ to a well-defined limit. We denote
this limit by ${:\\!Z_{L}^{i}\\!:}$ and we have
$\quad\sup_{L}\mathbb{E}\left[{\left\|{:\\!Z^{i}_{L}\\!:}\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}\right]<\infty.$
###### Proof.
The proof of the first statement is in [19, Lemma 3.2], and the polynomial-
weight case is [38, Theorem 5.1]. ∎
###### Lemma 3.9 (Wick powers of perturbations).
Let $\phi\in L^{p}(\nu_{L},B_{p,p}^{2\varepsilon})$, where $\varepsilon>0$ and
$p$ is large enough. Then
${:\\!(Z_{L}+\phi)^{j}\\!:}=\sum_{i=0}^{j}{:\\!Z^{i}_{L}\\!:}\;\phi^{j-i}.$
###### Proof.
It follows from properties of Hermite polynomials that
${:\\!(Z_{L,N}+\phi)^{j}\\!:}=\sum_{i=0}^{j}\binom{j}{i}{:\\!Z^{i}_{L,N}\\!:}\;\phi^{j-i}.$
Each term is well-defined as an element of
$B^{-\varepsilon}_{p/j,p/j}(\rho^{j+1})$ by Theorem 2.4; moreover the
multiplication is a continuous operation. The claim then follows by passing
$N\to\infty$. ∎
The following result lets us compute covariances of Wick powers by passing to
a Green’s function. For the proof, see e.g. [52, Theorem I.3].
As an application, we see that we can approximate the third Wick power by
continuous maps. We use this lemma to prove that sequences of periodic
solutions satisfy the PDEs also in the limit. The proof is somewhat technical,
and we leave it to Appendix A.
###### Theorem 3.10 (Wick’s theorem).
If $X$ and $Y$ are Gaussian, then
$\mathbb{E}\,[{:\\!X^{n}\\!:}\;{:\\!Y^{n}\\!:}]=n!\left(\mathbb{E}\,[XY]\right)^{n}.$
###### Lemma 3.11 (Approximation of Wick powers).
For every $\delta>0$ and $s>0$, there exists a continuous map
$f^{\delta}\colon H^{-s}(\rho)\to L^{2}(\rho)$ such that the following holds.
Let $Z$ (respectively $Z_{L}$) be sampled from the (periodic) Gaussian free
field and $\phi\in L^{p}(\mathbb{P},B_{p,p}^{\varepsilon}(\rho))$. Then
$\lim_{\delta\to
0}\mathbb{E}\,{\left\|f^{\delta}(Z+\phi)-{:\\!(Z+\phi)^{3}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho)}^{p}=0,$
and respectively with $Z$ replaced by $Z_{L}$.
### 3.2 Coupling of the $\phi_{4}$ measure and the GFF
We now turn to study the $\phi_{2}^{4}$ measure that will be the invariant
measure for our PDEs. We can define $\phi^{4}$ directly only in the periodic
case; we need to take a weak limit to get to infinite volume.
Let us first recall the definition and some basic results of weak convergence
of probability measures. These can be found in most probability textbooks; see
for example [31, Definition 13.12].
###### Theorem 3.12 (Weak convergence).
A sequence of probability measures $(\mu_{L})$ taking values in $\mathcal{X}$
is said to _converge weakly_ to $\mu$ if
$\lim_{L\to\infty}\int_{\Omega}f(\phi)\,\mathrm{d}\mu_{L}(\phi)=\int_{\Omega}f(\phi)\,\mathrm{d}\mu(\phi)\quad\text{for
all }f\in C_{b}(\mathcal{X};\mathbb{R}).$
If $\mathcal{X}$ is a Polish space, then the weak limit is unique.
###### Definition 3.13 (Tightness).
A family $(\mu_{L})_{L\in\mathbb{N}}$ of probability measures on $\mathcal{X}$
is called _tight_ if for any $\varepsilon>0$ there exists a compact set
$K_{\varepsilon}$ such that
$\sup_{L\in\mathbb{N}}\mu_{L}(\mathcal{X}\setminus K)<\varepsilon.$
###### Lemma 3.14 (Prokhorov’s theorem; [31, Theorem 13.29]).
Suppose that the sequence $(\mu_{L})$ is tight and defined on a metric space
$\mathcal{X}$. Then there is a subsequence $(\mu_{L_{k}})$ that converges
weakly to a limit measure $\mu$ on $\mathcal{X}$.
We will consider a sequence of $\phi^{4}_{2,L}$ measures over increasingly
large tori and show that it is tight over a polynomially weighted Besov space.
This will give us a weak limiting measure $\phi_{2}^{4}$.
###### Definition 3.15.
In finite volume $\Lambda_{L}$, the $\phi^{4}_{2,L}$ measure is given by
$\mathrm{d}\mu_{L}(\psi)\coloneqq
Z_{L}^{-1}\exp\left(-\lambda\int_{\Lambda_{L}}{:\\!\psi^{4}(x)\\!:}\,\mathrm{d}x\right)\,\mathrm{d}\nu_{L}(\psi),$
where $Z_{L}^{-1}$ is a normalization constant.
By Section 3.1 the Wick power ${:\\!\psi^{4}\\!:}$ makes sense as a
distribution $\nu_{L}$-almost surely, and one can show that the exponential
belongs to $L^{p}(\nu_{L})$ for any $p<\infty$ and $L<\infty$. For our
purposes, it is easier to consider $\mu_{L}$ as the invariant measure of the
stochastic quantization equation (see [38])
$\partial_{t}u+(m^{2}-\Delta)u+{:\\!u^{3}\\!:}=\xi,\quad u\in
C(\mathbb{R}_{+},H^{-\delta}(\Lambda_{L})).$ (SQE)
Here $\xi$ is space-time white noise, the Gaussian process valued in
$\mathcal{S}^{\prime}(\mathbb{R}\times\mathbb{R}^{2})$ with covariance
$\mathbb{E}\,[{\left\langle\xi,f\right\rangle}_{L^{2}(\mathbb{R}\times\mathbb{R}^{2})}{\left\langle\xi,g\right\rangle}_{L^{2}(\mathbb{R}\times\mathbb{R}^{2})}]={\left\langle
f,g\right\rangle}_{L^{2}(\mathbb{R}\times\mathbb{R}^{2})},$
where $f,g\in\mathcal{S}(\mathbb{R}\times\mathbb{R}^{2})$.
We will use (SQE) to control the $\phi_{2,L}^{4}$ measure in the limit
$L\to\infty$. That $\mu_{L}$ is indeed a stationary measure for this equation
was shown by Da Prato and Debussche [19]. We begin by decomposing the solution
as $u=Z+\phi$, where $Z$ is the Gaussian part that solves the stationary
equation
$\left\\{\begin{aligned} \partial_{t}Z+(m^{2}-\Delta)Z&=\xi,\\\
Z(0)&=Z_{0}\sim\text{GFF},\end{aligned}\right.$ (3.4)
and $\phi$ solves
$\left\\{\begin{aligned}
\partial_{t}\phi+(m^{2}-\Delta)\phi+{:\\!(Z+\phi)^{3}\\!:}&=0,\\\
\phi(0)&=u(0)-Z(0).\end{aligned}\right.$ (3.5)
We can take $(u,Z)$ to be jointly stationary solutions to (SQE) and (3.4) so
that $Z(t)\sim\text{GFF}$; see the beginning of Section 4.3 in [24]. In
particular the Wick power ${:\\!Z^{i}\\!:}$ is a well-defined random
distribution. Now multiplying (3.5) by $\rho\phi$ and integrating we obtain
$\partial_{t}\|\rho^{1/2}\phi\|^{2}_{L^{2}}+m^{2}\|\rho^{1/2}\phi\|^{2}_{L^{2}}+\|\rho^{1/2}\nabla\phi\|_{L^{2}}^{2}+\|\rho^{1/4}\phi\|^{4}_{L^{4}}=-G(Z,\phi),$
(3.6)
where
$\displaystyle G(Z,\phi)$
$\displaystyle=3\int\rho\;{:\\!Z^{3}\\!:}\;\phi\,\mathrm{d}x+3\int\rho\;{:\\!Z^{2}\\!:}\;\phi^{2}\,\mathrm{d}x$
$\displaystyle\qquad+\int\rho
Z\phi^{3}\,\mathrm{d}x+\int([\nabla,\rho]\phi)\nabla\phi\,\mathrm{d}x.$ (3.7)
Here $[\nabla,\rho]\coloneqq\nabla(\rho\phi)-\rho\nabla\phi=(\nabla\rho)\phi$
is the commutator of the weight and the gradient. In Appendix B we show that
${\left|G(Z,\phi)\right|}\leq
Q(Z)+\frac{1}{2}({\|\rho^{1/2}\phi\|}^{2}_{L^{2}}+{\|\rho^{1/2}\nabla\phi\|}_{L^{2}}^{2}+{\|\rho^{1/4}\phi\|}^{4}_{L^{4}}),$
(3.8)
where
$\sup_{L}\mathbb{E}\,[|Q(Z)|^{p}]<\infty\quad\text{for any }p<\infty.$
By moving common terms to the left-hand side of (3.6), we get the estimate
$\partial_{t}{\|\rho^{1/2}\phi\|}^{2}_{L^{2}}+\frac{1}{2}(m^{2}{\|\rho^{1/2}\phi\|}^{2}_{L^{2}}+{\|\rho^{1/2}\nabla\phi\|}_{L^{2}}^{2}+{\|\rho^{1/4}\phi\|}^{4}_{L^{4}})\leq
Q(Z).$ (3.9)
The time derivative term vanishes in expectation since $\phi=u-Z$ was assumed
to be stationary. Therefore we are left with
$\frac{1}{2}\mathbb{E}\,[(m^{2}{\|\rho^{1/2}\phi\|}^{2}_{L^{2}}+{\|\rho^{1/2}\nabla\phi\|}_{L^{2}}^{2}+{\|\rho^{1/4}\phi\|}^{4}_{L^{4}})]\leq\mathbb{E}\,Q(Z),$
(3.10)
which proves
$\sup_{L}\mathbb{E}\,{\|\rho^{1/2}\phi\|}^{2}_{H^{1}}<\infty,$
and thus tightness in $H^{1-\varepsilon}(\rho^{1/2+\varepsilon})$. We still
strengthen this in Section 3.3.
### 3.3 Wick powers of $\phi_{2}^{4}$
The bounds on the $\phi^{4}$ samples can be improved to exponential tails,
which then implies $L^{p}$ expectations for all $p$. We defer the proof of
this result to Appendix C.
###### Theorem 3.16 (Exponential tails).
There exists $\delta>0$ such that
$\int_{\Omega}\exp\left(\delta{\left\|W_{L}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho)}^{2}\right)\mathrm{d}\mu_{L}(W_{L})\lesssim
1,$
and the bound is uniform in $L$. The bound also holds in the limit $\mu$.
Since the nonlinearity in (NLW) is cubic, we will need the first three Wick
powers of the $\phi^{4}$ field. We construct and estimate the Wick powers of
$\phi_{2,L}^{4}$ uniformly in $L$, and thus in the $L\to\infty$ limit.
###### Theorem 3.17 (Wick powers of $\phi^{4}$).
Let $W_{L}=Z_{L}+\phi_{L}$ be sampled from $\phi_{2,L}^{4}$. Then
${:\\!W^{j}\\!:}$ is a well-defined random distribution for $j\leq 3$, and for
any $\varepsilon>0$ and $p<\infty$ we have
$\sup_{L}\mathbb{E}\,{\|{:\\!W_{L}^{j}\\!:}\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho^{2})}<\infty.$
Furthermore, if $W$ is sampled from the full-space $\phi_{2}^{4}$ measure,
then
$\mathbb{E}\,{\|{:\\!W^{j}\\!:}\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho^{2})}<\infty.$
###### Proof.
We only do the proof in the most difficult case $j=3$. The other cases are
analogous. Recall that we have
${:\\!W_{L}^{3}\\!:}=\sum_{i=0}^{3}\binom{3}{i}{:\\!Z_{L}^{i}\\!:}\;\phi_{L}^{3-i}.$
Now for $q=4/\varepsilon$ we can estimate
$\displaystyle{\left\|{:\\!Z_{L}^{i}\\!:}\;\phi_{L}^{3-i}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{2})}$
$\displaystyle\lesssim{\left\|{:\\!Z_{L}^{i}\\!:}\;\phi_{L}^{3-i}\right\|}_{B_{q,q}^{-\varepsilon/2}(\rho^{2})}$
$\displaystyle\lesssim{\left\|{:\\!Z_{L}^{i}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon/2}(\rho)}{\left\|\phi_{L}^{3-i}\right\|}_{B_{q,q}^{\varepsilon/2}(\rho)}$
$\displaystyle\leq{\left\|{:\\!Z_{L}^{i}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon/2}(\rho)}{\left\|\phi_{L}\right\|}^{3-i}_{B_{3q,3q}^{\varepsilon/2}(\rho^{1/3})}.$
(3.11)
The Gaussian part is bounded by Lemma 3.8. For the perturbation part, Theorem
3.16 implies that
$\mathbb{E}\,{\left\|W_{L}\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}<\infty$,
and we can decompose
$\sup_{L}\mathbb{E}\,{\left\|\phi_{L}\right\|}^{p}_{\mathcal{C}^{-\varepsilon/12}}\lesssim\sup_{L}\,(\mathbb{E}\,{\left\|W_{L}\right\|}^{p}_{\mathcal{C}^{-\varepsilon/12}(\rho)}+\mathbb{E}\,{\left\|Z_{L}\right\|}^{p}_{\mathcal{C}^{-\varepsilon/12}(\rho)})<\infty.$
This estimate provides $L^{p}$ regularity, whereas the estimate
$\mathbb{E}\,{\left\|\phi\right\|}^{2}_{H^{1}(\rho^{1/3})}<\infty$ from
Section 3.2 provides differentiability. We can interpolate between these two
through
$\displaystyle{\left\|\phi_{L}\right\|}_{B_{3q,3q}^{\varepsilon/2}(\rho^{1/3})}$
$\displaystyle\lesssim{\left\|\phi_{L}\right\|}^{\theta}_{\mathcal{C}^{-\varepsilon/12}(\rho^{1/3})}{\left\|\phi_{L}\right\|}^{1-\theta}_{H^{1}(\rho^{1/3})}$
$\displaystyle\lesssim{\left\|\phi_{L}\right\|}^{2\theta}_{\mathcal{C}^{-\varepsilon/12}(\rho^{1/3})}+{\left\|\phi_{L}\right\|}^{2-2\theta}_{H^{1}(\rho^{1/3})},$
where $\theta=1-\varepsilon/12$. As we substitute this back into (3.3), we
find that the final expectation is bounded. ∎
From Theorem 3.17 we can bootstrap a stronger statement for the coupling. The
perturbation $\phi$ is two derivatives more regular than $Z$, instead of just
one derivative as showed earlier.
###### Corollary 3.18 (Strong bound for regular part).
We can find random variables $Z_{L},\phi_{L}$ such that $Z_{L}\sim\nu_{L}$,
$Z_{L}+\phi_{L}\sim\mu_{L}$, and
$\sup_{L}\mathbb{E}\,\|\phi_{L}\|^{p}_{H^{2-\varepsilon}(\rho)}\lesssim 1.$
###### Proof.
Recall that from the stochastic quantization equation (SQE) we have
$\phi_{L}(t)=\int^{t}_{0}e^{-(t-s)\Delta}{:\\!(Z_{L}(s)+\phi_{L}(s))^{3}\\!:}\,\mathrm{d}s+e^{-t\Delta}\phi_{L}(0).$
So provided $p$ is large enough that $|t-s|^{(1-\varepsilon/2)p/(p-1)}\in
L^{1}$, we can use the smoothing effect of the heat operator ([38, Proposition
5]) to estimate
$\displaystyle\mathbb{E}\,\|\phi_{L}(t)\|^{p}_{H^{2-\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle\mathbb{E}\,{\left\|\int^{t}_{0}e^{-(t-s)\Delta}{:\\!(Z_{L}(s)+\phi_{L}(s))^{3}\\!:}\,\mathrm{d}s\right\|}^{p}_{H^{2-\varepsilon}(\rho)}+\mathbb{E}\,{\left\|e^{-t\Delta}\phi_{L}(0)\right\|}^{p}_{H^{2-\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle\mathbb{E}\left[\int^{t}_{0}\frac{{\left\|{:\\!(Z_{L}(s)+\phi_{L}(s))^{3}\\!:}\right\|}_{H^{-\varepsilon/2}(\rho)}}{{\left|t-s\right|}^{1-\varepsilon/2}}\,\mathrm{d}s\right]^{p}+\frac{\mathbb{E}\,{\left\|\phi_{L}(0)\right\|}^{p}_{H^{-\varepsilon/2}(\rho)}}{t^{1-\varepsilon/2}}$
$\displaystyle\lesssim\;$
$\displaystyle\int^{t}_{0}\mathbb{E}\,{\left\|{:\\!(Z_{L}(s)+\phi_{L}(s))^{3}\\!:}\right\|}^{p}_{H^{-\varepsilon/2}(\rho)}\,\mathrm{d}s+\frac{\mathbb{E}\,{\left\|\phi_{L}(0)\right\|}^{p}_{H^{-\varepsilon/2}(\rho)}}{t^{1-\varepsilon/2}}.$
Since $Z_{L}$ and $\phi_{L}$ are both stationary, we may choose $t$ as we
like. The integrand is then uniformly bounded by Theorem 3.17. ∎
In total we have obtained that
$\sup_{L}\mathbb{E}\,{\left\|\phi\right\|}^{p}_{H^{2-\varepsilon}(\rho)}<\infty$.
By the same compactness argument as above, $\operatorname{Law}(Z,\phi)$ is
tight on $H^{-\varepsilon}(\rho)\times
H^{2-\varepsilon}(\rho^{1+\varepsilon})$. In particular
$\mu_{L}=\operatorname{Law}(Z+\phi)$ is tight on
$H^{-\varepsilon}(\rho^{1+\varepsilon})$ and has a weakly converging
subsequence. We have thus proved the following:
###### Theorem 3.19 ($\phi^{4}_{2}$ as a weak limit).
Let $\rho$ be a sufficiently integrable polynomial weight. The measure
$\mu_{L}$ can be represented as
$\mu_{L}=\operatorname{Law}(Z_{L}+\phi_{L})$
where $Z_{L}$ is a GFF on $\Lambda_{L}$, and $\phi_{L}$ satisfies
$\sup_{L}\mathbb{E}\,\|\phi_{L}\|^{p}_{H^{2}(\rho)}<\infty$. Identifying
$Z_{L}+\phi_{L}$ with its periodic extension on $\mathbb{R}^{2}$ we have that
$\mu_{L}=\operatorname{Law}(Z_{L}+\phi_{L})$ is tight on
$H^{-\varepsilon}(\rho^{1+\varepsilon})$ and any limiting point $\mu$
satisfies
$\mu=\operatorname{Law}(Z+\phi)$
where $Z$ is a Gaussian free field on $\mathbb{R}^{2}$ and
$\mathbb{E}\,{\left\|\phi\right\|}^{p}_{H^{2}(\rho)}<\infty$.
###### Proof.
We know that the limit of $\operatorname{Law}(Z_{L})$ as $L\to\infty$ is a
Gaussian free field on $\mathbb{R}^{2}$; this follows for instance from the
convergence of the covariances. It remains to show that
$\mathbb{E}\,\|\phi\|^{p}_{H^{2-\varepsilon}(\rho)}<\infty$
but since $\|\phi\|^{2}_{H^{2}(\rho)}$ is lower semicontinuous on
$H^{2-\varepsilon}(\rho^{1+\varepsilon})$ we have by weak convergence
$\mathbb{E}\,[\|\phi\|^{p}_{H^{2}(\rho)}]\leq\liminf_{L\to\infty}\mathbb{E}\,[\|\phi_{L}\|^{p}_{H^{2}(\rho)}]<\infty.\qed$
###### Remark 3.20.
We were careful to state the preceding theorem for “any limiting point $\mu$”.
When the coupling parameter $\lambda$ is large enough, there exist
subsequences of $(\phi^{4}_{2,L})$ that converge to different weak limits.
This is one of the main complications in our study.
## 4 Invariance of periodic NLW
Let us now move on to solving the nonlinear wave equation. We fix a bounded
domain $\Lambda_{L}={[{-L},{L}]}^{2}$ and consider
$\left\\{\begin{aligned}
\partial_{tt}u(x,t)+(m^{2}-\Delta)u(x,t)&=-\lambda{:\\!u^{3}\\!:}(x,t),\\\
u(x,0)&=u_{0}(x),\\\
\partial_{t}u(x,0)&=u_{0}^{\prime}(x)\end{aligned}\right.$ (4.1)
on $\Lambda_{L}\times\mathbb{R}_{+}$. The initial value $u_{0}$ will be
sampled from a $\phi^{4}$ measure and the initial time derivative
$u_{0}^{\prime}$ from a white noise measure; as remarked at the beginning of
Section 3, we denote by $\mu$ now the product $(\phi^{4},\text{WN})$ measure.
The Wick ordering will always be taken with respect to the full space
covariance, even if we start from periodic initial data.
Thanks to the finite speed of propagation (formulated more precisely in
Theorem 4.1), boundary effects are visible only outside the ball $B(0,L-t)$.
By solving the equation in Fourier space, we can write the mild solution as
$u(x,t)={\mathcal{C}}_{t}u_{0}(x)+{\mathcal{S}}_{t}u_{0}^{\prime}(x)+\lambda\int_{0}^{t}[{\mathcal{S}}_{t-s}{:\\!u^{3}\\!:}](x)\,\mathrm{d}s,$
(4.2)
where we use the cosine and sine operators
${\mathcal{C}}_{t}=\cos((m^{2}-\Delta)^{1/2}t),\quad{\mathcal{S}}_{t}=\frac{\sin((m^{2}-\Delta)^{1/2}t)}{(m^{2}-\Delta)^{1/2}}.$
(4.3)
These are defined as Fourier multiplier operators. We see that
${\mathcal{C}}_{t}$ preserves the $H^{s}(\Lambda_{L})$ regularity of its
argument whereas ${\mathcal{S}}_{t}$ increases it by one.
We split the solution into nonlinear and linear parts $u=v+w$. Here $w$ solves
the linear equation with the given initial data, leaving $v$ to solve the
coupled equation
$\partial_{t}v(x,t)+(m^{2}-\Delta)v(x,t)=\lambda\,{:\\!(v+w)^{3}\\!:}$ (4.4)
with zero initial data. We will see that $v$ has one degree higher regularity
than $w$, and its growth is controlled by $w$. The final solution will exist
in $L^{p}([0,\tau];B^{-\varepsilon}_{p,p}(\Lambda_{L}))$ up to some short,
stochastic time $\tau$.
The result in this section was already proved by Oh and Thomann [46], and
stated without proof by Bourgain in 1999. The argument presented below
replaces the more specific Fourier restriction norm by a general Besov norm,
and includes the details on convergence of solutions.
### 4.1 Linear part
It is a basic property of the wave equation that all wave packets travel at a
fixed speed. This property applies to the Duhamel formulation (4.2) as well.
The solution operators are then also bounded in weighted spaces since the
weight does not change too much within a ball.
###### Lemma 4.1 (Finite speed of propagation).
If $(u_{0},u_{0}^{\prime})$ and $(v_{0},v_{0}^{\prime})$ coincide on $B(0,R)$,
then the corresponding linear wave equation solutions $u(t)$ and $v(t)$
coincide on $B(0,R-t)$.
###### Sketch of proof.
By linearity, we only need to show that
${\mathcal{C}}_{t}w_{0}(x)+{\mathcal{S}}_{t}w_{0}^{\prime}(x)$
vanishes on $B(0,R-t)$ if both $w_{0}$ and $w_{0}^{\prime}$ vanish inside
$B(0,R)$. By density, we can further assume $w_{0}$ and $w_{0}^{\prime}$ to be
smooth and compactly supported. The idea is to consider the local energy
$E(s)\coloneqq\int_{B(0,R-s)}m^{2}u(x,s)^{2}+{\left|\nabla
u(x,s)\right|}^{2}+{\left|\partial_{t}u(x,s)\right|}^{2}\,\mathrm{d}x.$
By a calculation with some vector analysis, we see that $\partial_{s}E(s)\leq
0$ up to time $t$, and $E(0)=0$ by assumption. See [22, Section 2.4.3] for
details. ∎
###### Lemma 4.2 (Boundedness of solution operators).
Let $t\leq T$. Then for $s\in\mathbb{R}$ and $f\in H^{s}(\rho)$ we have
${\left\|{\mathcal{C}}_{t}f\right\|}_{H^{s}(\rho)}\lesssim{\left\|f\right\|}_{H^{s}(\rho)}\quad\text{and}\quad{\left\|{\mathcal{S}}_{t}f\right\|}_{H^{s}(\rho)}\lesssim{\left\|f\right\|}_{H^{s-1}(\rho)}.$
###### Proof.
Denote by $K$ the convolution kernel of ${\mathcal{C}}_{t}$. As noted in Lemma
4.1, $K$ is supported in $B(0,T)$. It suffices to study the case $s=0$ as
${\mathcal{C}}_{t}$ commutes with derivatives. We see that ${\mathcal{C}}_{t}$
is bounded on $L^{2}$ with flat weight, since it is a Fourier multiplier with
bounded symbol. Then
$\displaystyle{\left\|{\mathcal{C}}_{t}f\right\|}_{L^{2}(\rho)}^{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}\rho^{2}(x)\left[\int_{\mathbb{R}^{2}}K(x-y)f(y)\,\mathrm{d}y\right]^{2}\,\mathrm{d}x$
$\displaystyle\lesssim\int_{\mathbb{R}^{2}}\left[\int_{\mathbb{R}^{2}}K(x-y)\rho(x-y)^{-1}\rho(y)f(y)\,\mathrm{d}y\right]^{2}\,\mathrm{d}x$
$\displaystyle\lesssim\int_{\mathbb{R}^{2}}\left[\int_{\mathbb{R}^{2}}K(x-y)\rho(y)f(y)\,\mathrm{d}y\right]^{2}\,\mathrm{d}x$
$\displaystyle={\left\|{\mathcal{C}}_{t}(\rho f)\right\|}_{L^{2}}^{2}$
$\displaystyle\lesssim{\left\|\rho f\right\|}_{L^{2}}^{2},$
We could estimate $\rho(x-y)^{-1}\leq C(T)$ since $K$ vanishes outside
$|x-y|\leq T$. For ${\mathcal{S}}_{t}$ the proof is identical, except that we
gain a derivative of regularity. ∎
In probabilistic terms, the linear part looks almost like the coupled
$\phi^{4}$ measure: there is an invariant Gaussian free field part and a more
regular term with conserved norm. This yields a moment bound, which we then
use to control the norm of the nonlinearity in the next section.
###### Lemma 4.3 (Distribution of linear part).
As we substitute $u_{0}=Z_{L}+\phi_{L}$, the linear part becomes
$w(\cdot,t)=\bigg{[}{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\bigg{]}+{\mathcal{C}}_{t}\phi_{L}.$
Here the term in brackets is GFF, whereas ${\mathcal{C}}_{t}\phi\in
H^{2-\varepsilon}(\rho)$ almost surely.
###### Proof.
The latter part follows from preservation of Sobolev regularity by the cosine
operator. To prove the first part, we need to compute the covariance. For any
test functions $\varphi$, $\psi$ we have
$\displaystyle\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}{\left\langle\psi,{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}\bigg{]}$
$\displaystyle=\;$
$\displaystyle\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{C}}_{t}Z\right\rangle}{\left\langle\psi,{\mathcal{C}}_{t}Z\right\rangle}\bigg{]}+\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}{\left\langle\psi,{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}\bigg{]}$
by independence of $Z$ and $u_{0}^{\prime}$. Because ${\mathcal{C}}_{t}$ is a
self-adjoint operator, the first term becomes
$\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{C}}_{t}Z\right\rangle}{\left\langle\psi,{\mathcal{C}}_{t}Z\right\rangle}\bigg{]}={\left\langle{\mathcal{C}}_{t}\varphi,\frac{{\mathcal{C}}_{t}\psi}{m^{2}-\Delta}\right\rangle}={\left\langle\varphi,\frac{\cos(m^{2}-\Delta)^{2}}{m^{2}-\Delta}\psi\right\rangle}.$
For the second term we have white noise covariance instead:
$\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}{\left\langle\psi,{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}\bigg{]}={\left\langle{\mathcal{S}}_{t}\varphi,{\mathcal{S}}_{t}\psi\right\rangle}={\left\langle\varphi,\frac{\sin(m^{2}-\Delta)^{2}}{m^{2}-\Delta}\psi\right\rangle}.$
Now the trigonometric identity $\sin^{2}+\cos^{2}=1$ implies
$\mathbb{E}\bigg{[}{\left\langle\varphi,{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}{\left\langle\psi,{\mathcal{C}}_{t}Z+{\mathcal{S}}_{t}u_{0}^{\prime}\right\rangle}\bigg{]}={\left\langle\varphi,\frac{1}{m^{2}-\Delta}\psi\right\rangle}.\qed$
###### Lemma 4.4 (Moment bounds for linear part).
Let $L$ belong to the convergent subsequence of Theorem 3.19. Let $w_{L}$ be
the $L$-periodic linear part started from data sampled from $L$-periodic
$(\phi^{4}_{2},\mathrm{WN})$ measure. For $j=1,2,3$ and $T>0$ we have the
moment bounds
$\mathbb{E}\,{\|{:\\!w_{L}^{j}\\!:}\|}_{L^{p}([0,T];\,B^{-\varepsilon}_{p,p}(\rho))}^{p}\lesssim_{p,T,\varepsilon}1.$
This also implies that
$\mathbb{E}\,{\|{:\\!w_{L}^{j}\\!:}\|}_{L^{p}([0,T];\,\mathcal{C}^{-2\varepsilon}(\rho))}^{p}\lesssim_{p,T,\varepsilon}1$
for $p$ sufficiently large. Both estimates are uniform in $L$.
###### Proof.
By Lemma 4.3 we can decompose $w_{L}=w_{\mathrm{st}}+\psi$ where
$\psi(t)={\mathcal{C}}_{t}\phi_{L}$ and $w_{\mathrm{st}}$ has distribution
$\nu_{L}$ for every time. We thus have by stationarity
$\int_{0}^{T}\mathbb{E}{\|{:\\!w_{\mathrm{st}}^{j}\\!:}\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}\,\mathrm{d}t=T\,\mathbb{E}{\left\|{:\\!(Z(0))^{j}\\!:}\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}.$
When $j=3$, we can expand the binomial and estimate
$\mathbb{E}{\left\|{:\\!w_{\mathrm{st}}^{3}\\!:}\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}+\mathbb{E}{\left\|{:\\!w_{\mathrm{st}}^{2}\\!:}\psi\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}+\mathbb{E}{\left\|w_{\mathrm{st}}\psi^{2}\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}+\mathbb{E}{\left\|\psi^{3}\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}$
at a fixed time. The middle terms can be expanded with Theorem 2.4 like in
$\displaystyle{\left\|{:\\!w_{\mathrm{st}}^{2}\\!:}\phi\right\|}_{B^{-\varepsilon}_{p,p}(\rho)}^{p}$
$\displaystyle\lesssim{\left\|{:\\!w_{\mathrm{st}}^{2}\\!:}\right\|}_{B^{-\varepsilon}_{2p,p}(\rho^{1/2})}^{p}{\left\|\phi\right\|}_{B^{2\varepsilon}_{2p,p}(\rho^{1/2})}^{p}$
$\displaystyle\lesssim
1+{\left\|{:\\!w_{\mathrm{st}}^{2}\\!:}\right\|}_{B^{-\varepsilon/2}_{2p,2p}(\rho^{1/2})}^{2p}{\left\|\psi\right\|}_{B^{2-\varepsilon}_{2p,2p}(\rho^{1/2})}^{2p}.$
The other cases $j=1$ and $j=2$ are analogous. Now we can deduce the claim
since the norm of $w_{\mathrm{st}}$ is in $L^{p}([0,T])$ by stationarity, and
$\psi$ is in $C([0,T],B^{\varepsilon}_{p,p})$ by Corollary 3.18 uniformly for
a sequence of $L\to\infty$.
The second claim follows from the embedding stated in Theorem 2.7 by choosing
$p\geq 2/\varepsilon$. ∎
We can now show that powers of the linear parts converge as $L\to\infty$. We
will use this result as we pass to the full space in Section 5.
###### Lemma 4.5 (Convergence of linear parts).
Let $1\leq p<\infty$. As $L\to\infty$, ${:\\!w^{i}_{L}\\!:}$ converges in
probability to ${:\\!w^{i}\\!:}$ in
$L^{p}([0,T],\mathcal{C}^{-\varepsilon}(\rho^{3}))$, where $w$ is a linear
solution started from initial data $Z+\phi$.
###### Proof.
The convergence of ${\mathcal{C}}_{t}\phi_{L}$ to ${\mathcal{C}}_{t}\phi$ in
$H^{2-\varepsilon}(\rho)$ follows from continuity of ${\mathcal{C}}_{t}$ in
$H^{2-\varepsilon}(\rho)$. We need to show that
${:\\!w^{i}_{{\mathrm{st}},L}\\!:}\to{:\\!w_{\mathrm{st}}^{i}\\!:}$ in
$L^{p}([0,T],\mathcal{C}^{-\varepsilon}(\rho^{3}))$. Then continuity of Besov
product from $\mathcal{C}^{-\varepsilon}\times H^{2-\varepsilon}$ to
$\mathcal{C}^{-\varepsilon}$ implies convergence of
$(w_{{\mathrm{st}}_{L}}+\phi_{L})^{3}$.
We have that $w_{{\mathrm{st}},L}\to w_{\mathrm{st}}$ in
$C([0,T],H^{-\varepsilon}(\rho))$ by continuity of the linear operators. Now
with $f^{\delta}$ as in Lemma 3.11 we have
$\displaystyle{:\\!w^{i}_{{\mathrm{st}},L}\\!:}-{:\\!w^{i}_{\mathrm{st}}\\!:}=\;$
$\displaystyle[{:\\!w^{i}_{{\mathrm{st}},L}\\!:}-f^{\delta}(w_{{\mathrm{st}},L})]$
$\displaystyle\quad+[f^{\delta}(w_{{\mathrm{st}},L})-f^{\delta}(w_{\mathrm{st}})]+[f^{\delta}(w_{\mathrm{st}})-{:\\!w^{i}_{\mathrm{st}}\\!:}].$
The middle term goes to $0$ since $f^{\delta}$ is continuous from
$H^{-\varepsilon}(\rho)$ to $\mathcal{C}^{-\varepsilon}(\rho^{3})$, and for
the first and last term we have by stationarity
$\mathbb{E}\bigg{[}\int_{0}^{T}{\left\|f^{\delta}(w_{\mathrm{st}})-{:\\!w^{i}_{\mathrm{st}}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{3})}^{p}\,\mathrm{d}s\bigg{]}=T{\left\|f^{\delta}(w_{\mathrm{st}}(0))-{:\\!w^{i}_{\mathrm{st}}(0)\\!:}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{3})}.$
This goes to $0$ uniformly in $L$ by Lemma 3.11. ∎
### 4.2 Fixed-point iteration
We now use the standard fixed-point argument for
$v(x,t)=-\lambda\int_{0}^{t}[{\mathcal{S}}_{t-s}{:\\!(v+w)^{3}\\!:}](x)\,\mathrm{d}s.$
(4.5)
As usual, we need to check boundedness and contractivity of the solution
operator. We do the iteration in the periodic Besov space
$L^{\infty}([0,\tau];\,H^{1-\varepsilon}(\Lambda_{L}))$. The weight must be
“flat” because we need it to be the same on both sides of the multiplicative
estimates.
This argument is completely deterministic. We control the growth of $v$ by
assuming bounds on the linear part $w$; these bounds will be verified by
stochastic estimates in Section 4.3.
###### Lemma 4.6 (Boundedness).
Let
$M=\max_{j=1,2,3}{\left\|{:\\!w^{j}\\!:}\right\|}_{L^{4}([0,1];\,\mathcal{C}^{-\varepsilon}(\rho))}$.
The operator
$(\mathcal{F}v)(x,t)\coloneqq-\lambda\int_{0}^{t}[{\mathcal{S}}_{t-s}{:\\!(v+w)^{3}\\!:}](x)\,\mathrm{d}s$
maps a ball of radius $R$ into a ball of radius
$C_{\Lambda}\lambda\tau^{1/2}M(1+R^{3})$ in the periodic space
$L^{\infty}([0,\tau];\,H^{1-\varepsilon}(\Lambda_{L}))$.
###### Proof.
We can commute the Fourier multiplier and apply Jensen’s inequality in
$\displaystyle{\left\|\mathcal{F}v\right\|}_{L^{\infty}_{\tau}H^{1-\varepsilon}(\Lambda_{L})}$
$\displaystyle=\lambda\sup_{0\leq
t\leq\tau}\left[\int_{\mathbb{R}^{2}}{\left|{\left\langle{\nabla}\right\rangle}^{1-\varepsilon}\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!(v+w)^{3}\\!:}\,\mathrm{d}s\right|}^{2}\,\mathrm{d}x\right]^{1/2}$
$\displaystyle\leq\lambda\tau^{1/2}\sup_{0\leq
t\leq\tau}\left[\int_{\mathbb{R}^{2}}\int_{0}^{t}{\left|{\left\langle{\nabla}\right\rangle}^{1-\varepsilon}{\mathcal{S}}_{t-s}{:\\!(v+w)^{3}\\!:}\right|}^{2}\,\mathrm{d}s\,\mathrm{d}x\right]^{1/2}$
$\displaystyle=\lambda\tau^{1/2}\left[\int_{0}^{\tau}\int_{\mathbb{R}^{2}}{\left|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}{:\\!(v+w)^{3}\\!:}\right|}^{2}\,\mathrm{d}s\,\mathrm{d}x\right]^{1/2}$
$\displaystyle=\lambda\tau^{1/2}{\left\|{:\\!(v+w)^{3}\\!:}\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}.$
In the second-to-last step we used the increase in Besov regularity from
${\mathcal{S}}_{t}$. We can now expand the binomial power by triangle
inequality and estimate each term separately. First,
${\left\|{:\\!w^{3}\\!:}\right\|}_{L^{2}H^{-\varepsilon}(\Lambda_{L})}\lesssim
L^{c}M$ by assumption and Jensen. The second term is estimated as
${\left\|{:\\!w^{2}\\!:}v\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}\lesssim{\left\|{:\\!w^{2}\\!:}\right\|}_{L^{4}_{\tau}\mathcal{C}^{-\varepsilon}(\Lambda_{L})}{\left\|v\right\|}_{L^{4}_{\tau}H^{2\varepsilon}(\Lambda_{L})},$
and for the third one we use Theorem 2.4 twice:
${\left\|wv^{2}\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}\lesssim{\left\|w\right\|}_{L^{4}_{\tau}\mathcal{C}^{-\varepsilon}(\Lambda_{L})}{\left\|v^{2}\right\|}_{L^{4}_{\tau}H^{2\varepsilon}(\Lambda_{L})}\lesssim
L^{c}M{\left\|v\right\|}_{L^{8}_{\tau}B^{3\varepsilon}_{4,4}(\Lambda_{L})}^{2}.$
We also perform the a similar multiplicative estimate for the $v^{3}$ term.
Thus we have estimated
$\displaystyle{\left\|{:\\!(v+w)^{3}\\!:}\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim\;$ $\displaystyle
L^{c}M\left[1+{\left\|v\right\|}_{L^{4}_{\tau}\mathcal{H}^{2\varepsilon}(\Lambda_{L})}+{\left\|v\right\|}_{L^{8}_{\tau}B^{3\varepsilon}_{4,4}(\Lambda_{L})}^{2}+{\left\|v\right\|}_{L^{12}_{\tau}B^{3\varepsilon}_{6,6}(\Lambda_{L})}^{3}\right],$
which yields the required bound after embedding $H^{1-\varepsilon}$ into
$B^{3\varepsilon}_{6,6}$ by Theorem 2.7. With the estimates above, this is
possible for $\varepsilon<1/12$. ∎
###### Lemma 4.7 (Contraction).
In the setting of Lemma 4.6, we also have
${\left\|\mathcal{F}v-\mathcal{F}\tilde{v}\right\|}_{L^{\infty}([0,\tau];\,H^{1-\varepsilon})}\lesssim
C_{\Lambda}\lambda\tau^{1/2}M(1+R^{2}){\left\|v-\tilde{v}\right\|}_{L^{\infty}([0,\tau];\,H^{1-\varepsilon})}.$
###### Proof.
We can begin as in Lemma 4.6 to get the upper bound
$\lambda\tau^{1/2}{\left\|{:\\!(v+w)^{3}\\!:}-{:\\!(\tilde{v}+w)^{3}\\!:}\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}.$
When we again expand the binomials, we get three terms to estimate (since the
$w^{3}$ terms cancel each other). First,
$\displaystyle{\left\|{:\\!w^{2}\\!:}(v-\tilde{v})\right\|}_{L^{2}_{\tau}H^{-\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim{\left\|{:\\!w^{2}\\!:}\right\|}_{L^{4}_{\tau}\mathcal{C}^{-\varepsilon}(\Lambda_{L})}{\left\|v-\tilde{v}\right\|}_{L^{4}_{\tau}H^{2\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim
M{\left\|v-\tilde{v}\right\|}_{L^{\infty}_{\tau}H^{1-\varepsilon}(\Lambda_{L})}.$
In the second term we additionally need to expand
$\displaystyle{\left\|v^{2}-\tilde{v}^{2}\right\|}_{L^{4}_{\tau}H^{2\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim{\left\|v-\tilde{v}\right\|}_{L^{4}_{\tau}B^{3\varepsilon}_{4,4}(\Lambda_{L})}{\left\|v+\tilde{v}\right\|}_{L^{4}_{\tau}B^{3\varepsilon}_{4,4}(\Lambda_{L})}$
$\displaystyle\lesssim
2R{\left\|v-\tilde{v}\right\|}_{L^{\infty}_{\tau}H^{1-\varepsilon}(\Lambda_{L})}.$
In the final term, the corresponding expansion is
$\displaystyle{\left\|v^{3}-\tilde{v}^{3}\right\|}_{L^{4}_{\tau}H^{2\varepsilon}}$
$\displaystyle=\;$
$\displaystyle{\left\|(v-\tilde{v})(v^{2}+v\tilde{v}+\tilde{v}^{2})\right\|}_{L^{4}_{\tau}H^{2\varepsilon}}$
$\displaystyle\lesssim\;$
$\displaystyle{\left\|v-\tilde{v}\right\|}_{L^{8}_{\tau}B^{3\varepsilon}_{4,4}}\left({\left\|v\right\|}_{L^{16}_{\tau}B^{4\varepsilon}_{8,8}}^{2}+{\left\|v\right\|}_{L^{16}_{\tau}B^{4\varepsilon}_{8,8}}{\left\|\tilde{v}\right\|}_{L^{16}_{\tau}B^{4\varepsilon}_{8,8}}+{\left\|\tilde{v}\right\|}_{L^{16}_{\tau}B^{4\varepsilon}_{8,8}}^{2}\right)$
$\displaystyle\lesssim\;$
$\displaystyle{\left\|v-\tilde{v}\right\|}_{L^{\infty}_{\tau}H^{1-\varepsilon}(\Lambda_{L})}R^{2}.$
All together, we get the claimed inequality for $\varepsilon$ small. ∎
###### Theorem 4.8.
The nonlinear equation (4.5) has a unique solution
$v\in L^{\infty}([0,\tau];\,H^{1-\varepsilon}(\Lambda_{L})).$
The norm of $v$ depends only on $M$ as given in Lemma 4.6, and the solution
time $\tau$ depends on both $M$ and the period length of $\Lambda_{L}$.
###### Proof.
It only remains to choose $R$ and $\tau$ such that
$\begin{cases}C_{\Lambda}\lambda\tau^{1/2}M(1+R^{3})\leq R,\\\
C_{\Lambda}\lambda\tau^{1/2}M(1+R^{2})\leq\frac{1}{2}.\end{cases}$
We can select $R=\max\\{1,M\\}$ and $\tau=C_{\Lambda}^{-2}(4\lambda
R^{3})^{-2}$. ∎
### 4.3 Extension to global time
The analysis of previous sections also applies to the truncated equation
$\left\\{\begin{aligned} \partial_{tt}u(x,t)+(m^{2}-\Delta)u(x,t)&=-\lambda
P_{N}{:\\!P_{N}u^{3}\\!:},\\\ u(x,0)&=P_{N}u_{0}(x),\\\
\partial_{t}u(x,0)&=P_{N}u_{0}^{\prime}(x)\end{aligned}\right.$ (4.6)
where $P_{N}$ truncates the Fourier series to terms with frequency at most
$2^{N}$ in absolute value.333Recall that we define the Besov space with a
full-space Fourier transform; the Fourier transform is a linear combination of
Dirac deltas in this case. The estimates are only changed by a constant factor
since the projection operators $P_{N}$ are bounded uniformly in Besov norm,
and the linear operators ${\mathcal{C}}_{t}$ and ${\mathcal{S}}_{t}$ do not
change the Fourier support.
The reason to pass to (4.6) is that the state space now consists of finitely
many Fourier modes. Because the equation is still Hamiltonian, a theorem of
Liouville automatically implies invariance of the corresponding Gibbs measure.
###### Definition 4.9 (Truncated Gibbs measure).
The measure $\mu_{L,N}$ is supported on the subset of
${\mathcal{H}}^{-\varepsilon}(\rho)$ that contains $2L$-periodic functions
Fourier-truncated to ${[{-2^{N}},{2^{N}}]}^{2}$, and is given by the density
$f(u_{0},u_{0}^{\prime})=\exp\left(-\lambda\int{:\\!P_{N}u^{4}_{0}\\!:}\,\mathrm{d}x\right)$
with respect to the periodic, truncated (Gaussian free field, white noise)
measure.
###### Theorem 4.10 (Local-in-time invariance).
The flow of (4.6) is well-defined up to time $\tau$, and the Fourier-truncated
measure $\mu_{L,N}$ is invariant under the flow.
###### Proof.
Existence of solution follows from the previous sections, and invariance of
measure from Liouville’s theorem. ∎
The invariance of measure allows us to probabilistically extend the solution
to arbitrary time. If the local time $\tau$ would be a conserved quantity, we
could simply restart the flow from $u(\tau)$ and get a solution up to time
$2\tau$. This is the case for $L^{2}$ solutions of (NLS). Such a conservation
law does not exist here, but the growth of Besov norm can be controlled in a
high-probability set. This is because the solutions at time $\tau$ are
distributed identically to the initial data.
###### Definition 4.11 (Bounded-moment set).
Let us recall that we denote by ${\mathcal{H}}^{-\varepsilon}(\rho)$ the space
of all initial data $H^{-\varepsilon}(\rho)\times H^{-1-\varepsilon}(\rho)$.
We define
$B_{M}\coloneqq\left\\{(u_{0},u_{0}^{\prime})\in{\mathcal{H}}^{-\varepsilon}(\rho)\colon{\|{:\\!w^{j}\\!:}\|}_{L^{4}([0,1];\;\mathcal{C}^{-\varepsilon/2}(\rho))}\leq
M\text{ for }j=1,2,3\right\\},$
where $w$ is the $L$-periodic linear solution to (4.1) with data
$(u_{0},u_{0}^{\prime})$.
Let us denote the truncated nonlinear flow by $\Phi_{N,t}$, and let $\tau$ be
the solution time from Theorem 4.8 with $M$ as above. Then the set $B_{M}$
contains initial data that lead to a well-defined solution in $[0,\tau]$.
Correspondingly, $\Phi_{N,\tau}^{-1}B_{M}$ contains such data that the
solution exists in $[-\tau,0]$; by intersection with $B_{M}$, we thus get a
solution in $[0,2\tau]$. By iterating this $m$ times, we get a set of initial
data that supports the flow up to time $T=m\tau$.
By overlapping these solution intervals, we can guarantee uniqueness of the
solution. The dependency on period length $L$ is contained in the solution
time $\tau$ as chosen in Theorem 4.8, and thus the next growth bound is
uniform in $L$. However, the number of intersections $m$ depends on $L$ and
moment bound $M$ via $\tau$.
###### Lemma 4.12 (Growth bound).
Let us define
$\mathcal{B}_{M,L,N}\coloneqq
B_{M}\cap\Phi_{N,\tau/2}^{-1}B_{M}\cap\cdots\cap\Phi_{N,\tau/2}^{-2m}B_{M}.$
For $\mu_{L,N}$-almost all $(u_{0},u_{0}^{\prime})\in\mathcal{B}_{M,L,N}$,
there exists a unique solution $u$ to Eq. (4.6) up to time $T=2m\tau$, and
${\left\|{:\\!u^{j}\\!:}\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}\lesssim
T^{1/2}M^{j}$
for $j=1,2,3$. The constant is independent of $N$ and $L$.
###### Proof.
Although the definition of $B_{M}$ uses the non-truncated linear equation, we
may pass to the truncated equation by
${\mathcal{C}}_{t}(P_{N}u_{0})=P_{N}({\mathcal{C}}_{t}u_{0})$ and boundedness
of $P_{N}$ on the periodic space.
That the solution is unique follows from the fact that the local solution
intervals overlap and each local solution is unique. It thus remains to verify
the moment bounds.
For $j=1$ the claim follows immediately from writing $u=v+w$; the nonlinear
part $v$ has $L^{\infty}([k\tau,(k+1)\tau],H^{1-\varepsilon}(\Lambda_{L}))$
norm bounded by $M$ in Theorem 4.8, whereas the linear part satisfies
${\left\|w\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}\lesssim
T^{1/4}M$
by Hölder and the definition of $B_{M}$. For $j=2$ we are to estimate
${\left\|{:\\!w^{2}\\!:}\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}+2{\left\|vw\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}+{\left\|v^{2}\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}.$
Here the only relevant difference is estimating
${\left\|vw\right\|}_{L^{2}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}\lesssim{\left\|v\right\|}_{L^{4}([0,T];\,H^{2\varepsilon}(\Lambda_{L}))}{\left\|w\right\|}_{L^{4}([0,T];\,H^{-\varepsilon}(\Lambda_{L}))}$
with Besov multiplication and Hölder. Thanks to regularity of $v$, we have
${\left\|v^{2}\right\|}_{L^{2}([0,T];\,H^{-2\varepsilon}(\Lambda_{L}))}\lesssim
T^{1/2}{\left\|v\right\|}_{L^{\infty}([0,T];\;H^{1-\varepsilon}(\Lambda_{L}))}^{2}\leq
T^{1/2}M^{2}.$
The case $j=3$ follows similarly. ∎
Moreover, this set of initial data has high probability. Here we use the
finite-dimensional invariance to bound the probabilities.
###### Lemma 4.13 (Data has high probability).
Given $k\in\mathbb{N}$, there exists $M_{k}$ such that
$\mu_{L,N}(\mathcal{B}_{M_{k},L,N})\geq 1-2^{-k}$. The value of $M_{k}$ may
depend on $L$ but not $N$.
###### Proof.
We may first use the triangle inequality and union bound to estimate the
probability of the complement (non-existence of local solution) as
$\displaystyle\mathbb{P}\left(\max_{\begin{subarray}{c}j=1,2,3\\\
k=0,\ldots,m\end{subarray}}{\|{:\\!w_{N}^{j}\\!:}\|}_{L^{4}([k\tau,k\tau+1];\;\mathcal{C}^{-\varepsilon}(\rho))}>M\right)$
$\displaystyle\leq\;$
$\displaystyle\sum_{j=1}^{3}\sum_{k=0}^{m}\mathbb{P}\left({\|{:\\!w_{N}^{j}\\!:}\|}_{L^{4}([k\tau,k\tau+1];\;\mathcal{C}^{-\varepsilon}(\rho))}>M\right).$
This in turn is bounded with Markov’s inequality and invariance:
$\sum_{k=0}^{m}\frac{\mathbb{E}\,{\|{:\\!w_{N}^{j}\\!:}\|}_{L^{4}([k\tau,k\tau+1];\;\mathcal{C}^{-\varepsilon}(\rho))}^{p}}{M^{p}}\lesssim
m\frac{\mathbb{E}\,{\|{:\\!w_{N}^{j}\\!:}\|}_{L^{p}([0,1];\;\mathcal{C}^{-\varepsilon}(\rho))}^{p}}{M^{p}}.$
The expectation is bounded by Lemma 4.4 for any large $p$; this estimate is
uniform in $N$. Now we substitute $m=T/\tau$ and $\tau=C_{L}M^{-C}$ from
Theorem 4.8. By choosing $p$ large enough, our final estimate is
$\mathbb{P}\left(w_{N}\notin\mathcal{D}_{M,N}\right)\lesssim_{L,p}TM^{C-p},$
which vanishes as $M$ is chosen to be large. ∎
### 4.4 Invariance of non-truncated measure
Let us use Lemma 4.13 to rename the sets of initial data defined above:
###### Definition 4.14 (High-probability set of data).
We define the set $\mathcal{D}_{k,L,N}$ to equal $\mathcal{B}_{M_{k},L,N}$
where $M_{k}$ is chosen with Lemma 4.13 such that
$\mu_{L,N}(\mathcal{D}_{k,L,N})\geq 1-2^{-k}$.
We can now take a limit of these sets and get a high-probability set of
initial data without any Fourier truncation. We follow here the argument of
Burq and Tzvetkov [17, Section 6]. We need to drop the regularity a bit in
order to use the compact embedding (Theorem 2.8).
To define the limit measure $\mu_{L}$, we remove the truncation in Definition
4.9. By construction, the measures $\mu_{L,N}$ and $\mu_{L}$ are all
absolutely continuous with respect to the non-truncated (GFF, WN) measure, and
$\mu_{L,N}\to\mu_{L}$ in total variation.
###### Theorem 4.15 (Limiting set of initial data).
We define a subset of ${\mathcal{H}}^{-\varepsilon}(\rho)$ by
$\mathcal{D}_{k,L}\coloneqq\left\\{(u_{0},u_{0}^{\prime})=\lim_{m\to\infty}(u_{0,N_{m}},u_{0,N_{m}}^{\prime})\in\mathcal{D}_{k,L,N_{m}}\text{
for some }N_{m}\to\infty\right\\},$
where the limit is taken in ${\mathcal{H}}^{-2\varepsilon}(\rho)$. Then
$\mu_{L}(\mathcal{D}_{k,L})\geq 1-2^{-k}$. The linear parts of solutions
started from $(u_{0},u_{0}^{\prime})\in\mathcal{D}_{k,L}$ satisfy the same
moment bound as in Lemma 4.13.
###### Proof.
It follows from the definition that
$\limsup_{N\to\infty}\mathcal{D}_{k,L,N}\subset\mathcal{D}_{k,L},$
and then Fatou’s lemma implies
$\displaystyle\mu_{L}(\mathcal{D}_{k,L})$
$\displaystyle\geq\mu_{L}\left(\limsup_{N\to\infty}D_{k,L,N}\right)$
$\displaystyle\geq\limsup_{N\to\infty}\mu_{L}\left(D_{k,L,N}\right)$
$\displaystyle=\limsup_{N\to\infty}\mu_{L,N}\left(D_{k,L,N}\right)$
$\displaystyle\geq 1-2^{-k}.$
Here the equality holds by convergence of $\mu_{L,N}\to\mu_{L}$ in total
variation. The bound for moments follows from continuity of
${\mathcal{S}}_{t}$ and ${\mathcal{C}}_{t}$ and the matching bound for
$\mathcal{D}_{k,L,N}$, uniform in $N$. ∎
To show invariance of the limiting measure as $N\to\infty$, we need to
approximate full solutions by Fourier-truncated solutions. The next lemma
gives convergence in a qualitative sense.
###### Lemma 4.16 (Limit solves NLW).
If $u_{m}\in L^{p}([0,T];\;H^{-\varepsilon}(\rho))$ are solutions to the
truncated equation (4.6) with data $(u_{0,N_{m}},u_{0,N_{m}}^{\prime})$ as in
Theorem 4.15, then $u(x,t)\coloneqq\lim_{m\to\infty}u_{m}(x,t)$ solves the
non-truncated equation (4.1) with the limiting initial data
$(u_{0},u_{0}^{\prime})\in\mathcal{D}_{k,L}$.
###### Proof.
As the solution operators
$({\mathcal{C}}_{t},{\mathcal{S}}_{t})\colon{\mathcal{H}}^{-\varepsilon}(\Lambda_{L})\to
H^{-\varepsilon}(\Lambda_{L})$ are continuous, the linear part converges:
$w(t)={\mathcal{C}}_{t}u_{0}+{\mathcal{S}}_{t}u_{0}^{\prime}=\lim_{m\to\infty}\left({\mathcal{C}}_{t}u_{0,N_{m}}+{\mathcal{S}}_{t}u_{0,N_{m}}^{\prime}\right).$
Let us then consider the integral part in (4.2). We need to show that
$\lim_{m\to\infty}\int_{0}^{t}{\mathcal{S}}_{t-s}\left(P_{N_{m}}{:\\!P_{N_{m}}u_{m}^{3}\\!:}-{:\\!u^{3}\\!:}\right)\\!(x,s)\,\mathrm{d}s=0.$
We write the inner term as
$P_{N_{m}}({:\\!P_{N_{m}}u_{m}^{3}\\!:}-{:\\!u^{3}\\!:})-(1-P_{N_{m}}){:\\!u^{3}\\!:}$
and note that
$\int^{T}_{0}{\left\|(1-P_{N_{m}}){:\\!u^{3}\\!:}\right\|}_{H^{-2\varepsilon}}\,\mathrm{d}s\leq\int^{T}_{0}2^{-N_{m}\varepsilon}{\left\|P_{>N_{m}}{:\\!u^{3}\\!:}\right\|}_{H^{-\varepsilon}}\,\mathrm{d}s$
goes to $0$ as $N_{m}\to\infty$ by the definition of Besov norm. Since
$P_{N_{m}}$ is bounded (uniformly in $N_{m}$) it is sufficient to show that
${:\\!P_{N_{m}}u_{N_{m}}^{3}\\!:}\to{:\\!u^{3}\\!:}\quad\text{in
}L^{1}([0,T];\;H^{-2\varepsilon}(\Lambda_{L})).$
We write $u_{N_{m}}=w_{N_{m}}+v_{N_{m}}$ where
${\left\|v_{N_{m}}\right\|}_{H^{1-\varepsilon}}\lesssim M^{3}$ by
construction. This means that we can extract a subsequence of $v_{N_{m}}$ with
a limit $v=u-w$.
Assume that ${:\\!P_{N_{m}}w^{j}_{N_{m}}\\!:}\to{:\\!w^{j}\\!:}$ for $j=1,2,3$
in $L^{p}([0,T],\mathcal{C}^{-\varepsilon}(\Lambda_{L}))$. Then
${:\\!(P_{N_{m}}u_{N_{m}})^{3}\\!:}\to{:\\!u^{3}\\!:}$ follows from
${:\\!(P_{N_{m}}u)^{3}\\!:}=\sum_{j=0}^{3}{:\\!(P_{N}w)^{j}\\!:}\,v^{3-j}$
and continuity of the products.
We observe that $(u_{0},u_{0}^{\prime})\to{:\\!w^{j}\\!:}$ is a measurable map
from $B_{p,p}^{-\varepsilon}\times B_{p,p}^{-1-\varepsilon}$ to
$L^{p}([0,T],\mathcal{B}^{-\varepsilon}_{p,p})$. Lusin’s theorem implies for
any $\delta>0$ there exists a set $A_{\delta}$ such that
$\nu_{L}(A_{\delta})<\delta$ and ${:\\!w^{j}\\!:}$ depends continuously on
$(u_{0},u_{0}^{\prime})$. Furthermore
${:\\!(P_{N}w)^{j}\\!:}\to{:\\!w^{j}\\!:}$ almost surely, at least up to
subsequence. By Egorov’s theorem we can find a set $\bar{A}_{\delta}$ such
that ${:\\!(P_{N}w)^{j}\\!:}\to{:\\!w^{j}\\!:}$ uniformly and
$\nu_{L}(\bar{A}_{\delta})<\delta$. Then on the complement of
$\tilde{A}\coloneqq A_{\delta}\cup\bar{A}_{\delta}$ we have that (note that
$w$ depends on the initial data $u_{0}$)
$\displaystyle\lim_{m\to\infty}{\left\|{:\\!(P_{N_{m}}w_{N_{m}})^{j}\\!:}-{:\\!w^{j}\\!:}\right\|}_{L^{p}([0,T],B^{-\varepsilon}_{p,p}(\Lambda_{L}))}$
$\displaystyle\leq\;$
$\displaystyle\lim_{m\to\infty}\sup_{u_{0}\in\bar{A}_{\delta}}{\left\|{:\\!(P_{N_{m}}w)^{j}\\!:}-{:\\!w^{j}\\!:}\right\|}_{L^{p}([0,T],B^{-\varepsilon}_{p,p}(\Lambda_{L}))}$
$\displaystyle\quad+{\|{:\\!w_{N_{m}}^{j}\\!:}-{:\\!w^{j}\\!:}\|}_{L^{p}([0,T],B^{-\varepsilon}_{p,p}(\Lambda_{L}))}$
Now the first term goes to $0$ by uniform convergence and the second by
continuity. This implies that on the complement of $\tilde{A}_{\delta}$ the
equation holds. Thus it holds also on $\cup_{\delta>0}\tilde{A}_{\delta}^{c}$,
which is a set of probability $1$. ∎
To quantify the convergence, we derive pointwise bounds in the bounded set
$\mathcal{D}_{k,L}$. These pointwise results follow from Fourier projections
in Besov spaces, although we need to drop the regularity of our target space
by $\varepsilon$. Again, this change is irrelevant since $\varepsilon$ is
arbitrarily small.
###### Theorem 4.17 (Invariance of finite-volume measure).
We have
$\lim_{N\to\infty}\sup_{A}{\left|\mu_{L}(A)-\mu_{L,N}(A)\right|}=0,$
where the supremum is taken over all measurable subsets of
${\mathcal{H}}^{-2\varepsilon}(\Lambda_{L})$. This implies that
$\mu_{L}(\Phi_{t}A)=\mu_{L}(A)$ for all $t\in{[{0},{T}]}$.
###### Proof.
The first statement extends total variation convergence to the product
measure, which holds since the first marginal converges in total variation. We
may assume $A$ to be a subset of $\mathcal{D}_{k,L}$ for $k$ large, since the
complement of $\mathcal{D}_{k,L}$ has probability $2^{-k}$. By the same
reasoning, we may also intersect with the set of probability $1-2^{-k}$ to be
defined below. Moreover, we can bound the difference
${\left|\mu_{L}(\Phi_{t}A)-\mu_{L}(A)\right|}$ by
${\left|\mu_{L,N}(\Phi_{t}^{N}P_{N}A)-\mu_{L,N}(P_{N}A)\right|}+\sum_{s\in\\{0,t\\}}{\left|\mu_{L}(\Phi_{s}A)-\mu_{L,N}(\Phi_{s}^{N}P_{N}A)\right|}.$
Here the first term vanishes by finite-dimensional invariance from Theorem
4.10.
The second term can be split as
${\left|\mu_{L}(\Phi_{s}A)-\mu_{L,N}(\Phi_{s}A)\right|}+{\left|\mu_{L,N}(\Phi_{s}A)-\mu_{L,N}(\Phi_{s}^{N}P_{N}A)\right|},$
where the first part vanishes uniformly in $A$ due to convergence in total
variation. The latter part is the measure of a symmetric difference, which can
be bounded by the measure of $B(0,R)\times H^{-1-2\varepsilon}(\Lambda_{L})$.
Here the radius of $B(0,R)\subset H^{-2\varepsilon}(\Lambda_{L})$ must satisfy
$R=\sup_{u_{0}\in
A}{\left\|\Phi_{s}u_{0}-\Phi_{s}^{N}P_{N}u_{0}\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}.$
This pointwise bound is crude but suffices since we assume
$(u_{0},u_{0}^{\prime})\in\mathcal{D}_{k,L}$. In particular, we do not need to
care about estimating the time derivatives, as the product measure of the
difference is already small by the first factor.
By the uniform bounds on $\mathcal{D}_{k,L}$, we know that the full solution
$u$ and the Fourier-truncated solution $u_{N}$ are well-defined up to time
$T$. We split the difference $(u-u_{N})(t)$ further into three parts in Lemmas
4.18, 4.19, and 4.20 below. Thanks to the uniform moment bounds given in
Theorem 4.15, we get
$\displaystyle R$ $\displaystyle=\sup_{u_{0}\in
A}{\left\|\Phi_{s}u_{0}-\Phi_{s}^{N}P_{N}u_{0}\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}$
$\displaystyle\lesssim 2^{-cN}(M^{3}+\exp(CM^{3}))+H_{N}\exp(CM^{3}).$
This bound is uniform in $A$ and the constant does not depend on $N$. The
first term vanishes as we take $N\to\infty$. Inside the set of probability
$1-2^{-k}$ from Lemma 4.20 we have
$0\leq H_{N}\leq\kappa_{N}\xrightarrow[N\to\infty]{}0,$
so the second term goes to $0$ as well. This means that the ball vanishes in
the limit, and correspondingly $\mu_{L}(B(0,R)\times
H^{-1-2\varepsilon}(\Lambda_{L}))\to 0$. ∎
The first two estimates are deterministic and use Besov space properties;
however, the low frequencies of nonlinear parts may have more complicated
interactions. The probabilistic terms are due to presence of the GFF. Again,
the moment bounds for GFF allow us to control the nonlinear part $v$ in sets
of arbitrarily high probability.
###### Lemma 4.18 (Fourier approximation, linear part).
The linear parts satisfy
${\left\|w(t)-w_{N}(t)\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}\leq
C_{k}2^{-\varepsilon N}$
with probability $1-2^{-k}$.
###### Proof.
Since the operators defining $w$ and $w_{N}$ are Fourier multipliers, the
difference can be written as a projection:
${\left\|w(t)-w_{N}(t)\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}={\left\|P_{>N}w(t)\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}.$
It now follows from the definition of Besov space that
${\left\|P_{>N}w(t)\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}\lesssim
2^{-\varepsilon N}{\left\|w(t)\right\|}_{H^{-\varepsilon}(\Lambda_{L})}.$
The bounds in Lemma 4.4 are $L^{p}$ in time and thus inapplicable here, but by
stationarity it is enough to consider $w(0)$. Since the moments of $\phi^{4}$
are bounded by Lemma 3.17, we can choose $C_{k}$ such that
$\mathbb{P}({\left\|w(t)\right\|}_{H^{-\varepsilon}(\Lambda_{L})}>C_{k})<2^{-k}$.
∎
###### Lemma 4.19 (Fourier approximation, high-frequency nonlinearity).
We have for all $t\leq T$ the estimate
${\left\|\int_{0}^{t}P_{>N}{\mathcal{S}}_{t-s}{:\\!u(x,s)^{3}\\!:}\,\mathrm{d}s\right\|}_{H^{-2\varepsilon}(\Lambda_{L})}\lesssim
2^{-\varepsilon N}M^{3}.$
###### Proof.
By Jensen
${\left\|\int_{0}^{t}P_{>N}{\mathcal{S}}_{t-s}{:\\!u(x,s)^{3}\\!:}\,\mathrm{d}s\right\|}_{H^{-2\varepsilon}}^{2}\leq
t\int_{0}^{t}{\left\|P_{>N}{\mathcal{S}}_{t-s}{:\\!u(x,s)^{3}\\!:}\right\|}_{H^{-2\varepsilon}}^{2}\,\mathrm{d}s.$
Now we can again use the boundedness and pay a little regularity to get
${\left\|P_{>N}{\mathcal{S}}_{t-s}{:\\!u(x,s)^{3}\\!:}\right\|}_{H^{-2\varepsilon}}^{2}\lesssim
2^{-2\varepsilon
N}{\left\|{:\\!u(x,s)^{3}\\!:}\right\|}_{H^{-1-\varepsilon}}^{2}.$
Now the estimate follows from Lemma 4.12. By continuity, the lemma holds also
in the limit. ∎
###### Lemma 4.20 (Fourier approximation, low-frequency nonlinearity).
There exist random variables
$H_{N}\colon{\mathcal{H}}^{-\varepsilon}(\Lambda_{L})\to\mathbb{R}_{+}$ such
that on $\mathcal{D}_{k,L}$ we have
$\left\|\int{\mathcal{S}}_{t-s}\big{[}{:\\!u^{3}\\!:}(s)-P_{N}{:\\!(P_{N}u_{N})^{3}\\!:}(s)\big{]}\,\mathrm{d}s\right\|_{H^{1-2\varepsilon}(\Lambda_{L})}\\!\\!\lesssim\exp(CM^{3})(2^{-\varepsilon
N}+H_{N}),$
and for any $k\in\mathbb{N}$ there exists a sequence $\kappa_{N}\rightarrow 0$
such that
$\nu(H_{N}\geq\kappa_{N})\leq 2^{-k}.$
###### Proof.
Let us rewrite the left-hand side as $\|v-v_{N}\|_{H^{1-2\varepsilon}}$, where
$v_{N}=\int_{0}^{t}{\mathcal{S}}_{t-s}P_{N}{:\\!(P_{N}u_{N}(s))^{3}\\!:}\,\mathrm{d}s,\quad\text{and}\quad
v=\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!u(s)^{3}\\!:}\,\mathrm{d}s.$
By Lemma 4.12 we have
$\|v_{N}(t)\|_{H^{1-\varepsilon}}=\left\|\int_{0}^{t}S_{t-s}P_{N}{:\\!(P_{N}u_{N})^{3}\\!:}(s)\,\mathrm{d}s\right\|_{H^{1-\varepsilon}}\lesssim
M^{3},$
and similarly for the limit $v$. We can split
$\displaystyle\|v-v_{N}\|_{H^{1-2\varepsilon}}$ $\displaystyle=$
$\displaystyle\left\|\int_{0}^{t}S_{t-s}\big{[}{:\\!u^{3}\\!:}(s)-P_{N}{:\\!(P_{N}u_{N})^{3}\\!:}(s)\big{]}\,\mathrm{d}s\right\|_{H^{1-2\varepsilon}}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\|{:\\!u^{3}\\!:}(s)-{:\\!(P_{N}u_{N})^{3}\\!:}(s)\|_{H^{-2\varepsilon}}\,\mathrm{d}s+\left\|\int_{0}^{t}P_{>N}{:\\!(P_{N}u_{N})^{3}\\!:}(s)\,\mathrm{d}s\right\|_{H^{1-2\varepsilon}}$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{t}\|{:\\!u^{3}\\!:}(s)-{:\\!(P_{N}u_{N})^{3}\\!:}(s)\|_{H^{-2\varepsilon}}\,\mathrm{d}s+2^{-\varepsilon
N}M^{3},$
where the second term is bounded as in Lemma 4.19. We expand the remaining
integrand to get
${:\\!u^{3}\\!:}-{:\\!(P_{N}u_{N})^{3}\\!:}=\sum_{j=0}^{3}\binom{3}{j}({:\\!w^{j}\\!:}v^{3-j}-{:\\!w^{j}_{N}\\!:}(P_{N}v_{N})^{3-j}),$
and further split it into
$\displaystyle{:\\!w^{j}\\!:}v^{3-j}-{:\\!w^{j}_{N}\\!:}(P_{N}v_{N})^{3-j}$
$\displaystyle=({:\\!w^{j}\\!:}-{:\\!w^{j}_{N}\\!:})(P_{N}v_{N})^{3-j}$
$\displaystyle\quad{}+{:\\!w^{j}\\!:}(v^{3-j}-(P_{N}v_{N})^{3-j}).$
Now the first term is bounded by
$\|({:\\!w^{j}\\!:}-{:\\!w^{j}_{N}\\!:})(P_{N}v_{N})^{3-j}\|_{H^{-2\varepsilon}}\lesssim\|({:\\!w^{j}\\!:}-{:\\!w^{j}_{N}\\!:})\|_{H^{-2\varepsilon}}M^{3},$
and in the nontrivial cases $j=1,2$ we can bound the second term by
$\displaystyle\|{:\\!w^{j}\\!:}(v^{3-j}-(P_{N}v_{N})^{3-j})\|_{H^{-2\varepsilon}}$
$\displaystyle\lesssim$
$\displaystyle\|{:\\!w^{j}\\!:}\|_{\mathcal{C}^{-\varepsilon}}\|v^{3-j}-(P_{N}v_{N})^{3-j}\|_{H^{2\varepsilon}}$
$\displaystyle\lesssim$
$\displaystyle\|{:\\!w^{j}\\!:}\|_{\mathcal{C}^{-\varepsilon}}\|v-P_{N}v_{N}\|_{H^{1-2\varepsilon}}M^{3}.$
By again separating the high-frequency part, the final factor equals
$\|v-P_{N}v_{N}\|_{H^{1-2\varepsilon}}\lesssim\|v-v_{N}\|_{H^{1-2\varepsilon}}+2^{-\varepsilon
N}M^{3}.$
Now setting
$H_{N}\coloneqq\sup_{j\leq
3}\int^{T}_{0}\|({:\\!w^{j}\\!:}-{:\\!w^{j}_{N}\\!:})(s)\|_{\mathcal{C}^{-\varepsilon}}\,\mathrm{d}s,$
we finally obtain
$\displaystyle\|(v-v_{N})(t)\|_{H^{1-2\varepsilon}}$ $\displaystyle\lesssim\;$
$\displaystyle M^{3}\bigg{[}2^{-\varepsilon
N}+H_{N}+\int_{0}^{t}\sum_{j=1}^{2}\|{:\\!w^{j}\\!:}(s)\|_{\mathcal{C}^{-\varepsilon}}\|(v-v_{N})(s)\|_{H^{1-2\varepsilon}}\,\mathrm{d}s\bigg{]}.$
Then Grönwall’s lemma gives that
$\displaystyle\|(v-v_{N})(t)\|_{H^{1-2\varepsilon}}$ $\displaystyle\lesssim
M^{3}(2^{-\varepsilon
N}+H_{N})\exp\left(\int^{T}_{0}\sum_{j=1}^{2}\|{:\\!w^{j}\\!:}(s)\|_{C^{-\varepsilon}}\mathrm{\,\mathrm{d}s}\right)$
$\displaystyle\lesssim\exp(CM^{3})(2^{-\varepsilon N}+H_{N}).\qed$
## 5 Global invariance of NLW
In any bounded region the behaviour of (NLW) only depends on the light cone,
and we are free to use the periodic solution theory. Within this bounded
region, it is impossible to distinguish between flows of different period
length (as long as the period is sufficiently long). We use this property to
pass the period length to limit.
In Section 5.1 we show that all statements about measurable events can be
reduced back to the periodic case. We then show in Section 5.2 that the
unperiodic flow can be approximated by periodic solutions started from
periodic data. The main results are finally proved in Section 5.3.
Let us first reference some more properties of weak convergence, in addition
to those defined in Section 3.2. We use these implicitly in the following.
###### Lemma 5.1 (Weak limits in product spaces; [7, Theorem 2.8]).
Assume that $\mathcal{X}\times\mathcal{X}^{\prime}$ is separable. Then
$(\mu_{L}\times\mu_{L}^{\prime})$ converges weakly to $\mu\times\mu^{\prime}$
if and only if $(\mu_{L})$ and $(\mu_{L}^{\prime})$ converge weakly to $\mu$
and $\mu^{\prime}$ respectively.
###### Lemma 5.2 (Skorokhod’s theorem; [7, Theorem 6.7]).
Suppose that $(\mu_{L})$ converge weakly to $\mu$ supported on a separable
space. Then there exist a common probability space $\tilde{\mathbb{P}}$ and
random variables $X_{L}$, $X$ such that $\text{Law}(X_{L})=\mu_{L}$,
$\text{Law}(X)=\mu$, and $X_{L}\to X$ almost surely.
### 5.1 Reduction to bounded domain
The Borel $\sigma$-algebra of $\mathbb{R}^{2}$ can be generated by compact
sets, or even just closed balls. We will show below an analogous result for
the Borel $\sigma$-algebra of ${\mathcal{H}}^{-2\varepsilon}(\rho)$: the
$\sigma$-algebra is generated by restrictions of distributions to bounded
domains.
###### Theorem 5.3 ($\sigma$-algebra generated by bounded-domain functions).
Let $s,s^{\prime}\in\mathbb{R}$, and let $\mathcal{A}^{s}$ be the family of
sets that where inclusion only depends on restrictions to compact domains:
$\mathcal{A}^{s}\coloneqq\left\\{A\subset H^{s}(\rho)\colon\exists\text{
compact }D\text{ s.t. }f\in A\Longleftrightarrow f|_{D}\in A\quad\forall f\in
H^{s}(\rho)\right\\}.$
That is, $\mathbf{1}_{A}(f)=g_{A}(f|_{D})$ for some
$g_{A}\colon{\mathcal{H}}^{-2\varepsilon}(D)\to\\{0,1\\}$. Then
1. 1.
the closed ball $\bar{B}=\bar{B}(f,R)\subset H^{s}(\rho)$ can be constructed
with $\sigma$-closed operations from sets in $\mathcal{A}^{s}$;
2. 2.
the Borel $\sigma$-algebra of $H^{s}(\rho)$ is a sub-$\sigma$-algebra of
$\sigma(\mathcal{A}^{s})$;
3. 3.
the Borel $\sigma$-algebra of $H^{s}(\rho)\times H^{s^{\prime}}(\rho)$ is a
sub-$\sigma$-algebra of
$\sigma(\mathcal{A}^{s})\times\sigma(\mathcal{A}^{s^{\prime}})$.
###### Proof.
By the definitions of Borel $\sigma$-algebra and product $\sigma$-algebra, it
is enough to verify the first statement for arbitrary $\bar{B}=\bar{B}(f,R)$
and $s\in\mathbb{R}$. We can write
$\displaystyle\bar{B}$ $\displaystyle=\left\\{g\in
H^{s}(\rho)\colon\int_{\mathbb{R}^{2}}\rho(x)^{2}{\left|(1-\Delta)^{s/2}(f-g)\right|}^{2}(x)\,\mathrm{d}x\leq
R^{2}\right\\}$ $\displaystyle=\left\\{g\in
H^{s}(\rho)\colon\int_{\mathbb{R}^{2}}\rho(x)^{2}{\left|\int_{\mathbb{R}^{2}}K_{s}(x-y)(f-g)(y)\,\mathrm{d}y\right|}^{2}\,\mathrm{d}x\leq
R^{2}\right\\}$ $\displaystyle=\limsup_{N\to\infty}\bigg{\\{}g\in
H^{s}(\rho)\colon\sum_{\ell,m,n=1}^{N}\int_{A_{\ell}}\\!\rho(x)^{2}\left[\int_{A_{m}}K_{s}(x-y)(f-g)(y)\,\mathrm{d}y\right]$
$\displaystyle\hskip
150.00023pt\left[\int_{A_{n}}K_{s}(x-y)(f-g)(y)\,\mathrm{d}y\right]\,\mathrm{d}x\leq
R^{2}\bigg{\\}}.$
Here we denote by $K_{s}$ the convolution kernel of $(1-\Delta)^{s/2}$ and by
$(A_{j})_{j\in\mathbb{N}}$ some partitioning of $\mathbb{R}^{2}$, e.g. by unit
squares. For finite $N$, the set thus depends on $K_{s}$, $f$, and $g$ only
inside the compact set $\cup_{m,n=1}^{N}\overline{(A_{m}+A_{n})}$. Since
$\limsup$ is a $\sigma$-closed operation, this proves that $\bar{B}$ can be
constructed from sets in $\mathcal{A}^{s}$. ∎
We need to combine this result with the definition of weak limit. If we a
priori assume the weak limit to exist, we can show uniqueness by testing
against a much smaller class of test functions. This strategy of adapting the
test functions to the specific model is very common; see the book of Ethier
and Kurtz [21, Section 3.4] for details.
###### Corollary 5.4 (Reduction to bounded domains).
Let $\mathcal{F}$ be the set of bounded Lipschitz functions
$\varphi\colon{\mathcal{H}}^{-2\varepsilon}(\rho)\to\mathbb{R}$ that depend
only on the restriction of argument to some compact domain: for any
$\varphi\in\mathcal{F}$, there exists a compact $D\subset\mathbb{R}^{2}$ such
that $\varphi(f)=\varphi(f|_{D})$ for all
$f\in{\mathcal{H}}^{-2\varepsilon}(\rho)$.
Assume that the sequence $(\mu_{L})$ has a weak limit $\mu^{*}$, and let $\mu$
be a Borel probability measure on ${\mathcal{H}}^{-2\varepsilon}(\rho)$. If
$\lim_{L\to\infty}\int_{\Omega}\varphi(f)\,\mathrm{d}\mu_{L}(f)=\int_{\Omega}\varphi(f)\,\mathrm{d}\mu(f)$
for all $\varphi\in\mathcal{F}$, then the weak limit $\mu^{*}$ equals $\mu$.
###### Proof.
Fix two distinct points $(f,f^{\prime})$ and $(g,g^{\prime})$ in
${\mathcal{H}}^{-2\varepsilon}(\rho)$. By the general theory of distributions,
there exist $\alpha,\beta\in C_{c}^{\infty}(\mathbb{R}^{2})$ such that
${\left\langle\alpha,f-g\right\rangle}\neq 0$ or
${\left\langle\beta,f^{\prime}-g^{\prime}\right\rangle}\neq 0$. Then
$\eta(f,f^{\prime})\coloneqq{\left|{\left\langle\alpha,f\right\rangle}\right|}+{\left|1+{\left\langle\alpha,f\right\rangle}\right|}+{\left|{\left\langle\beta,f^{\prime}\right\rangle}\right|}+{\left|1+{\left\langle\beta,f^{\prime}\right\rangle}\right|}$
depends only on the compact domain
$\operatorname{Supp}\alpha\cup\operatorname{Supp}\beta$, is continuous, takes
only non-negative values, and $h(f,f^{\prime})\neq h(g,g^{\prime})$.
Continuity follows from linearity and Theorem 2.5. Furthermore
$\tilde{\eta}(f,f^{\prime})\coloneqq\frac{\eta(f,f^{\prime})}{1+\eta(f,f^{\prime})}$
is bounded and composed of Lipschitz functions; thus it belongs to
$\mathcal{F}$.
As $(f,f^{\prime})$ and $(g,g^{\prime})$ were arbitrary, we have shown
$\mathcal{F}$ to separate points. Then [21, Theorem 3.4.5] implies that
$\mathcal{F}$ is a separating family, meaning that
$\int_{\Omega}\varphi(f)\,\mathrm{d}\mu^{*}(f)=\int_{\Omega}\varphi(f)\,\mathrm{d}\mu(f)\quad\text{for
all }\varphi\in\mathcal{F}$
implies $\mu^{*}=\mu$. The claim follows from the uniqueness of weak limit. ∎
### 5.2 Invariance of measure
Assuming that the period is large enough, the solution in a bounded domain $D$
is independent of the choice of periodization. However, the initial data
sampled from $\mu_{L}$ still depends on the period length. In this section we
quantify the convergence of solutions.
Let us first construct a probabilistic solution set associated with the region
$D\subset\mathbb{R}^{2}$. This argument is analogous to Theorem 4.13, but with
a twist: by Theorem 4.8 the growth bound in $D$ is independent of the
periodization, but the local solution time $\tau$ is not. However, at discrete
times $\\{k\tau\\}$ we can use the invariance of measure; this property is
qualitative and holds for all period lengths.
###### Theorem 5.5 (Limiting set of data, polynomial weight).
Fix a bounded domain $D\subset\mathbb{R}^{2}$. There exists a set
$\mathcal{E}_{k}\subset{\mathcal{H}}^{-2\varepsilon}(\rho)$ and $M_{k}>0$ such
that $\mu(\mathcal{E}_{k})>1-2^{-k}$ and for almost all
$(u_{0},u_{0}^{\prime})\in\mathcal{E}_{k}$ the flow $\Phi_{t}$ of (NLW) is
bounded by $CTM_{k}^{c}$ in $L^{2}([0,T]\;H^{-2\varepsilon}(D))$.
###### Proof.
Let us consider periodic initial data $(u_{0,L},u^{\prime}_{0,L})$ sampled
from $\mu_{L}$, and define the probabilistic data sets
$\mathcal{E}_{L,M}(D)\coloneqq\left\\{{\left\|\Phi_{L,t}u_{0,L}\right\|}_{L^{2}({[{0},{T}]};\;H^{-2\varepsilon}(D))}\leq
M\right\\}.$
We extend this definition as $\mathcal{E}_{\infty,M}$ for non-periodic data
sampled from $\mu$. By Skorokhod’s lemma we may assume that $u_{0,L}\to u_{0}$
almost surely. To estimate the probability of this limit, we first use Fatou’s
lemma:
$\mathbb{P}\left(\mathcal{E}_{\infty,M}\right)=\tilde{\mathbb{P}}\left(\mathcal{E}_{\infty,M}\right)\geq\tilde{\mathbb{P}}\left(\limsup_{L\to\infty}\mathcal{E}_{L,M}\right)\geq\limsup_{L\to\infty}\tilde{\mathbb{P}}\left(\mathcal{E}_{L,M}\right).$
Let us now bound the norm of the flow. We have
$D+B(0,T)\subset\Lambda_{R},\Lambda_{L}$ for $R$ sufficiently large and $L\geq
R$; thus the restriction norm is not able to distinguish between $\Phi_{R,t}$
and $\Phi_{L,t}$. By moving between these two flows, we can extract the
solution time $\tau$ to depend on $R$ (fixed) and not $L$ (divergent).
First, we have the following bound for any $\tau>0$:
$\displaystyle{\left\|\Phi_{L,t}u_{0,L}\right\|}_{L^{2}({[{0},{T}]};\;H^{-2\varepsilon}(D))}^{2}$
$\displaystyle={\left\|\Phi_{R,t}u_{0,L}\right\|}_{L^{2}({[{0},{T}]};\;H^{-2\varepsilon}(D))}^{2}$
$\displaystyle=\sum_{k=0}^{T/\tau}{\left\|\Phi_{R,t}u_{0,L}\right\|}_{L^{2}({[{k\tau},{(k+1)\tau}]};\;H^{-2\varepsilon}(D))}^{2}$
$\displaystyle=\sum_{k=0}^{T/\tau}{\left\|\Phi_{R,t}\Phi_{L,k\tau}u_{0,L}\right\|}_{L^{2}({[{0,\tau}]};\;H^{-2\varepsilon}(D))}^{2}.$
Let us show that the $R$-periodic flow is valid. Assume that
${\left\|{:\\!(\Phi_{\text{lin}}\Phi_{L,k\tau}u_{0,L})^{j}\\!:}\right\|}_{L^{2}({[{0,1}]};\;H^{-2\varepsilon}(D))}^{2}\leq
M$
for $j=1,2,3$ and all $k\leq T/\tau$, where $\tau$ is chosen as per Theorem
4.8 for the domain $\Lambda_{R}$ (fixed) and moment bound $M$. Then the local
growth bound in Lemma 4.12 gives that $\Phi_{R}$ is valid up to time $\tau$
and
${\left\|\Phi_{R,t}\Phi_{L,k\tau}u_{0,L}\right\|}_{L^{2}({[{0,\tau}]};\;H^{-2\varepsilon}(D))}\leq
2M.$
Correspondingly the full sum satisfies
${\left\|\Phi_{L,t}u_{0,L}\right\|}_{L^{2}({[{0},{T}]};\;H^{-2\varepsilon}(D))}^{2}\lesssim\frac{TM^{2}}{\tau}\lesssim
TM^{2+c}.$
The assumption made above is probabilistic, and as before our estimate for the
sum holds with probability at least
$1-\mathbb{P}\left(\max_{\begin{subarray}{c}j=1,2,3\\\
k=0,\ldots,T/\tau\end{subarray}}{\left\|{:\\!(\Phi_{\text{lin}}\Phi_{L,k\tau}u_{0,L})^{j}\\!:}\right\|}_{L^{2}({[{0,1}]};\;H^{-2\varepsilon}(D))}^{2}>M\right).$
It is here that we use the invariance of $\mu_{L}$ under $\Phi_{L}$. As in
Lemma 4.13, we bound the probability from below by
$1-CTM^{c}\frac{\mathbb{E}\,{\left\|{:\\!(\Phi_{\text{lin}}u_{0,L})^{j}\\!:}\right\|}_{L^{p}({[{0,1}]};\;H^{-2\varepsilon}(D))}^{p}}{M^{p}}.$
The expectation is bounded by Lemma 4.4 uniformly in $L$. Again we choose $p$
large enough to make the power on $M$ negative and then pass to $M$ large
enough. ∎
As we approximate full-space solutions by periodic ones, we also need to
estimate the error made. The following result is analogous to Lemma 4.20. In
contrast to Theorem 4.17, now we also need to estimate the time derivatives.
However, the estimate for time derivatives is bootstrapped from the one for
the $\phi^{4}$ component.
###### Lemma 5.6 (Stability, $\phi^{4}$ component).
Assume that
$\max_{j=1,2,3}{\left\|{:\\!w_{L}^{j}\\!:}\right\|}_{L^{4}({[0,1]};\mathcal{C}^{-2\varepsilon}(\rho))}\leq
M$
holds for all $L$ and in the limit. Fix a bounded domain
$D\subset\mathbb{R}^{2}$, and define the spatial cutoff $\chi_{t}$ as a
mollified indicator of $D$ and $\chi_{t-s}$ as that of $D+B(0,s+1)$. Then the
random solutions satisfy
${\left\|\chi_{t}(u_{L}-u)(t)\right\|}_{H^{-2\varepsilon}(\rho)}\lesssim{\left\|\chi_{0}(u_{0,L},u_{0,L}^{\prime})-\chi_{0}(u_{0},u_{0}^{\prime})\right\|}_{{\mathcal{H}}^{-2\varepsilon}(\rho)}+\exp(CM^{3})H_{L},$
where $H_{L}$ are random variables satisfying: for any $k\in\mathbb{N}$ there
exist $\kappa_{L}\to 0$ such that $\tilde{\mathbb{P}}(H_{L}\geq\kappa_{L})\leq
2^{-k}$.
###### Proof.
The first term on the right-hand side comes from continuity of the linear
solution operators (Lemma 4.2). For the nonlinear term, our goal is to show
$\chi_{t}\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!u(s)^{3}\\!:}\,\mathrm{d}s=\lim_{L\to\infty}\chi_{t}\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!u_{L}(s)^{3}\\!:}\,\mathrm{d}s.$
We can first apply finite speed of propagation to ${\mathcal{S}}_{t-s}$ to
reduce
$\displaystyle\int_{0}^{t}{\left\|\chi_{t}{\mathcal{S}}_{t-s}[{:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:}]\right\|}_{H^{1-2\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle\int_{0}^{t}{\left\|\chi_{t-s}[{:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:}]\right\|}_{H^{1-2\varepsilon}(\rho)}.$
We then perform the same manipulations as in Lemma 4.20, only replacing the
Fourier cutoff $N$ by the period length $L$ and keeping track of the weights.
Thanks to the bounded domain, we can always re-introduce $\rho$ such as in
$\displaystyle{\left\|{:\\!w^{2}(s)\\!:}(v-v_{L})(s)\chi_{t-s}\right\|}_{H^{-2\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle{\left\|\chi^{\prime}{:\\!w^{2}(s)\\!:}\right\|}_{\mathcal{C}^{-2\varepsilon}(1)}{\left\|\chi_{t-s}(v-v_{L})(s)\right\|}_{H^{1-2\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle{\left\|\chi^{\prime}\right\|}_{\mathcal{C}^{1}(\rho^{-1})}{\left\|{:\\!w^{2}(s)\\!:}\right\|}_{\mathcal{C}^{-2\varepsilon}(\rho)}{\left\|\chi_{t-s}(v-v_{L})(s)\right\|}_{H^{1-2\varepsilon}(\rho)},$
where $\chi^{\prime}$ is a smooth cutoff of some larger domain. We also define
the stochastic term
$H_{L}\coloneqq\sup_{j\leq
3}\int_{0}^{t}{\left\|({:\\!w^{j}\\!:}-{:\\!w_{L}^{j}\\!:})(s)\right\|}_{\mathcal{C}^{-2\varepsilon}(\rho)}\,\mathrm{d}s$
as a variation of that in Lemma 4.20. Thus we have bounded
$\displaystyle{\left\|\chi_{t}(v-v_{L})(t)\right\|}_{H^{1-2\varepsilon}(\rho)}$
$\displaystyle\lesssim\;$ $\displaystyle
M^{3}\left[H_{L}+\int_{0}^{t}\sum_{j=1}^{2}{\left\|{:\\!w^{j}\\!:}\right\|}_{\mathcal{C}^{-2\varepsilon}(\rho)}{\left\|\chi_{t-s}(v-v_{L})(t-s)\right\|}_{H^{1-2\varepsilon}(\rho)}\right],$
and again Grönwall gives
${\left\|\chi_{t}(v-v_{L}(t))\right\|}_{H^{1-2\varepsilon}(\rho)}\lesssim\exp(CM^{3})H_{L}.\qed$
###### Lemma 5.7 (Stability, time derivative).
Fix $k,\ell\in\mathbb{N}$, and assume that
$\lim_{L\to\infty}{\left\|\chi(u_{L}-u)(t)\right\|}_{H^{-2\varepsilon}(\rho)}=0,$
where $\chi$ is any cutoff. Then we have
${\left\|\chi\partial_{t}(u_{L}-u)(t)\right\|}_{H^{-1-2\varepsilon}(\rho)}\leq
2^{-\ell}$
in a set of probability at least $1-2^{-k}$.
###### Proof.
Let us first note that by passing to the mild formulation we have
$\lim_{s\to 0}\frac{u(t+s)-u(t)}{s}=\lim_{s\to
0}\left[\frac{1}{s}\int_{t}^{t+s}{\mathcal{S}}_{t-r}{:\\!u(r)^{3}\\!:}\,\mathrm{d}r+\frac{{\mathcal{C}}_{t+s}-{\mathcal{C}}_{t}}{s}u_{0}+\frac{{\mathcal{S}}_{t+s}-{\mathcal{S}}_{t}}{s}u_{0}^{\prime}\right].$
The first term converges almost surely to ${:\\!u(t)^{3}\\!:}$. The limit of
the last two terms is a bounded linear operator $\mathcal{L}$ from
${\mathcal{H}}^{-2\varepsilon}(\rho)$ to $H^{-1-2\varepsilon}(\rho)$, as can
be seen by considering the Fourier multiplier symbols.
We estimate the difference of Wick powers with Lemma 3.11:
${\left\|{:\\!u_{L}(t)^{3}\\!:}-{:\\!u(t)^{3}\\!:}\right\|}_{H^{-2\varepsilon}(\rho)}\leq
J_{L}+{\left\|f^{\delta}(u_{L}(t))-f^{\delta}(u(t))\right\|}_{H^{-2\varepsilon}(\rho)}+J_{\infty},$
where
$J_{L}\coloneqq{\left\|u_{L}(t)^{3}-f^{\delta}(u_{L}(t))\right\|}_{\mathcal{H}^{-2\varepsilon}(\rho)}$
and $J_{\infty}$ is defined analogously. By Lemma 3.11 the expectations of
$J_{L}$ and $J_{\infty}$ vanish as $\delta\to 0$; we get two $2^{-\ell}/3$
terms with probability $1-2^{-k}$ by fixing $\delta$ small enough. We then
bound the middle term by $2^{-\ell}/3$ by continuity of $f^{\delta}$, choosing
$L$ to be large enough. ∎
### 5.3 Invariance proof finished
###### Theorem 5.8 (Global invariance).
We have $\mu\circ\Phi_{t}=\mu$ for all $0\leq t\leq T$.
###### Proof.
We know a priori that the pushforward measure $\mu\circ\Phi_{t}$ exists as
$\Phi_{t}$ is a measurable map. (By Theorem 5.3, we only need to check
restrictions to bounded domains. There $\Phi_{t}$ is almost surely defined as
a composition of small-time periodic flows.)
By the weak limit and finite-volume invariance, we also have that for all
bounded and continuous
$f\colon{\mathcal{H}}^{-2\varepsilon}(\rho)\to\mathbb{R}$,
$\int_{{\mathcal{H}}}\\!f(u_{0},u_{0}^{\prime})\,\mathrm{d}\mu=\lim_{L\to\infty}\int_{{\mathcal{H}}}f(\Phi_{L,t}(u_{L,0},u_{L,0}^{\prime}))\,\mathrm{d}\mu_{L}.$
Since the weak limit is unique, we only need to show that
$\lim_{L\to\infty}\int_{{\mathcal{H}}}f(\Phi_{L,t}(u_{L,0},u_{L,0}^{\prime}))\,\mathrm{d}\mu_{L}=\int_{{\mathcal{H}}}f(\Phi_{t}(u_{0},u_{0}^{\prime}))\,\mathrm{d}\mu.$
Corollary 5.4 lets us assume that $f$ is Lipschitz and depends on the
restriction of its arguments to some $\Lambda_{R}$. We can further pass to a
common probability space by Lemma 5.2.
Let $\mathcal{F}_{L,M}$ be the intersection of those data sets such that both
the full and $L$-periodic flow have moment bound $M$ in $\Lambda_{R}$ by
Theorem 5.5, that $H_{L}\leq 2^{-\ell}$ in Lemma 5.6, and that the $2^{-\ell}$
bound in Lemma 5.7 holds. This set has probability at most $C2^{-k}$ when $M$
and $L$ are large enough. Let us recall that $M$ only depends on $R$ and not
$L$. We can then estimate
$\displaystyle\lim_{L\to\infty}\tilde{\mathbb{E}}\,{\left|f(\Phi_{L,t}(u_{L,0},u_{L,0}^{\prime}))-f(\Phi_{t}(\tilde{u}_{0},\tilde{u}_{0}^{\prime}))\right|}$
$\displaystyle\leq\;$
$\displaystyle\lim_{L\to\infty}\tilde{\mathbb{E}}\,{\left|\mathbf{1}_{\mathcal{F}_{L,M}}[f(\Phi_{L,t}(u_{L,0},u_{L,0}^{\prime}))-f(\Phi_{t}(\tilde{u}_{0},\tilde{u}_{0}^{\prime}))]\right|}+2^{-k}{\left\|f\right\|}_{\infty}$
$\displaystyle=\;$
$\displaystyle\lim_{L\to\infty}\tilde{\mathbb{E}}\,{\left|\mathbf{1}_{\mathcal{F}_{L,M}}[f(\Phi_{L,t}(\tilde{u}_{L,0},\tilde{u}_{L,0}^{\prime}))-f(\Phi_{L,t}(\tilde{u}_{0},\tilde{u}_{0}^{\prime}))]\right|}+2^{-k}{\left\|f\right\|}_{\infty}$
$\displaystyle\leq\;$
$\displaystyle\lim_{L\to\infty}\text{Lip}_{f}\tilde{\mathbb{E}}\,\mathbf{1}_{\mathcal{F}_{L,M}}{\left\|\chi[\Phi_{L,t}(\tilde{u}_{L,0},\tilde{u}_{L,0}^{\prime})-\Phi_{L,t}(\tilde{u}_{0},\tilde{u}_{0}^{\prime})]\right\|}_{{\mathcal{H}}^{-2\varepsilon}(\rho)}+2^{-k}{\left\|f\right\|}_{\infty}.$
Here we used respectively boundedness, that $f$ cannot distinguish between
$\Phi_{t}$ and $\Phi_{L,t}$ once $L\geq R+T$, and Lipschitz continuity. The
spatial cutoff $\chi$ depends on $f$ through $R$.
The two components of ${\mathcal{H}}^{-2\varepsilon}(\rho)$ are now estimated
with Lemmas 5.6 and 5.7: The first component is bounded by
$\lim_{L\to\infty}\tilde{\mathbb{E}}\,{\left\|\chi[(\tilde{u}_{L,0},\tilde{u}_{L,0}^{\prime})-(\tilde{u}_{0},\tilde{u}_{0}^{\prime})]\right\|}_{{\mathcal{H}}^{-2\varepsilon}(\rho)}+\exp(CM^{3})2^{-\ell}.$
The initial data converges almost surely and uniform integrability allows us
to commute the limit and expectation. By increasing $\ell$ and $L$ as
necessary, we can make this term arbitrarily small. This then allows us to
bound the second component. ∎
We can rephrase the invariance as a global existence and uniqueness result as
in the introduction. The same result holds for the second and third powers of
the solution since the corresponding $\phi^{4}$ moments are finite.
###### Proof of Theorem 1.1.
We have now showed that a limit exists in any compact region of
$\mathbb{R}^{2}$, and that the product measure $\mu$ is invariant under the
flow. The latter implies that
$\mathbb{E}\,{\left\|\Phi_{t}u_{0}\right\|}_{L^{2}([0,T];\;H^{-2\varepsilon}(\rho))}^{2}=\int_{0}^{T}\mathbb{E}\,{\left\|u_{0}\right\|}_{H^{-2\varepsilon}(\rho)}^{2}\,\mathrm{d}t\lesssim
T,$
meaning that the norm is almost surely finite. We still need to check that the
limit really solves (NLW).
The convergence of linear parts follows from Lemma 4.5. We only need to show
that
$v(t)=\int_{0}^{t}{\mathcal{S}}_{t-s}{:\\!u(s)^{3}\\!:}\,\mathrm{d}s.$
Equivalently, we can show that
$\lim_{L\to\infty}{\left\|v-v_{L}\right\|}_{H^{1-2\varepsilon}(\rho)}\lesssim\lim_{L\to\infty}\int_{0}^{t}{\left\|{\mathcal{S}}_{s-t}[{:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:}]\right\|}_{H^{1-2\varepsilon}(\rho)}\,\mathrm{d}s$
vanishes as $L\to\infty$. Since ${\mathcal{S}}_{t-s}$ is bounded by Lemma 4.2
and ${:\\!u^{3}\\!:}$ is assumed to belong to $H^{-2\varepsilon}(\rho)$, we
can use dominated convergence. Then we are left with estimating
$\lim_{L\to\infty}{\left\|\sum_{j=0}^{3}\binom{3}{j}\left[{:\\!w(s)^{j}\\!:}v(s)^{3-j}-{:\\!w_{L}(s)^{j}\\!:}v_{L}(s)^{3-j}\right]\right\|}_{H^{-2\varepsilon}(\rho)}.$
As $w_{L}$ converges in $L^{p}([0,T];H^{-2\varepsilon}(\rho))$ and $v_{L}$
converges in $C([0,T];H^{1-2\varepsilon}(\rho))$, and the product is a
continuous operator, we can pass $w_{L}$ and $v_{L}=u_{L}-w_{L}$ to the limit.
This shows that $v$ satisfies the mild formulation. ∎
## 6 Weak invariance of NLS
The situation is more complicated for the nonlinear Schrödinger equation
$\left\\{\begin{aligned}
i\partial_{t}u_{L}+(m^{2}-\Delta)u_{L}&=-\lambda{:\\!u_{L}^{3}\\!:},\\\
u_{L}(0)&\sim\phi_{2,L}^{4},\end{aligned}\right.$ (6.1)
on $\Lambda_{L}\times\mathbb{R}$. Invariance of the periodic $\phi^{4}_{2}$
measure under this equation was shown already by Bourgain [9] – see also the
more pedagogic review [45] – but our preceding extension argument is broken
for two reasons.
The linear solution operator
${\mathcal{T}}_{t}u\coloneqq\exp(it\Delta)u\coloneqq\mathcal{F}^{-1}\left[\exp(-it|\xi|^{2})\hat{u}(\xi)\right]$
(6.2)
does not increase the regularity of its argument. Therefore the mild solution
$u(t)={\mathcal{T}}_{t}u(0)+\lambda\int_{0}^{t}[{\mathcal{T}}_{t-s}{:\\!u(s)^{3}\\!:}](x)\,\mathrm{d}s$
(6.3)
is not amenable to the fixpoint argument of Section 4. Moreover, NLS does not
possess finite speed of propagation: wave packets propagate at a speed
proportional to their frequency squared. This means that the argument in
Section 5 is not applicable either.
However, if we can accept some loss of regularity, we can still use the
previous tightness argument. That allows us to approximate full-space
solutions by (a subsequence of) periodic solutions. This sense of invariance
was introduced by Albeverio and Cruzeiro [2] in the context of Navier–Stokes
equations.
###### Remark 6.1.
Unlike with the wave equation, here the initial data is complex-valued. Our
construction of the $\phi^{4}$ measure holds also in this setting and all
results from Section 3 carry over.
Compactness is given by a version of the usual embedding theorem for Hölder-
continuous functions:
###### Lemma 6.2 (Arzelà–Ascoli theorem).
The Hölder space $C^{\alpha}([0,T];H^{s}(\rho))$ is defined by the norm
${\left\|f\right\|}_{C^{\alpha}([0,T];\;H^{s}(\rho))}\coloneqq\sup_{0\leq
s\neq t\leq
T}\frac{{\left\|f(t)-f(s)\right\|}_{H^{s}(\rho)}}{{\left|t-s\right|}^{\alpha}}.$
Then the embedding
$C^{2\varepsilon}([0,T];\;H^{s}(\rho))\hookrightarrow
C^{\varepsilon}([0,T];\;H^{s-\varepsilon}(\rho^{1+\varepsilon}))$
is compact.
We first collect a lemma needed for the tightness proof. By giving up some
differentiability, we can get a dispersive estimate that is independent of
time.
###### Lemma 6.3 (Dispersive estimate).
Fix $1\leq p,q\leq\infty$, and let us endow $\mathbb{R}^{d}$ with weight
$\rho(x)=(1+{\left|x\right|}^{2})^{\alpha/2}$, where
${\left|\alpha\right|}>d$. The Schrödinger propagator ${\mathcal{T}}_{t}$ then
satisfies for all $s\in\mathbb{R}$ the estimate
${\left\|{\mathcal{T}}_{t}f\right\|}_{B^{s}_{p,q}(\rho)}\lesssim{\left\|f\right\|}_{B^{s+d}_{p,q}(\rho)}.$
###### Proof.
Let us first assume $\alpha>d$. We will estimate the $L^{p}$ norm inside
${\left\|{\mathcal{T}}_{t}f\right\|}_{H^{s}_{p,q}(\rho)}={\left\|2^{ks}{\left\|\Delta_{k}{\mathcal{T}}_{t}f\right\|}_{L^{p}(\rho)}\right\|}_{\ell^{q}_{k}}.$
We can write
$\Delta_{k}{\mathcal{T}}_{t}=\Delta_{k}\Delta_{k}^{\prime}{\mathcal{T}}_{t}$
where $\Delta_{k}^{\prime}$ is a smooth indicator of a larger annulus, given
by multiplier symbol $\varphi(2^{-k}\,\cdot\,)$. Let $K_{k}$ be the
convolution kernel of $\Delta_{k}^{\prime}{\mathcal{T}}_{t}$; by weighted
Young’s inequality [24, Lemma A.8] we then have
$\displaystyle{\left\|K_{k}\ast(\Delta_{k}f)\right\|}_{L^{p}(w)}$
$\displaystyle\leq{\left\|K_{k}\right\|}_{L^{1}(w^{-1})}{\left\|\Delta_{k}f\right\|}_{L^{p}(w)}$
$\displaystyle\leq{\left\|1\right\|}_{L^{1}(w^{-1})}{\left\|K_{k}\right\|}_{L^{\infty}}{\left\|\Delta_{k}f\right\|}_{L^{p}(w)}.$
Since $w^{-1}$ is integrable, the first factor is finite. The second factor
satisfies the scaling relation
${\left|\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}\varphi(2^{-k}\xi)e^{-it{\left|\xi\right|}^{2}}\,\mathrm{d}\xi\right|}\leq
2^{kd}\int_{\mathbb{R}^{d}}{\left|\varphi(\eta)\right|}\,\mathrm{d}\eta,$
where $\varphi$ is compactly supported. This yields the required bound. The
case $k=-1$ (ball in Fourier space) also gives a constant factor.
For $\alpha<-d$, we proceed by duality and skew-adjointness of the operator.
By Theorem 2.5 and the above we have
$\displaystyle{\left\|{\mathcal{T}}_{t}f\right\|}_{B^{s}_{p,q}(\rho)}$
$\displaystyle=\sup_{g}{\left\langle{\mathcal{T}}_{t}f,g\right\rangle}$
$\displaystyle\leq\sup_{g}{\left\|f\right\|}_{B^{s+d}_{p,q}(\rho)}{\left\|{\mathcal{T}}_{-t}g\right\|}_{B^{-s-d}_{p^{\prime},q^{\prime}}(\rho^{-1})}$
$\displaystyle\lesssim\sup_{g}{\left\|f\right\|}_{B^{s+d}_{p,q}(\rho)}{\left\|g\right\|}_{B^{-s}_{p^{\prime},q^{\prime}}(\rho^{-1})}$
$\displaystyle=\sup_{g}{\left\|f\right\|}_{B^{s+d}_{p,q}(\rho)},$
where the supremum is taken over all $g$ with
$B^{-s}_{p^{\prime},q^{\prime}}(\rho^{-1})$ norm equal to $1$. ∎
###### Remark 6.4.
We can also swap $p$ and $1$ in the weighted Young’s inequality to get
${\left\|{\mathcal{T}}_{t}f\right\|}_{B^{s}_{\infty,q}(\rho)}\lesssim{\left\|f\right\|}_{B^{s+d}_{1,q}(\rho)}.$
In the unweighted case this Besov estimate follows from Theorem 2.7 and
conservation of $L^{2}$ norm. These results are reminiscent of the standard
dispersive estimate
${\left\|{\mathcal{T}}_{t}f\right\|}_{L^{\infty}(\mathbb{R}^{d})}\lesssim{\left|t\right|}^{-d/2}{\left\|f\right\|}_{L^{1}(\mathbb{R}^{d})},$
and by interpolation one can find “exchange rates” between spatial regularity
and integrability in time. Interpolation of weighted Besov spaces is explored
in [35]; see especially Theorem 4 there.
###### Theorem 6.5 (Tightness).
The sequence of periodic solutions $u_{L}$ is tight in
$C^{\alpha-\varepsilon}([0,T];\;H^{-4-2\varepsilon}(\rho^{1+\varepsilon}))$
for any $\alpha<1/2$, $\varepsilon>0$, and integrable weight $\rho$.
###### Proof.
We will show that
$\sup_{L}\mathbb{E}\,{\|u_{L}\|}_{C^{\alpha}([0,T];\;H^{-4-\varepsilon}(\rho))}^{2}<\infty.$
This implies tightness in a slightly less regular space by Lemma 6.2.
From the mild formulation of the equation we obtain that
$u_{L}(t)-u_{L}(s)=({\mathcal{T}}_{t}-{\mathcal{T}}_{s})u_{L}(0)+\int^{t}_{s}{\mathcal{T}}_{t-l}{:\\!u_{L}(l)^{3}\\!:}\,\mathrm{d}r,$
so we will need to estimate
${\left\|\int^{t}_{s}{\mathcal{T}}_{t-r}{:\\!u_{L}(r)^{3}\\!:}\,\mathrm{d}r\right\|}_{H^{-4-\varepsilon}(\rho)}+{\left\|({\mathcal{T}}_{t}-{\mathcal{T}}_{s})u(0)\right\|}_{H^{-4-\varepsilon}(\rho)}.$
For the first term, we can use Cauchy–Schwarz to exchange the integrals:
$\displaystyle{\left\|\int_{s}^{t}{\mathcal{T}}_{t-r}{:\\!u_{L}(r)^{3}\\!:}\,\mathrm{d}r\right\|}_{H^{-4-\varepsilon}(\rho)}$
$\displaystyle\leq\;$
$\displaystyle{\left|t-s\right|}^{1/2}\left[\int_{s}^{t}{\left\|{\mathcal{T}}_{t-r}{:\\!u_{L}(r)^{3}\\!:}\right\|}_{H^{-4-\varepsilon}(\rho)}^{2}\,\mathrm{d}r\right]^{1/2}$
$\displaystyle\lesssim\;$
$\displaystyle{\left|t-s\right|}^{1/2}\left[\int_{0}^{T}{\left\|{:\\!u_{L}(r)^{3}\\!:}\right\|}_{H^{-2-\varepsilon}(\rho)}^{2}\,\mathrm{d}r\right]^{1/2}.$
Here we used the uniform bound from Lemma 6.3. The Wick power is bounded in
expectation by Theorem 3.17, and the bound is uniform in $L$.
For the second term we use the functional derivative
$(e^{-it\Delta}-e^{-is\Delta})f=\int_{s}^{t}(-i\Delta)e^{-ir\Delta}f\,\mathrm{d}r$
and fundamental theorem of calculus to compute
$\displaystyle{\left\|({\mathcal{T}}_{t}-{\mathcal{T}}_{s})u(0)\right\|}_{H^{-4-\varepsilon}(\rho)}$
$\displaystyle={\left\|\int_{s}^{t}\Delta{\mathcal{T}}_{r}u(0)\,\mathrm{d}r\right\|}_{H^{-4-\varepsilon}(\rho)}$
$\displaystyle\leq{\left|t-s\right|}^{1/2}\left[\int_{0}^{T}{\left\|\Delta{\mathcal{T}}_{r}u(0)\right\|}_{H^{-4-\varepsilon}(\rho)}^{2}\,\mathrm{d}r\right]^{1/2}$
$\displaystyle\leq{\left|t-s\right|}^{1/2}\left[\int_{0}^{T}{\left\|u(0)\right\|}_{H^{-\varepsilon}(\rho)}^{2}\,\mathrm{d}r\right]^{1/2}.$
Again the expectation is bounded. All of these estimates are uniform in $L$
and hold for all $t,s\in{[{0},{T}]}$. ∎
By changing the probability space with Skorokhod’s theorem (Lemma 5.2) we can
assume that $u_{L}\to u$ almost surely. We still need to establish that $u$
satisfies the limiting equation.
###### Theorem 6.6 (Limit solves NLS).
There exists a probability space $\tilde{\mathbb{P}}$ and random variable
$\tilde{u}\in
L^{2}(\tilde{\mathbb{P}},C^{\varepsilon}([0,T];H^{-4-\varepsilon}(\rho)))$
such that
$\tilde{u}(t)={\mathcal{T}}_{t}\tilde{u}(0)+\int^{t}_{0}{\mathcal{T}}_{t-s}{:\\!\tilde{u}(s)^{3}\\!:}\,\mathrm{d}s$
and $\operatorname{Law}(\tilde{u}(t))=\mu$ for all $t\in[0,T]$.
###### Proof.
For clarity we omit the tildes in the proof. In addition to almost sure
convergence of $u_{L}$ and ${:\\!u_{L}^{3}\\!:}$, we have by tightness
$u_{L}\to u\quad\text{in}\quad
L^{2}(\tilde{\mathbb{P}};\;C^{\varepsilon}([0,T];\;H^{-4-\varepsilon}(\rho))).$
Hölder continuity implies that we also have convergence of
$u_{L}(t)\to u(t)\quad\text{in}\quad
L^{2}(\tilde{\mathbb{P}};\;H^{-4-\varepsilon}(\rho))$
for all $t\in[0,T]$.
We repeat the approximation argument of Lemma 4.5. Let $f^{\delta}(u)$
approximate ${:\\!u^{3}\\!:}$ as in Lemma 3.11. Then
$\displaystyle\int_{0}^{t}{\mathcal{T}}_{t-s}({:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:})\,\mathrm{d}s$
$\displaystyle=\int_{0}^{t}{\mathcal{T}}_{t-s}({:\\!u(s)^{3}\\!:}-f^{\delta}(u(s)))\,\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}{\mathcal{T}}_{t-s}(f^{\delta}(u(s))-f^{\delta}(u_{L}(s)))\,\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}{\mathcal{T}}_{t-s}({:\\!u_{L}(s)^{3}\\!:}-f^{\delta}(u_{L}(s)))\,\mathrm{d}s.$
Here the first and last term vanish in $H^{-4-\varepsilon}(\rho)$ norm as
$\delta\to 0$. This follows directly from the approximation result and is
uniform in $L$.
To bound the second term, we use boundedness of $f^{\delta}$:
$\mathbb{E}\,{\left\|{\mathcal{T}}_{t-s}f^{\delta}(u(s))\right\|}_{H^{-2}(\rho)}\lesssim\mathbb{E}\,{\left\|f^{\delta}(u(s))\right\|}_{L^{2}(\rho)}\lesssim\mathbb{E}\,{\left\|u(s)\right\|}_{H^{-4-\varepsilon}(\rho)},$
and the same for $u_{L}$. This lets us use dominated convergence in
$\lim_{L\to\infty}\mathbb{E}\int_{0}^{t}{\mathcal{T}}_{t-s}(f^{\delta}(u(s))-f^{\delta}(u_{L}(s)))\,\mathrm{d}s=0.$
Thus for any fixed $\delta>0$ and $\kappa>0$, we have for all large $L$ that
$\mathbb{E}\int_{0}^{t}{\mathcal{T}}_{t-s}(f^{\delta}(u(s))-f^{\delta}(u_{L}(s)))\,\mathrm{d}s<\kappa.$
In summary, we first pass to $\delta$ small and then to $L$ large. This gives
that
$\lim_{L\to\infty}\mathbb{E}\int_{0}^{t}{\mathcal{T}}_{t-s}({:\\!u(s)^{3}\\!:}-{:\\!u_{L}(s)^{3}\\!:})\,\mathrm{d}s=0.$
By passing to a further subsequence, we then have almost sure convergence of
these nonlinear terms. This finishes the proof. ∎
## Appendix A Proof of Lemma 3.11
Let us state the lemma in a slightly different way. We remark that this lemma
is not used in the construction of ${:\\!(Z+\phi)^{3}\\!:}$, so the
expressions are valid.
###### Lemma A.1.
Let us fix a Fourier cutoff $\chi\in C^{\infty}_{c}(\mathbb{R}^{2})$ and
define $\chi_{\delta}(x)=\chi(\delta x)$. Let
$Z^{\delta}=\chi_{\delta}({\left\langle{\nabla}\right\rangle})Z$ and
$Z_{L}^{\delta}=\chi_{\delta}({\left\langle{\nabla}\right\rangle})Z_{L}$.
Define
$f^{\delta}(Z^{\delta})\;\coloneqq\;{:\\!(Z^{\delta})^{3}\\!:}_{\delta}\;\coloneqq\;(Z^{\delta})^{3}-3a_{\delta}Z^{\delta},$
where $a_{\delta}=\mathbb{E}\,[(Z^{\delta})^{2}]$. Then if $\phi\in
L^{p}(\mathbb{P},B_{p,p}^{\varepsilon}(\rho))$, we have
$\displaystyle\lim_{\delta\to
0}\mathbb{E}\,{\left\|({:\\!(Z+\phi)^{3}\\!:}-{:\\!(Z^{\delta}+\phi)^{3}\\!:}_{\delta})\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}=0,\text{
and}$ (A.1) $\displaystyle\lim_{\delta\to
0}\sup_{L}\mathbb{E}\,{\left\|({:\\!(Z_{L}+\phi)^{3}\\!:}-{:\\!(Z_{L}^{\delta}+\phi)^{3}\\!:}_{\delta})\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}=0.$
(A.2)
Since $f^{\delta}$ restricts the Fourier support of its argument to a bounded
interval, it follows that its image is in $L^{2}$ and the map is continuous.
We first establish (A.1) in the case $\phi=0$. In this case it is enough to
prove that
$\sup_{\delta}\mathbb{E}\,{\left\|f^{\delta}(Z)\right\|}^{p}_{\mathcal{C}^{-\varepsilon}(\rho)}<\infty,$
(A.3)
and that for any fixed $\varepsilon>0$ we have
$\lim_{\delta\to
0}\mathbb{E}\,{\left\|\chi_{\varepsilon}({\left\langle{\nabla}\right\rangle})({:\\!(Z^{\delta})^{3}\\!:}_{\delta}-{:\\!Z^{3}\\!:})\right\|}^{2}_{L^{2}(\rho)}=0,$
(A.4)
which will imply convergence in tempered distributions $\mathcal{S}^{\prime}$.
Let us first verify (A.3). By Besov embedding it is enough to establish
$\sup_{\delta}\mathbb{E}\,{\left\|{:\\!(Z^{\delta})^{3}\\!:}_{\delta}\right\|}^{p}_{B_{p,p}^{-\varepsilon/2}(\rho)}<\infty$
for $p$ large. We use the stationarity of $Z$, integrability of $\rho$, and
hypercontractivity to estimate
$\displaystyle\mathbb{E}\,{\left\|{:\\!(Z^{\delta})^{3}\\!:}_{\delta}\right\|}^{p}_{B_{p,p}^{-\varepsilon/2}(\rho)}$
$\displaystyle=\sum_{k\geq-1}2^{-kp\varepsilon/2}\int_{\mathbb{R}^{2}}\mathbb{E}\,{\left|\Delta_{k}{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(x)\right|}^{p}\rho(x)^{p}\,\mathrm{d}x$
$\displaystyle=\sum_{k\geq-1}2^{-kp\varepsilon/2}\int_{\mathbb{R}^{2}}\mathbb{E}\,{\left|\Delta_{k}{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(0)\right|}^{p}\rho(x)^{p}\,\mathrm{d}x$
$\displaystyle\lesssim\sum_{k\geq-1}2^{-kp\varepsilon/2}\left(\mathbb{E}\,{\left|\Delta_{k}\,{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(0)\right|}^{2}\right)^{p/2}.$
Now we can use Wick’s theorem to get
$\displaystyle\mathbb{E}\,|\Delta_{k}\,{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(0)|^{2}$
$\displaystyle=\;$
$\displaystyle\int_{\mathbb{R}^{2+2}}\mathbb{E}\,[K_{k}(x_{1})\,{:\\!(Z^{\delta})^{3}\\!:}(x_{1})\,K_{k}(x_{2})\,{:\\!(Z^{\delta})^{3}\\!:}(x_{2})]\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}$
$\displaystyle=\;$
$\displaystyle\int_{\mathbb{R}^{2+2}}K_{k}(x_{1})K_{k}(x_{2})\,G^{\delta}(x_{1},x_{2})^{3}\,\mathrm{d}x_{1}\,\mathrm{d}x_{2},$
where $K_{k}$ is the convolution kernel of $\Delta_{k}$ and
$G^{\delta}(x_{1},x_{2})\coloneqq\mathbb{E}\,[Z^{\delta}(x_{1})Z^{\delta}(x_{2})]$.
It is known from [23, Chapter 7.2] that
${\left|G^{\delta}(x_{1},x_{2})\right|}\lesssim(C+{\left|\log({\left|x_{1}-x_{2}\right|})\right|})\exp(-m^{2}|x_{1}-x_{2}|).$
This implies that that
$\sup_{\delta>0}{\left\|G^{\delta}(x_{1},x_{2})\right\|}_{L^{p}(\mathrm{d}x_{1}\,\mathrm{d}x_{2})}<\infty$
for any $p<\infty$. In particular
$\displaystyle{\left|\int_{\mathbb{R}^{2+2}}K_{k}(x_{1})K_{k}(x_{2})G^{\delta}(x_{1},x_{2})^{3}\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}\right|}$
$\displaystyle\lesssim{\left\|K_{k}(x_{1})K_{k}(x_{2})\right\|}_{L^{q}(\mathrm{d}x_{1}\,\mathrm{d}x_{2})}$
$\displaystyle\lesssim{\left\|K_{k}\right\|}^{2}_{L^{q}}$
for any $q>1$. Since $K_{k}$ has constant $L^{1}$ norm for all $k$ and its
$L^{\infty}$ norm scales as $2^{2k}$, we can take
${\left\|K_{i}(x_{1})\right\|}_{L^{q}}\lesssim 2^{k\varepsilon/3}$ by choosing
$q$ close enough to $1$. Substituting this back to the Besov norm, we get a
convergent geometric sum. This proves (A.3).
Let us then turn to (A.4). Let
$\bar{K}_{\varepsilon}=\mathcal{F}^{-1}\chi^{2}_{\varepsilon}$. We have
$\bar{K}_{\varepsilon}\in L^{\infty}$ since $\chi_{\varepsilon}^{2}$ is a
Schwartz function. Then we compute
$\displaystyle\mathbb{E}\,{\left\|\chi_{\varepsilon}({\left\langle{\nabla}\right\rangle})({:\\!(Z^{\delta})^{3}\\!:}_{\delta}-{:\\!Z^{3}\\!:})\right\|}^{2}_{L^{2}(\rho)}$
$\displaystyle=\;$
$\displaystyle\mathbb{E}\int\rho(x)^{2}\bar{K}(x-y)\,{:\\!Z^{3}\\!:}(x)\;{:\\!Z^{3}\\!:}(y)\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle+\mathbb{E}\int\rho(x)^{2}\bar{K}(x-y)\,{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(x)\;{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(y)\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle{}-2\mathbb{E}\int\rho(x)^{2}\bar{K}(x-y)\,{:\\!Z^{3}\\!:}(x)\;{:\\!(Z^{\delta})^{3}\\!:}_{\delta}(y)\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=\;$
$\displaystyle\int\rho(x)^{2}\bar{K}(x-y)G(x,y)^{3}\,\mathrm{d}x\,\mathrm{d}y+\int\rho(x)^{2}\bar{K}(x-y)G^{\delta}(x,y)^{3}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle{}-2\int\rho(x)^{2}\bar{K}(x-y)\tilde{G}^{\delta}(x,y)^{3}\,\mathrm{d}x\,\mathrm{d}y,$
where $\tilde{G}^{\delta}(x,y)=\mathbb{E}\,[Z^{\delta}(x)Z(y)]$. We will have
that the right-hand side goes to $0$ as soon as we can show that
$\lim_{\delta\to 0}G^{\delta}(x,y)=\lim_{\delta\to
0}\tilde{G}^{\delta}(x,y)=G(x,y)\quad\text{in }L^{1}(\mathbb{R}^{2}).$
Since
$\tilde{G}^{\delta}(x,y)=\int
K^{\delta}(x-z)\mathbb{E}\,[Z(z)Z(y)]\,\mathrm{d}x=\int
K^{\delta}(x-z)G(z,y)\,\mathrm{d}x,$
and $K^{\delta}$ goes to a Dirac measure, the convergence holds. The argument
for $G^{\delta}(x,y)$ is the same.
We still need to check the $\phi\neq 0$ case. Here
${:\\!(Z^{\delta}+\phi)^{3}\\!:}_{\delta}=\sum_{i=0}^{3}{:\\!(Z^{\delta})^{i}\\!:}_{\delta}\,\phi^{3-i}\longrightarrow\sum_{i=0}^{3}{:\\!(Z)^{i}\\!:}\,\phi^{3-i}={:\\!(Z+\phi)^{3}\\!:}$
where the convergence holds by convergence of
${:\\!(Z^{\delta})^{i}\\!:}_{\delta}$ in $\mathcal{C}^{-\varepsilon}(\rho)$
and continuity of the product.
For the periodic case we again start with $\phi=0$. We may pass from
${:\\!(\cdot)^{3}\\!:}_{\delta}$ to ${:\\!(\cdot)^{3}\\!:}_{\delta,L}$ where
${:\\!(\cdot)\\!:}_{\delta,L}=(\cdot)^{3}-a_{\delta,L}(\cdot)$ and
$a_{\delta,L}=\mathbb{E}Z^{2}_{\delta,L}$, since we can show in the same way
as in Lemma 3.7 that $\sup_{\delta,L}|a_{\delta,L}-a_{\delta}|<\infty$. Then
repeating the computations from the first case we arrive at having to estimate
${\left\|\int
K^{\delta}(\cdot-z)G_{L}(z,y)\,\mathrm{d}x-G_{L}(\cdot,y)\right\|}_{L^{1}}\lesssim\delta{\left\|G_{L}(\cdot,y)\right\|}_{W^{1,1}}$
and the right-hand side is known to be bounded uniformly in $L$ [23, Chapter
7.3]. The $\phi\neq 0$ case follows analogously as above.
## Appendix B Computations for Section 3.2
In order to prove Theorem 3.19, we needed to bound the absolute value of
$3\int_{\mathbb{R}^{2}}\rho\phi{:\\!Z^{3}\\!:}\,\mathrm{d}x+3\int_{\mathbb{R}^{2}}\rho\phi^{2}{:\\!Z^{2}\\!:}\,\mathrm{d}x+\int_{\mathbb{R}^{2}}\rho\phi^{3}Z\,\mathrm{d}x+\int_{\mathbb{R}^{2}}\phi(\nabla\rho\cdot\nabla\phi)\,\mathrm{d}x$
with a term like
$Q(Z)+\frac{1}{2}\left(m^{2}{\left\|\phi\right\|}_{L^{2}(\rho^{1/2})}^{2}+{\left\|\phi\right\|}_{H^{1}(\rho^{1/2})}^{2}+{\left\|\phi\right\|}_{L^{4}(\rho^{1/4})}^{4}\right),$
where $Q(Z)$ is bounded in expectation.
We bound each of the integrals in the following lemmas, selecting $Q(Z)$ to
consist of norms of Wick powers of $Z$. The norms have bounded expectation by
Lemma 3.8. In each proof we apply the multiplicative inequality (Theorem 2.4)
and Besov duality (Theorem 2.5). These calculations are originally due to
Mourrat and Weber [38].
###### Lemma B.1 ([24, Lemma A.7]).
Let $\rho_{1}$ and $\rho_{2}$ be polynomial weights and $s,\varepsilon>0$. We
have the following two estimates:
$\displaystyle{\left\|f^{2}\right\|}_{B^{s}_{1,1}(\rho_{1}\rho_{2})}$
$\displaystyle\lesssim{\left\|f\right\|}_{L^{2}(\rho_{1})}{\left\|f\right\|}_{H^{s+\varepsilon}(\rho_{2})}$
$\displaystyle{\left\|f^{3}\right\|}_{B^{s}_{1,1}(\rho_{1}^{2}\rho_{2})}$
$\displaystyle\lesssim{\left\|f\right\|}_{L^{4}(\rho_{1})}^{2}{\left\|f\right\|}_{H^{s+\varepsilon}(\rho_{2})}.$
###### Lemma B.2.
Assume that $\rho^{1/2}\in L^{1}(\mathbb{R}^{2})$ and $\varepsilon<1/4$. Then
for any $\delta>0$ there exists a constant $C>0$ that
${\left|\int_{\mathbb{R}^{2}}\rho\phi^{3}Z\,\mathrm{d}x\right|}\leq
C{\left\|Z\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{1/16})}^{8}+\delta\left({\left\|\phi\right\|}_{L^{4}(\rho^{1/4})}^{4}+{\left\|\phi\right\|}_{H^{1}(\rho^{1/2})}^{2}\right).$
###### Proof.
We first use duality and Lemma B.1 to estimate
$\displaystyle\int_{\mathbb{R}^{2}}{\left|\rho\phi^{3}Z\right|}\,\mathrm{d}x$
$\displaystyle\lesssim{\left\|\phi^{3}\right\|}_{B^{\varepsilon}_{1,1}(\rho^{15/16})}{\left\|Z\right\|}_{\mathcal{C}(\rho^{1/16})}$
$\displaystyle\lesssim{\left\|\phi\right\|}_{L^{4}(\rho^{4/16})}^{2}{\left\|\phi\right\|}_{H^{2\varepsilon}(\rho^{7/16})}{\left\|Z\right\|}_{\mathcal{C}(\rho^{1/16})}.$
Inside the middle Besov norm, we can trade off some weight via
${\left\|\rho^{7/16}\Delta_{j}\phi\right\|}_{L^{2}}\leq{\left\|\rho^{1/16}\right\|}_{L^{8}}{\left\|\rho^{6/16}\Delta_{j}\phi\right\|}_{L^{8/3}}.$
We can also increase the regularity from $2\varepsilon$ to $1/2$. This
simplifies the interpolation
${\left\|\phi\right\|}_{B^{1/2}_{8/3,8/3}(\rho^{3/8})}\lesssim{\left\|\phi\right\|}_{B^{0}_{4,\infty}(\rho^{1/4})}^{1/2}{\left\|\phi\right\|}_{B^{1}_{2,8/3}(\rho^{1/2})}^{1/2}\lesssim{\left\|\phi\right\|}_{L^{4}(\rho^{1/4})}^{1/2}{\left\|\phi\right\|}_{H^{1}(\rho^{1/2})}^{1/2}.$
We substitute this back into the original product. Finally, we use Young’s
product inequality twice to first extract
${\left\|Z\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{1/16})}$ and then to
separate the $L^{4}$ and $H^{1}$ norms. ∎
###### Lemma B.3.
Assume that $\rho^{1/4}\in L^{1}(\mathbb{R}^{2})$ and $\varepsilon<1/2$. Then
for every $\delta>0$ there exists a constant $C>0$ such that
${\left|\int_{\mathbb{R}^{2}}\rho\phi^{2}{:\\!Z^{2}\\!:}\,\mathrm{d}x\right|}\lesssim
C{\left\|{:\\!Z^{2}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{1/16})}^{4}+\delta\left({\left\|\phi\right\|}_{L^{4}(\rho^{1/4})}^{4}+{\left\|\phi\right\|}_{H^{1}(\rho^{1/2})}^{2}\right).$
###### Proof.
Again, duality and Lemma B.1 give
$\int_{\mathbb{R}^{2}}{\left|\rho\phi^{2}{:\\!Z^{2}\\!:}\right|}\,\mathrm{d}x\lesssim{\left\|\phi\right\|}_{L^{2}(\rho^{7/16})}{\left\|\phi\right\|}_{H^{2\varepsilon}(\rho^{1/2})}{\left\|{:\\!Z^{2}\\!:}\right\|}_{\mathcal{C}(\rho^{1/16})}.$
We can again trade off some weight in
${\|\rho^{7/16}\phi\|}_{L^{2}}\leq{\|\rho^{3/16}\|}_{L^{4/3}}{\|\rho^{1/4}\phi\|}_{L^{4}}.$
We can increase the regularity in the middle term make the weight larger in
the last term to make them match the statement. Young’s product inequality
finishes the proof. ∎
###### Lemma B.4.
Assume that $\rho^{1/4}\in L^{1}(\mathbb{R}^{2})$ and $\varepsilon<1$. Then
for every $\delta>0$ there exists a $C>0$ such that
${\left|\int_{\mathbb{R}^{2}}\rho\phi{:\\!Z^{3}\\!:}\,\mathrm{d}x\right|}\leq
C{\left\|{:\\!Z^{3}\\!:}\right\|}_{\mathcal{C}^{-\varepsilon}(\rho^{1/4})}^{2}+\delta{\left\|\phi\right\|}_{H^{1}(\rho^{1/2})}^{2}.$
###### Proof.
By duality
$\int_{\mathbb{R}^{2}}{\left|\rho\phi{:\\!Z^{3}\\!:}\right|}\,\mathrm{d}x\lesssim{\left\|\phi\right\|}_{B^{\varepsilon}_{1,1}(\rho^{3/4})}{\left\|{:\\!Z^{3}\\!:}\right\|}_{\mathcal{C}(\rho^{1/4})}.$
Then we do a series of tradeoffs in
${\left\|\phi\right\|}_{B^{\varepsilon}_{1,1}(\rho^{3/4})}\lesssim{\left\|\phi\right\|}_{B^{1}_{1,2}(\rho^{3/4})}\lesssim{\left\|\phi\right\|}_{B^{1}_{2,2}(\rho^{1/2})}$
and finish with Young’s product inequality. ∎
###### Lemma B.5.
Assume that $\rho^{1/4}\in L^{1}(\mathbb{R}^{2})$. Then there exists $C>0$
such that
${\left|\int_{\mathbb{R}^{2}}\phi(\nabla\rho\cdot\nabla\phi)\,\mathrm{d}x\right|}\leq
C+\delta\left({\left\|\phi\right\|}_{L^{4}(\rho^{1/4})}^{4}+{\left\|\phi\right\|}_{H^{1}(\rho^{1/2})}^{2}\right).$
###### Proof.
Let us observe that we can write the dot product components as
$(\partial_{j}\rho)(\partial_{j}\phi)=(\partial_{j}[1+x_{1}^{2}+x_{2}^{2}]^{-\alpha/2})(\partial_{j}\phi)=\frac{\alpha
x_{j}\rho(x)}{1+x_{1}^{2}+x_{2}^{2}}(\partial_{j}\phi)$
The factor in front is uniformly bounded by $\rho(x)$. Thus
$\displaystyle\int_{\mathbb{R}^{2}}{\left|\phi(x)(\nabla\rho\cdot\nabla\phi)(x)\right|}\,\mathrm{d}x$
$\displaystyle\leq\alpha\int_{\mathbb{R}^{2}}\rho(x){\left|\phi(x)\right|}{\left|\nabla\phi(x)\right|}\,\mathrm{d}x$
$\displaystyle\leq\alpha{\left\|\phi\right\|}_{L^{2}(\rho^{1/2})}{\left\|\nabla\phi\right\|}_{L^{2}(\rho^{1/2})}$
$\displaystyle\leq
C+\delta\left({\left\|\phi\right\|}_{L^{4}(\rho^{1/4})}^{4}+{\left\|\phi\right\|}_{H^{1}(\rho^{1/2})}^{2}\right),$
where we did again a weight–$L^{p}$ tradeoff and applied Young twice. ∎
## Appendix C Exponential tails
In order to prove the existence of Wick powers, we needed to establish
exponential tails of some weighted Besov norms for $\phi_{2,L}^{4}$ uniformly
in $L$. More concretely we will define the measure
$\bar{\mu}_{L}(A)=\frac{1}{Z_{L}}\int_{A}\exp(h({\|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}_{L^{p}(\rho)}))\,\mathrm{d}\mu_{L}(\psi),$
(C.1)
where $h\colon\mathbb{R}\to\mathbb{R}$ is a smooth function, constant near $0$
and growing linearly at infinity, and $Z_{L}$ is the associated normalization
constant. We will prove that $\sup_{L}Z_{L}<\infty$.
We begin with the following lemma. In finite volume the Gaussian tails of
$\mu_{L}$ are not difficult to establish; see [6, Section 3]. This _a priori_
bound means that the assumptions of the lemma are satisfied. The lemma then
makes the uniform bound easier to derive.
###### Lemma C.1 ([5, Lemma A.7]).
Let $(\Omega,F)$ be a measurable space and $\upsilon$ be a probability measure
on $\Omega$. Let $S\colon\Omega\mapsto\mathbb{R}$ be a measurable function
such that
$\exp(S)\in L^{1}(\mathrm{d}\upsilon).$
Define
$\,\mathrm{d}\nu_{S}=\frac{1}{\int\exp(S)\,\mathrm{d}\upsilon}\exp(S)\,\mathrm{d}\upsilon$.
Then
$\int\exp(S)\,\mathrm{d}\upsilon\leq\exp\left(\int
S(x)\,\mathrm{d}\upsilon_{S}\right).$
###### Proof.
Multiplying both sides of
$\,\mathrm{d}\nu_{S}=\frac{1}{\int\exp(S)\,\mathrm{d}\upsilon}\exp(S)\,\mathrm{d}\upsilon$
by $\exp(-S)$ and integrating we obtain
$\left(\int\exp(-S)\,\mathrm{d}\upsilon_{S}\right)\left(\int\exp(S)\,\mathrm{d}\upsilon\right)=1.$
Then it remains to apply Jensen’s inequality to the first factor. ∎
We choose
$S=h({\|\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}_{L^{p}})$
and $\upsilon=\mu_{L}$ in the lemma. Then the claim follows if we can find a
uniform estimate for
$\int_{H^{-\varepsilon}(\rho)}h({\|\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}_{L^{p}})\,\mathrm{d}\bar{\mu}_{L}.$
(C.2)
To do this we again use stochastic quantization. By the chain rule the
gradient operator of $h$ is
$\nabla_{\psi}\,h({\|\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}_{L^{p}})=\frac{h^{\prime}({\|\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}_{L^{p}})}{{\|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}^{p-1}_{L^{p}(\rho)}}(\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi)^{p-1}\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}.$
(C.3)
We can write the right-hand side via the adjoint of
$\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}$ as
$V(\psi)=\frac{h^{\prime}({\|\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}_{L^{p}})}{{\|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}^{p-1}_{L^{p}(\rho)}}\,(\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon})^{*}\\!\left[(\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi)^{p-1}\right].$
(C.4)
We then have the following lemma:
###### Lemma C.2.
The measure $\bar{\mu}_{L}$ is an invariant measure for the equation
$\partial_{t}\bar{u}+(m^{2}-\Delta)\bar{u}+{:\\!\bar{u}^{3}\\!:}=V(\bar{u})+\xi,$
where $\xi$ is space-time white noise.
###### Proof.
Note that $V$ is continuous on ${\mathcal{C}}^{-\delta}(\Lambda_{L})$. With
this in mind the proof becomes a minor modification of the proof of Da Prato
and Debussche [19, Section 4] and we omit it. ∎
Again performing the Da Prato–Debussche trick, i.e. decomposing
$\bar{u}=Z+\bar{\phi}$, we obtain that $\bar{\phi}$ satisfies
$\partial_{t}\bar{\phi}+(m^{2}-\Delta)\bar{\phi}+{:\\!(Z+\bar{\phi})^{3}\\!:}=V(Z+\bar{\phi}).$
(C.5)
We again test the equation with $\rho\bar{\phi}$ to obtain
$\partial_{t}\int\rho\bar{\phi}^{2}\,\mathrm{d}x+m^{2}\int\rho\bar{\phi}^{2}\,\mathrm{d}x+\int{\left|\nabla\bar{\phi}\right|}^{2}\,\mathrm{d}x+\int\bar{\phi}^{4}\,\mathrm{d}x+G(Z,\bar{\phi})=\int\rho
V(Z+\bar{\phi})\bar{\phi}\,\mathrm{d}x.$ (C.6)
From the definitions and Hölder’s inequality
$\displaystyle\int\rho V(Z+\bar{\phi})\bar{\phi}\,\mathrm{d}x$
$\displaystyle=\;$
$\displaystyle\frac{h^{\prime}({\|\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}\psi\|}_{L^{p}})}{{\|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}(Z+\bar{\phi})\|}^{p-1}_{L^{p}(\rho)}}\int(\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}(Z+\bar{\phi}))^{p-1}\rho{\left\langle{\nabla}\right\rangle}^{-\varepsilon}(\rho\bar{\phi})\,\mathrm{d}x$
$\displaystyle\lesssim\;$
$\displaystyle\frac{1}{{\|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}(Z+\bar{\phi})\|}^{p-1}_{L^{p}(\rho)}}{\|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}(Z+\bar{\phi})\|}_{L^{p}(\rho)}^{p-1}{\|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}(\rho\bar{\phi})\|}_{L^{p}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle{\|{\left\langle{\nabla}\right\rangle}^{-\varepsilon}(\rho\bar{\phi})\|}_{L^{p}(\rho)}$
$\displaystyle\lesssim\;$
$\displaystyle{\|\rho\|}_{H^{-\varepsilon}(\mathbb{R}^{2})}{\|\bar{\phi}\|}_{H^{1}(\rho)}.$
$\displaystyle\leq\;$ $\displaystyle
C+\frac{1}{2}{\|\bar{\phi}\|}^{2}_{H^{1}(\rho)}.$
We thus have that
$\int\rho
V(Z+\bar{\phi})\bar{\phi}\,\mathrm{d}x\leq\frac{1}{2}\left(m^{2}\int\rho\bar{\phi}^{2}+\int|\nabla\bar{\phi}|^{2}\,\mathrm{d}x\right)+C.$
(C.7)
We also apply the reasoning from Section 3.2 to the remainder term
$G(Z,\bar{\phi})$. Upon taking an expectation, the time derivative and white
noise integrals vanish.
This implies again the boundedness of $H^{1}$ norm. Sobolev embedding then
gives
$\sup_{L}\mathbb{E}\,[{\left\|\bar{\phi}\right\|}^{2}_{L^{p}(\rho)}]<\infty$,
which gives the statement for exponential tails in $L^{p}$ norm.
###### Corollary C.3.
As $p$ was arbitrary, we also have
$\sup_{L}\int\exp({\left\|\psi\right\|}_{\mathcal{C}^{-2\varepsilon}(\rho)})\,\mathrm{d}\mu_{L}<\infty.$
This implies that
$\int{\left\|\psi\right\|}^{p}_{\mathcal{C}^{-2\varepsilon}(\rho)}\,\mathrm{d}\mu_{L}$
is finite uniformly in $L$ for any $p<\infty$.
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|
# Timing-Based Backpropagation in Spiking Neural Networks Without Single-Spike
Restrictions
Kakei Yamamoto, Yusuke Sakemi, Kazuyuki Aihara Kakei Yamamoto is with the
University of Tokyo, Tokyo, Japan (e-mail: kakei@g.ecc.u-tokyo.ac.jp).Yusuke
Sakemi is with Research Center for Mathematical Engineering, Chiba Institute
of Technology, Narashino, Japan.Kazuyuki Aihara is with the International
Research Center for Neurointelligence (WPI-IRCN), The University of Tokyo
Institutes for Advanced Study, The University of Tokyo, Tokyo, Japan.
###### Abstract
We propose a novel backpropagation algorithm for training spiking neural
networks (SNNs) that encodes information in the relative multiple spike timing
of individual neurons without single-spike restrictions. The proposed
algorithm inherits the advantages of conventional timing-based methods in that
it computes accurate gradients with respect to spike timing, which promotes
ideal temporal coding. Unlike conventional methods where each neuron fires at
most once, the proposed algorithm allows each neuron to fire multiple times.
This extension naturally improves the computational capacity of SNNs. Our SNN
model outperformed comparable SNN models and achieved as high accuracy as non-
convolutional artificial neural networks. The spike count property of our
networks was altered depending on the time constant of the postsynaptic
current and the membrane potential. Moreover, we found that there existed the
optimal time constant with the maximum test accuracy. That was not seen in
conventional SNNs with single-spike restrictions on time-to-fast-spike (TTFS)
coding. This result demonstrates the computational properties of SNNs that
biologically encode information into the multi-spike timing of individual
neurons. Our code would be publicly available.
###### Index Terms:
Spiking neural networks (SNNs), supervised learning, backpropagation, temporal
coding, multi-spike.
## I Introduction
How does the human brain acquire intelligence? Neural systems in the brain
have various functions by forming very complex networks, and much remains a
mystery. One approach to this question is to use mathematical models that
represent the dynamics of biological neurons from a microscopic chemical
viewpoint. You can simulate and analyze spiking phenomena, and investigate the
functions of the biological neural system that the model mimics. These models,
which originated in the Hodgkin-Huxley model [1] and the FitzHugh-Nagumo model
[2, 3], can reproduce various types of neural behavior. Because of their
complexity, however, it is difficult to analyze the neural networks composed
by these neuron models. Therefore, motivated by the desire to understand the
efficient computational architecture underlying biological neural networks,
spiking neural networks (SNNs), which are constructed with neuron models
simplified for computation, have been researched. Among them, the leaky
integrated firing (LIF) neuron model [4] and its variants have been widely
studied due to their mathematical tractability. SNNs are biologically
plausible neural networks and differ significantly from general artificial
neural networks (ANNs) in that they encode information into spikes and
propagate the spikes through synapses. We focus on the dynamic temporal coding
properties specific to neural circuits and SNNs in terms of the encoding
scheme and the learning mechanism of the neural system. In the neural networks
of biological organisms, it was shown experimentally that not only the spike
firing rate but also the individual precise spike timing held information, and
that it was an important factor in achieving more efficient computation [5,
6]. In recent years, gradient methods for end-to-end supervised learning via
direct spiking have been becoming established, but a number of research
challenges remain due to the difficulties specific to SNNs.
One of the fundamental challenges in the supervised learning of SNNs using
gradient methods is the non-differentiability of the discontinuous binary
spike function. Various approaches have been developed and successfully
overcome this difficulty [7, 8, 9]. Most of those used in the recent research
can be divided into two categories depending on a state quantity, through
which the error signals are backpropagated: the rate-based, and the timing-
based method. First, the rate-based method calculates the derivatives between
time-evolving functions such as membrane potentials and spike functions, which
leads to excellent results in a variety of tasks [10, 11]. However, to achieve
the functional derivatives, the rate-based method requires the time
discretization of each function and the approximation of the gradient
computation by replacing discontinuous spike gradients with continuous
surrogate gradients (SG) during error backpropagation. In addition, there are
concerns about how accurately the transfer of information is approximated due
to errors introduced by these techniques of the rate-based method. Note that
the use of SGs, which are continuous real-valued functions, can be interpreted
as approximating the spike rate of each neuron with an activation function,
suggesting that the network is not efficiently dealing with exact spike
timing. Therefore, it is somewhat questionable whether it is appropriate as a
temporal-coding scheme.
In contrast, the timing-based method spearheaded by SpikeProp [12] provides a
solution for computing exact gradients using spike timing. What is more,
Mostafa [13] explicitly derives spike timing gradients for non-leaky IF neuron
models, leading to superior performance. Furthermore, Comsa et al. [14] and
Göltz et al. [15] extended the method to LIF neuron models that also account
for current leakage. The rate-based method backpropagates the gradient from
the postsynaptic current to the presynaptic membrane potential regardless of
the presence of spike firing events, whereas the timing-based method
backpropagates the gradient between layers via the spike timing. Therefore,
the timing-based method can avoid the problem of indifferentiability without
the rate-based coding scheme. This point suggests the potential of the
temporal-coding properties of timing-based methods. In general, timing-based
methods adopt time-to-first-spike (TTFS) as a coding method, which converts
the intensity of input information into absolute spike timing, and also they
impose single-spike restrictions on all neurons [16, 14, 15, 17]. These ideas
are supported by the physiological studies [18, 5] that argue that the first
spike retains more information than the later ones. On the other hand, we
define multi-spike SNNs as the networks in which individual neurons are
allowed to fire multiple times, in contrast to SNNs with single-spike
restrictions. It is clear that the single-spike restriction per neuron limits
the learning efficiency and information processing capacity of SNNs.
Furthermore, since biological neural networks are multi-spike, multi-spike is
essential for investigating the coding properties of SNNs. Some researches
tried on timing-based learning of multi-spike SNNs, but their results do not
indicate that multi-spike SNNs were effective enough to outperform SNNs with
single-spike restrictions [19, 20].
In response to the need to understand the characteristics of multi-spike SNN
models, we propose a new algorithm for timing-based error backpropagation of
multi-spike SNNs. To the best of our knowledge, this is the first study of
supervised learning of SNNs based on spike timing allowing multi-spike. It is
a natural extension of the timing-based method that originated from the
Mostafa [13] in that it adopted the LIF model as the neuron model and did not
involve the intervention of heuristic techniques. The contributions of this
paper are as follows:
* •
We propose a timing-based backpropagation for multi-spike SNNs with exact
gradients by constructing a learning algorithm via spike timing only.
* •
Using the MNIST dataset, a fundamental benchmark for image recognition, we
revealed for the first time the fundamental properties of multi-spike
dynamics, the relationship between the leakage time constant and the number of
spikes, and its performance improvement over single-spike restricted timing-
based methods.
* •
We propose a spike-count loss that can be used in gradient methods based
solely on spike timing, called a dead neuron penalty loss that pulls back
neurons that do not respond to any input pattern into learning.
We hope that these results will provide a solid foundation for research on
gradient learning methods for timing-based SNNs and serve as a good benchmark
for working with more complex networks and tasks.
## II Methods
We propose a learning algorithm for feedforward SNNs based solely on spike
timing. In this white paper, we employ the spike timing sequence as the output
of each layer. Note that we use spike timing sequences based on two different
representations of ”spike order” depending on the situation: local spike order
and global spike order. The former refers to the order in which spikes are
output from a single neuron. The local spike order-based spike timing of
neuron $i$ is defined as $t_{i}^{(m)}$, which means the spike timing when
neuron $i$ fires for the $m$-th time. The latter, on the other hand, is the
spike order consistently counted for all output spikes from all neurons in a
layer, which is an integration of the local spike order over all neurons in
the layer. The spike timing based on the global spike order is defined as
$\hat{t}^{(m)}$, which is the spike timing of the $m$-th spike in this layer.
When the total number of spikes in a layer is $M$ and the number of output
spikes for each neuron $i\in\\{1,\cdots,I\\}$ is $M_{i}$, these satisfy
$M=\sum_{i=1}^{I}M_{i}$.
### II-A Neuron Models
We use a leaky integrate-and-fire (LIF) neuron in continuous time. The neuron
model involves reset dynamics, which is induced as an output spike is
generated. Suppose that the local spike timing sequence of neuron
$i\in\\{1,\cdots,I\\}$ in the previous layer is
$\\{t_{i}^{(m)}\\}_{m=1}^{M_{i}}$. The differential equations satisfied by the
postsynaptic current $I_{j}(t)$ and the membrane potential $V_{j}(t)$ of
postsynaptic neuron $j\in\\{1,\cdots,J\\}$ at time $t\in\mathbb{R}_{\geq 0}$
are defined as
$\displaystyle S_{i}(t)$
$\displaystyle=\sum_{m=1}^{M_{i}}\delta(t-t_{i}^{(m)}),$ (1)
$\displaystyle\frac{\mathrm{d}I_{j}(t)}{\mathrm{d}t}$
$\displaystyle=-\frac{1}{\tau_{I}}I_{j}(t)+\beta_{I}\sum_{i=1}^{I}w_{ij}S_{i}(t),$
(2) $\displaystyle\frac{\mathrm{d}V_{j}(t)}{\mathrm{d}t}$
$\displaystyle=-\frac{1}{\tau_{V}}V_{j}(t)+\beta_{V}I_{j}(t),$ (3)
$\displaystyle V_{j}(t)$ $\displaystyle\leftarrow 0\quad,\text{when}\
V_{j}(t)=V_{th},$ (4)
where $S_{i}(t)$ is the spike sequence of the presynaptic neuron $i$, and
$\delta(t)$ denotes Dirac delta function. $w_{ij}\in\mathbb{R}^{I\times J}$ is
a synaptic weight from presynaptic neuron $i$ to postsynaptic neuron $j$.
$\tau_{I}\in\mathbb{R}_{>0}$ and $\tau_{V}\in\mathbb{R}_{>0}$ denote the decay
time constants for the postsynaptic current and the membrane potential,
respectively. $V_{th}$ is the threshold potential, which is fixed at $1.0$.
$\beta_{I}\in\mathbb{R}_{>0}$ and $\beta_{V}\in\mathbb{R}_{>0}$ are the scale
coefficients, but, for simplicity, we fix both $\beta$s to a $1$, hereafter.
Forward pass based on spike timing In this study, a LIF model with decaying
voltage and current with $\tau_{I},\tau_{V}<\infty$ are adopted, and we let
$\tau_{I}\neq\tau_{V}$ consistently. Assume here that neuron $j$ does not
fire, i.e., this neuron satisfies the initial values $I_{j}(0),V_{j}(0)=0$ at
time $t=0$, continues to receive spikes from the previous layer. In this case,
the postsynaptic potential of neuron $j$ at time $t$ and it’s current,
respectively, are obtained as convolution integral representations:
$\displaystyle I_{j}(t)$
$\displaystyle=\int_{0}^{t}\theta(t^{\prime})\mathcal{A}(t^{\prime})\left(\sum_{i=1}^{I}w_{ij}S_{i}(t-t^{\prime})\right)\mathrm{d}t^{\prime},$
(5) $\displaystyle V_{j}(t)$
$\displaystyle=\frac{\tau_{I}\tau_{V}}{\tau_{I}-\tau_{V}}\int_{0}^{t}\theta(t^{\prime})\mathcal{K}(t^{\prime})\left(\sum_{i=1}^{I}w_{ij}S_{i}(t-t^{\prime})\right)\mathrm{d}t^{\prime},$
(6) $\displaystyle\theta(t)$
$\displaystyle=\left\\{\begin{array}[]{ll}1&\text{if}\quad t\geq 0\\\
0&\text{otherwise}\end{array}\right.,$ (9)
where $\theta(t)$ is the step function and the kernel $\mathcal{K}$ and the
function $\mathcal{A},\mathcal{B}$ are defined as
$\displaystyle\mathcal{K}(t)$ $\displaystyle=\mathcal{A}(t)-\mathcal{B}(t),$
(10) $\displaystyle\mathcal{A}(t)$
$\displaystyle=\exp\left(-\frac{t}{\tau_{I}}\right),$ (11)
$\displaystyle\mathcal{B}(t)$
$\displaystyle=\exp\left(-\frac{t}{\tau_{V}}\right).$ (12)
That is, the membrane potential $V_{j}(t)$ is represented as the weighted
convolution integral of this kernel $\mathcal{K}$ with respect to all spikes
in the output of the previous layer.
Then we consider the neuronal dynamics after the postsynaptic neurons fire.
That is, it resets the membrane potential as an output spike is generated.
Note that the synaptic current is not reset at this time. Unlike in the
existing studies of SNNs using timing-based methods [13, 14, 15], where the
single-spike restrictions let the activity of the neuron totally stopped at
its own spike timing, the reset term results in multi-spike in this study.
Thereafter, $\hat{t}^{(m)}$ denotes the presynaptic spike timing at $m$-th in
the global spike order, and the intersynaptic weight from the presynaptic
neuron $i$ which fires at $m$-th in the global spike order to the postsynaptic
neuron $j$ is defined as $w_{ij}^{(m)}$. Assuming that time $t$ satisfies
$t\in[\hat{t}^{(m^{\prime})},\hat{t}^{(m^{\prime}+1)})$ and that a neuron $j$
newly fires during this interval at $t=t_{j}^{(n^{\prime}+1)}$. At any
$t\in[\hat{t}^{(m^{\prime})},\hat{t}^{(m^{\prime}+1)})\land[t_{j}^{(n^{\prime})},t_{j}^{(n^{\prime}+1)})$,
the synaptic current and membrane potential of neuron $j$ can be written
respectively as
$\displaystyle I_{j}(t)$
$\displaystyle=\sum_{m=1}^{m^{\prime}}w_{ij}^{(m)}\mathcal{A}(t-\hat{t}^{(m)}),$
(13)
$\displaystyle\begin{split}V_{j}(t)&=\frac{\tau_{I}\tau_{V}}{\tau_{I}-\tau_{V}}\sum_{m=1}^{m^{\prime}}w_{ij}^{(m)}\left[\mathcal{A}(t-\hat{t}^{(m)})\right.\\\
&-\left.\max\left(\mathcal{A}(t_{j}^{(n^{\prime})}-\hat{t}^{(m)}),\mathcal{B}(t_{j}^{(n^{\prime})}-\hat{t}^{(m)})\right)\mathcal{B}(t-t_{j}^{(n^{\prime})})\right].\end{split}$
(14)
For convenience, we define the leaky factor of the two different time
constants $\tau_{I,V}$ as $p=\tau_{I}/\tau_{V}$. Also, since spike timing only
appears in an exponential form such as $\exp(-t/\tau_{I}),\exp(-t/\tau_{V})$,
we can transform times $t$ to $z$ according to the definition of
$z:=\exp(t/\tau_{I})$, e.g., $\hat{z}^{(m)}:=\exp(\hat{t}^{(m)}/\tau_{I})$ and
$z_{j}^{(n)}:=\exp(t_{j}^{(n)}/\tau_{I})$. Both $I_{j},V_{j}$ at any
$z\in[\hat{z}^{(m^{\prime})},\hat{z}^{(m^{\prime}+1)})\land[z_{j}^{(n^{\prime})},z_{j}^{(n^{\prime}+1)})$
are multiplied by the spike timing sequence using
$\\{\hat{z}^{(m)}\\}_{m=1}^{M}$ as
$\displaystyle I_{j}(z)$
$\displaystyle=\sum_{m=1}^{m^{\prime}}w_{ij}^{(m)}\frac{\hat{z}^{(m)}}{z},$
(15)
$\displaystyle\begin{split}V_{j}(z)&=\frac{\tau_{I}\tau_{V}}{\tau_{I}-\tau_{V}}\\\
&\sum_{m=1}^{m^{\prime}}w_{ij}^{(m)}\left[\frac{\hat{z}^{(m)}}{z}-\frac{\hat{z}^{(m)}(\max\\{z_{j}^{(n^{\prime})},\hat{z}^{(m)}\\})^{p-1}}{(z)^{p}}\right],\end{split}$
(16)
$\displaystyle=\frac{\tilde{\mathcal{A}}_{j}^{(m^{\prime})}}{z}-\frac{\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})}}{(z)^{p}},$
(17)
where, for convenience, we use the following definitions:
$\displaystyle\tilde{\mathcal{A}}_{j}^{(m^{\prime})}$
$\displaystyle=\frac{\tau_{I}\tau_{V}}{\tau_{I}-\tau_{V}}\sum_{m=1}^{m^{\prime}}w_{ij}^{(m)}z^{(m)},$
(18) $\displaystyle\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})}$
$\displaystyle=\frac{\tau_{I}\tau_{V}}{\tau_{I}-\tau_{V}}\sum_{m=1}^{m^{\prime}}w_{ij}^{(m)}\hat{z}^{(m)}(\max\\{z_{j}^{(n^{\prime})},\hat{z}^{(m)}\\})^{p-1}.$
(19)
For each case, a solution for the spike time $z_{j}^{(n^{\prime}+1)}$, defined
by
$\displaystyle V_{j}(z)=V_{th}$ (20)
has to be found, sequentially. In the whole experiments in the study, let the
leaky factor $p=2$. Therefore, we can solve the next spike timing
$z_{j}^{(n^{\prime}+1)}$ can be solved explicitly and analytically by using
the coefficient parameters
$\tilde{\mathcal{A}}_{j}^{(m^{\prime})},\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})}$
based on the historical spike timing sequence of the previous layer
$\\{z^{(m)}\\}_{m=1}^{m^{\prime}}$ and its own previous spike timing
$\hat{z}_{j}^{(n^{\prime})}$ as
$\displaystyle z_{j}^{(n^{\prime}+1)}$
$\displaystyle=\frac{\tilde{\mathcal{A}}_{j}^{(m^{\prime})}-\sqrt{\left(\tilde{\mathcal{A}}_{j}^{(m^{\prime})}\right)^{2}-4V_{th}\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})}}}{2V_{th}}.$
(21)
The condition for the existence of $z_{j}^{(n^{\prime}+1)}$ is that each
coefficient parameter
$\tilde{\mathcal{A}}_{j}^{(m^{\prime})},\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})}$
and the root content of the equation (21) are non-negative and
$z_{j}^{(n^{\prime}+1)}\in[z^{(m^{\prime})},z^{(m^{\prime}+1)})$. When
actually implementing the algorithm, it can be computed independently for each
$m^{\prime},j$ in accordance with the definition of $z_{j}^{(n^{\prime}+1)}$.
The obtained spike timing sequences $\\{z_{j}^{(n^{\prime})}\\}_{n^{\prime}}$
for each neuron are finally merged into a unified spike timing sequence
$\hat{z}^{(n)}$ in the global spike order, and then subsequent layers can be
calculated in the same way.
Figure 1: Computational graph of a multi-spiking neural network (4-2-3).
Inputs and hidden units are spike units. The SNN has only two neurons in the
hidden layer and, in this case, neuron $1$ and neuron $2$ fire three times and
twice in a given period, respectively. (a) The forward pass is illustrated as
colored arrows: the green solid arrows indicate the propagation of spikes
through the synapse or the reset path, the green dashed arrows indicate
merging the spikes of each neuron in the global spike order. (b)
Backpropagation flow of computational for $\frac{\partial\mathcal{L}}{\partial
z_{1}^{(1)}}$ with orange arrows.
Fig. 1 shows the computational graph where edges start from the input spike
sequence to the loss function and its backward. In the hidden layer, the
computation of the spike timing is done independently for each neuron and
finally merged into a spike timing sequence based on the global spike order
for the entire layer (the dashed arrows). The arrows between spikes within the
same neuron represent the reset paths derived from their own most recent past
spike timing, which are induced by the reset of the membrane potential. When
calculating the hidden layer spike timing sequence from the input spike timing
sequence, the computational path from the input spike received earlier will
continue to affect the calculation of spike timing after that received time
via two types of the path: the postsynaptic currents that remain after resets,
and the latest spike timing of themselves. Note that, considering the
convolution integral representations above, the time range that a neuron is
affected by a previous single spike depends on the leakage time constant. In
the case of a multi-layered SNNs, the spike timing sequence based on the
global spike order of the previous hidden layer can be regarded as the input
spike sequence, and the spike timing sequence of the next layer can be
calculated just as the spike timing is calculated from the input spike
sequence in the figure. The second graph illustrates to computation for
$\frac{\partial\mathcal{L}}{\partial z_{1}^{(1)}}$ by backpropagation from the
bottom $\mathcal{L}$ (the orange arrows). Note that the gradient is propagated
backward without ignoring all the reset paths.
### II-B Input formats
The input format to the network is latency coding. The latency input consists
of the same number of input neurons as the dimension of the input data, and
each value is mapped to the spike timing of each input neuron in a one-to-one
fashion. Each input neuron fires only once per sample, and its exact spike
timing contains the information. Every input neuron has a one-way weighted,
fully connected, directed path to all neurons in the hidden layer, and spikes
carrying information at the spike timing are input as synaptic currents
through this path to each middle layer.
### II-C Loss functions
We adopt membrane potential loss as the loss function. The same number of
output neurons as a given number of output dimensions is placed in the final
layer, and the output from one neuron corresponds to one output dimension. The
threshold is set to infinite only for the output neurons, and they never fire.
The value of the membrane potential of each output neuron was observed at the
preset end time $t_{out}$, and this value was taken as the final output of
each neuron. In other words, the output membrane potential
$\textbf{v}_{out}:=\\{v_{o}\\}_{o=1}^{N}$ of this network is defined by the
output layer coefficient parameters
$\mathcal{A}_{o}^{(M)},\mathcal{B}_{o}^{(M,0)}$ as
$\displaystyle
v_{o}=\frac{\mathcal{A}_{o}^{(M)}}{z_{out}}-\frac{\mathcal{B}_{o}^{(M,0)}}{(z_{out})^{2}},$
(22)
where $z_{out}:=\exp(t_{out}/\tau_{I})$. For the multiclass classification
problem, the cross-entropy, which incorporates the correct answer labels into
the softmax of these output values, was used as the error function.
In this paper, we also introduce spike count penalty loss for efficient
learning. By incorporating this into the overall model loss function, the
number of spikes in each neuron can be freely manipulated even in the gradient
method for SNNs based only on spike timing. This penalty loss provides one
solution to the challenge that one of the difficulties in learning timing-
based SNNs is that it is unable to incorporate the spike counts directly into
the model because the timing-based error backpropagation has no state
variables other than the spike timing.
To incorporate the spike count penalty loss, the number of spikes for each
neuron that fires during the forward trial process is counted, and a flag
$\displaystyle\mathcal{P}:=\left\\{\mathcal{P}\in\\{0,\ 1\\}^{J}\ |\
\mathcal{P}_{j}=c(M_{j}),\ \forall j=1,\dots,J\right\\}$ (23)
is the vector for each neuron that meets the specified spike count condition,
where $M_{j}$ is the spike count of neuron $j$ and $J$ is the number of
neurons in the target hidden layer. $c:\mathbb{Z}_{\geq
0}\rightarrow\\{0,1\\}$ is a binary function that defines the spike count
condition. Let $\textbf{v}_{hidden}$ be a $J$-dimensional vector of the
measured values of each output neuron membrane potential at the end time of
the trial $t_{out}$, and then the spike count penalty loss,
$\displaystyle
L_{c}(\textbf{v}_{hidden};\mathcal{P})=\frac{1}{J}\mathcal{P}^{\top}(-\textbf{v}_{hidden}+V_{th}\textbf{1}_{J})$
(24)
is obtained, where $\textbf{1}_{J}$ is an $J$-dimensional 1-vector. By
incorporating this penalty loss, the spike counts of individual neurons can be
conditioned into the SNN timing-based learning algorithm.
In particular, by penalizing dead neurons that do not fire for any input
pattern, it is possible to suppress initial value dependence, which has been
considered difficult in timing-based methods, and to facilitate the
experimental design. In this case, we refer to the loss of spike count penalty
as ’dead neuron loss’. Importantly, by setting the conditional equation to
flag dead neurons, we can avoid the problem of neurons that do not fire
depending on the initial weight variable (since the timing-based method only
handles spike timing, neurons that never fire cannot be included in the
training, and all gradients will be zero, making them untrainable). This
reduces the dependence on the initial weight variables, which is a major
concern with timing-based methods, and makes SNN training design much easier.
The square of the norm of the output multiplied by a small factor is also
added to the overall loss to prevent output values from diverging. Although
learning is possible without this loss term, we added it in our experiments
because we observed that the loss function of the test increased in the middle
of learning, even though the accuracy increased.
In summary, for the obtained network membrane potential output $v_{o}$ and
target class indicator probability $q$, the error function $E$ is
$\displaystyle E(\textbf{v}_{out})$
$\displaystyle=-\sum_{o}q_{o}\log\text{softmax}(v_{o}),$ (25)
and the loss function $\mathfrak{L}(\textbf{v}_{out})$ for the whole model is
given by using the hyperparameters $\lambda,\ \sigma$ as
$\displaystyle\mathfrak{L}(\textbf{v}_{out})$
$\displaystyle=E(\textbf{v}_{out})+\lambda
L_{c}(\textbf{v}_{hidden};\mathcal{P})+\sigma\frac{1}{N_{out}}||\textbf{v}_{out}||_{2}^{2}.$
(26)
### II-D Memory- and computation-efficient algorithms
Training and inference of SNNs on ordinary Von Neumann type computers are
extremely computation- and memory-intensive due to the nature of the dynamics
in continuous time. Although research on SNNs has been accelerated in recent
years with the rapid development of computers, its computational complexity
still remains an obstacle to research on SNNs. In fact, this limitation has
made it difficult to handle large-scale datasets and models. To realize a
computationally efficient timing-based error backpropagation method that
allows multi-spike, we adopted a computation cycle based on the spike count
independent of the spike timing and introduced the two-stage partitioned
algorithm. First, the spike-count-based computation loop enables parallel
processing of calculations proportional to the input dimension and the number
of hidden neurons, which can be efficiently computed on GPUs and TPUs to
achieve a fast computation rate. Furthermore, though there is a trade-off
between computation time and memory resources, and if the input dimension or
model parameters are large, the burden on memory increases when covering a
large number of spikes, the two-stage cycle algorithm can dramatically reduce
the amount of memory required for computation. The sequence of processes of
the proposed algorithm with two-stage cycle processing is shown in Algorithm
1. In this algorithm, the iterator $j,m^{\prime}$ loops can be computed in
parallel.
Algorithm 1 Forward pass of multi-spiking neural networks
0: $\\{\hat{t}^{(m)}\\}_{m}$: Vector of input spike timing
0: $M$: Number of input spikes
0: $J$: Number of hidden neurons
0: $O$: Number of output neurons
0: $\\{w_{ij}\\}_{i,j},\\{w_{jo}\\}_{j,o}$: Set of weight matrix
0: $N_{1},\ N_{2}$: Maximum number of spikes to break the loop
0: $t_{out}$: Time to observe output membrane potential
0: $\tau_{I}$: Leakage time constant
0: $\\{v_{o}\\}_{o}$: Vector of output membrane potential Initialisation :
1: $\hat{z}^{(m)}\leftarrow\exp(\hat{t}^{(m)}/\tau_{I}),\ \forall
m\in\\{1,\cdots,M\\}$
2: $z_{j}^{(0)}[m^{\prime}]\leftarrow 0,\ \forall j\in\\{1,\cdots,J\\},\forall
m^{\prime}\in\\{1,\cdots,M\\}$
3: $z_{out}\leftarrow\exp(t_{out}/\tau_{I})$ 1st Loop Process :
4: for $n^{\prime}=0$ to $N_{1}-1$ do
5: for $j=1$ to $J$ (Parallel) do
6: for $m^{\prime}=1$ to $M$ (Parallel) do
7:
$\tilde{\mathcal{A}}_{j}^{(m^{\prime})}\leftarrow\tau_{I}\sum_{m=1}^{m^{\prime}}w_{ij}^{(m)}z^{(m)}$
8:
$\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})}\leftarrow\tau_{I}\sum_{m=1}^{m^{\prime}}w_{ij}^{(m)}\hat{z}^{(m)}\max\\{z_{j}^{(n^{\prime})},\hat{z}^{(m)}\\}$
9: if
$\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})},\tilde{\mathcal{A}}_{j}^{(m^{\prime})},\left(\tilde{\mathcal{A}}_{j}^{(m^{\prime})}\right)^{2}-4\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})}V_{th}\geq
0$ then
10:
$z_{j}^{(n^{\prime}+1)}[m^{\prime}]\leftarrow\frac{\tilde{\mathcal{A}}_{j}^{(m^{\prime})}-\sqrt{\left(\tilde{\mathcal{A}}_{j}^{(m^{\prime})}\right)^{2}-4\tilde{\mathcal{B}}_{j}^{(m^{\prime},n^{\prime})}V_{th}}}{2V_{th}}$
11: if $z_{j}^{(n^{\prime}+1)}[m^{\prime}]<\hat{z}^{(m^{\prime})}$ or
$z_{j}^{(n^{\prime}+1)}[m^{\prime}]>\hat{z}^{(m^{\prime}+1)}$ then
12: $z_{j}^{(n^{\prime}+1)}[m^{\prime}]\leftarrow\infty$
13: end if
14: else
15: $z_{j}^{(n^{\prime}+1)}[m^{\prime}]\leftarrow\infty$
16: end if
17: end for
18:
$z_{j}^{(n^{\prime}+1)}\leftarrow\min_{m^{\prime}}z_{j}^{(n^{\prime}+1)}[m^{\prime}]$
19: $\mathcal{A}_{o}\leftarrow\mathcal{A}_{o}+\beta_{I}\beta_{V}\tau_{I}\
w_{jo}\left(z_{j}^{(n^{\prime}+1)}\right)^{2},\ \forall o\in\\{1,\cdots,O\\}$
20: $\mathcal{B}_{o}\leftarrow\mathcal{B}_{o}+\beta_{I}\beta_{V}\tau_{I}\
w_{jo}\ z_{j}^{(n^{\prime}+1)},\ \forall o\in\\{1,\cdots,O\\}$
21: end for
22: end for2nd Loop Process :
23: for $n^{\prime}=N_{1}$ to $N_{2}-1$ do
24: for $j\in\\{j\ |\ z_{j}^{(N_{1})}<\infty\\}$ (Parallel) do
25: the same process as 1st Loop
26: end for
27: end for
28:
$v_{o}\leftarrow\frac{\mathcal{A}_{o}}{z_{out}}-\frac{\mathcal{B}_{o}}{(z_{out})^{2}},\
\forall o$
29: return $\\{v_{o}\\}_{o}$
## III Results
We use the MNIST benchmark to illustrate the high performance of our method
and the properties of SNNs acquired by multi-spike. Our training model is
characterized by the application of error backpropagation to a feed-forward
SNN with multi-spike, without using surrogate gradient or plasticity. The
weight variables were initialized according to a Gaussian distribution with
expectation and variance of $0.03$ and $0.3$, respectively, for all tasks.
There were $10$ output units since all experiments were solving MNIST
classification problems, and the inputs were composed of the same number of
input neurons and spikes as image dimensionality $784(=28\times 28)$. The
number of neurons in the hidden layer is consistently set to $400$, making a
two-layer network ($784-400-10$ network). Weights were updated using Adam [21]
and trained $100$ epochs with a learning rate of $0.001$. With respect to the
input, the MNIST dataset consists of $70000$ grayscale handwritten digit
images and correct answer labels consisting of any integer from $0-9$. Of
these, $60000$ were used for training and the remaining $10000$ for testing.
The pixel intensity in each dimension was scaled to a value between $[0,1]$
after inverting the black and white of the image and converted to input spike
timing between $t=0.0$ and $t=1.0$. This was split into mini-batches of $100$
each and trained. The output was the membrane potential of the output neuron
at the end time $t_{\text{end}}=1.0$. As a dead neuron loss, we flagged
neurons that only fire on input patterns less than $1/10$ of the mini-batch
size. Hyperparameters were set as $\lambda=0.01,\sigma=0.0001$ after grid-
searched, but their effect on learning performance was small unless taken to
extremes. Bias spiking was not used in any of the experiments. All programs
were implemented on PyTorch [22].
Figure 2: Time evolution of the membrane potential and postsynaptic current of
each neuron when a certain input pattern is given to the learned network. At
the top, a single round symbol indicates a single input spike. The second from
the top shows the membrane potential and postsynaptic current of a neuron
selected in the hidden layer, represented by solid and dotted lines,
respectively. The third figure from the top captures the temporal evolution of
the membrane potential of 20 neurons in the hidden layer. The bottom figure
illustrates the membrane potential of each of the 10 output neurons over time.
Fig. 2 shows the dynamics of neurons in the hidden layer and the output layer
when an input spike sequence was input to a trained SNN with $\tau_{I}=1.0$
under the above parameter settings. The number of spikes for each neuron
varies significantly, and it can be seen that the firing frequency of the same
neuron fluctuates along with the postsynaptic current within a given time
($t_{out}=1.0$ in this case).
First, we investigated how the number of spikes of SNNs can be altered under
multi-spike conditions. In the two-layer feedforward SNNs described above, the
entire computational process is based solely on the spike timing of each
spike. It is worth noting what distribution of the number of spikes per neuron
emerges when multi-spike is allowed.
Figure 3: Distributions of the number of spikes in SNNs (784-400-10) for
various leakage time constants $\tau_{V}=0.2,0.4,6.4$.
Fig. 3 shows histograms of the number of spikes of each hidden neuron, which
is averaged over ten different SNNs trained with different initial weights. As
this figure shows, the distribution depends on the magnitude of the current
and voltage leakage terms, i.e., the leakage time constant. Fig.3 shows that
the smaller $\tau_{I}$ is, the larger the mode of the distribution of the
number of spikes shifts toward. For $\tau_{I}=0.2$, the distribution shifts
unimodal to multimodal and the higher mode of the distribution appears
separated into non-$0$ areas. This indicates a bifurcation of all neurons in
an SNN into two unconsolidated classes, neurons that do not fire at all and
neurons that do fire at least once, which significantly hinders learning.
Figure 4: The relationship between the average number of spikes per neuron and
the leakage constant time $\tau_{I}$. the dashed horizon line denotes $1$,
which the SNN model with single-spike restrictions cannot exceed.
Fig. 4 shows the spike count per hidden neuron in each trained SNN when the
leakage time constant is varied, averaged over all hidden neurons. Fig. 4
indicates that the smaller the time constant of the synaptic leakage term,
i.e., the larger the leakage rate, under conditions in which multi-spike are
allowed, the more average number of neurons fire. By contrast, as the leakage
time constant increases, i.e., as the model asymptotically approaches a non-
leaky neuron model, the average number of spikes per neuron converges to a
certain value $\simeq 1.5$. With the single-spike restrictions, the average
number of spikes per neuron is equal to the density of the entire network, the
proportion of neurons that fire with respect to an input pattern. The figure
with single-spike restrictions indicates that when the leakage time constant
approaches close to zero, almost all neurons will fire. The SNNs with the
single-spike restrictions converge to 0.6 asymptotically (non-leaky models),
but unlike the multi-spike model, the curve is neither smooth nor monotonic.
In terms of sparsity on a per-neuron basis, the multi-spike SNNs are sparser
than the SNNs with single-spike restrictions over a wide range of leakage time
constants.
Figure 5: The relationship between learning performance and the leakage
constant time $\tau_{I}$.
Fig. 5 shows the relationship between leakage time constant and test accuracy
on the MNIST dataset. First, the multi-spike SNNs outperform the SNNs
restricted to single-spike in accuracy under any leakage time constant
condition. Common to both multi-spike and single-spike SNNs is that as the
leakage time constant is very close to zero accuracies deteriorate rapidly as
leakage increases. For both cases, there is a convergence value when the
leakage time constant is increased, converging to $98.23\%$ for multi-spike
and $97.96\%$ for single-spike, respectively. Note that there is a clear
optimum leakage time constant $\tau_{I}=0.8$ (with the test accuracy at
$98.43\pm 0.05\%$) only for the multi-spike case.
TABLE I: Performance of supervised learning for neural networks on MNIST tasks
by neuron type
Network | Coding | BP scheme | Spike type | Leakage | $\tau_{V}$ | $\tau_{I}$ | Test accuracy
---|---|---|---|---|---|---|---
784-800-10 (+ dropout)[23] | ANN | | | | | | $98.4\%$
784-800-10[24] | SNN (rate coding) | surrogate-gradient | multi | leaky | $\tau$ | $\tau$ | $98.64\%$
784-800-10[25] | SNN (rate coding) | surrogate-gradient | multi | leaky | $\tau$ | $\tau$ | $98.84\pm 0.02\%$
784-800-10 [13] | SNN (temporal coding) | timing-based | single | non-leaky | $\infty$ | $\tau$ | $97.55\%$
784-340-10 [14] | SNN (temporal coding) | timing-based | single | leaky | $\tau$ | $\tau$ | $97.96\%$
784-500-10 [17] | SNN (temporal coding) | timing-based | single | non-leaky | $\infty$ | $\infty$ | $97.99\pm 0.07\%$
784-350-10 [15] | SNN (temporal coding) | timing-based | single | leaky | $\tau$ | $2\tau$ | $97.2\pm 0.1\%$
256-246-10 [26] | SNN (temporal coding) | surrogate-gradient | multi | leaky | $\tau$ | $\tau$ | $97.5\pm 0.1\%$
784-400-10 [This work] | SNN (temporal coding) | timing-based | single | non-leaky* | $\tau$ | $2\tau$ | $97.96\pm 0.08\%$
784-400-10 [This work] | SNN (temporal coding) | timing-based | single | leaky | $\tau$ | $2\tau$ | $\bf{97.99\pm 0.06\%}$
784-400-10 [This work] | SNN (temporal coding) | timing-based | multi | non-leaky* | $\tau$ | $2\tau$ | $98.23\pm 0.07\%$
784-400-10 [This work] | SNN (temporal coding) | timing-based | multi | leaky | $\tau$ | $2\tau$ | $\bf{98.43\pm 0.05\%}$
*It can be regarded as non-leaky due to extreme conditions: $\tau=3.2$.
A comparison of the results obtained in this experiment with the MNIST test
accuracy of various existing 2-layer feedforward SNNs are summarized in Table
I. The error indications in our study are defined by the standard errors of
the results of 10 experiments performed with different initial weights. The
results show that in supervised learning of simple feedforward SNNs trained
with temporal coding, multi-spike SNNs in the timing-based method achieve the
highest accuracy over all existing methods. In addition, there is already no
significant difference when compared to SNN supervised learning based on rate
coding.
## IV Discussions
We performed supervised learning on both the existing single-spike-limited
SNNs per neuron and the multi-spike-allowed model in this study and compared
the properties of the error backpropagation method for SNNs based solely on
strict spike timing. In particular, the present study shows that omitting the
single-spike restriction and allowing multiple firings of a neuron improves
the performance of the standard MNIST benchmark under all leakage time
constant conditions. A notable feature of our method is that it uses the exact
spike timing in the backpropagation computation process, thus achieving ideal
temporal coding mediated by spikes on multi-spike SNNs in terms of gradient
computation. In related works on multi-spike SNNs, although types of SG
method, applied an appropriate regularization and achieved sparse, near-
temporal-coding learning [26, 27]. However, since these learning methods using
gradient approximation assume continuous spiking behaviors during
backpropagation, it is open to question whether they are strictly temporal-
coding learning from the viewpoint of gradient propagation. To the best of our
knowledge, our proposed algorithm, for the first time, achieved the supervised
learning of multi-spike SNNs with backpropagation of exact gradients based on
spike timing.
Our proposed model has advantages in precisely dealing with the reset
dynamics. The resetting behavior at the moment of firing is one of the factors
that significantly influence the information processing mechanism. In general,
studies addressing supervised learning of multi-spike SNNs have introduced
double-reset neuron models, which reset the membrane potential and also the
corresponding postsynaptic current at the same time due to its simplicity [10,
20]. However, there are concerns that the adoption of double-reset means that
any information input up to each spike timing is lost on each firing,
completely breaking the dependency between previous and subsequent spikes
within the same neuron, thus lacking the benefits of multi-spike. By contrast,
we reset only the membrane potential that has reached the threshold to the
baseline, $0$, while the postsynaptic current is not reset and retains its
pre-firing values. In this way, the past information received by the neuron
itself is propagated into the future via both its own past spike timing and
the postsynaptic current path. There is a non-negligible difference in the
network structure whether double-reset is employed or not, and we employed the
latter in this study.
Heuristic approximations and abbreviations in the error backpropagation
calculation process for SNNs are also major factors that compose the network
learning paradigm. In general, learning methods using surrogate gradients
ignore the error backpropagation paths led by resets of membrane potential,
what is called reset path, in multi-spike [28, 10, 20]. Backpropagating
through surrogate gradient results in the fact that the reset information is
propagated recursively by a number of times equal to the number of time steps,
even though the actual reset is instantaneous at the spike timing. Even though
the amount of the approximation error induced by a single surrogate derivative
is tolerable, the accumulated error can be huge because the number of time
steps is usually by far larger than the number of spikes [28, 20]. To avoid
this approximation error diverging is the main reason why they ignore the
reset paths. On the other hand, the timing-based method used in this study
does not use any approximation in the whole calculation process of
backpropagation, i.e., it realizes an exact gradient calculation. It should be
noted, however, that the timing-based method cannot capture the occurrence and
disappearance of sudden spikes caused by weight updates and the accompanying
changes in the number of spikes. The sudden spike changes pointed out here are
not spike changes that increase or decrease with continuous changes in
membrane potential in the time direction around the end time $t_{out}$, but
spike changes that occur with discontinuous membrane potential changes in the
middle of a given period $(0,t_{out})$. Although this is a challenge inherent
to the timing-based method as pointed out by SpikeProp [12], it was
experimentally shown that learning works well both for single-spike SNNs and
also for the multi-spike SNNs in this study. The impact of this sudden spike
variation is one of the issues that should be considered in the future.
Through this study, it is clear that the properties of the overall network
spike count are altered depending on the time constants of the membrane
potential and synaptic current, i.e., the synaptic leakage term. The results
basically show that the smaller the leakage time constant (i.e., the more
leakage), the higher the overall network spike count, which leads to higher
learning performance. This change in performance with increasing voltage
leakage is a phenomenon not seen in existing SNNs restricted to single-spike.
In the case of multi-spike SNNs trained through surrogate gradients, through
experiments of rate-coding with feedforward networks, [29] showed that the
leakage of membrane potentials and postsynaptic currents ensures the
robustness of the network, leading to high learning performance. In the
present timing-based method, a similar result may be derived from the
correlation between the increase in the number of spikes and the increase in
performance. In other words, it is possible that the increased rate-coding-
like quality induced by the increase in the number of spikes may contribute to
the superior generalization effect. In contrast, single-spike SNNs are clipped
at a maximum of $1$ in the number of spikes per neuron, so the amount of
leakage of the membrane potential should not affect their performance.
Considering that time constants smaller than the time scale of the experiment
($1\text{ms}$ in this case) would conversely cause a sharp decline in
performance, therefore, there are optimal values for the time constants of
membrane potential and synaptic current for multi-spike SNNs. In particular,
the amount of optimized leakage time constant can be characterized by the
magnitude of the task itself imposed on the SNNs and the local network
structure on which the time scale of the experiment depends. Optimizing the
learning capacity of the entire network through leakage cannot be discussed
only in terms of machine learning aspects. It has been studied and reported
from a physiological perspective that the leakage time constant of a neuron
plays an important role as an indicator to characterize that brain region and
that the diversity of time constants is key to the task-solving ability of
that neural networks [30, 16]. This is in line with the relationship derived
in this study between the optimal leakage time constant and the timescale of
the tasks.
## V Conclusion
In this study, we proposed a novel efficient backpropagation algorithm for
multi-spike SNN models and pursued the goal of achieving high algorithmic
performance and clarifying its learning characteristics compared to single-
spike SNN models. Moreover, we proposed novel learning techniques such as a
two-stage algorithm based on spike count and the addition of a dead neuron
penalty. The performance of our algorithm on the MNIST dataset outperformed
the state-of-the-art SNNs based on temporal coding and was confirmed to be on
par with the conventional SNNs and ANNs based on rate coding. Focusing on the
leakage time constant of the postsynaptic current and the membrane potential,
we investigated how this affected the characteristics of spike counts of SNNs
with multi-spike neurons. We also showed the existence of an optimal membrane
potential leakage rate that achieved maximum performance, around the time
scale of the task. Moreover, the phenomenon was not observed in training SNNs
with single-spike restrictions. Our multi-spike networks suggested the
possibility of supervised learning based on the spike timing of SNNs with a
biologically plausible recurrent topological structure. In the future, we
would like to pursue timing-based learning of more complex and large-scale
tasks and recurrent networks. Finally, we would like to add that our SNN
model, which learns with information encoded in the spike timing, can also
contribute to the construction of very energy-efficient systems including the
learning process [15, 17].
## Acknowledgments
This work was partially supported by SECOM Science and Technology Foundation,
JST PRESTO Grant Number JPMJPR22C5, the NEC Corporation, Moonshot R&D Grant
Number JPMJMS2021, AMED under Grant Number JP21dm0307009, Institute of AI and
Beyond of UTokyo, the International Research Center for Neurointelligence
(WPI-IRCN) at The University of Tokyo Institutes for Advanced Study (UTIAS),
JSPS KAKENHI Grant Number JP20H05921. Computational resource of AI Bridging
Cloud Infrastructure (ABCI) provided by National Institute of Advanced
Industrial Science and Technology (AIST) was used.
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# A High Availability Management Model based on VM Significance Ranking and
Resource Estimation for Cloud Applications
Deepika Saxena and Ashutosh Kumar Singh, D. Saxena and A.K. Singh are with the
Department of Computer Applications, National Institute of Technology,
Kurukshetra, India. E-mail<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract
Massive upsurge in cloud resource usage stave off service availability
resulting into outages, resource contention, and excessive power-consumption.
The existing approaches have addressed this challenge by providing multi-
cloud, VM migration, and running multiple replicas of each VM which accounts
for high expenses of cloud data centre (CDC). In this context, a novel VM
Significance Ranking and Resource Estimation based High Availability
Management (SRE-HM) Model is proposed to enhance service availability for
users with optimized cost for CDC. The model estimates resource contention
based server failure and organises needed resources beforehand for maintaining
desired level of service availability. A significance ranking parameter is
introduced and computed for each VM, executing critical or non-critical tasks
followed by the selection of an admissible High Availability (HA) strategy
respective to its significance and user specified constraints. It enables cost
optimization for CDC by rendering failure tolerance strategies for significant
VMs only instead of all the VMs. The proposed model is evaluated and compared
against state-of-the-arts by executing experiments using Google Cluster
dataset. SRE-HM improved the services availability up to 19.56% and scales
down the number of active servers and power-consumption up to 26.67% and
19.1%, respectively over HA without SRE-HM.
###### Index Terms:
Cloud computing, Cost optimization, Failure predictor, High Availability,
Resource management, VM Ranking.
## 1 Introduction
Nowadays, there is a strong trend towards "digitization in all data and all
data in digitization" across the globe, wherein cloud data centres (CDCs) have
emerged as a backbone of the entire information and communication technology.
It facilitates high availability of information technology (IT) resources and
maximum computing benefits at minimum capital investment to the end users. A
Cloud Service Providers (CSPs) is obliged to Service Level Agreement (SLA)
adherence by taking responsibility for their infrastructure and ensuring
availability and safety at all ends [1], [2], services and performance outage
occurs, stemming from a surge in resource utilization [3], [4], [5]. However,
the exponential growth of internet traffic and cloud resource demands impede
the delivery of cloud services with consistent high availability the whole
time. For instance, a recent survey of the COVID-19 pandemic witnessed
numerous spectacular incidents of cloud outages for several hours which have
overwhelmed the titan Cloud Service Providers (CSPs), including Zoom,
Microsoft Azure, Google Cloud Platform, Amazon Web Services, Salesforce, IBM
Cloud, etc. [6]. Such incidents prompt a critical question about cloud
technology and increase the obligation of high availablity (HA) on CSPs. Major
causes of cloud outages are failures of applications, VMs, and servers or
physical machines (PMs) which renders computing and storage instances
inaccessible [7], [8]. Such instances of cloud outages point to a critical
question on cloud technology concerning reliability and availability of
services for the end users. The existing approaches furnish highly available
services by applying proactive as well as reactive approaches [9], [10], [11].
Proactive techniques of failure handling rely on the prior knowledge of the
failure of applications and VMs and migrating such VMs. While the reactive
techniques include checkpointing, replication, retry, application
resubmission, etc. which are triggered at the occurrence of actual failure
detection [12]. Among the proactive approaches, the most reliable approach for
high availability is to maintain multiple replicas of VM instances while
engaging active physical machines in excess [13] and hiring multiple clouds
[5]. The major concern is that previous reactive approaches engage delay in
service provisioning while the proactive approaches account for high
operational cost.
Since the availability of the cloud resources is measured in terms of mean
time between failures (MTBF) and mean time to repair (MTTR) associated with
service uptime and downtime, respectively [14], [15]; the key solution for
improving services availability is to maximize MTBF and minimize MTTR i.e.,
time of recovery/repair by employing fault tolerance strategies along with
effective resource management. The proactive resource failure detection and
mitigation of VM and server failure enhances service uptime leading to higher
MTBF. Diversely, the MTTR can be minimized by appointing multiple replicas for
VM instances. Therefore, both proactive, as well as reactive mechanisms must
be applied efficiently to upgrade cloud services availability. A cloud
application being composed of numerous small sub-units or tasks makes it
highly expensive to provide redundant or alternative instances for all VMs
[3], [16], [17]. Moreover, it is not desirable to deliver highly available
services for non-critical tasks which either have minimal or no effect on
application execution. In this context, the best approach is to identify the
significance of different tasks beforehand and employ a selected optimal HA
strategy for their execution accordingly. However, this approach is entangled
with two primary challenges: (i) How to locate the importance of various tasks
in a complex cloud application? (ii) How to decide on an admissible HA
strategy for different tasks subject to their contribution to an application
execution and user-specified constraints?
### 1.1 Our Contribution
Contrary to the existing works, this paper proposes a novel Significance
Ranking and Resource Estimation based High Availability Management (SRE-HM)
model that encapsulates both proactive and reactive approaches to deliver
highly available cloud services with optimized operational costs. It locates
critical and non-critical tasks based on their characteristics, number and
frequency of invocations. Accordingly, a significance rank computation method
is developed for estimation of the importance of different VMs depending on
the characteristics of tasks assigned for execution. Thereafter, an optimal HA
strategy selection method is established corresponding to significance rank of
different VMs and user specified constraints to optimize the operational cost
for CSPs. Further, the proposed model imparts a prior estimation of resource
contention and mitigation of performance degradation by migrating VMs and load
balancing in real-time. The key contributions of the paper are fourfold:
* •
A Virtual High Availability Zone (VHAZ) comprising of several Virtual High
Availability Network (VHAN) is created where a new parameter ‘significance
rank’ of a VM is enumerated to locate importance of each VM in an application
execution.
* •
A method for selection of an admissible HA strategy is developed to curtail
the excessive power expenses while achieving high performance and service
availability experience for the end users.
* •
A real-time server failure predictor based on Long Short Term Memory (LSTM)
Neural Network is implemented to estimate resource-contention and mitigate its
worse effects by triggering VM shifting, proactively.
* •
Empirical simulation and performance evaluation of proposed model reveal that
it substantially improves high availability for cloud users with optimized
cost for cloud data centre.
Organization: Section 2 discusses related work with research gaps and Section
3 entails description of SRE-HM model. Sections 4, 5, and 6 discuss PM failure
estimation and handling, Significance rank computation, and HA strategy
selection, respectively. Section 7 presents operational design and complexity
analysis followed by the performance evaluation and comparison in Section 8.
Section 9 remarks the conclusion and future scope of the proposed model.
## 2 Related work
The related work pertaining to HA in cloud environment is distinguished into
following sub-sections: (i) Proactive approaches and (ii) Reactive approaches.
### 2.1 Proactive approaches
Marahatta et al. [18] proposed a deep neural network based failure predictor
to differentiate the future tasks on VMs into failure prone and non-failure
prone tasks. Accordingly, resources are allocated exclusively for failure and
non-failure prone tasks by replicating three consecutive failure-prone tasks
into three copies to be executed on different servers to prohibit overlapping
and redundant execution. The performance of the work is investigated using
Internet and Euler datasets which improve fault tolerance and energy
efficiency within CDC. Pinto et al. [19] have developed a Support Vector
Machine (SVM) based failure predictor for distributed computing Hadoop
clusters which is trained with a non-anomalous dataset during different
operation patterns like boot-up, shutdown, idle, task allocation, resource
distribution, etc. to detect and classify between normal and abnormal
situation. An online prediction based multi-objective load-balancing (OP-MLB)
framework is proposed in [3] for proactive estimation of server overload and
its alleviation by employing VM migration for improving service availability
and energy-efficiency of CDCs. The forthcoming load on VMs is estimated by
employing an online neural-network based prediction system to determine the
future resource utilization of the servers proactively. It helped to detect
server overloading which is tackled by migrating VMs of highest resource
capacity from overloaded server to an appropriate server machine. The VM
placement and migration are executed using a multi-objective algorithm for
minimization of power consumption.
Sharma et al. presented a failure aware and energy-efficient (FAEE) VM
placement scheme in [20] which predicted VM failure using an exponential
smoothing based forecasting technique. Thereafter, two fault tolerance methods
including VM migration and checkpointing are triggered to handle service
failure and upgrade service availability. This work was evaluated using
Grid5000 Failure Traffic Analysis (FTA) dataset. It concluded that the energy
efficiency and reliability of cloud systems can be significantly improved by
considering the failure characteristics of physical resources. Dabbagh et al.
[21] presented an integrated energy-efficient VM placement and migration
framework for the cloud data centre. It applied a Wiener filter based
predictor for estimation of the number of VM requests and the future resource
requirement. These predicted values are used to allow only the required number
of physical machines in active state and helps in achieving a substantial
energy saving and resource utilization. Nguyen et al. [22] addressed service
availability problem by adopting multiple usage prediction through multiple
linear regression for estimation of the relationship between the input
variables and the output for energy efficient data centres. This work
estimated overload or host failure with multiple usage prediction (OHD-MUP)
and underloaded host detection with multiple usage prediction (UHD-MUP) and
balanced load by migrating selected VMs from overloaded servers to energy-
efficient server. Multiple Additive Regression Trees with Gradient Boosting
(MART-GB) algorithm is utilized for Cloud Disk Error Forecasting (CDEF) in
[23] which ranked all disks according to degree of error-proneness to allow
shifting of existing VMs in real-time and assignment of new VMs on healthy
disks to improve service availability. Adamu et al. [24] have improved system
availability by proposing a Linear Regression (LR) Model and Support Vector
Machine (SVM) with a Linear Gaussian kernel for predicting hardware failures
in a real-time cloud environment. Sharma et al. [20] have predicted VM failure
using a forecasting technique based on exponential smoothing. This work has
utilized two fault tolerance methods including VM migration and checkpointing
to handle VM failure and improve service availability. It achieved a
significant improvement in energy efficiency and reliability by considering
failure characteristics of physical resources. A Fault Tolerant Elastic
Resource Management (FT-ERM) framework has been proposed in [25] to improve
availability of services by inducing fault management in servers and VMs. This
framework included an online failure predictor for proactive estimation of
failure-prone VMs where the operational status of server is monitored with the
help of a power analyser, resource estimator, and thermal analyser. The
failure-prone VMs are assigned to fault-tolerance unit consisting of decision
matrix and safe box to handle any outage beforehand while assuring high
availability of services for the cloud users.
### 2.2 Reactive Approaches
Zhang et al. [5] proposed a data hosting scheme called as CHARM which
collaborated two desired key functions to furnish HA. Firstly, multiple
suitable clouds and an optimal replication scheme for data storage and enhance
availability are selected. Secondly, VM migration and re-allocation is
triggered as per the variations in data access patterns and pricing of cloud
applications. This work enabled fine-grained decisions making regarding which
storage mode to be used and which clouds to be used for data placement. An HA-
aware scheduling algorithm named CHASE is proposed in [26] that considers and
analyses functionality requirements, redundancies between applications, and
real-time interdependencies in real-time. Its prototype was designed to
schedule components in real cloud environment while communicating with
OpenStack.
Wang et al. [7] developed a fault-tolerant elastic scheduling algorithms
(FESTAL) to provide a fault tolerant VM scheduling accompanied with
virtualization technology and an appropriate VM migration scheme with battery
back-up features. It addressed the issues of reliability, elasticity and
schedulability of virtualized clouds. It achieved both fault tolerance and
high performance in terms of resource utilization by employing an elastic
resource management and the fault-tolerant scheduling algorithms. Extensive
experiments based on the synthetic workloads and the real world traces
validated the performance of FESTAL. Zhu et al. [27] have utilized battery
back-up scheduling schemes to establish an algorithm for fault-tolerant
execution of scientific workflows (FASTER) by incorporating task allocation
and message transmission features that employed a backward shifting approach
for use of physical resources. A real-time workflow fault-tolerant model is
developed which extends the traditional primary backup model by including the
cloud characteristics. Using this model, a task allocation and message
transmission approach is developed to ensure fault tolerance during
information processing and workflow execution. This algorithm enabled full
utilization of idle resources and incorporated task overlapping and VM
migration for improved resource utilization. Further, it applied the
vertical/horizontal scaling-up technique to provision the resources for a
bursty workflows, and used a vertical scaling-down scheme to avoid unnecessary
and ineffective resource changes due to fluctuating workflow requests.
The existing works have attempted to provide HA either by proactive methods
such as resource failure prediction or reactive methods including multi-cloud
and running multiple replicas of a VM leading to high operational cost [28].
In contrast, SRE-HM furnishes both reactive as well as proactive approaches to
maximize MTBF and minimize MTTR and thereby upgrading the service availability
at both ends. Also, it reduces the major issue of providing replicas for all
the VMs which leads to high operational costs. SRE-HM estimates significance
ranks of VMs according to the characteristics of tasks executed on them
followed by a selection of the most optimal HA strategy corresponding to each
VM’s significance and user-specified constraints to enhance the HA experience
for users with an optimized operational cost for CSPs. Moreover, the proposed
model imparts a prior estimation of resource contention and mitigation of its
worse effects by migrating VMs and balancing load in real-time. The prediction
of resource usage assists in alleviating server over-/under-load while
reducing resource and power wastage. Table I shows the list of symbols with
their explanatory terms used throughout the paper.
TABLE I: Notations and their descriptions $U$: user; $M$: number of users;
$V$: virtual machine ;
---
$Q$: number of VMs; $PM$: physical machine; $\mathds{AV}$: Availability Score;
$P$: number of servers; $i$: index for servers; $j$: index for VMs;
$k$: index for User; $\omega_{kji}$: Mapping of $k^{th}$ user’s $j^{th}$ VM on
$i^{th}$ server;
$\mathchorus{RU}$: resource utilization; $\mathchorus{PW}$: power consumption;
$\mathchorus{N}$: VM as a node;
$r$: user’s request; $\mathds{R}$: resources; $\mathchorus{C}$: Cluster;
$Mem$: Memory; $C$: CPU;
$\mathds{N}$: Number of resources; $\eta$: probability of server failure;
$\mathchorus{E}_{ab}$: Edge between node a and b; $\mathchorus{fq}$: frequency
of invocation;
$\Omega$: mapping for HA selection; $\mathchorus{S}(\mathchorus{N}_{a})$:
Significance value of node ‘a’;
$\mathds{V}_{fp}$: Failure-prone VMs; $\mathchorus{F}$: Failure probability of
HA strategy;
## 3 SRE-HM Model
Consider a Cloud Service Provider (CSP) owns a Cloud Data Centre (CDC)
consisting of $K$ Clusters of Servers (CoS) {$C_{1}$, $C_{2}$, …, $C_{K}$}
$\in\mathds{C}$ employing $P$ physical machines (PM) {$PM^{k1}$, $PM^{k2}$, …,
$PM^{kp}$} $\in\mathds{PM}$, where $\forall k\in[1,K]$ as demostrated in Fig.
1. The PMs can be in any of the three possible states: active, inactive, or
failed due to hardware failure. Each active server deploys Resource Estimator
(RE) and $Q$ VMs such that {$V^{ki}_{1}$, $V^{ki}_{2}$, …, $V^{ki}_{q}$}
$\in\mathds{V}$ are hosted on $i^{th}$ server $PM^{ki}$ within cluster
$C_{k}$, where $\forall_{i}\in[1,P]$, $\forall_{k}\in[1,K]$.
Figure 1: SRE-HM Model
RE predicts resource contention or overloading condition on its associated
active server ($PM$) to mitigate its worse effects proactively (discussed in
Section 4). CDC employs Resource Management Unit (RMU) composed of two major
sub-units Request Scheduling Unit (RSU) and Response Retrieval Unit (RRU) for
distribution of users assigned job requests, retrieval of responses, and
efficient management of physical resources. Assume that $M$ Users {$U_{1}$,
$U_{2}$, …, $U_{M}$} $\in\mathds{U}$ submit job requests {$\lambda_{1}$,
$\lambda_{2}$, …, $\lambda_{M}$} to RSU for execution. Each request is a high
computing application in the form of Bag of Tasks (BoT) such as {$t^{m}_{1}$,
$t^{m}_{2}$, …, $t^{m}_{b}$} $\in\lambda_{m}:\forall m\in[1,M]$, where $b$ is
total number of tasks with a specific cost ($\mathchorus{EC}^{m}$) and
deadline of execution ($\mathchorus{T}^{m}$).
RSU distributes tasks {$t^{m}_{1}$, $t^{m}_{2}$, …, $t^{m}_{b}$}
$\in\lambda_{m}:\forall m\in[1,M]$ for execution among the VMs deployed on
different PMs with in CoS. Thereafter, the responses {$\delta_{1}$,
$\delta_{2}$, …, $\delta_{M}$} generated by VMs are assembled and fetched to
users {$U_{1}$, $U_{2}$, …, $U_{M}$} with the help of RRU. Both RMU and users
{$U_{1}$, $U_{2}$, …, $U_{M}$} send consecutive feedback to CoS and High
Availability Management Unit (HAMU) by evaluating the Availability Score
($\mathds{AV}_{score}$) using the following Eq. (1); where $\mathds{AV}_{g}$
is guaranteed availability (as defined in SLA terms and conditions) and
$\mathds{AV}_{o}$ is offered availability (actual availability of resources
experienced by the user).
$\mathds{AV}_{score}=\frac{\mathds{AV}_{g}-\mathds{AV}_{o}}{\mathds{AV}_{g}}\times
100$ (1)
The availability score ($\mathds{AV}_{score}$) is sent periodically as a
feedback for analysing the performance of currently adopted availability
strategies and further improve the status of high availability management
within CoS. The VMs executing tasks {$t^{m}_{1}$, $t^{m}_{2}$, …, $t^{m}_{b}$}
$\in\lambda_{m}:\forall m\in[1,M]$ of $m^{th}$ user have correlation among
them because of mutual data exchange for successful execution of $m^{th}$ job
request $\lambda_{m}$. Therefore, a virtual network of such collaborating VMs
is created depending on the scheduling of tasks (belonging to a common job
request) on different VMs which are allocated under the constraints of high
availability forming a Virtual High Availability Network (VHAN). The proposed
model introduces VHAN, Virtual High Availability Zone (VHAZ) and VM
Significance Rank (VSR) which are defined as follows:
###### Definition 1 (VHAN).
A virtual network of VMs $\\{V^{m}_{1}$, $V^{m}_{2}$, …, $V^{m}_{b}\\}$ having
heterogeneous resource $\\{\mathchorus{C}$, $\mathchorus{Mem}\\}$ capacities
executing inter-dependent tasks $\\{t^{m}_{1}$, $t^{m}_{2}$, …,
$t^{m}_{b}\\}\in\lambda_{m}:\forall m\in[1,M]$ deployed on different physical
machines by the CSP, are dedicated to provide high availability service to
user $U_{m}$.
###### Definition 2 (VHAZ).
A virtual zone is a collection of multiple VHANs confined to provide high
availability to the number of users $\\{U_{1}$, $U_{2}$, …, $U_{M}\\}$ by
employing selected optimal high availability strategies.
###### Definition 3 (VSR).
A significance rank assigned to a VM is an estimated value for a quantitive
measurement of its intendment based on the criticality and characteristics of
the task it holds, frequency of invocations, and number of invocations by
other VMs.
A VHAZ comprising of VHANs: $H_{1}$, $H_{2}$, …, $H_{M}$ dedicated for users
$U_{1}$, $U_{2}$, …, $U_{M}$, respectively is designed, where a set of VMs
{$V^{21}_{q}$, $V^{K1}_{q}$, $V^{1p}_{q-1}$, $V^{2p^{\prime}}_{1}$} $\in
H_{1}$; {$V^{1p}_{q}$, $V^{21}_{1}$, $V^{21}_{q-1}$, $V^{11}_{q}$,
$V^{2p^{\prime}}_{q}$} $\in H_{2}$ and so on. The different VMs {$V^{ki}_{1}$,
$V^{ki}_{2}$, …, $V^{ki}_{b}$} $\in H_{m}$: $\forall
k\in[1,K],i\in[1,q],m\in[1,M]$ constituting $m^{th}$ VHAN are physically
deployed on different PMs within CoS. The essential constraints that must be
satisfied for VM deployment on a PM are stated in Eq. (2), where
$\omega_{kji}$ represents a mapping $\omega_{kji}:C_{k}\times V_{j}\times
PM_{i}\in\\{1,0\\}$ such that $\omega_{kji}=1$ if VM $V_{j}$ is deployed on
server $PM_{i}$ within cluster $C_{k}$, else, it is $0$; $\forall
i\in[1,P],k\in[1,K]$; ${\mathds{R}}$ specifies resources viz., CPU
($\mathchorus{C}$) and memory ($\mathchorus{Mem}$) for assignment of VM
($V_{j}$) on $PM_{i}$.
$\displaystyle\sum_{j=1}^{Q}{V^{ki}_{j}\times{{\mathds{R}_{j}}}\times\omega_{kji}\leq
PM_{i}\times{{\mathds{R}_{i}}}}$ (2)
The resource utilization of all the VMs belonging to each VHAN is periodically
monitored and the probability of failure is estimated proactively with the
help of RE. The significance ranks of VMs (i.e., VSR) belonging to different
VHAN are estimated by employing VM Significance Rank Computation Unit (VSRCU)
which is discussed in detail in Section 5. VSR helps in determining the
valuable contributions of different VMs for the successful execution of a job
request $\lambda$ in a VHAN and thereby allows an optimal selection of a high
availability strategy from the pool encapsulating different HA strategies.
HAMU fetches information from VSRCU, VHAZ, and a pool of HA strategies
(Section 6) to decide and adopt the most suitable HA strategy for each VM in
VHAN to implement the best possible high availability environment for requests
execution.
## 4 PM Failure Estimation and Handling
The availability of physical machines is managed with the help of a Resource
Estimator (RE) dedicated to analyse the upcoming resource demand on a
particular server proactively. Fig. 2 portrays a complete mechanism for
managing the availability of PMs, where RE utilizes Long Short Term Memory
(LSTM) Neural Network for accurate prediction of expected resource usage on a
server in the future which is trained/re-trained periodically. LSTM neural
network is chosen for server failure prediction due to their high capability
of remembering information for a longer duration and extracting intuitive
patterns by finding useful correlations among them [29]. The physical machines
$PM^{1}$, $PM^{2}$, …, $PM^{p}$ employ an exclusive LSTM neural network based
resource predictor, optimized with the latest historical resource usage
information of VMs.
Figure 2: Mechanism for Physical Machine Availability
LSTM is a cell state, where previous block ($\mathchorus{S}^{t-1}$)
information flows to the current block ($\mathchorus{S}^{t}$). It is composed
of four neural network layers, where the first layer ($\mathchorus{G}_{1}$)
decides the amount of previous information of resource usage
($\mathchorus{cf}_{\mathchorus{RU}_{i}}^{t}$) to be passed using Eq. (3),
where $\mathchorus{W}$ is the weight matrix, $\mathchorus{b}$ is a bias value,
$Z_{Pr}^{\mathchorus{RU}^{t-1}}$ and $\mathchorus{RU}_{i}^{t}$ are previous
output and current input, respectively. The cell state is updated using two
network layers viz., $sigmoid$ layer ($\mathchorus{G}_{2}$) and $tanh$ layer.
$\mathchorus{G}_{2}$ decides the values to be updated ($\mathchorus{I}^{t}$)
using Eq. (4) and $tanh$ layer generates a new candidate values vector
($\hat{\mathchorus{S}}^{t}$) as stated in Eq. (5). Finally, Eq. (6) combines
both outputs to update cell state.
$\displaystyle\mathchorus{cf}_{\mathchorus{RU}_{i}}^{t}=\mathchorus{G}_{1}(\mathchorus{W}_{\mathchorus{cf}}\cdotp[Z_{Pr}^{\mathchorus{RU}^{t-1}},\mathchorus{RU}^{t}]+\mathchorus{b}_{\mathchorus{RU}})$
(3)
$\displaystyle\mathchorus{I}^{t}=\mathchorus{G}_{2}(\mathchorus{W}_{\mathchorus{I}}\cdot[Z_{Pr}^{\mathchorus{RU}^{t-1}},\mathchorus{RU}^{t}]+\mathchorus{b}_{\mathchorus{I}})$
(4)
$\displaystyle\hat{\mathchorus{S}}^{t}=\mathchorus{tanh}(\mathchorus{W}_{\mathchorus{S}}\cdot[Z_{Pr}^{\mathchorus{RU}^{t-1}},\mathchorus{RU}^{t}]+\mathchorus{b}_{\mathchorus{S}})$
(5)
$\displaystyle\mathchorus{S}^{t}=\mathchorus{cf}_{\mathchorus{RU}_{i}}^{t}\times\mathchorus{S}^{t}+\hat{\mathchorus{S}}^{t-1}\times\mathchorus{I}^{t}$
(6)
This LSTM based resource predictor periodically forecasts CPU
($Z_{Pr}^{\mathchorus{C}_{i}}$) and memory usage
($Z_{Pr}^{\mathchorus{Mem}_{i}}$) for $PM^{i}$ to be provided to RE for
estimation of the probability of failure ($\eta_{i}^{\ast}$) of the respective
PM using Eq. (7), where $\mathchorus{RU}_{i}^{Thr}$ is a threshold value of
resource utilization viz., $\mathchorus{C}$ and $\mathchorus{Mem}$ usage value
for $PM^{i}$ and $\forall_{i}\in[1,P]$. This threshold value is decided by RMU
in the real world cloud environment and it is set to 85% for the empirical
evaluation.
$\eta_{i}^{\ast}=\begin{cases}1,&{If(\mathchorus{RU}_{i}^{Thr}\geq\eta_{i})}\\\
0,&{\text{otherwise}}\end{cases}\quad\eta_{i}:\cup(Z_{Pr}^{\mathchorus{C}_{i}},Z_{Pr}^{\mathchorus{Mem}_{i}})$
(7)
A Physical Machine Management Unit (PMMU) is employed for proactive handling
of any resource-contention based failure by allowing VM migration. PMMU
gathers failure information of PMs as {$\eta_{1}^{\ast}$, $\eta_{2}^{\ast}$,
…, $\eta_{p}^{\ast}$} and alleviates them by migrating VMs before occurrence.
The largest size VMs is migrated (to avoid the occurrence of resource
congestion frequently) from the estimated overloaded PM to a suitable active
PM preferably rather than waking up an inactive PM for minimization of overall
power consumption in CDC. The inactive PMs are turned ON only when currently
activated PMs are unable to host the migrating VMs.
If the predicted information of resource usage of a VM is higher or lesser
than the current resource usage of that VM, then the respective over-
utilization or under-utilization of the PM is estimated. In case of probable
over-utilization of the PM, the chances of resource congestion increase which
indicates the probability of PM failure (i.e., $\eta^{\ast}$ becomes TRUE).
Accordingly, the status of VM (${V}_{j}^{{status}}$) turns into ‘1’ in Eq. (8)
which triggers the migration of the respective VM to alleviate the effect of
server over/under-load before their actual occurrence. The VM migration cost
($Mig_{cost}$) is computed using Eq. (9).
$\displaystyle{{V}_{j}}^{status}=\begin{cases}1&{If(\omega_{kji}\times\eta_{i}^{\ast}=1)}\\\
0&{\text{otherwise.}}\end{cases}$ (8)
$Mig_{cost}={(\sum{c_{mig.j}*(D(PM_{i},PM_{ii})\times
W(V_{mig}))})+\sum{n_{i}\times{d_{i}}}}$
(9)
where $D(PM_{i},PM_{ii})$ is the distance or number of hops covered by
$V_{mig}$ from source ($PM_{i}$) to destination server $PM_{ii}$,
{$i,ii\in[1,P],i\neq ii$}, $V_{mig}$, $W(V_{mig})$ =
$V_{mig}^{\mathchorus{C}}\times V_{mig}^{\mathchorus{M}}$ is the size of
migrating VM. The first term
$\sum{c_{mig.{ii}}*{D(PM_{i},PM_{ii})*W(V_{mig})}}$ specifies energy consumed
during VM migration. The second term $\sum{n_{i}*{d_{i}}}$ states energy
consumed in the server state transition, where if $j^{th}$ VM is placed at
${ii}^{th}$ server after migration then $c_{mig.{ii}}=1$, otherwise,
$c_{mig.{ii}}=0$. If the ${ii}^{th}$ server receives one or more VMs after
migration, then $n_{ii}=1$ else it is 0. Similarly, if $d_{ii}=0$ then
${ii}^{th}$ server is already active before migration, otherwise,
$d_{ii}=E_{tr}$ where $E_{tr}$ is the energy consumed in switching a server
from sleep to active state.
## 5 Significance Rank Computation
A group of inter-communicating VMs in a VHAN ($H_{m}$) executing a cloud
application: {$t^{m}_{1}$, $t^{m}_{2}$, …, $t^{m}_{n}$} $\in\lambda_{m}$:
$\forall m\in[1,M]$, is considered as a weighted directed graph (WDG), where a
VM represents a node ($\mathchorus{N}_{a}$) and invoking relation between VMs
is a directed edge ($\mathchorus{E}_{ab}$) from $\mathchorus{N}_{a}$ to
$\mathchorus{N}_{b}$ in WDG. A weight value ($\xi(\mathchorus{E}_{ab})$) is
assigned to each edge in WDG using Eq. (10), where $\mathchorus{fq}_{ab}$ is
the invocation frequency of node $\mathchorus{N}_{a}$ by node
$\mathchorus{N}_{b}$, $n$ is the number of nodes in WDG, $\mathchorus{C}_{a}$
determines whether $\mathchorus{N}_{a}$ is critical ($\mathchorus{C}$) or not,
which is $1$ for $\mathchorus{N}_{a}$ is a critical node, else it is $0$, and
if $\mathchorus{N}_{a}$ invokes $\mathchorus{N}_{b}$ then
$\mathchorus{fq}_{ab}=1$; otherwise, $\mathchorus{fq}_{ab}=0$.
$\xi(\mathchorus{E}_{ab})=\frac{\mathchorus{fq}_{ab}\times\mathchorus{C}_{a}}{\sum_{b=1}^{n}{\mathchorus{fq}_{ab}}}$
(10)
Hence, the edge $\mathchorus{E}_{ab}$ has a higher weight value if
$\mathchorus{N}_{a}$ is invoked more frequently by $\mathchorus{N}_{b}$ as
compared to other nodes invoked by $\mathchorus{N}_{b}$. WDG containing $n$
nodes is represented as $n\times n$ matrix $\xi$ established using Eq. (10)
subject to Eq. (11), where $\xi(\mathchorus{E}_{ab})=\frac{1}{n}$ if
$\mathchorus{N}_{a}$ has all the incoming edges only.
$\forall a,\sum_{b=1}^{n}{\xi(\mathchorus{E}_{ab})}=1$ (11)
Initially, a random value in the range [0, 1] is assigned to all the nodes of
WDG. All the nodes are differentiated on the basis of their characteristics
into a critical node ($\mathchorus{C}$) and a non-critical node
($\mathchorus{NC}$). Accordingly, the significance value of node
$\mathchorus{N}_{a}$ represented as $\mathchorus{S}(\mathchorus{N}_{a})$ is
estimated using either Eq. (12) or Eq. (13) for critical or non-critical
nodes, respectively;
$\displaystyle\mathchorus{S}(\mathchorus{N}_{a})=(1-\mathchorus{d}){\frac{\Psi}{|\mathchorus{C}|}+\mathchorus{d}{\sum_{g\in
Z(\mathchorus{N}_{a})}{\mathchorus{S}(\mathchorus{N}_{g})\xi(\mathchorus{E}_{ga})}}}$
(12)
$\displaystyle\mathchorus{S}(\mathchorus{N}_{a})=(1-\mathchorus{d}){\frac{1-\Psi}{|\mathchorus{NC}|}+\mathchorus{d}{\sum_{g\in
Z(\mathchorus{N}_{a})}{\mathchorus{S}(\mathchorus{N}_{g})\xi(\mathchorus{E}_{ga})}}}$
(13)
where $|\mathchorus{C}|$ and $|\mathchorus{NC}|$ are number of critical and
non-critical nodes, respectively such that
$|\mathchorus{C}|+|\mathchorus{NC}|=n$, $Z(\mathchorus{N}_{a})$ is a set of
nodes invoking $\mathchorus{N}_{a}$, $d$ is an adjusting parameter in the
range [0, 1]. The parameter $\Psi$ ($\frac{|\mathchorus{C}|}{n}\leq\Psi\leq
1$) determines the effectiveness of critical and non-critical components in
the estimation of VSR as when $\frac{|\mathchorus{C}|}{n}<\Psi\leq 1$, the
value of critical node $(1-\mathchorus{d}){\frac{\Psi}{|\mathchorus{C}|}}$ is
greater than non-critical node
$(1-\mathchorus{d}){\frac{1-\Psi}{|\mathchorus{NC}|}}$. If $\Psi$=1, then
value of the critical nodes is 1, and larger significant values are assigned
to critical nodes, while if $\Psi=\frac{|\mathchorus{C}|}{n}$, both the
critical and non-critical nodes show equal effects. Finally, the VMs are
ranked in the descending order of their significance values. VSRCU estimates
the significance value based ranks of VMs using an enhanced version of the
Weighted Page Rank algorithm [30] as mentioned above.
## 6 HA Strategy Selection
The user specifies several high availability constraints regarding response
time and total cost of application execution in the SLA to be requited during
actual operation. HAMU incorporates significance ranks of VMs
($\mathchorus{S}$), response time ($\mathchorus{T}$), failure probability of
$j^{th}$ HA strategy ($\mathchorus{F}_{j}$), and execution cost
($\mathchorus{EC}$) which yields a mapping $\Omega$ defined in Eq. (14) to
select most admissible HA strategy ($Y_{j}$) for VM ($V^{m}_{i}$) of $m^{th}$
user subject to constraints stated in Eqs. (15), (16), (17), where $D$ is
total number of HA strategies.
$\displaystyle\Omega:\sum_{j}^{D}\mathchorus{S}(V^{m}_{i})\times
Y_{j}\times\mathchorus{F}_{j}\quad\forall_{i}\in[1,n]\quad s.t.$ (14)
$\displaystyle\sum_{j}^{D}Y_{j}\times\mathchorus{EC}_{j}\leq
V^{m}_{i}\times\mathchorus{EC}^{m}_{i}\times U_{m},$ (15)
$\displaystyle\sum_{j}^{D}Y_{j}\times\mathchorus{T}_{j}\leq
V^{m}_{i}\times\mathchorus{T}^{m}_{i}\times U_{m},$ (16)
$\displaystyle\sum_{j}^{D}Y_{j}=1,\quad Y_{j}\in\\{0,1\\}$ (17)
In this work, Automatic recovery block (ARP), Multi-version programming (MVP),
and Parallel execution (PE) based HA strategies are adopted:
* •
ARP: It engages redundant standby images of a VM which are invoked in sequence
if currently an active image of VM fails. ARP failure probability
($\mathchorus{F}^{ARP}$) is computed using Eq. (18), which get fail only if
all the replicated images get fail, where $num$ is the number of replicated
images and $\mathchorus{F}^{ARP}_{i}$ is failure of $i^{th}$ VM.
$\mathchorus{F}^{ARP}=\prod_{i=1}^{num}\mathchorus{F}^{ARP}_{i}$ (18)
* •
MVP: It appoints multiple active images or versions of a VM instance
concurrently for the execution of a common task and the final output is
obtained by majority voting. Eq. (19) computes its failure probability, where
$num$ is a number of versions ($num$ is an odd number), and
$\mathchorus{f}(i)$ is the failure probability of alternative VM images. MVP
fails only if more than half of redundant VM images get fail.
$\mathchorus{F}^{MVP}=\sum_{i=\frac{num+1}{2}}^{num}\mathchorus{f}(i)$ (19)
* •
PE: It allows parallel execution of all $num$ versions of $i^{th}$ VM instance
and the first generated response is provided as final output. It fails only if
all the parallel executions get fail and its failure probability is computed
in Eq. (20)
$\mathchorus{F}^{PE}=\prod_{i=1}^{num}\mathchorus{F}^{PE}_{i}$ (20)
## 7 Operational Design and Complexity
The proposed model initializes a list of PMs, VMs, and users, where the users
submit requests for execution of applications to the resource manager module
which assigns these requests on different VMs placed on PMs as stated in
Algorithm 1. For each time-interval {$t1$, $t2$}, VMs resource utilization is
predicted using LSTM, which further helps to determine any overload or
resource-contention problem and mitigate this situation by VM migration,
proactively. The user becomes an ephemeral owner of a group of VMs engaged in
task execution belonging to the respective user’s application, thus creating a
VHAN. On the same line, for each user exclusive VHAN is generated, which are
later transformed into a WDG matrix. An optimal HA strategy is selected by
including only those strategies which satisfies user specified constraints,
among which a HA strategy with the least failure probability is selected as
the optimal strategy for the target VM.
1 Initialize: $List_{{\mathds{S}}}$, $List_{\mathds{V}}$, $List_{\mathds{U}}$;
2
3for _each time-interval $\\{t_{1},t_{2}\\}$_ do
4
5 Predict $\mathchorus{RU}$ on each VM on a PM and aggregate it to analyse
resource-contention ;
6 Migrate maximum size VM from overloaded PM by applying the steps for PMMU in
Section 4 ;
7
8 User submitted jobs are distributed into tasks to be executed on VMs
(arranged in VHAN) placed on different PMs while satisfying resource
constraints (Eq. (2));
9 Estimate VSR for each VM in VHAN by applying concepts mentioned in Section
5;
10 WDG in the form of $n\times n$ matrix is built from VHAN ;
11 for _( $j=1$; $j\leq D$; $j++$)_ do
12 if _Eq. ( 15) && Eq. (16) && Eq. (17) _ then
13 $x_{j}=\mathchorus{F}_{j}$;
14 end if
15 continue;
16
17 end for
18 Select HA strategy which has minimal failure probability among all $x_{j}$;
19
20 end for
21
Algorithm 1 SRE-HM Operational Summary
Step 1 initializes lists of servers, users, and VMs associated with different
users and consumes time-complexity $O(1)$. Let steps 2-15 repeat for $t$ time-
slots, where step 3 appoints LSTM-RNN for prediction of resource utilization
with complexity of $O(\mathchorus{h})$, where $\mathchorus{h}$ is length of
input sequence. Step 4 performs VM migration has $O(\mathchorus{m})$
complexity, where $\mathchorus{m}$ is a number of migrations. Steps 5-7 create
VHAN followed by WDG in the form of $n\times n$ matrix have $O(n^{2})$
computational complexity. Steps 8-13 iterate $D$ times with $O(D)$ complexity
while Step 14 shows $O(1)$ complexity. Hence, overall computational complexity
turn into $O(\mathchorus{h}\mathchorus{m}n^{2}Dt)$.
## 8 Performance Evaluation
### 8.1 SRE-HM Implementation
A SRE-HM prototype is configured with the collaboration of major modules
discussed below:
* •
VMs Resource Estimation: The future resource usage of different VMs hosted on
each server is predicted and their number and size are scaled accordingly.
Also, any overload or resource contention failure is detected based on
prediction and mitigated by VM migration proactively.
* •
VMs Allocation: The predicted VMs are assigned to the servers conforming to
the resource distribution constraints specified in Eq. (2).
* •
User’s Task submission: Each user submits different tasks along with their
deadline and cost of execution, to be assigned to VMs hosted on different
servers.
* •
Generation of WDG: A WDG is built for each user by establishing links and
invocations randomly among different VMs selected for task execution of the
user.
* •
Deciding Critical and Non-Critical VMs: Based on the number of invocations of
a VM, greater than a threshold value of invocation (which is set to 3 in the
experiments), are taken as ‘critical VMs’ and remaining VMs are ‘non-critical
VMs’.
* •
VM Ranks Computation: The VM ranks are estimated for critical and non-critical
VMs exclusively by following the steps mentioned in Section (5).
* •
HA Strategy Selection and execution: As per the computed ranks of VMs, failure
probability of different HA strategies, cost and deadline of task execution,
an admissible strategy with the least perhaps failure probability is selected
for each VM. This module is extensible, where any ranking algorithm and HA
strategies can be incorporated adaptively.
* •
Availability and Execution Cost Evaluation: Based on the above modules, the
availability is computed using Eq. (23) while the cost is estimated in terms
of resource utilization and power consumption computed as given below.
### 8.2 Parameters evaluation
The HA performance metrics including MTBF, MTTR, and average availability of
SRE-HM model are evaluated by considering lifecycle of a hypothetical service
as demonstrated in Fig. 3, wherein the variables $DT$, $UT$, and $TT$ are
downtime, uptime, and total time of a service, respectively.
Figure 3: Lifecycle of a service indicating uptime, outage and total time
A service may be either in uptime as illustrated by the variables $UT_{1}$,
$UT_{2}$, $UT_{3}$, and $UT_{4}$ or in downtime defined by variables $DT_{1}$,
$DT_{2}$, and $DT_{3}$. The term $Num_{\mathchorus{F}}$ represents a number of
failures of system (i.e., 3 in given Fig. 3); the MTBF and MTTR are evaluated
using Eqs. (21) and (22), respectively. MTTF and MTTR are stated as the
averages of uptime and downtime, respectively in these equations. Accordingly,
the average availability can be computed using Eq. (23), where
$Num_{\mathchorus{F}}$ is total number of failures, $\sum_{i=1}^{M}{UT_{i}}$
and $\sum_{i=1}^{M}{DT_{i}}$ represent total uptime and downtime of a service,
respectively experienced by $M$ users over time-interval {$t_{1}$, $t_{2}$}.
$MTBF=\int\limits_{\begin{subarray}{c}t_{1}\\\
\mathcal{}\end{subarray}}^{t_{2}}(\frac{\sum_{i=1}^{M}{UT_{i}}}{Num_{\mathchorus{F}}})dt$
(21) $MTTR=\int\limits_{\begin{subarray}{c}t_{1}\\\
\mathcal{}\end{subarray}}^{t_{2}}(\frac{\sum_{i=1}^{M}{DT_{i}}}{Num_{\mathchorus{F}}})dt$
(22) $A_{avg}=\frac{MTBF}{MTBF+MTTR}$ (23)
The resource utilization ($\mathchorus{RU}$) of data centre is estimated using
Eqs. (24) and (25), where $\mathds{N}$ is the number of resources,
$\mathchorus{RU}{C}$ and $\mathchorus{RU}{Mem}$ are CPU and memory of server.
If $k^{th}$ server $PM_{k}$ is active, hosting VMs ($\beta_{k}$ = 1),
otherwise, $\beta_{k}$ = 0.
$\mathchorus{RU}^{DC}=\frac{\sum_{k=1}^{P}{\mathchorus{RU}_{k}^{\mathchorus{C}}}+\sum_{k=1}^{P}{\mathchorus{RU}_{k}{Mem}}}{|\mathds{N}|\times\sum_{k=1}^{P}{\beta_{k}}}$
(24) $\mathchorus{RU}_{k}^{\mathds{R}}=\frac{\sum_{i=1}^{Q}{\omega_{ik}}\times
V_{i}^{\mathds{R}}}{S_{k}^{\mathds{R}}}\quad\forall_{k}\in\\{1,P\\},\mathds{R}\in\\{\mathchorus{C},\mathchorus{Mem}\\}$
(25)
Eq. (26) computes power consumption ($\mathchorus{PW}^{DC}$), where
${PW_{i}}^{max}$, ${PW_{i}}^{min}$, and ${PW_{i}}^{idle}$ are maximum,
minimum, and idle state power consumption, respectively of $i^{th}$ server.
$\mathchorus{PW}^{DC}=\sum_{i=1}^{P}{[{PW_{i}}^{max}-{PW_{i}}^{min}]\times{RU}+{PW_{i}}^{idle}}$
(26)
### 8.3 Experimental Set-up and Dataset
The simulation experiments are executed on a server machine assembled with two
Intel® Xeon® Silver 4114 CPU with 40 core processor and 2.20 GHz clock speed.
The server machine is deployed with 64-bit Ubuntu 16.04 LTS, having main
memory of 128 GB. The data centre environment included three different types
of servers and four types of VMs configuration shown in Tables II and III in
Python. The resource features like power consumption ($PW_{max},PW_{min}$),
MIPS, RAM and memory are taken from real server IBM [31] configuration where
$S_{1}$ is ‘ProLiantM110G5XEON3075’, $S_{2}$ is ‘IBMX3250Xeonx3480’ and
$S_{3}$ is ‘IBM3550Xeonx5675’. The VMs configuration is inspired from the VM
instances of Amazon website [32].
TABLE II: Server Configuration
Server | PE | MIPS | RAM(GB) | $PW_{max}$ | $PW_{min}$/$PW_{idle}$
---|---|---|---|---|---
$S_{1}$ | 2 | 2660 | 4 | 135 | 93.7
$S_{2}$ | 4 | 3067 | 8 | 113 | 42.3
$S_{3}$ | 12 | 3067 | 16 | 222 | 58.4
TABLE III: VM Configuration VM type | PE | MIPS | RAM(GB)
---|---|---|---
$v_{small}$ | 1 | 500 | 0.5
$v_{medium}$ | 2 | 1000 | 1
$v_{large}$ | 3 | 1500 | 2
$v_{Xlarge}$ | 4 | 2000 | 3
Dataset: Google Cluster Data (GCD) dataset is utilized for performance
estimation of SRE-HM and comparative approaches which contains resources viz.,
CPU, memory, disk I/O request and resource usage information of 672,300 jobs
executed on 12,500 servers for the period of 29 days [33]. The CPU and memory
utilization percentage of VMs are obtained from the given CPU and memory usage
percentage for each task in every five minutes over period of twenty-four
hours.
### 8.4 Results
Table IV reports the performance metrics: MTTR, MTBF, average availability
($\mathds{AV}$), accuracy of failure prediction ($Acu^{Pr}$), average number
of overloads ($\mathds{OV}$), power consumption ($\mathchorus{PW}$), Resource
utilization ($\mathchorus{RU}$), average number of VM migrations
($\mathchorus{Mig}$) achieved for varying percentage of failure-prone VMs
($\mathds{V}_{fp}$) over period of 500 minutes.
TABLE IV: Performance metrics for GCD workloads $\mathds{V}_{fp}$ | $T(min.)$ | $MTTR$ | $MTBF$ | $\mathds{AV}$ | $Acu^{Pr}$ | $\mathds{OV}$ | $\mathchorus{PW}$ (W) | $\mathchorus{RU}$ | $\mathchorus{Mig}\texttt{\\#}$
---|---|---|---|---|---|---|---|---|---
5 | 100 | 1.47 | 2757.14 | 99.91 | 99.3 | 5 | 8660.9 | 76.2 | 7
250 | 1.05 | 3900 | 99.96 | 98.8 | 7 | 8660.9 | 76.6 | 5
300 | 1.05 | 3900 | 99.96 | 98.6 | 7 | 8660.9 | 76.6 | 5
| 500 | 0.84 | 4900 | 99.98 | 99.6 | 8 | 8660.9 | 76.5 | 4
10 | 100 | 2.73 | 1438.46 | 99.84 | 97.9 | 11 | 8498.6 | 75.8 | 13
250 | 2.1 | 1900 | 99.85 | 98.4 | 9 | 8498.6 | 75.3 | 10
300 | 1.89 | 2122.22 | 99.81 | 95.5 | 11 | 8498.6 | 75.8 | 9
500 | 1.68 | 2400 | 99.93 | 98.7 | 8 | 8498.6 | 75.6 | 8
30 | 100 | 3.99 | 952.63 | 99.58 | 98.8 | 14 | 8601.7 | 77.8 | 19
200 | 3.57 | 1076.47 | 99.66 | 97.6 | 16 | 8601.7 | 76.9 | 17
300 | 3.15 | 1233.33 | 99.74 | 98.8 | 12 | 8601.7 | 76.7 | 15
500 | 2.94 | 1328.57 | 99.78 | 93.7 | 19 | 8601.7 | 76.8 | 14
80 | 100 | 4.62 | 809.09 | 99.43 | 97.9 | 15 | 8578.4 | 74.8 | 22
250 | 4.41 | 852.38 | 99.48 | 98.2 | 18 | 8578.4 | 75.1 | 21
300 | 3.99 | 952.63 | 99.58 | 98.2 | 18 | 8578.4 | 75.1 | 19
500 | 3.36 | 1150.00 | 99.76 | 98.8 | 18 | 8578.4 | 74.9 | 16
The values achieved for MTBF and MTTR depends on the number of failures
($Num_{\mathchorus{F}}$) as depicted in Eqs. (21) and (22), respectively. The
values of uptime ($UT$) are obtained by calculating the product of number of
successfully deployed VMs. The value of MTTR associated with a VM is 0.21
minutes which is reported in [34], [35]. Accordingly, the values of MTTR are
enumerated for different number of VM migrations that changes with the number
of unpredicted VM failures. The resultant values of availability are evaluated
using Eq. (23) depending on the MTBF and MTTR values recorded over time-
interval {$t_{1}$, $t_{2}$} i.e., 100 minutes for the observed experiments.
The availability for the Google Cluster workload is above 99% for all the
observed cases. It can be observed that with increasing number of
$\mathds{V}_{fp}$, the performance of SRE-HM (independent of time) is durable
where $Acu^{Pr}$, and $\mathchorus{RU}$ are greater than 97%, and 74.5%,
respectively while $\mathchorus{PW}$, and $\mathchorus{Mig}$ lesser than 8661
W and 22 VM migrations, respectively.
### 8.5 Comparison
The different versions of SRE-HM including SRE-HM ($S^{+}$), SRE-HM with only
critical VM ranking ($S^{\ast}$), Without SRE-HM ($S^{-}$), and SRE-HM without
VM ranking ($S^{\ast\ast}$) collaborated with three HA strategies ($arp$,
$mvp$, $pe$) are implemented and compared for intermediate result values and
performance metrics.
#### A PM Failure Estimation
Fig. 4(a) compares boxplots for the outcomes of PM failure estimation of SRE-
HM with state-of-the-arts: DNN [18], ES [20], SVM [19], GB [23]. It shows that
the upper, median, as well as lower quartiles have highest value of prediction
accuracy for SRE-HM prediction approach over all the compared methods. On the
same lines, the percentage of resource contention based PM failures (RCF) is
least for SRE-HM up to 72.2% against ES as depicted in Fig. 4(b).
(a) Failure Prediction Accuracy (b) PM Failures
Figure 4: PM Failure Estimation
#### B VM Ranking and HA Strategy Selection
Fig. 5 highlights the comparison of the inter-mediate outcomes achieved during
execution of SRE-HM and its various versions: SRE-HM ($S^{+}$), $S^{\ast}$,
$S^{-}$, $S^{\ast\ast}_{arp}$, $S^{\ast\ast}_{mvp}$, and $S^{\ast\ast}_{pe}$.
Fig. 5(a) shows values of significance ranks obtained for a randomly selected
VMs retrieved over a period of 500 minutes via experimental execution. It is
observed that significance ranks of a VM varies depending upon the number of
invocations between 0 and 1 over period of 500 minutes. The comparative
intermediate resultant values of HA strategy selection (%) for varying number
of failure-prone VMs (%) are shown in Fig. 5(b), where $S^{\ast\ast}_{arp}$,
$S^{\ast\ast}_{mvp}$, $S^{\ast\ast}_{pe}$ show straight lines i.e., 100 %
because these set-ups lack VM rank estimation and HA strategy selection and
executed with respect to single HA strategy only.
(a) VM Ranking (b) HA Strategies
Figure 5: HA Strategy Selection
#### C HA Analysis
The availability varies with the values of MTBF and MTTR, obtained during
online processing over time-interval {$t_{1}$, $t_{2}$}. The variations
observed (during experimental simulation) in the values of MTTR and MTBF are
shown in Figs. (6) and (7) for Google Cluster workload execution. It is to be
noticed that MTTR decreases when MTBF increases, which specifies a inverse
relation between them. The MTBF increases and MTTR decreases in the order:
$S^{+}$>$S^{\ast\ast}_{pe}$
>$S^{\ast\ast}_{mvp}$>$S^{\ast\ast}_{arp}$>$S^{\ast}$ > $S^{-}$. Accordingly,
Fig. 8 compares $S^{+}$, $S^{\ast}$, $S^{\ast\ast}_{arp}$,
$S^{\ast\ast}_{mvp}$, $S^{\ast\ast}_{pe}$, and ($S^{-}$), where the percentage
of availability is highest for $S^{+}$ and outperforms $S^{-}$ by 19.56% due
to proactive PM failure estimation and adoption of most appropriate HA
strategy on the basis of significance ranking of VMs.
Figure 6: MTTR Figure 7: MTBF Figure 8: Availability
#### D Power Consumption
The power consumption is shown in Fig. 9 which is lesser for $S^{+}$ up to
19.1% as compared with that of $S^{-}$. Further, it is observed that power
consumption of $S^{+}$ is nearly 0.6% higher as compared with
$S^{\ast\ast}_{arp}$ due to adoption of ARP HA strategy for each VM which has
lesser power consumption as compared with MVP, and PE strategies. However, at
the cost of 0.6% increased power consumption, the availability is improved up
to 5.31% by utilizing SRE-HM ($S^{+}$).
Figure 9: Power Consumption
#### E Cost Optimization
Fig. 10 compares the cost optimization in terms of resource utilization and
number of active servers, attained by different versions of SRE-HM viz., With
SRE-HM ($S^{+}$), $S^{\ast\ast}_{arp}$, $S^{\ast\ast}_{mvp}$,
$S^{\ast\ast}_{pe}$, with HA Without SRE-HM ($S^{-}$). Fig. 10 (a) reveals
that $S^{+}$ has highest RU up to 78.96% which is superior to
$S^{\ast\ast}_{pe}$ by 10%. The number of active PMs in case of $S^{+}$ are
reduced up to 26.67% as depicted in Fig. 10 (b) which is due to consolidation
of VMs by including ranking before HA strategy selection for each VM rather
than applying same strategy to all.
(a) Resource Utilization (b) Active Servers
Figure 10: Cost Optimization parameters
#### F SRE-HM v/s State-of-the-arts
SRE-HM model is compared with significant state-of-the-art works discussed in
Section 2 subject to different key characteristics and performance indicators
as shown in Table V. The description of various characteristics and values of
performance metrics are obtained from the published versions of these works
while some other values are computed based on the values of related metrics.
The comparison of different performance parameters are reported in subsequent
subsections which are based on the values obtained during experimental
evaluations of SRE-HM and the compared approaches.
TABLE V: Pandect comparison: SRE-HM v/s state-of-the-arts
Approaches | Key characteristics | Performance indicators
---|---|---
Failure Estimation | HA Approach | HA Strategy, Selection | Availability | Resource Utilization | Energy Consumption | Active PMs | VM migration
PEFS [18] | Deep NN | Replication | $\times$ | $\times$ | 70%-72% | 11500.47 | $\times$ | $\times$
FAEE [20] | Exp. Smooth. | checkpointing | $\checkmark$ | 82% | $\times$ | 9825.68 | $\times$ | $\checkmark$
HDCC [19] | SVM and LR | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
CDEF [23] | MART-GB | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
FESTAL [7] | $\times$ | Backup | $\checkmark$ | 95.08% | Varying | $\times$ | Varying | $\times$
FASTER [27] | $\times$ | Backup | $\checkmark$ | 98.5% | Varying | $\times$ | Varying | $\times$
OP-MLB [3] | DE-NN | $\times$ | $\times$ | $\times$ | 64.6% | 8801.67kWH | 45% | $\checkmark$
WP-SM [36] | Wiener Filter | $\times$ | $\times$ | $\times$ | $\times$ | 9001.67 kWH | 51.2% | $\checkmark$
OHD-MUP [37] | Multiple LR | $\times$ | $\times$ | $\times$ | $\times$ | 7701.67 kWH | 31.2% | $\checkmark$
CHARM [5] | $\times$ | Multi-cloud | $\checkmark$ | Varying | $\times$ | $\times$ | $\times$ | $\times$
CHASE [26] | $\times$ | scheduling | $\checkmark$ | 99.1% | $\times$ | $\times$ | $\times$ | $\times$
SRE-HM | LSTM-NN | VM Ranking | $\checkmark$ | 99.3% | 76.9% | 8660.7 kWH | 53% | $\checkmark$
## 9 Conclusion and Future Directions
A novel SRE-HM model is proposed which computes significance ranks of VMs.
Based on this ranking, user specified constraints, and failure probability, an
optimal HA strategy is chosen to provide admissible high availability
respective to each VM dedicated for execution of cloud applications. A
resource predictor detects resource contention prior to its occurrence and
alleviates it by migrating VMs from corresponding PM. Implementation and
performance evaluation of proposed model acknowledge that it substantially
improves the availability for users with optimized operational cost for data
centre. In future, the proposed model can be extended with more useful
characteristics of cloud applications to determine their critical nature
before siginificance rank computation and VM collaboration relation during
resource usage prediction to achieve more robust high availability at
optimized cost.
## Acknowledgment
This work is financially supported by National Institute of Technology,
Kurukshetra, India.
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| Deepika Saxena received her M.Tech degree in Computer Science and
Engineering from Kurukshetra University Kurukshetra, Haryana, India in 2014.
Currently, she is pursuing her Ph.D from Department of Computer Applications,
National Institute of Technology (NIT), Kurukshetra, India. Her major research
interests are Neural Networks, Evolutionary Algorithms, Resource Management
and Security in Cloud Computing.
---|---
| Ashutosh Kumar Singh is working as a Professor and Head in the Department
of Computer Applications, National Institute of Technology Kurukshetra, India.
He has research and teaching experience in various Universities of the India,
UK, and Malaysia. He received his PhD in Electronics Engineering from Indian
Institute of Technology, BHU, India and Post Doc from Department of Computer
Science, University of Bristol, UK. He is also Charted Engineer from UK. His
research area includes Verification, Synthesis, Design and Testing of Digital
Circuits, Data Science, Cloud Computing, Machine Learning, Security, Big Data.
He has published more than 330 research papers in different journals,
conferences and news magazines.
---|---
|
# Inferring Attack Relations for Gradual Semantics
Nir Oren
University of Aberdeen
Scotland
<EMAIL_ADDRESS>
&Bruno Yun
University of Aberdeen
Scotland
<EMAIL_ADDRESS>
###### Abstract
A gradual semantics takes a weighted argumentation framework as input and
outputs a final acceptability degree for each argument, with different
semantics performing the computation in different manners. In this work, we
consider the problem of attack inference. That is, given a gradual semantics,
a set of arguments with associated initial weights, and the final desirable
acceptability degrees associated with each argument, we seek to determine
whether there is a set of attacks on those arguments such that we can obtain
these acceptability degrees. The main contribution of our work is to
demonstrate that the associated decision problem, i.e., whether a set of
attacks can exist which allows the final acceptability degrees to occur for
given initial weights, is NP-complete for the weighted h-categoriser and
cardinality-based semantics, and is polynomial for the weighted max-based
semantics, even for the complete version of the problem (where all initial
weights and final acceptability degrees are known). We then briefly discuss
how this decision problem can be modified to find the attacks themselves and
conclude by examining the partial problem where not all initial weights or
final acceptability degrees may be known.
_K_ eywords Gradual Semantics $\cdot$ Argumentation $\cdot$ Complexity
## 1 Introduction
Abstract argumentation semantics associate a justification status with a set
of arguments based on interactions between arguments. Such interactions can
include inter-argument attacks [1], or preference-based defeats [2, 3, 4, 5,
6], or may consider both of these together with some supportive relationship
[7, 8, 9, 10]. Given such a framework, an argument may be considered justified
if it appears in one (resp. many or all) _extensions_ , where such extensions
– sets of non-conflicting arguments accepted together – are computed according
to the argumentation semantics, or based on the label assigned to the argument
(we refer the reader to [11] for an introduction to the most common
argumentation semantics).
Arguments often have associated weights, and different semantics have been
proposed which consider the weight associated with an argument [12, 8, 10,
13]. While some semantics developed in this context return whether an argument
is, or is not justified [12], many other _ranking-based_ semantics either
return a preference ordering over arguments by making use of the argument’s
initial weight or associate a numeric _final acceptability degree_ with each
argument in the framework. Such ranking-based semantics have been used to —
for example — identify irrationality in reasoning [14] by examining whether
the initial weights associated with an argument affect the argument’s final
acceptability degree in an appropriate manner (i.e., consistent with the
ranking-based semantics being used); refining the results obtained by
extension semantics [15, 16]; and applied to multi-agent settings [17].
The top portion of Fig. 1 shows the reasoning process used when reasoning
using gradual semantics. Here, a weighted argumentation framework consisting
of arguments, attacks and initial weights associated with arguments is
provided as input. A gradual semantics is then used to compute a _final
acceptability degree_ for each argument. In many semantics, these final
acceptability degrees are then used to compute a preference ordering over the
arguments.
More recent work has considered different combinations of inputs and outputs
to the problem. For example, [13] seeks to identify a set of initial weights
for arguments based on the final argument preference ordering and
argumentation semantics, while [18] determines the preferences between
arguments given argument justification status, semantics and argumentation
framework. In this paper, and as illustrated in the bottom portion of Fig. 1,
we consider the problems of determining whether a set of attacks between
arguments can be identified given specific argumentation semantics, the final
acceptability degrees, and the initial weights. As discussed further in
Section 5, we leave the problem of using preferences over arguments as input
for our problem as future work. While it is true that the problem we consider
is somewhat abstract and has limited real-world applications, it serves as a
departure point for potentially important applications of argumentation to
opponent modelling [19, 20] and preference elicitation [18].
Figure 1: The process (top) by which a gradual semantics is applied to compute
an acceptability preference between arguments and (bottom) the inverse problem
considered in this paper. The “final acceptability degrees" are now used as
input. That is, given arguments, initial weights, and desirable final
acceptability degrees, can we find a suitable set of attacks?
While other ranking-based semantics are described in the literature, we
consider three popular gradual semantics (the weighted h-categoriser, the
weighted card-based, and the weighted max-based semantics [21]). These three
semantics were chosen due to their similarity to one another, and due to the
way in which a similar approach can be used to solve the problem under
consideration when these semantics are used. We show that when given all final
acceptability degrees and initial weights, the problems for weighted
h-categoriser and weighted card-based semantics are both NP-complete, while
the problem can be solved in polynomial time for the weighted max-based
semantics.
The remainder of this paper is structured as follows. In Section 2, we provide
the necessary background to understand the remainder of the paper, introducing
the h-categoriser semantics. Section 3 formalises the decision problem of
whether attacks can be found for gradual semantics and then demonstrates that
it is NP-complete for some semantics. In Section 4 we discuss solvers for our
problem, and consider related and future work (including a partial version of
the problem) in Section 5.
## 2 Background
As discussed above, we situate our work in the context of _weighted_ abstract
argumentation frameworks (WAFs), which can be encoded as graphs with weighted
nodes. Each argument has an initial weight (also called “basic score” in [22])
from $[0,1]$. The smaller the initial weight of an argument, the weaker the
argument. The initial weight of an argument may represent different issues
like the likelihood degree of its premises [23], the degree of trust in its
source [24], or an aggregation of votes provided by users [25] among others.
In this paper, the origin of the weights and arguments is left unspecified.
Similarly, arguments and attacks are considered abstract notions.
###### Definition 1.
A _weighted argumentation framework_ is a triple
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ where $\mathcal{A}$ is a
_finite_ set of arguments, $\mathcal{D}\subseteq\mathcal{A}\times\mathcal{A}$
is a binary attack relation, and $w:\mathcal{A}\to[0,1]$ is a total weighting
function which associates an _initial weight_ between $0$ and $1$ to each
argument.
Given an argument $a\in\mathcal{A}$, we refer to $w(a)$ as $a$’s _initial
weight_. For any argument $a\in\mathcal{A}$, the set
$\\{b\in\mathcal{A}|(b,a)\in\mathcal{D}\\}$, denoted by $\mathtt{Att}(a)$,
contains all attackers of $a$. Similarly, we define
$\mathtt{Att}^{*}(a)=\\{b\in\mathcal{A}|(b,a)\in\mathcal{D}$ and $w(b)>0\\}$,
i.e., the set of attackers of $a$ with strictly positive weights. When the
current argumentation framework $\mathcal{F}$ is not clear from the context,
we will use the notation $\mathtt{Att}_{\mathcal{F}}$ and
$\mathtt{Att}^{*}_{\mathcal{F}}$ respectively.
A gradual semantics $\sigma$ takes as input a weighted argumentation framework
and outputs a function that associates a _final acceptability degree_ for each
argument111Often, the final step in using a gradual semantics involves using
this final acceptability degree to compute a preference ordering over
arguments.. Several such semantics have been described in the literature. In
this paper, we focus on three widely used gradual semantics introduced by
Amgoud et al., i.e. the weighted h-categoriser, the weighted max-based, and
the weighted card-based semantics, denoted $\sigma_{HC}$, $\sigma_{MB}$ and
$\sigma_{CB}$ respectively [21].
The weighted h-categoriser considers the initial weight of the argument as
well as the sum of the acceptability degrees of all its attackers to determine
the acceptability degree of the argument. In this semantics, all (non-zero)
attackers will be considered when determining the acceptability degree of an
argument and the number of attackers is not used.
###### Definition 2.
The weighted h-categoriser semantics, denoted $\sigma_{HC}$, is the function
that takes as input any weighted argumentation framework
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ and returns the function
$\sigma^{\mathcal{F}}_{HC}:A\to[0,1]$ such that for all
$a\in\mathcal{A},\sigma^{\mathcal{F}}_{HC}(a)=HC_{\infty}(a)$, where:
$HC_{i}(a)=\begin{cases}w(a)&\text{if }i=0\\\
\frac{w(a)}{1+\sum_{b\in\mathtt{Att}(a)}HC_{i-1}(b)}&\text{otherwise}\end{cases}$
The next semantics is the weighted max-based which considers the initial
weight of the argument as well as the highest acceptability degrees of its
attackers to determine the acceptability degree of the considered argument. In
this semantics, only the stronger attacker is considered and the number of
attackers is not used.
###### Definition 3.
The weighted max-based semantics, denoted $\sigma_{MB}$, is the function that
takes as input any weighted argumentation framework
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ and returns the function
$\sigma^{\mathcal{F}}_{MB}:A\to[0,1]$ such that for all
$a\in\mathcal{A},\sigma^{\mathcal{F}}_{MB}(a)=MB_{\infty}(a)$, where:
$MB_{i}(a)=\begin{cases}w(a)&\text{if }i=0\\\
\frac{w(a)}{1+max_{b\in\mathtt{Att}(a)}MB_{i-1}(b)}&\text{otherwise}\end{cases}$
The last semantics studied in this paper is the weighted card-based which
considers the initial weight of the argument, the number of its attackers as
well as the sum of the acceptability degrees of its attackers to determine the
acceptability degree of the considered argument. In this semantics, the number
of attackers is the most important factor to determine the acceptability
degree of an argument.
###### Definition 4.
The weighted card-based semantics, denoted $\sigma_{CB}$, is the function that
takes as input any weighted argumentation framework
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ and returns the function
$\sigma^{\mathcal{F}}_{CB}:A\to[0,1]$ such that for all
$a\in\mathcal{A},\sigma^{\mathcal{F}}_{CB}(a)=CB_{\infty}(a)$, where:
$CB_{i}(a)=\begin{cases}w(a)&\text{if }i=0\\\
\frac{w(a)}{1+|\mathtt{Att}^{*}(a)|+\frac{\sum_{b\in\mathtt{Att}^{*}(a)}CB_{i-1}(b)}{|\mathtt{Att}^{*}(a)|}}&\text{otherwise}\end{cases}$
Note that by convention, if $|\mathtt{Att}^{*}(a)|=0$, then we set
$CB_{i}(a)=\frac{\sum_{b\in\mathtt{Att}^{*}(a)}CB_{i-1}(b)}{|\mathtt{Att}^{*}(a)|}=0$.
We note in passing that the properties for these semantics have been
researched in depth by Amgoud et al. in [21]. Perhaps the most important
property, in the context of this paper, is the convergence of the
$\sigma_{HC},\sigma_{MB}$ and $\sigma_{CB}$ semantics for finite frameworks
(see Theorem 7, 12, 17 in [21]). A proof of convergence for a broader class of
semantics (including these three) was proved in the work of Oren et al. [26].
This means that for any given argumentation framework, semantics and initial
weights, the final acceptability degrees of all arguments can always be
computed.
###### Example 1.
An agent may believe the following arguments.
* $a_{0}$:
Tomatoes older than a week can go rotten; these tomatoes are a week and a half
old.
* $a_{1}$:
Tomatoes kept in the fridge (like these ones) can last longer than a week, and
so the tomatoes are not rotten.
* $a_{2}$:
My friend ate one of the tomatoes this morning, and it tasted fine, therefore
they are not rotten.
* $a_{3}$:
My friend is not very good at discriminating whether something is rotten by
taste, so the tomatoes might be rotten.
Here, $a_{0}$ is attacked by $a_{1}$ and $a_{2}$, while the latter is attacked
by $a_{3}$. The agent ascribes each argument with an initial weight encoding
the agent’s belief in the applicability or strength of the argument, i.e.,
$w(a_{0})=0.9$, $w(a_{1})=0.7$, $w(a_{2})=0.7$, and $w(a_{3})=0.6$. Fig. 2
illustrates this weighted argumentation framework
$\mathcal{F}=\langle\\{a_{0},a_{1},a_{2},a_{3}\\},\\{(a_{1},a_{0}),(a_{2},a_{0}),(a_{3},a_{2})\\},w\rangle$.
Figure 2: The weighted argumentation framework from Example 1.
We consider three different semantics in this paper (defined above); yielding
the final acceptability degrees in Table 1. In turn, a reasoner using the
weighted h-categoriser semantics $\sigma_{HC}$ or $\sigma_{CB}$ semantics
would have preferences $a_{1}\succ a_{3}\succ a_{2}\succ a_{0}$, while if it
were to use $\sigma_{MB}$ preferences would be $a_{1}\succ a_{3}\succ
a_{0}\succ a_{2}$. Note that while the $\sigma_{HC}$ and $\sigma_{CB}$
semantics yield similar preference orderings, the agent would be less sure of
its conclusions in the latter case (due to the smaller difference in weights).
Argument | $\sigma_{HC}^{\mathcal{F}}(a_{i})$ | $\sigma_{MB}^{\mathcal{F}}(a_{i})$ | $\sigma_{CB}^{\mathcal{F}}(a_{i})$
---|---|---|---
$a_{0}$ | 0.421 | 0.529 | 0.258
$a_{1}$ | 0.7 | 0.7 | 0.7
$a_{2}$ | 0.438 | 0.438 | 0.269
$a_{3}$ | 0.6 | 0.6 | 0.6
Table 1: Final acceptability degrees for each argument of Example 1 (Fig. 2)
for $\sigma_{HC},\sigma_{MB},$ and $\sigma_{CB}$.
## 3 Inferring Attacks
In this paper, we focus on the problem of identifying whether we can infer
suitable attacks between arguments given their initial weights and desirable
final acceptability degrees (as shown at the bottom of Fig. 1). However, we
ground most of our discussion in the closely related decision problem. Given a
gradual argumentation semantics, a set of arguments, and associated
information about these arguments (i.e., initial weights and some or all
desirable final acceptability degrees), can we determine the attacks between
arguments that — according to the semantics — lead to the desirable final
acceptability degrees from the initial weights.
In this section, we consider the _complete problem_ , where the initial
weights and desirable final acceptability degrees for _all_ arguments are
known. We formalise this problem as follows.
###### Problem 1.
The decision problem $\mathcal{DEC}^{x}_{c}$ is: _Given a set of arguments
$\mathcal{A}$, a gradual semantics $\sigma_{x}$, a weighting function $w$, and
a total desired final acceptability degree function $S:\mathcal{A}\to[0,1]$,
is there a set of attacks $\mathcal{D}\subseteq\mathcal{A}\times\mathcal{A}$
such that for all $a\in\mathcal{A},\sigma^{\mathcal{F}}_{x}(a)=S(a)$, where
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$?_
We investigate this decision problem for the three gradual semantics
$\sigma_{x}$, for $x\in\\{HC,CB,MB\\}.$
###### Proposition 1.
$\mathcal{DEC}_{c}^{MB}$ is polynomial and can be decided in $O(n)$ time and
space.
###### Proof.
We create the set $L=\\{1+S(b)\mid b\in\mathcal{A}\\}$, for which an $O(1)$
lookup can be performed. We can then use the following trivial algorithm to
solve $\mathcal{DEC}_{c}^{MB}$:
1:for all $a\in\mathcal{A}$ do
2: if ($w(a)=0$ and $S(a)\neq 0$) or ($w(a)\neq 0$ and $S(a)=0$) then
3: return False $\triangleright$ A zero initial weight cannot lead to a non-
zero final acceptability degree (and vice-versa).
4: end if
5: if $S(a)\neq 0$ and $w(a)/S(a)\notin L$ then
6: return False $\triangleright$ Contradiction by definition of the
$\sigma_{MB}$ semantics.
7: end if
8:end for
9:return True
∎
In the algorithm above, line 2 checks whether arguments with zero (resp. non-
zero) initial weights and non-zero (resp. zero) desired final acceptability
degrees exist; the presence of such arguments would mean that the weighted-
max-based semantics cannot be satisfied (line 3). Line 5 then checks the
existence of an attacking argument with a suitable acceptability degree (via a
lookup in the set $L$) to satisfy the weighted-max-based semantics in $O(1)$
time. Since lines 1-8 iterate over all arguments, the time complexity is
$O(n)$; similarly, storing $L$ takes $O(n)$ space.
$a_{0}$1$\frac{nm^{*}}{0.4T+nm^{*}}$$a_{1}$$\frac{0.4m_{1}}{nm^{*}}$$a_{2}$$\frac{0.4m_{2}}{nm^{*}}$$a_{n}$$\frac{0.4m_{n}}{nm^{*}}$$\vdots$$m_{1}$$m_{2}$$m_{n}$$T$
Figure 3: Graphical representation of the polynomial transformation (dashed
lines) of an $\mathcal{SSP}$ instance (rectangular nodes) into an instance of
$\mathcal{DEC}_{c}^{HC}$ (circular nodes). The initial weight of $a_{0}$ is 1
(shown in black above $a_{0}$), and of $a_{1}$ to $a_{n}$, is equal to their
final acceptability degree (where final acceptability degrees are shown in
blue). The initial weights of the latter arguments have thus been omitted.
###### Proposition 2.
$\mathcal{DEC}_{c}^{HC}$ is NP-complete.
###### Proof.
We show that $\mathcal{DEC}_{c}^{HC}$ is NP-complete using the subset-sum
problem ($\mathcal{SSP}$). Formally, $\mathcal{SSP}$ is defined as answering
the following question. Consider a multiset of positive numbers $M$ and a
number $T\in\mathbb{R}^{+}$, is there a subset $M^{\prime}\subseteq M$ such
that $\sum\limits_{m^{\prime}\in M^{\prime}}m^{\prime}=T$?
To prove $\mathcal{DEC}_{c}^{HC}$ is NP-complete we must demonstrate that it
is in NP and identify a polynomial time reduction from $\mathcal{SSP}$ to
$\mathcal{DEC}_{c}^{HC}$.
To demonstrate that $\mathcal{DEC}_{c}^{HC}$ belongs to NP we observe that the
certificate of the problem is a set of attacks. Given a set of attacks, we can
check in polynomial time whether
$S(a)=w(a)/(1+\sum_{b\in\mathtt{Att}(a)}S(b))$ for all $a\in\mathcal{A}$ (as
we have all initial weights and the desired final acceptability degrees),
therefore the problem is in NP.
Turning to the reduction, we begin by reducing $\mathcal{SSP}$ to
$\mathcal{DEC}_{c}^{HC}$. Let us assume we have a multi-set of positive
numbers $M=\\{m_{1},\dots,m_{n}\\}$ and $T\in\mathbb{R}^{+}$. We denote
$\sum_{i}m_{i}$ by $m^{*}$. We then create a set of $n+1$ arguments
$\mathcal{A}=\\{a_{0},a_{1},a_{2},\dots,a_{n}\\}$ such that for all
$i\in\\{1,\dots,n\\}$, $S(a_{i})=w(a_{i})=(0.4m_{i})/(nm^{*})$, and we set
$w(a_{0})=1$ and $S(a_{0})=nm^{*}/(0.4T+nm^{*})$. This transformation is
represented in Fig. 3.
###### Example 2.
Consider the $\mathcal{SSP}$ instance where $T=100$ and
$M=\\{23,94,1,37,40\\}$. Here, $n=5$ and $m^{*}=195$. Using our
transformation, we obtain the arguments, initial weights, and final
acceptability degrees shown in Table 2.
Argument | Initial | Desired final
---|---|---
| Weight | acceptability degree
$a_{0}$ | 1 | 0.96059
$a_{1}$ | 0.00944
$a_{2}$ | 0.03856
$a_{3}$ | 0.00041
$a_{4}$ | 0.01518
$a_{5}$ | 0.01641
Table 2: The arguments created from Example 2 using the reduction described
in Proposition 2.
* •
We now demonstrate that — using the above reduction — a solution to
$\mathcal{SSP}$ exists only if a solution to $\mathcal{DEC}_{c}^{HC}$ exists.
If there exists an $M^{\prime}\subseteq M$ such that
$\sum\limits_{m^{\prime}\in M^{\prime}}m^{\prime}=T$, then
$\mathcal{D}=\\{(f(m),a_{0})\mid m\in M^{\prime}\\}$ is a set of attacks such
that for all $a\in\mathcal{A},\sigma^{\mathcal{F}}(a)=S(a)$, where $f$
associates the corresponding argument and
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$. Indeed, we have for all
$i\in\\{1,\dots,n\\},\sigma^{\mathcal{F}}_{HC}(a_{i})=S(a_{i})$ since $a_{i}$
is not attacked and:
$\displaystyle\sigma^{\mathcal{F}}_{HC}(a_{0})$
$\displaystyle=\frac{1}{1+\sum_{m\in M^{\prime}}S(f(m))}$
$\displaystyle=\frac{1}{1+\frac{0.4T}{nm^{*}}}$
$\displaystyle=\frac{nm^{*}}{nm^{*}+0.4T}$ $\displaystyle=S(a_{0})$
* •
We now show the reduction in the other direction — a solution to
$\mathcal{DEC}_{c}^{HC}$ exists only if a solution to $\mathcal{SSP}$ can be
found via the above reduction. If there exists
$\mathcal{D}\subseteq\mathcal{A}\times\mathcal{A}$ such that for all
$b\in\mathcal{A},\sigma^{\mathcal{F}}(b)=S(b)$, then
$M^{\prime}=\\{(S(b)nm^{*})/0.4\mid(b,a_{0})\in\mathcal{D}\\}\subseteq M$ is
such that $\sum\limits_{m^{\prime}\in M^{\prime}}m^{\prime}=T$, where
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$. Indeed:
$\displaystyle\sum\limits_{m^{\prime}\in M^{\prime}}m^{\prime}$
$\displaystyle=\left(\sum\limits_{(b,a_{0})\in\mathcal{D}}S(b)\right)\frac{nm^{*}}{0.4}$
$\displaystyle=\left(\sum\limits_{(b,a_{0})\in\mathcal{D}}\sigma^{\mathcal{F}}_{HC}(b)\right)\frac{nm^{*}}{0.4}$
$\displaystyle=\left(\frac{1}{\frac{nm^{*}}{0.4T+nm^{*}}}-1\right)\frac{nm^{*}}{0.4}$
$\displaystyle=\left(\frac{0.4T}{nm^{*}}\right)\frac{nm^{*}}{0.4}$
$\displaystyle=T$
Note that $(a_{0},a_{0})\notin\mathcal{D}$. Indeed, if
$(a_{0},a_{0})\in\mathcal{D}$ then we have that
$\sigma_{HC}^{\mathcal{F}}(a_{0})=1/(1+\sigma_{HC}^{\mathcal{F}}(a_{0})+Y)$,
where $Y=\sum_{b\in
Att(a_{0})\setminus\\{a_{0}\\}}\sigma_{HC}^{\mathcal{F}}(b)$. Hence,
($\sigma_{HC}^{\mathcal{F}}(a_{0})^{2}+\sigma_{HC}^{\mathcal{F}}(a_{0})-1)/(-\sigma_{HC}^{\mathcal{F}}(a_{0}))=Y$
and since $Y\geq 0$, we conclude that
$0\leq\sigma_{HC}^{\mathcal{F}}(a_{0})\leq\frac{-1+\sqrt{5}}{2}\simeq 0.618$.
However, we get a contradiction as $S(a_{0})=\sigma_{HC}^{\mathcal{F}}(a_{0})$
and:
$\displaystyle\frac{T}{nm^{*}}$ $\displaystyle\leq 1$
$\displaystyle\frac{0.4T}{nm^{*}}$ $\displaystyle\leq 0.4$
$\displaystyle\frac{0.4T+nm^{*}}{nm^{*}}$ $\displaystyle\leq 1.4$
$\displaystyle S(a_{0})$ $\displaystyle\geq 0.714$
We have proved that $\mathcal{DEC}_{c}^{HC}$ is in NP and that $\mathcal{SSP}$
is polynomial time reducible to $\mathcal{DEC}_{c}^{HC}$ (the size of the
argumentation framework produced is polynomial with respect to the size of the
$\mathcal{SSP}$ instance), therefore $\mathcal{DEC}_{c}^{HC}$ is NP-complete.
∎
###### Example 3 (Cont’d Example 2).
We see that the subset sum problem has a solution (using values 23,37 and 40).
Analogously, a solution to $\mathcal{DEC}_{c}^{HC}$ exists by having arguments
$a_{1},a_{4}$ and $a_{5}$ attack argument $a_{0}$, as shown in Fig. 4.
$a_{0}$10.96059$a_{1}$0.00944$a_{2}$0.03856$a_{3}$0.00041$a_{4}$0.01518$a_{5}$0.01641239413740100
Figure 4: Representation of how a solution to an $\mathcal{SSP}$ instance
(represented with the gray rectangles) can be obtained from a solution to a
corresponding instance from $\mathcal{DEC}^{HC}_{c}$ (the attacks drawn). Blue
values for $a_{1}\ldots a_{5}$ indicate both inital weights and final
acceptability degrees, and denote final acceptability degrees for $a_{0}$,
which has an initial weight of 1.
###### Proposition 3.
$\mathcal{DEC}_{c}^{CB}$ is NP-complete.
###### Proof.
Similar to the previous proof, we show that $\mathcal{DEC}_{c}^{CB}$ is NP-
complete using the $k$-subset-sum problem ($k\mathcal{SSP}$). Formally,
$k\mathcal{SSP}$ is defined as answering the following question. Consider a
multiset of positive numbers $M$ and a number $T\in\mathbb{R}^{+}$, is there a
subset $M^{\prime}\subseteq M$ such that $\sum_{m^{\prime}\in
M^{\prime}}m^{\prime}=T$ and $|M^{\prime}|=k$?
We first observe that the certificate of the problem $\mathcal{DEC}_{c}^{CB}$
is a set of attacks. Given a set of attacks, we can check in polynomial time
whether
$S(a)=w(a)/(1+|\mathtt{Att}^{*}(a)|+\sum_{b\in\mathtt{Att}^{*}(a)}S(b)/|\mathtt{Att}^{*}(a)|)$
for all $a\in\mathcal{A}$, meaning that the problem is in NP.
We now reduce $k\mathcal{SSP}$ to $\mathcal{DEC}_{c}^{CB}$. Let us assume we
have a multi-set of positive numbers $M=\\{m_{1},\dots,m_{n}\\}$, $1\leq k\leq
n$, and $T\in\mathbb{R}^{+}$. We denote by $m^{*}=\max_{m_{i}\in M}m_{i}$ and
$u=\frac{\sqrt{k^{2}+2k+4/k+1}-k-1}{3}$ (note that $u\in]0,(2\sqrt{2}-2)/3]$
when $k\geq 1$). We create $n+1$ arguments
$\mathcal{A}=\\{a_{0},a_{1},a_{2},\dots,a_{n}\\}$ such that for all
$i\in\\{1,\dots,n\\},S(a_{i})=w(a_{i})=um_{i}/m^{*}$ and we set $w(a_{0})=1$
and $S(a_{0})=1/(1+k+Tu/(km^{*}))$.
* •
If there exists an $M^{\prime}\subseteq M$ such that $\sum_{m^{\prime}\in
M^{\prime}}m^{\prime}=T$ and $|M^{\prime}|=k$, then
$\mathcal{D}=\\{(f(m),a_{0})|m\in M^{\prime}\\}$ is a set of $k$ attacks such
that for all $a\in\mathcal{A},\sigma^{\mathcal{F}}(a)=S(a),$ where
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$. Indeed, we have for all
$i\in\\{1,\dots,n\\},\sigma_{CB}^{\mathcal{F}}(a_{i})=S(a_{i})$ since $a_{i}$
is not attacked and:
$\displaystyle\sigma_{CB}^{\mathcal{F}}(a_{0})$
$\displaystyle=\frac{1}{1+k+\frac{\sum_{m\in M^{\prime}}S(f(m))}{k}}$
$\displaystyle=\frac{1}{1+k+\frac{\frac{Tu}{m^{*}}}{k}}$
$\displaystyle=\frac{1}{1+k+\frac{Tu}{km^{*}}}$
* •
Assume we have a solution to $\mathcal{DEC}_{c}^{CB}$, i.e. we have
$\mathcal{D}\subseteq\mathcal{A}\times\mathcal{A}$ such that for all
$b\in\mathcal{A},\sigma^{\mathcal{F}}(b)=S(b)$. We know that $a_{0}$ is
attacked by $k$ arguments in $\mathcal{D}$ as its value lies between
$[1/(k+2),1/(k+1)]$ and that arguments $a_{1}$ to $a_{n}$ are not attacked
(except if they or their attackers have initial weights of $0$). We can build
$M^{\prime}=\\{S(b)*m^{*}/u|(b,a_{0})\in\mathcal{D}\\}\subseteq M$ such that
$|M^{\prime}|=k$ and $\sum_{m^{\prime}\in M^{\prime}}m^{\prime}=T$. Indeed:
$\displaystyle\sum_{m^{\prime}\in M^{\prime}}m^{\prime}$
$\displaystyle=\left(\sum_{(b,a_{0})\in\mathcal{D}}S(b)\right)\frac{m^{*}}{u}$
$\displaystyle=\left(\frac{Tu}{m^{*}}\right)\frac{m^{*}}{u}$ $\displaystyle=T$
We show that $(a_{0},a_{0})\notin\mathcal{D}$ by contradiction. Assume we have
this self-attack, then the maximum value with self attack possible for
$\sigma_{CB}^{\mathcal{F}}(a_{0})$ is determined by computing the unique
fixed-point of the function $f(x)=1/(1+k+x/k)$ which is:
$0<\frac{-k^{2}+\sqrt{k(k^{3}+2k^{2}+k+4)}-k}{2}\leq 1$
Moreover, it holds that:
$\displaystyle\frac{-k^{2}+\sqrt{k(k^{3}+2k^{2}+k+4)}-k}{2}$
$\displaystyle<\frac{1}{1+k+u}$
$\displaystyle<\frac{1}{1+k+\frac{Tu}{km^{*}}}$ $\displaystyle<S(a_{0})$
This is a contradiction with $S(a_{0})=\sigma_{CB}^{\mathcal{F}}(a_{0})$.
∎
To recap, our results show that $\mathcal{DEC}^{MB}_{c}$ is polynomial (and
indeed, can be solved in linear time), while $\mathcal{DEC}^{HB}_{c}$ and
$\mathcal{DEC}^{CB}_{c}$ are both NP-complete. This latter result was obtained
by a reduction to a variant of the subset-sum problem.
## 4 Identifying Solutions
Rather than simply considering the decision problem, it is useful — if
possible — to be able to infer the attacks induced by some set of initial
weights and final acceptability degrees. We consider each semantics
individually.
### 4.1 The Weighted Max-based Semantics
For the weighted max-based semantics, it is trivial to modify the algorithm
from Proposition 1 to infer a suitable set of attacks (when possible). Rather
than simply returning True or False, we return, by modifying lines 5 and 6,
any set of attacks $X$ that satisfy the condition that for all
$a\in\mathcal{A}$, there exists $(x,a)\in X$ such that $w(a)/S(a)=(1+S(x))$.
Thus, we note that this semantics does little to constrain the attacks in the
framework. If we discover that there is an attack from argument $a_{i}$ to
argument $a_{0}$ in a solution, then under the max-based semantics, we can
expand this solution by adding additional attacks from any argument $a_{j}$ to
$a_{0}$, where $S(a_{j})\leq S(a_{i})$, without affecting the result.
###### Proposition 4 (Solution Expansion).
Given a set of arguments $\mathcal{A}$, a weighting function
$w:\mathcal{A}\to[0,1]$, and a desirable final acceptability degree function
$S:\mathcal{A}\to[0,1]$. If $\mathcal{D}\subseteq\mathcal{A}\times\mathcal{A}$
is such that for all $a\in\mathcal{A}$, $\sigma_{MB}^{\mathcal{F}}(a)=S(a)$,
where $\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ then for all
$a\in\mathcal{A},\sigma_{MB}^{\mathcal{F}^{\prime}}(a)=S(a)$, where
$\mathcal{F}^{\prime}=\langle\mathcal{A},\mathcal{D}^{\prime},w\rangle,\mathcal{D}^{\prime}=\mathcal{D}\cup
X$ and $X\subseteq\\{(a_{i},a_{j})\mid\exists(a_{k},a_{j})\in\mathcal{D}$ and
$S(a_{i})\leq S(a_{k})\\}\cup\\{(a_{i},a_{j})\mid w(a_{j})=0\\}$. We say that
$\mathcal{D}^{\prime}$ is an expansion of $\mathcal{D}$.
Similarly, if we have a set of attacks that achieve the desirable final
acceptability degrees for all arguments and there is an attack from $a_{i}$ to
$a_{0}$, then under the max-based semantics, we can contract this solution by
removing attacks from any other argument $a_{j}$ to $a_{0}$, where
$S(a_{j})\leq S(a_{i})$.
###### Proposition 5 (Solution Contraction).
Given a set of arguments $\mathcal{A}$, a weighting function
$w:\mathcal{A}\to[0,1]$, and a desirable final acceptability degree function
$S:\mathcal{A}\to[0,1]$. If $\mathcal{D}\subseteq\mathcal{A}\times\mathcal{A}$
is such that for all $a\in\mathcal{A}$, $\sigma_{MB}^{\mathcal{F}}(a)=S(a)$,
where $\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ then for all
$a\in\mathcal{A},\sigma_{MB}^{\mathcal{F}^{\prime}}(a)=S(a)$, where
$(a_{i},a_{0})\in\mathcal{D},\mathcal{F}^{\prime}=\langle\mathcal{A},\mathcal{D}^{\prime},w\rangle,\mathcal{D}^{\prime}=\mathcal{D}\setminus
X$ and $X\subseteq\\{(a_{j},a_{0})\in\mathcal{D}\mid a_{j}\neq a_{i}$ and
$S(a_{j})\leq S(a_{i})\\}\cup\\{(a_{j},a_{0})|w(a_{0})=0\\}$. We say that
$\mathcal{D}^{\prime}$ is a contraction of $\mathcal{D}$ (on
$(a_{i},a_{0})\in\mathcal{D}$).
Figure 5: The argumentation framework of Example 4.
###### Example 4.
Let us consider $\mathcal{A}$, $w$, $\mathcal{D}$ as shown in Figure 5 and the
desirable final acceptability degree function $S$ such that $S(a_{0})=0.43$,
$S(a_{1})=0.30$, $S(a_{2})=0.58$, and $S(a_{3})=0.30$. A contraction
$\mathcal{D}^{\prime}$ of $\mathcal{D}$ on $(a_{2},a_{2})$ will remove the
attacks $(a_{0},a_{2}),(a_{1},a_{2})$, and $(a_{3},a_{2})$ from $\mathcal{D}$
without changing the acceptability degrees under the max-based semantics.
Likewise, an expansion $\mathcal{D}^{\prime\prime}$ of $\mathcal{D}^{\prime}$
can be obtained by adding the attack $(a_{0},a_{2})$ as this does not change
the acceptability degrees. This is represented in Fig. 6.
Figure 6: Representation of
$\mathcal{F}^{\prime}=\langle\mathcal{A},\mathcal{D}^{\prime},w\rangle$
(left), where $\mathcal{D}^{\prime}$ is a contraction of $\mathcal{D}$ on
$(a_{2},a_{2})$ and
$\mathcal{F}^{\prime\prime}=\langle\mathcal{A},\mathcal{D}^{\prime\prime},w\rangle$
(right), where $\mathcal{D}^{\prime\prime}$ is an expansion of
$\mathcal{D}^{\prime}$.
Once a solution for the max-based semantics is found, we can reach all the
other solutions by expanding this initial solution once and then successively
contracting it.
###### Proposition 6.
Given a set of arguments $\mathcal{A}$, a weighting function
$w:\mathcal{A}\to[0,1]$, and a desirable final acceptability degree function
$S:\mathcal{A}\to[0,1]$. If
$\mathcal{D},\mathcal{D}^{\prime}\subseteq\mathcal{A}\times\mathcal{A}$ are
such that for all $a\in\mathcal{A}$, $\sigma_{MB}^{\mathcal{F}}(a)=S(a)$,
$\sigma_{MB}^{\mathcal{F}^{\prime}}(a)=S(a)$ where
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ and
$\mathcal{F}^{\prime}=\langle\mathcal{A},\mathcal{D}^{\prime},w\rangle$ then
there exists $\mathcal{D}^{*}\subseteq\mathcal{A}\times\mathcal{A}$, such that
$\mathcal{D}^{*}$ is an expansion of $\mathcal{D}$, and a sequence of sets of
attacks $(\mathcal{D}_{1},\mathcal{D}_{2},\dots,\mathcal{D}_{n})$, where
$\mathcal{D}_{1}=\mathcal{D}^{*}$, $\mathcal{D}_{n}=\mathcal{D}^{\prime}$ and
for all $1\leq i\leq n-1,\mathcal{D}_{i+1}$ is a contraction of
$\mathcal{D}_{i}$.
###### Proof.
Let us consider $\mathcal{A}=\\{a_{1},a_{2},\dots,a_{n}\\}$ and two arbitrary
solutions
$\mathcal{D},\mathcal{D}^{\prime}\subseteq\mathcal{A}\times\mathcal{A}$ for
$\mathcal{DEC}_{c}^{MB}$. We prove this proposition by construction. By
definition, for all
$a\in\mathcal{A},\sigma_{MB}^{\mathcal{F}}(a)=\sigma_{MB}^{\mathcal{F}^{\prime}}(a)=S(a)$,
where $\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ and
$\mathcal{F}^{\prime}=\langle\mathcal{A},\mathcal{D}^{\prime},w\rangle$. This
means that for all $a\in\mathcal{A}$, such that $w(a)\neq
0,\max_{b\in\mathtt{Att}_{\mathcal{F}}(a)}S(b)=\max_{b\in\mathtt{Att}_{\mathcal{F}^{\prime}}(a)}S(b)$.
Thus, $\mathcal{D}^{*}=\mathcal{D}\cup\mathcal{D}^{\prime}$ is an expansion of
$\mathcal{D}$. Then, for all $1\leq i\leq n$, it holds that
$\mathcal{D}_{i+1}=\mathcal{D}_{i}\setminus\\{(a_{j},a_{i})\in\mathcal{D}|(a_{j},a_{i})\notin\mathcal{D}^{\prime}\\}$
is a contraction of $\mathcal{D}_{i}$, where
$\mathcal{D}_{1}=\mathcal{D}^{*}$. It is clear that
$\mathcal{D}_{n+1}=\mathcal{D}^{\prime}$ as all attacks in
$\mathcal{D}\setminus\mathcal{D}^{\prime}$ have been removed from
$\mathcal{D}^{*}$. ∎
### 4.2 The Weighted H-Categoriser Semantics
Let us consider the case of the weighted h-categoriser semantics. Given an
input $\mathcal{A}=\\{a_{1},\dots,a_{n}\\}$, an initial weight function $w$,
and a desirable final acceptability degree function $S$; we seek a set of
attacks $\mathcal{D}\subseteq\mathcal{A}\times\mathcal{A}$ such that for all
$a\in\mathcal{A},\sigma_{HC}^{\mathcal{F}}(a)=S(a)$, where
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$.
Recall from the definition of the weighted h-categoriser semantics that any
solution must mean that the following equation holds.
$S(a_{i})=\frac{w(a_{i})}{1+\sum_{b\in\mathtt{Att}(a_{i})}S(b)}$
Moreover, if $w(a_{i})\neq 0$, we can rearrange the previous equation to
obtain:
$\sum_{b\in\mathtt{Att}(a_{i})}S(b)=\frac{w(a_{i})-S(a_{i})}{S(a_{i})}$
For the complete problem, we are given $w(a_{i})$ and $S(a_{i})$ for all
$a_{i}\in\mathcal{A}$, allowing us to compute
$\sum_{b\in\mathtt{Att}(a_{i})}S(b)$ via the above equation. In other words,
for every argument $a_{i}$, we can compute a target value
$T=\sum_{b\in\mathtt{Att}(a_{i})}S(b)$ for which the final acceptability
degrees of all its attackers must sum up to. By considering the multi-set
$M=\\{S(a_{j})\mid a_{j}\in\mathcal{A}\\}$, we must solve a single subset-sum
problem to identify the attackers for a single argument $a_{i}$. Extending
this to all $n=|\mathcal{A}|$ arguments in our framework, we can therefore
identify all attacks in the framework by solving $n$ versions of the subset-
sum problem. Algorithm 1 provides the pseudo-code for this approach.
1:$\mathcal{A}$ a set of arguments
2:$w:\mathcal{A}\to[0,1]$ the initial weights for each argument
3:$S:\mathcal{A}\to[0,1]$ the desired final acceptability degrees for each
argument
4:function SolveHC($\mathcal{A},w,S$)
5: $D=\\{\\}$
6: $M\leftarrow\\{S(b)|b\in\mathcal{A}\\}$
7: for all $a\in\mathcal{A}$ do
8: if ($w(a)=0$ and $S(a)\neq 0$) or ($w(a)\neq 0$ and $S(a)=0$) then
9: return False
10: end if
11: if $w(a)\neq 0$ and $w(a)\neq S(a)$ then
12: $T\leftarrow\frac{w(a)-S(a)}{S(a)}$
13: if SSP($T,M$) = False then
14: return False
15: end if
16: $D=D\cup\\{(b,a)|S(b)\in$SSP($T,M$)$\\}$
17: end if
18: end for
19: return $D$
20:end function
Algorithm 1 Procedure to solve $\mathcal{DEC}_{c}^{HC}$. SSP (line 13, 16)
calls a subset-sum solver which returns the elements of $M$ that sum up to
$T$, or False if no such elements can be found.
We note in passing that the standard version of the subset-sum problem assumes
that $T$ and elements of $M$ are all integers. If we assume that all initial
weights and final acceptability degrees are rational, we can easily transform
our computed $T$ and $M$ values to integers. The most common — dynamic
programming — approach to solving standard subset-sum can run in pseudo-
polynomial time. However, the integers we obtain become very large very
quickly, making this approach impractical. Future work will consider using
state-of-the-art techniques for solving subset-sum over real numbers instead,
e.g. the recent FPTAS of Costandin [27]. In addition, having transformed our
problem to an instance of the subset-sum problem opens up the possibility of
transforming the problem to other NP-complete problems for which efficient
solvers exist, such as satisfiability.
Moreover, once a set of attacks is found to be a solution, we can obtain other
solutions by replacing the attacks to a single argument $x$ by other attacks
to this argument so that the sum of the degree of the attackers remains the
same. Given the simplicity of this proposition, we do not provide a proof for
it here.
###### Proposition 7.
Consider two weighted argumentation frameworks
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$,
$\mathcal{F}^{\prime}=\langle\mathcal{A},\mathcal{D}^{\prime},w\rangle$, and a
desirable final acceptability degree function $S:\mathcal{A}\to[0,1]$ such
that $\sigma^{\mathcal{F}}_{HC}(a)=S(a)$ for any $a\in\mathcal{A}$. If all of
the following hold
* •
$x\in\mathcal{A}$
* •
$Z\subseteq\mathcal{A}\times\\{x\\}$
* •
$\sum_{(z,x)\in Z}S(z)=(w(x)-S(x))/S(x)$ if $w(x)\neq 0$
* •
$\mathcal{D}^{\prime}=(\mathcal{D}\cap(\mathcal{A}\times(a\setminus\\{x\\})))\cup
Z$
then for all $a\in\mathcal{A},\sigma^{\mathcal{F}^{\prime}}_{HC}(a)=S(a)$.
### 4.3 The Weighted Cardinality-based Semantics
We can adopt a similar approach for $\sigma_{CB}$ as was done for
$\sigma_{HC}$. We begin by assuming that there are $k$ attacks from arguments
with non-zero initial weight against an argument $a_{i}$. Any solution must
mean that the following equation holds:
$S(a_{i})=\frac{w(a_{i})}{1+k+\frac{1}{k}\sum_{b\in\mathtt{Att}(a_{i})}S(b)}$
If $w(a_{i})\neq 0$, rearranging gives us the equation:
$\sum_{b\in\mathtt{Att}(a_{i})}S(b)=-\frac{k(S(a_{i})k+S(a_{i})-w(a_{i}))}{S(a_{i})}$
Again, all values on the right-hand side of the equation are given for the
complete problem given, allowing us to compute the value of the left-hand side
for each argument. Assuming a non-zero number of attacks, while we could
(naively) iterate over all $k=1\ldots n$ to identify the number of attacks
against an argument, and thereby determine $k$, we observe that if
$S(b)=\epsilon>0$, $\lim{\epsilon}\to 0$, for all $b\in\mathtt{Att}(a_{i})$
the semantic equation reduces to the following.
$S(a_{i})=\frac{w(a_{i})}{1+k}$
Solving for $k$, we see that:
$k=\frac{w(a_{i})-S(a_{i})}{S(a_{i})}$
Now assume instead that $S(b)=1$ for all $b\in\mathtt{Att}(a_{i})$. Performing
the same operations means that:
$k=\frac{w(a_{i})-2S(a_{i})}{S(a_{i})}$
These two equations serve as an upper and lower bound for $k$. Since $k$ must
be an integer, we can avoid the need to iterate over the number of attacks by
computing $k$ as follows.
$k=\lfloor\frac{w(a_{i})-S(a_{i})}{S(a_{i})}\rfloor=\lceil\frac{w(a_{i})-2S(a_{i})}{S(a_{i})}\rceil$
The pseudo-code for this approach is described in Algorithm 2.
1:$\mathcal{A}$ a set of arguments
2:$w:\mathcal{A}\to\mathbb{Q}$ the initial weights for each argument
3:$S:\mathcal{A}\to\mathbb{Q}$ the final acceptability degree for each
argument
4:function SolveCB($\mathcal{A},w,S$)
5: $D=\\{\\}$
6: $M\leftarrow\\{S(b)|b\in\mathcal{A}\\}$
7: for all $a\in\mathcal{A}$ do
8: if ($w(a)=0$ and $S(a)\neq 0$) or ($w(a)\neq 0$ and $S(a)=0$) then
9: return False
10: end if
11: if $w(a)\neq 0$ and $w(a)\neq S(a)$ then
12: $k\leftarrow\lfloor\frac{w(a)-S(a)}{S(a)}\rfloor$
13: $T\leftarrow-\frac{k(S(a)k+S(a)-w(a))}{S(a)}$
14: if kSSP($T,M,k$) = False then
15: return False
16: end if
17: $D\leftarrow D\cup\\{(b,a)|S(b)\in$kSSP($T,M,k$)$\\}$
18: end if
19: end for
20: return $D$
21:end function
Algorithm 2 Procedure to solve $\mathcal{DEC}_{c}^{CB}$. kSSP (line 14, 17)
calls a subset-sum solver which returns $k$ elements of $M$ which sum up to
$T$.
Since $k$ is known, the $k\mathcal{SSP}$ solver called in Algorithm 2 could
potentially be more efficient than that used in the h-categoriser semantics.
Analogous to the result from Proposition 7, under the cardinality semantics,
once a set of attacks is found to be a solution, we can obtain other solutions
by replacing the $k$ attacks on a single argument $x$ by another set of $k$
attacks to this argument so that the sum of the final acceptability degree of
the attackers remains the same. Given the simplicity of this proposition, we
do not provide a proof for it here.
###### Proposition 8.
Given two weighted argumentation frameworks
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle,\mathcal{F}^{\prime}=\langle\mathcal{A},\mathcal{D}^{\prime},w\rangle$,
and a desirable final acceptability degree function $S:\mathcal{A}\to[0,1]$
such that for all $a\in\mathcal{A}$, $\sigma_{CB}^{\mathcal{F}}(a)=S(a)$. If
all the following hold:
* •
$x\in\mathcal{A}$
* •
$Z\subseteq\mathcal{A}\times\\{x\\}$, and
* •
$\mathcal{D}^{\prime}=\left(\mathcal{D}\cap(\mathcal{A}\times(\mathcal{A}\setminus\\{x\\}))\right)\cup
Z$,
* •
$|Z|=\lfloor\frac{w(a)-S(a)}{S(a)}\rfloor$ and $\sum_{(z,x)\in
Z}S(z)=-\frac{k(S(x)k+S(x)-w(x))}{S(x)}$ if $w(x)\neq 0$.
then for all $a\in\mathcal{A}$, it is the case that
$\sigma_{CB}^{\mathcal{F}^{\prime}}(a)=S(a)$
Propositions 4, 5, and 6 demonstrate that there are typically a large number
of non-minimal argumentation frameworks for the weighted max-based semantics.
For the weighted h-categoriser semantics, attacks can be substituted as long
as the sum of the final acceptability degrees of the attacking arguments match
(see Proposition 7), while for weighted cardinality semantics, attacks can
only be substituted by a set of attackers of the same size which final
acceptability degree sums to the same value (see Proposition 8). We also note
that, for all semantics, any argument with an initial weight of 0 can clearly
have any number of attacks against it.
## 5 Discussion and Future Work
To this point, we have considered only the complete problem of attack
inference in gradual argumentation frameworks, and we now briefly discuss the
partial problem. Here, we are given a semantics, a set of arguments, a partial
mapping between arguments and initial weights, and a partial mapping between
arguments and final acceptability degrees, and we must determine whether
attacks (and initial weights which were not provided) can be identified which
make the given final acceptability degrees consistent with the semantics.
Since the complete problem is a special case of this partial problem, and
since we can still verify correctness in polynomial time, the NP-completeness
results of Section 3 still apply to the $CB$ and $HC$ semantics. The behaviour
of the $MB$ version of the partial problem — which was polynomial in the
complete problem — is more challenging to characterise. While it is easy to
show that there are polynomial instances of the partial problem (e.g., when
all initial weights are given and all but one final acceptability degree is
provided), we strongly believe that the general case is NP-complete. We leave
proof of this result as an element of future work. Other directions for
research include providing a mapping from subset-sum to other problems for
which solvers exist (e.g., satisfiability) and evaluating the performance of
such solvers222An implementation of our algorithms using an optimised depth-
first-search SSP solver can be found at
https://github.com/jhudsy/Gradual_Attack_Inference. We do not provide
experimental data as the performance of our solver demonstrates the
exponential growth of the underlying subset-sum problem. Systems with more
than $\sim$13 arguments can only rarely be solved in reasonable time using our
implementation.. Argumentation researchers have found that this approach has
often yielded excellent results [28]. In addition, it may be interesting to
evaluate such solvers on different (given) underlying graph topologies.
Finally, we focused on three gradual semantics in this paper. However, many
other weighted semantics have been proposed (e.g., [29, 30, 31, 32, 25]), and
extending the current work to these, and in particular, to probabilistic
semantics [33, 34] would be interesting.
We can consider another inverse problem, namely the computation of semantics
for a given weighted argumentation framework and set of final acceptability
degrees or preferences over arguments. We note in passing that this problem is
trivial to solve in polynomial time, as one simpy needs to check for
consistency between the inputs (weighted argumentation framework) and outputs
(final acceptability degrees or preferences) for each semantics.
Turning to related work, researchers have examined the notion of _argument
realisability_ in abstract argumentation frameworks [35], seeking to identify
an attack relation that yields a specific labelling or extension. Recent work
by Mumford et al. [36] has shown that — for complete semantics and IN/OUT
labellings — the problem is NP-complete, whereas it can be solved in
polynomial time if UNDEC labellings are also allowed. The work of Skiba et al.
[37] focuses on whether — for a given ranking and ranking-based semantics — we
can find an unweighted argumentation framework such that the selected ranking-
based semantics induces the ranking when applied to the argumentation
framework. They show that the above problem is true for a number of ranking-
based semantics, including burden-based and discussion-based semantics [38],
the simple product semantics on social abstract frameworks [25, 31], and the
probabilistic graded semantics [39, 40]. Another related area to the current
work is _argument synthesis_ [41], where the attack relation must satisfy some
positive and negative constraints, but where the problem is not fully
constrained. There are still several open questions regarding the time
complexity of these types of problems, even for the case of abstract
argumentation semantics [36].
More generally, our work can be seen as a type of inverse argumentation
problem. [13, 26] has examined one such inverse problem in the context of the
gradual semantics, seeking to identify a set of initial weights given a
semantics, a set of arguments, attacks and final acceptability degrees. In the
context of abstract argumentation, such inverse problems have examined
inferring preferences from justified arguments [42, 18], and has applications
in the context of belief revision [43].
The current paper focuses on the complexity of the underlying decision problem
and inferring attacks given the semantics, initial weights, and final
acceptability degrees. This — in a sense — is unrealistic, and limits the
applications of the current work. In most uses of gradual semantics, one
considers the preferences obtained from final acceptability degrees rather
than the final acceptability degrees themselves, as the latter are not
(normally) exposed within an application. Thus, as future work, we intend to
consider the problem of inferring attacks from a semantics, initial weights,
and preferences over arguments. We believe that the complexity of this problem
for a given semantics will be similar to that obtained in the current work.
Given this more general problem one could — for example — determine whether an
agent is rational given the initial weights they assign to a set of arguments,
a semantics, and a resultant preference ordering over the arguments. Other
applications, as discussed above, include opponent modelling [19] in the
context of dialogue and preference elicitation.
## 6 Conclusions
We have considered the problem of inferring an attack relation given a gradual
semantics and all initial weights and final acceptability degrees for an
argumentation framework. We have shown that for the weighted max-based
semantics, this problem can be solved in linear time, but that the obtained
solution is (typically) not unique. For the h-categoriser and cardinality
semantics — where solutions have no redundancies — the problem becomes NP-
complete. Our proofs are based on a reduction to variants of the subset-sum
problem, and efficient solvers for this problem can be applied to our work,
facilitating its application in domains such as opponent modelling. Finally,
we have focused on the complete problem, and there are exciting avenues for
future research dealing with its partial form.
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# The complex organic molecular content in the L1517B starless core
A. Megías,1 I. Jiménez-Serra,1 J. Martín-Pintado,1 A. I. Vasyunin,2 S.
Spezzano,3 P. Caselli,3 G. Cosentino 4 and S. Viti 5,6
1 Centro de Astrobiología (CAB), CSIC-INTA, Carretera de Ajalvir, km 4, 28805,
Torrejón de Ardoz, Spain
2 Ural Federal University, Kuybysheva st. 48, 620002, Ekaterinburg, Russian
Federation
3 Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1,
85748, Garching, Germany
4 Chalmers University of Technology, SE41296, Gothenburg, Sweden
5 Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA, Leiden, the
Netherlands
6 Department of Physics and Astronomy, University College London, Gower
Street, London, WC1E 6BT, United Kingdom E-mail: amegias @ cab.inta-csic.es
(Accepted 2022 November 22. Received 2022 November 15; in original form 2022
August 30. DOI: https://doi.org/10.1093/mnras/stac3449.)
###### Abstract
Recent observations of the pre-stellar core L1544 and the younger starless
core L1498 have revealed that complex organic molecules (COMs) are enhanced in
the gas phase toward their outer and intermediate-density shells. Our goal is
to determine the level of chemical complexity toward the starless core L1517B,
which seems younger than L1498, and compare it with the other two previously
studied cores to see if there is a chemical evolution within the cores. We
have carried out 3 mm high-sensitivity observations toward two positions in
the L1517B starless core: the core’s centre and the position where the
methanol emission peaks (at a distance of $\sim$5000 au from the core’s
centre). Our observations reveal that a lower number of COMs and COM
precursors are detected in L1517B with respect to L1498 and L1544, and also
show lower abundances. Besides methanol, we only detected CH3O, H2CCO, CH3CHO,
CH3CN, CH3NC, HCCCN, and HCCNC. Their measured abundances are $\sim$3 times
larger toward the methanol peak than toward the core’s centre, mimicking the
behaviour found toward the more evolved cores L1544 and L1498. We propose that
the differences in the chemical complexity observed between the three studied
starless cores are a consequence of their evolution, with L1517B being the
less evolved one, followed by L1498 and L1544. Chemical complexity in these
cores seems to increase over time, with N-bearing molecules forming first and
O-bearing COMs forming at a later stage as a result of the catastrophic
depletion of CO.
###### keywords:
astrochemistry – ISM: abundances – ISM: molecules – ISM: individuals: L1517B
††pubyear: 2023††pagerange: The complex organic molecular content in the
L1517B starless core–C
## 1 Introduction
In Astrochemistry, complex organic molecules (or COMs) are defined as carbon-
bearing compounds with at least 6 atoms in their structure (Herbst & van
Dishoeck, 2009). It was initially believed that COMs could only form on dust
grains in the presence of a source of heat via hydrogenation, atom addition
and radical-radical reactions (Watanabe & Kouchi, 2002; Garrod et al., 2008).
Indeed, COMs were firstly detected in relatively hot sources with $T\gtrsim$
100 K, such as massive hot cores (Hollis et al., 2000; Hollis et al., 2006;
Belloche et al., 2008; Belloche et al., 2013) or low-mass warm cores (hot
corinos; Bottinelli et al., 2004; Jørgensen et al., 2013). However, the
detection of several COMs in the gas phase in starless/pre-stellar cores and
dark cloud cores (e.g., Marcelino et al., 2007; Öberg et al., 2010; Bacmann et
al., 2012; Cernicharo et al., 2012; Vastel et al., 2014; Jiménez-Serra et al.,
2016; Jiménez-Serra et al., 2021; Scibelli & Shirley, 2020) indicates that
there must be another chemical pathway that allows the presence of COMs in the
gas phase at temperatures as low as 10 K. In the last years, there have been
different proposals to explain the formation of these COMs under these
conditions. However, the chemical pathways giving rise to COMs in cold cores
is not well understood yet, and more observations are needed to constrain the
proposed models (Rawlings et al., 2013; Vasyunin & Herbst, 2013; Balucani et
al., 2015; Vasyunin et al., 2017; Holdship et al., 2019; Jin & Garrod, 2020;
Punanova et al., 2022). For example, Scibelli et al. (2021) studied the
starless core L1521E, claiming that COMs are not only formed by gas-phase
reactions, bus also by surface reactions on dust grains. Other authors suggest
that cosmic rays induce desorption from icy mantles on dust grains (Sipilä et
al., 2021; Redaelli et al., 2021). It is remarkable that methanol (CH3OH),
which plays a central role in the formation for larger COMs, is detected
systematically in starless cores (Scibelli & Shirley, 2020), and thought to
form on the surface of dust grains, partially returning into the gas phase
upon reactive desorption (Garrod et al., 2006, 2007; Vasyunin et al., 2017).
Figure 1: $\mathrm{H}_{2}$ column density map obtained from Herschel/Spire
data at 0.25, 0.35 and 0.50 mm, using the same procedure as the one used for
L1544 in Spezzano et al. (2016). Crosses indicate the positions observed in
the core: the dust peak (in black) and the methanol peak (in red). The beam
size for each filter is 17.9, 24.2 and 35.4 arcseconds (0.25, 0.35 and 0.50
mm, respectively), although the images of the two first filters were smoothed
to the resolution of 0.50 mm, which is marked with a circle at the bottom-left
corner of the image. The positions of the dust and methanol peaks are obtained
from Tafalla et al. (2004); Tafalla et al. (2006). The beam sizes were
retrieved from the Spire Handbook:
https://www.cosmos.esa.int/web/herschel/spire-overview.
In order to understand how COMs form under the cold conditions of pre-
stellar/starless cores, Jiménez-Serra et al. (2016); Jiménez-Serra et al.
(2021) investigated the radial distribution of large COMs as a function of
radius in the L1544 and L1498 starless cores. Methanol tends to show a ring-
like morphology circumventing the dust continuum emission (see Tafalla et al.,
2006; Bizzocchi et al., 2014; Spezzano et al., 2016; Punanova et al., 2022).
Since mapping the emission of larger COMs in starless cores requires large
amounts of telescope time, Jimenez-Serra et al. (2016, 2021) observed two
positions within these cores: the centre, defined by the position of the dust
peak, and the location where methanol peaks. The latter position is
representative of an outer, intermediate-density shell located at radii
between $\sim$4000 and $\sim$11000 au from the core centre in L1544 and L1498,
respectively. Several COM precursors have been found toward both cores such as
tricarbon monoxide (CCCO) and cyanoacetylene (HCCCN), but more complex
molecules like acetaldehyde (CH3CHO), methyl formate (CH3OCHO) or
dimethylether (CH3OCH3) have only been detected toward L1544, which is at a
more advanced stage of evolution than L1498 (see Jiménez-Serra et al., 2016;
Jiménez-Serra et al., 2021). However, N-bearing COMs are detected toward L1498
with abundances close to those measured toward L1544.
In this work, we present a similar study carried out toward another starless
core, L1517B, believed to be at an even earlier stage of evolution than L1498
based on the deuterium fractionation and the absence of infall motions (Crapsi
et al., 2005). The goal is to compare chemical complexity measured toward
L1517B with that observed in L1544 and L1498, and to establish if the observed
trend of increasing chemical complexity is due to evolution (see Jiménez-Serra
et al., 2016; Jiménez-Serra et al., 2021). These cores, located in the Taurus
molecular cloud complex, show different observational signatures that provide
information about their stage of gravitational collapse. L1544 is classified
as a pre-stellar core because it shows clear evidence of gravitational
collapse toward its innermost regions as probed by the emission of N2H+
(Caselli et al., 2002; Redaelli et al., 2019). Its high deuterium
fractionation ($N_{\mathrm{N_{2}D^{+}}}/N_{\mathrm{N_{2}H^{+}}}=0.23\pm
0.04$), CO depletion factor ($f_{\mathrm{CO}}=14\pm 3$; Crapsi et al., 2005;
Redaelli et al., 2019), and central H2 density ($n_{\mathrm{H_{2}}}\sim
10^{7}$ cm-3; Caselli et al., 2022), are also consistent with this idea. L1498
presents signatures of infall motions in the outer envelope of the core as
revealed by the asymmetries observed in the line profiles of CS (Tafalla et
al., 2004) but its deuterium fractionation, CO depletion factor and central H2
density are lower than those measured toward L1544
($N_{\mathrm{N_{2}D^{+}}}/N_{\mathrm{N_{2}H^{+}}}=0.04\pm 0.01$,
$f_{\mathrm{CO}}=7.5\pm 2.5$, and $n_{\mathrm{H_{2}}}\simeq 9.4\times 10^{4}$
cm-3; Crapsi et al., 2005; Tafalla et al., 2004). For L1517B, the CO depletion
factor and deuterium fractionation are similar to those of L1498,
$f_{\mathrm{CO}}=9.5\pm 2.8$ and
$N_{\mathrm{N_{2}D^{+}}}/N_{\mathrm{N_{2}H^{+}}}$ = 0.06 $\pm$ 0.01 (Crapsi et
al., 2005). However, although L1517B shows a slightly higher central H2
density ($n_{\mathrm{H_{2}}}\simeq 2.2\times 10^{5}$ cm-3), the absence of
infall motions either toward the innermost regions or toward the envelope
(Tafalla et al., 2004), suggests that L1517B is at a younger evolutionary
stage than L1498 and L1544. Hence, L1517B, located at a distance of 159 pc
(Galli et al., 2019), would be the dynamically youngest of the three cores.
Table 1: Frequency ranges, velocity resolution and RMS noise level of our observations. Frequency | Resolution | RMS noise (mK)
---|---|---
(GHz) | (km s-1) | Dust peak | Methanol peak
78.2–80.0 | 0.18 | 4.8 | 3.9
81.4–83.3 | 0.18 | 4.2 | 3.5
83.3–85.2 | 0.17 | 2.8 | 3.3
86.7–88.5 | 0.17 | 3.0 | 3.5
93.5–95.7 | 0.15 | 4.1 | 3.4
95.8–96.8 | 0.15 | 10.8 | 10.9
97.1–99.0 | 0.15 | 4.1 | 3.6
99.1–100.9 | 0.15 | 3.0 | 3.5
102.4–104.2 | 0.14 | 4.1 | 4.7
109.2–111.0 | 0.13 | 10.1 | 11.0
Figure 2: Sample of the COM and COM precursor lines observed toward L1517B’s
dust continuum peak and their corresponding fits obtained with Madcuba (LTE,
in red) and Radex (non-LTE, in orange). See Sections 2 and 3.1 for more
details.
The paper is organized as follows. In Section 2, we describe the observations
carried out toward L1517B. In Section 3 we present the results of the analysis
of the COMs and COM precursor emission and report the values of the derived
excitation temperatures, column densities and molecular abundances. Section 4
compares the abundances obtained toward L1517B with those measured toward
L1498 and L1544. We also compare our results with those reported by Nagy et
al. (2019) and Scibelli et al. (2021) toward the young L1521E starless core.
This core has been found to be rich in COM emission despite its youth. In
Section 5, we present the modelling of the COMs and COM precursor chemistry of
the L1517B core, and compare the model predictions with the observations. In
Section 6, we compare the column densities ratios between the N-bearing
species HC3N and CH3CN derived toward L1517B, L1498, L1544, and L1521E, with
those obtained in protostellar systems, protoplanetary discs, and comets, and
discuss the observed discrepancies. Finally, in Section 7, we summarise our
conclusions.
## 2 Observations
The observations of the L1517B starless core were carried out from September
$30^{\mathrm{th}}$ to October $4^{\mathrm{th}}$ of 2020, with the Instituto de
Radioastronomía Milimétrica (IRAM) 30 m telescope (Granada, Spain). As for
L1544 and L1498, we observed two positions within the L1517B core: the
location of the dust peak and the position where the emission of methanol
peaks. These two positions have the equatorial coordinates (in J2000 system)
$\alpha$ = $4^{\mathrm{h}}55^{\mathrm{m}}17\aas@@fstack{s}6$, $\delta$ =
$30\degree 37^{\prime}44^{\prime\prime}$ for the dust continuum peak, and
$\alpha$ = $4^{\mathrm{h}}55^{\mathrm{m}}15\aas@@fstack{s}7$, $\delta$ =
$30\degree 38^{\prime}04^{\prime\prime}$ for the position of the methanol peak
(see Tafalla et al., 2004; Tafalla et al., 2006). The latter is located
$\sim$32′′ away from the dust peak (see Fig. 1), which corresponds to
$\sim$5000 au at a distance of 159 pc.
The high-sensitivity 3 mm spectra were obtained in frequency-switching mode
using a frequency throw of 7.14 MHz. The EMIR E090 receivers were tuned at
84.37 GHz and 94.82 GHz with rejections of $\geq$10 dB. The observed frequency
ranges are shown in Table 1. To identify possible weak spurious features in
the observed spectra, we carried out part of the observations by shifting
slightly the central frequencies by $\pm$20 MHz (see also Jiménez-Serra et
al., 2016). We used the narrow mode of the FTS spectrometer that provided a
spectral resolution of 49 kHz, equivalent to 0.13–0.18 km s-1 at 3 mm. Typical
system temperatures ranged between 75–110 K and the telescope beam size was
22–31$\,{}^{\prime\prime}$ between 78 and 111 GHz; as the beams are almost
gaussian and the dust and methanol peaks of L1517B are located $\sim$32′′ away
($>2\,\upsigma$), the contamination between both positions should be
negligible (less than 5 percent). The spectra were calibrated in units of
antenna temperature, $T_{\mathrm{A}}^{*}$, and converted into main beam
temperature, $T_{\mathrm{mb}}$, by using beam efficiencies of 0.81 at 79–101
GHz and of 0.78 at 102–111
GHz.111https://publicwiki.iram.es/Iram30mEfficiencies/ The root mean square
(RMS) noise level of the original observations ranged between 3 and 11 mK for
both observing positions (see Table 1), having similar values for each
frequency range than the ones obtained by Jiménez-Serra et al. (2016);
Jiménez-Serra et al. (2021) for L1544 and L1498.
Figure 3: Sample of the COM lines observed toward L1517B’s methanol peak and
their corresponding fits obtained with Madcuba (LTE, in red) and Radex (non-
LTE, in orange). See Sections 2 and 3.1 for more details. Acetaldehyde
(CH3CHO) transitions are quite noisy, but we could actually detect it because
we observed several ones. It seems that acetonitrile (CH3CN) may show a
transition, but we could not properly fit the two covered transitions neither
with Madcuba nor Radex; plus, the signal to noise is quite low (below 2 for
the shown transition and below 1 for the other one).
Our observations have covered the transitions of both O-bearing and N-bearing
COMs and COM precursors, summarised in Table blue2. For O-bearing species, we
have targeted methanol (CH3OH), methoxy (CH3O), tricarbon monoxide (CCCO),
ketene (H2CCO), formic acid (t-HCOOH), acetaldehyde (CH3CHO), methyl formate
(CH3OCHO), dimethylether (CH3OCH3), cyclopropenone (c-C3H2O), and propynal
(HCCCHO). As N-bearing COMs and precursors, we have observed cyanoacetylene
(HCCCN), isocyanoacetilene (HCCNC), vinyl cyanide (CH2CHCN), acetonitrile
(CH3CN), and methyl isocyanide (CH3NC).
The raw spectra have been analysed and reduced with Class, from the package
Gildas,222https://www.iram.fr/IRAMFR/GILDAS/ as well as with a Python pipeline
written specifically for this
purpose.333https://github.com/andresmegias/gildas-class-python/ This pipeline
fits baselines to the spectra using an iterative method that first masks the
strongest lines using sigma-clips and then applies rolling medians and rolling
averages, interpolating the masked regions with third order splines. Finally,
we used the software Madcuba (Martín et al., 2019) to search for the molecular
transitions of all targeted species, and to carry out the fitting of the
molecular line profiles under the assumption of local thermodynamic
equilibrium (LTE). The physical parameters derived in the fitting are the
molecular column density ($N_{\mathrm{obs}}$), excitation temperature
($T_{\mathrm{ex}}$), linewidth ($\Delta v$) and LSR radial velocity
($v_{\mathrm{LSR}}$). For this, we used the tool Slim (Spectral Line
Identification and Modelling) of Madcuba, employing the Cologne Database for
Molecular Spectroscopy (CDMS; Endres et al., 2016) and the Jet Propulsion
Laboratory (JPL) molecular catalogue (Pickett et al., 1998). In addition, for
the cases of CH3OH, CH3CN, and HCCCN, we used the non-LTE code Radex (Van der
Tak et al., 2007) to do the fitting of the lines.
## 3 Results
### 3.1 Detected transitions
Figs. 2 and 3 show the observed spectra of some representative transitions of
the COM and COM precursors detected toward L1517B, while Table blue2 lists all
the transitions covered in our observations with their derived line
parameters. The targeted transitions are the same as in Jiménez-Serra et al.
(2016) and Jiménez-Serra et al. (2021), and correspond to those expected to be
the brightest for an excitation temperature of $\sim$10 K.
Besides methanol, our spectra reveal the detection of other COMs and COM
precursor species in L1517B: CH3O, H2CCO, CH3CHO, HCCCN, HCCNC, CH3CN, and
CH3NC, although acetaldehyde (CH3CHO) was only detected toward the methanol
peak and acetonitrile (CH3CN) and methyl isocyanide (CH3NC) were only detected
toward the dust peak. More complex molecules such as CH3OCH3 or CH3OCHO were
not detected within our noise levels. All detected lines lie above the
$3\,\upsigma$ level in integrated intensity (area), where $1\,\upsigma$ is
calculated as $\Delta T(\Delta v\thinspace\updelta v)^{1/2}$, where $\Delta T$
is the RMS noise level, $\Delta v$ is the line width, and $\updelta v$ is the
velocity resolution of the spectrum (see Table 1). We are confident about the
detection of the transitions since their derived radial velocities correspond
to the $v_{\mathrm{LSR}}$ of the source ($\sim$5.8 km s-1). Moreover, except
for CH3OH A, CH3NC and HCCNC, we have measured at least two transitions above
the 3 $\upsigma$ level in integrated intensity for the detected species,
stressing the identification of the species since the linewidths are narrow
($\sim\,$0.3–0.5 km s-1). The level of line confusion at the targeted RMS
noise level is also very low, as commonly found in starless/pre-stellar cores.
For the non-detections, we provide upper limits to the integrated intensities
by using 3 $\Delta T(\Delta v\thinspace\updelta v)^{1/2}$.
Table 2: Most of the COMs and COM precursors transitions covered in our L1517B
observations and their derived line parameters.
| | | Dust peak | Methanol peak
---|---|---|---|---
Species | Line | Frequency | Area a | Linewidth | LSR velocity b | S/N c | Area a | Linewidth | LSR velocity b | S/N c
| | (MHz) | (mK km s-1) | (km s-1) | (km s-1) | | (mK km s-1) | (km s-1) | (km s-1) |
CH3OH | $2_{0,2}\rightarrow 1_{0,1}$ A | 96741.371 | 206.1 $\pm$ 1.9 | 0.278 $\pm$ 0.004 | 5.785 $\pm$ 0.002 | 109 | 241.1 $\pm$ 2.0 | 0.277 $\pm$ 0.005 | 5.788 $\pm$ 0.002 | 121
| $2_{1,2}\rightarrow 1_{1,1}$ A | 95914.310 | < 7 | … | … | … | < 7 | … | … | …
| $2_{1,1}\rightarrow 1_{1,0}$ A | 97582.798 | < 2.4 | … | … | … | < 2.4 | … | … | …
| $2_{1,2}\rightarrow 1_{1,1}$ E | 96739.358 | 150.4 $\pm$ 1.9 | 0.262 $\pm$ 0.008 | 5.786 $\pm$ 0.003 | 80 | 181.6 $\pm$ 2.0 | 0.262 $\pm$ 0.003 | 5.785 $\pm$ 0.003 | 92
| $2_{0,2}\rightarrow 1_{0,1}$ E | 96744.545 | 9.7 $\pm$ 1.9 | 0.244 $\pm$ 0.012 | 5.84 $\pm$ 0.04 | 5.1 | 11.2 $\pm$ 2.0 | 0.23 $\pm$ 0.08 | 5.78 $\pm$ 0.03 | 5.6
| $2_{1,1}\rightarrow 1_{1,0}$ E | 96755.501 | < 7 | … | … | … | < 7 | … | … | …
CH3O d | $F=1\rightarrow 0,\,\Lambda=-1$ | 82455.980 | $<4$ | … | … | … | $<3$ | … | … | …
| $F=2\rightarrow 1,\,\Lambda=-1$ | 82458.252 | $5.4\pm 1.2$ | $0.41\pm 0.08$ | $5.81\pm 0.03$ | 4.5 | $4.2\pm 0.9$ | $0.39\pm 0.08$ | $5.82\pm 0.03$ | 4.7
| $F=2\rightarrow 1,\,\Lambda=+1$ | 82471.825 | $5.4\pm 1.2$ | $0.41\pm 0.08$ | $5.81\pm 0.03$ | 4.5 | $4.2\pm 0.9$ | $0.39\pm 0.08$ | $5.82\pm 0.03$ | 4.7
| $F=1\rightarrow 0,\,\Lambda=+1$ | 82524.180 | $<4$ | … | … | … | $<3$ | … | … | …
CCCO | $10\rightarrow 9$ | 96214.813 | $<7$ | … | … | … | $<7$ | … | … | …
t-HCOOH | $1_{1,1}\rightarrow 0_{0,0}$ | 87926.863 | $<1.9$ | … | … | … | $<2.2$ | … | … | …
H2CCO | $4_{1,3}\rightarrow 3_{1,2}$ o- | 81586.299 | $11.1\pm 1.3$ | $0.53\pm 0.10$ | $5.89\pm 0.04$ | 8.5 | $12.6\pm 1.2$ | $0.70\pm 0.25$ | $5.95\pm 0.11$ | 10
| $5_{1,5}\rightarrow 4_{1,4}$ o- | 100094.510 | $9.9\pm 0.9$ | $0.53\pm 0.10$ | $5.89\pm 0.04$ | 11 | $12.6\pm 1.2$ | $0.70\pm 0.25$ | $5.95\pm 0.11$ | 10
| $4_{0,4}\rightarrow 3_{0,3}$ p- | 80832.189 | $21\pm 4$ | $0.53\pm 0.10$ | $5.89\pm 0.04$ | 5.2 | $17\pm 4$ | $0.70\pm 0.25$ | $5.95\pm 0.11$ | 4.2
CH3OCHO | $7_{2,6}\rightarrow 6_{2,5}$ A | 84454.754 | $<2.0$ | … | … | … | $<2.5$ | … | … | …
CH3OCH3 | $3_{1,3}\rightarrow 2_{0,2}$ EE | 82650.180 | $<3$ | … | … | … | $<2.4$ | … | … | …
| $4_{1,4}\rightarrow 3_{0,3}$ EE | 99326.000 | $<2.0$ | … | … | … | $<2.2$ | … | … | …
CH3CHO | $5_{1,4}\rightarrow 4_{1,3}$ A | 98900.944 | $<3$ | … | … | … | $4.0\pm 0.8$ | $0.21\pm 0.05$ | $5.765\pm 0.018$ | 5.0
| $5_{0,5}\rightarrow 4_{0,4}$ A | 95963.459 | $<7$ | … | … | … | $<6$ | … | … | …
| $4_{1,3}\rightarrow 3_{1,2}$ A | 79150.166 | $<3$ | … | … | … | $3.9\pm 0.7$ | $0.21\pm 0.05$ | $5.765\pm 0.018$ | 5.6
| $4_{1,3}\rightarrow 3_{1,2}$ E | 79099.313 | $<3$ | … | … | … | $3.9\pm 0.7$ | $0.21\pm 0.05$ | $5.765\pm 0.018$ | 5.6
c-C3H2O | $6_{1,6}\rightarrow 5_{1,5}$ | 79483.519 | $<3$ | … | … | … | $<3$ | … | … | …
HCCCHO | $9_{0,9}\rightarrow 8_{0,8}$ | 83775.816 | $<2.0$ | … | … | … | $<2.5$ | … | … | …
CH3CN | $6_{0}\rightarrow 5_{0}$ | 110383.500 | 6.4 $\pm$ 1.9 | 0.13 $\pm$ 0.13 | 5.902 $\pm$ 0.023 | 3.4 | < 6 | … | … | …
| $6_{1}\rightarrow 5_{1}$ | 110381.372 | 11.8 $\pm$ 1.9 | 0.24 $\pm$ 0.08 | 5.81 $\pm$ 0.03 | 6.2 | < 6 | … | … | …
| $6_{2}\rightarrow 5_{2}$ | 110374.989 | < 6 | … | … | … | < 6 | … | … | …
CH3NC | $5_{0}\rightarrow 4_{0}$ | 100526.541 | $3.6\pm 0.7$ | $0.4\pm 0.3$ | $5.87\pm 0.13$ | 5.1 | $<2.2$ | … | … | …
CH2CHCN | $9_{0,9}\rightarrow 8_{0,8}$ | 84945.988 | $<2.0$ | … | … | … | $<2.5$ | … | … | …
| $9_{1,9}\rightarrow 8_{1,8}$ | 83207.496 | $<3$ | … | … | … | $<2.4$ | … | … | …
| $10_{0,10}\rightarrow 9_{0,9}$ | 94276.625 | $<3$ | … | … | … | $<2.4$ | … | … | …
HCCCN | $9\rightarrow 8$ | 81881.461 | 693.6 $\pm$ 1.0 | 0.395 $\pm$ 0.001 | 5.824 $\pm$ 0.001 | 694 | 346.7 $\pm$ 0.9 | 0.383 $\pm$ 0.002 | 5.816 $\pm$ 0.001 | 385
| $11\rightarrow 10$ | 100076.392 | 286.5 $\pm$ 0.7 | 0.318 $\pm$ 0.001 | 5.842 $\pm$ 0.001 | 409 | 96.3 $\pm$ 0.8 | 0.290 $\pm$ 0.003 | 5.837 $\pm$ 0.001 | 120
HCCNC | $8_{8}\rightarrow 7_{8}$ | 79484.131 | $16.1\pm 1.1$ | $0.31\pm 0.15$ | $5.83\pm 0.07$ | 15 | $6.9\pm 0.9$ | $0.29\pm 0.16$ | $5.86\pm 0.06$ | 7.7
| $10_{10}\rightarrow 9_{10}$ | 99354.250 | $6.8\pm 0.7$ | $0.31\pm 0.15$ | $5.83\pm 0.07$ | 9.7 | $<2.3$ | … | … | …
Line profiles were fitted using Madcuba, except for methanol (CH3OH),
cyanoacetilene (HCCCN) and acetonitrile (CH3CN), where we used Class (see
Section 3.2 for details). (a) Uncertainties in the line area are calculated as
$\Delta T(\Delta\nu\,\updelta v)^{1/2}$, with $\Delta T$ the RMS noise nevel,
$\Delta v$ the line width, and $\updelta v$ the velocity resolution of the
spectrum. Similarly, upper limits are calculated as 3 $\Delta T(\Delta
v\,\updelta v)^{1/2}$. (b) LSR stands for _local standard of rest_. (c) This
refers to the signal to noise ratio in integrated intensity area. If a certain
value has no uncertainty, this means that it had to be fixed so that Madcuba
could fit the LTE model. (d) Hyperfine components of the $N=1\rightarrow
0,\,K=0,\,J=3/2\rightarrow 1/2$ transition of CH3O.
From Figs. 2 and 3, we find that the line intensities vary for each position:
for the N-bearing species (CH3CN, HCCNC, and HCCCN), the emission is brighter
toward the core’s centre with respect to the methanol peak. On the contrary,
for O-bearing species (CH3OH and H2CCO) we find that the emission level is
similar for both positions. Note that in the cases of CH3OH, CH3CN and HCCCN
the fits were obtained with Radex instead of Madcuba. In these cases the lines
were also fitted with independent gaussians using Class, but they are not
shown in Figs. 2 and 3, as they are independent fits for each transition and
there are only made to obtain the parameters of the lines shown in Table
blue2.
Table 3: Excitation/kinetic temperatures (_$T$_), column densities
(_$N_{\mathrm{obs}}$_), and abundances (_$\chi_{\mathrm{obs}}\thinspace$_) of
COMs and COM precursors toward the dust and methanol peaks in L1517B.
| Dust peak | Methanol peak
---|---|---
Molecule | _$\boldsymbol{T}\thinspace(\mathrm{K})$_ | _$\boldsymbol{N}_{\mathrm{\mathbf{obs}}}\thinspace(\mathrm{cm}{}^{-2})$_ | _$\boldsymbol{\chi}_{\mathbf{\mathrm{\mathbf{obs}}}}\thinspace$_ | _$\boldsymbol{T}\thinspace(\mathrm{K})$_ | _$\boldsymbol{N}_{\mathrm{\mathbf{obs}}}\thinspace(\mathrm{cm}{}^{-2})$_ | _$\boldsymbol{\chi}_{\mathrm{\mathbf{obs}}}\thinspace$_
CH3OH A | $10.0$ | $(4.29\pm 0.11)\cdot 10^{12}$ | $1.23_{-0.16}^{+0.21}\cdot 10^{-10}$ | $10.0$ | $(4.96\pm 0.12)\cdot 10^{12}$ | $5.2_{-0.5}^{+0.6}\cdot 10^{-10}$
CH3OH E | $10.0$ | $4.59_{-0.25}^{+2.63}\cdot 10^{12}$ | $1.5_{-0.3}^{+0.7}\cdot 10^{-10}$ | $10.0$ | $(5.62\pm 0.20)\cdot 10^{12}$ | $5.9_{-0.6}^{+0.7}\cdot 10^{-10}$
CH3OH | $10.0$ | $8.9_{-0.3}^{+2.6}\cdot 10^{12}$ | $2.7_{-0.5}^{+0.7}\cdot 10^{-10}$ | $10.0$ | $(1.058\pm 0.023)\cdot 10^{13}$ | $1.10_{-0.10}^{+0.12}\cdot 10^{-9}$
CH3O | $10\pm 4$ | $(2.8\pm 0.9)\cdot 10^{11}$ | $8.0_{-2.5}^{+2.8}\cdot 10^{-12}$ | $7\pm 4$ | $(1.8\pm 0.6)\cdot 10^{11}$ | $1.9_{-0.6}^{+0.7}\cdot 10^{-11}$
CH3OCHO | $7.7$ | $<9\cdot 10^{11}$ | $<4\cdot 10^{-11}$ | $9.7$ | $<1.0\cdot 10^{12}$ | $<1.5\cdot 10^{-10}$
CH3OCH3 | $7.7$ | $<1.0\cdot 10^{12}$ | $<5\cdot 10^{-11}$ | $9.7$ | $<5\cdot 10^{11}$ | $<7\cdot 10^{-11}$
CH3CHO | $7.7$ | $<2.5\cdot 10^{11}$ | $<1.2\cdot 10^{-11}$ | $9.7$ | $(2.1\pm 0.4)\cdot 10^{11}$ | $2.2_{-0.4}^{+0.5}\cdot 10^{-11}$
t-HCOOH | $7.7$ | $<3\cdot 10^{12}$ | $<1.5\cdot 10^{-10}$ | $9.7$ | $<1.9\cdot 10^{12}$ | $<3\cdot 10^{-10}$
c-C3H2O | $7.7$ | $<1.5\cdot 10^{10}$ | $<7\cdot 10^{-13}$ | $9.7$ | $<5\cdot 10^{10}$ | $<7\cdot 10^{-12}$
H2CCO | $7.7\pm 1.0$ | $(8.4\pm 1.3)\cdot 10^{11}$ | $2.4_{-0.4}^{+0.6}\cdot 10^{-11}$ | $10\pm 3$ | $(7.2\pm 1.9)\cdot 10^{11}$ | $7.6_{-2.0}^{+2.2}\cdot 10^{-11}$
CCCO | $7.7$ | $<2.4\cdot 10^{11}$ | $<1.1\cdot 10^{-11}$ | $9.7$ | $<1.2\cdot 10^{12}$ | $<1.8\cdot 10^{-10}$
HCCCHO | $7.7$ | $<1.6\cdot 10^{11}$ | $<8\cdot 10^{-12}$ | $9.7$ | $<1.9\cdot 10^{11}$ | $<3\cdot 10^{-11}$
HCCNC | $6.7$ | $(2.4\pm 1.0)\cdot 10^{11}$ | $(7\pm 3)\cdot 10^{-12}$ | $5.3$ | $(1.9\pm 0.8)\cdot 10^{11}$ | $1.9_{-0.8}^{+0.9}\cdot 10^{-11}$
CH2CHCN | $6.7$ | $<4\cdot 10^{10}$ | $<1.8\cdot 10^{-12}$ | $5.3$ | $<3\cdot 10^{10}$ | $<5\cdot 10^{-12}$
CH3NC | $6.7$ | $(1.6\pm 0.9)\cdot 10^{10}$ | $(5\pm 3)\cdot 10^{-13}$ | $5.3$ | $<2.3\cdot 10^{10}$ | $<3\cdot 10^{-12}$
CH3CN | $10.0$ | $(2.1\pm 0.6)\cdot 10^{11}$ | $6.0_{-1.9}^{+2.1}\cdot 10^{-12}$ | $10.0$ | $<2.0\cdot 10^{11}$ | $<3\cdot 10^{-11}$
HCCCN | $10.0$ | $(5.16\pm 0.05)\cdot 10^{12}$ | $1.48_{-0.18}^{+0.25}\cdot 10^{-10}$ | $10.0$ | $3.72_{-0.13}^{+0.14}\cdot 10^{12}$ | $(3.9\pm 0.4)\cdot 10^{-10}$
Temperatures ($T$) refer to excitation temperatures ($T_{\mathrm{ex}}$) for
all the species except for methanol (CH3OH), cyanoacetilene (HCCCN) and
acetonitrile (CH3CN), where they refer to kinetic temperatures
($T_{\mathrm{kin}}$). We used Madcuba to derive the molecular parameters from
the observations except for methanol, cyanoacetilene and acetonitrile, where
we used Radex. Molecular abundances were calculated using an $\mathrm{H}_{2}$
column density of $(3.5\pm 0.5)\times 10^{22}$ cm-2 for the dust continuum
peak and of $(9.6\pm 1.0)\times 10^{21}$ cm-2 for the position of the methanol
peak. For the non-detections and also for some detections, we had to fix the
excitation temperature ($T_{\mathrm{ex}}$) so that Madcuba could fit the
column density ($N_{\mathrm{obs}}$).
### 3.2 Molecular column densities and excitation temperatures
#### 3.2.1 LTE analysis
Using the tool Slim from Madcuba, we performed the LTE fitting to the observed
line profiles for each species, obtaining the parameters shown in Table blue2.
As in most cases we detected several lines of the same molecular species, this
fitting allows us to obtain its column density ($N_{\mathrm{obs}}$) and
excitation temperature ($T_{\mathrm{ex}}$). These parameters are also shown in
Table blue3.
However, there were cases in which the Madcuba fitting algorithm did not
converge. This can be due because the signal-to-noise ratio is not high
enough, or also because we only have one transition detected. In these cases,
we had to fix one or more parameters, so that the algorithm could converge.
Similarly, for non-detections we had to fix the excitation temperature so that
Madcuba could fit an upper limit for the column density.
In general, when we needed to fix the temperature, we used the fitted value
for H2CCO for oxygen-bearing species, and the fitted one for HCCCN for
nitrogen-bearing species, as these are molecules with several transitions
which also have high signal-to-noise ratio. These values are, for H2CCO, 7.7 K
and 9.7 K for the dust and methanol peaks, respectively, and, for HCCCN, 6.7 K
and 5.3 K. For cases where we had to fix the central velocity of the line, we
used a value of $v_{\mathrm{LSR}}$ = 5.8 km s-1.
For methyl isocyanide (CH3NC) toward the core centre and isocyanoacetylene
(HCCNC) toward the methanol peak, we only detected 1 transition, so we had to
fix the excitation temperature to $T_{\mathrm{ex}}$ = 6.7 K for CH3NC and to
$T_{\mathrm{ex}}$ = 5.3 K for HCCNC (which are the corresponding values for
HCCCN at the dust and methanol peaks, respectively).
For ketene (H2CCO), we calculated the ortho-to-para ratio (we have detected
two ortho transitions and one para transition), by carrying out a separate fit
of the transitions and by dividing the resulting column densities of the ortho
and para species. We obtain an ortho-to-para ratio of $1.0_{-0.3}^{+0.4}$ for
the position of the dust peak and $0.9_{-0.4}^{+0.5}$ for the methanol peak.
This ratio is different to what has been previously observed towards the
molecular cloud TMC-1 and the pre-stellar core L1689B, where the ratio was
$\sim$3 (Ohishi et al., 1991; Bacmann et al., 2012). An exact value of 3 would
correspond to the statistical ratio due to the nuclear spin degeneracy.
However, lower values are also possible for the lowest temperatures, like in
the case of formaldehyde (H2CO), where an ortho-to-para ratio of 1 would
indicate that this molecule is formed under thermal equilibrium at a
temperature of $\sim$10 K (Kahane et al., 1984). We note, however, that we
have only detected two ortho lines and one para line of ketene, and therefore
the derived ortho-to-para ratios can be subject to significant uncertainties.
#### 3.2.2 Non-LTE analysis
For some cases, in particular for methanol (CH3OH), cyanoacetilene (HCCCN) and
acetonitrile (CH3CN), we have decided to use the non-LTE code Radex (Van der
Tak et al., 2007) to infer the column density $N_{\mathrm{obs}}$ of the
species instead of Madcuba. In the case of methanol and acetonitrile, we did
it because Madcuba could not fit properly all the transitions, and the errors
were greater than $3\,\upsigma$. For the case of cyanoacetilene (HCCCN), we
opted for Radex because it allows us to estimate the H2 number density, as we
have two transitions. As input parameters for these calculations, we assumed
the linewidth obtained with the Madcuba fit ($\sim$0.28 km s-1) and a kinetic
temperature of 10 K (as inferred for all radii across the L1517B starless
core; see Tafalla et al. (2004) and Section 5 below). Apendix A describes the
procedure used to derive the phyisical properties and their uncertainties from
the modeling.
In the case of CH3OH, we have to distinguish between the A and E species,
making a separate fit for each molecule. We used the collisional coefficients
with H2 given by Rabli & Flower (2010). In addition to deriving the column
densities reported in Table blue3, we also derived the H2 number density from
the observations of methanol E, as this parameter can be determined by the
intensity ratio of the detected lines. Unfortunately, for the case of methanol
A we only detected one transition so the column density was estimated by
assuming the H2 number density obtained by Tafalla et al. (2004): $2.20\times
10^{5}$ cm-3 for L1517B core centre and $1.23\times 10^{5}$ cm-3 for the
location of the methanol peak. For methanol E we could derive the H2 number
density, although the uncertainties are high, obtaining values of
$5_{-4}^{+8}\times 10^{4}$ cm-3 for the core’s centre and
$2.0_{-0.6}^{+0.9}\times 10^{5}$ cm-3 for the methanol peak. These values are
consistent, within the uncertainties, with those derived by Tafalla et al.
(2004), as we have to take into account the possible uncertainties in the
determination of the density profiles from the absorption coefficient of the
dust. As for the excitation temperatures, they are close to the assumed
kinetic temperature of 10 K, consistent with those expected for the derived H2
densities, although this is biased by the fact that the lowest temperature for
which we have collisional rates is 10 K.
For CH3CN, we fixed the H2 number density to the values from Tafalla et al.
(2004) (as the signal-to-noise ratio was not high enough to get a good fit).
For the core’s centre, one of the lines could be fitted within $2\,\upsigma$,
providing a better fit than the one by Madcuba. For the methanol peak, we did
not detect any transition, so we obtained an upper limit to its column density
from Radex.
For HCCCN, we have used the two transitions to derive the column density and
the H2 number density towards both positions. We used the collisonal
coefficients with H2 given by Faure et al. (2016). Fig. 12 shows the results
obtained from the non-LTE analysis for the core’s centre. As expected the
derived H2 densities and column densities are related. We obtained the HCCCN
column densities shown in Table blue3 and H2 densities of
$1.80_{-0.04}^{+0.04}\times 10^{5}$ cm-3 and $5.56_{-0.18}^{+0.19}\times
10^{4}$ cm-3 for the dust and methanol peaks, respectively. Although these
results present lower uncertainties than those of methanol, our H2 densities
are still consistent with the prediction by Tafalla et al. (2004) taking into
account the beam size of our observations and the uncertainties in the
determination of the density from dust emission, as we previously discussed.
In the following sections, we use the values from Tafalla et al. (2004)
because they are more direct than the derivation through Radex, and because
the authors assume the same dust emissivity ($\kappa=0.005$ cm2 g-1 for 1.2
mm) as that considered by Crapsi et al. (2005) for inferring the CO depletion
factor of L1517B, which has been employed in our astrochemical simulations
(Section 5).
### 3.3 Molecular abundances
Once we know the column densities of COMs and COM precursors detected toward
L1517B, we can compute the molecular abundances of these species by using the
H2 column densities measured toward the positions of the dust and methanol
peaks (Table blue3). For the position of the dust peak, the H2 column density
of $(3.5\pm 0.5)\times 10^{22}$ cm-2 was obtained using the 1.2 mm continuum
emission observed by Tafalla et al. (2004) with the Mambo 1 mm bolometer array
at IRAM 30 m telescope. For the position of the methanol peak, however, we
employed the data from the Herschel space telescope at 0.25, 0.35 and 0.5 mm
assuming a dust optical depth index of $\beta=1.5$ (Spezzano et al., 2016);
the derived H2 column density is $(9.6\pm 1.0)\times 10^{21}$ cm-2. As
discussed in Jiménez-Serra et al. (2021), this is the best procedure to probe
the total H2 column density respectively toward the innermost regions (with
bolometers at 1.2 mm) and outer shells (with Herschel) in starless cores.
Figure 4: Bar plot of the abundances of different COMs and COM precursors
measured for the cores L1517B, L1498 and L1544, toward the dust peak (lighter
colors) and the methanol peak (darker colors) for each of them. Upper limits
are indicated both with arrows at the top of the corresponding bar and with
stripes. Abundances for L1544 are taken from Jiménez-Serra et al. (2016) and
Jiménez-Serra et al. (2021), while for L1498 they are taken from Jiménez-Serra
et al. (2021). Note that this is not a cumulative bar plot, so the upper edge
of each bar, for both colors (that is, for both the centre and the methanol
peak, for each core), indicates the abundance of the corresponding species.
From Table blue3, we can see that the abundance of the detected species is
enhanced toward the position of the methanol peak with respect to the core’s
centre. For methanol (CH3OH), its abundance is enhanced by a factor of
$\sim$4, while for the rest of detected species in both positions (CH3O,
H2CCO, HCCNC, and HCCCN) the factor of enhancement is $\sim$3\. Acetaldehyde
(CH3CHO) is only detected in the methanol peak, the ratio would be
$\gtrsim$2\. As for acetonitrile (CH3CN) and methyl isocianide (CH3NC) we only
find lines toward the dust peak, and the ratio would be $\lesssim\,$6\.
Therefore, for these three molecules, the lower/upper limits for the factor of
enhancement toward the methanol peak are consistent with those of the
molecules detected in both positions. Additionally, the ratio of the
abundances of methanol A and E species toward the dust and methanol peak is,
respectively, $0.91_{-0.32}^{+0.08}$ and $0.88_{-0.04}^{+0.04}$, which is
similar to what has been observed in the L1544 and L1498 cores (Jiménez-Serra
et al., 2016; Jiménez-Serra et al., 2021) and to the expected values of
$\sim\,$0.7–1.0 (Wirström et al., 2011).
## 4 Comparison with other cores
In this Section, we compare our results obtained for the starless core L1517B
with those of the L1544 and L1498 cores (see Jiménez-Serra et al., 2016;
Jiménez-Serra et al., 2021) and also with those of the L1521E core (see Nagy
et al., 2019; Scibelli et al., 2021).
### 4.1 Comparison with L1498 and L1544
For L1517B, L1498 and L1544, the same molecular species have been observed
toward two positions (i.e., the dust peak and the methanol peak for each of
them), which allows us to obtain information about the radial distribution of
the abundances of COM and COM precursors within the cores.
Fig. 4 presents the abundances of the COM and COM precursor molecules observed
toward these three cores and for the two measured positions. The abundances
measured toward the dust and methanol peaks are plotted respectively in yellow
and red for the L1544 pre-stellar core (Jiménez-Serra et al., 2016), in light
and dark blue for the L1498 starless core (Jiménez-Serra et al., 2021), and in
light and dark purple for the L1517B starless core (this work).
Consistently with L1498 and L1544, the abundances of the species detected
toward L1517B tend to be higher toward the position of the methanol peak than
toward the dust peak (t-HCOOH toward L1544 is an exception). Fig. 4 also shows
that the L1517B core presents a lower number of detections than L1544 and a
similar number to L1498. For the N-bearing species, we have detected only
HCCCN and HCCNC toward both positions of L1517B from a total of 5 targeted
species, although we have found CH3CN and CH3NC toward the dust peak. A
significant difference with respect to L1498 and L1544 is that vinyl cyanide
(CH2CHCN) is not detected toward L1517B, showing upper limits to its abundance
that are factors $\gtrsim$10 lower than the abundances measured toward L1498
and L1544. Following the same trend, for O-bearing species only four COMs and
COM precursor species (CH3OH, CH3O, CH3CHO, and H2CCO) out of 10 have been
detected toward L1517B, and in the case of CH3CHO it is only detected toward
the methanol peak.
If we compare in detail L1517B and L1498, we can see that the first one has a
level of chemical complexity somewhat lower than L1498. Firstly, the
abundances of the COMs and COM precursors detected toward L1517B tend to show
lower abundances than (or are comparable to) those measured toward L1498.
Secondly, if we count the total number of detections for any position, we
obtain 13 detections for L1517B and 14 for L1498, which may seem a small
difference. However, if we consider the detections and non-detections for each
position, the lower level of chemical complexity for the L1517B starless core
becomes more apparent.
To quantitatively measure this we have calculated, for each pair of cores, the
fraction of molecules with the same state of detection (detected or not
detected). That is, for each source, we build a vector with one entry for each
molecule and position (15 molecules $\times$ 2 positions, 30 entries), whose
value is 1 if there is a detection and 0 if not. Then, we build another vector
with the same number of entries, that are 1 if the two input entries (of the
two cores) have the same value and 0 otherwise. Finally, we sum up all the
elements of this vector and divide it by the number of entries, in order to
normalize it. We call the resulting number the fractional similarity, whose
values lie between 0 and 1. In this way, if for each position and molecule
both entries have the same value toward both sources (either a detection or a
non-detection), the fractional similarity is 1. On the contrary, if for each
position and molecule the entries are different, the resulting similarity is
0. If we compute the fractional similarity between L1517B and L1498 we obtain
$0.70\pm 0.03$.444The value used for the uncertainty is $1/30\simeq 0.03$,
which is the difference in the fractional similarity produced by one entry
that is different in the two vectors. This indicates that, although the two
cores are similar in terms of number of detections, they are truly different
regarding their level of chemical complexity since the similarity, clearly
less than 1, is produced by 9 different entries in the vectors (that is, in
the detections for each molecule and position).
For the sake of completeness, we have also computed the fractional similarity
between L1544 and L1517B, which is $0.57\pm 0.03$, and between L1544 and
L1498, which is $0.43\pm 0.03$. This indicates that the three cores show a
significantly different level of chemical complexity in terms of detections
and non detections, and the two more similar cores would be L1498 and L1517B.
Table 4: Weighted geometric mean of the abundance with respect to
$\mathrm{H}_{2}$, $\left\langle\chi_{\mathrm{obs}}\right\rangle$, and weighted
mean molecular mass, $\left\langle m_{\mathrm{molec}}\right\rangle$ for the
targeted species in L1517B, L1498, and L1544.
Source | $\boldsymbol{\left\langle\chi_{\mathrm{obs}}\right\rangle}$ $\left(\times 10^{-12}\right)$ | $\boldsymbol{\left\langle m_{\mathrm{molec}}\right\rangle}$ $\left(\mathrm{g/mol}\right)$
---|---|---
L1517B | dust peak | $7.1_{-1.5}^{+1.5}$ | $15_{-3}^{+3}$ | $45.5_{-0.3}^{+0.3}$ | $45.61_{-0.27}^{+0.20}$
meth. peak | $23_{-5}^{+5}$ | $45.7_{-0.4}^{+0.3}$
L1498 | dust peak | $10.5_{-2.1}^{+2.1}$ | $32_{-5}^{+5}$ | $46.18_{-0.24}^{+0.18}$ | $46.25_{-0.17}^{+0.15}$
meth. peak | $53_{-10}^{+10}$ | $46.32_{-0.24}^{+0.23}$
L1544 | dust peak | $19_{-4}^{+4}$ | $30_{-4}^{+4}$ | $46.8_{-0.4}^{+0.3}$ | $46.46_{-0.21}^{+0.15}$
meth. peak | $42_{-7}^{+7}$ | $46.13_{-0.19}^{+0.16}$
The weights for the mean abundance are the molecular masses for each molecule.
The weights for the mean molecular mas are the abundance for each molecule in
logarithmic scale. The values on the second columns for each magnitude are the
arithmetic means of the magnitudes for both positions. To deal with
uncertainty propagation and upper limits we made simulations with statistical
distributions (see Appendix B).
We have used two other methods to evaluate the level of chemical complexity
toward the three cores. First, we have computed for each source and position
the mean molecular mass of the species weighted by the abundances in a
logarithmic scale.555We used logarithmic weights due to the range of the
values of our abundances, which encloses several orders of magnitude. This can
be viewed as an estimate of the level of complexity of each source and
position, as COMs tend to have more atoms with increasing complexity, and thus
a greater molecular mass. From Table 4, we find that the lowest value of the
mean molecular mass is obtained for L1517B, while the greatest one is for
L1544, with L1498 being close to it. This would indicate that L1517B presents
the lowest level of chemical complexity of the three cores, followed by L1498
and L1544. Additionally, we also computed for each source and position the
geometric mean of the abundance of the species weighted by the molecular
mass.666We used the geometric mean due to the range of the values of our
abundances. The resulting value represents a measure of the amount of the more
complex species in the core, as we are giving larger weights in the mean to
more complex species. As shown by Table 4, L1517B shows the lowest mean
abundance, followed by L1498 and L1544, which present barely the same
quantity. All of this indicates that the level of chemical complexity in the
L1517B starless core seems to be lower than that of L1498, and of L1544.
From all these results, we suggest that L1517B is in a less chemically evolved
stage than L1498 and L1544. This is supported by the deuterium fractionation
of L1517B, similar to that of L1498, but L1517B shows no evidence of infall
motions (Crapsi et al., 2005; Tafalla et al., 2004). Additionally, L1517B
shows the lowest level of chemical complexity within the three cores,
presenting only the simplest COMs and COM precursors and with lower abundances
in most of the cases.
If we take into account the possible time evolution between the three cores
(with L1517B being the youngest and L1544 the oldest), we propose a scenario
in which the N-bearing molecules would form first in starless cores. Then, as
the cores evolve, they would accrete gas from the surrounding molecular cloud,
becoming denser and yielding a strong depletion of carbon monoxide (CO). At
the pre-stellar core stage, the catastrophic depletion of CO takes place,
triggering the formation of O-bearing COMs and COM precursors as observed in
L1544 (Vasyunin et al., 2017). Therefore, chemical complexity would increase
over time. In the next section, we explore this scenario by modelling the
chemistry of N-bearing and O-bearing COMs and COM precursors toward L1517B.
### 4.2 Comparison with L1521E
Figure 5: Bar plot of the abundances of different COMs and COM precursors
measured for the young starless cores L1517B (in light and dark purple) and
L1521E (in light and dark green). For L1517B, the values are as in Figure 4.
For L1521E, the abundances shown in dark green refer to the bulk COM emission
observed by Scibelli et al. (2021) toward L1521E with a beam size of
$\sim$70′′. The large beam includes not only the dust peak position but also
the methanol ring (peak located at 22–29$\,{}^{\prime\prime}$ from L1521E’s
dust peak). Abundances from L1521’s dust peak are shown in light green and are
taken from Nagy et al. (2019). Upper limits are indicated with both arrows and
stripes.
L1521E is another well-studied starless core located in the Taurus molecular
cloud. Like L1517B, L1521E is also considered to be young. Indeed, L1521E
shows no evidence of infall motions, and has a central density of $2.7\times
10^{5}$ cm-3 (Tafalla & Santiago, 2004). Its CO depletion factor is
$f_{\mathrm{CO}}=4.3\pm 1.6$ (Nagy et al., 2019), lower than those of L1517B
and L1498. Although its deuterium fractionation
$N_{\mathrm{N_{2}D^{+}}}/N_{\mathrm{N_{2}H^{+}}}$ has not been measured yet,
several authors agree in the fact that this core is young (e.g., Tafalla &
Santiago 2004; Kong et al. 2015; Scibelli et al. 2021). Nagy et al. (2019)
observed the molecular line emission toward the dust peak of this core
(including some COMs and COM precursors) using the IRAM 30 m telescope. More
recently, Scibelli et al. (2021) detected four COMs toward L1521E (CH3OCHO,
CH3OCH3, CH3CHO, and CH2CHCN) using the 12 m ARO (Arizona Radio Observatory)
telescope, with a much larger beam size ($\sim$70′′ versus
22–31$\,{}^{\prime\prime}$ for the IRAM 30 m telescope). This implies that
while the observations of Nagy et al. (2019) would cover the location of just
the dust peak, the observations of Scibelli et al. (2021) would cover both the
dust peak and the methanol ring, as its methanol peak is found
22–29$\,{}^{\prime\prime}$ away from the core center (Nagy et al., 2019). We
thus refer to the observations of Scibelli et al. (2021) as those performed
toward the bulk of the core.
Among the 15 targeted species measured toward L1517B, there are 9 of them that
have also been targeted toward L1521E: six toward the dust peak and four for
the bulk of the core (with just one molecule in common for both positions).
Fig. 5 reports the abundances of the common species observed toward L1517B and
L1521E. For the data under the bulk label, we use the values derived by
Scibelli et al. (2021) assuming two source size cases: the lower limits of the
error bars refer to the values obtained assuming a beam filling factor of 1,
while the error bars upper limits refer to the abundances obtained with the
best-fit source size of 35′′. Note, however, that vinyl cyanide (CH2CHCN) is
an exception since its abundance was only derived assuming a beam filling
factor of 1 (the source size for this molecule could not be estimated;
Scibelli et al., 2021).
In general, there is a good agreement between the abundances measured toward
the dust peak position of the two cores (see light purple and light green bars
in Fig. 5), with the sole exception of cyclopropenone (c-C3H2O). In contrast,
the differences are bigger between the bulk abundances of L1521E and the
abundances measured toward the dust and methanol peaks of L1517B. However, if
we take into account the uncertainties with the error bars upper/lower limits,
the abundances of L1521E’s bulk are largely consistent with those measured
toward both positions of L1517B, with the exception of vinyl cyanide
(CH2CHCN). A possible explanation for this would be that L1521E is more
chemically evolved than L1517B, lying closer to L1498 than to L1517B.
Alternatively, although these cores are located in the same molecular cloud
complex, the local physical and chemical properties of their environment could
differ, causing significant differences in their final chemical composition.
Studies of the COM chemical content toward a larger sample of starless cores
are needed to establish whether differences in the environment affect the COM
chemical evolution in starless cores.
We finally note that calculations of the fractional similarity and of the
geometric means of the abundance and of the molecular mass for L1521E, cannot
be performed since the number of data available for this core is rather small.
## 5 Modelling the formation of COMs and COM precursors in L1517B
### 5.1 The model
We have modelled the chemistry of COMs and COM precursors toward the starless
core L1517B by using the 0D gas-grain chemical code Monaco (Vasyunin & Herbst,
2013; Vasyunin et al., 2017), which has been successfully applied previously
to the starless cores L1544 and L1498 (Jiménez-Serra et al., 2016; Jiménez-
Serra et al., 2021). Our goal is to compare the observed molecular abundances
with those predicted by the model, and to infer the radial distribution of
COMs and COM precursors toward L1517B and its age.
Figure 6: H2 gas density and temperature profiles used for modelling the
chemistry of COMs and COM precursors toward L1517B. These radial profiles are
taken from Tafalla et al. (2004). The vertical gray dashed line indicates the
distance of the observed methanol peak with respect to the centre of the core.
Figure 7: Evolution of the simulated CO depletion factor over time. The blue
line and shaded region show the value $f_{\mathrm{CO}}=9.8\pm 2.8$, derived by
Crapsi et al. (2005), while the gray line indicates the best-fit age for
L1517B. Figure 8: Radial distribution of the modelled abundances of the COMs
and COM precursors observed for L1517B. Colour curves represent the modelled
abundances, while dots and vertical arrows represent the observed abundances,
and their upper limits, toward the two observed positions in L1517B, the dust
and methanol peaks. The position of some of the dots (observed COMs and COM
precursor abundances) has been slightly shifted to enhance visibility. The
dotted-dashed gray line indicates the position of the observed methanol peak,
while the gray dashed line indicates the position of the methanol peak
according to the chemical simulations.
The Monaco chemical code is a rate equations-based, three-phase (gas, ice
surface, ice bulk) numerical model that provides the evolution of the
fractional abundances of atomic and molecular species with time. To obtain the
radial distribution of the abundances of COMs and COM precursors in L1517B,
the code is run for a grid of distances on the density and temperature profile
of L1571B, assuming physical parameters to be constant in time. The physical
structure of the L1517B starless core is obtained from the H2 gas density
profile by Tafalla et al. (2004), and the gas kinetic temperature distribution
inferred by the same authors using NH3 observations. The assumed H2 gas
density profile for L1517B is:
$\indent n(r)\;=\;\frac{2.2\times
10^{5}\mathrm{\;cm}^{-3}}{1+\left(\frac{r}{35"}\right)^{2.5}}\;\;\mathrm{,}$
(1)
where $r$ is the angular distance (in arcseconds). As for the temperature
radial profile, we have used a constant temperature of $T$ = 10 K (see Tafalla
et al., 2004). Fig. 6 shows both profiles as a function of distance to the
centre of the core. In our model, we assume that the dust and gas temperatures
are equal.
Given the low temperatures in starless cores ($T$ $\simeq$ 10 K), the
precursors of COMs are formed on the surface of dust grains via hydrogenation
processes. Once formed, a small fraction of these compounds are non-thermally
desorbed from dust grains into the gas phase, where they undergo gas-phase
chemical reactions that yield the observed COMs (Vasyunin & Herbst, 2013;
Vasyunin et al., 2017). As non-thermal desorption processes, the code includes
UV photo-desorption, cosmic ray-induced desorption (Hasegawa & Herbst, 1993)
and reactive (chemical) desorption (Minissale et al., 2016). The chemical
network for the O-bearing and N-bearing COMs and precursors is the same as the
one used for the L1498 starless core (see the details in Jiménez-Serra et al.,
2021). As initial abundances, the model employs the results of a simulation of
a diffuse cloud model with constant gas density of 100 cm-3 and gas
temperature of 20 K for 107 years, using the low metals initial abundances EA1
from Wakelam & Herbst (2008).
### 5.2 Comparing simulations and observations
Using the initial conditions explained in Section 5.1, we have simulated the
chemical evolution of L1517B by running the Monaco code for $10^{6}$ years. To
constrain the age of the core, we have used the CO depletion factor of
$f_{\mathrm{CO}}$ = 9.5 $\pm$ 2.8 measured by Crapsi et al. (2005) toward
L1517B, and the location of the methanol peak at a distance of 5000 au.
Interestingly, our model cannot reproduce simultaneously the location of the
methanol peak at 5000 au and the CO depletion factor of $\sim\,$7–12. In Fig.
7, we show the evolution of the predicted CO depletion factor over time and
its observed value. Note that the Monaco code considers the telescope beam
size when calculating the CO depletion factor. By comparing the model results
with the CO and COMs and COM precursor observations, the best agreement is
reached after 1.5 $\times$ 105 years. The CO depletion factor at this time is
$\sim$7.3, which is on the lower edge of the estimated interval for the CO
depletion factor (see Fig. 7).
At this time of chemical evolution, however, the location of the methanol peak
in the model does not match the observed one toward L1517B. This is clearly
shown in Fig. 8, where we present the radial distribution of the abundance of
COMs and COM precursors modelled for L1517B, together with the measured values
and upper limits (see dots and vertical arrows). As shown in Fig. 8, the
location of the methanol peak in the model is at $\sim$15000 au away from the
dust peak (vertical dashed line), which is a factor of 3 further away that
observed (at $\sim$5000 au; see dashed-dotted vertical line).
The origin of this discrepancy likely arises from the fact that the central H2
gas density toward L1517B is $2.2\times 10^{5}$ cm-3, i.e. significantly lower
than measured toward the L1544 pre-stellar core. If this density were higher,
as in L1544 ($\sim$107 cm-3; Caselli et al., 2022), the CO depletion factor
(9.5 $\pm$ 2.8) would be reached earlier in the simulations, enabling a better
match between the modelled and the observed location of the methanol peak at
5000 au. This would also help reconciling the age of the L1517B starless core
predicted by the model (of $\sim$1.5$\times$105 years) with the one expected
from observations. As already mentioned, L1517B does not show evidence for gas
accretion either toward the innermost regions or from the outer envelope, as
observed toward L1498 or L1544, which suggests that L1517B is at an earlier
stage of evolution, or at least, at a similar evolutionary stage as L1498
(note that they both have similar central H2 gas densities and deuterium
fractionation values; Tafalla et al., 2004; Crapsi et al., 2005). Another
possible explanation of this discrepancy would be a prominent asphericity of
the core, although from Fig. 1 it seems that L1517B is rather spherical.
For the abundances of COMs and COM precursors, Table blue5 compares the values
between the modelled and observed abundances toward the positions of the dust
and methanol peaks in L1517B. All the modelled and observed abundances agree
within a factor of 10 except for CH3CN in the core’s center and HC3N in both
positions. Actually, at earlier times of the simulations, where the simulated
methanol peak is closer to the observed one, abundances of those two species
are closer to the observed ones. In any case, we should take into account that
astrochemical models entail intrinsic uncertainties derived from uncertainties
from some of the constants used in the simulation, such as the chemical
reaction rates (Vasyunin et al., 2004; Wakelam et al., 2005, 2010).
Comparing the predicted age for L1517B ($1.5\times 10^{5}$ yr) with those
predicted for L1498 and L1544, we find that, contrary to what we thought,
L1517B would not be the youngest core, as the predicted ages for L1498 and
L1544 are, respectively, $9.8\times 10^{4}$ yr and $1.6\times 10^{5}$ yr
(Jiménez-Serra et al., 2016; Jiménez-Serra et al., 2021). Therefore, according
to the chemical ages based on similar initial conditions, L1517B would be more
evolved than L1498 and less than L1544. However, although L1517B shows some
concentration of H2 at its centre, there is no clear sign of current
contraction motions toward this core. This, together with the observed level
of chemical complexity, seems to suggest that L1517B is less evolved that
L1544 and L1498. Note also that our chemical modelling does not include the
dynamical evolution of the core and, thus, the chemical age derived by our
model does not necessarily coincide with the actual dynamical age of the core.
In fact, if the initial conditions of the cores were different, the final COM
chemical composition of the cores would be significantly altered as compared
to our chemical modelling.
For L1521E, Scibelli et al. (2021) also used the Monaco code to estimate the
age of the core (of $\sim$6 $\times 10^{4}$ yr). However, they could only
reproduce the observed abundances of acetaldehyde (CH3CHO), out of a total of
five modelled COMs (with differences in the predicted abundances by factors of
20-160). Therefore, additional aspects in the chemical modelling of these
young starless cores such as different initial conditions and/or dynamical
evolution of the cores, may be required.
## 6 Evolution of the HC3N$\,/\,$CH3CN abundance ratio across low-mass star
formation
Table 5: Observed and modelled abundances ($\chi_{\mathrm{obs}}$,
$\chi_{\mathrm{mod}}$) for several COMs and COM precursors in L1517B.
| Dust peak | Methanol peak
---|---|---
Species | $\boldsymbol{\chi}_{\mathrm{\mathbf{obs}}}$ | $\boldsymbol{\chi}_{\mathrm{\mathbf{mod}}}$ | agreement | $\boldsymbol{\chi}_{\mathrm{\mathbf{obs}}}$ | $\boldsymbol{\chi}_{\mathrm{\mathbf{mod}}}$ | agreement
CH3OH | $2.7_{-0.5}^{+0.7}\times 10^{-10}$ | $1.21\times 10^{-9}$ | + | $1.11_{-0.10}^{+0.12}\times 10^{-9}$ | $2.23\times 10^{-9}$ | +
CH3O | $8.0_{-2.5}^{+2.9}\times 10^{-12}$ | $3.0\times 10^{-12}$ | + | $1.9_{-0.6}^{+0.7}\times 10^{-11}$ | $6.6\times 10^{-12}$ | +
CCCO | $<\thinspace 8\times 10^{-12}$ | $1.12\times 10^{-11}$ | + | $<\thinspace 1.3\times 10^{-10}$ | $2.05\times 10^{-11}$ | +
t-HCOOH | $<\thinspace 1.0\times 10^{-10}$ | $8.6\times 10^{-11}$ | + | $<\thinspace 2.0\times 10^{-10}$ | $1.63\times 10^{-10}$ | +
H2CCO | $2.4_{-0.5}^{+0.6}\times 10^{-11}$ | $1.12\times 10^{-10}$ | + | $7.6_{-2.0}^{+2.3}\times 10^{-11}$ | $2.17\times 10^{-10}$ | +
CH3OCH3 | $<\thinspace 3\times 10^{-11}$ | $2.32\times 10^{-13}$ | + | $<\thinspace 5\times 10^{-11}$ | $5.2\times 10^{-13}$ | +
CH3CHO | $<\thinspace 8\times 10^{-12}$ | $2.09\times 10^{-12}$ | + | $2.2_{-0.4}^{+0.5}\times 10^{-11}$ | $4.0\times 10^{-12}$ | +
HCCCHO | $<\thinspace 5\times 10^{-12}$ | $1.62\times 10^{-13}$ | + | $<\thinspace 2.1\times 10^{-11}$ | $3.2\times 10^{-13}$ | +
CH3CN | $6.0_{-1.9}^{+2.1}\times 10^{-12}$ | $4.0\times 10^{-13}$ | - | $<\thinspace 3\times 10^{-11}$ | $7.0\times 10^{-13}$ | +
CH2CHCN | $<\thinspace 1.2\times 10^{-12}$ | $1.46\times 10^{-14}$ | + | $<\thinspace 3\times 10^{-12}$ | $2.9\times 10^{-14}$ | +
HCCCN | $1.48_{-0.18}^{+0.24}\times 10^{-10}$ | $1.04\times 10^{-11}$ | - | $3.9_{-0.4}^{+0.4}\times 10^{-10}$ | $1.85\times 10^{-11}$ | -
The modelled and measured abundances agree (+) or not (-) within a factor of
10.
Recent observational campaigns have been devoted to investigate the chemical
composition in COMs (both O-bearing and N-bearing) towards all stages in the
process of star formation. In this Section, we compare the information
available for several sources for two of these COMs and COM precursors, the
N-bearing species cyanoacetylene (HC3N) and acetonitrile (CH3CN), with the
abundances obtained towards our limited sample of starless cores (L1544,
L1498, L1517B, and L1521E). We have selected these two molecular species
because they have been measured systematically in the past few years across a
variety of objects, from Class 0/I protostars (Bergner et al., 2017) to Class
II protoplanetary discs (see the Maps ALMA large program; e.g., Ilee et al.,
2021), and comets (Biver et al., 2021).
Fig. 9 reports the values of the column density ratio of HC3N$\,/\,$CH3CN for
the starless cores L1544, L1498 and L1517B (for both positions measured in
these cores), the starless core L1521E in the dust peak, twelve Class 0/I
protostellar systems (studied by Bergner et al., 2017), four Class II
protoplanetary discs (studied by Ilee et al., 2021), and two comets (studied
by Biver et al., 2021, and references therein). To obtain this ratio, we used
the column densities for both molecules, except for the comets, for which we
employed the abundances of both molecules with respect to water. From Fig. 9,
it seems that there is a trend for the starless cores to present much higher
HC3N$\,/\,$CH3CN ratios (of $\sim\,$200–700) than those measured toward Class
0/I protostars and Class II protoplanetary discs, and toward comets (in this
case, their HC3N$\,/\,$CH3CN abundance ratios reach values even lower than 1).
The core L1521E would be an exception, as its ratio is similar to those of the
protostars. In any case, the observed general trend suggests that the observed
HC3N$\,/\,$CH3CN abundance ratio decreases when starless cores enter into the
Class 0 phase, remaining approximately constant until the Class II
protoplanetary phase, and droping by two orders of magnitude for comets.
The observed range of values for the HC3N$\,/\,$CH3CN ratio could be explained
by the fact that it is difficult to form HC3N at late evolutionary stages in
the process of low-mass star formation. Indeed, HC3N is mainly a gas-phase
product whose formation is favoured by the low-densities and cold temperatures
found in the outer envelopes of pre-stellar systems and protostars (see
Section 5.1 in Bergner et al., 2017, for more details). In the same work, the
authors report similar CH3CN abundances for different kinds of sources (young
stellar objects, comets, hot cores, and hot corinos), which would explain the
decreasing trend in the HC3N$\,/\,$CH3CN abundance ratio with time.
## 7 Conclusions
The high-sensitivity spectra obtained for the starless core L1517B reveal a
chemical complexity a bit poorer than that observed in the core L1498, and
quite lower than in L1544, as it only presents a few and simple O-bearing and
N-bearing species, such as CH3O, H2CCO, CH3CN, and HCCCN, and in general with
lower abundances than in L1498 and L1544. Quantitatively, the geometric mean
of the abundance of the targeted species in L1517B is about half than in L1498
and L1544. Similarly to these latter cores, the targeted molecules are more
abundant by a factor of $\sim$3 towards the methanol peak of L1517B, located
at $\sim$5000 au from the core’s centre.
We have also modelled the chemical evolution of L1517B and, except for HCCCN
and CH3CN, our simulations agree with the observed abundances and upper limits
within a factor of 10. The model also predicts the observed enhancement of the
abundances as we move further away from the core’s centre, although we
overestimate the distance of the methanol peak by a factor of 3. Our model
suggests that L1517B is chemically older than L1498, but the absence of infall
motions and its relatively poor chemical complexity makes us think that this
core is at an earlier evolutionary stage than L1498. Actually, our model would
predict a younger age for L1517B than L1498 if the central density of the core
were higher.
We propose a scenario in which N-bearing molecules are formed first, followed
by O-bearing molecules once the catastrophic depletion of CO takes place.
Furthermore, complexity increases with time, with bigger molecules being
formed at later stages of the core, as observed in L1544. We note, however,
that the starless core L1521E does not seem to follow this trend, although the
sample of molecules studied is smaller than for the other three cores, and the
beam of the ARO observations is larger. More observational studies of starless
and pre-stellar cores are needed in order to clearly determine the influence
of the environment, initial conditions and dynamics on the chemical evolution
of the cores.
Finally, we have also studied the HC3N$\,/\,$CH3CN abundance ratio measured in
sources at different stages in the formation of a low-mass stellar system,
observing a decreasing trend over time, which could be explained by the
adverse conditions of the late stages of the low-mass star formation to form
HC3N.
Figure 9: Abundance ratio between cyanoacetylene (HC3N) and acetonitrile
(CH3CN) for different sources: the starless cores L1517B, L1498 and L1544
(with both positions, the dust and methanol peaks), the starless core L1521E
in the dust peak, 12 Class 0/I protostars, 4 Class II proto-planetary discs,
and 2 comets (46P and 67P). The references for each source are: L1517B, this
work; L1498, Jiménez-Serra et al. (2021); L1544, Jiménez-Serra et al. (2016);
L1521E, Nagy et al. (2019); Class 0/I protostellar systems, Bergner et al.
(2017); Class II protoplanetary discs, Ilee et al. (2021); comets, Biver et
al. (2021). The points of the plot as well as the original data used to obtain
the ratios can be found in Appendix C.
## Acknowledgements
We thank an anonymous referee for their valuable comments and suggestions,
which improved the quality of the paper. We also thank J. Aguirre and M.
Fernández-Ruz for their input about ways to measure complexity. A.M.
aknowledges support from grant PRE2019-091471 under project MDM-2017-0737-19-2
(Unidad de Excelencia «María de Maeztu» – Centro de Astrobiología, CSIC-INTA)
funded by the Spanish Ministry of Science and Innovation / State Agency of
Research, MCIN/AEI/10.13039/501100011033, and by ‘ERDF, A way of making
Europe’. I.J.-S. and J.M.-P. aknowledge support from grant
PID2019-105552RB-C41 by the Spanish Ministry of Science and Innovation / State
Agency of Research, MCIN/AEI/10.13039/501100011033, and by ‘ERDF, A way of
making Europe’. The work by A.V. is supported by the Russian Ministry of
Science and Higher Education via the State Assignment contract FEUZ-2020-0038.
IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).
## Data Availability
The original spectra used in this work will be shared on reasonable request to
the corresponding author. Most of the codes used to process and visualize the
data can be downloaded from GitHub.777https://github.com/andresmegias/gildas-
class-python/,
and the same link but changing the name of the repository (gildas-class-
python) by madcuba-slim-scripts, radex-python, or richvalues.
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## Appendix A Derivation of the methanol column density and its uncertainty
with Radex
In this Section, we explain the procedure we have followed to determine the
column densities of methanol, acetonitrile and cyanoacetilene, and their
uncertainties, using the non-LTE molecular excitation code, Radex (Van der Tak
et al., 2007). It is a radiative transfer model that allows us to predict the
line intensity of the selected molecule from the following parameters:
* •
Molecule. It has to be included in the Radex molecule list, therefore having
collisional cross-section derived, experimentally or theoretically.
* •
Spectral range. Frequency range for the transitions to be predicted for the
selected molecule.
* •
Excitation conditions.
* –
Molecular hydrogen number density, $n_{\mathrm{H_{2}}}$.
* –
Kinetic temperature of the molecule, $T_{\mathrm{kin}}$.
* –
Background temperature, $T_{\mathrm{bg}}$.
* •
Radiative transfer parameters.
* –
Column density, $N$.
* –
Line width, $\Delta v$.
For each species, the results of the calculation would be a set of intensity
lines, $\\{\tilde{I_{i}}\\}$, where $i$ is the index of the line. In our case,
the molecules are methanol (CH3OH) A and E, acetonitrile (CH3CN), and
cyanoacetilene (HC3N), and we have observations for their lines in a specific
spectral range (see Table blue2), so we have a set of observed intensities,
$\\{I_{i}\\}$. We have an uncertainty for each line (the RMS noise), so we
will also have a set of uncertainties $\\{\Delta I_{i}\\}$. We use the
frequency range of the observed lines, plus a little margin, for the Radex
calculations. The rest of the parameters except the column density (and
optionally the H2 number density) are known (see Section 3.2 for more
details):
* •
Molecular hydrogen number density (not fixed for the case of methanol E and
cyanoacetilene):
* –
Core’s centre: $n_{\mathrm{H_{2}}}$ = $2.20\times 10^{5}$ cm-3 (Tafalla et
al., 2004).
* –
Methanol peak: $n_{\mathrm{H_{2}}}$ = $1.24\times 10^{5}$ cm-3 (Tafalla et
al., 2004).
* •
Background temperature: $T_{\mathrm{bg}}$ = 2.73 K (temperature of the cosmic
microwave background).
* •
Kinetic temperature: $T_{\mathrm{kin}}$ = 10 K (Tafalla et al., 2004).
* •
Line width: $\Delta v\sim 0.3$ km s-1 (we choose for each case the result of
the Madcuba fit).
Our goal is to optimize the column density and optionally also the H2 number
density (when more than one transition is available) to minimize the
difference between the observed lines, $\\{I_{i}\\}$, and the predicted ones,
$\\{\hat{I_{i}}\\}$, taking into account the uncertainties, $\\{\Delta
I_{i}\\}$. To measure that difference, we define a loss function,
$\mathcal{L}$, which we choose to be the chi-square ($\upchi^{2}$) error:
$\indent\mathcal{L}\;=\;\sum_{i}{\left(\frac{\hat{I}_{i}-I_{i}}{\Delta
I_{i}}\right)^{2}}\,\,.$ (2)
In this way we are measuring the quadratic error weighted with the squared
inverse of the uncertainty, so that a difference between the observations and
the model contributes less to the loss if the uncertainty on the line
intensity is greater.
Figure 10: Histogram of the distribution of losses obtained from the
intensities of the observed lines and their uncertainties, for the case of
methanol E in the dust peak of L1517B. The scale in the horizontal axis is
symmetrical-logarithmic with threshold 1, that is, it is linear from 0 to 1
and logarithmic beyond 1. The blue line on the left shows the value of the
loss without taking into account the uncertainties of the observed lines
(Equation 2), which is almost 0; and the blue dashed line indicates the 68.27
percentile.
Once we have properly defined our loss function, we can minimize it with
respect to the column density for each of our 8 cases (four species, methanol
A and E, cyanoacetilene, and acetonitrile; and two positions, dust and
methanol peaks) and also with respect to the H2 number density for methanol E
and cyanoacetilene. We did this with Python,888https://www.python.org/ using
the methods Cobyla and Nelder-Mead within the function minimize from the
library SciPy.999https://scipy.org/
Finally, we have to estimate the uncertainty for the optimized value/s of our
parameter/s. To do so, we take a threshold for the loss, greater than the
minimized value, which defines a lower and an upper uncertainty. In order to
find a proper threshold, we create $10^{5}$ sets of variations of the observed
intensities, $\\{\\{I_{i}\\}_{j}\\}$, so that the value of each intensity,
$I_{i,j}$, comes from a normal distribution with mean $I_{i}$ and standard
deviation $\Delta I_{i}$. Then, we calculate the loss for the optimized model
with equation 2 for each set of intensities $\\{I_{i}\\}_{j}$, obtaining a
distribution of losses. We choose the threshold so that the loss values lower
than such a threshold are the 68.27 percent of the total values (similarly to
the definition of $1\,\upsigma$ confidence interval, but for an asymmetric
distribution with a minimum value). The threshold thus corresponds to the
68.27 percentile (see Fig. 10).
Then, we have to explore our parameter space around the minimized values in
order to find when the loss function reaches our threshold value. If we only
fit the column density, we only have to calculate the loss function in the
surroundings of the minimized value until we reach the threshold value.
However, if we also want to minimise the H2 number density, we have to explore
a bidimensional parameter space. We opted to make an adaptive grid that starts
calculating the loss values at the surroundings of the obtained minimum and
continues enlarging its size until it encloses all the loss values minor to
the threshold with a reasonable margin. This way, we obtained Figs. 11 and 12.
Figure 11: Plot of the loss ($\mathcal{L}$) versus the column density ($N$)
and the H2 number density ($n_{\mathrm{H_{2}}}$) for a range of values around
the optimized values that minimize the loss function, for the case of methanol
E in the dust peak of L1517B. The gray lines indicate the optimized values for
$N$ and $n_{\mathrm{H_{2}}}$. The color map is composed by two ranges of
colors in order to highlight the points in which the loss ($\mathcal{L}$) is
lower than the threshold value (marked in blue in the color bar), which will
define the uncertainties for both optimized parameters. Figure 12: Same as
Fig. 11, but for cyanoacetilene (HC3N) in the dust peak of L1517B.
Following this procedure, we have written a Python script that uses Radex and
analyses our data in an automated
way.101010https://github.com/andresmegias/radex-python/ In this way, we
obtained the column density values for methanol, acetonitrile and
cyanoacetilene in L1517B shown in Table 3.
## Appendix B Propagation of uncertainties and treatment of upper limits
Let’s consider a group of $n$ variables $\\{x_{i}\\}$, with central values
$\mu_{i}$ and uncertainties $\sigma_{i}$. This means that the probability
density function (PDF) of the variable $x_{i}$ is centered around $\mu_{i}$
with a width of the order of $\sigma_{i}$, so that the $1\,\upsigma$
confidence interval (68.27 percent) is ($\mu_{i}-\sigma_{i}$,
$\mu_{i}+\sigma_{i}$). To propagate the uncertainties through a function $f$
applied to the variables $\\{x_{i}\\}$, we can draw a sample of a large number
of values $(\gtrsim\,10^{4}\sqrt{n})$ of each variable $x_{i}$, apply the
function to each of the elements of the samples, and then obtain a central
value and an uncertainty for the resulting distribution.
To do so, we need two things: an appropiate PDF for converting each variable
$x_{i}$ to a distribution of values, and a proper method to obtain a central
value and an uncertainty from the resulting distribution. For the last task,
we can use the mean or the median as the central value,111111We prefer to use
the median, as it is more robust to outliers. and the $1\,\upsigma$ confidence
interval with respect to it (that includes 68.27 % of the distribution) to
obtain the lower and upper uncertainties. As for the PDF, if the domain of the
variable $x$ is ($-\infty$, $\infty$), a proper function is the normal
distribution:
$\indent
f(x)\;=\;\frac{1}{\sqrt{\uptau}\;\sigma}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right)\;\;,$
(3)
with $\uptau\equiv 2\uppi$. However, if the domain of the variable is not
($-\infty$, $\infty$), this function would be incorrect. Therefore, we have to
use another function as the PDF.
Let’s suppose a domain ($b_{1}$, $b_{2}$). We define the left and right
amplitudes, $a_{1}$ and $a_{2}$, as the distances between the limits of the
domain and the median, that is, $a_{j}=|b_{j}-\mu|$ for $j=1,2$. Now, as these
amplitudes can be different, we will split our desired PDF in two halfs, one
for $x\leq\mu$ and other for $x>\mu$. Then, we will use an amplitude $a$ as a
reference, which must be greater than the uncertainty, $a>\sigma$. If the
amplitude is quite greater than the uncertainty, $a\gg\sigma$, a good PDF
would be just the normal distribution truncated to the domain ($b_{1}$,
$b_{2}$). However, for amplitudes closer to the uncertainty, it would be
considerably incorrect, as the truncation shifts the median of the
distribution and modifies the confidence intervals, and thus the
uncertainties. To fix this, we make the following variable change:
$\indent
x-\mu\;\;\rightarrow\;\;\tilde{x}-\mu\;\equiv\;\frac{4}{\uptau}\,a\,\tan\left(\frac{\uptau}{4}\;\frac{x-\mu}{a}\right)\,\,.$
(4)
Using this new variable $\tilde{x}$ with a normal distribution, we are able to
compress the original domain of ($-\infty$, $\infty$) to ($-a$, $a$). However,
we get two disadvantages: firstly, the normalization constant is now
different, and secondly, the relation between the parameter of the standard
deviation of the original normal distribution and the $1\,\upsigma$ confidence
interval (from which we define the uncertainty, $\sigma$) is now different;
therefore, we should rename the standard deviation of the original normal
distribution to $s$, which we will call width. The first change is not a
problem, as the PDF will be normalized computationally for each case. As for
the second one, we have characterized computationally the relation between
$\sigma$ and $s$. It happens that $s/\sigma$ decreases with $a/\sigma$, having
$s/\sigma\simeq 2.65$ when $a/\sigma$ = 2 and $s/\sigma\rightarrow 1$ when
$a/\sigma\rightarrow\infty$. We have then a relation between the width, the
uncertainty and the amplitude, that is, $s=s(\sigma,a)$.
Therefore, the resulting PDF would be the following:
$\indent
f(x)\;\propto\;\exp\left(-\frac{1}{2}\left(\frac{\frac{4}{\uptau}a\tan\left(\frac{\uptau}{4}\frac{x-\mu}{a}\right)}{s(\sigma,a)}\right)^{2}\right)\;\;.$
(5)
We observed that $s/\sigma$ increases quasi-exponentially as $a/\sigma$
approaches a minimum value of $\sim\,$1.47. Moreover, we discovered that if
$a/\sigma\lesssim 1.7$, the estimated relation $s(\sigma,a)$ starts to be
incorrect because of the increasing dispersion in $s$. Therefore, we must use
another PDF for the cases in which $a/\sigma\lesssim 1.7$. Actually, for that
limit case the shape of the PDF is almost a uniform distribution between $-a$
and $a$. We can model this PDF as a trapezoidal function with a rectangular
core and triangles in the edges. By doing so, one can easily demonstrate that
this kind of PDF would only work if $a/\sigma$ is greater than the inverse of
the integrated area corresponding to the $1\,\upsigma$ confidence interval
($\sim\,$0.683); that is, $a/\sigma\gtrsim 1.46$, which is consistent with the
found asymptote of the calculated relationship $s(\sigma,a)$.
Figure 13: Probability density functions for a normal distribution and two
bounded normal distributions with diferent domains.
We then choose to only use this distribution when $a\geq 2\sigma$, for the
sake of simplicity and following a conservative approach. For $a\gg\sigma$,
this PDF tends to a normal PDF, so we use this general PDF instead of a
truncated gaussian for this regime (see Figure 13). We call this distribution
a bounded normal distribution.
Finally, for $\sigma<a<2\sigma$, we use a shifted and mirrored lognormal
distribution whose PDF would be:
$\indent
f(x)\;=\;\frac{1}{\sqrt{\uptau}\;\,a\left|\mathrm{ln}\left(1-\frac{\sigma}{a}\right)\right|}\;\frac{1}{1-\frac{|x-\mu|}{a}}\\\
\times\;\,\exp\left(-\frac{1}{2}\left(\frac{\mathrm{ln}\left(1-\frac{|x-\mu|}{a}\right)}{\mathrm{ln}\left(1-\frac{\sigma}{a}\right)}\right)^{2}\right)\;\;\;.$
(6)
This PDF meets our requirements: the median is equal to $\mu$ and the
uncertainty associated with the $1\,\upsigma$ confidence interval is $\sigma$.
We call this distribution a mirrored lognormal distribution.
We have defined our PDFs for the case of a variable $x$ with a central value
$\mu$ and an uncertainty $\sigma$, building the final PDF with two halfs with
amplitude $a=a_{j}$ for $j=1,2$. In case we had lower and upper uncertainties,
$\sigma_{1}$ and $\sigma_{2}$, we should just replace $\sigma$ by $\sigma_{1}$
for the left half of the PDF and by $\sigma_{2}$ for the right half.
Table C1. Column densities of HC3N and CH3CN ($N$) and abundance ratio of this
molecules ($\chi_{\mathrm{HC_{3}N}}\,/\,\chi_{\mathrm{CH_{3}CN}}$) for the
starless cores L1517B, L1498, L1544, and L1521E, 12 protostellar systems, 4
protoplanetary discs, and 2 comets.
Source | $\boldsymbol{N_{\mathrm{HC_{3}N}}}\,(\mathrm{cm}{}^{-2})$ | $\boldsymbol{N_{\mathrm{CH_{3}CN}}}\,(\mathrm{cm}{}^{-2})$ | $\boldsymbol{\chi_{\mathrm{HC_{3}N}}\thinspace/\thinspace\chi_{\mathrm{CH_{3}CN}}}$
---|---|---|---
starless cores | L1517B | dust peak | $(5.16\thinspace\pm\thinspace 0.05)\times 10^{13}$ | $(2.1\thinspace\pm\thinspace 0.6)\times 10^{11}$ | $250_{-50}^{+100}$
methanol peak | $3.72_{-0.13}^{+0.14}\times 10^{12}$ | $<\thinspace 2.0\times 10^{11}$ | $>\thinspace 17$
L1498 | dust peak | $(1.6\thinspace\pm\thinspace 0.3)\times 10^{13}$ | $<\thinspace 8\times 10^{10}$ | $>\thinspace 100$
methanol peak | $(2.2\thinspace\pm\thinspace 1.0)\times 10^{13}$ | $(1.0\thinspace\pm\thinspace 0.1)\times 10^{11}$ | $220_{-100}^{+100}$
L1544 | dust peak | $(1.0\thinspace\pm\thinspace 0.3)\times 10^{14}$ | $(1.5\thinspace\pm\thinspace 0.2)\times 10^{11}$ | $670_{-200}^{+230}$
methanol peak | $(4.2\thinspace\pm\thinspace 0.4)\times 10^{13}$ | $<\thinspace 9.1\times 10^{10}$ | $>\thinspace 310$
L1521E | dust peak | $(8.4\thinspace\pm\thinspace 2.5)\times 10^{12}$ | $(4.8\thinspace\pm\thinspace 1.4)\times 10^{11}$ | $18_{-6}^{+9}$
protostellar systems | B1-a | $(4.2\thinspace\pm\thinspace 1.2)\times 10^{12}$ | $(4.9\thinspace\pm\thinspace 1.1)\times 10^{11}$ | $9_{-3}^{+3}$
B1-c | $(4\thinspace\pm\thinspace 3)\times 10^{12}$ | $(3.5\thinspace\pm\thinspace 0.6)\times 10^{11}$ | $12_{-9}^{+11}$
B5 IRS1 | $(3.2\thinspace\pm\thinspace 1.2)\times 10^{12}$ | $<\thinspace 1.7\times 10^{11}$ | $>\thinspace 5$
IRAS 03235 | $(3.9\thinspace\pm\thinspace 1.0)\times 10^{12}$ | $(1.6\thinspace\pm\thinspace 0.9)\times 10^{11}$ | $25_{-11}^{+31}$
IRAS 03245 | $(4\thinspace\pm\thinspace 3)\times 10^{12}$ | $(1.9\thinspace\pm\thinspace 1.5)\times 10^{11}$ | $19_{-15}^{+69}$
IRAS 03271 | $(1.7\thinspace\pm\thinspace 1.4)\times 10^{12}$ | $(1.9\thinspace\pm\thinspace 1.1)\times 10^{11}$ | $9_{-7}^{+14}$
IRAS 23238 | $(3.2\thinspace\pm\thinspace 2.6)\times 10^{12}$ | $(1.4\thinspace\pm\thinspace 0.3)\times 10^{11}$ | $22_{-18}^{+21}$
L1014 IRS | $(4.4\thinspace\pm\thinspace 3.6)\times 10^{11}$ | $<\thinspace 6\times 10^{10}$ | $>\thinspace 0.7$
L1455 IRS3 | $(6\thinspace\pm\thinspace 5)\times 10^{11}$ | $<\thinspace 9\times 10^{10}$ | $>\thinspace 0.7$
L1455 SMM1 | $(2.4\thinspace\pm\thinspace 1.9)\times 10^{12}$ | $<\thinspace 1.1\times 10^{11}$ | $>\thinspace 2.2$
L1489 IRS | $(3.5\thinspace\pm\thinspace 2.4)\times 10^{12}$ | $<\thinspace 1.4\times 10^{11}$ | $>\thinspace 0.3$
SVS 4-5 | $(1.1\thinspace\pm\thinspace 0.3)\times 10^{13}$ | $(5.2\thinspace\pm\thinspace 0.9)\times 10^{11}$ | $21_{-6}^{+8}$
protoplanetary discs | GM Aur | $1.9_{-0.4}^{+0.4}\times 10^{13}$ | $2.1_{-0.1}^{+0.2}\times 10^{12}$ | $8.8_{-1.9}^{+2.0}$
As 209 | $2.9_{-0.5}^{+0.5}\times 10^{13}$ | $1.7_{-0.2}^{+0.2}\times 10^{12}$ | $17_{-3}^{+4}$
HD 163296 | $7_{-2}^{+3}\times 10^{13}$ | $2.3_{-0.2}^{+0.2}\times 10^{12}$ | $32_{-9}^{+11}$
MWC 480 | $8_{-3}^{+4}\times 10^{13}$ | $3.5_{-0.2}^{+0.2}\times 10^{12}$ | $22_{-8}^{+11}$
Source | $\boldsymbol{\boldsymbol{\chi_{\mathrm{HC_{3}N}}\thinspace/\thinspace\chi_{\mathrm{H_{2}O}}}}$ | $\boldsymbol{\boldsymbol{\chi_{\mathrm{CH_{3}CN}}\thinspace/\thinspace\chi_{\mathrm{H_{2}O}}}}$ | $\boldsymbol{\chi_{\mathrm{HC_{3}N}}\thinspace/\thinspace\chi_{\mathrm{CH_{3}CN}}}$
comets | 46P | $<\thinspace 3\times 10^{-5}$ | $(1.7\thinspace\pm\thinspace 0.01)\times 10^{-4}$ | $<\thinspace 0.21$
67P | $4\times 10^{-6}$ | $5.9\times 10^{-5}$ | $0.068$
References for each source: L1517B: this work; L1498: Jiménez-Serra et al.
(2021); L1544: Jiménez-Serra et al. (2016); L1521E: Nagy et al. (2019);
protostellar systems: Bergner et al. (2017); protoplanetary discs: Ilee et al.
(2021); comets: Biver et al. (2021).
Lastly, we should also address the case of a variable with an upper/lower
limit or even a finite interval. Let’s consider an interval ($x_{1}$,
$x_{2}$), which may be finite or infinite. If it is finite, we choose a
uniform distribution between $x_{1}$ and $x_{2}$ as the corresponding PDF. But
if it is infinite, we choose a log-uniform distribution with finite thresholds
for 0 and $\pm\infty$, which we set to $\pm 10^{-90}$ and $\pm 10^{90}$. For
example, for an interval of ($-100$, $\infty$), we would build a sample
$\\{x_{-}\\}$ from a uniform distribution between $-90$ and $2$ and a sample
$\\{x_{+}\\}$ from a uniform distribution between $-90$ and $90$. Our final
distribution would be the joining of the samples of $\\{-10^{\\{x_{-}\\}}\\}$
and $\\{10^{\\{x_{+}\\}}\\}$.
To sum up, if we have a set of variables with central values $\mu_{i}$ and
uncertainties $\sigma_{1}$, $\sigma_{2}$, we first build distributions
$\\{x_{i}\\}$ using the mentioned PDFs. Then, we apply the function to the
distributions, $f(\\{x_{i}\\})$, obtaining a new distribution. Finally, we use
an algorithm to detect if the distribution corresponds to an interval (that
can be an upper/lower limit) or a defined value with uncertainties, and derive
the corresponding parameters.
Using this approach, we have written a Python library that allows to deal with
values with uncertainties and upper/lower limits, performing the uncertainty
propagation automatically.121212https://pypi.org/project/richvalues/ We have
used it throughout the calculations of this paper, e.g., for obtaining the
values shown in Fig. 9.
## Appendix C Abundance ratio of cianoacetylene and acetonitrile
Table blueC1 shows the abundance ratio of cyanoacetylene (HC3N) and
acetonitrile (CH3CN) through different sources, from starless cores to comets.
The propagation of the uncertainties and the upper limits in the column
density is done through simulations using statistical distributions (see
Appendix B).
|
# Peculiarities of gender disambiguation and ordering of non-English authors’
names for Economic papers beyond core databases ††thanks: This work was
supported in part (OM) by the National Research Foundation of Ukraine, project
No. 2020.01/0338.
O. Mryglod Institute for Condensed Matter Physics of the National Academy of
Sciences of Ukraine, 1 Svientsitskii St., 79011 Lviv, Ukraine S. Nazarovets
Borys Grinchenko Kyiv University, 18/2 Bulvarno-Kudriavska Str., 04053 Kyiv,
Ukraine S. Kozmenko University of Social Sciences Spoleczna Akademia Nauk, 9
Sienkiewicza St., 90–113 Łódź, Poland
###### Abstract
This paper presents the results of further exploration of Crossref data
related to Ukrainian Economics research (the first part can be found in
[Mryglod, O., Nazarovets, S. & Kozmenko, S. (2021) Scientometrics, 126, 8187.
https://doi.org/10.1007/s11192-021-04079-7]).
Purpose: To supplement the quantitative portrait of Ukrainian Economics
discipline with the results of gender and author ordering analysis at the
level of individual authors, special methods of working with bibliographic
data with a predominant share of non-English authors are used. The properties
of gender mixing, the likelihood of male and female authors occupying the
first position in the authorship list, as well as the arrangements of names
are studied.
Design/methodology/approach: A data set containing bibliographic records
related to Ukrainian journal publications in the field of Economics is
constructed using Crossref metadata. Partial semi-automatic disambiguation of
authors’ names is performed. First names, along with gender-specific ethnic
surnames, are used for gender disambiguation required for further comparative
gender analysis. Random reshuffling of data is used to determine the impact of
gender correlations. To assess the level of alphabetization for our data set,
both Latin and Cyrillic versions of names are taken into account. Findings:
The lack of well-structured metadata and the poor use of digital identifers
lead to numerous problems with automatization of bibliographic data pre-
processing, especially in the case of publications by non-Western authors. The
described stages for working with such specific data help to work at the level
of authors and analyse, in particular, gender issues. Despite the larger
number of female authors, gender equality is more likely to be reported at the
individual level for the discipline of Ukrainian Economics. The tendencies
towards collaborative or solo-publications and gender mixing patterns are
found to be dependent on the journal: the differences for publications indexed
in Scopus and/or Web of Science databases are found.
It has also been found that Ukrainian Economics research is characterized by
rather a non-alphabetical order of authors.
Research limitations: Only partial authors’ name disambiguation is performed
in a semi-automatic way. Gender labels can be derived only for authors
declared by full First names or gender-specific Last names.
Practical implications: The typical features of Ukrainian Economic discipline
can be used to perform a comparison with other countries and disciplines, to
develop an informed-based assessment procedures at the national level. The
proposed way of processing publication data can be borrowed to enrich metadata
about other research disciplines, especially for non-English speaking
countries.
Originality/value: To our knowledge, this is the first large-scale
quantitative study of Ukrainian Economic discipline. The results obtained are
valuable not only at the national level, but also contribute to general
knowledge about Economic research, gender issues and authors’ names ordering.
An example of the use of Crossref data is provided, while this data source is
still less used due to a number of drawbacks. Here, for the first time,
attention is drawn to the explicit use of the features of the Slavic authors’
names. Keywords: scholarly metadata, economics, Crossref, OUCI, Ukraine,
gender, non-Western authors
## 1 Introduction
This paper contains the results that are part of a more general study Mryglod
. (2021), the purpose of which is to perform a large-scale quantitative
analysis of the Ukrainian Economics discipline using the publication data
predominantly beyond Web of Science and Scopus databases, where currently more
than 150 Ukrainian journals are indexed while the National List of recognized
scientific journals111https://mon.gov.ua/ua/nauka/nauka/atestaciya-kadriv-
vishoyi-kvalifikaciyi/naukovi-fahovi-vidannya includes almost 1,500 titles.
Motivated by the fact that Ukrainian research is still understudied because of
its poor representation in core databases Aksnes Sivertsen (2019) (and this is
especially true for Social Sciences and Humanities – SSH), we have made an
attempt to provide a quantitative portrait of one of Ukrainian SSH disciplines
using the Crossref database as an alternative data source. Our interest in
Ukraine is natural, as all three authors are Ukrainians and therefore
motivated to contribute to a more transparent and evidence-based management of
national research. Nevertheless, we also believe that this is an interesting
case study that contributes to the better understanding of the research
process in the developing countries of Eastern Europe, countries with a
special historical heritage. Ukraine is characterised by non-English speaking
and Cyrillic writing; this is especially true for the analysis of SSH.
Economics is chosen as one of the most ‘visible’ SSH disciplines, which is
often considered as a transitional between the ‘hard’ and ‘social’ sciences
Cainelli . (2012); Mryglod (2012).
The quantitative analysis at the level of publications based on the Crossref
data is rather straightforward. For example, an estimation of the number of
authors per paper can be done even without sophisticated data pre-processing.
However, the consideration of individual publication histories at the level of
authors or gender analysis requires name disambiguation. Obviously, this task
is a big challenge, especially when dealing with non-Western names
Treeratpituk Giles (2012); Gomide . (2017); Kim . (2021). Since the majority
of publications in the Ukrainian Economic discipline are related to local
authors Mryglod . (2021)), it is natural to find mainly Ukrainian first and
last names in our data set. Although such peculiarities as the use of middle
name or a prespecified order of parts in composite names Treeratpituk Giles
(2012); Gomide . (2017) are not typical for Ukraine, a huge problem of
transliteration (e.g., see also Müller . (2017)) still exists. But there is
another side of the coin: the so-called ethnicity can be used to improve
gender disambiguation. Therefore, along with the initial motivation to
contribute to the quantitative description of Ukrainian Economics discipline,
a special emphasis is made on the methods of processing such specific
bibliographic data.
The research questions here are related both to the methods of data pre-
processing and to the results of the analysis of these data:
* RQ1:
What peculiarities of Ukrainian authors’ names have to be taken into account
during the process of name and gender disambiguation?
* RQ2:
What gender proportion is typical for the Ukrainian Economics discipline, and
how can it be compared with similar results for other data sets (countries)?
* RQ3:
What level of alphabetization characterizes the Ukrainian Economics
discipline, and is it possible to identify any gender-related distinctions?
Answering the main research questions, this paper serves also as another
evidence of the usefulness of Crossref data as a potential source for
bibliometric analysis. Economic publications are considered in many other
studies, where the data from Web of Science or Scopus databases are exploited,
see Schläpfer (2010); Vaio Weisdorf (2009); Zhao . (2016); Wei (2018); Truc .
(2021). And this is reasonable in order to assess top-impact output and reveal
the research front in Economics. However, if the rest of the entire picture is
needed, the potential of other sources such as Crossref can be efficiently
used. In this context, Ukraine has an advantage – a special interface called
Open Ukrainian Citation Index (OUCI) was developed a few years ago. It
provides a possibility to extract structured Crossref metadata related to all
journals published in Ukraine Cheberkus Nazarovets (2019). Moreover, all these
journals are labeled by subject category according to the Ukrainian national
classification scheme. While this data source is not as comprehensive as a
national current research information system could be, it provides a unique
opportunity to supplement knowledge about research output of Ukraine.
The paper is organized as follows: the description of our data set is provided
in Section 2; the applied name disambiguation procedure is described in
Section 3; Section 4 describes the peculiarities of gender disambiguation
procedure for our data and contains the results of gender analysis. The author
name ordering for the Ukrainian Economics discipline is studied in Section 5;
the final discussion can be found in the last Section.
## 2 Our data
This paper is a continuation of authors’ previous work Mryglod . (2021), which
analyses Crossref data for publications in Ukrainian journal papers in the
field of Economics. The same principle of collecting data is used here, i.e.,
Crossref publication records related to Ukrainian Economics journals. A not-
for-profit membership organization Crossref222https://www.crossref.org/
collects metadata for publications with registered DOI (Digital Object
Identifier) numbers. Each record contains basic bibliometric elements required
for DOI registration (i.e., title, publication dates, authors, source title,
volume and issue number, etc.). In addition, Crossref encourages its
depositors to enrich metadata with authors’ affiliations, ORCID numbers,
abstracts, lists of references, funding information, etc. These metadata are
publically open, license-free and distributed through Crossref tools and APIs.
Since 2018, DOI registration is required for any research paper to be
officially recognized in Ukraine333See the Order of the Ministry of Education
and Science of Ukraine N32 (2018, January 15).
https://zakon.rada.gov.ua/laws/show/z0148-18, therefore, Crossref metadata can
be considered as a usefull source of information about the published outputs
related to Ukrainian research. Moreover, a special web-interface – Open
Ukrainian Citation Index (OUCI)444https://ouci.dntb.gov.ua/en/ – was developed
to efficiently import these data. In particular, a number of search filters
allowing classification of journals by their specialty are implemented into
OUCI. In addition, information about current indexing of each journal in
Scopus and Web of Science is provided.
The topical relevance of each journal is defined using the ‘‘Speciality’’
search filter. The following specialities are considered to be related to the
Economics field (a similar subject classification is used by the State
Attestation Commission of Ukraine555Official web-page of the State Attestation
Commission of Ukraine: https://mon.gov.ua/ua/tag/atestatsiya-kadriv-vishchoi-
kvalifikatsii: Economics; Tax and Accounting Policy; Finance, Banking and
Insurance; Management; Marketing; Business, Entrepreneurship and Stock
Markets; Public Administration; and International Economic Relations. Only
journals in the National List of recognized scientific journals are
considered. To exclude multidisciplinary editions, journal disciplines (upper
classification level) are limited to the following list: Social and Behavioral
Sciences; Management and Administration; Public Management and Administration;
International Relations.
In addition to the data available from Scopus and Web of Science, Crossref
provides an important piece of the puzzle required to build the full picture
of Ukrainian Economics research. The results presented in this paper are based
on the updated data set: data collection is performed at the end of February
2021. Altogether, 25,933 records for papers published in Ukrainian Economics
journals between 2002 and 2020 were collected (the annual publication
statistics is low before 2012 and rapidly increases afterwards: 97% of records
correspond to the period 2013–2020). The imported records contain the
following fields: Publication year; Journal ISSN number; DOI; Publisher;
Title; Authors’ names; Number of DOI-to-DOI citations (if information is
provided by Crossref depositors); Journal is indexed in Scopus Yes/No; Journal
is indexed in Web of Sciences Yes/No (up-to-date information in the last two
fields is added by OUCI).
## 3 Name disambiguation problem
While data analysis at the level of papers is performed in Mryglod . (2021),
many interesting questions can be put at the level of authors. To give an
example, typical individual productivity or authors’ collaboration patterns
have to be known to set benchmarks for comparing, assessing or detecting
examples of unusual publishing behaviour. What is also important, authors’
gender is typically (and in this work) inferred from the given names.
Therefore, gender label cannot be assigned if only initials are specified
instead of the full name. However, merging various records related to the same
person, allows us to enlarge the statistics of papers with genderized authors.
For example, gender can be defined for KAFKA S. (?) merged with KAFKA SOFIYA
(Female). However, the widely-known problem of name disambiguation appears if
unique digital identifiers are not commonly used. With only names, it is
impossible to guarantee that two identically written names correspond to the
same person. The uncertainty is higher if only initials are used. But
everything is even more complicated in the case of publications by authors who
are not native English speakers. It is possible to find numerous alternative
transliterations of Cyrillic names for the same authors in our data set.
Moreover, speaking of Ukrainian names, one should take into account the
tradition of ‘‘translating’’ given names, and sometimes even last names, into
Russian. For example, an author Bosovskaya can be also mentioned as Bosovska;
Orlovskaya – Orlovska; Mostenskaya – Mostenska. Many of the first names can be
transliterated to English using Ukrainian or Russian Cyrillic versions; some
of the most used are: Mykola – Nikolay, Oleksandr – Aleksandr, Kateryna –
Ekaterina, Olena – Elena. It can be instantly noted that the names in these
pairs correspond to different initials: M – N, O – A, K – E, O – E,
respectively. The space of possible alternatives is also expanded by using
different short versions of names: Olena $\rightarrow$ Lena, Oleksiy
$\rightarrow$ Alex, Anastasiya $\rightarrow$ Nastya, Tetyana $\rightarrow$
Tanya, etc. Sometimes, the same name can be written in many ways, each of them
is automatically recognized as a separate name. Moreover, metadata for
Ukrainian Economics journals can be deposited not only in English, but also in
Ukrainian (Russian). Last but not least, inaccurate usage of Latin and
Cyrillic alphabets is a problem. The homoglyphs – letters that look the same
on a screen but coded differently – are used arbitrarily. After all, 40
versions of the name Eugen are found in the data set. Taken together, all
these peculiarities of metadata of Ukrainian (non-native English) publications
complicate the process of disambiguating the names of authors.
Due to the numerous nuances listed above, it is too difficult to perform a
full disambiguation procedure automatically. That can be done only partially
and only in a semi-automatic way. The following criteria and approaches are
used:
* •
Identical names found in different papers are considered as related to one
person, since the limited data set that corresponds to a particular subject
area is studied. The assumption that there is a low probability of duplicated
names within our data set is confirmed by manually checking randomly selected
records. Of course, exceptions are possible. Authors’ records with identical
first and second names are separated if both appear in the same paper.
* •
The existence of common co-authors for two authors is considered as an
argument to merge corresponding records.
* •
A manually created list of Ukrainian given names together with ‘‘synonymical’’
forms (Latin and Cyrillic) was used to find candidates for merging666The list
is available online, https://doi.org/10.6084/m9.figshare.13580297. A gender
label is initially assigned to each name (manually). A few examples to
demonstrate the variety of names are shown below.
$\bullet$ ALEKSEI; ALEKSEY; ALEKSII; ALEXEI; ALEXEJ; ALEXEY; AЛЕКСЕЙ; ALEKCEY;
OLEKSEY; OLEKSII; OLEKSIY; OLEKSYI; OLEKSІI; OLEKSІY; OLEXIJ; OLEXIY; АLEXEI;
АЛЕКСЕЙ; ОЛЕКСІЙ; ОLEKSII; ALEKSY; OLEKCII; ОLEXII; OLEXII; ALEXSEY; OLEKSIJ;
ОLEKSIY; – Male
$\bullet$ CHRISTINA; CHRISTINE; CHRYSTYNA; CRISTINA; HRISTINE; KHRYSTYNA;
KRISTINA; KRISTINE; KRISTYNA; KRYSTYNA; ХРИСТИНА; КРИСТИНА; КРІСТІНА;
KHRISTINA; KRISZTINA; – Female
* •
The records are merged if no contradictions appear. To give an example, all
names from the following list: ‘‘KAFKA S.М.; KAFKA SOFIYA; КАФКА С.М.; KAFKA
S.M.; KAFKA SOFIIA; KAFKA S.’’ are merged to get a single author’s record. But
an ambiguity exists for authors from the list ‘‘VERHUN А.; VERGUN ANDRIJ
IVANOVYCH; VERHUN A.; VERHUN ANTONINA; VERHUN ANDRIJ’’: VERHUN A.777It seems
like the same name appears twice, since Cyrillic ‘А’ and Latin ‘A’ are used
for the author’s initial. corresponds either to ANDRIJ or to ANTONINA – no
merging is performed in this case.
The list of authors’ names was processed using the own Python code to find the
list of candidates for merging and to mark them as more or less probable. The
final merging was manually confirmed using the results of this preliminary
automatic procedure. Additional manual checks were performed for particular
cases, where candidates are considered as important ‘‘players’’ due to a large
number of publications or co-authors. Merging was not performed for the pairs
where ambiguity remains, but even so, the initial set of 31.5 thousand
authors’ records was reduced to 23,094.
## 4 Gender disambiguation and analysis
As mentioned before, the gender label for an author is inferred from his/her
given name. Since the majority of authors are from Ukraine Mryglod . (2021),
Slavic first names are predominantly found in our data set. Besides the list
of Slavic names manually labeled by gender, free web resource
Genderize888Genderize.io — Determine the gender of a first name.
https://genderize.io/#overview was partially used to detect gender for non-
Slavic names999Only results for the names occurring at least 10 times with a
probability of more than 0.9 were taken into account.. Thus, 54.5% of 23,094
author records were marked by gender: 7,748 (33.5%) females and 4,865 (21%)
males. According to this, females appear in our data set approximately 1.59
times more often.
Some typical endings of Slavic surnames can be considered as gender-specific.
Author records genderized on the previous step were used as a validation
subset in order to check whether surnames’ endings are distinctive enough for
our data set. The gender of 19.85% females was repeatedly recognized using
gender-specific endings of last names: ‘‘OVA’’, ‘‘EVA’’, ‘‘ОВА’’, ‘‘ЕВА’’,
‘‘АЯ’’, ‘‘AYA’’, ‘‘AIA’’, ‘‘INA’’, ‘‘ІНА’’. In their turn, 18.19% of males
were recognized using the following list of endings: ‘‘OV’’, ‘‘EV’’, ‘‘ОВ’’,
‘‘ЕВ’’, ‘‘ЄВ’’, ‘‘YI’’, ‘‘YJ’’, ‘‘KY’’, ‘‘KII’’, ‘‘KIJ’’. As it can be seen,
such an approach is almost equally accurate for both genders: males are
slightly less recognizable. Only 8 (0.1%) female and 7 (0.14%) male authors
are re-marked incorrectly. Such a few examples can be considered as
exceptions: male surnames with female-like endings DIBROVA, СОВА (SOVA),
ELLAIA, JAYA, GLOVA, GECHBAIA, BOVA and female surnames with male-like endings
KYI, MYSHELOV, GLAMBOSKY, LAZANYI, BOKII, URSAKII, SUSHYI, IVANOV. Although
gender can be assigned incorrectly to a particular person, we believe that the
results are statistically correct.
Thus, gender labels were inferred from surnames for an additional 1,260 female
and 828 male authors. Altogether, we continue with 63.7% genderized authors’
names. And having in mind that the number of male authors is slightly
underestimated, one can state that 1.5 times more female authors are found.
Our finding is in line with the statement in Larivière . (2013), where Ukraine
is mentioned among other ‘‘countries with lower scientific output’’ that are
characterized by more prevalent female authorship. But our research is not
cross-disciplinary, it is initially related to the Economics area. Moreover,
while Web of Science data were used in Larivière . (2013), we exploit Crossref
as a data source in our work. The remaining question is how different can be
results obtained for top journals indexed in authoritative databases such as
Scopus or Web of Science and for data beyond these sources.
It is shown that gender disparities are disciplinary-dependent Nicola
D’Agostino (2021). The Economics discipline is considered rather as a male-
dominated one101010Gender in the global research landscape.
https://www.elsevier.com/__data/assets/pdf_file/0008/265661/ElsevierGenderReport_final_for-
web.pdf Liu . (2020); Bayer Rouse (2016). For example, 20.3% female versus
63.4% male authors were found for an economics-related data set analyzed in
Maddi Gingras (2021)111111The gender is undetectable for the rest authors.. A
similar proportion was reported in Liu . (2020): ‘‘The proportion of men is
2.45 times higher than that of women’’. On the contrary, more female authors
are found in our data121212Of course, one has to remember that different
datasets are used in these different case studies.. Thus, it is natural to
expect more papers from female authors (at least one author is recognized as
female) – 76.3% than from male authors – 48.4%. But let’s look more deeply
into the individual contributions by female and male authors, as it was
suggested in Huang . (2020). The conclusion that ‘‘female and male authors are
largely indistinguishable when it comes to the number of publications per
year’’ supported also by results presented in Liu . (2020) is relevant to our
data: authors of both genders publish approximately the same number of papers
per year on average. To be more precise, 1.28 papers per year on average are
published by male authors, and 1.34 by female authors.
Another interesting relevant issue is the analysis of gender mixing and the
patterns for forming authorship teams. The annual change of shares of papers
classified according to gender of authors for our data is shown in Fig. 1.
Five categories are defined here for 17,352 out of 25,933 papers (the rest
cannot be classified due to the lack of gender information about authors):
papers authored by a single person are labeled as _F solo_ or _M solo_ in
correspondence to the author’s gender (F stands for female and M for male); _F
coll_ and _M coll_ labels are used only for papers where gender for all
authors is defined and the same (solo-gender collaboration); if not all
authors are labeled by gender, but at least one female and one male are found,
the paper is attributed to the _MIX coll_ category (cross-gender
collaboration). Since not all authors in the data set are labeled by gender,
the gender spectrum of papers can be considered as not conclusive. However,
the same rule is applied here to data that correspond to different years,
therefore, the tendencies are considered as informative.
Figure 1: Annual dynamics of shares of papers classified according to the
gender of authors: a single female author (_F solo_); two or more authors,
only females (_F coll_); two or more authors, at least one female and at least
one male (_MIX coll_); two or more authors, only males (_M coll_); a single
male author (_M solo_). The time window between 2013 and 2020 is chosen for
visualization due to small annual statistics (less than 100 papers per year)
before 2013.
As Fig. 1 shows, many papers correspond to the decreasing but still largest
category of publications by a single female author. On the one hand, this is
in line with the conclusion in Boschini Sjögren (2007), where over-
representation of single female authors is reported. On the other hand, it was
already mentioned that the number of papers by females is expected to be
larger in principle, simply due to the larger number of female authors.
Indeed, for the same data set, if authors’ gender labels ‘F’, ‘M’ and
‘undefined’ are randomly reshuffled first, the share of papers in the _F solo_
category $\approx 34.8\%$ is very close (even slightly larger) to the real
value ($\approx 33.8\%$), see Table 1. Again, gender equality can be found at
the individual level. The similar average share of solo-authored papers per
author’s portfolio is found for both genders (approximately 24% if all
genderized authors are taken into account; 28% (F) and 30% (M) if only
genderized authors with at least 2 papers are considered). The similar
conclusion is found in Kwiek Roszka (2021) for Polish authors in the field of
Economics – the share of solo-publications in an individual portfolio is
approximately equal for both male and female authors. According to this, a
curious paradox can be observed: While the majority of papers in the Economics
discipline are not collaborative Mryglod . (2021), at the individual level,
only each third (fourth if authors with a single paper are considered) paper
is written without co-authors. No gender differences are also found in terms
of the average number of authors per individual collaborative papers.
The homophily of co-authorship groups with respect to authors’ gender in the
Economics field was discussed in Boschini Sjögren (2007). At first glance, our
results deny this conclusion. The cross-gender category is the second-largest
one among the gender-labeled papers in our data, see Fig. 1 and Table 1.
However, it is easy to show that this share is smaller than expected. And
indeed, solo-gender collaborative papers by female authors are over-
represented in the real data set in comparison to reshuffled data. It is
interesting to see that the shares of cross-gender papers and collaborative
papers by male authors are remarkably higher for papers that are indexed in
Scopus and/or WoS, see Table 1. To some extent, this is in agreement with the
conclusion about the tendency to comparatively lower gender homophily in
higher-impact journals in Holman Morandin (2019). Such gender mixing can be
seen in a very positive way, since gender is one of the most important
dimensions of team diversity, which, in its turn, is often considered as a
powerful catalyst for creativity, see, e.g., Farhoomand Drury (2001); Reynolds
Lewis (2017); Liao (2010) and references therein.
Table 1: Distribution of papers according to authors’ gender: a single female
author (_F solo_); two or more authors, only females (_F coll_); two or more
authors, at least one female and at least one male (_MIX coll_); two or more
authors, only males (_M coll_); a single male author (_M solo_).
| Entire data set | Reshuffled∗ | Not indexed in Scopus and/or WoS | Indexed in Scopus and/or WoS
---|---|---|---|---
| # records | Share | Share | # records | Share | # records | Share
All papers | 25933 | – | – | 22683 | – | 3250 | –
All papers assigned to one of the categories | 17352 | 100% | (13785 records on average) | 15422 | 100% | 1930 | 100%
F solo | 5871 | 33.8% | 34.8% | 5684 | 36.9% | 187 | 9.7%
F coll | 3085 | 17.8% | 10.4% | 2841 | 18.4% | 244 | 12.6%
MIX coll | 4283 | 24.7% | 28.5% | 3236 | 21% | 1047 | 54.3%
M coll | 816 | 4.7% | 4.1% | 598 | 3.9% | 218 | 11.3%
M solo | 3297 | 19% | 22.2% | 3063 | 19.8% | 234 | 12.1%
∗Averages for 10 versions of the original set of publications labeled by
categories after random reshuffling of authors’ gender labels are provided.
## 5 Authors ordering
Another aspect of forming collaboration teams – the ordering of authors – can
be studied using our data. The position of the author’s name in a list, which
is ordered neither randomly nor alphabetically, can be considered as a basis
for credit allocation. In this case, it is also reasonable to investigate
correlation between authors’ gender and their roles (i.e., positions in co-
authorship lists). For example, the statistics of female first-authored
journal articles are studied in Thelwall Mas-Bleda (2020).
Economics is often considered as one of the fields where alphabetization is
common (see, e.g., Frandsen Nicolaisen (2010); Waltman (2012); Levitt Thelwall
(2013); Kuld O’Hagan (2017)), although the alphabetization rate in the economy
has declined somewhat over the past decade Wohlrabe Bornmann (2022). Moreover,
top Economic journals are characterized by the share of alphabetized articles
that is even higher compared to other Economic journals (70% vs. 60%,
correspondingly) Levitt Thelwall (2013); Kuld O’Hagan (2017). Such a way of
ordering can be interpreted as a declaration of equal authors’ contribution.
At the same time, the first position can still be perceived as special by
external assessors. The so-called ‘‘alphabetical discrimination’’ is discussed
in Einav Yariv (2006); Kuld O’Hagan (2017): The staff members of a top U.S.
economic department whose surnames start with letters from the first part of
the alphabet are found to be more tenured. In some sense, this can be seen as
a consequence of the Thomas theorem Bornmann Marx (2020): even a groundless
consideration of the first author as the principal one can cause the further
advantage in an academic career. Therefore, the following two questions are
addressed further: (i) Is there any gender preference for the first position
in a co-authorship list? and (ii) Can we state that the alphabetical ordering
of authors is typical for the Ukrainian Economics discipline, in general?
_Gender of first authors._ Due to the larger share of female authors, it is
expected also to get a larger share of collaborative papers, where the first
positions in co-authorship lists are occupied by female authors. But it is
interesting to calculate the probability of being on the first position in a
collaborative list for each author. The individual publication records for
both male and female authors with at least three collaborative papers are
analysed for this purpose: the share of collaborative papers, where the given
author is in the first position, are counted. According to our results, the
probability to occupy first position is equal to 0.48 for authors of both
genders. In this context, rather gender equality is found.
_Alphabetization._ The alphabetical or non-alphabetical order of authors is
determined by their last names. For the authors with identical last names,
initials are considered. Since the majority of author names hypothetically are
Slavs, their alphabetization can be performed in two ways: for their original
names written in Cyrillic or for the corresponding versions in Latin.
Therefore, the possible alphabetical order of each sequence of names is
checked twice: for the names as they are in metadata and for their
transliterations. To give an example, the names in the list _‘‘MARTYNENKO
VALENTYNA; ZAMOTA IRINA’’_ are found to be _ordered_ by Latin alphabet (_M
ARTYNENKO; ZAMOTA_), but their transliterated versions are _not ordered_ by
Cyrillic alphabet: (_М АРТИНЕНКО; ЗАМОТА_).
The more authors in the collaboration list, the less the probability of
accidental alphabetical authorship is (see, e.g., Waltman (2012); Kuld O’Hagan
(2017)). Since small co-authorship lists are dominant for the Economics
discipline, in many cases one cannot be sure whether the names are sorted
alphabetically intentionally or unintentionally. But non-alphabetical sorting
is an unambiguous indicator of other priority scheme usage. Therefore, we
count the share of papers with authors’ names ordered neither by Latin nor
Cyrillic alphabets. Corresponding numbers for different publication samples
are provided in Table 2. For example, it can be seen that in almost a half of
all collaborative articles, authors are not sorted by the last names.
Moreover, knowing the exact numbers of authors in the rest of the papers, one
can suggest that another 21.3% of the papers are sorted in alphabetic order
accidentally131313Considering all collaborative papers in the initial data
set, one can find that the total number of alphabetically sorted papers is
6,641. The probability for a paper to be unintentionally ordered
alphabetically depends on the number of authors $n$: $P(n)=1/n!$.
Correspondingly, the number of papers that are intentionally alphabetized
might be smaller: $5,194(1-1/2!)=2,597$ duo-authored papers;
$1,284(1-1/3!)=1,070$ trio-authored papers; $135(1-1/4!)\approx 129$ quarto-
authored papers; $\ldots$. Waltman (2012); Kuld O’Hagan (2017). Therefore, we
conclude that the level of alphabetization of authors’ names in Ukrainian
Economic papers turned out to be lower than it was reported for other
publication sets. Moreover, the share of non-alphabetized articles indexed in
Scopus or Web of Science databases is even larger. One can only speculate
about the reasons for such feature of Ukrainian Economics research. While the
first position in the list of authors is not encouraged officially, still it
is considered as more beneficial due to its greater visibility (it is a common
practice to mention just the first author to refer to the co-authored
publication) and its special perception within a number of disciplines. The
name of the first author appears at the beginning of the reference.
Table 2: The numbers of non-alphabetically ordered papers and estimated volume of potentially alphabetized papers (adjusted values) for different data samples. | Number of papers (collaborative only) | Number (%) of papers with definitely non-alphabetical ordering of authors’ names | Adjusted number (%) of papers that are potentially alphabetically ordered by intention
---|---|---|---
Entire data sets | 13237 | 6596(49.8%) | 3824 (28.9%)
Papers marked by gender categories | 8184 | 4057 (49.6%) | 2384 (29.1%)
Papers indexed in Scopus and/or WoS | 2592 | 1672 (64.5%) | 601 (23.2%)
Papers NOT indexed in Scopus and/or WoS | 10645 | 4924 (46.3%) | 3223 (30.3%)
Cross-gender papers | 4283 | 2369 (55.3%) | 1169 (27.3%)
Solo-gender papers | 3901 | 1688 (43.3%) | 1215 (31.1%)
## 6 Discussion and conclusions
The analysis of Ukrainian journals within the Economics discipline, started in
the previous work Mryglod . (2021), is continued in this study. Revealing the
typical features of this particular segment of scholar literature is important
for solving many practical issues related to the development of assessment
procedures at the national level. However, another goal of this work is to
reinforce the call for complete and qualitative metadata. Crossref database is
used here to describe one of SSH disciplines for poorly studied European
countries. Publication metadata related to Ukrainian Economic journals are
collected from the Crossref database. An attempt was made to conduct an
analysis with an emphasis on gender effects at the level of individual
authors. However, the procedure for disambiguating authors’ names can be done
only partially. A number of peculiarities of processing author names related
to the usage of Cyrillic and local traditions of parallel usage of different
forms of names and even surnames are highlighted. Since the gender of an
author is inferred from the full first name, even partial merging of authors’
records allows one to increase the statistics of publications with authors
labeled by gender. Moreover, a manually created list of gender-specific
endings for Slavic last names was used to enlarge the number of genderized
authors. Altogether, 63.7% of 23,094 author records were labeled by gender,
and the number of female authors is found to be 1.5 times larger than male
authors. This result contradicts with the statements about the masculinum
nature of Economics research. Alternatively, female dominance in Ukrainian
Economics research may be considered as a hint about its specific thematic
spectrum. According to Thelwall . (2019), keywords related to qualitative and
exploratory methods are statistically associated with female scholar authors,
while other keywords related to quantitative methods are more related to male
authors. In some sense, such sensitivity of gender representation to the topic
selection is in line with the conclusion in West . (2013): considerable
differences in this context were observed for different Economics subfields.
This reinforces our previous conclusions about the specific patterns of
collaborativeness in Ukrainian Economics research. Still, an important caution
exists: this study is one of the rare examples where data beyond
internationally recognized databases is used. Therefore, the guess about
different nature of locally-oriented and internationally-oriented topics
chosen for Economics research remains relevant.
Gender mixing is analysed to find the evidence that gender plays a role when
forming collaboration teams. All papers labeled according to five gender-
related categories (solo-publications by males; solo-publications by females;
solo-gender collaborations of males; solo-gender collaborations of females;
and cross-gender collaboration) are considered. While only one third141414One
fourth if authors with single papers are considered as well of individual
papers are written without coauthors and there is a tendency towards more
collaborative papers Mryglod . (2021), the share of solo-publications remains
high. One third of all papers are found to be solo-publications authored by
female authors. The corresponding share for male authors is slightly smaller
than expected. Finally, while the share of cross-gender teams is larger than
the shares of solo-gender teams (see Fig. 1), the results compared with
randomly reshuffled data indicate that the share of cross-gender teams is
considerably higher than it can be expected only for publications in the
journal indexed in Scopus and/or WoS, see Table 1.
It is shown that the level of alphabetization of authors’ names in Ukrainian
Economic papers is comparatively low. This is especially true for articles
indexed in Scopus or Web of Science databases.
Interestingly, different results of gender mixing are found for papers
published in journals indexed in Scopus or Web of Science, compared to the
rest of publications. Remarkably, while the largest share of papers solo-
authored by female authors is expected due to the greater general number of
female authors, this category of papers indexed in the international databases
is the least represented one. Most papers in internationally recognized
journals are characterized by cross-gender collaboration. This can be seen as
the manifestation of the so-called reactivity of the Ukrainian Economics
discipline (see Aistleitner . (2019); Sasvári . (2019)). One can speculate
about the adaptive publishing behaviour: a different publishing or even
research strategy is chosen depending on the level of recognition and audience
of the target journal. The similar conclusion can be drawn for different
shares of papers where authors are listed alphabetically. Interestingly, while
a high level of alphabetization is found for the Economics discipline in
general and even higher for Economic publications in internationally
recognized journals, the opposite pattern is observed in Ukrainian Economics
research.
To conclude, the results of another case study is presented. Besides the
findings specifically related to Ukrainian research, some key aspects related
to the processing of non-English metadata are highlighted. It is worth
emphasizing once more that many complications become irrelevant if unique
digital identifiers are commonly used.
## Acknowledgements
While the results reported in the paper are obtained before the massive
russian invasion to Ukraine, authors would like to thank all Ukrainian
defenders for the possibility to finalise and publish this work. The authors
thank anonymous peer reviewers for improving the paper.
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|
# Joint Neural Architecture and Hyperparameter Search for Correlated Time
Series Forecasting
Xinle Wu, Dalin Zhang∗, Miao Zhang, Chenjuan Guo, Bin Yang∗, Christian S.
Jensen
Department of Computer Science, Aalborg University, Denmark xinlewu, dalinz,
miaoz, cguo, byang<EMAIL_ADDRESS>
(2023)
###### Abstract.
Sensors in cyber-physical systems often capture interconnected processes and
thus emit correlated time series (CTS), the forecasting of which enables
important applications. The key to successful CTS forecasting is to uncover
the temporal dynamics of time series and the spatial correlations among time
series. Deep learning-based solutions exhibit impressive performance at
discerning these aspects. In particular, automated CTS forecasting, where the
design of an optimal deep learning architecture is automated, enables
forecasting accuracy that surpasses what has been achieved by manual
approaches. However, automated CTS solutions remain in their infancy and are
only able to find optimal architectures for predefined hyperparameters and
scale poorly to large-scale CTS. To overcome these limitations, we propose
SEARCH, a joint, scalable framework, to automatically devise effective CTS
forecasting models. Specifically, we encode each candidate architecture and
accompanying hyperparameters into a joint graph representation. We introduce
an efficient Architecture-Hyperparameter Comparator (AHC) to rank all
architecture-hyperparameter pairs, and we then further evaluate the top-ranked
pairs to select a final result. Extensive experiments on six benchmark
datasets demonstrate that SEARCH not only eliminates manual efforts but also
is capable of better performance than manually designed and existing
automatically designed CTS models. In addition, it shows excellent scalability
to large CTS.
correlated time series forecasting, neural architecture search
*corresponding authors: D. Zhang<EMAIL_ADDRESS>and B. Yang<EMAIL_ADDRESS>
††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: the 2023 International Conference on Management
of Data, June 18–23, 2023, Seattle, WA, USA.; 15.00††price: ;††isbn:
978-1-4503-XXXX-X/18/06††ccs: Information systems Spatial-temporal
systems††ccs: Information systems Data mining
## 1\. Introduction
Many systems, including societal infrastructures such as transportation
systems, electricity grids, and sewage systems (Rajkumar et al., 2010),
include cyber-physical components that encompass multiple sensors that each
emit a time series, resulting in multiple time series that are often
correlated. For example, inductive-loop detectors in a vehicular
transportation system measure the time-varying traffic volume at different
road locations, and measurements along the same or nearby roads often
correlate. The forecasting of future values from correlated time series often
has important applications(Wu et al., 2019). For example, accurate forecasting
of traffic volumes can facilitate the prediction of congestion and near-future
travel times, in turn enabling, e.g., more effective vehicle routing.
The key to successful correlated time series forecasting is the ability to
capture both the temporal dependencies among historical values of each time
series and the spatial correlations across different time series. Leveraging
the powerful feature extraction capabilities of deep learning models,
different neural architectures, called ST-blocks, have been proposed to
capture spatio-temporal (ST) dependencies to enable accurate forecasting.
Traditionally, human experts have designed ST-blocks and have chosen
accompanying hyperparameter settings manually. However, this is a resource-
intensive endeavor for which human expertise is ill-suited.
A more recent approach is to automate the design of effective ST-blocks (Pan
et al., 2021; Wu et al., 2022). Figure 1(a) outlines a typical automated
framework. A search space, represented as a supernet, contains a massive
number of possible ST-blocks (Figure 1(a) left), any subnet of which is a
candidate ST-block (Figure 1(a) right). Nodes in the supernet and subnets
represent latent representations, and the directed edges between them
represent different operators (e.g., convolution, graph convolution, and
Transformer). In a supernet, the transformation from node $h_{i}$ to node
$h_{j}$ is a weighted sum of all candidate operators, while in a subnet only
one operator between each node pair is kept. The goal is to learn the
operator-associated weights $\\{\alpha_{i}\\}$, upon which an optimal subnet
is obtained by picking the operator with the highest weight between every two
nodes.
(a) Existing Supernet-based Framework
(b) The SEARCH Framework
Figure 1. Comparsion between exisitng supernet-based framework vs. the
proposed SEARCH framework. Hyperparameter $C$ indicates the number of nodes in
an ST-block, and $H$ is the size of hidden representations. Edges of different
colors represent different operators.
Although existing automated CTS forecasting methods achieve design automation
and are capable of better performance than manually designed models, they
still suffer from three main limitations.
(1) Lack of support for joint search for architectures and hyperparameters.
When training supernets, existing automated CTS forecasting methods rely on
predefined hyperparameters, including architectural hyperparameters (e.g., the
number of latent representations, i.e., nodes, in an ST-block and the size of
a latent representation) and training hyperparameters (e.g., the dropout
rate). Depending on the hyperparameter settings, the same architecture can
yield markedly different performance. In spite of this, existing solutions
rely on an expert to select proper hyperparameters settings, which may well
lead to the choice of a suboptimal architecture and which renders the
framework “semi-automated.” Alternatively, it is possible to combine a
hyperparameter optimization method (e.g., grid search or Bayesian
optimization) sequentially with an existing automated architecture search
method (Pan et al., 2021; Wu et al., 2022) to achieve a two-step, automated
approach. However, the running time would be excessive as existing automated
architecture search needs to be performed each time a hyperparameter set is
sampled. Rather, an efficient, joint architecture and hyperparameter search
scheme is called for.
(2) Poor scalability. Existing automated CTS forecasting methods usually
suffer from poor scalability, as an entire supernet must reside in memory
during training, which may cause memory overflow in large-scale CTS settings
(Pan et al., 2021; Wu et al., 2022). Specifically, the memory cost of the
neural operators that compose a supernet increases rapidly with the number of
time series $N$ and the number of historical timestamps $P$ in time series.
For example, the memory cost of commonly used one-dimensional convolutional
neural networks (1D CNNs, T-operators) and Graph Convolution Networks (GCNs,
S-operators) grows linearly with $N$ and $P$, and the memory cost of
Transformers (S/T-operators) grows quadratically with $P$ and $N$. Since there
are $O$ operators between each node pair in a supernet, the memory cost of a
supernet is approximately $O$ times that of a subnet, i.e., an actual CTS
forecasting model. Taking Figure 1(a) as an example, the supernet has three
types of candidate operators between each pair of nodes, i.e., 1D CNN, GCN,
and Transformer (represented by different colored lines). The memory cost of
the supernet is thus about three times that of one of its subnets. When
increasing $N$ or $P$, the memory usage of a supernet goes overflow more than
a subnet, which limits the scalability of neural architecture search.
(3) One-time use. Existing automated CTS forecasting methods train a supernet
for each specific dataset from scratch, which is costly (Pan et al., 2021; Wu
et al., 2022). However, given the possible similarities between different CTS
forecasting datasets, reusing knowledge learned from previous search on
relevant datasets may significantly reduce the search time on a new dataset.
Therefore, developing a transferable automated framework holds potential to
increase search efficiency.
We propose SEARCH, a Scalable and Efficient joint ARChitecture and
Hyperparameter search framework to address the above limitations. First, we
design a joint search space that contains a wide variety of Architecture-
Hyperparameter (arch-hyper) combinations, and then aim at finding the optimal
arch-hyper in this search space, thus addressing the first limitation. For
example, with existing supernet based methods, the number $C$ of nodes in an
ST-block must be set before searching. The example in Figure 1(a) can only
search for ST-blocks with $C=4$ nodes. In contrast, our joint search space
considers multiple values for $C$, e.g., {4, 5, 6, 7, 8} in Figure 1(b), and
allows searching for ST-blocks with different numbers of nodes.
Second, to avoid using a supernet that consumes extensive memory and limits
scalability, we propose a novel Architecture Hyperparamter Comparator (AHC) to
rank candidate arch-hypers from the joint search space. Given encodings of two
candidate arch-hypers, the AHC estimates a binary value, indicating which
arch-hyper has the best accuracy. Thus, AHC is able to estimate a ranking of
candidate arch-hypers. We then train the top-K ranked arch-hypers and select
the arch-hyper that gives the highest forecasting accuracy. Because the AHC is
implemented with a lightweight graph neural network and the input to the AHC
is the encodings of arch-hypers instead of CTS data, its performance is
independent of $P$ and $N$, making it more scalable than existing supernet-
based frameworks.
Third, instead of learning a new AHC on each unseen dataset from scratch, we
propose to transfer the AHC trained on one dataset to unseen datasets, thereby
improving the training efficiency on the unseen datasets, thus addressing the
third limitation.
To the best of our knowledge, this is the first study that enables joint
search for architectures and corresponding hyperparameter settings for
correlated time series forecasting. Specifically, we make the following
contributions:
* (1)
We propose a novel search space for correlated time series forecasting to
facilitate joint search for architectures and hyperparameter settings.
* (2)
A memory-efficient Architecture-Hyperparameter Comparator (AHC) is proposed to
rank arch-hyper candidates that encompasses an easy-to-obtain proxy metric to
generate pseudo-labels to train an AHC in a denoising manner, thereby
improving search efficiency.
* (3)
We propose a method that is able to quickly adapt a trained AHC to a new
dataset, thus significantly improving the AHC training efficiency on new
datasets.
* (4)
Extensive experiments on six benchmark datasets show that SEARCH is able to
efficiently find better-performing CTS forecasting models compared to state-
of-the-art manual and automatic methods.
## 2\. Preliminaries
### 2.1. Problem Settings
Correlated Time Series (CTS). We denote $N$ correlated time series (CTS) by
$\bm{\mathcal{X}}\in\mathbb{R}^{N\times T\times F}$, where each time series
contains $T$ timestamps and has an $F$-dimension feature vector at each
timestamp. The feature vectors in the $i$-th time series
$\bm{X}^{(i)}\in\mathbb{R}^{T\times F}$ $\subset\bm{\mathcal{X}}$, $1\leq
i\leq N$, are correlated with previous feature vectors in the time series as
well as feature vectors in other time series. Therefore, it is natural to
model a CTS as a graph $G=(V,E,A)$, where vertex set $V$ represents the set of
time series, edge set $E$ represents correlation relationships between time
series, and adjacency matrix $A\in\mathbb{R}^{N\times N}$ captures the
strengths of the relationships between time series. $A$ is usually predefined
based on the distances of the sensors that generate the time series, or
learned adaptively.
Correlated Time Series Forecasting. We consider multi-step and single-step
correlated time series forecasting, both of which have important applications
in the real world (Shih et al., 2019; Lai et al., 2018; Wu et al., 2020; Bai
et al., 2020). Given the feature vectors of the past $P$ time steps of
$\bm{\mathcal{X}}$, the goal of multi-step CTS forecasting is to predict the
feature vectors of the future $Q$ time step, with $Q>1$; and the goal of
single-step CTS forecasting is to predict the vector at the $Q$-th future time
step, where $Q\geq 1$. Formally, we define the multi-step CTS forecasting as
follows:
(1)
$\displaystyle{(\bm{\hat{X}}_{t+P+1},\bm{\hat{X}}_{t+P+2},...,\bm{\hat{X}}_{t+P+Q})}=\mathcal{F}(\bm{X}_{t+1},\bm{X}_{t+2},\ldots,\bm{X}_{t+P};G),$
and we define the single-step CTS forecasting as follows:
(2)
$\displaystyle\bm{\hat{X}}_{t+P+Q}=\mathcal{F}(\bm{X}_{t+1},\bm{X}_{t+2},\ldots,\bm{X}_{t+P};G),$
where $\bm{X}_{t}\in\mathbb{R}^{N\times F}$ denotes the feature vectors of all
time series at timestamp $t$; while $\bm{\hat{X}}$ represents the predicted
feature vectors, and $\mathcal{F}$ is a CTS forecasting model.
Problem Definition. The goal is to automatically build an optimal ST-block
$\mathcal{F}^{\ast}$ from a predefined combined architecture-hyperparameter
search space $\mathcal{S}$ that minimizes the forecasting error on a
validation dataset $\mathcal{D}_{val}$. Mathematically, the objective function
can be stated as the following equation:
(3)
$\displaystyle\mathcal{F}^{\ast}=\mathit{argmin}_{\mathcal{F}\in\mathcal{S}}~{}\mathit{ErrorMetric}(\mathcal{F},\mathcal{D}_{val})$
### 2.2. Neural Forecasting Models
As summarized in Figure 2, the common framework of manually designed neural
CTS forecasting models has three components: an input module, an ST-backbone,
and an output module. The input and output modules usually consist of one or
two fully-connected layers that encode the input time series and decode
extracted spatiotemporal features to forecasting values, respectively.
The ST-backbone is the core component of a CTS forecasting model. It consists
of $B$ ST-blocks. The $B$ ST-blocks can be connected using different
topologies, with sequential stacking being a simple yet effective topology and
that we use as an example in Figure 2. An ST-block captures the spatial
correlations between time series and the temporal dependencies in individual
time series. There are thus two categories of operators in an ST-block,
S-operators (e.g., GCNs) and T-operators (e.g., Transformer), for extracting
spatial and temporal features, respectively. The specific types of
S/T-operators and their connections are critical to the success of a CTS
forecasting model.
Figure 2. An example CTS forecasting model.
### 2.3. Existing Automated Methods
Since the input and output modules of neural CTS forecasting models are
simple, typically consisting of one or two fully-connected layers, and only
involve a few design choices, such as the choice of embedding dimension,
existing automated CTS forecasting frameworks (Wu et al., 2022; Pan et al.,
2021) focus on the design of ST-blocks. Specifically, automated frameworks
typically start by designing a search space that encompasses a wide variety of
architectures for ST-blocks. The search space is represented by a directed
acyclic graph (DAG) (Figure 1(a) left), dubbed a supernet, with $C$ nodes and
a number of edges. Each node $h_{i}$, $0\leq i\leq C-1$, denotes a latent
representation. Each node pair $(h_{i},h_{j})$, has $|\mathcal{O}|$ directed
edges from $h_{i}$ to $h_{j}$, $i<j$, corresponding to $|\mathcal{O}|$
candidate S/T-operators, where $\mathcal{O}$ is a predefined candidate
operator set consisting of S/T-operators, such as CNN, GCN, and Transformer.
The goal of existing automated frameworks is to select operators and
connections that minimize validation errors to obtain ST-blocks and a
consequent CTS forecasting model. To achieve this, a vector
$\alpha^{(i,j)}\in\mathbb{R}^{|\mathcal{O}|}$ is introduced to weigh the edges
between each node pair $(h_{i},h_{j})$. These vectors then reflect the
importance of the edges and are to be learned during training. Then the
transformation from node $h_{i}$ to node $h_{j}$ is formulated as a weighted
sum of all edges, i.e., operators:
(4) $\displaystyle
f^{(i,j)}=\sum\limits_{o\in\mathcal{O}}\frac{exp(\alpha_{o}^{(i,j)})}{\sum_{o^{\prime}\in\mathcal{O}}exp(\alpha_{o^{\prime}}^{(i,j)})}o(h_{i}),$
where $\alpha_{o}^{(i,j)}$ is the weight of operator $o\in\mathcal{O}$, and
$o(\cdot)$ represents the transform function of operator $o$. Then, the latent
representation of node $h_{j}$ is obtained by summing all the transformations
from its predecessor nodes:
(5) $\displaystyle h_{j}=\sum\limits_{i<j}f^{(i,j)}$
This way, a supernet can be trained on the target CTS forecasting dataset
using gradient descent to learn both the neural operator parameters and
architecture parameter $\alpha$. After training, an optimal ST-block is
derived by removing the unimportant edges from the supernet, retaining only
one edge between each node pair and at most two incoming edges for each node
(Figure 1(a) right).
For existing automated frameworks, hyperparameters such as the number $C$ of
nodes in an ST-block need to be predefined (e.g., C is set to 4 in Figure
1(a)). In other words, existing frameworks do not support jointly searching
for architectures and hyperparameters. In addition, existing automated
frameworks consume substantial memory since very large supernets must reside
in memory during training. Furthermore, existing automated frameworks start
from scratch for each new dataset, which is inefficient.
## 3\. scalable and Efficient Joint Search
Our framework enables search for an optimal ST-block, i.e., an optimal
combination of a neural architecture and a set of accompanying
hyperparameters. Figure 1(b) offers an overview of the proposed automated CTS
forecasting framework. In order to support joint search, we first design a
joint search space (Section 3.1) containing various candidate [architecture,
hyperparameters] pairs, each of which is denoted as an arch-hyper.
We then propose a novel search framework that leverages an Architecture-
Hyperparameter Comparator (AHC) (Section 3.2) to achieve a predicted accuracy
based ranking of all arch-hypers in the joint search space. The AHC is
implemented as a neural network that takes the encodings of two candidate
arch-hypers as input and produces a binary label indicating which input arch-
hyper has the higher accuracy. To this end, training the AHC requires a large
number of labeled samples of the form ($ah_{1}$, $ah_{2}$, $y$), where
$ah_{1}$ and $ah_{2}$ are arch-hypers and $y$ is a binary label. Intuitively,
obtaining the labels requires completely training the two input arch-hypers,
which is computationally expensive. To reduce cost, we propose an easy-to-get
proxy metric (Section 3.3) to generate pseudo-labels and further train the AHC
in a noise reduction manner to reduce the negative impact of the pseudo-
labels. Once an arch-hyper ranking is obtained, we pick the top-$K$ arch-
hypers for full training, and we finally select the optimal arch-hyper with
the highest validation accuracy. Furthermore, we propose a simple yet
effective transfer method (Section 3.4) to avoid learning an AHC from scratch
on each new dataset, which is achieved by transferring an AHC that is trained
on one dataset to a new dataset. Thus, only few AHC training samples from the
new dataset are required to finetune the pretrained AHC, which significantly
improves the search efficiency without compromising the accuracy compared to
training an AHC from scratch on the new dataset.
### 3.1. Joint Search Space
We focus on the automated design of ST-blocks that extract spatial and
temporal features and thus are the core components of CTS forecasting models.
The joint search space considers two aspects of ST-blocks: 1) the
architecture, including operators and their connections, and 2) the
hyperparameters, including architecture-related structural hyperparameters
(e.g., the hidden dimension) and optimization-related training hyperparameters
(e.g., the dropout rate). Next, we introduce the search spaces of the
architecture and hyperparameters in turn, and then show how to combine these
into a joint search space.
#### 3.1.1. Architecture Search Space
In an ST-block, S/T-operators extract spatial/temporal features, and the
connections between operators control the information flow.
Candidate operators. By empirically analyzing manually designed CTS
forecasting models and the search spaces of existing automated CTS forecasting
frameworks, we include two compelling candidate T-operators. The Gated Dilated
Causal Convolution (GDCC) (Wu et al., 2019; Pan et al., 2021; Wu et al., 2022)
can effectively capture short-term temporal dependencies. In contrast,
Informer (INF-T) (Zhou et al., 2021), which is a variant of the Transformer,
excels at learning long-term temporal dependencies.
We also include two S-operators for extracting two sorts of spatial features.
The Diffusion Graph Convolution Network (DGCN) (Li et al., 2018), as
demonstrated in many popular CTS forecasting studies (Wu et al., 2019; Pan et
al., 2021; Wu et al., 2022), is effective at capturing static spatial
correlations. In addition, the Informer (INF-S) (Zhou et al., 2021) included
due to its strength at discovering dynamic spatial correlations.
We also include an “identity” operator to support skip-connections between
nodes. In this way, we obtain a candidate operator set $O$ composed of the
above five operators. The framework can easily accommodate additional
operators. Specifically, to add a new operator, we first include the operator
in the candidate operator set $O$. Then, we sample some arch-hypers that
include the new operator and use them to generate additional clean and noisy
samples to retrain the AHC (see Section 3.2). The clean and noisy samples
collected before can be reused when retraining the AHC, and AHC training is
quite efficient (see Section 3.3).
Topological connections. After selecting the candidate operators, we consider
the possible topological connections among the operators within an ST-block.
An ST-block can be represented as a directed acyclic graph (DAG) $G_{d}$
(e.g., Figure 3 left) with $C$ nodes, where each node $h_{i}$ represents a
feature representation and each edge represents an operator $O_{i}$. We
propose the following topological connection rules to generate candidate ST-
blocks. (1) There is at most one edge from node $h_{i}$ to node $h_{j}$, and
no edge is allowed from node $h_{j}$ to node $h_{i}$, where $i<j$. This is to
form the forward flow of a neural network. (2) The operator on an edge is
selected from the chosen candidate set, including the identity operator.
#### 3.1.2. Hyperparameter Search Space
We consider two kinds of hyperparameters: structural hyperparameters and
training hyperparameters. Table 1 summarizes the hyperparameters in the
hyperparameter search space and also lists their possible values. The
framework can easily include additional hyperparameters as well as expanded
ranges of values for existing hyperparameters.
Structural hyperparameters relate to the specific structure of an ST-block,
including the number $B$ of ST-blocks in a backbone, the number $C$ of nodes
in an ST-block, the hidden dimension $H$ of S/T-operators, the output
dimension $I$, and the output mode $U$ of an ST-block. Larger $B$, $C$, $H$,
and $I$ generally result in more expressive ST-blocks but also yield models
that are more prone to overfitting on small datasets. The output mode $U$ is a
binary value indicating which node in an ST-block produces the output. We
consider two alternative modes: one takes the last node $h_{C-1}$ as the
output, like AutoCTS (Wu et al., 2022), and the other takes the sum of all
nodes $h_{1}$, $h_{3}$, …, $h_{C-1}$ as the output, like Graph WaveNet (Wu et
al., 2019).
Training hyperparameters include the dropout rate $\delta$, which can be used
to alleviate overfitting when training a deep CTS model. The value of $\delta$
can be 0 or 1, which means dropout is used or not used. A set of chosen
hyperparameter values from the hyperparameter search space can be represented
as a $r$-dimensional vector ($r=6$ in this paper). For example, in Table 1,
[2, 5, 32, 64, 0, 0] is a possible hyperparameter vector.
Table 1. The Hyperparameter Search Space. _Hyperparameters_ | _Possible values_
---|---
B (number of ST-blocks) | {2, 4, 6}
C (number of nodes in an ST-block) | {5, 7}
H (hidden dimension) | {32, 48, 64}
I (output dimension) | {64, 128, 256}
U (output mode) | {0, 1}
$\delta$ (dropout) | {0, 1}
#### 3.1.3. Encoding of the Joint Search Space
Having designed the architecture and hyperparameter search spaces, we combine
them to construct a joint search space to support the search for an optimal
arch-hyper. Performing a naive combination is infeasible as the two search
spaces have different types of encodings (i.e., DAGs vs. vectors). We
therefore choose to design the joint search space as a joint dual DAG. This
encoding of the joint search space is easy to explore due to its efficient
representation.
We first convert the original DAG $G_{d}$ of an architecture (Figure 3 left)
in the architecture search space into its dual graph $G_{d}^{\ast}$ (Figure 3
middle), where nodes represent operators and edges represent information flow.
This dual form facilitates learning of the representation of an arch-hyper
using graph neural networks. Then, we add a new “Hyper” node that represents
the hyperparameter setting of the architecture to the dual DAG (Figure 3
middle). The “Hyper” node connects to all other nodes. This way, we can use a
single DAG $G_{a}$ (an arch-hyper graph) to represent a complete ST-block
containing both the candidate architecture and the hyperparameters, as shown
in the middle of Figure 3.
We use an adjacency matrix $A_{a}$ and a feature matrix $F_{a}$ to encode an
arch-hyper graph $G_{a}$. We consider a $G_{a}$ with $n+1$ nodes ($n=5$ in
Figure 3 middle), where $n$ nodes represent operators and one node represents
the hyperparameter settings. An adjacency matrix
$A_{a}\in\mathbb{R}^{(n+1)\times(n+1)}$ reflects the topology information of
$G_{a}$, where the binary value of an entry ($i$, $j$) indicates whether there
is information flow between these two nodes. We also add self-connections to
all nodes. A feature matrix $F_{a}\in\mathbb{R}^{(n+1)\times D}$ is also
included that contains operator information of each node in an arch-hyper
graph. For the “Hyper” node, the original feature is an $r$-dimensional vector
from the hyperparameter search space. We first employ min-max normalization to
normalize the original feature of the “Hyper” node and then convert the
normalized feature into a $D$-dimensional embedding:
(6) $\displaystyle F_{h}=norm(H_{o})W_{c},$
where $H_{o}\in\mathbb{R}^{r}$ is the original feature vector of the “Hyper”
node, $W_{c}\in\mathbb{R}^{r\times D}$ is a learnable matrix, and
$F_{h}\in\mathbb{R}^{D}$ is the embedding of the “Hyper” node. For the other
$n$ nodes (i.e., the operator nodes), we first embed each operator with an
one-hot embedding and then introduce a learnable matrix that converts the one-
hot embeddings of all operator nodes into an embedding matrix. Formally,
(7) $\displaystyle F_{e}=H_{e}W_{e},$
where $H_{e}\in\mathbb{R}^{n\times|O|}$ and $F_{e}\in\mathbb{R}^{n\times D}$
are the one-hot embeddings and the transformed embedding matrix of the $n$
nodes, respectively; further, $W_{e}\in\mathbb{R}^{|O|\times D}$ is the
learnable matrix, and $|O|$ is the number of candidate operator types in the
architecture search space ($|O|=4$ in Figure 3). The final feature matrix
$F_{a}\in\mathbb{R}^{(n+1)\times D}$ is the concatenation of the embeddings of
the “Hyper” node and the operator nodes, i.e.,
$F_{a}=concatenate(F_{h},F_{e})$. This way, each arch-hyper in the joint
search space can be encoded as an adjacency matrix $A_{a}$ and a feature
matrix $F_{a}$. The above learnable parameters $W_{c}$ and $W_{e}$ are learned
together with the model parameters of the AHC.
Figure 3. Architecture DAG, Arch-hyper Graph, and its adjacency matrix and
feature matrix representation. $o_{i}$ represents an operator.
### 3.2. Architecture-Hyperparameter Comparator
We propose a novel search framework to find the best arch-hyper in the joint
search space. Specifically, we rank arch-hypers according to pairwise
comparisons, enabled by an architecture-hyperparameter comparator (AHC), and
the top-ranked arch-hypers are selected as the final search result. Unlike in
the supernet-based search framework, where a large supernet that embeds all
candidate models must reside in memory, in the proposed search framework only
a lightweight AHC needs to reside in memory during search. This addresses the
second limitation of existing automated CTS frameworks, poor scalability.
To rank arch-hypers and to avoid evaluating all candidate pairs, many existing
studies build an accuracy estimator, which can be a neural network. The
accuracy estimator needs to be trained using a large volume of ($ah$, $R(ah)$)
samples, where $ah$ is an arch-hyper and $R(ah)$ is the validation accuracy of
a full trained $ah$. This way, the accuracy of all candidate arch-hypers in
the search space can be estimated, and top-ranked arch-hypers are then
selected. However, it is very time-consuming to collect a large amount of
($ah$, $R(ah)$) samples to train an accuracy estimator, since it is required
to fully train many $ah$ to get their validation accuracy $R(ah)$. Further,
absolute accuracy is not necessary to achieve the ranking of $ah$; instead,
the relative comparison result of two arch-hypers is sufficient to obtain
their ranking.
In light of the above considerations, we propose to use a comparator to
achieve the relative accuracy relation of two candidate arch-hypers.
Specifically, we design an AHC that takes the embeddings of two arch-hypers
($ah_{1}$, $ah_{2}$) as input and outputs a binary value $y$, indicating which
arch-hyper may have higher validation accuracy. Since the binary relation
facilitates obtaining a linear ordering, we can use the AHC to obtain the
predicted-accuracy based ranking of arch-hypers in the search space. Although
it is difficult to achieve a fully accurate ranking in practice, we
hypothesize that a high-accurate AHC can produce an ordering that approximates
the true linear ordering of arch-hypers in the search space, and we offer
experimental evidence of this in Section 4.2.6.
Given $a$ measured ($ah$, $R(ah)$) pairs, we can build $a(a-1)$ training
samples for AHC in the form of ($ah_{1}$, $ah_{2}$, $y$) by pairing every two
of ($ah$, $R(ah)$) pairs, where $y$ is a binary value indicating which arch-
hyper has higher accuracy, thus alleviating the issue of requiring a large
amount of training samples. A well trained AHC can then be used to compare all
($ah_{1}$, $ah_{2}$) pairs from the joint search space to obtain the magnitude
relation between each pair and thus the ranking of candidate arch-hypers.
Figure 4. Architecture-Hyperparameter Comparator.
Figure 4 shows the proposed AHC. The input to the AHC is a pair of arch-hypers
($ah_{1}$, $ah_{2}$), each of which is encoded by the adjacency matrix and
feature matrix of its joint graph as described in Section 3.1. Considering the
powerful ability of Graph Isomorphism Networks (GINs) to distinguish any two
graphs, we leverage GINs to encode the arch-hyper as a compact continuous
embedding. Formally, given the adjacency matrix $A_{a}$ and the feature matrix
$F_{a}$ of an arch-hyper graph, the corresponding GIN can be expressed
recursively as follows:
(8) $\displaystyle GIN(A_{a},F_{a})=H^{(L)}$ (9) $\displaystyle
H^{(k)}=MLP^{(k)}((1+\epsilon^{(k)})\cdot H^{(k-1)}+AH^{(k-1)}),k=1,2,...,L,$
where $L$ is the number of GIN layers, $H^{(0)}=F_{a}$, $\epsilon$ is a
trainable bias, and $MLP$ is a multi-layer perceptron. To simplify the
representation of an arch-hyper, we use the latent representation of the
“Hyper” node from $H^{(L)}$ as the representation of the entire arch-hyper,
denoted as $l_{a}$, since the “Hyper” node connects to all other nodes in the
arch-hyper graph.
We encode two input arch-hypers $ah_{1}$ and $ah_{2}$ as $l_{a}$ and
$l_{a}^{\prime}$, respectively, using the same GIN and then concatenate them
in the feature dimension:
(10) $\displaystyle L_{a}=concatenate(l_{a},l_{a}^{\prime}).$
Finally, we feed $L_{a}$ into a classifier that is composed of a fully-
connected (FC) layer and a Sigmoid function $\sigma(\cdot)$. We use 0.5 as the
threshold to force the output of the classifier to only be 0 (less than the
threshold) or 1 (larger than the threshold). Formally,
(11) $\displaystyle output=\mathbbm{1}(\sigma(FC(L,w_{l}))\geq 0.5),$
where $w_{l}$ is the parameter matrix of the fully-connected layer, and
$\mathbbm{1}(\cdot)$ is an indicator function with $\mathbbm{1}(\mathrm{X})=1$
if $\mathrm{X}$ is true, and $\mathbbm{1}(\mathrm{X})=0$ otherwise. We say
that $R(ah)\geq R^{\prime}(ah)$ holds if the output of the AHC is 1, and that
$R(ah)<R^{\prime}(ah)$ holds if the output of the AHC is 0. We optimize
$W_{c}$, $W_{e}$, and the parameters of the AHC with Binary Cross-Entropy
(BCE) loss.
### 3.3. Training AHC with Noisy Proxies
Training a reliable AHC requires a large number of training samples of the
form $(ah_{1},ah_{2},y)$, where $ah_{1}$ and $ah_{2}$ are two arch-hypers, and
$y=\mathbbm{1}(R(ah_{1})\geq R(ah_{2}))$ with $R(ah)$ being the forecasting
accuracy of an arch-hyper $ah$ on the validation set. However, it is costly to
achieve $R(ah)$ since it is required to train an $ah$ from scratch to achieve
its forecasting accuracy. Furthermore, we only need to compare $R(ah_{1})$ and
$R(ah_{2})$ so an exact forecasting accuracy is not essential. Therefore, a
reasonable and cheaper alternative is a lightweight proxy $R^{\prime}(\cdot)$
that when $R^{\prime}(ah_{1})\geq R^{\prime}(ah_{2})$ holds, $R(ah_{1})\geq
R(ah_{2})$ holds with high confidence. In other words, we aim to find a proxy
$R^{\prime}(\cdot)$ that can generate pseudo-labels
$y^{\prime}=\mathbbm{1}(R^{\prime}(ah_{1})\geq R^{\prime}(ah_{2}))$ to
approximate the true label $y$ for $(ah_{1},ah_{2})$.
This way, we are supposed to design a computation-efficient
$R^{\prime}(\cdot)$, so as easily obtain a large number of noisy training
samples $(ah_{1},ah_{2},y^{\prime})$. We call them noisy training samples
because $y^{\prime}$ may be incorrectly labelled sometimes. In this
subsection, we first propose an effective proxy $R^{\prime}(\cdot)$ and then
describe how to train the AHC in a noise reduction manner to reduce the
negative effects introduced by the noisy training samples produced by the
proxy.
#### 3.3.1. Proxy performance metric
Recent studies have proposed several proxy metrics to quickly measure the
performance of neural networks without full training, such as the number of
parameters (nparam) (Ning et al., 2021), Synflow (Abdelfattah et al., 2020),
Snip (Abdelfattah et al., 2020), and neural tangent kernel (NTK) score (Chen
et al., 2021a). These metrics can be used to generate a pseudo-label
$y^{\prime}$ for each $(ah_{1},ah_{2})$ pair, and consequently generate noisy
training samples of the form $(ah_{1},ah_{2},y^{\prime})$. However, these
proxy metrics are tailored for the computer vision domain, whose operators and
model architectures are very different from ours, and thus may not be suitable
for CTS forecasting. Therefore, the noisy training samples
$(ah_{1},ah_{2},y^{\prime})$ generated by these proxies can be severely
mislabeled and too noisy to achieve a reliable AHC.
Here, we propose a computation-efficient proxy metric specifically for CTS
forecasting, which we later empirically demonstrate to provide a better
approximation to $y$ than the aforementioned proxy metrics. During
experiments, we notice that if the validation accuracy of an arch-hyper is
higher than another in the first few epochs, then its final validation
accuracy is very likely to be higher as well. Based on this observation, we
propose to train an arch-hyper for only $k$ ($k\leq 5$) epochs and use the
validation accuracy of the under-trained model as a proxy. We call the
proposed performance proxy metric as early-validation proxy that:
(12) $\displaystyle
R^{\prime}(ah)=\mathit{ErrorMetric}(\mathcal{F}(ah)_{k},\mathcal{D}_{val}),$
where $\mathcal{F}(ah)_{k}$ is the CTS forecasting model under arch-hyper
setting $ah$ with only $k$ epochs training.
Next, we conduct experiments on two CTS forecasting datasets, PEMS04 and
PEMS08, to demonstrate that the early-validation proxy is superior to existing
performance proxy metrics. Specifically, for each dataset, we collect $M$
($M$=50 in this experiment) ($ah$, $R(ah)$) samples by fully training each
$ah$, and then pair up them to build $M\times(M-1)$ $(ah_{1},ah_{2},y)$
samples, where $y=\mathbbm{1}(R(ah_{1})\geq R(ah_{2}))$. Then we use the
proposed and existing proxy metrics to generate pseudo-label $y^{\prime}$ for
each $(ah_{1},ah_{2})$, and count how many samples are wrongly labeled. We
propose the pair-wise ranking accuracy (PRA) to measure the performance of
each proxy that:
(13) $\displaystyle PRA(R^{\prime})=\frac{\sum_{m=1,m\neq
n}^{M}\sum_{n=1}^{M}\Delta(R^{\prime}(ah_{n}),R^{\prime}(ah_{m}))}{M(M-1)},$
where $\Delta(R^{\prime}(ah_{n}),R^{\prime}(ah_{m}))$ is 1 when
$R^{\prime}(ah_{n})\geq R^{\prime}(ah_{m})$ and $R(ah_{n})\geq R(ah_{m})$, or
0 when $R^{\prime}(ah_{n})<R^{\prime}(ah_{m})$ and $R(ah_{n})<R(ah_{m})$,
otherwise $\Delta(R^{\prime}(ah_{n}),R^{\prime}(ah_{m}))$ is 0.
The pair-wise ranking accuracy for different proxies is shown in Table 2. It
is clear that the proposed proxy metric has significantly higher accuracy than
existing proxy metrics on both datasets. Thus, the proposed proxy metric is
more suitable for generating noisy training samples for the AHC.
Table 2. Comparison between the proposed and existing proxy metrics on pair-wise ranking accuracy. | NTK | Synflow | Snip | nParam | Ours(k=5)
---|---|---|---|---|---
PEMS04 | 0.64 | 0.57 | 0.49 | 0.59 | 0.80
PEMS08 | 0.69 | 0.65 | 0.50 | 0.63 | 0.82
#### 3.3.2. Training AHC with denoising algorithms
While we can use the early-validation proxy metric to generate a large number
of noisy training samples in a highly efficient manner, it is far from enough
to achieve a reliable AHC. This is because some noisy samples are wrongly
labeled, and the AHC trained with these noisy samples may not be accurate. To
minimize the impact of training with noisy samples, we fully train a few
additional arch-hypers to obtain clean training samples of the form
$(ah_{1},ah_{2},y)$ and then train the AHC with both noisy and clean samples
in a denosing manner. Formally, given a noisy training set $S_{1}$ containing
$|S_{1}|$ samples $(ah_{1},ah_{2},y^{\prime})$ and a clean training set
$S_{2}$ containing $|S_{2}|$ samples $(ah_{1},ah_{2},y)$
($|S_{1}|\gg|S_{2}|$), our goal is to train an AHC with $S_{1}\bigcup S_{2}$
better than training with only $S_{1}$ or $S_{2}$.
There are many alternative methods for training with both noisy and clean
labels and have been shown to achieve promising performance on a variety of
tasks (Zhang et al., 2020; Shu et al., 2019; Ren et al., 2018; Kang et al.,
2022). We adopt a commonly used denoising training method, fine-tuning, in our
framework. Note that our framework supports the use of other denoising
training methods, and we use fine-tuning in this paper for its simplicity and
effectiveness. Specifically, we first randomly sample $L_{1}$ arch-hypers from
the joint search space and use the proposed proxy metric to obtain the proxy
score $R^{\prime}(ah)$ for each arch-hyper $ah$. Then we pair up these arch-
hypers to produce $L_{1}(L_{1}-1)$ noisy samples of the form
$(ah_{1},ah_{2},y^{\prime})$. Next, we randomly sample $L_{2}$
($L_{2}<<L_{1}$) arch-hypers and train them completely to obtain the
validation accuracy $R(ah)$ for each arch-hyper $ah$. Then we also pair up
these arch-hypers to produce $L_{2}(L_{2}-1)$ clean samples of the form
$(ah_{1},ah_{2},y)$. Note that the processes of collecting noisy and clean
samples are independent and can be highly parallelized. After collecting noisy
and clean samples, we first warm up the AHC for $k_{t}$ epochs using the noisy
samples, and then use the clean samples to finetune the AHC until convergence.
### 3.4. Search Strategy and AHC Transfer
#### 3.4.1. Search strategy
When a well-trained AHC is achieved, the process for searching the optimal
arch-hyper starts. Since the search space is enormous, it is inefficient to
compare all candidate hyper-archs using the AHC to get the optimal one. To
address this issue, we first shrink the joint search space by removing
obviously ineffective arch-hypers based on domain knowledge. Specifically, we
remove the arch-hypers that do not contain either spatial or temporal
operators since existing works (Wu et al., 2019, 2020) demonstrate that
considering only temporal or spatial dependencies leads to poor forecasting
performance. After that, we consider a heuristic approach, e.g., evolutionary
algorithm (Guo et al., 2020), to find the best arch-hyper in the shrunk joint
search space. Specifically, we first sample $K_{s}$ arch-hypers, which are
paired up to produce $K_{s}(K_{s}-1)/2$ comparison pairs of the form
$(ah_{1},ah_{2})$, and the descending ranking of the $K_{s}$ arch-hypers can
be easily obtained based on the comparative performance determined by the
trained AHC. Then, we select the top $k_{p}$ from the $K_{s}$ arch-hypers in
descending order as the initial population. Each arch-hyper has crossover and
mutation probability $p_{1}$ and $p_{2}$, respectively, when generating new
offspring in each evolution step. The offspring are added to the population,
and the learned AHC is used to compare arch-hypers in the population and to
remove inferior arch-hypers to keep the population size at $k_{p}$. Lastly, we
choose the top-$K$ arch-hypers from the population to collect their exact
forecasting accuracy and pick the one with the highest accuracy as the final
searched ST-block. The reason we choose top-$K$ arch-hypers instead of the
top-1 is that the achieved ranking is not perfectly aligned with the true
ranking, thus the top-1 arch-hyper may not be the best arch-hyper. In
addition, we also note that a random search strategy could replace the
evolutionary algorithm for simplicity (Bender et al., 2018).
#### 3.4.2. Transfer a well-trained AHC
A practical goal of deep learning for real-world tasks is to learn features
that are transferable. For example, a pretrained model (e.g., ResNet50) on
ImageNet is commonly used as the initialization for finetuning on downstream
tasks or different datasets. Naturally, we expect that a well-trained AHC can
be an effective initialization for searching on unseen datasets. Although an
arch-hyper performs differently on different CTS forecasting datasets, we
observe that the relative performance of two models depends not only on the
data, but also on the arch-hyper itself. For example, Graph WaveNet (Wu et
al., 2019) consistently outperforms STGCN (Yu et al., 2018) on different CTS
forecasting datasets, demonstrating that the knowledge of the superiority of
Graph WaveNet keeps consistent across datasets. In light of the above
observations, designing an automated framework with transferability is
essential and feasible for practical CTS forecasting applications. Unlike
existing automated CTS forecasting frameworks with poor transferability, our
well-trained AHC can be transferred to unseen datasets to further improve the
efficiency of the proposed framework.
Specifically, given a well-trained AHC $\mathcal{N}_{s}$ on a source dataset
$D_{s}$ (e.g., we used $L_{1}$ noisy samples and $L_{2}$ clean samples for
training), a new well-trained AHC $\mathcal{N}_{t}$ on a target dataset
$D_{t}$ can be obtained by transferring $\mathcal{N}_{s}$ from $D_{s}$ to
$D_{t}$ with much less arch-hyper samples $ah_{t}$ on $D_{t}$:
(14) $\mathcal{N}_{t}(D_{t})\xleftarrow[]{ah_{t}}\mathcal{N}_{s}(D_{s}),$
where $ah_{t}$ consists of $z_{1}$ noisy samples and $z_{2}$ clean samples,
and $z_{1}\ll L_{1},z_{2}\ll L_{2}$. The transfer process is particularly
implemented using the fine-tuning technique. Our AHC transfer strategy can
overcome the one-time use shortcoming of existing automated frameworks and
significantly improve efficiency. We experimentally demonstrate that the
proposed transfer method can significantly improve search efficiency without
loss of search accuracy compared to training our framework on new datasets
from scratch. The complete search strategy, including the optional transfer,
is shown in Algorithm 1.
Algorithm 1 Search Algorithm
Input: target CTS dataset $\mathcal{D}$, joint search space $\Omega$, AHC
$\mathcal{N}_{s}$ on a source dataset $\mathcal{D}_{s}$ (optional),
$\\{L_{1},L_{2},z_{1},z_{2}\\}$ s.t. $L_{1}\gg L_{2}$, $L_{1}\gg z_{1}$,
$L_{2}\gg z_{2}$;
Output: optimal arch-hyper $ah^{*}$
1:Split $\mathcal{D}$ into $\mathcal{D}_{train}$, $\mathcal{D}_{val}$, and
$\mathcal{D}_{test}$
2:if $\mathcal{N}_{s}$ exists:
3: Randomly sample $s_{z_{1}}=\\{ah\in\Omega\\}$, $|s_{z_{1}}|=z_{1}$
4: Generate $S_{z_{1}}=\\{(ah_{1},ah_{2},y^{\prime})|ah_{*}\in s_{z_{1}}\\}$,
$|S_{z_{1}}|=z_{1}(z_{1}-1)$
5: Randomly sample $s_{z_{2}}=\\{ah\in\Omega\\}$, $|s_{z_{2}}|=z_{2}$
6: Generate $S_{z_{2}}=\\{(ah_{1},ah_{2},y)|ah_{*}\in s_{z_{2}}\\}$,
$|S_{z_{2}}|=z_{2}(z_{2}-1)$
7: while not converged do
8: $\mathcal{N}_{t}$=fine-tuning($\mathcal{N}_{s},{S_{z_{1}}},{S_{z_{2}}}$)
9: end
10:else
11: Randomly sample $s_{1}=\\{ah\in\Omega\\}$, $|s_{1}|=L_{1}$
12: Generate $S_{1}=\\{(ah_{1},ah_{2},y^{\prime})|ah_{*}\in s_{1}\\}$,
$|S_{1}|=L_{1}(L_{1}-1)$
13: Randomly sample $s_{2}=\\{ah\in\Omega\\}$, $|s_{2}|=L_{2}$
14: Generate $S_{2}=\\{(ah_{1},ah_{2},y)|ah_{*}\in s_{2}\\}$,
$|S_{2}|=L_{2}(L_{2}-1)$
15: Initialize $\mathcal{N}_{t}$
16: for t = 1, …, $k_{t}$ do
17: Train $\mathcal{N}_{t}$ with $S_{1}$.
18: end for
19: while not converged do
20: Train $\mathcal{N}_{t}$ with $S_{2}$
21: end
22:end if
23:Shrink $\Omega$ to $\Omega_{s}$ with domain knowledge
24:Heuristic search and rank $ah$ $\in$ $\Omega_{s}$ using the trained AHC.
25:Train and evaluate the top-$K$ $ah\in\Omega_{s}$ on $\mathcal{D}_{train}$
and $\mathcal{D}_{val}$, respectively
26:return The optimal $ah^{*}$ that yields the highest validation accuracy
## 4\. Experiments
We conduct experiments on four CTS datasets for multi-step forecasting and two
CTS datasets for single-step forecasting. The results demonstrate that our
proposed framework successfully addresses the three limitations of previous
work with higher forecasting accuracy, lower memory consumption, and faster
search process.
### 4.1. Experimental Settings
#### 4.1.1. Datasets
To enable easy and fair comparisons, we use widely adopted benchmark datasets
for single- and multi-step forecasting experiments. We consider both kinds of
forecasting for two reasons. First, the literature on CTS forecasting often
considers both of these—thus, considering both enables fair comparisons.
Second, the two kinds of forecasting focus on evaluating different aspects of
the abilities of CTS models, with multi-step forecasting typically focusing on
evaluating the “average” ability to capture long-term and short-term temporal
dependencies, while single-step forecasting focuses on evaluating the ability
to capture either long-term or short-term dependencies, depending on the time
length of a single step.
Multi-step forecasting:
* •
PEMS03, PEMS04, PEMS07 and PEMS08 (Song et al., 2020): These four datasets
record the traffic flow in four different regions of California, and are
collected from the Caltrans Performance Measurement System (PeMS). Each
dataset contains traffic flow data of hundreds of roads, and the traffic flow
readings are aggregated into 5-minute windows, resulting in 12 data points per
hour. We construct an adjacency matrix reflecting the correlation among roads
for each dataset based on pairwise road network distances (Song et al., 2020).
We summarize the statistics of the four datasets in Table 3, where $N$
represents the number of time series, $T$ represents the total number of
timestamps, and “Split Ratio” refers to the ratio between train, validation,
and test sets. We consider two different settings of input ($P$) and
forecasting ($Q$) lengths, i.e., $P$-12/$Q$-12 and $P$-48/$Q$-48,
corresponding to short- and long-term CTS forecasting, respectively.
Single-step forecasting:
* •
Solar-Energy (Lai et al., 2018): This dataset contains the solar power
production records collected from 137 PV plants in the Alabama State.
* •
Electricity (Lai et al., 2018): This dataset contains the electricity
consumption records collected from 321 clients.
We also summarize the statistics of the two datasets in Table 3. Following
existing literature (Lai et al., 2018; Shih et al., 2019; Wu et al., 2020), we
use the past 168 timestamps to predict the future 1 timestamp, where the
future 1 timestamp can be the 3rd or 24th timestamp. Note that the single-step
datasets do not include spatial information that enables computing the
distances among sensors. Thus, there is no predefined distance-based adjacency
matrix, which also means that the strengths of spatial correlations between
time series in the single-step datasets are unknown in advance.
Table 3. Dataset statistics. Split ratio refers to the train-validation-test split ratio. P/Q refers to the input length and the forecasting length. Dataset | $N$ | $T$ | Split Ratio | $P$/$Q$ | $P$/$Q$
---|---|---|---|---|---
PEMS03 | 358 | 26,208 | 6:2:2 | 12/12 | 48/48
PEMS04 | 307 | 16,992 | 6:2:2 | 12/12 | 48/48
PEMS07 | 883 | 28,224 | 6:2:2 | 12/12 | 48/48
PEMS08 | 170 | 17,856 | 6:2:2 | 12/12 | 48/48
Solar-energy | 137 | 52,560 | 6:2:2 | 168/1 (3rd) | 168/1 (24th)
Electricity | 321 | 26,304 | 6:2:2 | 168/1 (3rd) | 168/1 (24th)
#### 4.1.2. Evaluation Metrics
To compare the performance of different CTS forecasting models, we follow
previous studies (Yu et al., 2018; Li et al., 2018; Wu et al., 2019; Bai et
al., 2020; Wu et al., 2020) to use mean absolute error (MAE), root mean
squared error (RMSE), and mean absolute percentage error (MAPE) for estimating
the accuracy of multi-step forecasting, and use Root Relative Squared Error
(RRSE) and Empirical Correlation Coefficient (CORR) for estimating the
accuracy of single-step forecasting. For MAE, RMSE, MAPE, and RRSE, smaller
values are better, while larger values for CORR are better. To evaluate the
performance of the AHC, we use Spearman’s rank correlation coefficient
($\rho$) to measure the similarity of the rankings produced by the AHC to the
true ranking of arch-hypers.
#### 4.1.3. Baselines
In recent works, we have seen that automated methods often outperform manually
designed methods, thus we compare the proposed framework with two best-
performing manually designed CTS forecasting models and two state-of-the-art
automated methods. The results of the baselines are obtained by running the
originally released source code.
* $\bullet$
MTGNN: A multivariate time series forecasting model, which employs mix-hop
graph convolution and dilated inception convolution to build ST-blocks (Wu et
al., 2020).
* $\bullet$
AGCRN: Adaptive graph convolutional recurrent network, which employs 1D GCN
and GRU to build ST-blocks (Bai et al., 2020).
* $\bullet$
AutoSTG: A supernet-based automated CTS forecasting framework, which employs
DGCN and 1D convolution to build the search space, and introduces meta
learning to learn the weights of neural operators (Pan et al., 2021). However,
it relies on the coordinates of sensors to build the attributed graph to work,
which are not available on the evaluation datasets, so we replace them with
the ordinal number of time series during experiment.
* $\bullet$
AutoCTS: A supernet-based automated CTS forecasting framework, which focuses
on selecting the optimal set of neural operators to build the search space (Wu
et al., 2022).
#### 4.1.4. Implementation Details
We have implementation for both preparing AHC and training the CTS forecasting
model.
Setting up the AHC. Since we allow ST-blocks to have different numbers of
nodes, the shape of the adjacency matrix $A_{a}$ may be different for
different arch-hypers. Thus, we pad the size of the adjacency matrix to 14
with zero paddings. We set the number $L$ of layers of the GIN to 4, with
$D=128$ hidden units in each layer. For training the AHC, we use Adam (Kingma
and Ba, 2014) with a learning rate of 0.0001 and a weight decay of 0.0005 as
the optimizer. The batch size is set to 8. For pretraining the AHC on the
source dataset, we first use $L_{1}=2000$ arch-hypers to generate noisy
samples to train the AHC for $k_{t}=10$ epochs, with an early stopping
patience of 3 epochs; then we use $L_{2}=150$ arch-hypers to generate clean
samples to finetune the AHC for 10 epochs, with an early stopping patience of
3 epochs. For transferring the AHC to a target dataset, we only use
$z_{1}=100$ arch-hypers to generate noisy samples and $z_{2}=5$ arch-hypers to
generate clean samples to finetune a trained AHC for 3 epochs.
We set the crossover and mutation probability $p_{1}$ and $p_{2}$ to 0.8 and
0.2, respectively, and the population size $k_{p}$ to 10. Lastly, we choose
the top-3 arch-hypers from the population.
Setting up CTS forecasting models. We use MAE as the training objective to
train CTS forecasting models, and use Adam with a learning rate of 0.001 and a
weight decay of 0.0001 as the optimizer. The batch size is set to 64. For
generating clean samples, we set the training epochs to 100. For generating
noisy samples, we set the training epochs $k$ to 5.
Reproducibility. To support the reproducibility, we provide the source code
and links to the datasets in the supplementary materials. Source code will be
released upon acceptance. We conduct all experiments on multiple Nvidia Quadro
RTX 8000 GPUs.
### 4.2. Experimental Results
#### 4.2.1. Performance Comparison
Table 4 presents the performance comparison between our framework and the
baselines on the four multi-step CTS forecasting datasets. To demonstrate the
robustness of our framework to different source datasets for AHC transfer, we
report the results of two specific settings Ours-1 and Ours-2, which use
PEMS08 and PEMS04 as the source datasets, respectively. Since the baselines do
not manually tune hyperparameters under the $P$-48/$Q$-48 setting, for fair
comparison, we conduct grid-search for them to find the best hidden dimension
$H$ and the output dimension $I$, and also include the hyperparameter setting
they use under the $P$-12/$Q$-12 setting. To facilitate observation, we use
bold and underline to highlight the best and the second best results,
respectively. We summarize main observations as follows.
Table 4. Performance Comparison of Multi-step Forecasting. Data | Metric | $P$-12/$Q$-12 | $P$-48/$Q$-48
---|---|---|---
MTGNN | AGCRN | AutoSTG | AutoCTS | _Ours-1_ | _Ours-2_ | MTGNN | AGCRN | AutoSTG | AutoCTS | _Ours-1_ | _Ours-2_
PEMS03 | MAE | 15.10 | 15.89 | 17.97 | 14.71 | 14.60 | 14.59 | 20.66 | 19.84 | 22.46 | 19.61 | 18.37 | 18.44
RMSE | 25.93 | 28.12 | 28.47 | 24.54 | 24.35 | 24.21 | 34.13 | 33.93 | 37.57 | 32.93 | 30.86 | 30.77
MAPE | 15.67% | 15.38% | 18.08% | 14.39% | 13.85% | 14.02% | 24.31% | 18.56% | 21.32% | 19.80% | 17.69% | 17.76%
PEMS04 | MAE | 19.32 | 19.83 | 20.46 | 19.13 | 18.97 | 18.95 | 24.56 | 22.83 | 26.17 | 23.69 | 22.68 | 22.74
RMSE | 31.57 | 32.26 | 32.18 | 30.44 | 30.36 | 30.31 | 37.15 | 36.20 | 41.07 | 36.45 | 35.28 | 35.35
MAPE | 13.52% | 12.97% | 13.77% | 12.89% | 12.81% | 12.75% | 19.35% | 15.09% | 18.02% | 18.04% | 15.93% | 15.82%
PEMS07 | MAE | 22.07 | 21.31 | 26.77 | 20.93 | 20.65 | 20.77 | 25.74 | 24.90 | 38.56 | 25.49 | 23.74 | 23.53
RMSE | 35.80 | 35.06 | 41.63 | 33.69 | 33.54 | 33.49 | 40.59 | 41.48 | 62.63 | 40.33 | 38.69 | 38.46
MAPE | 9.21% | 9.13% | 11.63% | 8.90% | 8.81% | 8.76% | 11.78% | 10.68% | 18.14% | 11.63% | 10.69% | 10.57%
PEMS08 | MAE | 15.71 | 15.95 | 16.23 | 14.82 | 14.68 | 14.72 | 20.37 | 19.44 | 21.25 | 18.85 | 17.68 | 17.73
RMSE | 24.62 | 25.22 | 25.72 | 23.64 | 23.46 | 23.43 | 30.75 | 31.40 | 34.59 | 29.13 | 27.95 | 27.98
MAPE | 10.03% | 10.09% | 10.25% | 9.51% | 9.41% | 9.45% | 16.69% | 13.38% | 14.39% | 15.08% | 12.60% | 12.47%
First, our framework, either Ours-1 or Ours-2, achieves consistent state-of-
the-art forecasting accuracy on all datasets and $P$/$Q$ settings. In
particular, our framework outperforms two existing automated methods, AutoSTG,
which employs different S/T operators and a different search algorithm, and
AutoCTS, which employs the same S/T operators but a different search
algorithm.
Second, our framework achieves more significant improvements on the
$P$-48/$Q$-48 setting than on the $P$-12/$Q$-12 setting. This is because the
architectures and hyperparameters of the baselines are specifically designed
for the $P$-12/$Q$-12 setting in the original studies. And under the
$P$-48/$Q$-48 setting, although we conduct grid-search to find the best
hyperparameters for the baselines, it does not yield competitive arch-hypers.
In contrast, our proposed joint search framework can find high-performance
arch-hypers under different $P$/$Q$ settings. This result indicates the
necessity of joint search architecture and hyperparameters and the advances of
our proposed framework in this context.
Third, the results of Ours-1 and Ours-2 are overall comparable. This
demonstrates that our AHC transfer method is effective regardless which
dataset is used as the source dataset. Besides, the results using the
transferred AHC (ours-1 on PEMS04 and ours-2 on PEMS08) and the originally
trained AHC (ours-2 on PEMS04 and ours-1 on PEMS08) are closely analogous,
demonstrating that our AHC transfer method can improve the search efficiency
without sacrificing accuracy.
Table 5 presents the performance comparison between our framework and the
baselines on the two single-step CTS forecasting datasets, where the source
datasets of Ours-1 and Ours-2 remain PEMS08 and PEMS04, respectively. Since we
use the same experimental settings, the results of MTGNN and AutoCTS in Table
5 are obtained from the original papers. Next, as AGCRN and AutoSTG do not
report results on the single-step datasets, for a fair comparison, we conduct
a grid search to determine the best settings for two key hyperparameters: the
hidden dimension $H$ and the output dimension $I$. We keep the settings of
other hyperparameters as in the original papers. Note that AutoSTG relies on a
predefined adjacency matrix for spatial graph convolution operators. As such a
matrix is not available in the single-step CTS datasets, we drop the spatial
graph convolution operators in AutoSTG’s search space when running it on the
single-step CTS datasets. Also note that none of the remaining baselines and
our framework require predefined adjacency matrices, as they encompass self-
adaptive adjacency matrices to learn correlations among time series. We
observe similar trends for single-step forecasting and the multi-step
forecasting. Specifically, our framework achieves the consistently best
performance across all datasets and forecasting timestamps, suggesting the
success of our joint search; our searched models with different AHC source
datasets, ours-1 and ours-2, have comparable results, indicating that our AHC
transfer method is robust to different source datasets.
Table 5. Performance Comparison of Single-step Forecasting. Data | Solar-Energy | Electricity
---|---|---
Models | Metric | 3 | 24 | 3 | 24
MTGNN | RRSE | 0.1778 | 0.4270 | 0.0745 | 0.0953
CORR | 0.9852 | 0.9031 | 0.9474 | 0.9234
AGCRN | RRSE | 0.1830 | 0.4602 | 0.1033 | 0.0994
CORR | 0.9846 | 0.9016 | 0.8854 | 0.9073
AutoSTG | RRSE | 0.2094 | 0.5066 | 0.1188 | 0.0998
CORR | 0.9811 | 0.8611 | 0.9070 | 0.8846
AutoCTS | RRSE | 0.1750 | 0.4143 | 0.0743 | 0.0947
CORR | 0.9855 | 0.9085 | 0.9477 | 0.9239
_Ours-1_ | RRSE | 0.1657 | 0.3980 | 0.0736 | 0.0921
CORR | 0.9875 | 0.9152 | 0.9483 | 0.9253
_Ours-2_ | RRSE | 0.1663 | 0.4009 | 0.0732 | 0.0935
CORR | 0.9886 | 0.9138 | 0.9487 | 0.9244
#### 4.2.2. Scalability Study
We compare the scalability of our framework and existing automated frameworks
and plot the results in Figure 5. The PEMS07 dataset is used for experiments
because it has the largest number of time series $N$, which is most suitable
for observing how the memory cost increases with $N$. Specifically, we first
fix the number $P$ of input timestamps to 12, and vary
$N\in\\{100,200,300,400,500,600,700,800\\}$ to observe the scalability w.r.t.
$N$; then, we fix $N$ to 50, and vary $P\in\\{12,24,36,48,60,72,84,96\\}$ to
observe the scalability w.r.t. $P$.
In Figure 5(a), the memory usage of AutoSTG and AutoCTS grows rapidly with
$N$, while the memory usage of our framework is a constant and does not change
with $N$. Similarly, as shown in Figure 5(b), the memory usage of AutoSTG and
AutoCTS grows rapidly with $P$, while the memory usage of our framework is a
small constant and does not change with $P$. This demonstrates that the
proposed framework is more scalable than existing automated frameworks. The
reason is that the AHC is implemented with a lightweight graph neural network
and is independent of CTS datasets. While the memory cost of existing
automated frameworks is positively correlated with $N$ and $P$ of CTS
datasets. The high scalability of our framework allows to work on various
sizes of CTS datasets (i.e., a large range of $N$ and $P$), while AutoSTG and
AutoCTS may fail because their memory cost goes overflow more easily.
(a) Memory w.r.t. N
(b) Memory w.r.t. P
Figure 5. Scalability Comparison on the PEMS07 dataset.
#### 4.2.3. Transferability Study
We study the effectiveness of the proposed AHC transfer approach and summarize
the results in Table 6. The PEMS04 and PEMS08 datasets are used as each
other’s source and target datasets. We build three variants: (1) w/ transfer
that uses the proposed AHC transfer approach with $z_{1}=100$
$(ah,R^{\prime}(ah))$ pairs and $z_{2}=5$ $(ah,R(ah))$ pairs for fine-tuning;
(2) w/o transfer that trains an AHC directly on the target dataset using
$z_{1}=100$ $(ah,R^{\prime}(ah))$ pairs and $z_{2}=5$ $(ah,R(ah))$ pairs
without transfer; (3) random that randomly selects 3 arch-hypers from the
joint search space and trains them completely to get the best one. To
eliminate the bias caused by the sampling procedure, we first determine five
random seeds. For w/ transfer and w/o transfer, we select arch-hypers to
generate noisy and clean samples under each random seed setting (with the same
settings for $z_{1}$ and $z_{2}$). For random, we select 3 arch-hypers under
each random seed setting. For each variant, we collect the results obtained
with the five different random seeds and report their mean and standard
deviation.
We can observe that for both datasets the w/o transfer is significantly worse
than the w/ transfer with almost the same performance as the random variant.
It demonstrates that without AHC transfer, a small number of
$(ah,R^{\prime}(ah))$ and $(ah,R(ah))$ pairs is far from enough to achieve a
reliable AHC and the search based on such an AHC is just like random.
Conversely, given a AHC trained on a source dataset, we can easily obtain a
reliable AHC on a target dataset by fine-tuning it with a small number of
$(ah,R^{\prime}(ah))$ and $(ah,R(ah))$ pairs, thus ensuring high training
efficiency.
Table 6. Transferability Study on the $P$-12/$Q$-12 Multi-step Forecasting (mean$\pm$standard deviation). Target | Metric | w/ transfer | w/o transfer | random
---|---|---|---|---
Dataset
PEMS04 | MAE | 18.93$\pm$0.08 | 19.44$\pm$0.16 | 19.36$\pm$0.19
RMSE | 30.29$\pm$0.11 | 30.62$\pm$0.25 | 30.64$\pm$0.21
MAPE | 12.76%$\pm$0.13% | 14.08%$\pm$0.28% | 14.39%$\pm$0.33%
PEMS08 | MAE | 14.64$\pm$0.09 | 15.38$\pm$0.24 | 15.32$\pm$0.22
RMSE | 23.41$\pm$0.15 | 24.19$\pm$0.19 | 24.14$\pm$0.26
MAPE | 9.44%$\pm$0.11% | 10.26%$\pm$0.31% | 10.68%$\pm$0.36%
#### 4.2.4. Ablation Studies
We conduct ablation studies to investigate the effectiveness of each key
component of the proposed framework. We report results for all datasets and
settings in Tables 7 to 9, where the source dataset of Ours-1 remains PEMS08,
and similar trends can be observed when using Ours-2, i.e., using PEMS04 as
the source dataset. We compare our framework with the following variants.
* $\bullet$
w/o joint only searches for architectures. We use a fixed hyperparameter
setting (4, 5, 32, 256, 1, 0), which is commonly used in previous studies (Wu
et al., 2019, 2022), and perform our framework to search for the best
architecture.
* $\bullet$
w/o clean does not train an AHC with clean samples but only with noisy
samples;
* $\bullet$
w/o noisy does not train an AHC with noisy samples but only with clean
samples;
* $\bullet$
w/o denoising does not train an AHC with the denoising method. Instead, it
blends clean and noisy samples for AHC training.
Table 7. Ablation studies, $P$-12/$Q$-12 Data | Metric | _Ours-1_ | w/o | w/o | w/o | w/o
---|---|---|---|---|---|---
joint | clean | noisy | denosing
PEMS03 | MAE | 14.60 | 14.81 | 14.62 | 14.74 | 14.67
RMSE | 24.35 | 24.69 | 24.39 | 24.52 | 24.22
MAPE | 13.85% | 14.43% | 14.05% | 14.38% | 14.12%
PEMS04 | MAE | 18.97 | 19.32 | 19.11 | 19.24 | 19.01
RMSE | 30.36 | 30.59 | 30.38 | 30.50 | 30.34
MAPE | 12.81% | 13.05% | 12.85% | 13.12% | 12.91%
PEMS07 | MAE | 20.65 | 20.98 | 20.89 | 20.95 | 20.83
RMSE | 33.54 | 33.84 | 33.68 | 33.72 | 33.60
MAPE | 8.81% | 8.97% | 8.92% | 9.06% | 8.85%
PEMS08 | MAE | 14.68 | 15.07 | 14.83 | 14.89 | 14.77
RMSE | 23.46 | 24.01 | 23.58 | 23.76 | 23.53
MAPE | 9.41% | 9.79% | 9.64% | 9.81% | 9.52%
Table 8. Ablation studies, $P$-48/$Q$-48. Data | Metric | _Ours-1_ | w/o | w/o | w/o | w/o
---|---|---|---|---|---|---
joint | clean | noisy | denosing
PEMS03 | MAE | 18.37 | 19.57 | 18.65 | 18.97 | 18.82
RMSE | 30.86 | 32.31 | 31.15 | 31.60 | 31.02
MAPE | 17.69% | 20.13% | 18.19% | 19.78% | 18.00%
PEMS04 | MAE | 22.68 | 23.75 | 22.86 | 23.44 | 22.92
RMSE | 35.28 | 36.67 | 35.41 | 35.84 | 35.37
MAPE | 15.93% | 17.64% | 16.18% | 16.93% | 16.02%
PEMS07 | MAE | 23.74 | 25.19 | 23.94 | 24.46 | 23.89
RMSE | 38.69 | 39.95 | 38.73 | 39.31 | 38.57
MAPE | 10.69% | 11.87% | 10.75% | 11.04% | 10.62%
PEMS08 | MAE | 17.68 | 18.94 | 18.02 | 18.31 | 17.82
RMSE | 27.95 | 29.18 | 28.22 | 28.54 | 28.15
MAPE | 12.60% | 14.29% | 12.85% | 13.51% | 12.73%
Table 9. Ablation studies, single-step datasets. Data | Metric | _Ours-1_ | w/o | w/o | w/o | w/o
---|---|---|---|---|---|---
joint | clean | noisy | denosing
Solar-Energy | RRSE | 0.1657 | 0.1762 | 0.1669 | 0.1691 | 0.1677
(3) | CORR | 0.9875 | 0.9848 | 0.9881 | 0.9869 | 0.9874
Solar-Energy | RRSE | 0.3980 | 0.4138 | 0.4033 | 0.4051 | 0.4010
(24) | CORR | 0.9152 | 0.9087 | 0.9118 | 0.9106 | 0.9132
Electricity | RRSE | 0.0736 | 0.0752 | 0.0741 | 0.0748 | 0.0739
(3) | CORR | 0.9483 | 0.9441 | 0.9478 | 0.9462 | 0.9480
Electricity | RRSE | 0.0921 | 0.0956 | 0.0942 | 0.0954 | 0.0938
(24) | CORR | 0.9253 | 09223 | 0.9236 | 0.9228 | 0.9242
From Tables 7 to 9, we can observe that (1) our proposed framework
consistently outperforms all variants on all evaluation metrics that every
tested component is essential to the success of the proposed framework; (2)
w/o joint is much worse than Ours-1 that it is necessary to jointly search for
architectures and hyperparameters to ensure satisfactory performance; (3) both
w/o clean and w/o noisy are worse than Ours-1 that both noisy samples and
clean samples are keys to a reliable AHC thus a competitive searched model;
Clean samples are essential to alleviating the bias produced by noisy samples,
while noisy samples can efficiently contribute to achieving sufficient
training samples; (4) w/o denoising is worse than our framework, demonstrating
that it is necessary to train the AHC in a noise reduction manner, e.g., fine-
tuning in ours.
(a) PEMS03 dataset, $P$-12/$Q$-12
(b) PEMS08 dataset, $P$-48/$Q$-48
Figure 6. Case Study of Searched ST-blocks on Different Target Datasets and
$P$/$Q$ Settings.
#### 4.2.5. Efficiency Study
We compare the search time of our framework with that of existing automated
CTS forecasting frameworks in Table 10. For our framework, we consider the
transfer setting and count the search time on target datasets. This is because
training AHC on the source dataset is usually a one-time job, so the
corresponding search cost is not important.
It is obvious that our framework requires significantly less search time than
AutoSTG and AutoCTS on the four datasets, which demonstrates the efficiency of
our framework.
Table 10. Search Time Comparison in GPU hours for $P$-48/$Q$-48 Multi-step Forecasting. Target Dataset | AutoSTG | AutoCTS | Ours-1 | Ours-2
---|---|---|---|---
PEMS03 | 584.3 | 892.6 | 190.4 | 184.5
PEMS04 | 316.6 | 538.2 | 102.1 | -
PEMS07 | 1153.9 | 2382.2 | 447.7 | 429.8
PEMS08 | 261.4 | 473.8 | - | 53.5
#### 4.2.6. Impact of $L_{1}$, $L_{2}$, $z_{1}$, and $z_{2}$
In the pre-training phase on the source dataset, we collect $L_{1}$ pairs of
$(ah,R^{\prime}(ah))$ and $L_{2}$ pairs of $(ah,R(ah))$ to generate noisy and
clean training samples for the AHC, respectively. In the transfer phase, we
use $z_{1}$ pairs of $(ah,R^{\prime}(ah))$ and $z_{2}$ pairs of $(ah,R(ah))$
to generate noisy and clean training samples to finetune the AHC. The number
of $L_{1}$, $L_{2}$, $z_{1}$, and $z_{2}$ affects the accuracy of the AHC. It
is expected that more clean and noisy samples lead to a more accurate AHC and
thus better search results. We evaluate the impact of $L_{1}$, $L_{2}$,
$z_{1}$, and $z_{2}$ on the PEMS08 dataset under the $P$-48/$Q$-48 setting. To
remove the bias caused by the sampling procedure, for each ($L_{1}$, $L_{2}$)
or ($z_{1}$, $z_{2}$) setting, we train and test our framework five times
using different $(ah,R^{\prime}(ah))$ or $(ah,R(ah))$ pairs chosen with five
different random seeds, and we report the results in the format
“mean$\pm$standard deviation”.
Table 11. Impact of $L_{1}$ and $L_{2}$ (mean$\pm$standard deviation). $\rho$ is the Spearman’s rank correlation. ($L_{1}$, $L_{2}$) | MAE | RMSE | MAPE | $\rho$
---|---|---|---|---
(0, 150) | 18.79$\pm$0.11 | 29.16$\pm$0.13 | 14.34%$\pm$0.16% | 0.78$\pm$0.02
(1000, 0) | 18.42$\pm$0.09 | 28.69$\pm$0.16 | 14.22%$\pm$0.17% | 0.80$\pm$0.02
(1000, 100) | 18.28$\pm$0.08 | 28.51$\pm$0.12 | 13.54%$\pm$0.13% | 0.83$\pm$0.01
(1000, 150) | 18.10$\pm$0.06 | 28.29$\pm$0.09 | 13.35%$\pm$0.10% | 0.85$\pm$0.01
(2000, 100) | 17.83$\pm$0.09 | 27.92$\pm$0.15 | 13.08%$\pm$0.14% | 0.87$\pm$0.02
(2000, 150) | 17.71$\pm$0.07 | 27.90$\pm$0.12 | 12.56%$\pm$0.12% | 0.89$\pm$0.01
Table 12. Impact of $z_{1}$ and $z_{2}$ (mean$\pm$standard deviation). $\rho$ is the Spearman’s rank correlation. $(z_{1},z_{2})$ | MAE | RMSE | MAPE | $\rho$
---|---|---|---|---
(0, 0) | 18.76$\pm$0.05 | 28.92$\pm$0.08 | 14.27%$\pm$0.09 | 0.79$\pm$0.00
(100, 5) | 17.76$\pm$0.09 | 28.04$\pm$0.12 | 12.52%$\pm$0.10% | 0.89$\pm$0.01
(500, 10) | 17.64$\pm$0.09 | 27.81$\pm$0.11 | 12.90%$\pm$0.15% | 0.89$\pm$0.01
(1000, 20) | 17.85$\pm$0.10 | 27.86$\pm$0.12 | 12.82%$\pm$0.13% | 0.89$\pm$0.01
From Table 11, it is observed that with either increasing $L_{1}$ or $L_{2}$,
the achieved performance increases. In particular, when $L_{1}=2000$ and
$L_{2}=150$, the trained AHC achieves a high $\rho$, which demonstrates that
the trained AHC can produce a ranking close to the true ranking of arch-hypers
by performing pairwise comparison. Considering that the cost of a clean sample
is about 20 times that of a noisy sample (100 training epochs vs 5 training
epochs), and $\rho(1000,0)>\rho(0,150)$, the proposed proxy improves the
sample efficiency by a factor of about 3.
From Table 12 we can observe that: (1) $(z_{1}=0,z_{2}=0)$ gets the worst
performance and the worst AHC but the performance is still competitive
compared to baselines (see Table 4). This demonstrates that good arch-hypers
can be found even if we directly use the AHC trained on the source dataset,
while further fine-tuning the AHC with a small number of noisy and clean
samples leads to better performance; (2) with $z_{1}$ larger than 100, and
$z_{2}$ larger than 5, the performance saturates so we set
$(z_{1}=100,z_{2}=5)$ as our default setting.
#### 4.2.7. Case Study
We show two representative searched ST-blocks (arch-hypers) on different
target datasets and settings in Figure 6. Figure 6(a) is searched on the
PEMS03 dataset under the $P$-12/$Q$-12 setting, with PEMS08 as the source
dataset; Figure 6(b) is searched on the PEMS08 dataset under the $P$-48/$Q$-48
setting, with PEMS04 as the source dataset.
We can see obvious differences between the two ST-blocks. In particular, the
two ST-blocks have different hyperparameters: numbers of nodes $C$ (7 vs. 5),
output modes $U$ (1 vs. 0), and output dimensions $I$ (128 vs. 256); for the
architectures, they have totally different connections and operators between
node pairs. These observations show that different datasets and settings
prefer different arch-hypers, and our framework can successfully capture such
preferences. The reasons why different CTS datasets prefer different arch-
hypers are as follows. First, different CTS datasets exhibit different
temporal and spatial patterns, and different arch-hypers may be good at
capturing different temporal and spatial patterns. Second, different CTS
datasets contain different numbers of time series.
## 5\. Related Work
### 5.1. Manual Model Design.
There are many manually designed models for forecasting correlated time series
(Faloutsos et al., 2019; Park et al., 2020; Wang et al., 2020; Lai et al.,
2018; Shih et al., 2019; Wu et al., 2019; Bai et al., 2020; Wu et al., 2020;
Cirstea et al., 2021). The biggest difference between these works is that they
design different ST-blocks, which determines their ability to capture temporal
and spatial dependencies. Graph WaveNet (Wu et al., 2019) employs Diffusion
GCN and gated dilated causal convolution to capture spatial and temporal
dependencies, respectively, and sequentially stacks the two operators to build
ST-blocks. AGCRN (Bai et al., 2020) employs enhanced Chebyshev GCN and GRU to
capture spatial and temporal dependencies, respectively. MTGNN (Wu et al.,
2020) uses mix-hop graph convolution and dilated inception convolution to
capture spatial and temporal dependencies, respectively. We are inspired by
these manually designed models when designing the architecture search space.
### 5.2. Automated Model Design.
Recently, several automated frameworks have been proposed for time series
forecasting. AutoAI-TS (Shah et al., 2021) provides an automated pipeline for
time series forecasting, and it is able to select the most appropriate
forecasting model from a set of existing forecasting models for a specific
data set and forecasting setting. In contrast, the proposed SEARCH framework
aims at automatically designing novel CTS forecasting models. AutoST (Li et
al., 2020) divides a city’s time series into grids based on longitude and
latitude, and forecasts CTS data using grid-based images containing values for
each timestamp as input. This work is not applicable to our problem because
the CTS dataset we use lacks latitude and longitude information, so grid-based
images cannot be constructed. AutoSTG (Pan et al., 2021) and AutoCTS (Wu et
al., 2022) are the two studies most relevant to this paper. AutoSTG designs a
search space for CTS forecasting, and introduces meta-learning to learn the
weights of architectures. AutoCTS focuses on designing a compact and effective
search space for CTS forecasting, and achieves the best accuracy on multiple
benchmark datasets. However, both methods are supernet-based, thus are not
scalable and do not support joint search for architectures and
hyperparameters.
There is also a large body of studies aim at automatically designing neural
architectures for various tasks (Liu et al., 2018; Zoph and Le, 2017; El et
al., 2020; Li et al., 2021b, a). BRP-NAS (Dudziak et al., 2020) and CTNAS
(Chen et al., 2021b) perform pairwise architecture comparisons to explore the
search space to search for the best architecture. However, both methods do not
support joint search for architectures and hyperparameters, and thus fail to
address the first limitation. AutoHAS (Dong et al., 2020) and FBNetV3 (Dai et
al., 2021) are two studies that aim to automatically search for the best
combination of architectures and hyperparameters. However, AutoHAS is also a
supernet-based method, which has poor scalability and thus fails to address
the second limitation. Furthermore, both methods are very inefficient. For
example, FBNetV3 spends more than 10K GPU hours searching for the best arch-
hyper, so it cannot solve the third limitation.
## 6\. Conclusion
We present a scalable and efficient joint search framework to automatically
design high-performance ST-blocks for CTS forecasting. In particular, we first
design a joint search space containing massive arch-hypers, each of which is a
combination of an architecture and a hyperparameter setting. Next, we propose
an AHC-based search strategy to explore the search space to find the best
arch-hyper. To improve the sample efficiency and accuracy of the AHC, we
propose a proxy metric to generate noisy training samples and train the AHC in
a noise reduction manner to alleviate the negative effects of wrongly labeled
training samples. Besides, we propose a transfer method to further improve the
efficiency of the framework, allowing us to transfer a pretrained AHC to
unseen datasets to reduce the need for samples to train the AHC. Comprehensive
experiments on six commonly used correlated time series forecasting datasets
demonstrate the effectiveness, scalability, and efficiency of the proposed
framework. As future work, it is of interest to design more accurate proxy
metrics and denosing algorithms to further improve the efficiency of the
proposed framework. In addition, the proposed AHC can be used in other domains
such as computer vision and natural language processing, because the proposed
joint search space, i.e., the arch-hyper graph, can also represent models used
in other domains. We leave this as future work. Next, the proposed framework
only searches for the optimal arch-hyper based on the accuracy metric, while
metrics such as latency and energy consumption may also need to be considered
when deploying CTS forecasting models; thus, it is also of interest to design
multi-objective CTS forecasting frameworks.
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# Spin Torque Oscillator and Magnetization Switching in Double-Barrier Rashba
Zeeman Magnetic Tunnel Junction
Saumen<EMAIL_ADDRESS>Arindam
<EMAIL_ADDRESS>Reeta<EMAIL_ADDRESS>and
Nimisha<EMAIL_ADDRESS>Department of Physics, Dibrugarh
University, Dibrugarh 786 004, Assam, India
###### Abstract
In this work, we have studied the spin torque based magnetization oscillations
and switching in presence of Rashba - Zeeman (RZ), Ruderman - Kittel - Kasuya
- Yoside (RKKY) and Dzyaloshinskii - Moriya (DM) interactions in a double-
barrier RZ$|$Heavy Metal (HM)$|$RZ magnetic tunnel junction (MTJ). The system
has stable magnetization oscillations and can work as an oscillator or a
switcher for a significant difference in the strength of RKKY and DM
interaction under suitable spin transfer torque (STT). For the proposed system
with same order of RKKY and DM interaction, a nonlinear characteristic of the
magnetization oscillation is observed. However, this nonlinearity of
oscillations can be reduced by an external magnetic field or considering a
material with suitable RZ interaction. In addition to this, our study reveals
the magnetization switching can be tuned by using suitable STT. A dependence
of switching time on layer thickness is also observed. Also, the switching
speed increases with the thickness for systems having either same order of
RKKY and DM interaction or dominated by RKKY interaction. An opposite
characteristic is seen when DM interaction dominates over RKKY interaction.
###### pacs:
72.25.Dc, 72.25.-b, 75.78.-n, 75.75.−c, 85.75.-d
## I Introduction
Spin torque oscillators (STOs) are nano-sized electronic devices based on
magnetic tunnel junctions (MTJ) that have recently received considerable
attention due to their wide range of applications slonczewski1 ; slonczewski2
; berger ; locatelli ; grollier ; torrejon ; kudo ; tsunegi ; romera ;
acharjee1 ; taniguchi ; johansen . STOs are essentially magnetoresistive
stacks, where the polarized spin current generates the spin torque
slonczewski1 ; slonczewski2 ; berger . Thus, it leads to self-sustaining
magnetization oscillations in the free layer. This characteristic of STOs can
be utilized to use them as microwave generators, field sensors, phased array
antennae etc locatelli ; grollier ; torrejon ; kudo ; tsunegi ; romera ;
acharjee1 ; taniguchi ; johansen . Another essential phenomenon witnessed in
an MTJ is magnetization switching, which is the backbone of current non-
volatile magnetic memories katine ; kiselev ; kubota ; krivorotov1 ;
krivorotov2 .
Usually, a conventional MTJ consists of a tunnel barrier sandwiched between
two ferromagnetic layers, namely, pinned and free layers. However, such
arrangements have non-adequate thermal stability below $40$ nm li . This can
be achieved by using double interface MTJs, typically an arrangement of a
heavy metal (HM) between two ferromagnetic layers choi ; sato ; li . The
FM$|$HM$|$FM arrangement is vital in enhancing the spin-orbit coupling (SOC)
and generating Ruderman - Kittel - Kasuya - Yoside (RKKY) interaction. Due to
the presence of RKKY interaction, the magnetizations of the two layers are
ferromagnetically coupled parkin ; li . So they can behave like identical
layers. Moreover, the strong SOC of HM can also even induce Dzyaloshinskii -
Moriya (DM) interaction, which is an antisymmetric exchange coupling,
dzyaloshinsky ; moriya . Recent studies have shown that RKKY interactions may
suppress the detrimental effects of DM interactions in STT switching li . Thus
it is noteworthy to understand the interplay of SOC, RKKY and DM interaction
in STT magnetization dynamics.
Recently discovered Rashba - Zeeman (RZ) effect in Ag2Te / Cr2O3 composite can
provide some novel features that are not found in pure Rashba or Zeeman
systems tao1 . These materials have some unique characteristics and also have
the ability to trigger insulator to conductor via an exchange field tao1 and
can also be used as spin filters xiao ; wojcik ; acharjee4 . In general, the
Rashba SOC (RSOC) is an antisymmetric SOC responsible for splitting of energy
sub-bands rashba ; liu11 ; zhang111 ; chico ; ganguly1 ; ganguly2 ; fouladi1 ;
zhang121 ; liu111 ; lashell ; acharjee ; acharjee2 ; acharjee3 ; cavigilia ;
ishizaka . The impact of RSOC on STOs is worth mentioning, which shows the
utilization of RSOC in magnetization switching and also its contribution to
self-oscillations in STOs. Moreover, the RSOC of the multilayer systems can
substantially increase the size of the STO phase johansen ; duan . So, it is
necessary to understand the role of RSOC and its interplay with RKKY and DMI
in STOs and magnetization switching. Though STOs and magnetization switching
had been studied in double-barrier MTJs earlier, the introduction of RZ
material as a free layer can significantly change auto oscillation and
magnetization switching conditions. Moreover, STOs and magnetization switching
were not studied in the same frame earlier. Thus in this work, we consider a
double-barrier Rashba Zeeman Magnetic Tunnelling Junction (RZ-MTJ) where we
consider an HM sandwiched between two RZs.
The organization of this paper are as follows: we present a minimal theory to
study magnetization dynamics and switching in double interface RZ-MTJ in
section II. In Section III, we present the results of our work by considering
the effect of RKKY, DMI, RSOC and other important parameters like external
magnetic field. We have analysed the effect of the parameter on magnetization
dynamics and switching too in this section. Finally, a brief summary of our
work is presented in Section IV.
## II Minimal Theory
The schematic representation of a double-barrier RZ based MTJ is shown in Fig.
1. In general a double-barrier MTJ is composed of three magnetic layers viz.
reference layer, free/storage layer and control layer. The reference and
control layers act as polarisers whose magnetizations can be controlled
independently of the free layer coelho . We have considered an FM reference
layer and an FM control layer with a RZ$|$HM$|$RZ composite free layer for our
analysis. A Spin Transfer Torque (STT) is induced in the free layer because of
the FM polarisers li ; coelho as shown in Fig. 1. We choose the easy axis
anisotropy along the y-direction, while an external magnetic field is
considered along the z-direction in our analysis. The time evolution of the
magnetization in RZ layers can be studied by using two coupled Landau -
Lifshitz - Gilbert - Slonczewski (LLGS) equations.
$\partial_{t}\mathbf{m}_{j}=-\gamma\mathbf{m}_{j}\times(\mathbf{H}_{\text{eff}}+\mathbf{H}^{\text{R}}_{j})+\alpha_{j}\mathbf{m}_{j}\times\partial_{t}\mathbf{m}_{j}+\mathbf{\mathcal{T}}_{j}^{\text{STT}}$
(1)
where, the indices $j=1,2$ corresponds to first and second RZ layers
respectively. The parameter $\gamma$ is the gyromagnetic ratio and
$\alpha_{j}$ is the Gilbert damping parameter of the RZ layers while
$\mathbf{H}_{\text{eff}}$ is the effective field of the system can be obtained
using the effective Hamiltonian $\mathcal{H}_{\text{eff}}$ of the system
$\mathcal{H}_{\text{eff}}=\text{K}_{\text{exc}}(\nabla\mathbf{m}_{j})^{2}-\mathbf{D}_{12}.(\mathbf{m}_{1}\times\mathbf{m}_{2})+\mathcal{J}_{12}(1-\mathbf{m}_{1}.\mathbf{m}_{2})\\\
-\text{H}_{\text{ext}}(\hat{z}.\mathbf{m}_{j})+h_{0}\mathbf{\sigma}_{j}.\hat{m}_{j}$
(2)
where, $\text{K}_{\text{exc}}$ incorporate the exchange coupling between the
FM and RZ layers, $\mathbf{D}_{12}$ is the DMI vector,
$\mathcal{J}_{12}=\frac{\sigma_{exc}}{\Delta_{12}}$ is the ratio of bilinear
exchange coefficient between two surfaces with discretion cell dimension li .
Here, $\text{H}_{\text{ext}}$ is the external magnetic field strength and the
last term of Eq. (2) gives the contribution of Zeeman energy with $h_{0}$
being the strength of Zeeman energy. The Rashba field
$\mathbf{H}^{\text{R}}_{j}$ in Eq. (1) is given by johansen
$\mathbf{H}^{\text{R}}_{j}=-\frac{1}{1+\beta^{2}}\frac{\alpha_{\text{R}}m_{e}\mathcal{P}_{j}j_{0}}{eM_{0}\hbar\mu_{0}}(\hat{y}\times\hat{z})$
(3)
where, $\alpha_{\text{R}}$ is the RSOC strength with $\mathcal{P}_{j}$ being
the polarization of the current $j_{0}$ through the RZ-layers. $M_{0}$ is the
saturation magnetization and $\beta$ is the adiabatic damping parameter. It is
to be noted that the anisotropy of the system can be incorporated by
considering an easy-axis anisotropy field
$\mathbf{H}_{\text{an}}=\frac{1}{2}\frac{\text{K}_{\text{an}}m_{y}}{M_{0}}\hat{y}$
in Eq. (1) acharjee1 .
Figure 1: Schematic illustration of the proposed double-barrier MTJ consisting
of Rashba-Zeeman (RZ)$|$Heavy metal (HM)$|$Rashba-Zeeman (RZ) hybrid as a
composite free layer sandwiched between Ferromagnetic (FM) reference and
control layers. The reference and control layers act as polarisers whose
magnetizations can be controlled independently of the free layer. As a result
an STT is induced in the RZ$|$HM$|$RZ composite free layer. $\mathbf{m}_{1}$
and $\mathbf{m}_{2}$ represent the magnetizations while $t_{1}$ and $t_{2}$
are the thickness of the RZ1 and RZ2 layers respectively. The magnetization of
FM layer is denoted by $\mathbf{m}$.
Figure 2: (a) - (f) Oscillation trajectories of $\textbf{m}_{1}$ (blue) and
$\textbf{m}_{2}$ (red) for different values of $\tau_{1}$ and $\tau_{2}$
considering K${}_{\text{R}}$ = $0.1$K${}_{\text{DM}}$. Plots (g) - (l)
represent the time evolution of the magnetization component $\text{m}_{x1}$
(red) and $\text{m}_{x2}$ (blue). The variation of switching time with
$(\tau_{1},\tau_{2})$ and $(t_{1},t_{2})$ are shown in plots (m), (p) and (n),
(r) respectively. The Fourier transform of magnetization for (o)
$(\tau_{1},\tau_{2})=(0.65,0.15)$ and (r) $(\tau_{1},\tau_{2})=(-0.65,-0.2)$
respectively.
The last term of Eq. (1) corresponds to the spin-transfer torques
$\mathbf{\mathcal{T}}_{j}^{\text{STT}}$ acting on the magnetization
$\mathbf{m}_{j}$ and can be given by johansen
$\mathbf{\mathcal{T}}_{1}^{\text{STT}}=-\tau_{1}\left[\mathbf{m}_{1}\times(\mathcal{P}_{0}\mathbf{m}-\mathcal{P}_{1}\mathbf{m}_{2})\times\mathbf{m}_{1}\right]$
(4)
$\mathbf{\mathcal{T}}_{2}^{\text{STT}}=-\tau_{2}\mathcal{P}_{1}(\mathbf{m}_{2}\times\mathbf{m}_{1}\times\mathbf{m}_{2})$
(5)
where, $\tau_{i}=\frac{\gamma\hbar j_{1}}{2eM_{0}\mu_{0}t_{i}}$ with the index
$i=1,2$ corresponds to first and second RZ layers respectively and measured in
$Jm^{5}/A^{4}s$. Here, $t_{i}$ are the thickness while $\mathcal{P}_{0}$,
$\mathcal{P}_{1}$ are the polarization of the current current in the
respective RZ layers. Here, $\mathbf{m}$ is the magnetization of the fixed FM
layer of our system. Using Eqs. (2)-(4) in Eq. (1), we obtained six non linear
coupled first order differential equations:
$m_{\text{x1}}^{\prime}[t]=\frac{\gamma}{1+\alpha^{2}}\\{\alpha\mathcal{H}_{\text{eff1}}(m_{\text{y1}}^{2}+m_{\text{z1}}^{2})+\mathcal{H}_{\text{eff2}}(m_{\text{z1}}\\\
-\alpha m_{\text{x1}}m_{\text{y1}})-\mathcal{H}_{\text{eff3}}(\alpha
m_{\text{x1}}m_{\text{z1}}+m_{\text{y1}})\\}+\mathcal{P}\tau_{1}[(\alpha(m_{\text{x1}}^{2}+m_{\text{y1}}^{2}\\\
+m_{\text{z1}}^{2})\\{m_{\text{y1}}(m_{\text{z2}}-1)-m_{\text{y2}}m_{\text{z1}}\\}+m_{\text{x1}}\\{-m_{\text{y1}}m_{\text{y2}}\\\
-m_{\text{z1}}(m_{\text{z2}}-1)\\}+m_{\text{x2}}(m_{\text{y1}}^{2}+m_{\text{z1}}^{2})]$
(6)
$m_{\text{y1}}^{\prime}[t]=\frac{\gamma}{1+\alpha^{2}}\\{-\mathcal{H}_{\text{eff1}}(\alpha
m_{\text{x1}}m_{\text{y1}}+m_{\text{z1}})+\alpha\mathcal{H}_{\text{eff2}}(m_{\text{x1}}^{2}\\\
+m_{\text{z1}}^{2})+\mathcal{H}_{\text{eff3}}(m_{\text{x1}}-\alpha
m_{\text{y1}}m_{\text{z1}})\\}+\mathcal{P}\tau_{1}[(-\alpha(m_{\text{x1}}^{2}+m_{\text{y1}}^{2}\\\
+m_{\text{z1}}^{2})\\{m_{\text{x1}}(m_{\text{z2}}-1)-m_{\text{x2}}m_{\text{z1}}\\}-m_{\text{x1}}m_{\text{x2}}m_{\text{y1}}+m_{\text{x1}}^{2}m_{\text{y2}}\\\
-m_{\text{y1}}m_{\text{z1}}m_{\text{z2}}+m_{\text{y1}}m_{\text{z1}}+m_{\text{y2}}m_{\text{z1}}^{2})]$
(7)
$m_{\text{z1}}^{\prime}[t]=\frac{\gamma}{1+\alpha^{2}}\\{\mathcal{H}_{\text{eff1}}(m_{\text{y1}}-\alpha
m_{\text{x1}}m_{\text{z1}})-\mathcal{H}_{\text{eff2}}(m_{\text{x1}}\\\ +\alpha
m_{\text{y1}}m_{\text{z1}})+\alpha\mathcal{H}_{\text{eff3}}(m_{\text{x1}}^{2}+m_{\text{y1}}^{2})\\}+\mathcal{P}\tau_{1}[\alpha(m_{\text{x1}}^{2}+m_{\text{y1}}^{2}\\\
+m_{\text{z1}}^{2})(m_{\text{x1}}m_{\text{y2}}-m_{\text{x2}}m_{\text{y1}})-m_{\text{x1}}m_{\text{x2}}m_{\text{z1}}+m_{\text{x1}}^{2}m_{\text{z2}}\\\
-m_{\text{x1}}^{2}-m_{\text{y1}}m_{\text{y2}}m_{\text{z1}}+m_{\text{y1}}^{2}m_{\text{z2}}-m_{\text{y1}}^{2})]$
(8)
$m_{\text{x2}}^{\prime}[t]=\frac{\gamma}{1+\alpha^{2}}\\{\alpha\mathcal{H}_{\text{eff4}}(m_{\text{y2}}^{2}+m_{\text{z2}}^{2})+\mathcal{H}_{\text{eff5}}(m_{\text{z2}}\\\
-\alpha m_{\text{x2}}m_{\text{y2}})-\mathcal{H}_{\text{eff6}}(\alpha
m_{\text{x2}}m_{\text{z2}}+m_{\text{y2}})\\}+\mathcal{P}\tau_{2}[-(m_{\text{y2}}^{2}\\\
+m_{\text{z2}}^{2})(m_{\text{x1}}-\alpha m_{\text{y1}}m_{\text{z2}}+\alpha
m_{\text{y2}}m_{\text{z1}})+\alpha
m_{\text{x2}}^{2}(m_{\text{y1}}m_{\text{z2}}\\\
-m_{\text{y2}}m_{\text{z1}})+m_{\text{x2}}(m_{\text{y1}}m_{\text{y2}}+m_{\text{z1}}m_{\text{z2}})]$
(9)
$m_{\text{y2}}^{\prime}[t]=\frac{\gamma}{1+\alpha^{2}}\\{-\mathcal{H}_{\text{eff4}}(\alpha
m_{\text{x2}}m_{\text{y2}}+m_{\text{z2}})+\alpha\mathcal{H}_{\text{eff5}}(m_{\text{x2}}^{2}\\\
+m_{\text{z2}}^{2})+\mathcal{H}_{\text{eff6}}(m_{\text{x2}}-\alpha
m_{\text{y2}}m_{\text{z2}})\\}+\mathcal{P}\tau_{2}[\alpha(m_{\text{x2}}^{2}+m_{\text{y2}}^{2}\\\
+m_{\text{z2}}^{2})(m_{\text{x2}}m_{\text{z1}}-m_{\text{x1}}m_{\text{z2}})+m_{\text{x1}}m_{\text{x2}}m_{\text{y2}}-m_{\text{x2}}^{2}m_{\text{y1}}\\\
-m_{\text{y1}}m_{\text{z2}}^{2}+m_{\text{y2}}m_{\text{z1}}m_{\text{z2}})]$
(10)
$m_{\text{z2}}^{\prime}[t]=\frac{\gamma}{1+\alpha^{2}}\\{\mathcal{H}_{\text{eff4}}(m_{\text{y2}}-\alpha
m_{\text{x2}}m_{\text{z2}})-\mathcal{H}_{\text{eff5}}(m_{\text{x2}}\\\ +\alpha
m_{\text{y2}}m_{\text{z2}})+\alpha\mathcal{H}_{\text{eff6}}(m_{\text{x2}}^{2}+m_{\text{y2}}^{2})\\})+\mathcal{P}\tau_{2}[-\alpha(m_{\text{x2}}^{2}+m_{\text{y2}}^{2}\\\
+m_{\text{z2}}^{2})(m_{\text{x2}}m_{\text{y1}}-m_{\text{x1}}m_{\text{y2}})+m_{\text{x1}}m_{\text{x2}}m_{\text{z2}}-m_{\text{x2}}^{2}m_{\text{z1}}\\\
+m_{\text{y1}}m_{\text{y2}}m_{\text{z2}}-m_{\text{y2}}^{2}m_{\text{z1}})]$
(11)
where,
$\displaystyle\mathcal{H}_{\text{eff1}}$
$\displaystyle=-D_{1}t_{1}K_{\text{DM}}+h_{0}-K_{\text{R}}m_{\text{x2}}-\alpha_{\text{R}},$
$\displaystyle\mathcal{H}_{\text{eff2}}$
$\displaystyle=-D_{1}t_{1}K_{\text{DM}}+h_{0}-K_{\text{R}}m_{\text{y2}}$
$\displaystyle\mathcal{H}_{\text{eff3}}$
$\displaystyle=K_{\text{an}}m_{\text{z1}}-\text{H}_{\text{extz}}m_{\text{z1}}-D_{1}t_{1}K_{\text{DM}}+h_{0}-K_{\text{R}}m_{\text{z2}}$
$\displaystyle\mathcal{H}_{\text{eff4}}$
$\displaystyle=-D_{2}t_{2}K_{\text{DM}}+h_{0}-K_{\text{R}}m_{\text{x1}}-\alpha_{\text{R}}$
$\displaystyle\mathcal{H}_{\text{eff5}}$
$\displaystyle=-D_{2}t_{2}K_{\text{DM}}+h_{0}-K_{\text{R}}m_{\text{y1}}$
$\displaystyle\mathcal{H}_{\text{eff6}}$
$\displaystyle=K_{\text{an}}m_{\text{z2}}-\text{H}_{\text{extz}}m_{\text{z2}}-D_{2}t_{2}K_{\text{DM}}+h_{0}-K_{\text{R}}m_{\text{z1}}$
Figure 3: (a) - (f) Oscillation trajectories of $\textbf{m}_{1}$ (blue) and
$\textbf{m}_{2}$ (red) for different values of $\tau_{1}$ considering
$\tau_{2}=0.4$ and K${}_{\text{R}}$ = K${}_{\text{DM}}$ = $1$. Plots (g) - (l)
represent the time evolution of the magnetization component $\text{m}_{x1}$
(red) and $\text{m}_{x2}$ (blue). Plots (m) - (p) are the Fourier transform of
the magnetization corresponding to the plots (g) - (j). The variation of
switching time with $(\tau_{1},\tau_{2})$ and $(t_{1},t_{2})$ are shown in
plots (q) and (r).
## III Results and Analysis
We investigate the magnetization dynamics and switching behaviour
quantitatively by solving Eqs. (6)-(11) numerically. We set the layer
thickness, $t_{1}:t_{2}=4:1$ and their respective polarizations as
$\mathcal{P}_{0}=\mathcal{P}_{1}=0.5$ for all our analysis taniguchi . This
makes an asynchronism between $m_{x1}$ and $m_{x2}$ even in presence of RKKY.
Moreover, other characteristics of magnetization dynamics like chaotic
oscillations, magnetization switching etc can also be investigated under such
condition. We also have explored different ratios of layer thickness and their
impact on magnetization switching in our analysis. The RZ strength are set as
$\alpha_{\text{R}}=3\times 10^{-10}$ eV.m and $h_{0}=0.7$ for all our analysis
in Figs. 2 \- 4. Furthermore, we set the DM coefficients $(D_{1},D_{2})$
$=(-0.43,0.43)$ are measured in $mJ/m^{2}$ for all our analysis li . The
opposite sign of $D_{1}$ and $D_{2}$ are considered due to the opposite
chirality of the induced DM interactions in RZ${}_{1}|$HM and HM$|$RZ2
interfaces.
### III.1 Interplay of RKKY with DM interaction
#### III.1.1 For systems with RKKY $<$ DM interaction
The magnetization oscillation and switching has been studied in Fig. 2 for
K${}_{\text{R}}$ = $0.1$K${}_{\text{DM}}$ for different choices of
$(\tau_{1},\tau_{2})$. We found that the system display magnetization
switching in both positive and negative spin torque conditions, which are
characterized by the positive and negative sign of $\tau_{1}$ and $\tau_{2}$.
Though the precession of the magnetic moments $m_{x1}$ and $m_{x2}$ are found
to be identical for $(\tau_{1},\tau_{2})$ = $(1.5,0.15)$, but magnetization
switching of $m_{x2}$ is observed around $\tau_{1}=0.01$ as seen from Figs.
2(b) and (h). Moreover, the system is found to have stable magnetization
oscillation in $m_{x1}$ and $m_{x2}$ for $\tau_{1}=0.65$, indicated by the
Figs. 2(c) and (i). The sustained oscillations can be confirmed by a sharp
peak around $250$ GHz in the Fourier transform spectra of $m_{x1}$ and
$m_{x1}$. Thus for $\tau_{1}=0.65$ and $\tau_{2}=0.15$, the moments display a
stable oscillation. It is to be noted that auto oscillation and the
synchronization of the magnetizations can be measured by calculating GMR or
TMR of the MTJ , where the resistance of the multilayer is related to
magnetization through $\mathbf{m}_{1}$ and $\mathbf{m}_{2}$ as taniguchi
$\mathcal{R}=\frac{\mathcal{R}_{\text{AP}}+\mathcal{R}_{\text{P}}}{2}-\frac{\mathcal{R}_{\text{AP}}-\mathcal{R}_{\text{P}}}{2}\mathbf{m}_{1}.\mathbf{m}_{2}$
(12)
where, $\mathcal{R}_{\text{P}}$ ($\mathcal{R}_{\text{AP}}$) is the resistance
of the system corresponding to parallel (antiparallel) orientations of
magnetization and the vector product $\mathbf{m}_{1}.\mathbf{m}_{2}$ can be
termed as MR of the system. The synchronization of the magnetization
$\mathbf{m}_{1}$ and $\mathbf{m}_{2}$ can be understood via two sharp MR peaks
around $0$ and $500$ GHz as shown in Fig. 2(o). A similar characteristic of MR
is also observed in Fig. 2(r) for negative biasing condition. The switching of
$m_{x1}$ for STT, $(\tau_{1},\tau_{2})=(-0.9,-0.2)$, is due to the opposite
biasing of RZ1 layer. However, the system can also behave like an oscillator
even in negative biasing conditions for $\tau_{1}=-0.65$ as seen from Figs.
2(f) and (l).
The variation of switching speed with $\tau_{2}$ for both positive and
negative values of $\tau_{1}$ is studied in Figs. 2(m) and (p) respectively.
The switching time (ST) can be numerically obtained by calculating the time
corresponding to $97.5\%$ of the saturation magnetization. It should be noted
that for $\tau_{1}=0.01$, the ST gradually decreases for low values of
$\tau_{2}$, but it remains constant in moderate and high $\tau_{2}$ regions.
Moreover, the switching speed decreases for $\tau_{1}=0.05$ and $0.1$. In this
condition, the maximum ST is found at $\tau_{2}=0.5$ and $0.7$, respectively.
An identical but opposite characteristic of magnetization switching is also
observed for negative $\tau_{1}$ values, as seen from Fig. 2(p). However, in
this case, a delayed switching of $m_{x1}$ is observed for lower values of
$\tau_{1}$, which may be due to the opposite current biasing of the RZ1 layer.
Although the magnetization switching can occur for different
$(\tau_{1},\tau_{2})$ combinations as seen from Figs. 2(m) and 2(p), yet the
interplay of STT with layer thickness and switching can be understood by
exploring the variation of ST with layer thickness. Figs. 2(n) and 2(q)
display the switching speed for ($\tau_{1}$, $\tau_{2}$) = $(0.01,0.15)$ and
($\tau_{1}$, $\tau_{2}$) = $(-0.9,-0.2)$ respectively. For ($\tau_{1}$,
$\tau_{2}$) = $(0.01,0.15)$, the ST is found to be nearly same for all
$(t_{1},t_{2})$. Nevertheless, the maximum switching speed is found for the
thickness $t_{1}=2$nm and $t_{2}=1$nm. The ST gradually increases with the
increase in $t_{1}$ and $t_{2}$ for ($\tau_{1}$, $\tau_{2}$) = $(-0.9,-0.2)$
as seen from Fig. 2(q). Moreover, the switching speed slowed down with the
increase in the thickness of the RZ2 layer, as seen from Fig. 2(q).
Figure 4: (a) - (f) Oscillation trajectories of $\textbf{m}_{1}$ (blue) and
$\textbf{m}_{2}$ (red), (g) - (l) the time evolution of the magnetization
component $\text{m}_{x1}$ (red) and $\text{m}_{x2}$ (blue) for different
values of $\tau_{1}$ considering $\tau_{2}=0.4$ and
K${}_{\text{R}}=10$K${}_{\text{DM}}$. Plots (m) - (o) and (r) are the Fourier
transform of the magnetization component, while plots (p) and (q) depicts the
variation of switching time with $\tau_{i}$ and $t_{i}$ respectively.
#### III.1.2 For systems with RKKY $\sim$ DM interaction
The magnetization oscillations of the system are found to be very sensitive to
$\tau_{1}$ As the RKKY interaction is of the order of DMI, as seen from Figs.
3(a) and (g). For $\tau_{1}=0.1$, the synchronization of $\mathbf{m}_{1}$ and
$\mathbf{m}_{2}$ cease to exist. In this condition, $\mathbf{m}_{1}$ precesses
about the z-axis while $\mathbf{m}_{2}$ about the x-axis. However, the
oscillations are associated with multiple frequencies, as seen in Fig. 3(m).
With a little increase in $\tau_{1}$ to $0.2$, the system executes non-linear
oscillations as seen from Figs. 3(b) and (h). This nonlinearity is further
enhanced as $\tau_{1}$ increases to $0.3$ as indicated by Figs. 3(c) and (i).
These non-linear oscillations can be characterized by the multiple frequencies
in the Fourier spectra in 3(n) and (o). This may be due to the same order of
RKKY and DM interaction which destabilizes the oscillations. However,
sustained magnetization oscillations at about $250$ GHz can be achieved with a
further increase in $\tau_{1}$ to $0.7$ as suggested by Figs. 3(d) and (j).
The switching characteristics can be studied by choosing $\tau 1=1.0$ and
$-0.7$ as observed from Figs. 3(e), (f), (k) and (l). The ST vs $\tau_{2}$ for
different $\tau_{1}$ has been studied in Fig. 3(q). It is seen that the rapid
switching can be obtained by considering low values of $\tau_{1}$. It should
be noted that the ST is identical with Fig. 2, but rapid switching is obtained
for low values of $\tau_{1}$. Moreover, rapid switching is obtained for
$t_{1}=t_{2}=4$ nm. This indicates that by controlling the bias, current
magnetization oscillations and switching can be achieved even when RKKY
interaction is of the order DM interaction.
#### III.1.3 For systems with RKKY $>$ DM interaction
In Fig. 4, we study the magnetization dynamics for
K${}_{\text{R}}=10$K${}_{\text{DM}}$ with different choices of $\tau_{1}$
considering $\tau_{2}=0.4$. The system executes sustained oscillations about
the frequency $150$ GHz for $\tau_{1}=0.01$ and $-0.7$ as seen from Figs.
4(a), (m), (f), and (r). However, for $\tau_{1}=0.01$, the system stabilizes
after a short time, as indicated by Fig. 4(g). The switching characteristics
are observed for $\tau_{1}=0.1$ and $0.8$. However, it is to be noted that in
this case, the magnetization reversal of $\mathbf{m}_{2}$ is observed as
suggested by Figs. 4(b), (d), (h) and (j). With increase in $\tau_{1}$ to
$1.5$, a rapid switching of $\mathbf{m}_{2}$ is seen in Fig. 4(e) and (k). For
moderate $\tau_{1}$, the system executes non-linear oscillations as seen from
Fig. 4(c), (i) and (o). The dependence of ST with $\tau_{1}$ and $\tau_{2}$ is
identical to that of Fig. 3. However, a gradual decrease in ST is observed
with the rise in $t_{1}$ and $t_{2}$ as indicated by Fig. 4(p) and (q). Thus,
the non-linear characteristics of the magnetization oscillations significantly
decreases for K${}_{\text{R}}>$ K${}_{\text{DM}}$ as compared to
K${}_{\text{R}}\sim$ K${}_{\text{DM}}$. Moreover, the frequency of
oscillations also decreases in this scenario.
We observe that for RKKY $<$ DM interaction, the system behaves like a
switcher for $0.01<\tau_{1}<0.6$, oscillator for $0.6<\tau_{1}<0.7$, while it
remains unswitched for $0.7<\tau_{1}<1$ in case of $\tau_{2}>0$. When
$\tau_{2}<0$, the system remains unswitched for $-0.6<\tau_{1}<0$ while
behaves as an oscillator for $-0.65<\tau_{1}<-0.6$ and as a switcher for
$-1<\tau_{1}<-0.65$ as seen from Fig. 2. For the systems with RKKY $\sim$ DM
interaction, it shows unswitched characteristics for $-0.7<\tau_{1}<0.2$,
behaves as an oscillator for $0.6<\tau_{1}<0.7$ and as a switcher for
$0.7<\tau_{1}<1$ considering $\tau_{2}=0.4$. Moreover, under this condition,
the system display chaotic oscillations for $0.2<\tau_{1}<0.6$ as seen from
Fig. 3. Likewise, for the systems with RKKY $>$ DM interaction, oscillatory
characteristics are seen for $-0.7<\tau_{1}<0.1$, chaotic oscillations are
found for $0.1<\tau_{1}<0.4$ and the system remain unswitched for
$0.4<\tau_{1}<1.5$ considering $\tau_{2}=0.4$ as seen from Fig. 4.
Figure 5: Oscillation trajectories of $\textbf{m}_{1}$ (blue) and
$\textbf{m}_{2}$ (red) for (a) $\alpha_{\text{R}}=4\times 10^{-10}$ eV.m, (b)
$\alpha_{\text{R}}=4.6\times 10^{-10}$ eV.m and (c) $\alpha_{\text{R}}=5\times
10^{-10}$ eV.m considering $\tau_{1}=\tau_{2}=0.4$ and K${}_{\text{R}}$ = $10$
K${}_{\text{DM}}$. Plots (d) - (f) are the time evolution of the magnetization
component $\text{m}_{y1}$ (red) and $\text{m}_{y2}$ (blue) while plots (g) -
(i) are the Fourier transform of the magnetization.
Figure 6: Oscillation trajectories of $\textbf{m}_{1}$ (blue) and
$\textbf{m}_{2}$ (red) for (a) $H_{\text{ext}}=10^{-3}$T, $\tau_{1}=0.7$ (b)
$H_{\text{ext}}=0.1$T, $\tau_{1}=0.7$ and (c) $H_{\text{ext}}=0.1$T,
$\tau_{1}=-0.7$ considering $\tau_{2}=0.4$ and K${}_{\text{R}}$ = $10$
K${}_{\text{DM}}$. Plots (d) - (f) are the time evolution of the magnetization
component $\text{m}_{x1}$ (red) and $\text{m}_{x2}$ (blue) while plots (g) -
(i) are the Fourier transform of the magnetization.
Figure 7: Density plot of $m_{y}$ with $\alpha_{\textbf{R}}$ and $t$ for
$h_{0}=0.01$ (top), $h_{0}=0.7$ (middle) and $h_{0}=0.01$ (bottom) with
different choices of K${}_{\text{R}}$ and K${}_{\text{DM}}$. The plots (a)-(c)
in the left panel are for K${}_{\text{R}}=0.1K_{\text{DM}}$, (d)-(f) in the
middle panel are for K${}_{\text{R}}=$ K${}_{\text{DM}}$ while (g)-(i) in the
right panel are for K${}_{\text{R}}=10$K${}_{\text{DM}}$.
### III.2 Effect of RSOC
Even though magnetization dynamics in conventional ferromagnetic MTJs have
been extensively explored, the impact of SOC is yet to be understood. In
general, SOC is responsible for splitting the spin-up and spin-down sub-bands.
However, SOC can also significantly interplay with RKKY and DM interaction
and, thus, affect the magnetization oscillations and reversal. Hence, in Fig.
5, we investigated the oscillations for different RSOC considering
$h_{0}=0.7$, $\tau_{1}=\tau_{2}=0.4$ and K${}_{\text{R}}$ =
$10$K${}_{\text{DM}}$. The system undergoes non-linear oscillations for
$\alpha_{\text{R}}=4\times 10^{-10}$ eV.m as seen from Figs. 5(a) and (d).
This can be characterized through the multiple oscillation frequencies from
Fig. 5(g). The disappearance of non-linear oscillations for
$\alpha_{\text{R}}=4.6\times 10^{-10}$ eV.m indicates stable magnetization
oscillations as seen from Figs. 5(b) and (e). This can be confirmed from the
sharp peak around $350$ GHz in Fig. 5(h). With the further rise in
$\alpha_{\text{R}}=5\times 10^{-10}$ eV.m, both $\mathbf{m}_{1}$ and
$\mathbf{m}_{2}$ precess about the z-axis and decays slowly with an increase
in time as seen from Figs. 5(c) and (f). The synchronized nature of these
oscillations indicates the dominance of RKKY interaction in this system. So,
RSOC can significantly impact RKKY and DM interactions, which indeed control
the oscillations.
### III.3 Effect of external magnetic field
To understand the interplay of an external magnetic field on RZ material, we
study the magnetization dynamics and oscillations under low and moderate
external magnetic field in Fig. 6, considering $\tau_{2}=0.4$,
$\alpha_{\text{R}}=3\times 10^{-10}$ eV.m, $h_{0}=0.7$ and K${}_{\text{R}}$ =
10K${}_{\text{DM}}$. The system executes stable magnetization oscillations
around frequency $180$ GHz for $H_{\text{ext}}=10^{-3}$T as seen from Figs.
6(a), (d) and (g). This is because the impact of DM interaction and the
external magnetic field is minimized for strong K${}_{\text{R}}$. However, as
the magnetic field is increased to $0.1$T, the synchronization of $m_{x1}$ and
$m_{x2}$ due to K${}_{\text{R}}$ is broken. Thus, the system executes
nonlinear oscillations as seen from Figs. 6(b) and (e). These nonlinear
oscillations can be understood through the Fourier transform of the
magnetization component in Fig. 6(h). Though the non-linear characteristics of
the magnetization oscillations are found for $\tau_{1}=-0.7$, the oscillations
stabilize around the time range $\sim(3-8)$ ns for $\tau_{1}=0.7$ as seen from
Fig. 6(f). As indicated by the Fourier spectra in Fig. 6(i), the magnetization
display nonlinear oscillations under low low-frequency condition. This
indicates that the oscillations not only depend upon the biasing current but
can also be controlled through a suitable external magnetic field.
### III.4 Interplay of RKKY, DM interaction with RSOC
Though the interplay of RKKY and DM interactions have been investigated for
STT switching and oscillation in various previous works li ; emori ; pizzini ;
ryu ; baumgartner ; cao ; boulle ; sampaio ; jang but the impact of RZ effect
is yet to be understood. Fig. 7 depicts the magnetization dynamics in presence
of STT for double-barrier RZ-MTJ under different $h_{0}$ and K${}_{\text{R}}$
and K${}_{\text{DM}}$. It is observed that the magnetization oscillation under
K${}_{\text{R}}$ = 0.1 K${}_{\text{DM}}$ is quite similar with that of
K${}_{\text{R}}$ = 10 K${}_{\text{DM}}$ for corresponding choices of $h_{0}$
as seen from Figs. 7(a)-(c) and Figs. 7(g)-(i). We observe that with the
increase in RSOC the magnetization oscillation also increases for low
($h_{0}=0.01$) and moderate ($h_{0}=0.7$) Zeeman strength. It is to be noted
that the magnetization oscillation decays too rapidly as
$\alpha_{\text{R}}\rightarrow 0$ for low values of $h_{0}$ while an
oscillatory decay is observed at moderate $h_{0}$. This is due to the reason
that under low RZ energy, the Gilbert damping dominates the oscillations.
However, the oscillations are still present in the system for suitable STT.
The oscillations are too prominent under strong Zeeman condition even at low
RSOC as seen from Fig. 7(c) as the required energy can be achieved from the
exchange field. The oscillations slow down with time under moderate RSOC and
high Zeeman energy. A non-linear characteristic of the magnetization
orientation is observed as K${}_{\text{R}}$ = K${}_{\text{DM}}$ as seen from
the Figs. 7(d)-(f). This non-linear characteristics of $m_{y}$ is too
prominent for $h_{0}$ = $0.7$ under moderate RSOC. This is due to the
inconsistency of anisotropy of the corresponding magnetic layers at
K${}_{\text{R}}$ = K${}_{\text{DM}}$ li . However, this inconsistency can be
eliminated by using Zeeman field strength $h_{0}$ = $3$ as seen from Fig. 7(f)
which can be used as a good oscillator.
## IV Conclusions
In summary, in this work the magnetization oscillations and switching have
been investigated in the double-barrier RZ$|$HM$|$RZ MTJ in presence of
Rashba-Zeeman, RKKY and DM interactions for different STTs. We observed that
the system behaves as a good oscillator or a switcher when there exist a
significant difference in the strength of RKKY and DM interaction. A nonlinear
characteristics of the magnetization oscillation is observed as the order of
RKKY interaction matches with the DM interaction. This double interface MTJ
can also behave like a switcher for low and high STT when RKKY interaction is
weaker than DM interaction. A similar characteristics can be seen for systems
having strong RKKY interaction which signifies the interplay of RSOC on RKKY
and DM interaction in RZ$|$HM$|$RZ MTJ. Though the magnetization oscillations
are mostly nonlinear for systems having RKKY $\sim$ DMI, the magnetization
switching can still be tuned using suitable combinations of STT and RSOC. The
oscillations are found to be sensitive to the coupling between external
magnetic field and RZ interaction. The nonlinearity of oscillations can be
significantly reduced by using suitable combination of external magnetic field
and RZ interaction. It is noted that there exist a dependence of switching
time on layer thickness. The switching speed increases with thickness of the
layers for systems having RKKY $\geq$ DM interactions while an opposite
characteristics is seen for systems having RKKY $<$ DM interaction. Thus by
choosing materials with suitable RZ, RKKY and DM interactions, a double-
barrier RZ$|$HM$|$RZ MTJ can be realized as a switcher or an oscillator using
suitable STT.
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|
# Strichartz estimates for Maxwell equations in media: the fully anisotropic
case
Robert Schippa<EMAIL_ADDRESS>and Roland Schnaubelt
<EMAIL_ADDRESS>Department of Mathematics, Karlsruhe Institute of
Technology, Englerstrasse 2, 76131 Karlsruhe, Germany
###### Abstract.
We prove Strichartz estimates for Maxwell equations in media in the fully
anisotropic case with Hölder-continuous coefficients. To this end, we use the
FBI transform to conjugate the problem to phase space. After reducing to a
scalar estimate by means of a matrix symmetrizer, we show oscillatory integral
estimates for a variable-coefficient Fourier extension operator. The
characteristic surface has conical singularities for any non-vanishing time
frequency. Combined with energy estimates, we improve the local well-posedness
for certain fully anisotropic quasilinear Maxwell equations.
###### Key words and phrases:
Maxwell equations, Strichartz estimates, quasilinear wave equation, rough
coefficients, half wave equation, FBI transform
###### 2020 Mathematics Subject Classification:
Primary: 35L45, 35B65, Secondary: 35Q61.
## 1\. Introduction and main results
Maxwell equations are the fundamental principles of electro-magnetism. In
media they relate _displacement and electric field_
$(\mathcal{D},\mathcal{E}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{3}\times\mathbb{R}^{3}$
and _magnetic and magnetizing field_
$(\mathcal{B},\mathcal{H}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{3}\times\mathbb{R}^{3}$
via
(1)
$\left\\{\begin{array}[]{rlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H}-\mathcal{J}_{e},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=\rho_{e},\quad(t,x^{\prime})\in\mathbb{R}\times\mathbb{R}^{3},\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E}-\mathcal{J}_{m},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=\rho_{m},\end{array}\right.$
where
$(\rho_{e},\rho_{m}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}\times\mathbb{R}$
denote the _electric and magnetic charges_ and
$(\mathcal{J}_{e},\mathcal{J}_{m}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{3}\times\mathbb{R}^{3}$
the _electric and magnetic current_. (Space-time coordinates are written as
$x=(t,x^{\prime})=(x_{0},x_{1},\ldots,x_{d})\in\mathbb{R}\times\mathbb{R}^{d}$
and the dual variables in Fourier space as
$\xi=(\tau,\xi^{\prime})=(\xi_{0},\xi_{1},\ldots,\xi_{d})\in\mathbb{R}\times\mathbb{R}^{d}$.)
We refer to the physics’ literature for further explanations [4, 9]. We remark
that magnetic charges and currents are hypothetical, but included in the
analysis to highlight symmetry between electric and magnetic field.
To obtain a closed system in (1), one has to link the fields via material
laws. In the linear case, we consider the pointwise laws
(2)
$\mathcal{D}(x)=\varepsilon(x)\mathcal{E}(x),\quad\mathcal{B}(x)=\mu(x)\mathcal{H}(x)$
for the permittivity and permeability $\varepsilon,\mu\in
C^{s}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ with $0<s\leq
1$, whose values are supposed to be symmetric and uniformly elliptic, i.e.,
(3) $\exists\,\lambda,\Lambda>0:\,\forall v\in\mathbb{R}^{3},\,\forall
x\in\mathbb{R}^{4}:\;\quad\lambda|v|^{2}\leq
A^{ij}(x)v_{i}v_{j}\leq\Lambda|v|^{2},\;\;A\in\\{\varepsilon,\mu\\}.$
In the above display sum convention is in use.
In the previous work [15], the first author considered the partially
anisotropic case with
$\varepsilon(x)=\text{diag}(\varepsilon_{1}(x),\varepsilon_{2}(x),\varepsilon_{2}(x))$
and $\mu(x)\equiv\mu_{0}$. In [15], the Maxwell system was diagonalized to a
system of half-wave equations, for which the sharp Strichartz range for the
three dimensional wave equation could be recovered in the isotropic case
$\varepsilon_{1}(x)=\varepsilon_{2}(x)$. In the partially anisotropic case,
$\varepsilon_{1}(x)\neq\varepsilon_{2}(x)$, it remains an open question
whether solutions to Maxwell equations satisfy Strichartz estimates as for
(half-)wave equations in the $C^{2}$-case. However, for Lipschitz coefficients
the first author recovered Strichartz estimates with derivative loss as for
wave equations with Lipschitz coefficients (cf. [22]).
In two space dimensions [17] we established sharp Strichartz estimates in the
fully anisotropic case
$\varepsilon(x)=\begin{pmatrix}\varepsilon^{11}(x)&\varepsilon^{12}(x)\\\
\varepsilon^{21}(x)&\varepsilon^{22}(x)\end{pmatrix},\quad\mu(x)=\mu_{0}(x)$
via diagonalization without imposing additional assumptions on $\varepsilon$.
In three dimensions the fully anisotropic case (see (5)), Strichartz estimates
have not been studied before. In this case the diagonalization procedure
introduces singularities in the conjugation matrices, and we opt for a
different approach. The fully anisotropic case appears in the study of biaxial
crystals (cf. [5, 8]). Moreover, when treating nonlinear material laws
$(\varepsilon(\mathcal{E}),\mu(\mathcal{H}))$ one needs Strichatz estimates
for differentiated fields, see Section 5. However, these fields only fullfil a
Maxwell system with modified coefficients, which are matrix-valued even if the
original $(\varepsilon(\mathcal{E}),\mu(\mathcal{H}))$ are scalar and thus
isotropic.
We work under the following assumption on $\varepsilon$ and $\mu$, which can
be described as uniform anisotropic, possibly off-diagonal material law.
###### Assumption 1.
Let
$\varepsilon,\mu:\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}_{\text{sym}}^{3\times
3}$ satisfy (3) and suppose that there is $\Phi\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ with111In $\Phi$
we deviate from the usual matrix index notation and consider $\varphi_{i}$ as
column vectors.
$\Phi=\begin{pmatrix}\varphi_{1}&\varphi_{2}&\varphi_{3}\end{pmatrix}=\begin{pmatrix}\varphi_{11}&\varphi_{21}&\varphi_{31}\\\
\varphi_{12}&\varphi_{22}&\varphi_{32}\\\
\varphi_{13}&\varphi_{23}&\varphi_{33}\end{pmatrix},\quad\Phi^{t}(x)\Phi(x)=1_{3\times
3},$
and $\varepsilon^{d},\mu^{d}\in
C(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ with
$\varepsilon^{d}(x)=\text{diag}(\varepsilon^{1}(x),\varepsilon^{2}(x),\varepsilon^{3}(x))\text{
\ and \ }\mu^{d}(x)=\text{diag}(\mu^{1}(x),\mu^{2}(x),\mu^{3}(x))$
such that
$\varepsilon^{d}=\Phi^{t}\varepsilon\Phi,\quad\mu^{d}=\Phi^{t}\mu\Phi,$
and $\varepsilon^{d}$, $\mu^{d}$ satisfy
$\exists c>0:\,\forall x\in\mathbb{R}^{4}:\,\forall i\neq
j:\,\big{|}\frac{\varepsilon^{i}(x)}{\mu^{i}(x)}-\frac{\varepsilon^{j}(x)}{\mu^{j}(x)}\big{|}\geq
c.$
We let
(4) $P(x,D)=\begin{pmatrix}-\partial_{t}(\varepsilon\cdot)&\nabla\times\\\
\nabla\times&\partial_{t}(\mu\cdot)\end{pmatrix}$
such that (1) is concisely written as
$P(x,D)\begin{pmatrix}\mathcal{E}\\\
\mathcal{H}\end{pmatrix}=\begin{pmatrix}\mathcal{J}_{e}\\\
-\mathcal{J}_{m}\end{pmatrix},\quad\nabla\cdot(\varepsilon\mathcal{E})=\rho_{e},\;\nabla\cdot(\mu\mathcal{H})=\rho_{m}.$
We write $u=(\mathcal{E},\mathcal{H})$ and $\rho_{em}=(\rho_{e},\rho_{m})$.
As noted above in the isotropic and to some extent in the partially
anisotropic case, by [16] the Maxwell system possesses Strichartz estimates as
the scalar wave equation. However, in the fully anisotropic case already for
constant coefficients
$\varepsilon=\text{diag}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})$,
$\mu=\text{diag}(\mu_{1},\mu_{2},\mu_{3})$ satisfying
(5)
$\frac{\varepsilon_{1}}{\mu_{1}}\neq\frac{\varepsilon_{2}}{\mu_{2}}\neq\frac{\varepsilon_{3}}{\mu_{3}}\neq\frac{\varepsilon_{1}}{\mu_{1}},$
Liess [11] showed that the dispersive properties in the charge-free case
$\nabla\cdot\mathcal{D}=\nabla\cdot\mathcal{B}=0$ are only the same as for
wave equations in two dimensions, namely
(6)
$\|S_{1}^{\prime}(\mathcal{E},\mathcal{H})(t)\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim(1+|t|)^{-\frac{1}{2}}\|(\mathcal{E},\mathcal{H})(0)\|_{L^{1}(\mathbb{R}^{3})}.$
Here we use a homogeneous dyadic decomposition
$(S^{\prime}_{\lambda})_{\lambda\in 2^{\mathbb{Z}}}$ of spatial frequencies,
and $(S_{\lambda})_{\lambda\in 2^{\mathbb{Z}}}$ denotes a decomposition of
space-time frequencies, see (18).
This surprising behavior is connected to the shape of the characteristic
surface depending on the eigenvalues of $\varepsilon$ and $\mu$, which has
been discussed in [16, Section 2.3]. In [12] the first author and R. Mandel
proved the existence of solutions to the time-harmonic Maxwell equations in
the fully anisotropic case under the assumption that $\varepsilon$ and $\mu$
are commuting. To the best of the authors’ knowledge, this was the first time
the condition (5) appeared explicitly in the literature. In this case the
characteristic surface of Maxwell equations ceases to be regular as the union
of two ellipsoids (as in the (partially) anisotropic case). Instead it becomes
the singular Fresnel surface, see [12] for a detailed quantitative analysis.
The properties of the Fresnel surface are recalled in Section 3. We further
note that the characteristic surface can have more singularities if
$\varepsilon$ and $\mu$ do not commute, [3].
The decreased dispersive effect for the Fresnel surface forces us to consider
the wave admissibility condition in two dimensions:
(7) $\frac{2}{p}+\frac{1}{q}\leq\frac{1}{2},\quad 2\leq p,q\leq\infty.$
By (6) and the interpolation argument of Keel–Tao [7], for $(p,q)$ satisfying
(7) the solutions $(\mathcal{E},\mathcal{H})$ to homogeneous Maxwell equations
with constant coefficients fulfill the Strichartz estimate
$\||D^{\prime}|^{-\rho}(\mathcal{E},\mathcal{H})\|_{L_{t}^{p}(\mathbb{R},L_{x^{\prime}}^{q}(\mathbb{R}^{3}))}\lesssim\|(\mathcal{E},\mathcal{H})(0)\|_{L^{2}(\mathbb{R}^{3})}$
in the charge-free case. We give the details and local-in-time estimates in
the case of non-vanishing charges in Section 2. In the following we denote
space-time Lebesgue spaces by $L_{t}^{p}L_{x^{\prime}}^{q}=L^{p}L^{q}$ for the
sake of brevity, where $L^{p}=L^{p}L^{p}$. In the above display the derivative
loss is denoted by $\rho$ which satisfies the scaling condition
(8) $\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}.$
Furthermore, we use fractional derivatives given as Fourier multipliers by
$(|D|^{\alpha}f)\widehat{(}\xi)=|\xi|^{\alpha}\hat{f}(\xi),\;(|D^{\prime}|^{\alpha}f)\widehat{(}\xi)=|\xi^{\prime}|^{\alpha}\hat{f}(\xi),\;(\langle
D\rangle^{\alpha}f)\widehat{(}\xi)=\langle\xi\rangle^{\alpha}\hat{f}(\xi),\quad\text{etc.},$
where $\langle x\rangle=(1+|x|^{2})^{1/2}$.
We first show the following Strichartz estimates for rough coefficients in the
diagonal case.
###### Theorem 1.1.
Let $T>0$ and $\delta>0$, and suppose that $\varepsilon,\mu\in
C^{s}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}_{\text{sym}}^{3\times 3})$
with $0<s\leq 1$ and
$\|\partial(\varepsilon,\mu)\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}<\infty$
satisfy Assumption 1 with $\Phi\equiv 1$. Let $P$ be as in (4). Then the
estimate
$\displaystyle\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{2p}-\delta}u\|_{L^{p}(0,T;L^{q}(\mathbb{R}^{3}))}$
$\displaystyle\lesssim\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L_{x^{\prime}}^{2}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}$
(9) $\displaystyle\quad+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}}$
holds provided that $p,q$ satisfy (7), $q\neq 2$, and $\rho$ is given by (8).
The implicit constant depends on $\delta$, $T$,
$\|(\varepsilon,\mu)\|_{C^{s}}$, and
$\|\partial(\varepsilon,\mu)\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}$.
We stress that both the different admissibility conditions and the appearance
of the charges highlight the effect of the system character of the Maxwell
equations, compared to the case of scalar wave equations treated in [21, 22,
23].
To establish (1.1), we first show
(10)
$\||D|^{-\rho+\frac{s-2}{4}}u\|_{L^{p}L^{q}}\lesssim_{\|(\varepsilon,\mu)\|_{C^{s}}}\|u\|_{L^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{1}{2}+\frac{s-2}{4}}\rho_{em}\|_{L^{2}}$
(with a modification at the endpoint $q=\infty$) if also
$\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$ and $q>p$ via phase space analysis, see
Theorem 3.2. The core step in the argument is an estimate of a Fourier
extension operator on the Fresnel surface $S$, where we can reduce to scalar
problem using a symmetrizer. The singularities of $S$ are handled by a dyadic
scaling around them on annuli in Fourier space. Here we also use ideas from
[12] in the time-harmonic case with constant coefficients. The degeneracy at
$|\xi^{\prime}|\gg|\xi_{0}|$ of the main symbol of the Maxwell system is
treated using the charges in a separate microlocal argument.
We note that the derivative loss $\rho+\frac{2-s}{4}$ in (10) is the same as
in [21] for scalar wave equations. We actually improve the loss in (1.1), but
so far we cannot reach the sharp regularity loss established in [22, 23] for
the wave equation or [17, 15] in the 2D or isotropic Maxwell case. In contrast
to these works up to now we cannot use deeper properties of the Hamilton flow
of the problem because of the strong system character in the fully isotropic
case. In these papers also the case $s\in[1,2]$ and different norms on the
right-hand side of (1.1) have been treated. We plan to tackle these issues in
future work.
Starting from (10) we reduce the regularity loss by passing through short-time
Strichartz estimates. In Proposition 3.8 we show the endpoint estimate for
$p=4$ and $q=\infty$ on finite-time intervals
$\displaystyle\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L_{T}^{4}L^{\infty}_{x^{\prime}}}$
$\displaystyle\lesssim\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{\frac{s-2}{8}-\frac{1}{2}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}$
$\displaystyle\quad+\|\langle
D^{\prime}\rangle^{\frac{s-2}{8}-\frac{1}{2}}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}}$
with implicit constant depending on $\|(\varepsilon,\mu)\|_{C^{1}}$, $T$ and
$\delta$, where we set
$L^{p}_{T}L^{q}_{x^{\prime}}=L^{p}(0,T;L^{q}_{x^{\prime}})$. The different
form of the charge term stems from an argument involving Duhamel’s formula,
see (3.3) and also [15, 17]. Theorem 1.1 then follows by interpolating the
above display with the standard energy estimate
$\|u\|_{L^{\infty}_{T}L^{2}_{x^{\prime}}}\lesssim_{\|(\varepsilon,\mu)\|_{C^{1}},T}\|u_{0}\|_{L^{2}}+\|Pu\|_{L^{1}_{T}L_{x^{\prime}}^{2}}.$
We note that (10) is shown for $q>6$ and that our method of proof breaks down
for $(p,q)$ closer to the energy point $(p,q)=(\infty,2)$. Broadly speaking,
our arguments show that we can prove Strichartz estimates for characteristic
surfaces $S=\\{\tilde{q}(\xi_{0})=0\\}$ with isolated degeneracies
$\tilde{q}(\xi_{0})=0$, $\nabla\tilde{q}(\xi_{0})=0$,
$\det(\partial^{2}\tilde{q}(\xi_{0}))\neq 0$.
Second, we establish the following variant in the possibly non-diagonal case.
###### Theorem 1.2.
Let $T>0$, $\delta>0$, and suppose that $\varepsilon,\mu\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}_{\text{sym}}^{3\times 3})$
satisfy Assumption 1. Let $P$ be as in (4). Then the estimate
$\displaystyle\|\langle
D^{\prime}\rangle^{-\rho-\frac{1}{2p}-\delta}u\|_{L^{p}(0,T;L^{q}(\mathbb{R}^{3}))}$
$\displaystyle\lesssim\|u_{0}\|_{L^{2}}+\|Pu\|_{L^{1}_{T}L_{x^{\prime}}^{2}}$
$\displaystyle\quad+\|\langle
D^{\prime}\rangle^{-5/8}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-5/8}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}}$
holds provided that $(p,q)$ satisfy (7), $q\neq 2$, and $\rho$ is given by
(8). The implicit constant depends on $\delta$, $T$, and
$\|(\varepsilon,\mu)\|_{C^{1}}$.
To show this theorem, one uses the orthogonality transformation $\Phi$ to pass
to a Maxwell-type system with the diagonal coefficents $\varepsilon^{d}$ and
$\mu^{d}$ from Assumption 1, but with modified differential operators in
$x^{\prime}$ depending on $\Phi$. This system can be treated similar to the
system arising in the proof of Theorem 1.1.
We believe that the global existence and regularity of eigenvectors as stated
in Assumption 1 is non-trivial. Hence, we devote Section 6 to a discussion of
existence and regularity of eigenvectors for parameter-dependent matrices. It
turns out that if the eigenvalues are uniformly separated and the parameter
domain is simply connected, the eigenvectors are as regular as the eigenvalues
and admit global parametrisations. The results of Section 6 imply the
following proposition, which leads to a corollary to Theorem 1.2 stated below.
###### Proposition 1.3.
Suppose that $\mu\equiv 1$ and $\varepsilon\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3}_{\text{sym}})$
satisfies (3) and
$\exists c>0:\,\forall
x\in\mathbb{R}^{4}:\,|\varepsilon_{i}(x)-\varepsilon_{j}(x)|\geq c>0,\quad
i\neq j,$
where $(\varepsilon_{i}(x))_{i=1,2,3}$ are the eigenvalues of
$\varepsilon(x)$. Then $(\varepsilon,\mu)$ fulfill Assumption 1.
###### Corollary 1.4.
Let $\varepsilon$ and $\mu$ be like in Proposition 1.3. Then the Strichartz
estimates (1.1) hold true under the assumptions of Theorem 1.2.
In Section 5 we apply the Strichartz estimates from Theorem 1.1 to improve the
local well-posedness theory of quasilinear Maxwell equations
(11)
$\left\\{\begin{array}[]{rlrlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=0,&\quad\mathcal{D}(0)\\!\\!\\!\\!&=\mathcal{D}_{0}\in
H^{s}(\mathbb{R}^{3};\mathbb{R}^{3}),\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=0,&\quad\mathcal{B}(0)\\!\\!\\!\\!&=\mathcal{B}_{0}\in
H^{s}(\mathbb{R}^{3};\mathbb{R}^{3}),\end{array}\right.$
with small initial fields and such that $\varepsilon(\mathcal{E})$ has
uniformly separated eigenvalues. For sake of simplicity, we suppose that
$\mu=1_{3\times 3}$. Contrary to the isotropic Kerr case analyzed in [17, 15],
we cannot automatically deduce energy estimates by symmetrization. It turns
out that this requires additional symmetries of the permittivity. We shall
rewrite $\mathcal{D}=\varepsilon(\mathcal{E})\mathcal{E}$ to
$\mathcal{E}=\psi(\mathcal{D})\mathcal{D}$, which is possible for small fields
under mild assumptions on $\varepsilon$ by invoking the implicit function
theorem. In this form, we can phrase the condition for the existence of a
symmetrizer as
(12)
$\varepsilon^{ijk}\big{(}\frac{\partial}{\partial\mathcal{D}_{j}}\psi(\mathcal{D})_{k\ell}\big{)}\mathcal{D}^{\ell}=0$
for $i\in\\{1,2,3\\}$ and summation over $j,k\in\\{1,2,3\\}$, where
$\varepsilon^{ijk}$ denotes the Levi-Civita symbol. (Such a condition was also
used in [10].) We refer to the paragraph before Proposition 5.3 for discussion
of examples like
$\varepsilon=\text{diag}(\varepsilon_{0}^{1},\varepsilon_{0}^{2},\varepsilon_{0}^{3})+\text{diag}(\alpha_{1}|\mathcal{E}_{1}|^{2},\alpha_{2}|\mathcal{E}_{2}|^{2},\alpha_{3}|\mathcal{E}_{3}|^{2}).$
It seems plausible that permittivities obtained like this can model biaxial
crystals with nonlinear electric response.
We show the following theorem, which improves the local well-posedness via
Strichartz estimates by $1/9$ derivatives compared to energy arguments. It is
the first result of this kind for cases of fully anisotropic Maxwell systems.
This improvement is smaller as in [23] or [15, 17]. (For instance, for scalar
quasilinear wave equations one gains $1/3$ derivatives compared to energy
arguments by the results in [23].) Strichartz estimates with smaller
regularity loss would imply better results, but it is unclear what can be
achieved in the fully anisotropic Maxwell case.
###### Theorem 1.5.
Let $\varepsilon_{i}\in C^{\infty}(\mathbb{R};\mathbb{R})$ and let
$\varepsilon(\mathcal{E})=\mathrm{diag}(\varepsilon_{1}(\mathcal{E}_{1}),\varepsilon_{2}(\mathcal{E}_{2}),\varepsilon_{3}(\mathcal{E}_{3}))$
be uniformly positive definite with uniformly separated eigenvalues, i.e.,
there are $c,\delta>0$ such that for any $\mathcal{E}\in\mathbb{R}^{3}$ with
$|\mathcal{E}|\leq\delta$ we have
$|\varepsilon_{i}(\mathcal{E})-\varepsilon_{j}(\mathcal{E})|\geq c>0.$
Write $\mathcal{D}=\varepsilon(\mathcal{E})\mathcal{E}$ as
$\mathcal{E}=\psi(\mathcal{D})\mathcal{D}$ for
$\|\mathcal{D}\|_{L^{\infty}}\leq\delta$ and assume that $\psi$ satisfies
(12). Then there is $\delta^{\prime}>0$ such that (11) is locally well-posed
for $s>2+\frac{7}{18}$ for initial data
$\|u_{0}\|_{H^{s}}\leq\delta^{\prime}$.
For the proof we use Strichartz estimates to improve on Sobolev embedding.
Under the above symmetry assumptions, we find energy estimates
$\|(\mathcal{E},\mathcal{H})(t)\|_{H^{s}}\lesssim
e^{c(A)\int_{0}^{t}\|\partial(\mathcal{E},\mathcal{H})(s)\|_{L_{x^{\prime}}^{\infty}}ds}\|(\mathcal{E},\mathcal{H})(0)\|_{H^{s}}$
with
$A=\|(\mathcal{E},\mathcal{H})\|_{L_{T}^{\infty}L_{x^{\prime}}^{\infty}}$. An
estimate of
$\|\partial(\mathcal{E},\mathcal{H})\|_{L_{T}^{1}L_{x^{\prime}}^{\infty}}$
based on Sobolev embedding would lead to a regularity of $s>5/2$. We want to
use non-trivial Strichartz estimates for
$\|\partial(\mathcal{E},\mathcal{H})\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}$
provided that the coefficients are in $C^{\alpha}_{x}$. The Hölder norm is
again controlled by a spatial Sobolev norm and we can close the argument for
$s>3/2+\alpha$, if the derivative loss in the Strichartz estimates _for
$\partial(\mathcal{E},\mathcal{H})$_ is controlled in the regularity $H^{s}$.
This leads to the condition $s>\frac{3}{2}+\frac{8}{9}$. In the next step, we
control the $L^{2}$-norm of differences of solutions
$v=(\mathcal{E}^{1},\mathcal{H}^{1})-(\mathcal{E}^{2},\mathcal{H}^{2})$ by
$\|v(t)\|_{L^{2}}\lesssim e^{c(A)\int_{0}^{t}B(s)ds}\|v(0)\|_{L^{2}}$
with
$A=\|(\mathcal{E}^{1},\mathcal{H}^{1})\|_{L^{\infty}_{T}L^{\infty}_{x^{\prime}}}+\|(\mathcal{E}^{2},\mathcal{H}^{2})\|_{L^{\infty}_{T}L_{x^{\prime}}^{\infty}}$
and
$B(s)=\|(\mathcal{E}^{1},\mathcal{H}^{1})(s)\|_{L^{\infty}_{x^{\prime}}}+\|(\mathcal{E}^{2},\mathcal{H}^{2})(s)\|_{L_{x^{\prime}}^{\infty}}$.
By the same argument as above, we obtain Lipschitz continuous dependence in
$L^{2}$ for initial data in regularity $H^{s}$ if $s>\frac{3}{2}+\frac{8}{9}$.
To find continuous dependence to hold in $H^{s}$, we use the frequency
envelope approach due to Tao [20]. (See also [6] for an exposition, and [17]
for a previous application in the context of Maxwell equations.) This does not
yield uniform continuous dependence, which cannot be expected for a
quasilinear hyperbolic problem.
_Outline of the paper._ In Section 2 we revisit Strichartz estimates in the
constant-coefficient case for fully anisotropic Maxwell equations. In Section
3 we prove Strichartz estimates for diagonal variable coefficients in the
uniformly fully anisotropic case as stated in Theorem 1.1. In Section 4 we
reduce the more general case of Assumption 1 to the case of diagonal
permittivity and permeability and prove Theorem 1.2. In Section 5 we argue how
the Strichartz estimates for rough coefficients improve the local well-
posedness theory for quasilinear Maxwell equations. In Section 6 we show the
global existence and regularity of eigenvectors in case of separated
eigenvalues of parameter-dependent matrices in simply connected domains.
## 2\. Strichartz estimates in the constant-coefficient case
In this section we revisit the Strichartz estimates in the fully anisotropic
case for constant coefficients. For the remainder of the section, let
(13)
$\varepsilon=\text{diag}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}),\quad\mu=\text{diag}(\mu_{1},\mu_{2},\mu_{3})\quad\text{with
\ }\varepsilon_{i},\ \mu_{j}\in\mathbb{R}_{>0}.$
By the symmetries of Maxwell equations (cf. [12]), we can reduce the general
case of positive definite $\varepsilon,\mu\in\mathbb{R}^{3\times
3}_{\text{sym}}$ to (13) provided that $\varepsilon$ and $\mu$ commute. We
supplement the linear Maxwell equations
(14)
$\left\\{\begin{array}[]{rlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=\rho_{e},\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=\rho_{m}\end{array}\right.$
on $\mathbb{R}\times\mathbb{R}^{3}$ with the constant-coefficient material
laws
(15)
$\mathcal{D}=\varepsilon\mathcal{E},\quad\mathcal{B}=\mu\mathcal{H},\quad\text{where
\eqref{eq:DiagonalConstantCoefficients} is true.}$
In the charge-free case, dispersive time-decay of solutions to (14) was proved
by Liess [11]. Here we deduce Strichartz estimates also allowing for charges.
The first observation is that non-trivial charges in (14) inhibit global-in-
time Strichartz estimates
$\||D^{\prime}|^{-\rho}u\|_{L^{p}(\mathbb{R};L^{q}_{x^{\prime}})}\lesssim\|u(0)\|_{L^{2}_{x^{\prime}}}$
with $u=(\mathcal{E},\mathcal{H})$,
$\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$, and
$\frac{2}{p}+\frac{1}{q}\leq\frac{1}{2}$.
Indeed, given nonzero
$(\rho_{e},\rho_{m})\in\dot{H}^{\gamma}\times\dot{H}^{\gamma}$, let
$(\Phi_{1},\Phi_{2})$ solve the elliptic equations
(16)
$\nabla\cdot(\varepsilon\nabla\Phi_{1})=\rho_{e},\qquad\nabla\cdot(\mu\nabla\Phi_{2})=\rho_{m}.$
Then (14) possesses the stationary solution $(\mathcal{E},\mathcal{H})(0)=u$
with $\mathcal{E}(0)=\nabla\Phi_{1}$ and $\mathcal{H}(0)=\nabla\Phi_{2}$. We
have $(\Phi_{1},\Phi_{2})\in\dot{H}^{\gamma+2}$ and
$u(0)\in\dot{H}^{\gamma+1}$. However, the solution does not decay, and hence
global-in-time Strichartz estimates for (14) are not possible. To obtain such
estimates in the charge-free case, we use the following dispersive estimate
due to Liess [11, Theorem 1.3].
###### Proposition 2.1.
Let $u=(\mathcal{E},\mathcal{H})$ solve (14) with (15) and
$\rho_{e}=\rho_{m}=0$. We then have
(17)
$\|S_{1}^{\prime}u(t)\|_{L^{\infty}_{x^{\prime}}}\lesssim(1+|t|)^{-\frac{1}{2}}\|u(0)\|_{L^{1}_{x^{\prime}}}.$
To define the needed dyadic frequency decomposition, let $\chi\in
C^{\infty}_{c}(\mathbb{R};\mathbb{R}_{\geq 0})$ be a radially decreasing with
$\chi(x)=1$ for $|x|\leq 1$ and $\chi(x)=0$ for $|x|\geq 2$. We set
(18)
$\begin{split}(S^{\prime}_{\lambda}f)\widehat{(}\xi)&=(\chi(\|\xi^{\prime}\|/\lambda)-\chi(\|\xi^{\prime}\|/2\lambda))\hat{f}(\xi),\\\
(S_{\lambda}f)\widehat{(}\xi)&=(\chi(\|\xi\|/\lambda)-\chi(\|\xi\|/2\lambda))\hat{f}(\xi)\end{split}$
for $\lambda\in 2^{\mathbb{Z}}$. Moreover, we write
$\displaystyle S_{0}^{\prime}$ $\displaystyle=1-\sum_{\lambda\in
2^{\mathbb{N}_{0}}}S^{\prime}_{\lambda},\quad S_{0}=1-\sum_{\lambda\in
2^{\mathbb{N}_{0}}}S_{\lambda},\qquad S_{\geq
1}^{\prime}=1-S^{\prime}_{0},\quad S_{\geq 1}=1-S_{0},$ $\displaystyle
S^{\prime}_{\sim M}$
$\displaystyle=\sum_{\lambda=M/8}^{8M}S^{\prime}_{\lambda},\quad S_{\sim
M}=\sum_{\lambda=M/8}^{8M}S_{\lambda}.$
By Littlewood-Paley decomposition, rescaling, and the Keel–Tao interpolation
argument [7, Theorem 1.2], we find the desired global estimates.
###### Theorem 2.2.
Let $u=(\mathcal{E},\mathcal{H})$ be a solution to (14) with (15) and
$\rho_{e}=\rho_{m}=0$. Then the global Strichartz estimates
(19)
$\||D^{\prime}|^{-\rho}u\|_{L^{p}(\mathbb{R};L^{q}_{x^{\prime}})}\lesssim\|u(0)\|_{L^{2}_{x^{\prime}}}$
hold with $\rho$ given by (8) and $(p,q)$ satisfying (7) and $q<\infty$.
###### Remark 2.3.
For $q=\infty$, estimate (19) is true for the homogeneous Besov space:
$\|u\|_{L_{t}^{p}(\mathbb{R};\dot{B}^{-\rho}_{q,2}(\mathbb{R}^{3}))}\lesssim\|u(0)\|_{L^{2}_{x^{\prime}}}.$
To lighten the discussion in the following, we suppose $q<\infty$. See also
Remark 3.3.
We next show local-in-time estimates involving charges.
###### Theorem 2.4.
Let $\varepsilon$, $\mu$, $\rho$, $p$, and $q$ be as in Theorem 2.2. We then
have
(20) $\|\langle
D^{\prime}\rangle^{-\rho}u\|_{L^{p}_{T}L^{q}_{x^{\prime}}}\lesssim
T^{\frac{1}{p}}(\|u(0)\|_{L^{2}}+\|\rho_{em}\|_{H^{\frac{1}{p}-1}_{x^{\prime}}}).$
###### Proof.
We begin with estimating the low frequencies
$\|\langle
D^{\prime}\rangle^{-\rho}S^{\prime}_{0}u\|_{L^{p}_{T}L^{q}_{x^{\prime}}}\lesssim
T^{\frac{1}{p}}\|u(0)\|_{L^{2}_{x^{\prime}}}$
by Hölder in time and Bernstein’s inequality.
For the estimate of the high frequencies, we split the initial data into
stationary and dispersive components. Let $(\Phi_{1},\Phi_{2})$ be the
solutions to the elliptic equations in (16). Since we confine to high
frequencies, we can replace the homogeneous with inhomogeneous norms. We have
the stationary solution $u_{\text{stat}}(t)=S^{\prime}_{\geq
1}(\nabla\Phi_{1},\nabla\Phi_{2})$ and we write
$S^{\prime}_{\geq 1}(\mathcal{E},\mathcal{H})(0)=S^{\prime}_{\geq
1}(\nabla\Phi_{1},\nabla\Phi_{2})+S^{\prime}_{\geq
1}(\mathcal{E},\mathcal{H})_{\text{disp}}(0).$
We denote the solution emanating from the last term by $u_{\text{disp}}$,
which is clearly charge free. Taking the Fourier transform of
$\nabla\cdot(\varepsilon\mathcal{E}(0))=\nabla\cdot(\varepsilon\nabla\Phi_{1})$,
we obtain
$\frac{i\xi_{k}\varepsilon^{km}\hat{\mathcal{E}}_{m}(0,\xi)}{\xi_{i}\xi_{j}\varepsilon^{ij}}=\hat{\Phi}_{1}(\xi).$
Plancherel’s theorem thus yields
$\|\nabla\Phi_{1}\|_{L^{2}}\lesssim\|\mathcal{E}(0)\|_{L^{2}}$ and likewise
$\|\nabla\Phi_{2}\|_{L^{2}}\lesssim\|\mathcal{H}(0)\|_{L^{2}}$. Hence
$\|u_{\text{disp}}(0)\|_{L^{2}}\lesssim\|(\mathcal{E},\mathcal{H})(0)\|_{L^{2}}$.
By linearity, we have $S^{\prime}_{\geq 1}u=u_{\text{disp}}+u_{\text{stat}}$
and Theorem 2.2 implies
$\displaystyle\||D^{\prime}|^{-\rho}S^{\prime}_{\geq
1}u\|_{L^{p}_{T}L^{q}_{x^{\prime}})}$
$\displaystyle\leq\||D^{\prime}|^{-\rho}S^{\prime}_{\geq
1}u_{\text{disp}}\|_{L^{p}_{T}L^{q}_{x^{\prime}}}+\||D^{\prime}|^{-\rho}S^{\prime}_{\geq
1}u_{\text{stat}}\|_{L^{p}_{T}L^{q}_{x^{\prime}}}$
$\displaystyle\lesssim\|S^{\prime}_{\geq
1}u(0)\|_{L^{2}_{x^{\prime}}}+T^{\frac{1}{p}}\|(\rho_{e},\rho_{m})\|_{H^{\frac{1}{p}-1}_{x^{\prime}}}.$
Inequality (20) follows. ∎
We turn to the inhomogeneous problem
(21)
$\left\\{\begin{array}[]{rlrlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H}-\mathcal{J}_{e},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=\rho_{e},&\quad\mathcal{D}(0)\\!\\!\\!\\!&=\mathcal{D}_{0},\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E}-\mathcal{J}_{m},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=\rho_{m},&\quad\mathcal{B}(0)\\!\\!\\!\\!&=\mathcal{B}_{0}.\end{array}\right.$
We write $u=(\mathcal{E},\mathcal{H})$ for solutions to (21) as well as
$\mathcal{J}=(\mathcal{J}_{e},\mathcal{J}_{m})$ and
$\nabla\cdot\mathcal{J}=(\nabla\cdot\mathcal{J}_{e},\nabla\cdot\mathcal{J}_{m})$.
###### Theorem 2.5.
Let $u=(\mathcal{E},\mathcal{H})$ be a solution to (21), $(\rho,p,q)$ and
$(\tilde{\rho},\tilde{p},\tilde{q})$ satisfy (7) and (8) with
$q,\tilde{q}<\infty$. For $T<\infty$, then the estimates
(22) $\begin{split}\|\langle
D^{\prime}\rangle^{-\rho}u\|_{L^{p}([0,T],L^{q}(\mathbb{R}^{3}))}&\lesssim
T^{\frac{1}{p}}(\|u(0)\|_{L^{2}}+\|\rho_{em}(0)\|_{H^{\frac{1}{p}-1}})\\\
&\quad+T^{\frac{1}{p}}\big{(}\|\mathcal{J}\|_{L^{1}([0,T],L^{2})}+\|\nabla\cdot\mathcal{J}\|_{L^{1}([0,T],H^{\frac{1}{p}-1})}\big{)}.\end{split}$
hold. If $\rho_{em}=0$ and $\nabla\cdot\mathcal{J}=0$, we have the global
estimates
(23)
$\||D^{\prime}|^{-\rho}u\|_{L^{p}(\mathbb{R};L^{q}(\mathbb{R}^{3}))}\lesssim\|u(0)\|_{L^{2}(\mathbb{R}^{3})}+\||D^{\prime}|^{\tilde{\rho}}\mathcal{J}\|_{L^{\tilde{p}^{\prime}}L^{\tilde{q}^{\prime}}}.$
###### Proof.
We denote the propagator for the free solution by $U(t)$, so that we can write
$(\mathcal{D},\mathcal{B})(t)=U(t)(\mathcal{D},\mathcal{B})(0)-\int_{0}^{t}U(t-s)\mathcal{J}(s)ds$
by Duhamel’s formula. Since $\|\langle
D^{\prime}\rangle^{s}u\|_{L^{2}}\sim\|\langle
D^{\prime}\rangle^{s}(\mathcal{D},\mathcal{B})\|_{L^{2}}$, we can apply
Theorem 2.4 with initial time $s$ and Minkowski’s inequality to find (22) to
hold. Inequality (23) is a consequence of the dispersive estimate (17) and
Keel-Tao interpolation [7, Theorem 1.2]. ∎
## 3\. Proof of Strichartz estimates in the fully anisotropic case with
diagonal material coefficients
In the following we consider diagonal permittivity and permeability
(24)
$\varepsilon=\text{diag}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}),\quad\mu=\text{diag}(\mu_{1},\mu_{2},\mu_{3}),\;$
which are supposed to satisfy the uniform ellipticity condition
(25) $\exists\lambda,\Lambda>0:\forall
x\in\mathbb{R}^{4}:\,\lambda\leq\nu_{i}(x)\leq\Lambda\text{ for }i=1,2,3\text{
and }\nu\in\\{\varepsilon,\mu\\}.$
Furthermore, we require the following separability condition, which guarantees
uniformity of the curvature bounds for the characteristic surfaces (cf.
Assumption 1):
(26) $\exists c>0:\forall x\in\mathbb{R}^{4}:\forall i\neq
j:\,\big{|}\frac{\varepsilon_{i}(x)}{\mu_{i}(x)}-\frac{\varepsilon_{j}(x)}{\mu_{j}(x)}\big{|}\geq
c.$
In the following we show Strichartz estimates for Maxwell equations with the
rough material laws as above
(29)
$\displaystyle\left\\{\begin{array}[]{rlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H}-\mathcal{J}_{e},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=\rho_{e},\qquad
x\in\mathbb{R}^{4},\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E}-\mathcal{J}_{m},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=\rho_{m}.\end{array}\right.$
We use the pointwise material laws (2). Setting
$P(x,D)=\begin{pmatrix}-\partial_{t}(\varepsilon\cdot)&\nabla\times\\\
\nabla\times&\partial_{t}(\mu\cdot)\end{pmatrix},\quad
u=\begin{pmatrix}\mathcal{E}\\\
\mathcal{H}\end{pmatrix},\quad\rho_{em}=\begin{pmatrix}\rho_{e}\\\
\rho_{m}\end{pmatrix},$
the Maxwell system (29) becomes
$P(x,D)u=\begin{pmatrix}\mathcal{J}_{e}\\\
-\mathcal{J}_{m}\end{pmatrix},\quad\nabla\cdot\mathcal{D}=\rho_{e},\quad\nabla\cdot\mathcal{B}=\rho_{m}.$
We first show the basic energy estimate which will be used later in an
interpolation argument.
###### Proposition 3.1.
Let $T>0$, $\varepsilon$, $\mu\in
C(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3})$ satisfy (25) with
$\partial_{t}(\varepsilon,\mu)\in L^{1}_{T}L^{\infty}_{x^{\prime}}$. We then
have the estimate
(30)
$\|u\|_{L^{\infty}_{T}L^{2}_{x^{\prime}}}\lesssim\|u_{0}\|_{L^{2}}+\|Pu\|_{L_{T}^{1}L^{2}}.$
###### Proof.
We introduce an equivalent norm on $L^{2}(\mathbb{R}^{3})$ by
$\|u\|^{2}_{E^{0}}=\Big{\langle}u,\begin{pmatrix}\varepsilon&0\\\
0&\mu\end{pmatrix}u\Big{\rangle}.$
For homogeneous solutions we compute
$\begin{split}\partial_{t}\big{(}\int_{\mathbb{R}^{3}}\mathcal{E}.\mathcal{D}+\mathcal{H}.\mathcal{B}\big{)}&=\int_{\mathbb{R}^{3}}(\partial_{t}\mathcal{E}).\mathcal{D}+\mathcal{E}.(\partial_{t}\mathcal{D})+(\partial_{t}\mathcal{H}).\mathcal{B}+\mathcal{H}.(\partial_{t}\mathcal{B})\\\
&=:I+II+III+IV.\end{split}$
We find
$II+IV=\int_{\mathbb{R}^{3}}\mathcal{E}.(\nabla\times\mathcal{H})-(\nabla\times\mathcal{E}).\mathcal{H}=0$
by the symmetry of the curl-operator. Note that
$\begin{split}(\partial_{t}\varepsilon)\mathcal{E}+\varepsilon\partial_{t}\mathcal{E}=\nabla\times\mathcal{H}&\iff\partial_{t}\mathcal{E}=\varepsilon^{-1}\nabla\times\mathcal{H}-\varepsilon^{-1}(\partial_{t}\varepsilon)\mathcal{E},\\\
(\partial_{t}\mu)\mathcal{H}+\mu\partial_{t}\mathcal{H}=-\nabla\times\mathcal{E}&\iff\partial_{t}\mathcal{H}=-\mu^{-1}(\nabla\times\mathcal{E})-\mu^{-1}(\partial_{t}\mu)\mathcal{H},\end{split}$
which yields
$\begin{split}I+III&=\int(\varepsilon^{-1}(\nabla\times\mathcal{H})).\mathcal{D}-\mu^{-1}(\nabla\times\mathcal{E}).\mathcal{B}-\varepsilon^{-1}((\partial_{t}\varepsilon)\mathcal{E}).\mathcal{D}-\mu^{-1}((\partial_{t}\mu)\mathcal{H}).\mathcal{B}\\\
&=-\int(\partial_{t}\varepsilon)\mathcal{E}.\mathcal{E}+\mathcal{H}.(\partial_{t}\mu)\mathcal{H}.\end{split}$
We obtain
$|I+III|\lesssim\|\partial_{t}(\varepsilon,\mu)\|_{L^{\infty}_{x^{\prime}}}\|u\|^{2}_{E^{0}},$
which leads to the estimate
$\partial_{t}\|u\|^{2}_{E^{0}}\lesssim\|(\partial_{t}\varepsilon,\partial_{t}\mu)\|_{L_{x^{\prime}}^{\infty}}\|u\|^{2}_{E^{0}}.$
Grønwall’s lemma and the equivalence of the norms imply
(31) $\|u(t)\|^{2}_{L^{2}}\lesssim
e^{C\|(\partial_{t}\varepsilon,\partial_{t}\mu)\|_{L_{t}^{1}L_{x^{\prime}}^{\infty}}}\|u(0)\|^{2}_{L^{2}}.$
For the proof of the inhomogeneous estimate we consider the evolution of the
variables $(\mathcal{D},\mathcal{B})$. By (25), for vanishing currents
$\mathcal{J}_{em}=0$ we have
$\|(\mathcal{D},\mathcal{B})(t)\|_{L^{2}}\lesssim
e^{C\int_{0}^{t}\|\partial_{t}(\varepsilon,\mu)\|_{L^{\infty}_{x^{\prime}}}ds}\|(\mathcal{D},\mathcal{B})(0)\|_{L^{2}}.$
Let $U(t,s)$ be the propagator of the homogeneous Maxwell equations
$\left\\{\begin{array}[]{cl}\partial_{t}(\mathcal{D},\mathcal{B})\\!\\!\\!\\!&=Q(\mathcal{D},\mathcal{B}),\\\
(\mathcal{D},\mathcal{B})(s)\\!\\!\\!\\!&=(\mathcal{D},\mathcal{B})_{0}\in
L^{2},\end{array}\right.$
The claim now follows from Duhamel’s formula, writing
$(\mathcal{D},\mathcal{B})(t)=U(t,0)(\mathcal{D},\mathcal{B})_{0}-\int_{0}^{t}U(t,s)(\mathcal{J}_{e},\mathcal{J}_{m})(s)ds$
and applying Minkowski’s inequality and (31). ∎
We next show the following local-in-time Strichartz estimates.
###### Theorem 3.2.
Let $0<s\leq 1$, $\varepsilon,\mu\in C^{s}(\mathbb{R}^{4};\mathbb{R}^{3\times
3})$ be as in (24) satisfying (25) and (26). Suppose that
$\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$, $2\leq
p<q\leq\infty$, and $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$. We find the
following estimate to hold:
(32)
$\||D|^{-\rho+\frac{s-2}{4}}u\|_{(L^{p}L^{q})_{2}}\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|u\|_{L^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{1}{2}+\frac{s-2}{4}}\rho_{em}\|_{L^{2}}.$
###### Remark 3.3.
We consider the Besov norm with mixed coefficients defined by
$\|f\|_{(L^{p}L^{q})_{2}}=\big{(}\sum_{\lambda\in
2^{\mathbb{N}_{0}}\cup\\{0\\}}\|S_{\lambda}f\|_{L^{p}L^{q}}^{2}\big{)}^{\frac{1}{2}}$
on the left hand-side because at the endpoint $q=\infty$ the Littlewood-Paley
square-function estimate fails. For $q<\infty$ we can simplify (32) to
$\||D|^{-\rho+\frac{s-2}{4}}u\|_{L^{p}L^{q}}\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|u\|_{L^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{1}{2}+\frac{s-2}{4}}\rho_{em}\|_{L^{2}}.$
We prove Theorem 3.2 by phase space analysis close to the endpoint. To this
end, we apply the FBI transform (cf. [21, 22, 23]) to change into phase space
after
* •
reducing to high frequencies,
* •
spatial and dyadic frequency localisation of the solution,
* •
frequency truncation of $\varepsilon(x)$ and $\mu(x)$.
After these reductions, as in [17, 15] we arrive at the following proposition.
###### Proposition 3.4.
Let $\lambda\gg\lambda_{0}$ with $\lambda_{0}$ depending on the ellipticity of
$\varepsilon$ and $\mu$ and $\|(\varepsilon,\mu)\|_{C^{s}_{x}}$ for $0<s\leq
1$. Let $P_{\lambda}$ denote the operator $P$ with coefficients frequency
truncated at frequencies $<\lambda/8$ and let $u$ be essentially supported in
the unit cube. Under asumptions of Theorem 3.2, estimate (32) follows from
(33)
$\begin{split}\lambda^{-\rho+\frac{s-2}{4}}\|S_{\lambda}u\|_{L^{p}L^{q}}&\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|S_{\lambda}u\|_{L^{2}}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}u\|_{L^{2}}\\\
&\quad+\lambda^{-\frac{1}{2}+\frac{s-2}{4}}\|S_{\lambda}\rho_{em}\|_{L^{2}}.\end{split}$
The estimate corresponds to Strichartz estimates from [21] for wave equations
with rough coefficients _with integrability exponents as in the two-
dimensional case_. The reason that we cannot hope for the Strichartz range of
the wave equation in three dimensions is that the characteristic surface is no
longer regular, but has conical singularities. In [22, 23] Tataru used the
regularity of the Hamilton flow to recover the Euclidean Strichartz estimates
locally-in-time for $C^{2}$-coefficients and coefficients with
$\partial^{2}g\in L_{t}^{1}L_{x^{\prime}}^{\infty}$. In the present context it
is not clear how to take advantage of the Hamilton flow due to the strong
system character. For this reason, we do not recover the derivative loss over
scaling as for scalar wave equations with Hölder-continuous coefficients in
two space dimensions proved in [22]. (Also note that the results in [22] were
shown to be essentially optimal in [18]).
In a first step we show the above reduction result.
### 3.1. Proof of Proposition 3.4
Choosing $\lambda_{0}$ large enough, the uniform ellipticity is preserved
because
$\|\kappa_{\gtrsim\lambda}\|_{L^{\infty}_{x}}\lesssim\lambda^{-s}\|\kappa_{\gtrsim\lambda}\|_{C^{s}_{x}},\quad\kappa\in\\{\varepsilon;\mu\\}.$
Using Sobolev’s embedding, we first estimate low frequencies by
$\||D|^{-\rho+\frac{s-2}{4}}S_{0}u\|_{L^{p}L^{q}}\lesssim\||D|^{\frac{1}{2}+\frac{s-2}{4}}S_{0}u\|_{L^{2}}\lesssim\|S_{0}u\|_{L^{2}}.$
It remains to show the estimate
$\displaystyle\||D|^{-\rho+\frac{s-2}{4}}S_{\geq 1}u\|_{(L^{p}L^{q})_{2}}$
$\displaystyle\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|u\|_{L^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}}$
$\displaystyle\quad+\||D|^{-\frac{1}{2}+\frac{s-2}{4}}\rho_{em}\|_{L^{2}}$
for high frequencies. To deduce this inequality from (33) by means of the
definition of $(L^{p}L^{q})_{2}$, we have to bound
$\|P_{\lambda}S_{\lambda}u\|_{L^{2}}$ by $\|S_{\lambda}Pu\|_{L^{2}}$ plus
summable error terms. We write
$\tilde{S}_{\lambda}=\sum_{\mu=\lambda/8}^{8\lambda}S_{\mu}$ and
$P_{\sim\lambda}$ for the operator $P$ with coefficients frequency localized
near $\lambda$. We then obtain
$\displaystyle\|\tilde{S}_{\lambda}P_{\lambda}S_{\lambda}u\|_{L^{2}}$
$\displaystyle\leq\|\tilde{S}_{\lambda}PS_{\lambda}u\|_{L^{2}}+\|\tilde{S}_{\lambda}P_{\sim\lambda}S_{\lambda}u\|_{L^{2}}$
$\displaystyle\leq\|S_{\lambda}Pu\|_{L^{2}}+\|\tilde{S}_{\lambda}[P,S_{\lambda}]u\|_{L^{2}}+\|\tilde{S}_{\lambda}P_{\sim\lambda}S_{\lambda}u\|_{L^{2}}.$
Since $P$ is in divergence form, we can factor out the derivatives and
estimate
$\displaystyle\|\tilde{S}_{\lambda}[P,S_{\lambda}]u\|_{L^{2}}$
$\displaystyle\lesssim\lambda\|[(\varepsilon,\mu),S_{\lambda}]u\|_{L^{2}}\lesssim\lambda^{1-s}\|(\varepsilon,\mu)\|_{C^{s}}\|u\|_{L^{2}},$
$\displaystyle\|\tilde{S}_{\lambda}P_{\sim\lambda}S_{\lambda}u\|_{L^{2}}$
$\displaystyle\lesssim\lambda^{1-s}\|\lambda^{s}(\tilde{S}_{\lambda}(\varepsilon,\mu))S_{\lambda}u\|_{L^{2}}\lesssim\lambda^{1-s}\|(\varepsilon,\mu)\|_{C^{s}}\|u\|_{L^{2}}.$
In the last step, we used a well-known kernel estimate. This finishes the
proof of Proposition 3.4. $\hfill\Box$
In the following we tacitly suppose $\varepsilon$ and $\mu$ are truncated to
frequencies $<\lambda/8$ to lighten the notation.
### 3.2. Applying the FBI transform
In this section we prove (33). In the following we use the FBI transform to
reduce to an estimate in phase space. We refer to [17, Section 2] for further
explanations.
For $\lambda\in 2^{\mathbb{Z}}$, the FBI transform of an integrable function
$f:\mathbb{R}^{m}\to\mathbb{C}$ is defined by
$\begin{split}T_{\lambda}f(z)&=C_{m}\lambda^{\frac{3m}{4}}\int_{\mathbb{R}^{m}}e^{-\frac{\lambda}{2}(z-y)^{2}}f(y)dy,\quad
z=x-i\xi\in T^{*}\mathbb{R}^{m}\simeq\mathbb{R}^{2m},\\\
C_{m}&=2^{-\frac{m}{2}}\pi^{-\frac{3m}{4}}.\end{split}$
The FBI transform is an isometric mapping
$T_{\lambda}:L^{2}(\mathbb{R}^{m})\to L^{2}_{\Phi}(T^{*}\mathbb{R}^{m})$,
where $\Phi(z)=e^{-\lambda\xi^{2}}$. But $T_{\lambda}$ is not surjective
because $T_{\lambda}f$ is holomorphic (which motivates to write $z=x-i\xi$).
The adjoint in $L^{2}_{\Phi}$ provides the inversion formula
$T^{*}_{\lambda}T_{\lambda}f=f$ with
$T_{\lambda}^{*}F(y)=C_{m}\lambda^{\frac{3m}{4}}\int_{\mathbb{R}^{2m}}e^{-\frac{\lambda}{2}(\bar{z}-y)^{2}}\Phi(z)F(z)dxd\xi.$
Originally, the FBI transform was used in the context of microlocal analysis
to obtain an expansion of analytic symbols. Tataru reversed the logic in [21,
22] to find expansions of rough symbols $a\in C^{s}_{x}C^{\infty}_{c}$, which
satisfy suitable error bounds. Let $a\in C^{s}_{x}C^{\infty}_{c}$ with
$a(x,\xi)=0$ for $\xi\notin B(0,2)\backslash B(0,1/2)$ and
$\tilde{a}^{s}_{\lambda}=\sum_{|\alpha|+|\beta|<s}(\partial_{\xi}-\lambda\xi)^{\alpha}\frac{\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)}{|\alpha|!|\beta|!(-i\lambda)^{|\alpha|}\lambda^{|\beta|}}(\frac{1}{i}\partial_{x}-\lambda\xi)^{\beta}.$
For $s\leq 1$, we have $\tilde{a}^{s}_{\lambda}=a$. Tataru proved the
following approximation theorem in [22, Theorem 5] and [21, Theorem 2].
###### Theorem 3.5.
Suppose that $a\in C^{s}_{x}C^{\infty}_{c}$. Set
$a_{\lambda}(x,\xi)=a(x,\xi/\lambda)$ and
$R^{s}_{\lambda}=T_{\lambda}a_{\lambda}(x,D)-\tilde{a}^{s}_{\lambda}(x,\xi)T_{\lambda}$.
Then,
(34) $\|R^{s}_{\lambda}\|_{L^{2}\to
L^{2}_{\Phi}}\lesssim\lambda^{-\frac{s}{2}}.$
Moreover, if $\partial a\in L^{2}_{x_{0}}L^{\infty}_{x^{\prime}}$, then
(35) $\|R^{s}_{\lambda}\|_{L^{\infty}_{x_{0}}L^{2}_{x^{\prime}}\to
L^{2}_{\Phi}}\lesssim\lambda^{-\frac{1}{2}}.$
We return to the proof of (33). Write $P_{\lambda}S_{\lambda}u=J_{\lambda}$
and $T_{\lambda}S_{\lambda}u=v_{\lambda}$,
$T_{\lambda}(J_{\lambda}/\lambda)=f_{\lambda}$. An application of Theorem 3.5
yields
$p(x,\xi)v_{\lambda}=f_{\lambda}+g_{\lambda}\quad\text{with \
}\|g_{\lambda}\|_{L^{2}_{\Phi}}\lesssim_{\|(\varepsilon,\mu)\|_{C^{s}}}\lambda^{-\frac{s}{2}}\|S_{\lambda}u\|_{L^{2}}$
and
$p(x,\xi)=\begin{pmatrix}-i\xi_{0}\varepsilon&i\mathcal{C}(\xi^{\prime})\\\
i\mathcal{C}(\xi^{\prime})&i\xi_{0}\mu\end{pmatrix}\in\mathbb{C}^{6\times 6}.$
In the above display we denote
$\mathcal{C}(\xi^{\prime})=(-\varepsilon^{ijk}\xi_{k})_{ij}$ with
$(\varepsilon^{ijk})$ the Levi-Civita symbol such that $(\nabla\times
f)\widehat{(}\xi)=i\mathcal{C}(\xi^{\prime})\hat{f}(\xi)$. The estimate (33)
becomes
$\lambda^{-\rho+\frac{s-2}{4}}\|T_{\lambda}^{*}v_{\lambda}\|_{L^{p}L^{q}}\lesssim(1+\|(\varepsilon,\mu)\|_{C^{s}})\|v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{s}{2}}\|f_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S_{\lambda}\rho_{em}\|_{L^{2}}$
by invoking (34). It thus suffices to prove
(36)
$\|T_{\lambda}^{*}v_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|p(x,\xi)v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{em}\|_{L^{2}}).$
By non-stationary phase, we can suppose that
$\text{supp}(v_{\lambda})\subseteq B(0,1)\times\\{1/4\leq|\xi|\leq 4\\}=:U$
(up to a negligible error). In the following we reduce the vector-valued
estimate to a scalar one. For this purpose we adapt arguments from the
analysis of the constant-coefficient time-harmonic case given in [12]. Let
(37)
$\tilde{q}(x,\xi)=-\xi_{0}^{2}q(x,\xi):=-\xi_{0}^{2}(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(x,\xi)+q_{1}(x,\xi)),$
where
$\displaystyle q_{0}(x,\xi)$
$\displaystyle=\xi_{1}^{2}\big{(}\frac{1}{\varepsilon_{2}\mu_{3}}+\frac{1}{\mu_{2}\varepsilon_{3}}\big{)}+\xi_{2}^{2}\big{(}\frac{1}{\varepsilon_{1}\mu_{3}}+\frac{1}{\mu_{1}\varepsilon_{3}}\big{)}+\xi_{3}^{2}\big{(}\frac{1}{\varepsilon_{1}\mu_{2}}+\frac{1}{\varepsilon_{2}\mu_{1}}\big{)},$
$\displaystyle q_{1}(x,\xi)$
$\displaystyle=\frac{1}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}\mu_{1}\mu_{2}\mu_{3}}(\varepsilon_{1}\xi_{1}^{2}+\varepsilon_{2}\xi_{2}^{2}+\varepsilon_{3}\xi_{3}^{2})(\mu_{1}\xi_{1}^{2}+\mu_{2}\xi_{2}^{2}+\mu_{3}\xi_{3}^{2}).$
The dependence of $\varepsilon$ and $\mu$ on $x$ is suppressed in the above
display for the sake of brevity.
###### Proposition 3.6.
For the proof of (36) under the assumptions of Proposition 3.4, it suffices to
show the estimate
(38)
$\|T_{\lambda}^{*}w_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|w_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|q(x,\xi)w_{\lambda}\|_{L^{2}_{\Phi}})$
for $w_{\lambda}:\mathbb{R}^{4}\times\mathbb{R}^{4}\to\mathbb{C}$ with
$\text{supp}(w_{\lambda})\subseteq[-1,1]^{4}\times\\{(\xi_{0},\xi^{\prime})\in\mathbb{R}\times\mathbb{R}^{3}:|\xi_{0}|\sim|\xi^{\prime}|\sim
1\\}$.
###### Proof.
In the first step, we separate the evolution of the $\mathcal{E}$\- and
$\mathcal{H}$-variables by passing to a second order system. We multiply $p$
with the symmetrizer $\sigma$
$\sigma(x,\xi)=\begin{pmatrix}-i\xi_{0}\varepsilon^{-1}&i\varepsilon^{-1}\mathcal{C}(\xi^{\prime})\mu^{-1}\\\
i\mu^{-1}\mathcal{C}(\xi^{\prime})\varepsilon^{-1}&i\xi_{0}\mu^{-1}\end{pmatrix}\in\mathbb{C}^{6\times
6}$
(cf. [12, Proposition 1.2]), obtaining
(39) $\sigma(x,\xi)p(x,\xi)=\begin{pmatrix}M_{E}(x,\xi)-\xi_{0}^{2}&0\\\
0&M_{H}(x,\xi)-\xi_{0}^{2}\end{pmatrix}.$
Here we set
$\displaystyle
M_{E}(x,\xi)=-\varepsilon^{-1}(x)\mathcal{C}(\xi^{\prime})\mu^{-1}(x)\mathcal{C}(\xi^{\prime}),\quad
M_{H}(x,\xi)=-\mu^{-1}(x)\mathcal{C}(\xi^{\prime})\varepsilon^{-1}(x)\mathcal{C}(\xi^{\prime}).$
Moreover, in [12, p. 1835, Eq. (13)] it was pointed out that
$q(x,\xi)=\det(M_{E}(x,\xi^{\prime})-\xi_{0}^{2})=\det(M_{H}(x,\xi^{\prime})-\xi_{0}^{2}).$
Let $v_{\lambda}=(v_{\lambda,1},v_{\lambda,2})$ denote the grouping into
$3$-vectors. By boundedness of the entries of $\sigma(x,\xi)$ for $\xi\in U$,
it suffices to prove
$\|T_{\lambda}^{*}v_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|\sigma(x,\xi)p(x,\xi)v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{em}\|_{L^{2}})$
which can be written as
$\displaystyle\|T_{\lambda}^{*}v_{\lambda,1}\|_{L^{p}L^{q}}$
$\displaystyle\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|(M_{E}(x,\xi^{\prime})-\xi_{0}^{2})v_{\lambda,1}\|_{L^{2}_{\Phi}}$
(40)
$\displaystyle\quad+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{e}\|_{L^{2}}),$
$\displaystyle\|T_{\lambda}^{*}v_{\lambda,2}\|_{L^{p}L^{q}}$
$\displaystyle\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda,2}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|(M_{H}(x,\xi^{\prime})-\xi_{0}^{2})v_{\lambda,2}\|_{L^{2}_{\Phi}}$
(41)
$\displaystyle\quad+\lambda^{-\frac{1}{2}}\|S_{\lambda}\rho_{m}\|_{L^{2}}).$
The estimates will be proved in the regions
* (1)
$\\{1\sim|\xi_{0}|\gg|\xi^{\prime}|\\}$,
* (2)
$\\{1\sim|\xi^{\prime}|\gg|\xi_{0}|\\}$, for which we have to consider the
charges,
* (3)
$\\{1\sim|\xi_{0}|\sim|\xi^{\prime}|\\}$.
The region $\\{1\sim|\xi_{0}|\gg|\xi^{\prime}|\\}$ is easy to handle. Here we
have
$\|(M_{X}(x,\xi^{\prime})-\xi_{0}^{2})v_{\lambda}\|_{L^{2}_{\Phi}}\gtrsim\|v_{\lambda}\|_{L^{2}_{\Phi}},\quad
X\in\\{E;H\\},$
and so the claim follows by Sobolev embedding via
$\|T_{\lambda}^{*}v_{\lambda,i}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}\|v_{\lambda,i}\|_{L^{2}_{\Phi}}\lesssim\lambda^{\rho+\frac{1}{2}+\frac{s}{4}}\|(M_{X}(x,\xi^{\prime})-\xi_{0}^{2})v_{\lambda}\|_{L^{2}_{\Phi}}.$
For the second region $\\{|\xi_{0}|\ll|\xi^{\prime}|\sim 1\\}$ the charges
come into play. We decompose
$v_{\lambda,1}=v_{\lambda,1}^{s}+v_{\lambda,1}^{p}$ with
$v_{\lambda,1}^{p}=\frac{(\xi^{\prime}.\varepsilon^{\lambda}v_{\lambda,1})\xi^{\prime}}{|\xi^{\prime}|^{2}_{\varepsilon}},\quad|\xi^{\prime}|^{2}_{\varepsilon}=\xi^{\prime}.\varepsilon^{\lambda}\xi^{\prime},$
again indicating the frequency cutoff of the coefficients temporarily. The
contribution of $v_{\lambda,1}^{p}$ is estimated by Sobolev embedding:
$\|T_{\lambda}^{*}v_{\lambda,1}^{p}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}\|v_{\lambda,1}^{p}\|_{L^{2}_{\Phi}}\lesssim\lambda^{\rho+\frac{1}{2}}\|(\frac{\xi^{\prime}}{|\xi^{\prime}|_{\varepsilon}}.\varepsilon^{\lambda}v_{\lambda,1})\|_{L^{2}_{\Phi}}.$
Theorem 3.5 and commutator estimates yield
$\|\frac{\xi^{\prime}}{|\xi^{\prime}|_{\varepsilon}}.(\varepsilon^{\lambda}v_{\lambda,1})\|_{L^{2}_{\Phi}}\lesssim\lambda^{-\frac{s}{2}}\|u_{\lambda}\|_{L^{2}}+\lambda^{-1}\|S_{\lambda}\nabla^{\prime}.(\varepsilon^{\lambda}\mathcal{E})\|_{L^{2}}.$
By expanding
$\varepsilon^{\lambda}=\varepsilon+(\varepsilon-\varepsilon^{\lambda})$, we
find
$\begin{split}\lambda^{-1}\|S_{\lambda}\nabla^{\prime}.(\varepsilon^{\lambda}\mathcal{E})\|_{L^{2}}&\lesssim\lambda^{-1}\|S_{\lambda}\nabla^{\prime}.(\varepsilon\mathcal{E})\|_{L^{2}}+\|(\varepsilon-\varepsilon^{\lambda})\tilde{S}_{\lambda}\mathcal{E}\|_{L^{2}}\\\
&\lesssim\lambda^{-1}\|S_{\lambda}\rho_{e}\|_{L^{2}}+\lambda^{-s}\|\tilde{S}_{\lambda}\mathcal{E}\|_{L^{2}}.\end{split}$
This is an acceptable estimate for $v_{\lambda,1}^{p}$ in terms of $\rho_{e}$
and $\mathcal{E}$.
We turn to $v_{\lambda,1}^{s}$ noting that
$\xi^{\prime}.(\varepsilon^{\lambda}v_{1,\lambda}^{s})=0$. In the following we
show that
$|(M_{E}(x,\xi)-\xi_{0}^{2})v_{1}^{\lambda}|\gtrsim|v_{1}^{\lambda}|$ for
$\xi^{\prime}.(\varepsilon^{\lambda}v_{1}^{\lambda})=0$ and
$\\{|\xi_{0}|\ll|\xi^{\prime}|\\}$. We mit again the superscript $\lambda$ of
the coefficients. For $\xi_{0}\neq 0$ we have $q(x,\xi)\neq 0$. Let
$w=(M_{E}-\xi_{0}^{2})v_{1}$. We have to show that
$|(M_{E}-\xi_{0}^{2})^{-1}w|\lesssim|w|$
noting that $\xi^{\prime}.(\varepsilon w)=0$. In [12, Section 2.1] the matrix
$(M_{E}-\xi_{0}^{2})$ was inverted by
$(M_{E}-\xi_{0}^{2})^{-1}=q^{-1}(x,\xi)\text{adj}(M_{E}-\xi_{0}^{2})=\frac{-1}{\xi_{0}^{2}(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(x,\xi)+q_{1}(x,\xi))}Z_{\varepsilon,\mu}\frac{\varepsilon}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}},$
wher the components of $Z=Z_{\varepsilon,\mu}$ are given by
$\displaystyle\begin{aligned}
Z_{11}(\xi)&=\xi_{1}^{2}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}})-\xi_{0}^{2}(\frac{\varepsilon_{2}}{\mu_{2}}\xi_{1}^{2}+\frac{\varepsilon_{3}}{\mu_{3}}\xi_{1}^{2}+\frac{\varepsilon_{2}}{\mu_{1}}\xi_{2}^{2}+\frac{\varepsilon_{3}}{\mu_{1}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{2}\varepsilon_{3},\\\
Z_{12}(\xi)&=Z_{21}(\xi)=\xi_{1}\xi_{2}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}}-\xi_{0}^{2}\frac{\varepsilon_{3}}{\mu_{3}}),\\\
Z_{13}(\xi)&=Z_{31}(\xi)=\xi_{1}\xi_{3}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}}-\xi_{0}^{2}\frac{\varepsilon_{2}}{\mu_{2}}),\\\
Z_{22}(\xi)&=\xi_{2}^{2}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}})-\xi_{0}^{2}(\frac{\varepsilon_{1}}{\mu_{2}}\xi_{1}^{2}+\frac{\varepsilon_{3}}{\mu_{3}}\xi_{2}^{2}+\frac{\varepsilon_{1}}{\mu_{1}}\xi_{2}^{2}+\frac{\varepsilon_{3}}{\mu_{2}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{1}\varepsilon_{3},\\\
Z_{23}(\xi)&=Z_{32}(\xi)=\xi_{2}\xi_{3}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}}-\xi_{0}^{2}\frac{\varepsilon_{1}}{\mu_{1}}),\\\
Z_{33}(\xi)&=\xi_{3}^{2}(\frac{\xi_{1}^{2}}{\mu_{2}\mu_{3}}+\frac{\xi_{2}^{2}}{\mu_{1}\mu_{3}}+\frac{\xi_{3}^{2}}{\mu_{1}\mu_{2}})-\xi_{0}^{2}(\frac{\varepsilon_{1}}{\mu_{3}}\xi_{1}^{2}+\frac{\varepsilon_{2}}{\mu_{3}}\xi_{2}^{2}+\frac{\varepsilon_{1}}{\mu_{1}}\xi_{3}^{2}+\frac{\varepsilon_{2}}{\mu_{2}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{1}\varepsilon_{2}.\end{aligned}$
It is a crucial observation that $Z_{\varepsilon,\mu}$ will be applied to
vectors $v$ with $\xi^{\prime}.v=0$. This reduces to
$Z_{\varepsilon,\mu}^{\text{eff}}$, having the coefficcients
$\displaystyle\begin{aligned}
Z^{\text{eff}}_{11}(\xi)&=-\xi_{0}^{2}(\frac{\varepsilon_{2}}{\mu_{2}}\xi_{1}^{2}+\frac{\varepsilon_{3}}{\mu_{3}}\xi_{1}^{2}+\frac{\varepsilon_{2}}{\mu_{1}}\xi_{2}^{2}+\frac{\varepsilon_{3}}{\mu_{1}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{2}\varepsilon_{3},\\\
Z^{\text{eff}}_{12}(\xi)&=Z^{\text{eff}}_{21}(\xi)=-\xi_{1}\xi_{2}\xi_{0}^{2}\frac{\varepsilon_{3}}{\mu_{3}},\\\
Z^{\text{eff}}_{13}(\xi)&=Z^{\text{eff}}_{31}(\xi)=-\xi_{0}^{2}\frac{\varepsilon_{2}}{\mu_{2}}\xi_{1}\xi_{3},\\\
Z^{\text{eff}}_{22}(\xi)&=-\xi_{0}^{2}(\frac{\varepsilon_{1}}{\mu_{2}}\xi_{1}^{2}+\frac{\varepsilon_{3}}{\mu_{3}}\xi_{2}^{2}+\frac{\varepsilon_{1}}{\mu_{1}}\xi_{2}^{2}+\frac{\varepsilon_{3}}{\mu_{2}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{1}\varepsilon_{3},\\\
Z^{\text{eff}}_{23}(\xi)&=Z^{\text{eff}}_{32}(\xi)=-\xi_{2}\xi_{3}\xi_{0}^{2}\frac{\varepsilon_{1}}{\mu_{1}},\\\
Z^{\text{eff}}_{33}(\xi)&=-\xi_{0}^{2}(\frac{\varepsilon_{1}}{\mu_{3}}\xi_{1}^{2}+\frac{\varepsilon_{2}}{\mu_{3}}\xi_{2}^{2}+\frac{\varepsilon_{1}}{\mu_{1}}\xi_{3}^{2}+\frac{\varepsilon_{2}}{\mu_{2}}\xi_{3}^{2})+\xi_{0}^{4}\varepsilon_{1}\varepsilon_{2}.\end{aligned}$
We can write $Z^{\text{eff}}=\xi_{0}^{2}\tilde{Z}^{\text{eff}}$ with
$\tilde{Z}^{\text{eff}}$ having bounded entries for $|\xi_{0}|\lesssim 1$.
This gives
$\displaystyle(M_{E}(x,\xi)-\xi_{0}^{2})^{-1}w$
$\displaystyle=q^{-1}(x,\xi)\text{adj}(M_{E}-\xi_{0}^{2})w$
$\displaystyle=\frac{-1}{(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(x,\xi)+q_{1}(x,\xi))}\tilde{Z}^{\text{eff}}_{\varepsilon,\mu}\frac{\varepsilon}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}w.$
For $|\xi_{0}|\ll|\xi^{\prime}|$ or $|\xi_{0}|\gg|\xi^{\prime}|$, we have
$|\xi_{0}^{4}-\xi_{0}^{2}q_{0}(\xi)+q_{1}(\xi)|\sim 1\text{ provided that
}\max(|\xi_{0}|,|\xi^{\prime}|)\sim 1.$
Consequently, for $|\xi_{0}|\ll|\xi^{\prime}|\sim 1$, it follows
$\|(M_{E}(x,\xi)-\xi_{0}^{2})v\|_{L^{2}_{\Phi}(U)}\gtrsim\|v\|_{L^{2}_{\Phi}}.$
This shows estimate (3.2) in the second region. (41) follows by exchanging the
roles of $\varepsilon$ and $\mu$.
We turn to the third region. Again we focus on (3.2), for which we prove
$\|T_{\lambda}^{*}v_{\lambda,1}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|(M_{E}-\xi_{0}^{2})v_{\lambda,1}\|_{L^{2}_{\Phi}})$
for $v_{\lambda}$ supported in $B(0,1)\times\\{|\xi_{0}|\sim|\xi^{\prime}|\sim
1\\}$. This inequality is a consequence from (38). Indeed, for any component
$v_{\lambda,1,k}$, $k=1,2,3$, the estimate (38) yields
$\|T_{\lambda}^{*}v_{\lambda,1,k}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda,1,k}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|(Z_{\varepsilon,\mu}\frac{\varepsilon}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}(M_{E}-\xi_{0}^{2})v_{\lambda,1})_{k}\|_{L^{2}_{\Phi}})$
because
$Z_{\varepsilon,\mu}\frac{\varepsilon}{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}(M_{E}-\xi_{0}^{2})=q(x,\xi)$.
The assertion easily follows. ∎
It thus suffices to show for scalar $v_{\lambda}\in L^{2}_{\Phi}$ supported in
$\\{\frac{1}{4}\leq|\xi|\leq 4\\}$ that the estimate
(42)
$\|T_{\lambda}^{*}v_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{2-s}{4}}\|v_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{2+s}{4}}\|q(x,\xi)v_{\lambda}\|_{L^{2}_{\Phi}})$
holds true. We consider the operator
$W_{\lambda}=T_{\lambda}^{*}\frac{a(x,\xi)}{\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|}$
with $a\in C^{\infty}_{c}$ localizing to
$B(0,1)\times\\{\frac{1}{4}\leq|\xi|\leq 4\\}$. Hence it is enough to
establish
$\|W_{\lambda}\|_{L_{\Phi}^{2}\to
L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}.$
By the $TT^{*}$-argument, we can likewise prove the estimate
$\|\tilde{V}_{\lambda}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim\lambda^{1+2\rho},\quad\tilde{V}_{\lambda}=T_{\lambda}^{*}\frac{a^{2}(x,\xi)\Phi(\xi)}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|)^{2}}T_{\lambda}.$
For this estimate it is crucial to understand the curvature properties of
$\\{\xi\in\mathbb{R}^{4}:q(x,\xi)=0\\}$. Since $q(x,\xi)$ is $4$-homogeneous
in $\xi$, we have
$q(x,\xi)=\xi_{0}^{4}q(x,1,\xi^{\prime}/\xi_{0}).$
So it suffices to consider the Fresnel surface
$S=\\{\xi^{\prime}\in\mathbb{R}^{3}:q(x,1,\xi^{\prime})=0\\}$. The Fresnel
surface is classical and has been studied since the 19th century (cf. [2, 8,
5]). We refer to detailed computations laid out in [12] and recall properties
of the Fresnel surface $S$. We can consider $S$ as a union of three components
$S=S_{1}\cup S_{2}\cup S_{3}$ with
* •
$S_{1}$ describes a regular bounded surface with Gaussian curvature bounded
from below and above,
* •
$S_{2}$ is a neighbourhood of the Hamilton circles, which is regular, but has
one principal curvature vanishing along the Hamilton circles,
* •
$S_{3}$ is a neighbourhood of four conical singularities.
We redenote $a^{2}(x,\xi)$ by $a(x,\xi)$ for the sake of brevity, and split
$a=a_{1}+a_{2}+a_{3}$ and the operator
$V_{\lambda}=V_{\lambda,1}+V_{\lambda,2}+V_{\lambda,3}$ according to the above
components. We establish the three estimates separately, which is meant to
highlight a different dispersive behavior. To see that the uniformity of the
curvature bounds follows from the separability condition (26), we revisit the
analysis of the Fresnel surface provided in [12, Section 3].
First, the analysis of Fresnel’s surface for $|\xi_{0}|\sim|\xi^{\prime}|\sim
1$
$S_{\xi_{0}}=\\{\xi^{\prime}\in\mathbb{R}^{3}:p(\xi_{0},\xi^{\prime})=-\xi_{0}^{2}(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(\xi^{\prime})+q_{1}(\xi^{\prime}))=0\\}$
is reduced to the standard form $\mu_{1}=\mu_{2}=\mu_{3}=\xi_{0}=1$ by means
of the substitution
$\lambda_{i}=\frac{\xi_{i}}{\xi_{0}\sqrt{\mu_{i+1}\mu_{i+2}}}\qquad(i=1,2,3).$
Note that the change of coordinates is as smooth as $\mu$. We use cyclic
notation in the following, i.e., $\mu_{4}:=\mu_{1}$ and $\mu_{5}:=\mu_{2}$,
likewise for $\varepsilon_{i}$. In the following let
$\varepsilon_{i}:=\frac{\varepsilon_{i}}{\mu_{i}}$ (the right-hand side refers
to the original quantities).
We shall analyze
$S^{*}=\\{\lambda\in\mathbb{R}^{3}:1-q_{0}^{*}(\lambda)+q_{1}^{*}(\lambda)=0\\}$
with $q_{0}^{*}$ and $q_{1}^{*}$ denoting the reduced polynomials $q_{0}$ and
$q_{1}$ for $\mu_{1}=\mu_{2}=\mu_{3}=1$. We find that the coordinates of the
four singular points are given by
$\lambda_{j}^{2}=\frac{\varepsilon_{j+2}(\varepsilon_{j}-\varepsilon_{j+1})}{\varepsilon_{j}-\varepsilon_{j+2}},\quad\lambda_{j+1}=0,\quad\lambda_{j+2}^{2}=\frac{\varepsilon_{j}(\varepsilon_{j+2}-\varepsilon_{j+1})}{\varepsilon_{j+2}-\varepsilon_{j}}$
for $\varepsilon_{j+1}\in\langle\varepsilon_{j},\varepsilon_{j+2}\rangle$,
which denotes
$\varepsilon_{j+1}\in[\varepsilon_{j},\varepsilon_{j+2}]\cup[\varepsilon_{j+2},\varepsilon_{j}]$.
By uniform ellipticity and the separation condition (26), the singular points
vary as smoothly in $x\in\mathbb{R}^{4}$ as $(\varepsilon,\mu)$.
To analyze the regularity of the curvature in $(\varepsilon,\mu)$, we recall
the parametrization of the regular part of the Fresnel surface given by
$s=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2},\quad\varepsilon_{1}\lambda_{1}^{2}+\varepsilon_{2}\lambda_{2}^{2}+\varepsilon_{3}\lambda_{3}^{2}=\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}t^{-1},$
and
$\alpha(s,t)=\frac{(t-s)\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}{s^{2}t-(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3})st+(\varepsilon_{1}\varepsilon_{2}+\varepsilon_{1}\varepsilon_{3}+\varepsilon_{2}\varepsilon_{3})t-\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}.$
In $(s,t)$ coordinates, we have for the Gaussian curvature
$K(s,t)=\frac{(\alpha(s,t)-\varepsilon_{1})(\alpha(s,t)-\varepsilon_{2})(\alpha(s,t)-\varepsilon_{3})}{\alpha(s,t)(s-\varepsilon_{1})(s-\varepsilon_{2})(s-\varepsilon_{3})}$
and for the mean curvature
$\begin{split}K_{m}(s,t)&=-\frac{1}{2}\big{(}\frac{s}{\sqrt{\alpha}}K(s,t)-\frac{1}{\sqrt{\alpha}}\big{(}\frac{(\alpha-\varepsilon_{1})(\alpha-\varepsilon_{2})}{(s-\varepsilon_{1})(s-\varepsilon_{2})}+\frac{(\alpha-\varepsilon_{2})(\alpha-\varepsilon_{3})}{(s-\varepsilon_{2})(s-\varepsilon_{3})}\\\
&\qquad+\frac{(\alpha-\varepsilon_{1})(\alpha-\varepsilon_{3})}{(s-\varepsilon_{1})(s-\varepsilon_{3})}\big{)}\big{)}.\end{split}$
The limiting case $s=t$ corresponds to singular points. For separated $s$ and
$t$ we have $\alpha\geq d>0$ and the uniformity of the curvature bounds
follows. For the analysis of the curvature close to the singular points, we
rescale a dyadic decomposition away from the singular points to unit distance
(see below), for which the previous arguments apply.
We now conclude the proof of the Strichartz estimates in Theorem 3.2 for sharp
Strichartz exponents close to the endpoint. Note that the following
proposition implies Strichartz estimates for the contribution of
$V_{\lambda,1}$ like in (42) for the wider range
$\frac{1}{p}+\frac{1}{q}\leq\frac{1}{2}$ and $2<p<q\leq\infty$ by Sobolev
embedding. We recall the defintion of $V_{\lambda,i}$ in (46) below.
###### Proposition 3.7.
Let $\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$ and
$2<p<q\leq\infty$. The estimate
(43) $\|V_{\lambda,i}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim\lambda^{1+2\rho}$
holds true, if
* •
$i=1$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$,
* •
$i\in\\{2,3\\}$ and $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$.
For $i=1,2$, the assumption $2<p<q\leq\infty$ is not necessary because we can
interpolate with (43) for $p=\infty$ and $q=2$. We show this case. In the
proof we will see that if fails for $i=3$ at the singular points because these
satisfy $\partial_{\xi}q(x,\xi_{x})=0$.
After factoring out the FBI transform in the $x^{\prime}$ variables by virtue
of its $L^{2}$-boundedness, we it remains to prove the boundedness of
(44)
$\lambda^{-1}T_{\lambda,0}^{*}a_{i}(x,\xi)\Phi(\xi)\delta_{q=0}T_{\lambda,0}:L_{t}^{1}L_{x^{\prime},\xi^{\prime}}^{2}\to
L_{t}^{\infty}L_{x^{\prime},\xi^{\prime}}^{2}$
for the restricted FBI transform $T_{\lambda,0}:L^{2}(\mathbb{R}^{n})\to
L^{2}_{\Phi}(\mathbb{R}^{2}\times\mathbb{R}^{n-1})$ given by
$T_{\lambda,0}f(x_{0},\xi_{0},y^{\prime})=2^{-\frac{1}{2}}\pi^{-\frac{3}{4}}\lambda^{\frac{3}{4}}\int
e^{-\frac{\lambda}{2}(z-y_{0})^{2}}f(y_{0},y^{\prime})dy_{0}$
with $\Phi(x_{0},\xi_{0},x^{\prime})=e^{-\lambda|\xi_{0}|^{2}}$. (Compare the
proof of Proposition 3.7 below.) Hence, to establish (44), we have to show the
boundedness of the kernel
$K(y_{0},\bar{y}_{0})=\lambda^{\frac{1}{2}}\int_{\mathbb{R}}dx_{0}e^{-\frac{\lambda}{2}(x_{0}-y_{0})^{2}}e^{-\frac{\lambda}{2}(x_{0}-\bar{y}_{0})^{2}}\int_{\mathbb{R}}d\xi_{0}e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}a_{0}(x_{0},\xi_{0})\delta_{q=0}.$
It is enough to check the hyperbolicity condition
(45) $\partial_{\xi_{0}}q(x,\xi)\neq 0\quad\text{for \ }a_{i}(x,\xi)\neq
0,\;\;i=1,2.$
Indeed, from (45) follows
$\big{|}\int_{\mathbb{R}}d\xi_{0}e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}a_{0}(x_{0},\xi_{0})\delta_{q=0}\big{|}\lesssim
1$
and
$\int_{\mathbb{R}}dx_{0}e^{-\frac{\lambda}{2}(x_{0}-y_{0})^{2}}e^{-\frac{\lambda}{2}(x_{0}-\bar{y}_{0})^{2}}\lesssim\lambda^{-\frac{1}{2}},$
yielding (43) for $(p,q)=(\infty,2)$. For the remaining proof of (45), we
consider
$\partial_{\xi_{0}}q=4\xi_{0}^{3}-2\xi_{0}q_{0}=0.$
This implies $q_{0}=2\xi_{0}^{2}$. Since $q(\xi_{0},\xi^{\prime})=0$, we find
$q_{1}=\xi_{0}^{4}$ and so $q_{1}=(\frac{q_{0}}{2})^{2}$. By homogeneity of
$q$, it is enough to consider $\xi_{0}=1$. Then we use again the
parametrisations of [12]:
$\displaystyle s=\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2},$
$\displaystyle\qquad\varepsilon_{1}\xi_{1}^{2}+\varepsilon_{2}\xi_{2}^{2}+\varepsilon_{3}\xi_{3}^{2}=\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}t^{-1},$
$\displaystyle q_{1}^{*}(\xi)=st^{-1},$ $\displaystyle\qquad
q_{0}^{*}(\xi)=1+st^{-1}.$
Let $b=st^{-1}$. From $q_{1}=(\frac{q_{0}}{2})^{2}$ follows
$b=\big{(}\frac{1+b}{2}\big{)}^{2}$ and hence $(b-1)^{2}=0$. In the regular
part of $S$ we have $s<t$ or $t<s$ corresponding to the two sheets of $S$,
where $t=s$ formally corresponds to the singular points. This ensures
hyperbolicity within $\text{supp}(a_{1})\cup\text{supp}(a_{2})$. However,
within $\text{supp}(a_{3})$ the argument fails. We can thus prove (43) only
close to the endpoint, which yields the corresponding Strichartz estimates.
###### Proof of Proposition 3.7.
For $i=1,2$, the result is shown by a straight-forward adaptation of Tataru’s
arguments [21]. We shall be brief, but repeat the argument because in the most
difficult case $i=3$ we will elaborate on it. To prove a uniform bound for the
family of operators
(46)
$\lambda^{-1-2\rho}V_{\lambda,i}=\lambda^{-1-2\rho}T_{\lambda}^{*}\frac{a_{i}(x,\xi)\Phi(\xi)}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|)^{2}}T_{\lambda}:L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q},$
we use the integrability of the weight
$\int\frac{1}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q|)^{2}}dq\lesssim
1$
and can reduce to level sets $\delta_{q(x,\xi)=c}$ by foliation. It is enough
to consider $c=0$ as $\\{\xi\in\mathbb{R}^{4}:q(x,\xi)=0\\}$ is a regular
surface within the supports of $a_{i}$, $i=1,2$, and the supports can be
properly foliated.
For the proof of the non-endpoint estimate (43) in case $2<p<q\leq\infty$, we
consider the analytic family of operators
$V_{\lambda,i}^{\theta}=\lambda^{-\theta(2\rho+1)}\frac{e^{(\theta-1)^{2}}}{\Gamma(1-\theta)}T_{\lambda}^{*}a_{i}(x,\xi)\Phi(\xi)q^{-\theta}(x,\xi)T_{\lambda}$
such that $\lambda^{-1-2\rho}V_{\lambda,i}=V_{\lambda,i}^{1}$.
The estimate follows from interpolation between $\lambda$-independent bouns of
$\displaystyle V_{\lambda,i}^{\theta}:L^{2}\to L^{2},\qquad\qquad\quad\
\,\Re\theta=0,$ $\displaystyle
V_{\lambda,i}^{\theta}:L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty},\qquad\Re\theta=\theta_{1},$
where $p_{1},\theta_{1}$ are chosen so that the following points are
collinear:
(47)
$\big{(}\frac{1}{2},\frac{1}{2},0\big{)},\qquad\big{(}\frac{1}{p},\frac{1}{q},1\big{)},\qquad\big{(}\frac{1}{p_{1}},0,\theta_{1}\big{)}.$
The $L^{2}$-estimate is immediate from the $L^{2}$-mapping properties of the
FBI transform. For the $L^{p_{1}^{\prime}}L^{1}\to L^{p_{1}}L^{\infty}$ bound,
we compute the kernel $K^{\theta}_{\lambda,i}$ of $V^{\theta}_{\lambda,i}$ as
$\begin{split}K^{\theta}_{\lambda,i}(y,\bar{y})&=\lambda^{6-\theta(2\rho+1)}\int_{\mathbb{R}^{4}}dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\\\
&\quad\times\frac{e^{(\theta-1)^{2}}}{\Gamma(1-\theta)}\int_{\mathbb{R}^{4}}d\xi
e^{i\lambda(y-\bar{y}).\xi}q^{-\theta}(x,\xi)a_{i}(x,\xi).\end{split}$
We have the well-known oscillatory integral estimate (cf. [19, Chapter IX,
Section 1.2.3])
(48)
$\big{|}\frac{e^{(\theta-1)^{2}}}{\Gamma(1-\theta)}\int_{\mathbb{R}^{d}}e^{iy.\xi}a_{i}(x,\xi)q^{-\theta}(x,\xi)d\xi\big{|}\lesssim(1+|y|)^{(\Re\theta-1)-\frac{k_{i}}{2}},$
where $k_{1}=2$ and $k_{2}=1$ are the numbers of principal curvatures of the
regular surface $\\{q(x,\xi)=0\\}$222The surface is regular in $\xi$ within
the support of $a_{i}$, $i=1,2$. bounded from below. Estimate (43) now follows
in the same spirit as in [21]. We give the details only for $i=2$ because the
case $i=1$ is covered in [21, pp. 363-364] verbatim. For $i=2$, we find the
kernel bound
$|K^{\theta}_{\lambda,i}(y,\bar{y})|\lesssim\lambda^{4-\Re\theta(2\rho+1)}(1+\lambda|y-\bar{y}|)^{\Re\theta-\frac{3}{2}}.$
The relations
$\theta-\frac{3}{2}=-\frac{2}{p}-\frac{1}{q},\quad
4-\theta(2\rho+1)=\frac{2}{p}+\frac{6}{q},$
hold for the first two points in (47). So they hold for the third, which gives
(49) $\theta_{1}-\frac{3}{2}=-\frac{2}{p_{1}},\quad
4-\theta_{1}(2\rho+1)=\frac{2}{p_{1}}.$
Consequently, we have proved the kernel estimate
$|K^{\theta}_{\lambda,i}(y,\bar{y})|\lesssim|y-\bar{y}|^{-\frac{2}{p_{1}}},\qquad\Re\theta=\theta_{1}.$
The Hardy–Littlewood–Sobolev inequality then yields (43) for $i=2$.
We turn to the most difficult case $i=3$. Since there are four isolated
singular points $\xi_{x}^{\prime}$, it suffices to estimate the contribution
of each one separately. We make the decomposition $a_{3}(x,\xi)=\sum_{k\geq
0}a_{3,k}(x,\xi)$ where
$a_{3,k}(x,1,\xi^{\prime})=a_{3,0}(x,1,(\xi^{\prime}-\xi^{\prime}_{x})2^{k})$
localizes smoothly to a $2^{-k}$-annulus around $\xi_{x}^{\prime}$. This
cutoff is extended $1$-homogeneously in $\xi_{0}$, i.e.,
$a_{3,k}(x,\xi_{0},\xi^{\prime})=\xi_{0}a_{3,k}(x,1,\xi^{\prime}/\xi_{0})$.
This is possible because $|\xi_{0}|\in[1/4,4]$. The kernel $K_{k}$ of
$A_{\lambda,k}=T_{\lambda}^{*}\frac{a_{3,k}(x,\xi)\Phi(\xi)}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|)^{2}}T_{\lambda}$
is given by
$K_{k}(y,\bar{y})=\lambda^{6-(2\rho+1)}\int_{\mathbb{R}^{4}}dxe^{-\frac{\lambda(y-x)^{2}}{2}}e^{-\frac{\lambda(\bar{y}-x)^{2}}{2}}\int_{\mathbb{R}^{4}}d\xi\frac{e^{i\lambda\xi.(y-\bar{y})}a_{3,k}(x,\xi_{0},\xi^{\prime})}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi)|)^{2}}.$
By homogeneity of $q(x,\xi_{0},\xi^{\prime})$ and
$a_{3}(x,\xi_{0},\xi^{\prime})$ in $\xi_{0}$, we find
$\frac{a_{3,k}(x,\xi_{0},\xi^{\prime})a_{0}(\xi_{0})}{(\lambda^{-\frac{s}{4}}+\lambda^{\frac{s}{4}}|q(x,\xi_{0},\xi^{\prime})|)^{2}}=\frac{\xi_{0}a_{3,k}(x,1,\tilde{\xi}^{\prime})a_{0}(\xi_{0})}{\xi_{0}^{8}(\lambda^{-\frac{s}{4}}\xi_{0}^{-4}+\lambda^{\frac{s}{4}}|q(x,1,\tilde{\xi}^{\prime})|)^{2}}$
with $\tilde{\xi}^{\prime}=\xi^{\prime}/\xi_{0}$ and a suitable smooth cutoff
$a_{0}$.
Furthermore, we carry out a Taylor expansion of $q(x,1,\cdot)$ around
$\xi_{x}^{\prime}$ to find
$q(x,1,\xi^{\prime}+\xi_{x}^{\prime})=\frac{\langle\partial^{2}_{\xi^{\prime}\xi^{\prime}}q(x,1,\xi_{x}^{\prime})\xi^{\prime},\xi^{\prime}\rangle}{2}+O(|\xi^{\prime}|^{3}),$
recalling that
$q(x,1,\xi_{x}^{\prime})=0,\quad\nabla_{\xi^{\prime}}q(x,1,\xi_{x}^{\prime})=0.$
We write
$\begin{split}q(x,1,\tilde{\xi}^{\prime})=q(x,1,2^{-k}\xi^{\prime}+\xi_{x}^{\prime})&=2^{-2k}\big{(}\frac{\langle\partial^{2}_{\xi^{\prime}\xi^{\prime}}q(x,1,\xi_{x}^{\prime})\xi^{\prime},\xi^{\prime}\rangle}{2}+O(2^{-k}|\xi^{\prime}|^{3})\big{)}\\\
&=:2^{-2k}q_{k}(x,1,\xi^{\prime}).\end{split}$
For the analysis of the kernel we perform the change of variables
$\tilde{\xi}^{\prime}=\tilde{\tilde{\xi}}^{\prime}+\xi_{x}^{\prime}$ and
$\xi^{\prime}=2^{k}\tilde{\tilde{\xi}}^{\prime}$ to find
$\begin{split}K_{k}(y,\bar{y})&=\lambda^{6-(2\rho+1)}\\!\\!\int_{\mathbb{R}^{4}}\\!\\!dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\\!\\!\int_{\mathbb{R}}\\!d\xi_{0}\frac{e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}e^{i\lambda\xi_{0}\xi_{x}^{\prime}.(y^{\prime}-\bar{y}^{\prime})}a_{0}(\xi_{0})}{\xi_{0}^{4}}\\\
&\quad\times\int_{\mathbb{R}^{3}}d\xi^{\prime}\frac{2^{-3k}a_{3,0}(x,1,\xi^{\prime})e^{i\lambda
2^{-k}\xi_{0}\xi^{\prime}.(y^{\prime}-\bar{y}^{\prime})}}{(\lambda^{-\frac{s}{4}}\xi_{0}^{-4}+\lambda^{\frac{s}{4}}2^{-2k}|q_{k}(x,1,\xi^{\prime})|)^{2}}\\\
&=\lambda^{6-(2\rho+1)}\\!\\!\int_{\mathbb{R}^{4}}\\!\\!dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\\!\\!\int_{\mathbb{R}}\\!d\xi_{0}\frac{e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}e^{i\lambda\xi_{0}\xi_{x}^{\prime}.(y^{\prime}-\bar{y}^{\prime})}a_{0}(\xi_{0})}{\xi_{0}^{4}}\\\
&\quad\times\int_{\mathbb{R}^{3}}d\xi^{\prime}\frac{2^{k}a_{3,0}(x,1,\xi^{\prime})e^{i\lambda
2^{-k}\xi_{0}\xi^{\prime}.(y^{\prime}-\bar{y}^{\prime})}}{(\lambda^{-\frac{s}{4}}\xi_{0}^{-4}2^{2k}+\lambda^{\frac{s}{4}}|q_{k}(x,1,\xi^{\prime})|)^{2}}.\end{split}$
The weight
$(2^{2k}\lambda^{-\frac{s}{4}}\xi_{0}^{-4}+\lambda^{\frac{s}{4}}|q_{k}(x,1,\xi^{\prime})|)^{-2}$
is integrable with respect to the level sets, and we have
$\int_{0}^{\infty}\frac{dq}{(2^{2k}\lambda^{-\frac{s}{4}}+q\lambda^{\frac{s}{4}})^{2}}\lesssim
2^{-2k}.$
After foliation over level sets $q(x,\xi)=c$, we are thus left with the
operator having the kernel
$\displaystyle K^{1}_{k}(y,\bar{y})$
$\displaystyle=\lambda^{6-(2\rho+1)}\\!\\!\int_{\mathbb{R}^{4}}dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\\!\\!\int_{\mathbb{R}}d\xi_{0}\frac{e^{i\lambda\xi_{0}.(y-\bar{y})}e^{i\lambda\xi_{0}\xi_{x}^{\prime}.(y^{\prime}-\bar{y}^{\prime})}a_{0}(\xi_{0})}{\xi_{0}^{4}}$
$\displaystyle\quad\times\int_{\mathbb{R}^{3}}d\xi^{\prime}2^{-k}a_{3,0}(x,1,\xi^{\prime})e^{i\lambda
2^{-k}\xi_{0}\xi^{\prime}.(y^{\prime}-\bar{y}^{\prime})}\delta_{\tilde{q}_{k}(x,1,\xi^{\prime})=0}.$
We estimate it by interpolation using the analytic family of operators
$V^{\theta}_{k,\lambda}$ given by the kernels
(50)
$\begin{split}K^{\theta}_{\lambda,k}&=\lambda^{6-\theta(2\rho+1)}\int_{\mathbb{R}^{4}}dxe^{-\frac{\lambda}{2}(y-x)^{2}}e^{-\frac{\lambda}{2}(\bar{y}-x)^{2}}\int_{\mathbb{R}}d\xi_{0}e^{i\lambda\xi_{0}.(y_{0}-\bar{y}_{0})}\frac{e^{i\lambda\xi_{0}\xi_{x}^{\prime}.(y^{\prime}-\bar{y}^{\prime})}a_{0}(\xi_{0})}{\xi_{0}^{4}}\\\
&\quad\times\frac{e^{(\theta-1)^{2}}}{\Gamma(1-\theta)}\int_{\mathbb{R}^{3}}d\xi^{\prime}2^{-k}a_{3,0}(x,1,\xi^{\prime})e^{i\lambda
2^{-k}\xi_{0}\xi^{\prime}.(y^{\prime}-\bar{y}^{\prime})}(q_{k}+i0)^{-\theta}.\end{split}$
We show+ that the corresponding operators uniformly bounded on the spaces
$\begin{split}&V^{\theta}_{\lambda,k}:L^{2}\to
L^{2}\qquad\quad\;\quad\;\Re\theta=0,\\\
&V^{\theta}_{\lambda,k}:L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty}\quad\Re\theta=\theta_{1},\end{split}$
where $p_{1}$, $\theta_{1}$ are chosen so that the points
$\big{(}\frac{1}{2},\frac{1}{2},0\big{)},\quad\big{(}\frac{1}{p},\frac{1}{q},1\big{)},\quad\big{(}\frac{1}{p_{1}},0,\theta_{1}\big{)}$
are collinear.
* •
Estimate of $V^{\theta}_{\lambda,k}:L^{2}\to L^{2},\quad\Re\theta=0$: After
inverting the changes of variables and using the $L^{2}$-boundedness of the
FBI transform, we find
(51) $\|V^{\theta}_{\lambda,k}\|_{L^{2}\to L^{2}}\lesssim 2^{2k}.$
* •
Estimate of $V^{\theta}_{\lambda,k}:L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty}$, $\Re\theta=\theta_{1}$: Inequality (48) implies the
kernel estimates
$|K_{\lambda,k}^{\theta}|\lesssim\lambda^{4-\theta(2\rho+1)}2^{-k}(1+\lambda
2^{-k}|y^{\prime}-\bar{y}^{\prime}|)^{\Re\theta-\frac{3}{2}}.$
By the relations (49) and provided that
$|y_{0}-\bar{y}_{0}|\lesssim|y^{\prime}-\bar{y}^{\prime}|$, this gives
$|K^{\theta}_{\lambda,k}|\lesssim
2^{\frac{2k}{p_{1}}-k}|y-\bar{y}|^{-\frac{2}{p_{1}}}\leq
2^{\frac{2k}{p_{1}}-k}|y_{0}-\bar{y}_{0}|^{-\frac{2}{p_{1}}}.$
Since $p_{1}\geq 4$, we can apply the Hardy–Littlewood–Sobolev inequality On
the other hand, if $|y_{0}-\bar{y}_{0}|\gg|y^{\prime}-\bar{y}^{\prime}|$, we
can integrate by parts in $\xi_{0}$ in (50), which gains factors
$(\lambda|y_{0}-\bar{y}_{0}|)^{-1}$. So we obtain
$\|V^{\theta}_{k,\lambda}\|_{L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty}}\lesssim 2^{\frac{2k}{p_{1}}-k}.$
Interpolation yields
$\|V_{k,\lambda}^{\theta}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim\|V^{\theta}_{k,\lambda}\|_{L^{2}\to
L^{2}}^{\alpha}\|V^{\theta}_{k,\lambda}\|_{L^{p_{1}^{\prime}}L^{1}\to
L^{p_{1}}L^{\infty}}^{1-\alpha}$
with
$\big{(}\frac{1}{p},\frac{1}{q},1\big{)}=\alpha\big{(}\frac{1}{2},\frac{1}{2},0\big{)}+(1-\alpha)\big{(}\frac{1}{p_{1}},0,\theta_{1}\big{)}\quad\Longrightarrow\quad\alpha=\frac{2}{q}.$
These identities imply $\theta_{1}=\frac{1/2}{1/2-1/q}$ and
$(2k/p_{1})(1-2/q)=k(2/p-2/q)$, and thus the operator norm
$\|V_{k,\lambda}^{\theta}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim
2^{k\big{(}\frac{6}{q}-1\big{)}}2^{\frac{2k}{p_{1}}(1-\alpha)}\lesssim
2^{k\big{(}\frac{3}{q}-\frac{1}{2}\big{)}}.$
This is summable for $q>6$, but since $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$
and $p<q$ this is always the case under the assumptions. ∎
Note that (51) reflects the lack of hyperbolicity at the singular point. For
this purpose, we only prove the bounds at the level of the scalar equation
close to the endpoint. We remark that in two dimensions, it suffices to prove
the (allowed) endpoint estimate $L_{t}^{4}L_{x}^{\infty}$. We included the
above arguments to show that the proof remains valid for a characteristic
surface $\tilde{S}=\\{q(\xi)=0\\}$ in higher dimensions with the degeneracy
$q(\xi_{0})=0$, $\nabla q(\xi_{0})=0$, and $\det(\partial^{2}q(\xi_{0}))\neq
0$.
### 3.3. Short-time Strichartz estimates
In the estimate (32) proved so far, there is no gain in the regularity of the
homogeneous part $\|u\|_{L^{2}}$ by the Strichartz estimate over Sobolev
embedding. Nonetheless, by passing through short-time Strichartz estimates we
show the claimed improvement (see also Bahouri–Chemin [1]). In the following
we recast the smoothing in the inhomogeneous term $Pu$ as derivative gain by
using frequency dependent time localization.
###### Proposition 3.8.
Let $(\varepsilon,\mu)\in C^{s}_{x}$ for $0<s\leq 1$ and
$\partial(\varepsilon,\mu)\in L_{t}^{2}L_{x^{\prime}}^{\infty}$. Suppose that
the estimate
(52)
$\||D|^{-\rho+\frac{s-2}{4}}u\|_{(L_{t}^{4}L^{\infty}_{x^{\prime}})_{2}}\lesssim_{C}\|u\|_{L_{x}^{2}}+\||D|^{\frac{s-2}{2}}Pu\|_{L^{2}_{x}}+\||D|^{\frac{s-2}{4}-\frac{1}{2}}\rho_{em}\|_{L^{2}_{x}}$
holds true with
$C=C(\|(\varepsilon,\mu)\|_{C^{s}_{x}},\|\partial(\varepsilon,\mu)\|_{L_{t}^{2}L_{x^{\prime}}^{\infty}})$.
Let $\delta>0$ and $T>0$. We then obtain the inquality
(53) $\begin{split}\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L^{4}(0,T;L^{\infty}_{x^{\prime}})}&\leq
C(\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{\frac{s-2}{8}-\frac{1}{2}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}\\\
&\quad+\|\langle
D^{\prime}\rangle^{\frac{s-2}{8}-\frac{1}{2}}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}})\end{split}$
with
$C=C(\|(\varepsilon,\mu)\|_{C_{x}^{s}},\|\partial(\varepsilon,\mu)\|_{L_{t}^{2}L_{x^{\prime}}^{\infty}},T,\delta)$.
###### Proof.
An application of (52) with $P_{\lambda}$ (recall this is the Maxwell operator
with coefficients truncated at frequencies $\leq\lambda/8$) on $S_{\lambda}u$
yields
(54)
$\lambda^{-\rho+\frac{s-2}{4}}\|S_{\lambda}u\|_{L_{t}^{4}L_{x^{\prime}}^{\infty}}\lesssim\|S_{\lambda}u\|_{L_{x}^{2}}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}u\|_{L^{2}_{t}L_{x^{\prime}}^{2}}+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S_{\lambda}\rho_{em}\|_{L^{2}}.$
For the proof of (53), we use Minkowski’s inequality to split
$\begin{split}\|\langle
D^{\prime}&\rangle^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\\\
&\leq\sum\nolimits_{\lambda}\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}S_{\lambda}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\\\
&\leq\sum\nolimits_{\lambda}\big{[}\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}+\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}S_{\lambda}^{\ll\tau}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\big{]},\end{split}$
where $S_{\lambda}^{\ll\tau}$ projects to the space-time frequencies
$\\{\lambda\sim|\tau|\gg|\xi^{\prime}|\\}$. In this region the operator $P$ is
elliptic, which gains one derivative for Lipschitz coefficients. The summand
can then be bounded by Sobolev embedding. We refer to the proof of [17,
Corollary 1.7] for details on this estimate if $\partial(\varepsilon,\mu)\in
L^{2}_{x_{0}}L^{\infty}_{x^{\prime}}$.
For the first term we want to apply (54). Let $\chi\in C^{\infty}_{c}(-2,2)$
be radially decreasing with $\chi(t)=1$ for $t\in[-1,1]$. Note that passing to
$\chi(\lambda^{\frac{2-s}{2}}t-n)u$ does not essentially change the Fourier
support in time because the inverse Fourier transform of
$\chi_{\lambda^{\frac{2-s}{2}}}(t)=\chi(\lambda^{\frac{2-s}{2}}t-n)$ is
essentially supported in a $\lambda^{\frac{2-s}{2}}$-ball. We suppress
dependence on $n$ in the following due to uniformity of the estimates. We will
apply (54) to $S^{\prime}_{\lambda}\chi_{\lambda^{\frac{2-s}{2}}}u$ with
$I_{\lambda}=\text{supp}(\chi_{\lambda^{\frac{2-s}{2}}})$ and
$I_{\lambda}^{*}=\\{\chi_{\lambda^{\frac{2-s}{2}}}=1\\}$. At first, applying
$\partial_{t}$ to $\chi_{\lambda^{\frac{2-s}{2}}}$ in
$[P_{\lambda},\chi_{\lambda^{\frac{2-s}{2}}}]$ we derive
$\lambda^{\frac{s-2}{2}}\|P_{\lambda}\chi_{\lambda^{\frac{2-s}{2}}}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{x}^{2}}\lesssim\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}.$
This gives
$\begin{split}\lambda^{-\rho+\frac{s-2}{4}-\frac{\delta}{2}}\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{4}(I_{\lambda}^{*};L_{x^{\prime}}^{\infty})}&\lesssim\|S_{\lambda}^{\prime}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}\\\
&\quad+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S^{\prime}_{\lambda}\rho_{em}\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}.\end{split}$
An application of Hölder’s inequality on
$\|S_{\lambda}^{\prime}u\|_{L_{t}^{2}(I_{\lambda};L_{x^{\prime}}^{2})}$ yields
$\begin{split}\lambda^{-\rho+\frac{s-2}{4}-\frac{\delta}{2}}\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{4}(I_{\lambda}^{*};L^{\infty}_{x^{\prime}})}&\lesssim\lambda^{\frac{s-2}{4}}\|S^{\prime}_{\lambda}u\|_{L_{t}^{\infty}(I_{\lambda};L^{2}_{x^{\prime}})}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}\\\
&\quad+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S^{\prime}_{\lambda}\rho_{em}\|_{L_{t}^{2}(I_{\lambda};L^{2}_{x^{\prime}})}.\end{split}$
We sum this estimate over the $T\lambda^{\frac{2-s}{2}}$ disjoint intervals
$I_{\lambda}^{*}$ partitioning $I=(0,T)$ in $\ell^{4}$ such that
$\begin{split}\lambda^{-\rho+\frac{s-2}{4}-\frac{\delta}{2}}\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{T}^{4}(I;L_{x^{\prime}}^{\infty})}&\lesssim_{T}\lambda^{\frac{2-s}{8}+\frac{s-2}{4}}\|S^{\prime}_{\lambda}u\|_{L_{t}^{\infty}L^{2}_{x^{\prime}}}+\lambda^{\frac{s-2}{2}}\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}L^{2}_{x^{\prime}}}\\\
&\quad+\lambda^{\frac{s-2}{4}-\frac{1}{2}}\|S^{\prime}_{\lambda}\rho_{em}\|_{L_{x}^{2}}.\end{split}$
Because of $\ell^{2}\hookrightarrow\ell^{4}$ we do not lose powers of
$\lambda$ when summing $P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u$ and
$S^{\prime}_{\lambda}\rho_{em}$. By commutator arguments using
$\|\partial(\varepsilon,\mu)\|_{L^{2}_{t}L_{x^{\prime}}^{\infty}}\lesssim 1$,
cf. (4.4) in [23], we find
$\|P_{\lambda}S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{2}L_{x^{\prime}}^{2}}\lesssim\|S_{\lambda}^{\prime}Pu\|_{L_{t}^{2}L_{x^{\prime}}^{2}}+\|S^{\prime}_{\lambda}u\|_{L_{t}^{\infty}L_{x^{\prime}}^{2}}$
and hence,
$\displaystyle\lambda^{-\rho+\frac{s-2}{8}-\frac{\delta}{2}}\|S_{\lambda}S^{\prime}_{\lambda}u\|_{L_{t}^{4}(I;L^{\infty}_{x^{\prime}})}$
$\displaystyle\lesssim_{T}\|S^{\prime}_{\lambda}u\|_{L^{\infty}_{t}L^{2}_{x^{\prime}}}+\lambda^{\frac{3(s-2)}{8}}\|Pu\|_{L^{2}_{t}L^{2}_{x^{\prime}}}$
$\displaystyle\quad+\lambda^{\frac{s-2}{8}-\frac{1}{2}}\|S^{\prime}_{\lambda}\rho_{em}\|_{L_{x}^{2}}.$
Summing over $\lambda$, we find the estimate
(55) $\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L_{t}^{4}(I;L^{\infty}_{x^{\prime}})}\lesssim_{T,\delta,X}\|u\|_{L_{t}^{\infty}L^{2}_{x^{\prime}}}+\|Pu\|_{L^{2}_{t}L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\rho_{em}\|_{L^{2}_{x}}$
with
$X=\|(\varepsilon,\mu)\|_{C^{s}_{x}}+\|\partial(\varepsilon,\mu)\|_{L_{t}^{2}L_{x^{\prime}}^{\infty}}$.
(Note that we can treat frequencies $\lambda\sim|\tau|\gg|\xi^{\prime}|$ as
above.)
The $L_{t}^{2}L_{x^{\prime}}^{2}$-norms are changed to
$L_{t}^{1}L_{x^{\prime}}^{2}$-norms by the energy estimate and Duhamel’s
formula: An application of (55) to homogeneous solutions gives
$\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L_{t}^{4}(I;L^{\infty}_{x^{\prime}})}\lesssim_{T,\delta,X}\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}.$
Now we write the general function $u$ by Duhamel’s formula
$u(t)=U(t,0)u_{0}+\int_{0}^{t}U(t,s)(Pu)(s)ds$ and note that
$\nabla\cdot((Pu)_{1}(s),(Pu)_{2}(s))=(-\partial_{t}\rho_{e}(s),\partial_{t}\rho_{m}(s))$.
We find after applying Minkowski’s inequality
$\displaystyle\|\langle
D^{\prime}\rangle^{-\rho+\frac{s-2}{8}-\delta}u\|_{L_{t}^{4}(I;L^{\infty}_{x^{\prime}})}$
$\displaystyle\lesssim_{T,\delta,X}\|u_{0}\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}$
(56) $\displaystyle\quad+\|\langle
D^{\prime}\rangle^{-\frac{1}{2}+\frac{s-2}{8}}\partial_{t}\rho_{em}\|_{L^{1}_{T}L^{2}_{x^{\prime}}}.$
The proof is complete. ∎
Interpolation with the energy estimate from Proposition 3.1
$\|u\|_{L^{\infty}_{T}L^{2}_{x^{\prime}}}\lesssim\|u(0)\|_{L^{2}_{x^{\prime}}}+\|Pu\|_{L^{1}_{T}L^{2}_{x^{\prime}}}$
yields Theorem 1.1 on the sharp line $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$.
The general case follows from Sobolev embedding. In the next section we
generalize the result in the Lipschitz case to diagonalizable coefficients.
## 4\. Reducing to the case of diagonal material laws
We now prove Theorem 1.2 by transforming permittivity and permeability to
diagonal matrices. In the following we consider $\varepsilon,\mu\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3}_{\text{sym}})$.
### 4.1. Orthogonality transformations
Suppose there is $\Phi\in C^{1}(\mathbb{R}^{4};\mathbb{R}^{3\times 3})$ with
(57) $\Phi^{t}(x)\Phi(x)=1_{3\times
3},\quad\varepsilon^{d}=\Phi^{t}\varepsilon\Phi,\quad\mu^{d}=\Phi^{t}\mu\Phi.$
Write $\Phi=(\varphi_{1}\;\varphi_{2}\;\varphi_{3})$. We use the
transformations
$\tilde{\mathcal{E}}=\Phi^{-1}\mathcal{E},\quad\tilde{\mathcal{H}}=\Phi^{-1}\mathcal{H},\quad\tilde{\mathcal{J}}_{k}=\Phi^{-1}\mathcal{J}_{k},\;k\in\\{e,m\\}$
to reduce the analysis to the case that $\varepsilon$ and $\mu$ are diagonal
matrices. If $\mathcal{E}$, $\mathcal{H}$, $\mathcal{J}$ satisfy (29), then
$\tilde{\mathcal{E}}$, $\tilde{\mathcal{H}}$, $\mathcal{\tilde{J}}$ satisfy
the system
(58)
$\left\\{\begin{array}[]{cl}\partial_{t}(\varepsilon^{d}\tilde{\mathcal{E}})&=\Phi^{t}\nabla\times(\Phi\tilde{\mathcal{H}})-(\partial_{t}\Phi)\varepsilon^{d}\tilde{\mathcal{E}}-\tilde{\mathcal{J}}_{e},\\\
\partial_{t}(\mu^{d}\tilde{\mathcal{H}})&=-\Phi^{t}\nabla\times(\Phi\tilde{\mathcal{E}})-(\partial_{t}\Phi)\mu^{d}\tilde{\mathcal{H}}-\tilde{\mathcal{J}}_{m},\\\
\rho_{e}&=\nabla\cdot(\Phi\varepsilon^{d}\tilde{\mathcal{E}}),\quad\rho_{m}=\nabla\cdot(\Phi\mu^{d}\tilde{\mathcal{H}}).\end{array}\right.$
We write $\tilde{u}=(\tilde{\mathcal{E}},\tilde{\mathcal{H}})$. The curl
transforms as follows. We denote
$\nabla\times
f=i\mathcal{C}(D)f,\quad\mathcal{C}(\xi^{\prime})=(-\varepsilon^{ijk}\xi_{k})_{ij},\quad\mathcal{C}(\xi^{\prime})v=\xi^{\prime}\times
v.$
The leading order term of $\Phi^{t}(\nabla\times(\Phi\cdot))$ can be written
as
$\begin{split}\begin{pmatrix}\varphi_{1}^{t}\\\ \varphi_{2}^{t}\\\
\varphi_{3}^{t}\end{pmatrix}\begin{pmatrix}\xi^{\prime}\times\varphi_{1}&\xi^{\prime}\times\varphi_{2}&\xi^{\prime}\times\varphi_{3}\end{pmatrix}&=(\varphi_{i}\cdot(\xi^{\prime}\times\varphi_{j}))_{ij}=(\xi^{\prime}\cdot(\varphi_{j}\times\varphi_{i}))_{ij}\\\
&=(-\varepsilon^{ijk}\varphi_{k}(x)\cdot\xi^{\prime})_{ij}=:(\mathcal{C}(\eta))_{ij}.\end{split}$
We let $\eta_{k}(x,\xi)=\varphi_{k}(x)\cdot\xi^{\prime}$ and333It is important
to work with operators in divergence form. We highlight this by the notation
$a(D,x)$. $\eta_{k}(D,x)=\sum_{j}\partial_{j}(\varphi_{kj}(x)\cdot)$.
Consequently,
$\big{(}\Phi^{t}(\nabla\times(\Phi\cdot))\big{)}_{ij}=(-\varepsilon^{ijk}\varphi_{k}(x)\cdot\nabla_{x^{\prime}})_{ij}+b_{ij}(x)$
with $\|b_{ij}\|_{L^{\infty}}\lesssim\|\Phi\|_{C^{1}}$. In the following we
use that $\varepsilon^{d}$ and $\mu^{d}$ are diagonal as in Assumption 1:
(59)
$\varepsilon^{d}(x)=\text{diag}(\varepsilon_{1}^{d}(x),\varepsilon^{d}_{2}(x),\varepsilon^{d}_{3}(x)),\quad\mu^{d}(x)=\text{diag}(\mu^{d}_{1}(x),\mu_{2}^{d}(x),\mu^{d}_{3}(x)).$
For the electric charge we note that
$\rho_{e}=\nabla\cdot(\varepsilon\mathcal{E})=\nabla\cdot(\Phi\varepsilon^{d}\tilde{u}^{1})=\sum_{j,k=1}^{3}\partial_{j}(\varphi_{kj}\varepsilon_{k}^{d}\tilde{u}_{k}^{1})=:\eta(D,x)\cdot\varepsilon^{d}\tilde{u}^{1}.$
For the magnetic charge we find similarly
$\rho_{m}=\eta(D,x)\cdot(\mu^{d}\tilde{u}^{2}).$
We define
(60)
$\tilde{\rho}_{e}=\eta(D,x)\cdot(\varepsilon^{d}\tilde{u}^{1}),\quad\tilde{\rho}_{m}=\eta(D,x)\cdot(\mu^{d}\tilde{u}^{2}).$
It follows
$\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\Phi^{t}(\nabla\times(\Phi(\cdot)))\\\
\Phi^{t}(\nabla\times(\Phi(\cdot)))&\partial_{t}(\mu^{d}\cdot)\end{pmatrix}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\mathcal{C}(\eta(D,x))\\\
\mathcal{C}(\eta(D,x))&\partial_{t}(\mu^{d}\cdot)\end{pmatrix}+R(x)$
with $R(x)\in L^{\infty}$. Let
(61)
$\tilde{P}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\mathcal{C}(\eta(D,x))\\\
\mathcal{C}(\eta(D,x))&\partial_{t}(\mu^{d}\cdot)\end{pmatrix}.$
We have the following lemma:
###### Lemma 4.1.
Let $\Phi\in C^{1}(\mathbb{R}^{4};\mathbb{R}^{3\times 3})$ satisfy (57) and
$\varepsilon^{d}$, $\mu^{d}$ be like in (59). Set
$\tilde{u}=(\tilde{\mathcal{E}},\tilde{\mathcal{H}})=(\Phi^{-1}\mathcal{E},\Phi^{-1}\mathcal{H})$,
$\tilde{\mathcal{J}}_{k}=\Phi^{-1}\mathcal{J}_{k}$, $k\in\\{e,m\\}$,
$\varepsilon^{d}=\Phi^{t}\varepsilon\Phi$, $\mu^{d}=\Phi^{t}\mu\Phi$, and
$\tilde{P}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\mathcal{C}(\eta(D,x))\\\
\mathcal{C}(\eta(D,x))&\partial_{t}(\mu^{d}\cdot)\end{pmatrix};\quad\tilde{\rho}_{em}=(\eta(D,x)(\varepsilon^{d}\tilde{u}^{1}),\eta(D,x)(\mu^{d}\tilde{u}^{2})).$
Suppose that
(62)
$\||D|^{-\rho-\frac{1}{4}}\tilde{u}\|_{(L^{4}L^{\infty})_{2}}\lesssim\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}_{x}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}.$
Then, for $0<\delta<\frac{1}{4}$, the following estimate holds:
(63)
$\||D|^{-\rho-\frac{1}{4}-\delta}u\|_{(L^{4}L^{\infty})_{2}}\lesssim\|u\|_{L^{2}}+\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{3}{4}}\rho_{em}\|_{L^{2}}.$
The implicit constants depend on $\|(\Phi,\Phi^{t})\|_{C^{1}}$,
$\|(\varepsilon,\mu)\|_{C^{1}}$.
###### Proof.
The low frequencies can be treated like in the proof of Proposition 3.4 by
Sobolev embedding. For $0<\delta<\frac{1}{4}$, we thus obtain
$\|S_{0}|D|^{-\rho-\frac{1}{4}-\delta}u\|_{L^{4}L^{\infty}}\lesssim\|u\|_{L^{2}}.$
For high frequencies $M\gtrsim 1$, we write
$\begin{split}\|S_{M}|D|^{-\rho-\frac{1}{4}-\delta}u\|_{(L^{4}L^{\infty})_{2}}&\sim
M^{-\rho-\frac{1}{4}-\delta}\|S_{M}(\Phi^{t}\tilde{u})\|_{L^{4}L^{\infty}}\\\
&\lesssim M^{-\rho-\frac{1}{4}-\delta}\sum_{1\leq K\leq
M}\|S_{M}S^{\prime}_{K}(\Phi^{t}\tilde{u})\|_{L^{4}L^{\infty}}\\\
&\quad+M^{-\rho-\frac{1}{4}-\delta}\|S_{M}S^{\prime}_{0}(\Phi^{t}\tilde{u})\|_{L^{4}L^{\infty}}.\end{split}$
The second term is estimated
$M^{-\frac{3}{2}-\delta}\|S_{M}S^{\prime}_{0}(\Phi^{t}u)\|_{L^{4}L^{\infty}}\lesssim
M^{-1}\|\Phi^{t}u\|_{L^{2}}$
by Bernstein’s inequality, with easy summation in $M$. Hence, it suffices to
estimate the spatial frequencies, which are greater than $1$. To this end, we
also use paraproduct decompositions both in space and space-time frequencies.
First, we write (with $X=L_{x_{0}}^{4}L^{\infty}_{x^{\prime}}$)
$\displaystyle\|S_{M}S^{\prime}_{K}(\Phi^{t}\tilde{u})\|_{X}$
$\displaystyle\leq\|S_{M}((S^{\prime}_{\sim K}\Phi^{t})(S^{\prime}_{\ll
K}\tilde{u}))\|_{X}+\|S_{M}(S^{\prime}_{K}((S^{\prime}_{\gtrsim
K}\Phi^{t})(S^{\prime}_{\gtrsim K}\tilde{u}))\|_{X}$ (64)
$\displaystyle\quad+\|S_{M}((S^{\prime}_{\ll K}\Phi^{t})(S^{\prime}_{\sim
K}\tilde{u}))\|_{X}.$
Bernstein’s inequality implies for the first term in (4.1)
$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}\|S_{M}(S^{\prime}_{K}\Phi^{t}S^{\prime}_{\ll
K}\tilde{u})\|_{L_{t}^{4}L_{x^{\prime}}^{\infty}}\big{)}^{2}\\\
&\lesssim\sum_{M\geq 1}M^{-2\rho-\frac{1}{2}}M^{\frac{1}{2}}\big{(}\sum_{K\leq
M}K^{\frac{3}{2}}\|S^{\prime}_{K}\Phi^{t}S^{\prime}_{\ll
K}\tilde{u}\|_{L^{2}_{t,x^{\prime}}}\big{)}^{2}\\\ &\lesssim\sum_{M\geq
1}M^{-\frac{5}{2}}\big{(}\sum_{K\leq
M}K^{\frac{1}{2}}\|\nabla_{x^{\prime}}S^{\prime}_{K}\Phi^{t}\|_{L^{\infty}}\big{)}^{2}\|\tilde{u}\|^{2}_{L^{2}}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}.\end{split}$
The second term in (4.1) can be estimated likewise:
$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}\|S_{M}S^{\prime}_{K}(S^{\prime}_{\gtrsim K}\Phi^{t}S^{\prime}_{\gtrsim
K}\tilde{u}\big{)}\|_{L_{t}^{4}L_{x^{\prime}}^{\infty}}\big{)}^{2}\\\
&\lesssim\sum_{M\geq 1}M^{-2\rho}\big{(}\sum_{1\leq K\leq
M}K^{\frac{3}{2}}\|S^{\prime}_{\gtrsim
K}\Phi^{t}\|_{L^{\infty}_{x^{\prime}}}\|S^{\prime}_{\gtrsim
K}\tilde{u}\|_{L^{2}_{x}}\big{)}^{2}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}.\end{split}$
For the last term in (4.1), we additionally make the paraproduct decomposition
in space-time frequencies
$\displaystyle\|S_{M}(S^{\prime}_{\ll
K}\Phi^{t}S^{\prime}_{K}\tilde{u})\|_{X}$
$\displaystyle\leq\|(S_{M}S^{\prime}_{\ll K}\Phi^{t})(S_{\ll
M}S^{\prime}_{\sim K}\tilde{u})\|_{X}$ (65)
$\displaystyle\quad+\|S_{M}(S_{\gtrsim M}S^{\prime}_{\ll K}\Phi^{t}S_{\gtrsim
M}S^{\prime}_{\sim K}\tilde{u})\|_{X}$ $\displaystyle\quad+\|(S_{\ll
M}S^{\prime}_{\ll K}\Phi^{t})(S_{\sim M}S^{\prime}_{\sim K}\tilde{u})\|_{X}.$
The first and second term in (4.1) can be estimated using Bernstein’s
inequality. For the first term in (4.1) we find
$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}\|(S_{M}S^{\prime}_{\ll K}\Phi^{t})(S_{\ll M}S^{\prime}_{\sim
K}\tilde{u})\|_{L^{4}L^{\infty}}\big{)}^{2}\\\ &\lesssim\sum_{M\geq
1}M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}K^{\frac{3}{2}}M^{\frac{1}{4}}\|S_{M}\Phi^{t}\|_{L^{\infty}}\|\tilde{u}\|_{L^{2}}\big{)}^{2}\\\
&\lesssim\sum_{M\geq
1}M^{-3}M^{3}M^{\frac{1}{2}}\|S_{M}\Phi^{t}\|_{L^{\infty}}^{2}\|\tilde{u}\|_{L^{2}}^{2}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}.\end{split}$
The estimate of the second term in (4.1) is given by
$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}}\big{(}\sum_{1\leq K\leq
M}\|S_{M}(S_{\gtrsim M}S^{\prime}_{\ll K}\Phi^{t}S_{\gtrsim M}S^{\prime}_{\sim
K}\tilde{u}\big{)}\|_{L^{4}L^{\infty}}\big{)}^{2}\\\ &\lesssim\sum_{M\geq
1}M^{-3}M^{\frac{3}{2}+\epsilon}\|\Phi\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}\lesssim\|\Phi\|_{C^{1}}^{2}\|\tilde{u}\|_{L^{2}}^{2}.\end{split}$
For the third term in (4.1), we find
$\begin{split}\sum_{M\geq 1}&M^{-2\rho-\frac{1}{2}-2\delta}\big{(}\sum_{1\leq
K\leq M}\|(S_{\ll M}S^{\prime}_{\ll K}\Phi^{t})(S_{\sim M}S^{\prime}_{\sim
K}\tilde{u})\|_{L^{4}L^{\infty}}\big{)}^{2}\\\
&\lesssim\|\Phi^{t}\|_{L^{\infty}}^{2}\sum_{M\geq
1}M^{-2\delta}\big{(}\sum_{1\leq K\leq M}\|\langle
D\rangle^{-\rho-\frac{1}{4}}S_{\sim M}S^{\prime}_{\sim
K}\tilde{u}\|_{L^{4}L^{\infty}}\big{)}^{2}\\\
&\lesssim\|\Phi^{t}\|_{L^{\infty}}^{2}(\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}})^{2}.\end{split}$
In the last step we used the hypothesis. It remains to show
$\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}\lesssim\|u\|_{L^{2}}+\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{3}{4}}\rho_{em}\|_{L^{2}}.$
We first note that the definitions imply
$\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}\lesssim\|u\|_{L^{2}}+\||D|^{-\frac{3}{4}}\rho_{em}\|_{L^{2}}.$
We turn to the estimate of the second term. By the above we can write
$\tilde{P}\tilde{u}=P^{\prime}\tilde{u}+Ru$ with
$P^{\prime}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}\cdot)&\Phi^{t}(\nabla\times(\Phi\cdot))\\\
\Phi^{t}\nabla\times(\Phi\cdot)&\partial_{t}(\mu^{d}\cdot)\end{pmatrix},\qquad
R\in L^{\infty}.$
Moreover, $P^{\prime}\tilde{u}=\tilde{R}\tilde{u}+\Phi^{t}\mathcal{J}$, where
we write $\Phi^{t}$ instead of $\text{diag}(\Phi^{t},\Phi^{t}).$ Since
$\tilde{P}$ is in divergence form, the low frequencies satisfy with implicit
constant depending on the coefficients:
$\||D|^{-\frac{1}{2}}S_{0}(\tilde{P}\tilde{u})\|_{L^{2}}\lesssim\|\tilde{u}\|_{L^{2}}.$
For the high frequencies, we plug in the above relations to find
$\begin{split}\||D|^{-\frac{1}{2}}S_{\geq
1}(\tilde{P}\tilde{u})\|_{L^{2}_{x}}&\lesssim\||D|^{-\frac{1}{2}}S_{\geq
1}(\tilde{R}\tilde{u}+Ru+\Phi^{t}Pu)\|_{L^{2}}\\\
&\lesssim(\|\tilde{R}\|_{L^{\infty}}+\|R\|_{L^{\infty}})\|u\|_{L^{2}}+\||D|^{-\frac{1}{2}}S_{\geq
1}(\Phi^{t}Pu)\|_{L^{2}_{x}}.\end{split}$
For the last term we again use a paraproduct decomposition, writing
(66)
$\begin{split}\|S_{M}(\Phi^{t}Pu)\|_{L_{x}^{2}}&\leq\|(S_{M}\Phi^{t})S_{\ll
M}(Pu)\|_{L^{2}_{x}}+\|(S_{\ll M}\Phi^{t})S_{M}(Pu)\|_{L^{2}_{x}}\\\
&\quad+\|\sum_{K\gtrsim M}S_{K}\Phi^{t}S_{K}(Pu)\|_{L^{2}_{x}}\end{split}$
for $M\geq 1$. The first term in (66) is estimated by
$\begin{split}\sum_{M\geq 1}M^{-1}\|S_{M}\Phi^{t}S_{\ll
M}(Pu)\|_{L^{2}}^{2}&\lesssim\sum_{M\geq
1}M^{-1}\|S_{M}\Phi^{t}\|_{L^{\infty}}^{2}\sum_{K\ll
M}\|S_{K}(Pu)\|_{L^{2}}^{2}\\\ &\lesssim\sum_{M\geq
1}M^{-1}\|S_{M}\Phi^{t}\|^{2}_{L^{\infty}}\\!\sum_{1\leq K\ll
M}\\!K\||D|^{-\frac{1}{2}}S_{K}(Pu)\|_{L^{2}}^{2}\\\ &\lesssim\sum_{M\geq
1}\|S_{M}\Phi^{t}\|_{L^{\infty}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}\\\
&\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}.\end{split}$
Similarly, for the second term in (66) we obtain
$\sum_{M\geq 1}M^{-1}\|S_{\ll
M}\Phi^{t}S_{M}(Pu)\|_{L^{2}}^{2}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}.$
Finally, we estimate the third term in (66) by
$\displaystyle\sum_{M\geq 1}$ $\displaystyle M^{-1}\sum_{K\geq
M}\|S_{K}\Phi^{t}S_{K}(Pu)\|_{L^{2}}^{2}$ $\displaystyle\lesssim\sum_{M\geq
1}M^{-1}\sum_{K\gtrsim
M}K\|S_{K}\Phi^{t}\|_{L^{\infty}}^{2}K^{-1}\|S_{K}(Pu)\|_{L^{2}}^{2}$
$\displaystyle\lesssim\sum_{M\geq
1}M^{-1}\|\Phi^{t}\|_{C^{1}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}\lesssim\|\Phi^{t}\|_{C^{1}}^{2}\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}^{2}.\qed$
### 4.2. Proof of Strichartz estimates for the transformed equation
The proof of Theorem 1.2 in the general case revolves around the proof of
analogs of (32), which is
(67)
$\||D|^{-\rho-\frac{1}{4}}\tilde{u}\|_{(L^{p}L^{q})_{2}}\lesssim(1+\|(\varepsilon^{d},\mu^{d})\|_{C^{1}})\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}.$
Using Lemma 4.1, this inequality allows us to complete the proof of Theorem
1.2.
###### Proposition 4.2.
Let $\varepsilon^{d}$, $\mu^{d}\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3};\mathbb{R}^{3\times 3}_{sym})$ be
diagonal matrices, which are uniformly elliptic. Let $\tilde{P}$,
$\tilde{\rho}_{e}$, and $\tilde{\rho}_{m}$ be and in (60) and (61). Then we
find (67) to hold for $2\leq p<q\leq\infty$,
$\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$, and
$\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$.
In the following we apply the analysis of Section 3 with the role of the
partial derivatives $\partial_{k}$ presently played by the differential
operator $\eta_{k}=\varphi_{k}^{\leq\lambda}\cdot\nabla_{x^{\prime}}$. By
dyadic frequency localization and the usual commutator estimates, (67) follows
from
$\lambda^{-\rho-\frac{1}{4}}\|S_{\lambda}\tilde{u}\|_{L^{p}L^{q}}\lesssim\|S_{\lambda}u\|_{L^{2}}+\lambda^{-\frac{1}{2}}\|\tilde{P}_{\lambda}S_{\lambda}u\|_{L^{2}}+\lambda^{-\frac{3}{4}}\|S_{\lambda}\tilde{\rho}_{em}\|_{L^{2}}.$
Above $\tilde{P}_{\lambda}$ denotes the frequency truncated version at
frequencies $<\lambda/8$ of $\tilde{P}$ given by
$\tilde{P}_{\lambda}=\begin{pmatrix}-\partial_{t}(\varepsilon^{d}_{\lambda}\cdot)&\mathcal{C}(\eta_{\leq\lambda}(D,x))\\\
\mathcal{C}(\eta_{\lambda}(D,x))&\partial_{t}(\mu^{d}_{\lambda}\cdot)\end{pmatrix}.$
In $\eta_{\leq\lambda}$ the coefficients of $\varphi$ are also frequency
truncated at frequencies $\lambda/8$. The frequency truncation is suppressed
in the following to lighten the notation.
We apply the FBI transform and let
$v_{\lambda}=T_{\lambda}\tilde{u}_{\lambda}$ and
$T_{\lambda}(\tilde{\mathcal{J}}/\lambda)=f_{\lambda}$. For
$C^{1}$-coefficients an application of Theorem 3.5 yields
$\tilde{p}(x,\xi)v_{\lambda}=f_{\lambda}+g_{\lambda}\quad\text{with \
}\|g_{\lambda}\|_{L^{2}_{\Phi}}\lesssim_{\|(\varepsilon^{d},\mu^{d})\|_{C^{1}}}\lambda^{-1/2}\|S_{\lambda}u\|_{L^{2}}$
and
$\tilde{p}(x,\xi)=\begin{pmatrix}-i\xi_{0}\varepsilon^{d}&i\mathcal{C}(\eta(x,\xi))\\\
i\mathcal{C}(\eta(x,\xi))&i\xi_{0}\mu^{d}\end{pmatrix}\in\mathbb{C}^{6\times
6}.$
The estimate (67) becomes
$\lambda^{-\rho-\frac{1}{4}}\|T_{\lambda}^{*}\tilde{v}_{\lambda}\|_{L^{p}L^{q}}\lesssim(1+\|(\varepsilon^{d},\mu^{d})\|_{C^{1}})\|\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{1}{2}}\|\tilde{f}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-\frac{3}{4}}\|S_{\lambda}\tilde{\rho}_{em}\|_{L^{2}}.$
We recast this as
(68)
$\|T_{\lambda}^{*}\tilde{v}_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{\frac{1}{4}}\|\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{\frac{3}{4}}\|\tilde{p}(x,\xi)\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-\frac{1}{2}}\|S_{\lambda}\tilde{\rho}_{em}\|_{L^{2}}).$
Next, we reduce (68) to a scalar estimate. For this purpose, corresponding to
(37) we let
$\tilde{q}(x,\xi)=-\xi_{0}^{2}(\xi_{0}^{4}-\xi_{0}^{2}q_{0}(x,\eta^{\prime}(\xi))+q_{1}(x,\eta^{\prime}(\xi))),$
with $q_{0}$ and $q_{1}$ defined like in Section 3 and define the symmetrizer
$\tilde{\sigma}(x,\xi)=\begin{pmatrix}-i\xi_{0}\varepsilon_{d}^{-1}&i\varepsilon_{d}^{-1}\mathcal{C}(\eta(x,\xi))\mu_{d}^{-1}\\\
i\mu_{d}^{-1}\mathcal{C}(\eta(x,\xi))\varepsilon_{d}^{-1}&i\xi_{0}\mu_{d}^{-1}\end{pmatrix}\in\mathbb{C}^{6\times
6},$
obtaining
$\tilde{\sigma}(x,\xi)\tilde{p}(x,\xi)=\begin{pmatrix}M_{E}(x,\eta)-\xi_{0}^{2}&0\\\
0&M_{H}(x,\eta)-\xi_{0}^{2}\end{pmatrix},$
compare (39). We state the announced reduction.
###### Proposition 4.3.
For the proof of (68) under the assumptions of Proposition 4.2, it suffices to
show the estimate
(69)
$\|T_{\lambda}^{*}w_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|w_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|\tilde{q}(x,\xi)w_{\lambda}\|_{L^{2}_{\Phi}}).$
###### Proof.
Clearly, (68) follows from
$\|T_{\lambda}^{*}\tilde{v}_{\lambda}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|\tilde{\sigma}(x,\xi)\tilde{p}(x,\xi)\tilde{v}_{\lambda}\|_{L^{2}_{\Phi}}+\lambda^{-1/2}\|S_{\lambda}\tilde{\rho}_{em}\|_{L^{2}}).$
For this purpose, we show that
(70)
$\|T^{*}_{\lambda}\tilde{v}_{\lambda,1}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|\tilde{v}_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|(M_{E}(x,\eta)-\xi_{0}^{2})\tilde{v}_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{-1/2}\|S_{\lambda}\tilde{\rho}_{e}\|_{L^{2}}),$
(71)
$\|T_{\lambda}^{*}\tilde{v}_{\lambda,2}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|\tilde{v}_{\lambda,2}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|(M_{H}(x,\eta)-\Xi_{0}^{2})\tilde{v}_{\lambda,2}\|_{L^{2}_{\Phi}}+\lambda^{-1/2}\|S_{\lambda}\tilde{\rho}_{m}\|_{L^{2}}).$
In the following it becomes relevant that $|\eta^{\prime}|\sim|\xi^{\prime}|$.
We prove the estimates in the regions
* (1)
$\\{|\xi_{0}|\gg|\eta^{\prime}|\\}$,
* (2)
$\\{|\xi_{0}|\ll|\eta^{\prime}|\\}$, for which we take the generalized charges
into account,
* (3)
$\\{|\xi_{0}|\sim|\eta^{\prime}|\\}$.
The first region is handled like in the proof of Proposition 3.6 using
$|\eta^{\prime}|\sim|\xi^{\prime}|$. For the estimate in the second region, we
focus on (70) because (71) can be proved in a similar way. We decompose
$\tilde{v}_{\lambda,1}=\tilde{v}^{s}_{\lambda,1}+\tilde{v}_{\lambda,1}^{p}$
with
$\tilde{v}^{p}_{\lambda,1}=\frac{(\eta^{\prime}.\tilde{\varepsilon}^{\lambda}\tilde{v}_{\lambda,1})\eta^{\prime}}{|\eta^{\prime}|^{2}_{\varepsilon^{d}}}.$
The contribution of $\tilde{v}^{p}_{\lambda,1}$ is treated by Sobolev
embedding
$\|T_{\lambda}^{*}\tilde{v}^{p}_{\lambda,1}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}\|\tilde{v}^{p}_{\lambda,1}\|_{L^{2}_{\Phi}}\lesssim\lambda^{\rho+\frac{1}{2}}\|(\eta^{\prime}.\varepsilon^{d}_{\lambda}\tilde{v}_{\lambda,1})\|_{L^{2}_{\Phi}}.$
By Theorem 3.5 and a commutator estimate, we find
$\|\eta^{\prime}.(\varepsilon^{d}_{\lambda}v_{\lambda,1})\|_{L^{2}_{\Phi}}\lesssim\lambda^{-\frac{1}{2}}\|v_{\lambda,1}\|_{L^{2}}+\|S_{\lambda}\frac{\eta^{\prime}(x,D)}{\lambda}(\varepsilon^{d}\tilde{\mathcal{E}})\|_{L^{2}}.$
Secondly, it follows like in the proof of Proposition 3.6 by the same
algebraic relations, replacing $\xi^{\prime}$ with $\eta^{\prime}$, that
$|(M_{E}(x,\eta^{\prime})-\xi_{0}^{2})v_{1}^{\lambda}|\gtrsim|v_{1}^{\lambda}|$
for $\eta^{\prime}.(\varepsilon^{d}_{\lambda}\tilde{v}^{\lambda}_{1})=0$ and
$\\{|\xi_{0}|\ll|\xi^{\prime}|\\}$. The details are omitted.
Similarly, by replacing $\eta^{\prime}$ with $\xi^{\prime}$, we argue that the
estimate
$\|T_{\lambda}^{*}v_{\lambda,1}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|v_{\lambda,1}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|(M_{E}-\xi_{0}^{2})v_{\lambda,1}\|_{L^{2}_{\Phi}})$
holds true provided that
$\|T_{\lambda}^{*}v_{\lambda,1,k}\|_{L^{p}L^{q}}\lesssim\lambda^{\rho}(\lambda^{1/4}\|v_{\lambda,1,k}\|_{L^{2}_{\Phi}}+\lambda^{3/4}\|(Z_{\varepsilon,\mu}\frac{\varepsilon^{d}}{\varepsilon^{d}_{1}\varepsilon^{d}_{2}\varepsilon^{d}_{3}}(M_{E}-\xi_{0}^{2})v_{\lambda,1})_{k}\|_{L^{2}_{\Phi}}),$
but this is (69). ∎
To prove (69), we modify the arguments of Section 3. We consider the operator
$\tilde{W}_{\lambda}=T_{\lambda}^{*}\frac{a(x,\xi)}{\lambda^{\frac{1}{4}}+\lambda^{\frac{3}{4}}|\tilde{q}(x,\xi)|},$
for which we shall prove
$\|\tilde{W}_{\lambda}\|_{L^{2}\to
L^{p}L^{q}}\lesssim\lambda^{\rho+\frac{1}{2}}.$
By the $TT^{*}$-argument, we can likewise prove the estimate
$\tilde{V}_{\lambda}=T_{\lambda}^{*}\frac{a^{2}(x,\eta)\Phi(\xi)}{(\lambda^{-\frac{1}{4}}+\lambda^{\frac{1}{4}}|\tilde{q}(x,\xi)|)^{2}}T_{\lambda}.$
Again we have to understand the curvature properties of
$\\{\xi\in\mathbb{R}^{4}:\,\tilde{q}(x,\xi)=0\\}$. Note that
$\tilde{q}(x,\xi)$ is $4$-homogeneous in $\xi$, i.e.,
$\tilde{q}(x,\xi_{0},\xi^{\prime})=\xi_{0}^{4}\tilde{q}(x,1,\xi^{\prime}/\xi_{0}).$
This reduces again to the analysis of the surface
$\tilde{S}=\\{\xi^{\prime}\in\mathbb{R}^{3}:\tilde{q}(x,1,\xi^{\prime})=0\\}$.
By the change of variables
$\eta_{k}^{\prime}=\varphi_{k}^{\leq\lambda}(x)\cdot\xi^{\prime}$ we can
reduce to the Fresnel surface
$S=\\{\xi^{\prime}\in\mathbb{R}^{3}:q(x,1,\xi^{\prime})=0\\}$. Like in Section
3 we split replace $a^{2}(x,\xi)$ by $a(x,\xi)$ and split it into
$a=a_{1}+a_{2}+a_{3}$ according to the regions of the Fresnel surface $S$.
Correspondingly, we decompose
$\tilde{V}_{\lambda}=\tilde{V}_{\lambda,1}+\tilde{V}_{\lambda,2}+\tilde{V}_{\lambda,3}$.
We arrive at the following:
###### Proposition 4.4.
Let $\rho=3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}$ and $2<p\leq
q\leq\infty$. The estimate
$\|\tilde{V}_{\lambda,i}\|_{L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}}\lesssim\lambda^{1+2\rho}$
holds true, if
* •
$i=1$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}$,
* •
$i\in\\{2,3\\}$ and $\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$.
###### Proof.
For fixed $x\in[-1,1]^{4}$ we observe that the change of variables
(72) $\eta_{k}^{\prime}=\varphi_{k}^{\leq\lambda}(x)\cdot\xi^{\prime}$
is non-degenerate for $\lambda\gg 1$. Indeed,
$|J\eta^{\prime}|=|\det(\varphi_{1}^{\leq\lambda},\varphi_{2}^{\leq\lambda},\varphi_{3}^{\leq\lambda})|\sim
1$.
Hence, for $i=1,2$, a uniform bound for the family of operators
$\lambda^{-1-2\rho}T_{\lambda}^{*}\frac{a_{i}(x,\eta)\Phi(\xi)}{(\lambda^{-\frac{1}{4}}+\lambda^{\frac{1}{4}}|\tilde{q}(x,\xi)|)^{2}}T_{\lambda}:L^{p^{\prime}}L^{q^{\prime}}\to
L^{p}L^{q}$
follows from the integrability of the weight
$\int\frac{1}{(\lambda^{-\frac{1}{4}}+\lambda^{\frac{1}{4}}|\tilde{q}|)^{2}}d\tilde{q}\lesssim
1,$
by which we can reduce to level sets $\delta_{\tilde{q}(x,\xi)=c}$ by
foliation. Note that within $\text{supp}(a_{1})\cup\text{supp}(a_{2})$ the
surface $\\{\xi\in\mathbb{R}^{4}:\tilde{q}(x,\xi)=0\\}$ is a regular surface.
By the regular change of variables (72) the curvature properties are
inherited.
For the more involved case of neighbourhoods of the conical singularities, the
additional dyadic decomposition is carried out in $\eta$. After rescaling to
unit distance of the singularity, one can then proceed as in Proposition 3.7.
∎
### 4.3. Conclusion of the proof of Theorem 1.2
With the estimates for $\tilde{V}_{\lambda,i}$ at disposal, the propositions
of the previuous subsection yield the Strichartz estimates
$\||D|^{-\rho-\frac{1}{4}}\tilde{u}\|_{(L^{4}L^{\infty})_{2}}\lesssim(1+\|\partial(\varepsilon^{d},\mu^{d})\|_{L^{\infty}})\|\tilde{u}\|_{L^{2}}+\||D|^{-\frac{1}{2}}\tilde{P}\tilde{u}\|_{L^{2}}+\||D|^{-\frac{3}{4}}\tilde{\rho}_{em}\|_{L^{2}}.$
These estimates transpire to the original quantities by the considerations in
Section 4.1, leading to
$\||D|^{-\rho-\frac{1}{4}}u\|_{L_{t}^{4}L^{\infty}_{x^{\prime}}}\lesssim(1+\|\partial(\varepsilon,\mu)\|_{L^{\infty}})\|u\|_{L^{2}}+\||D|^{-\frac{1}{2}}Pu\|_{L^{2}}+\||D|^{-\frac{3}{4}}\rho_{em}\|_{L^{2}}.$
We can now conclude like in Section 3.3. Proposition 3.8 implies
$\begin{split}\|\langle
D^{\prime}\rangle^{-\rho-\frac{1}{8}-\delta}u\|_{L_{t}^{4}(0,T;L^{\infty}_{x^{\prime}})}&\leq
C(\|u_{0}\|_{L^{2}}+\|Pu\|_{L_{T}^{1}L^{2}_{x^{\prime}}}\\\ &\qquad+\|\langle
D^{\prime}\rangle^{-\frac{5}{8}}\rho_{em}(0)\|_{L^{2}_{x^{\prime}}}+\|\langle
D^{\prime}\rangle^{-\frac{5}{8}}\partial_{t}\rho_{em}\|_{L^{1}_{t}L^{2}_{x^{\prime}}})\end{split}$
with $C=C(X,T,\delta)$, $X=\|\partial(\varepsilon,\mu)\|_{L^{\infty}}$.
Interpolation with Proposition 3.1 yields Theorem 1.1 for
$\frac{2}{p}+\frac{1}{q}=\frac{1}{2}$. The general case follows from Sobolev
embedding. $\hfill\Box$
## 5\. Application to quasilinear Maxwell equations
With the Strichartz estimates at disposal, we can improve local well-posedness
results for quasilinear Maxwell equations in the fully anisotropic case. This
section is devoted to the proof of Theorem 1.5. By local well-posedness we
mean existence, uniqueness, and continuous dependence of the data-to-solution
mapping. We focus on proving estimates in rough norms for smooth solutions,
whose existence we take for granted. The data-to-solution mapping can then be
extended by continuity to rough initial data via standard arguments. We also
refer to our previous work [17].
We consider
(73)
$\left\\{\begin{array}[]{rlrlrl}\partial_{t}\mathcal{D}\\!\\!\\!\\!&=\nabla\times\mathcal{H},&\quad\nabla\cdot\mathcal{D}\\!\\!\\!\\!&=0,&\quad\mathcal{D}(0)\\!\\!\\!\\!&=\mathcal{D}_{0}\in
H^{s}(\mathbb{R}^{3};\mathbb{R}^{3}),\\\
\partial_{t}\mathcal{B}\\!\\!\\!\\!&=-\nabla\times\mathcal{E},&\quad\nabla\cdot\mathcal{B}\\!\\!\\!\\!&=0,&\quad\mathcal{B}(0)\\!\\!\\!\\!&=\mathcal{B}_{0}\in
H^{s}(\mathbb{R}^{3};\mathbb{R}^{3}),\end{array}\right.$
with fields
(74) $\|\mathcal{E}\|_{L^{\infty}}+\|\mathcal{B}\|_{L^{\infty}}\leq\delta,$
$\delta$ to be specified later, and constitutive relations
$\varepsilon=\varepsilon(\mathcal{E})\in\mathbb{R}^{3\times
3}_{\text{sym}},\quad\mu=1_{3\times 3}$
such that $\varepsilon(\mathcal{E})$ is uniformly elliptic and has uniformly
separated eigenvalues for $|\mathcal{E}|\leq\delta$. We use this smallness
condition to guarantee these two crucial properties of the permittivities, it
is not used in the wellposedness analysis given below.
We let $u=(\mathcal{D},\mathcal{B})$ in the following because these are the
variables of which the time-derivative is given. The proof of the theorem
follows along the general scheme of [15, 17]: Set $A=\sup_{0\leq
t^{\prime}\leq t}\|u(t^{\prime})\|_{L^{\infty}_{x^{\prime}}}$ and
$B(t)=\|\nabla_{x^{\prime}}u(t)\|_{L^{\infty}_{x^{\prime}}}$.
* •
We show energy estimates to hold for $s\geq 0$:
(75) $E^{s}(u(t))\lesssim E^{s}(u(0))e^{C\int_{0}^{t}B(s)ds}$
with $E^{s}(u)\approx_{A}\|u\|_{H^{s}}$.
* •
For differences of solutions we prove $L^{2}$-bounds depending on the
$H^{s}$-norm $v=u^{1}-u^{2}$ of the initial data:
$\|v(t)\|_{L^{2}}\lesssim\|v_{0}\|_{L^{2}}\text{ for }0\leq t\leq
T=T(\|u^{i}(0)\|_{H^{s}}).$
* •
We use the frequency envelope approach due to Tao [20] (see also [6]) to show
continuous dependence (but no locally uniform continuous dependence).
To improve on the threshold $s>\frac{5}{2}$ dictated by the energy method, we
want to estimate $\|\partial_{x}u\|_{L_{T}^{1}L_{x^{\prime}}^{\infty}}$ by
Strichartz estimates. This is carried out via a bootstrap argument. For $u$ a
solution in $H_{x^{\prime}}^{\frac{3}{2}+\alpha}$, the coefficients of the
quasilinear Maxwell equation are $\alpha$ Hölder-continuous and in the scope
of Theorem 1.1. If the provided Strichartz estimates can control
$\|\partial_{x}u\|_{L_{T}^{1}L_{x^{\prime}}^{\infty}}$ in terms of the
$H_{x^{\prime}}^{\frac{3}{2}+\alpha}$-norm, we can close the iteration. To
find $\alpha$, we equate the derivative loss $\rho+1$ of the Strichartz
estimates with $\frac{3}{2}+\alpha$, obtaining
$3\big{(}\frac{1}{2}-\frac{1}{q}\big{)}-\frac{1}{p}+\frac{2-\alpha}{2p}+1=\frac{3}{2}+\alpha,\quad
q=\infty,\;p=4.$
This gives $\alpha=\frac{8}{9}$ and suggests that the argument closes for
initial data with regularity $s>\frac{5}{2}-\frac{1}{9}$.
One new ingredient compared to previous works is to find the energy norm for
(73), which is carried out in detail below. It turns out that in the fully
anisotropic case introducing a suitable energy norm requires more care than in
the isotropic case considered in [17, 15]. We recall the following fact from
[15].
###### Proposition 5.1 (Existence of energy estimates).
Suppose that there is symmetric $C(u)$ such that
(76) $\mathcal{A}^{j}(u)^{t}C(u)=C(u)\mathcal{A}^{j}(u),$
and $C(u)$ is uniformly elliptic. Then, we have
$E^{s}(u):=\langle\langle D^{\prime}\rangle^{s}u,C(u)\langle
D^{\prime}\rangle^{s}u\rangle\approx_{A}\|u\|^{2}_{H^{s}}$
and (75) holds, for $s\geq 0$.
To this end, we rewrite Maxwell equations in conservative form
$\partial_{t}u=\mathcal{A}^{j}(u)\partial_{j}u$. However, to state (73) in
conservative form, we have to recast
$\mathcal{E}=\psi(\mathcal{D})\mathcal{D}$, for which we can formulate a
necessary and sufficient condition on the existence of symmetrizers. Note that
$\psi(\mathcal{D})$ is symmetric for symmetric $\varepsilon(\mathcal{E})$.
###### Proposition 5.2.
There exists
(77) $C(u)=\begin{pmatrix}C^{1}(u)&0\\\ 0&1_{3\times
3}\end{pmatrix}\in\mathbb{R}^{6\times 6}_{sym}$
with $C^{1}(u)\in\mathbb{R}^{3\times 3}$ that satisfies (76) if and only if
(78)
$\varepsilon^{ijk}\partial_{j}\psi(\mathcal{D})_{k\ell}\mathcal{D}_{\ell}=0$
for $i\in\\{1,2,3\\}$ with summation over $k,j\in\\{1,2,3\\}$.
###### Proof.
We write for $j=1,2,3$
$\mathcal{A}^{j}(u)=\begin{pmatrix}0&A_{1}^{j}(u)\\\
A_{2}^{j}(u)&0\end{pmatrix};\quad
A_{1}^{j}(x),A_{2}^{j}(x)\in\mathbb{R}^{3\times 3}$
with $(A_{1}^{j})_{mn}=-\varepsilon_{jmn}$. For the special form (77) the
equation (76) translates to
(79) $(A_{1}^{j})^{t}C^{1}=A^{j}_{2}.$
We compute $A^{j}_{2}$ from
$-(\nabla\times(\psi(\mathcal{D})\mathcal{D}))_{i}=(A_{2}^{j}(\mathcal{D})\partial_{j}\mathcal{D})_{i},$
where we have
$\begin{split}(\nabla\times(\psi(\mathcal{D})\mathcal{D}))_{i}&=\varepsilon^{ijk}\partial_{j}(\psi(\mathcal{D})\mathcal{D})_{k}\\\
&=\varepsilon^{ijk}\partial_{j}(\psi(\mathcal{D})_{km}\mathcal{D}_{m})\\\
&=\varepsilon^{ijk}[\psi(\mathcal{D})_{km}\partial_{j}\mathcal{D}_{m}]+\varepsilon^{ijk}\partial_{\ell}\psi(\mathcal{D})_{km}(\partial_{j}\mathcal{D}_{\ell})\mathcal{D}_{m}.\end{split}$
We define the matrices
$\mathcal{B}^{j}_{im}=-\varepsilon^{ijk}\psi(\mathcal{D})_{km},\quad\mathcal{C}^{j}_{im}=-\varepsilon^{ijk}\partial_{m}\psi(\mathcal{D})_{kl}\mathcal{D}_{l}$
and let $A_{2}^{j}=\mathcal{B}^{j}+\mathcal{C}^{j}$. We obtain
$\displaystyle\mathcal{B}^{1}$ $\displaystyle=\begin{pmatrix}0&0&0\\\
\psi(\mathcal{D})_{31}&\psi(\mathcal{D})_{32}&\psi(\mathcal{D})_{33}\\\
-\psi(\mathcal{D})_{21}&-\psi(\mathcal{D})_{22}&-\psi(\mathcal{D})_{23}\end{pmatrix},$
$\displaystyle\mathcal{B}^{2}$
$\displaystyle=\begin{pmatrix}-\psi(\mathcal{D})_{31}&-\psi(\mathcal{D})_{32}&-\psi(\mathcal{D})_{33}\\\
0&0&0\\\
\psi(\mathcal{D})_{11}&\psi(\mathcal{D})_{12}&\psi(\mathcal{D})_{13}\end{pmatrix},$
$\displaystyle\mathcal{B}^{3}$
$\displaystyle=\begin{pmatrix}\psi(\mathcal{D})_{21}&\psi(\mathcal{D})_{22}&\psi(\mathcal{D})_{23}\\\
-\psi(\mathcal{D})_{11}&-\psi(\mathcal{D})_{12}&-\psi(\mathcal{D})_{13}\\\
0&0&0\end{pmatrix}.$
Moreover,
$\begin{split}\mathcal{C}^{1}&=\begin{pmatrix}0&0&0\\\
\partial_{1}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}&\partial_{2}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}&\partial_{3}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}\\\
-\partial_{1}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}&-\partial_{2}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}&-\partial_{3}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}\end{pmatrix},\\\
\mathcal{C}^{2}&=\begin{pmatrix}-\partial_{1}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}&-\partial_{2}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}&-\partial_{3}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell}\\\
0&0&0\\\
\partial_{1}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}&\partial_{2}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}&\partial_{3}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}\end{pmatrix},\\\
\mathcal{C}^{3}&=\begin{pmatrix}\partial_{1}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}&\partial_{2}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}&\partial_{3}\psi(\mathcal{D})_{2\ell}\mathcal{D}_{\ell}\\\
-\partial_{1}\psi(\mathcal{D}_{1\ell}\mathcal{D}_{\ell}&-\partial_{2}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}&-\partial_{3}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}\\\
0&0&0\end{pmatrix}.\end{split}$
Note that
$A_{1}^{1}=\begin{pmatrix}0&0&0\\\ 0&0&-1\\\
0&1&0\end{pmatrix},\;A_{1}^{2}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
-1&0&0\end{pmatrix},\;A_{1}^{3}=\begin{pmatrix}0&-1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix}.$
This yields
$\begin{split}(A_{1}^{1})^{t}C^{1}&=\begin{pmatrix}0&0&0\\\
C^{1}_{31}&C^{1}_{32}&C^{1}_{33}\\\
-C^{1}_{21}&-C^{1}_{22}&-C^{1}_{23}\end{pmatrix},\;(A_{1}^{2})^{t}C^{1}=\begin{pmatrix}-C^{1}_{31}&-C^{1}_{32}&-C^{1}_{33}\\\
0&0&0\\\ C^{1}_{11}&C^{1}_{12}&C^{1}_{13}\end{pmatrix},\\\
(A_{1}^{3})^{t}C^{1}&=\begin{pmatrix}C^{1}_{21}&C^{1}_{22}&C^{1}_{23}\\\
-C^{1}_{11}&-C^{1}_{12}&-C^{1}_{13}\\\ 0&0&0\end{pmatrix}.\end{split}$
A comparison of the coefficients in (79) yields (78) by symmetry of
$\psi(\mathcal{D})$. For instance, we find
$\begin{split}((A^{1}_{1})^{t}C^{1})_{21}&=C^{1}_{31}=\psi(\mathcal{D})_{31}+\partial_{1}\psi(\mathcal{D})_{3\ell}\mathcal{D}_{\ell},\\\
((A^{2}_{1})^{t}C^{1})_{31}&=C^{1}_{13}=\psi(\mathcal{D})_{13}+\partial_{3}\psi(\mathcal{D})_{1\ell}\mathcal{D}_{\ell}.\end{split}$
This gives
$\varepsilon^{1jk}\partial_{j}\psi(\mathcal{D})_{k\ell}\mathcal{D}_{\ell}=0$.
The other conditions are found likewise. ∎
We remark that this assumption can be difficult to verify starting with
$\varepsilon=\varepsilon(\mathcal{E})$. It turns out that the seemingly
natural ansatz
$\varepsilon(\mathcal{E})=\text{diag}(\varepsilon_{0}^{1},\varepsilon_{0}^{2},\varepsilon_{0}^{3})+\text{diag}(\alpha_{1},\alpha_{2},\alpha_{3})|\mathcal{E}|^{2}$
for
$\varepsilon_{0}^{1}\neq\varepsilon_{0}^{2}\neq\varepsilon_{0}^{3}\neq\varepsilon_{0}^{1}$
fails if $\alpha_{i}\neq\alpha_{j}$ for some $i\neq j$. An admissible choice
of $\varepsilon$ is
(80)
$\varepsilon(\mathcal{E})=\text{diag}(\varepsilon_{0}^{1},\varepsilon_{0}^{2},\varepsilon_{0}^{3})+\text{diag}(\alpha_{1}|\mathcal{E}_{1}|^{2},\alpha_{2}|\mathcal{E}_{2}|^{2},\alpha_{3}|\mathcal{E}_{3}|^{2})$
with $\varepsilon_{0}^{i}>0$. In this case, we have
$\mathcal{D}_{i}=(\varepsilon_{0}^{i}+\alpha_{i}|\mathcal{E}_{i}|^{2})\mathcal{E}_{i}$,
which admits inversion as
$\mathcal{E}_{i}=\psi_{i}(\mathcal{D}_{i})\mathcal{D}_{i}\text{ and
}\psi_{i}(0)=(\varepsilon_{0}^{i})^{-1}.$
This allows for verification of (78). Instead of starting with
$\varepsilon=\varepsilon(\mathcal{E})$, one can make the ansatz
(81)
$\psi(\mathcal{D})_{ij}=\varepsilon^{0}_{ij}+\alpha_{ij}\mathcal{D}_{i}\mathcal{D}_{j},$
for which it is easy to see that (78) requires symmetry of
$(\varepsilon^{0}_{ij})_{ij}$ and $(\alpha_{ij})_{ij}$. Then, by the implicit
function theorem, for small fields
(82) $\|\mathcal{E}\|_{L^{\infty}}+\|\mathcal{D}\|_{L^{\infty}}\leq\delta$
we can rewrite $\mathcal{D}=\varepsilon(\mathcal{E})\mathcal{E}$ with
$\varepsilon(\mathcal{E})=\psi(\mathcal{D}(\mathcal{E}))^{-1}$. Clearly,
$\varepsilon(\mathcal{E})$ is symmetric as an inverse of a symmetric matrix,
and the uniform anisotropy and ellipticity are true for small
$\|\mathcal{E}\|_{L^{\infty}}$. Taking $\delta$ smaller than in (82), if
necessary, specifies $\delta$ in (74).
By the assumptions of Theorem 1.5, the energy estimate
(83)
$\|u(t)\|_{H^{s}}\lesssim_{\delta}\|u(0)\|_{H^{s}}e^{C\int_{0}^{t}B(s)ds}$
is true for $\|u\|_{L_{t,x^{\prime}}^{\infty}}\leq\delta$. In the next step we
use Strichartz estimates to show a priori estimates for $s>2+\frac{7}{18}$.
Note that by Sobolev embedding (83) yields a priori estimates for
$s>\frac{5}{2}$.
###### Proposition 5.3.
Under the assumptions of Theorem 1.5, the a priori estimate
$\sup_{t\in[0,T]}\|u(t)\|_{H^{s}}\lesssim\|u(0)\|_{H^{s}}$
holds for $s>2+\frac{7}{18}$, $T=T(\|u_{0}\|_{H^{s}})$ and
$\|u_{0}\|_{H^{s}}\leq\delta$ for some $\delta>0$.
###### Proof.
Suppose that the smooth solution $u=(\mathcal{D},\mathcal{H})$ exists for
$[0,T_{0}]$. Let $0<T\leq T_{0}$. The proof follows from bootstrapping the
energy estimate (83)
$E^{s}(u(t))\lesssim_{A}e^{C\int_{0}^{T}B(s)ds}E^{s}(u(0))$
and the Strichartz estimate
(84)
$\|\partial_{x^{\prime}}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\lesssim\|\langle
D^{\prime}\rangle^{s}u\|_{L_{T}^{\infty}L_{x}^{2}}$
for $s\geq s_{0}>2+\frac{7}{18}$. The constant in the Strichartz estimate has
to be uniform provided that $T$,
$\|\partial_{x}u\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}$,
$\|u\|_{C^{\frac{8}{9}+\sigma}_{x}}$ are bounded and $s\geq s_{0}$.
We have to establish (84), for which we make use of commutator arguments.
These only apply after changing Maxwell equations to non-divergence form. Let
$\varepsilon(\mathcal{E})=\text{diag}(\varphi_{1}(\mathcal{E}_{1}),\varphi_{2}(\mathcal{E}_{2}),\varphi_{3}(\mathcal{E}_{3}))$
and denote
$\tilde{\varepsilon}(\mathcal{E})=\text{diag}(\tilde{\varepsilon}_{1}(\mathcal{E}_{1}),\tilde{\varepsilon}_{2}(\mathcal{E}_{2}),\tilde{\varepsilon}_{3}(\mathcal{E}_{3}))\quad\text{with
\
}\tilde{\varepsilon}_{i}(\mathcal{E})=\varphi_{i}(\mathcal{E}_{i})+\varphi_{i}^{\prime}(\mathcal{E}_{i})\mathcal{E}_{i}.$
Note that
$\partial_{x}(\varepsilon(\mathcal{E})\mathcal{E})=\tilde{\varepsilon}(\mathcal{E})\partial_{x}\mathcal{E}$.
For the proof of (84) we have to show
$\|\partial_{x^{\prime}}(\mathcal{E},\mathcal{H})\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\lesssim\|(\mathcal{E},\mathcal{H})\|_{L_{T}^{\infty}H^{s}_{x^{\prime}}}\quad\text{for
\ }s\geq s_{0}>\frac{3}{2}+\frac{8}{9}.$
Via a continuity argument, we shall control
$\|\langle
D^{\prime}\rangle^{1+\kappa}(\mathcal{E},\mathcal{H})\|_{L_{T}^{4+\beta}L_{x^{\prime}}^{q}}\lesssim\|(\mathcal{E},\mathcal{H})\|_{L_{T}^{\infty}H^{s}_{x^{\prime}}}\quad\text{for
\ }s\geq s_{0}>\frac{3}{2}+\frac{8}{9},$
$q$ large enough, and $(4+\beta,q)$ being a sharp Strichartz pair. The
slightly larger exponent in $L_{t}^{p}$ allows us to bring powers of $T$ into
play. The positive number $\kappa$ is chosen such that
$\|f\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim\|\langle
D^{\prime}\rangle^{\kappa}f\|_{L^{q}(\mathbb{R}^{3})}$ by Sobolev embedding.
We consider the Maxwell system
$\left\\{\begin{array}[]{rlrl}\partial_{t}(\tilde{\varepsilon}\mathcal{E})\\!\\!\\!\\!&=\nabla\times\mathcal{H},&\quad\nabla\cdot(\tilde{\varepsilon}\mathcal{E})\\!\\!\\!\\!&=\rho_{e},\\\
\partial_{t}\mathcal{H}\\!\\!\\!\\!&=-\nabla\times\mathcal{E},&\quad\nabla\cdot\mathcal{H}\\!\\!\\!\\!&=0,\end{array}\right.$
and denote
$\tilde{P}=\begin{pmatrix}-\partial_{t}(\tilde{\varepsilon}\cdot)&\nabla\times\\\
\nabla\times&\partial_{t}\end{pmatrix},\quad
v=(v_{1},v_{2}):\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{R}^{3}\times\mathbb{R}^{3}.$
We can choose $\kappa,\beta,\sigma>0$, $\gamma\in(0,\frac{1}{9})$, and
$q<\infty$ such that $W^{\kappa,q}_{x^{\prime}}\hookrightarrow
L^{\infty}_{x^{\prime}}$, $\frac{3}{2}+\frac{8}{9}+\gamma+\sigma\leq s_{0}$
and such that the proof of Theorem 1.1, see (55), yields
(85) $\|\langle
D^{\prime}\rangle^{-\hat{s}}v\|_{L_{T}^{4+\beta}L_{x^{\prime}}^{q}}\lesssim_{T_{0},\beta,C}\|v\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}+\|\tilde{P}v\|_{L_{T}^{2}L_{x^{\prime}}^{2}}+\|\nabla\cdot(\tilde{\varepsilon}v_{1})\|_{L_{T}^{2}L_{x^{\prime}}^{2}}$
for $\hat{s}+\kappa\geq s_{0}-1$, $\alpha=\frac{8}{9}+\gamma$ and
$C=\|\partial_{x}\tilde{\varepsilon}\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}+\|\tilde{\varepsilon}\|_{C_{x}^{\frac{8}{9}+\gamma}}$.
By Moser estimates, we have
$\|\tilde{\varepsilon}\|_{C^{\alpha}_{x}}\lesssim_{\|\mathcal{E}\|_{L^{\infty}}}\|\mathcal{E}\|_{C^{\alpha}_{x}}$.
To estimate the space-time Hölder regularity, write
$\|(\mathcal{E},\mathcal{H})\|_{C_{x}^{\alpha}}\lesssim\|(\mathcal{E},\mathcal{H})\|_{C^{\alpha}_{t}L^{\infty}_{x^{\prime}}}+\|(\mathcal{E},\mathcal{H})\|_{L_{t}^{\infty}C_{x^{\prime}}^{\alpha}}.$
We have
$\|(\mathcal{E},\mathcal{H})\|_{L_{t}^{\infty}C_{x^{\prime}}^{\alpha}}\lesssim\|(\mathcal{E},\mathcal{H})\|_{L_{t}^{\infty}H_{x^{\prime}}^{\frac{3}{2}+\alpha+\sigma}},\quad\|(\mathcal{E},\mathcal{H})\|_{C^{\alpha}_{t}L^{\infty}_{x^{\prime}}}\lesssim\|(\mathcal{E},\mathcal{H})\|_{L_{t}^{\infty}H_{x^{\prime}}^{\frac{3}{2}+\alpha+\sigma}}.$
The first estimate is immediate from Sobolev embedding. The latter estimate
follows for $\alpha\in\\{0,1\\}$ from Sobolev embedding and the equation. Then
we interpolate to obtain bounds in $C^{\alpha}L^{\infty}$ with
$\alpha\in(0,1)$.
We pass to non-divergence form in the Maxwell system setting
$\eta(x,D)=\begin{pmatrix}\tilde{\varepsilon}_{1}\partial_{1}&\tilde{\varepsilon}_{2}\partial_{2}&\tilde{\varepsilon}_{3}\partial_{3}\end{pmatrix}\quad\text{
and
}\quad\tilde{P}^{\prime}=\begin{pmatrix}-\tilde{\varepsilon}\partial_{t}&\nabla\times\\\
\nabla\times&\partial_{t}\end{pmatrix}.$
By Hölder’s inequality and distributing derivatives, (85) gives
$\|\langle
D^{\prime}\rangle^{-\tilde{s}}v\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\lesssim\|\langle
D^{\prime}\rangle^{-\hat{s}}v\|_{L_{T}^{4+\beta}L_{x^{\prime}}^{q}}\lesssim\|v\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}+\|\tilde{P}^{\prime}v\|_{L_{T}^{2}L_{x^{\prime}}^{2}}+\|\eta(x,D)v_{1}\|_{L_{T}^{2}L_{x^{\prime}}^{2}}.$
with $\tilde{s}=\hat{s}+\kappa\geq s_{0}-1$. We apply these inequalities to
$v=\langle D^{\prime}\rangle^{\tilde{s}+1}(\mathcal{E},\mathcal{H})$,
obtaining
$\displaystyle\|\langle
D^{\prime}\rangle(\mathcal{E},\mathcal{H})\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}$
$\displaystyle\lesssim\|\langle
D^{\prime}\rangle^{\tilde{s}+1}(\mathcal{E},\mathcal{H})\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}+\|\tilde{P}^{\prime}(\langle
D^{\prime}\rangle^{\tilde{s}+1}(\mathcal{E},\mathcal{H}))\|_{L_{T}^{2}L_{x^{\prime}}^{2}}$
(86) $\displaystyle\quad+\|\eta(x,D)(\langle
D^{\prime}\rangle^{\tilde{s}+1}\mathcal{E})\|_{L_{T}^{2}L_{x^{\prime}}^{2}},$
$\displaystyle\|\langle
D^{\prime}\rangle^{1+\kappa}(\mathcal{E},\mathcal{H})\|_{L_{T}^{4+\beta}L_{x^{\prime}}^{q}}$
$\displaystyle\lesssim\|\langle
D^{\prime}\rangle^{\tilde{s}+1}(\mathcal{E},\mathcal{H})\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}+\|\tilde{P}^{\prime}(\langle
D^{\prime}\rangle^{\tilde{s}+1}(\mathcal{E},\mathcal{H}))\|_{L_{T}^{2}L_{x^{\prime}}^{2}}$
(87) $\displaystyle\quad+\|\eta(x,D)(\langle
D^{\prime}\rangle^{\tilde{s}+1}\mathcal{E})\|_{L_{T}^{2}L_{x^{\prime}}^{2}}.$
To conclude the argument, we prove commutator estimates for the last two
terms. Here we need to have operators in non-divergence form. For solutions to
(73), we note that $\tilde{P}^{\prime}(\mathcal{E},\mathcal{H})$ and
$\eta(x,D)\mathcal{E}$ vanish since
$\partial_{t}(\varepsilon\mathcal{E})=\tilde{\varepsilon}\partial_{t}\mathcal{E}$
and $\nabla\cdot(\varepsilon\mathcal{E})=\eta(x,D)\mathcal{E}$.
We turn to the proof of the first commutator estimate and compute
$\|\tilde{P}^{\prime}(\langle
D^{\prime}\rangle^{\tilde{s}+1}(\mathcal{E},\mathcal{H}))\|_{L_{T}^{2}L_{x^{\prime}}^{2}}\lesssim\|\langle
D^{\prime}\rangle^{\tilde{s}+1}\tilde{P}^{\prime}(\mathcal{E},\mathcal{H})\|_{L_{T}^{2}L_{x^{\prime}}^{2}}+\|[\tilde{\varepsilon},\langle
D^{\prime}\rangle^{\tilde{s}+1}]\partial_{t}\mathcal{E}\|_{L_{T}^{2}L_{x^{\prime}}^{2}}.$
Here and below the implicit constants depend on
$\|(\mathcal{E},\mathcal{H})\|_{L^{\infty}_{x}}$. The Kato-Ponce commutator
estimate at fixed times and Hölder’s inequality imply
$\displaystyle\|[\tilde{\varepsilon},\langle
D^{\prime}\rangle^{\tilde{s}+1}]\partial_{t}\mathcal{E}\|_{L_{T}^{2}L_{x^{\prime}}^{2}}$
$\displaystyle\lesssim\|\langle
D^{\prime}\rangle^{\tilde{s}+1}\tilde{\varepsilon}\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}\|\partial_{t}\mathcal{E}\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}$
$\displaystyle\quad+\|\partial_{x}\tilde{\varepsilon}\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}\|\langle
D^{\prime}\rangle^{\tilde{s}}\partial_{t}\mathcal{E}\|_{L_{T}^{4}L_{x^{\prime}}^{2}}.$
By Moser estimates, we have
$\|\langle
D^{\prime}\rangle^{\tilde{s}+1}\tilde{\varepsilon}\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}\lesssim\|\langle
D^{\prime}\rangle^{\tilde{s}+1}\mathcal{E}\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}.$
For the second term we rewrite the Maxwell equation as
$\partial_{t}\mathcal{E}=\tilde{\varepsilon}^{-1}\nabla\times\mathcal{H}$ and
estimate by the fractional Leibniz rule
$\displaystyle\|\langle D^{\prime}\rangle^{\tilde{s}}$
$\displaystyle(\tilde{\varepsilon}^{-1}\nabla\times\mathcal{H})\|_{L_{T}^{4}L_{x^{\prime}}^{2}}$
$\displaystyle\lesssim\|\langle
D^{\prime}\rangle^{\tilde{s}}(\tilde{\varepsilon}^{-1})\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}\|\nabla\times\mathcal{H}\|_{L_{T}^{4}L_{x^{\prime}}^{\infty}}+\|\tilde{\varepsilon}^{-1}\|_{L_{x}^{\infty}}\|\langle
D^{\prime}\rangle^{\tilde{s}}\nabla\times\mathcal{H}\|_{L_{T}^{4}L_{x^{\prime}}^{2}}.$
Using again the fractional Leibniz rule, Moser estimates and the Maxwell
system, we derive (with $w=(\mathcal{E},\mathcal{H})$)
$\displaystyle\|\tilde{P}^{\prime}(\langle
D^{\prime}\rangle^{\tilde{s}+1}w)\|_{L_{T}^{2}L_{x^{\prime}}^{2}}$
$\displaystyle\lesssim\|\langle
D^{\prime}\rangle^{\tilde{s}+1}w\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}\|\langle
D^{\prime}\rangle w\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}+\|\langle
D^{\prime}\rangle w\|_{L^{4}_{T}L_{x^{\prime}}^{\infty}}$ (88)
$\displaystyle\qquad\cdot\big{(}\|\langle
D^{\prime}\rangle^{\tilde{s}}w\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}\|\langle
D^{\prime}\rangle w\|_{L^{4}_{T}L_{x^{\prime}}^{\infty}}+\|\langle
D^{\prime}\rangle^{\tilde{s}+1}w\|_{L_{T}^{4}L_{x^{\prime}}^{2}}\big{)}.$
We turn to the last term in (86), which leads to
$\|\eta(x,D)(\langle
D^{\prime}\rangle^{\tilde{s}+1}\mathcal{E})\|_{L_{T}^{2}L_{x^{\prime}}^{2}}\leq\|\langle
D^{\prime}\rangle^{\tilde{s}+1}(\eta(x,D)\mathcal{E})\|_{L_{T}^{2}L_{x^{\prime}}^{2}}+\|[\tilde{\varepsilon}^{i},\langle
D^{\prime}\rangle^{\tilde{s}+1}]\partial_{i}\mathcal{E}\|_{L_{T}^{2}L_{x^{\prime}}^{2}}.$
By the Kato-Ponce commutator estimate at fixed times and Hölder’s inequality,
we obtain
(89) $\begin{split}\|[\tilde{\varepsilon}^{i},\langle
D^{\prime}\rangle^{\tilde{s}+1}]\partial_{i}\mathcal{E}\|_{L_{T}^{2}L_{x^{\prime}}^{2}}&\lesssim\|\langle
D^{\prime}\rangle^{\tilde{s}+1}\tilde{\varepsilon}\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}\|\partial_{i}\mathcal{E}\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}\\\
&\qquad+\|\partial\tilde{\varepsilon}^{i}\|_{L_{T}^{2}L_{x^{\prime}}^{\infty}}\|\langle
D^{\prime}\rangle^{\tilde{s}}\partial_{i}\mathcal{E}\|_{L_{T}^{\infty}L_{x^{\prime}}^{2}}.\end{split}$
This estimate can be handled as in (5).
We turn to the continuity argument. Let $F(T)=\|\langle
D^{\prime}\rangle^{1+\beta}(\mathcal{E},\mathcal{H})\|_{L_{T}^{4+\beta}L_{x^{\prime}}^{q}}$
and $E^{s}(T)=\sup_{t\in[0,T]}\|u(t)\|_{H^{s}}$ with
$s=\tilde{s}+1>2+\frac{7}{18}$. Inequalities (86)–(89) and (83) imply
$\left\\{\begin{array}[]{cl}F(T)\\!\\!\\!\\!&\lesssim
E^{s}(T)+T^{\frac{1}{4}}F(T)E^{s}(T)+T^{\frac{\beta}{4(4+\beta)}}F(T)^{2}E^{s}(T)+T^{\frac{1}{4}}E^{s}(T),\\\
E^{s}(T)\\!\\!\\!\\!&\lesssim
e^{T^{\frac{1}{4}}F(T)}\|u(0)\|_{H^{s}}.\end{array}\right.$
At this point a continuity argument allows us to choose
$T^{*}=T(\|u_{0}\|_{H^{s}})$ such that $F(T^{*})\lesssim
E^{s}(T^{*})\lesssim\|u(0)\|_{H^{s}}$. The proof is complete. ∎
We give the statement on the $L^{2}$-bound for differences of solutions.
###### Proposition 5.4.
Let $u^{i}=(\mathcal{E}^{i},\mathcal{H}^{i})$, $i=1,2$ be solutions to (73)
under the assumptions of Theorem 1.5 with finite $A$ and $B$, and set
$v=u^{1}-u^{2}$. Then the estimate
$\|v(t)\|^{2}_{L^{2}}\lesssim
e^{c(A)\int_{0}^{t}B(s^{\prime})ds^{\prime}}\|v(0)\|^{2}_{L^{2}}$
with
$A=\|u^{1}\|_{L_{x}^{\infty}}+\|u^{2}\|_{L_{x}^{\infty}}\text{ and
}B(s)=\|\partial_{x^{\prime}}u^{1}(s)\|_{L^{\infty}_{x^{\prime}}}+\|\partial_{x^{\prime}}u^{2}(s)\|_{L^{\infty}_{x^{\prime}}}$
holds true. Moreover, if $s>\frac{3}{2}+\frac{8}{9}$, there is a time
$T=T(\|u^{i}(0)\|_{H^{s}})$ such that $T$ is lower semicontinuous and
$\sup_{t\in[0,T]}\|v(t)\|_{L^{2}}\lesssim_{\|u^{i}(0)\|_{H^{s}}}\|v(0)\|_{L^{2}}.$
The proof is an obvious modification of the proof of [17, Proposition 6.2].
Lastly, continuous dependence is proved using frequency envelopes (cf. [17,
6]). We refer to [17, pp. 358ff] for the details. This finishes the proof of
Theorem 1.5.
## 6\. Existence and regularity of eigenvectors for self-adjoint matrices
In this section we prove regularity results for eigenvectors of differentiably
varying self-adjoint matrices.
### 6.1. Local existence and regularity
We start with local existence and regularity of eigenvectors of differentiably
varying self-adjoint matrices provided that the eigenvalue is simple. This is
well-known in the literature, but included for the sake of completeness.
###### Lemma 6.1.
Let $k,n,l\in\mathbb{N}$, and $U\subseteq\mathbb{R}^{k}$ be open. Suppose that
$A\in C^{l}(U;\mathbb{R}^{n\times n})$ and that $\lambda_{z}$ is a simple
eigenvalue of $A(z)$ for some $z\in U$. Then, there is
$V\subseteq\mathbb{R}^{k}$ open with $z\in V$ and $v\in
C^{l}(V;\mathbb{R}^{n})$, $\lambda\in C^{l}(V;\mathbb{R})$ such that
$\lambda(z)=\lambda_{z}$, $\|v\|_{2}=1$ and $A(x)v(x)=\lambda(x)v(x)$ for any
$x\in V$.
###### Proof.
The claim follows by applying the implicit function theorem (IFT) to
$F(x,v,\lambda)=\begin{pmatrix}(A(x)-\lambda)v\\\
\|v\|^{2}_{2}-1\end{pmatrix}.$
Note that by assumption there are exactly two vectors $w\in\mathbb{R}^{n}$
with $F(z,w,\lambda_{z})=0$. We compute
(90) $\partial_{(v,\lambda)}F(x,v,\lambda)=\begin{pmatrix}A(x)-\lambda&2v\\\
2v^{t}&0\end{pmatrix}.$
To apply the IFT, we prove for
$(u,\lambda)\in\ker(\partial_{(v,\lambda)}F(z,w,\lambda_{z}))$ that
$(u,\lambda)=0$. By the second line of (90), $\langle u,w\rangle=0$. By
multiplying $(A(z)-\lambda_{z})u-\lambda w=0$ with $(A(z)-\lambda_{z})$, we
find $(A(z)-\lambda_{z})^{2}u=0$. This implies $u=\alpha w$, which is easy to
see after changing to Jordan normal form. Hence, $u=0$ as $u\perp w$. Thus,
the IFT applies and yields eigenpairs as smooth as the matrix. ∎
We do not know how to generalize the argument to eigenspaces of higher
dimensions. Possibly, one can consider maps
$F:\mathbb{R}^{k}\times Gr(2,n)\to Gr(n),\quad F(x,E)=(A(x)-\lambda)E$
to at least show regular dependence of the eigenspaces.
### 6.2. Globalizing non-degenerate eigenpairs
It is unclear how to construct a mapping
$\tilde{F}:U\to(\mathbb{R}^{n})^{k}\in C^{l}$ from $F:U\to Gr(k,n)\in C^{l}$
in the general case. Indeed, we have the following counterexample:
###### Example 1.
We remark that globalization of the local solutions is not always possible as
this example
$A(\xi)=\begin{pmatrix}\xi_{1}&\xi_{2}\\\ \xi_{2}&-\xi_{1}\end{pmatrix}$
for $\xi\in\mathbb{S}^{1}$ shows. For $\|\xi\|_{2}=1$, $A(\xi)$ has the
eigenvalues $\pm 1$. We identify $\mathbb{R}^{2}\equiv\mathbb{C}$, parametrize
$\xi\in\mathbb{S}^{1}$ as $\xi=e^{i\varphi}$, denote $z\in\mathbb{C}$ and
$A(\xi)$ as mapping $z\mapsto e^{i\varphi}\bar{z}$ with $e^{i\varphi}$. Then,
denoting the eigenvector to $1$ by $z(\varphi)$, we have
$e^{i\varphi}\bar{z}=z(\varphi)\Rightarrow e^{i\varphi}=z^{2}(\varphi).$
But the square root cannot be continuously extended to the unit disk.
In case of non-degenerate eigenvalues on simply connected domains, we can glue
together the local solutions to obtain a global solution. We use the results
of Rheinboldt [14], whose terminology is repeated here for convenience.
Let $X$, $Y$ be topological spaces with the Hausdorff property. For
simplicity, we consider only relations $\phi\subseteq X\times Y$ with
$D(\phi)=X$.
###### Definition 6.2.
A relation $\phi\subseteq X\times Y$ is said to be a _local mapping relation_
, or to have the _local mapping property_ , if for each $(x_{0},y_{0})\in\phi$
there exist (relatively) open neighbourhoods $U(x_{0})$ of $x_{0}$ and
$V(y_{0})$ of $y_{0}$ such that the restriction
$\varphi=\phi\cap(U(x_{0})\times V(y_{0}))$ is a continuous mapping from
$U(x_{0})$ into $V(y_{0})$.
For $Q\subseteq X$, $\mathcal{P}(Q)$ denotes the set of all possible
continuous paths $p:[0,1]\to Q$. $p,q\in\mathcal{P}(Q)$ are called equal,
$p\equiv q$, if $p(\tau(t))=q(t)$, $t\in[0,1]$, where $\tau:[0,1]\to[0,1]$,
$\tau(0)=0$, $\tau(1)=1$ is a continuous, strictly monotone mapping.
In our context, we consider the local mapping relation relating
$x\in\mathbb{R}^{m}$ with the normalized eigenvectors $\|v(x)\|_{2}=1$, of
which there are exactly two if the eigenspace is one-dimensional. The local
continuity follows from Lemma 6.1.
To prove existence of global solutions derived from a local mapping relation,
Rheinboldt introduced the continuation and path-lifting property.
###### Definition 6.3.
A relation $\phi\subseteq X\times Y$ is said to have the continuation property
for the subset $\mathcal{P}_{X}\subseteq\mathcal{P}(X)$ if for any
$p\in\mathcal{P}_{X}$ and any continuous function $q:[0,\hat{t})\subseteq J\to
Y$ with $p(t)\phi q(t)$ for any $t\in[0,\hat{t})$ there exists a sequence
$(t_{k})\subseteq[0,\hat{t})$ with $\lim_{k\to\infty}t_{k}=\hat{t}$ such that
$\lim_{k\to\infty}q(t_{k})=\hat{y}$ and $p(\hat{t})\phi\hat{y}$.
Observe that for Example 1, the continuation property fails. We turn to the
second definition:
###### Definition 6.4.
A relation $\phi\subseteq X\times Y$ is said to have the path-lifting property
for a set $\mathcal{P}_{Y}\subseteq\mathcal{P}(X)$ if for any
$p\in\mathcal{P}_{X}$ and any $y_{0}\in\phi[p(0)]$, there exists a path
$q\in\mathcal{P}(Y)$ such that $q(0)=y_{0}$ and $p(t)\phi q(\tau(t))$. We call
$q$ a lifting of $p$ through $y_{0}$ and write $p\phi q$.
One of Rheinboldt’s main results is the equivalence of the path-lifting and
continuation property:
###### Theorem 6.5.
Let $\phi\subseteq X\times Y$ be a local mapping relation. Then $\phi$ has the
path-lifting property for $\mathcal{P}_{X}\subseteq\mathcal{P}(X)$ if and only
if $\phi$ has the continuation property for $\mathcal{P}_{X}$.
Next, we suppose that $X$ is $\mathcal{P}_{X}$-simply connected if it is path-
connected under $\mathcal{P}_{X}$ and if any two paths of $\mathcal{P}_{X}$
with the same endpoints are $\mathcal{P}_{X}$-homotopic. In this case, we have
the following global result.
###### Theorem 6.6.
Let $\phi\subseteq X\times Y$ be a local mapping relation with the path-
lifting property for $\mathcal{P}_{X}\subseteq\mathcal{P}(X)$ and suppose that
$X$ is $\mathcal{P}_{X}$-simply connected and locally $\mathcal{P}_{X}$-path-
connected. Then, there exists a family of continuous mappings $F_{\mu}:X\to Y$
such that $x\phi F_{\mu}(x)$ and $\phi[x]=\big{(}F_{\mu}(x)\big{)}_{\mu\in
M}$.
By considering local mapping relations for the eigenvectors, we would like to
apply the above theorem to find global parametrizations of eigenvectors. It
turns out that the following is sufficient to yield the path-lifting property:
###### Definition 6.7.
A relation $\phi\subseteq X\times Y$ with $D(\phi)=X$ is called a _covering
relation_ if for each $x\in X$ there exists an open neighbourhood $U(x)$ such
that $\phi[U(x)]=\bigcup_{\mu\in M}V_{\mu}$, where the $V_{\mu}$ are disjoint
open sets in $R(\phi)$ and for each $\mu\in M$ the restriction
$\varphi_{\mu}=\phi\cap(U(x)\times V_{\mu})$ is a continuous mapping from
$U(x)$ into $V_{\mu}$. $U(x)$ is called an admissible neighbourhood of $x$.
The above certainly applies to our eigenvector mappings as the normalized
eigenvectors are antipodal. Hence, the following theorem can be applied:
###### Theorem 6.8.
A covering relation $\phi\subseteq X\times Y$ is a local mapping relation with
the path-lifting property for $\mathcal{P}(X)$.
This guarantees the path-lifting property and by Theorem 6.5 the continuation
property. We can prove the following global result on eigenvector
parametrizations.
###### Theorem 6.9.
Let $U\subseteq\mathbb{R}^{k}$ be simply connected and $A\in
C^{l}(U;\mathbb{R}^{n\times n})$ such that for any $x\in U$, $A(x)$ is
symmetric. Suppose that there is $\lambda\in C(U;\mathbb{R})$ such that
$\lambda(x)$ is a simple eigenvalue of $A(x)$ for any $x\in U$. Then, there is
$F\in C^{l}(U;\mathbb{R}^{n})$ such that for any $x\in U$, $\|F(x)\|_{2}=1$
and $A(x)F(x)=\lambda(x)F(x)$.
###### Proof.
Lemma 6.1 yields $\lambda\in C^{l}(U;\mathbb{R})$. Moreover, Lemma 6.1 shows
that $\phi\subseteq U\times\mathbb{R}^{n}$ with $x\phi w(x)$ for $w(x)$ the
two normalized eigenvectors of $A(x)$ is a covering relation. Hence, Theorem
6.8 yields the path-lifting property and by Theorem 6.5 the continuation
property. Applying Theorem 6.6 finishes the proof. ∎
We remark that even in the case of higher dimensional eigenspaces, we have
global parametrizations $\lambda\in C^{1}(U;\mathbb{R})$ of the eigenvalues,
e.g., by Theorem 1.1 (E) in [13] for $l\geq 2$. Theorem 6.9 also yields that,
if $A\in C^{l}(U;\mathbb{R}^{n\times n})$, where $U$ is simply connected, such
that for any $x\in U$, $A(x)$ is symmetric with simple eigenvalues, we find
the eigenvalues to be given by functions $\lambda_{1},\ldots,\lambda_{n}\in
C^{l}(U;\mathbb{R})$ and corresponding eigenvectors given by
$v_{1},\ldots,v_{n}\in C^{l}(U;\mathbb{R}^{n})$.
## Acknowledgements
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) Project-ID 258734477 – SFB 1173.
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|
# Further Studies on Open Well-filtered Spaces
Chong Shen Xiaoyong Xi Dongsheng Zhao School of Science
Beijing University of Posts and Telecommunications
Beijing, China School of Mathematics and Statistics
Yancheng Terchers University
Yancheng, Jiangsu Province, China Mathematics and Mathematics Education,
National Institute of Education
Nanyang Technological University
1 Nanyang Walk, Singapore
###### Abstract
The open well-filtered spaces were introduced by Shen, Xi, Xu and Zhao to
answer the problem whether every core-compact well-filtered space is sober. In
the current paper we explore further properties of open well-filtered spaces.
One of the main results is that if a space is open well-filtered, then so is
its upper space (the set of all nonempty saturated compact subsets equipped
with the upper Vietoris topology). Some other properties on open well-filtered
spaces are also studied.
###### keywords:
well-filtered space, sober space, core-compact space, locally compact space,
open well-filtered space
††journal: Electronic Notes in Theoretical Informatics and Computer
Science††volume: 2††thanks: This work was supported by the National Natural
Science Foundation of China (1210010153, 12071188) and Jiangsu Provincial
Department of Education (21KJB110008).††thanks: Email:
<EMAIL_ADDRESS>Email<EMAIL_ADDRESS>Email:
<EMAIL_ADDRESS>
## 1 Introduction
The open well-filtered spaces were introduced in [6] and used to give an
alternative and more natural proof of the conjectured conclusion that every
core compact and well-filtered space is sober. In [6], the authors actually
proved a stronger conclusion: every core-compact and open well-filtered space
is sober, as every well-filtered space is open well-filtered and the converse
conclusion is not true in general. By [8] and [10], one knows that a space is
well-filtered if and only if its upper space is well-filtered. It is thus
natural to wonder whether a similar result holds for open well-filteredness.
In this paper we shall give a partial answer to this problem and prove that if
a space is open well-filtered then so is its upper space. A number of new
results on open well-filtered spaces will also be presented here.
## 2 Preliminaries
We first recall some basic definitions and results that will be used in the
paper.
Let $P$ be a poset. A nonempty subset $D$ of $P$ is _directed_ (resp.,
_filtered_) if every two elements of $D$ have an upper (resp., lower) bound in
$D$. A poset $P$ is a _directed complete poset_ , or a _dcpo_ for short, if
for any directed subset $D\subseteq P$, the supremum $\bigvee D$ exists.
For any subset $A$ of a poset $P$,
$\mathord{\uparrow}A=\\{y\in P:\exists x\in A,x\leq y\\},{\mbox{ and
}}\mathord{\downarrow}A=\\{y\in P:\exists x\in A,y\leq x\\}.$
In particular, for each $x\in X$, we write
$\mathord{\uparrow}x=\mathord{\uparrow}\\{x\\}$ and
$\mathord{\downarrow}x=\mathord{\downarrow}\\{x\\}$.
For $x,y\in P$, $x$ is _way-below_ $y$, denoted by $x\ll y$, if for any
directed subset $D$ of $P$ with $\bigvee D$ existing, $y\leq\bigvee D$ implies
$x\leq d$ for some $d\in D$. Denote
$\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadrightarrow$}}x=\\{y\in P:x\ll
y\\}$ and
$\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}x=\\{y\in P:y\ll
x\\}$. A poset $P$ is _continuous_ , if for any $x\in P$, the set
$\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}x$ is directed and
$x=\bigvee\mathord{\rotatebox[origin={c}]{90.0}{$\twoheadleftarrow$}}x$. A
continuous dcpo is also called a _domain_.
A subset $U$ of a poset $P$ is _Scott open_ if (i) $U=\mathord{\uparrow}U$ and
(ii) for any directed subset $D$ of $P$ with $\bigvee D$ existing, $\bigvee
D\in U$ implies $D\cap U\neq\emptyset$. All Scott open subsets of $P$ form a
topology on $P$, called the _Scott topology_ on $P$ and denote by $\sigma(P)$.
The space $\Sigma P=(P,\sigma(P))$ is called the _Scott space_ of $P$.
For a $T_{0}$ space $X$, the specialization order $\leq$ on $X$ is defined by
$x\leq y$ iff $x\in{\rm cl}_{X}(\\{y\\})$, where ${\rm cl}_{X}$ is the closure
operator of $X$. Clearly, ${\rm cl}_{X}(\\{y\\})=\mathord{\downarrow}y$.
In what follows, if no otherwise specified, the partial order on a $T_{0}$
space will mean the specialization order.
###### Remark 2.1.
[1, 2] For each poset $(P,\leq_{P})$, the specialization order on $\Sigma P$
coincides with $\leq_{P}$.
For any $T_{0}$ space $X$, we shall often use $\mathcal{O}(X)$ to denote the
topology of $X$ (the collection of all open sets of $X$). For any subset $A$
of $X$, the _saturation_ of $A$, denoted by $Sat_{X}(A)$, is defined by
$Sat_{X}(A)=\bigcap\\{U\in\mathcal{O}(X):A\subseteq U\\},$
or equivalently, $Sat_{X}(A)=\mathord{\uparrow}A$ with respect to the
specialization order (see [2, Proposition 4.2.9]). A subset $A$ of $X$ is
_saturated_ if $A=Sat_{X}(A)$.
For any $U,V\in\mathcal{O}(X)$, we write $U\ll V$ for that $U$ is way-below
$V$ in the poset $(\mathcal{O}(X),\subseteq)$. Using a similar proof to that
of the Alexander’s Subbase Lemma (see [2, Theorem 4.4.29]), we can obtain the
following result.
###### Lemma 2.2.
Let $X$ be a $T_{0}$ space, $\mathcal{S}$ be a subbase for $\mathcal{O}(X)$,
and $U,V\in\mathcal{O}(X)$. Then $U\ll V$ if and only if one can extract a
finite subcover of $U$ from any cover $\\{U_{i}:i\in I\\}\subseteq\mathcal{S}$
of $V$.
###### Definition 2.3.
[6] Let $X$ be a $T_{0}$ space.
1. (1)
A subfamily $\mathcal{F}\subseteq\mathcal{O}(X)$ is called _$\ll$ -filtered_,
denoted by $\mathcal{F}\subseteq_{flt}\mathcal{O}(X)$, if for any
$U_{1},U_{2}\in\mathcal{F}$, there exists $U_{3}\in\mathcal{F}$ such that
$U_{3}\ll U_{1}$ and $U_{3}\ll U_{2}$.
2. (2)
$X$ is called _open well-filtered_ if for each $\ll$-filtered family
$\mathcal{F}\subseteq\mathcal{O}(X)$ and $U\in\mathcal{O}(X)$,
$\bigcap\mathcal{F}\subseteq U$ implies that $V\subseteq U$ for some
$V\in\mathcal{F}.$
###### Proposition 2.4.
[6] If $X$ is an open well-filtered space and $\\{U_{i}:i\in I\\}$ is a
$\ll$-filtered family of nonempty open sets, then $\bigcap\\{U_{i}:i\in I\\}$
is a nonempty compact saturated set.
###### Definition 2.5.
[1, 2] A $T_{0}$ space $X$ is called _core-compact_ if for each $x\in X$ and
$U\in\mathcal{O}(X)$ such that $x\in U$, there exists $V\in\mathcal{O}(X)$
such that $x\in V\ll U$.
###### Remark 2.6.
[1, 2] A $T_{0}$ space $X$ is core-compact if and only if the poset
$(\mathcal{O}(X),\subseteq)$ is continuous.
###### Theorem 2.7.
[6] Every core-compact open well-filtered space is sober.
###### Definition 2.8.
[1, 2] A $T_{0}$ space $X$ is called _well-filtered_ if for any filtered
family $\\{Q_{i}:i\in I\\}$ of compact saturated subsets of $X$ and any open
set $U\subseteq X$, $\bigcap_{i\in I}Q_{i}\subseteq U$ implies
$Q_{i_{0}}\subseteq U$ for some $i_{0}\in I$.
###### Definition 2.9.
[1, 2] A nonempty subset $A$ of a topological space $X$ is called
_irreducible_ if for any closed sets $F_{1}$, $F_{2}$ of $X$, $A\subseteq
F_{1}\cup F_{2}$ implies $A\subseteq F_{1}$ or $A\subseteq F_{2}$. A $T_{0}$
space $X$ is called _sober_ , if for any irreducible closed set $F$ of $X$,
there is a (unique) point $x\in X$ such that $F={\rm cl}_{X}(\\{x\\})$.
The following result on irreducible sets in product spaces will be used in the
sequel.
###### Proposition 2.10.
[2, Proposition 8.4.7] Let $\\{X_{i}:i\in I\\}$ be a family of topological
spaces. The irreducible closed sets in $\prod_{i\in I}X_{i}$ are exactly the
sets of the form $\prod_{i\in I}C_{i}$, where each $C_{i}$ is an irreducible
closed set in $X_{i}$ ($i\in I$).
The relations among well-filtered spaces, open well-filtered spaces and sober
spaces are shown below:
soberwell-filteredopen well-filtered core-compact
The following lemma will be used in the sequel.
###### Lemma 2.11.
[6, Lemma 2.4] Let $X$ be a $T_{0}$ space, $\\{U_{i}:i\in I\\}$ be a
$\ll$-filtered family of open sets of $X$, and $F$ be a closed set of $X$. If
$F\cap U_{i}\not=\emptyset$ for all $i\in I$, then there is a minimal closed
set $F_{0}\subseteq F$ such that $F_{0}\cap U_{i}\not=\emptyset$ for all $i\in
I$. In addition, this $F_{0}$ is irreducible.
## 3 A new characterization for open well-filtered spaces
In [7, 9], a characterization for well-filtered spaces by means of KF-sets is
obtained. We now prove a similar characterization for open well-filtered
spaces.
###### Definition 3.1.
A nonempty subset $A$ of a $T_{0}$ space $X$ is called an _open well-filtered
set_ , or _OWF-set_ for short, if there exists a $\ll$-filtered family
$\\{U_{i}:i\in I\\}\subseteq\mathcal{O}(X)$ such that ${\rm cl}(A)$ is a
minimal closed set that intersects all $U_{i}$, $i\in I$.
Denote by $\mathsf{OWF}(X)$ the set of all closed OWF-subsets of $X$.
###### Remark 3.2.
1. (1)
A subset of a topological space is an OWF-set if and only if its closure is an
OWF-set.
2. (2)
By Lemma 2.11, every OWF-set is irreducible.
###### Theorem 3.3.
Let $X$ be a $T_{0}$ space. Then the following statements are equivalent:
1. (1)
$X$ is open well-filtered;
2. (2)
$\forall A\in\mathsf{OWF}(X)$, there exists a unique $x\in X$ such that
$A={\rm cl}(\\{x\\})$.
###### Proof 3.4.
(1) $\Rightarrow$ (2). Let $A\in\mathsf{OWF}(X)$. Then there exists
$\\{U_{i}:i\in I\\}\subseteq_{flt}\mathcal{O}(X)$ such that $A$ is a minimal
closed set that intersects all $U_{i}$, $i\in I$. Since $X$ is open well-
filtered, it follows that $\bigcap_{i\in I}U_{i}\cap A\neq\emptyset$. Choose
one $x\in\bigcap_{i\in I}U_{i}\cap A$. Then ${\rm cl}(\\{x\\})\subseteq A$,
and it is a closed set that intersects all $U_{i}$, $i\in I$. By the
minimality of $A$, we have that $A={\rm cl}(\\{x\\})$. The uniqueness of $x$
is determined by the $T_{0}$ separation of $X$.
(2) $\Rightarrow$ (1). Let $\\{U_{i}:i\in I\\}\subseteq_{flt}\mathcal{O}(X)$
and $U\in\mathcal{O}(X)$ such that $\bigcap_{i\in I}U_{i}\subseteq U$. We need
to show that $U_{i}\subseteq U$ for some $i\in I$. If, on the contrary, that
$U_{i}\nsubseteq U$ for all $i\in I$, then by Lemma 2.11, there exists a
minimal (irreducible) closed set $A\subseteq X\setminus U$ that intersects all
$U_{i}$, $i\in I$. Thus $A\in\mathsf{OWF}(X)$. By the assumption, there exists
a unique $x\in X$ such that $A={\rm cl}(\\{x\\})$. For each $i\in I$, since
${\rm cl}(\\{x\\})\cap U_{i}=A\cap U_{i}\neq\emptyset$, it follows that $x\in
U_{i}$, which implies that $x\in\bigcap_{i\in I}U_{i}\subseteq U$. Thus $x\in
U\cap A$, which contradicts $A\subseteq X\setminus U$. This contradiction
completes the proof.
###### Example 3.5.
Let $\mathbb{N}^{+}$ be the set of all positive integers, $\mathbb{N}^{+}_{\rm
cof}$ be the space of $\mathbb{N}^{+}$ with the co-finite topology (the open
sets are $\emptyset$ and all the complements of finite subsets of
$\mathbb{N}^{+}$), and let $\mathbb{N}^{+}_{\alpha}$ be the Alexandoff space
of $\mathbb{N}^{+}$ (the open sets are the upper subsets of $\mathbb{N}^{+}$
with the usual order of numbers). We have the following claims:
1. (c1)
It is trivial to check that each subset of $\mathbb{N}^{+}$ is compact in both
$\mathbb{N}^{+}_{\rm cof}$ and $\mathbb{N}^{+}_{\alpha}$. We then deduce that
$U\ll V$ iff $U\subseteq V$ for any open sets $U,V$ in $\mathbb{N}^{+}_{\rm
cof}$ or $\mathbb{N}^{+}_{\alpha}$.
2. (c2)
Neither $\mathbb{N}^{+}_{cof}$ nor $\mathbb{N}^{+}_{\alpha}$ is open well-
filtered.
For each $n\in\mathbb{N}^{+}$,
$U_{n}=\mathbb{N}^{+}\setminus\\{1,2,3,\ldots,n\\}$ is a nonempty open set in
both $\mathbb{N}^{+}_{\rm cof}$ and $\mathbb{N}^{+}_{\alpha}$. From (c1), it
follows that the family $\\{U_{n}:n\in\mathbb{N}^{+}\\}$ is $\ll$-filtered,
but $\bigcap_{n\in\mathbb{N}^{+}}U_{n}=\emptyset$. By Proposition 2.4, we
obtain (c2).
The following example shows that a saturated subspace of an open well-filtered
space need not be open well-filtered.
###### Example 3.6.
Let $\mathbb{J}=\mathbb{N}^{+}\times(\mathbb{N}^{+}\cup\\{\omega\\})$ be the
Johnstone’s dcpo [2, 3], which is ordered by
$(m,n)\leq(m^{\prime},n^{\prime})$ iff either $m=m^{\prime}$ and $n\leq
n^{\prime}$, or $n^{\prime}=\omega$ and $n\leq m^{\prime}$ (refer to Figure
1).
Figure 1: The Johnstone’s dcpo $\mathbb{J}$
We have the following conclusions.
1. (r1)
$\Sigma\mathbb{J}$ is open well-filtered.
Note that $\forall U,V\in\sigma(\mathbb{J})$, $U\ll V$ iff $U=\emptyset$ (see
[2, Exercise 5.2.15]), which implies that each $\ll$-filtered family
$\mathcal{F}$ of $\sigma(\mathbb{J})$ is equal to $\\{\emptyset,U\\}$, where
$U$ is an arbitrary Scott open set in $\mathbb{J}$. This means there exists no
OWF-set in $\Sigma\mathbb{J}$. By Theorem 3.3, we deduce that
$\Sigma\mathbb{J}$ is open well-filtered.
2. (r2)
The set of maximal points $\mathbb{N}^{+}\times\\{\omega\\}$, as a saturated
subspace of $\Sigma\mathbb{J}$, is homeomorphic to $\mathbb{N}^{+}_{cof}$ of
Example 3.5, and thus is not open well-filtered.
From (r1), we have that $\mathord{\downarrow}x$ (which is exactly the closure
of $\\{x\\}$ in $\Sigma\mathbb{J}$) is not an OWF-set for each $x\in J$. We
then deduce that the closure of singletons need not be OWF-sets. From (r2), it
follows that the saturated subspace of an open well-filtered spaces need not
be open well-filtered.
###### Remark 3.7.
From Example 3.6, we deduce that if each $\ll$-filtered family of open sets in
a space $X$ contains the empty set, then $X$ must be open well-filtered.
The following example shows that neither the closed subspace nor the retract
of an open well-filtered space is open well-filtered in general.
###### Example 3.8.
[6, Example 4.13] Let $P=\mathbb{J}\cup\mathbb{N}^{+}$, where $\mathbb{J}$ is
the Johnstone’s dcpo. For any $x,y\in P$, define $x\leq y$ if one of the
following conditions holds (refer to Figure 2):
1. (i)
$x,y\in\mathbb{N}^{+}$ and $x\leq y$ in $\mathbb{N}^{+}$ with the usual
ordering;
2. (ii)
$x,y\in\mathbb{J}$ and $x\leq y$ in $\mathbb{J}$;
3. (iii)
$x\in\mathbb{N}^{+}$ and $y=(x,\omega)$.
Figure 2: The poset $P$ of Example 3.8
We have the following conclusions.
1. (1)
$\Sigma P$ is open well-filtered (see [6, Example 4.13] for details).
2. (2)
$\mathbb{N}^{+}$ is a Scott closed subset of $P$, and as a subspace of $P$, is
homeomorphic to $\mathbb{N}^{+}_{\alpha}$, hence is not open well-filtered (by
Example 3.5 (2)).
3. (3)
$\mathbb{N}^{+}$, as a subspace of $P$, is a retract of $P$.
Let $e:\mathbb{N}^{+}\longrightarrow P$ be the identity embedding, and define
$r:P\longrightarrow\mathbb{N}^{+}$ as follows: $\forall x\in P$, $\forall
n\in\mathbb{N}^{+}$, $r(x)=n$ iff
$\left\\{\begin{array}[]{lll}x\in\mathord{\downarrow}(1,\omega),&\mbox{ when
}n=1;\\\
x\in\mathord{\downarrow}(n,\omega)\setminus\mathord{\downarrow}(n-1,\omega),&\mbox{
when }n\geq 2.\end{array}\right.$
Since $\mathbb{N}^{+}$ is a closed set in $\Sigma P$, we have that $e$ is
continuous, and note that
$r^{-1}(\mathord{\downarrow}n)=\mathord{\downarrow}(n,\omega)$ is Scott closed
for each $n\in\mathbb{N}^{+}$, which implies that $r$ is a continuous mapping.
It is clear that the composition $r\circ e$ is the identity mapping on
$\mathbb{N}^{+}$. Thus (3) holds.
From above (1)–(3), we deduce that the closed subspace or the retract of an
open well-filtered space need not be open well-filtered.
###### Proposition 3.9.
Let $X$ be a core-compact space. Then every irreducible set in $X$ is an OWF-
set.
###### Proof 3.10.
Suppose that $A$ is an irreducible subset of $X$. Let
$\mathcal{F}=\\{U\in\mathcal{O}(X):A\cap U\neq\emptyset\\}.$
_Claim 1:_ $\mathcal{F}\subseteq_{flt}\mathcal{O}(X)$.
Let $U_{1},U_{2}\in\mathcal{F}$. Then $A\cap U_{1}\neq\emptyset$ and $A\cap
U_{2}\neq\emptyset$, implying that $A\cap U_{1}\cap U_{2}\neq\emptyset$. Take
$x\in A\cap U_{1}\cap U_{2}$. Since $X$ is core-compact, there exists
$U_{3}\in\mathcal{O}(X)$ such that $x\in U_{3}\ll U_{1}\cap U_{2}$. Note that
$x\in U_{3}\cap A\neq\emptyset$, implying that $U_{3}\in\mathcal{F}$. Thus
$\mathcal{F}$ is $\ll$-filtered.
_Claim 2:_ ${\rm cl}(A)$ is a minimal closed set that intersects all members
of $\mathcal{F}$.
Suppose $B$ is a closed set such that $B\subseteq{\rm cl}(A)$ and $B\cap
U\neq\emptyset$ for all $U\in\mathcal{F}$. We need to prove ${\rm
cl}(A)\subseteq B$. Otherwise, ${\rm cl}(A)\nsubseteq B$, which implies
$A\cap(X\setminus B)\neq\emptyset$. Thus $X\setminus B\in\mathcal{F}$, which
contradicts that $B$ intersects all members of $\mathcal{F}$. Therefore, ${\rm
cl}(A)=B$.
All this shows that ${\rm cl}(A)\in\mathsf{OWF}(X)$.
The following result is immediate by using Proposition 3.9.
###### Corollary 3.11.
[4, 9] Every core-compact open well-filtered space is sober.
The following example shows that the continuous image of an OWF-set need not
be an OWF-set.
###### Example 3.12.
Let $X$ be the Alexandroff space of the Johnstone’s dcpo $\mathbb{J}$ (whose
open sets are the upper sets), and let $f:X\longrightarrow\Sigma\mathbb{J}$ be
the identity mapping. Clearly, $f$ is a continuous mapping. Note that for any
$x\in\mathbb{J}$, since $X$ is locally compact (hence core-compact), by
Proposition 3.9, $\mathord{\downarrow}x$ is an OWF-set in $X$, but it is not
an OWF-subset of $\Sigma\mathbb{J}$ by Example 3.6 (r1).
For a $T_{0}$ space $X$ and $Y\in\mathcal{O}(X)$, if $U,V\in\mathcal{O}(Y)$,
then we have that $U,V\in\mathcal{O}(X)$, and that $U\ll V$ in
$(\mathcal{O}(Y),\subseteq)$ if and only if $U\ll V$ in
$(\mathcal{O}(X),\subseteq)$. Using this fact, one can prove the following
result easily.
###### Proposition 3.13.
Every open subspace of an open well-filtered space is also open well-filtered.
###### Definition 3.14.
[2] A $T_{0}$ space $X$ is called _core-coherent_ if for any
$U,V,W\in\mathcal{O}(X)$, $U\ll V$ implies that $U\cap W\ll V\cap W$.
Example 3.6 shows that singletons need not be OWF-sets. As a remedy, we have
the following result.
###### Theorem 3.15.
Let $X$ be a core-coherent space. Then the following statements are
equivalent:
1. (1)
$X$ is core-compact;
2. (2)
all irreducible sets in $X$ are OWF-sets;
3. (3)
the singletons are OWF-sets.
###### Proof 3.16.
By Proposition 3.9, that (1) $\Rightarrow$ (2) is clear, and since singletons
are irreducible, we obtain that (2) $\Rightarrow$ (3). Now we prove (3)
$\Rightarrow$ (1).
Let $x\in X$ and $U$ be an open neighborhood of $x$. Since $\\{x\\}$ is an
OWF-set, there exists a $\ll$-filtered family $\\{U_{i}:i\in
I\\}\subseteq\mathcal{O}(X)$ such that ${\rm cl}(\\{x\\})$ is a minimal closed
set that intersects all $U_{i}$, $i\in I$. Fix an $i_{0}\in I$. It follows
that $x\in U_{i_{0}}$. Then there exists ${i_{1}}\in I$ such that $x\in
U_{i_{1}}\ll U_{i_{0}}$. Since $X$ is core-coherent, it holds that $x\in U\cap
U_{i_{1}}\ll U\cap U_{i_{0}}\subseteq U$. Therefore, $X$ is core-compact.
It is easy to verify the following lemma.
###### Lemma 3.17.
Let $f:X\longrightarrow Y$ be a continuous open mapping between $T_{0}$
spaces, and $U,V\in\mathcal{O}(X)$. If $U\ll V$, then $f(U)\ll f(V)$.
Regarding the product spaces, we are still not able to prove that the product
of two open well-filtered spaces is a open well-filtered space. Here is a
result for some special spaces.
###### Theorem 3.18.
For each $T_{0}$ space $X$, the product $X\times\Sigma\mathbb{J}$ is open
well-filtered.
###### Proof 3.19.
Let $U,V\in\mathcal{O}(X\times\Sigma\mathbb{J})$ such that $U\ll V$. We prove
that $U=\emptyset$. Note that the projection $p_{2}$ is a continuous open
mapping, so $p_{2}(U),p_{2}(V)\in\sigma(\mathbb{J})$ and by Lemma 3.17,
$p_{2}(U)\ll p_{2}(V)$. Thus $p_{2}(U)=\emptyset$, which implies that
$U=\emptyset$. By Remark 3.7, $X\times\Sigma\mathbb{J}$ is an open well-
filtered space.
The above theorem indicates that the open well-filteredness of the product of
spaces does not imply the open well-filteredness of the factor spaces. In the
following, we will show that the open well-filteredness of the product of
finite $T_{0}$ spaces implies that one of the factor spaces is open well-
filtered.
It also is trivial to verify the following lemma.
###### Lemma 3.20.
Let $\\{X_{k}:1\leq k\leq n\\}$ be a finite family of $T_{0}$ spaces, and
$U_{k},V_{k}\in\mathcal{O}(X)$ such that $U_{k}\ll V_{k}$ for $1\leq k\leq n$.
Then $\prod_{1\leq k\leq n}U_{k}\ll\prod_{1\leq k\leq n}V_{k}$.
###### Theorem 3.21.
Let $\\{X_{k}:1\leq k\leq n\\}$ be a finite family of $T_{0}$ spaces, and $X$
be their product space. If $X$ is an open well-filtered space, then there is
$k_{0}$ ($1\leq k_{0}\leq n$) such that $X_{k_{0}}$ is open well-filtered.
###### Proof 3.22.
Suppose on the contrary that every $X_{k}$ is not open well-filtered, $1\leq
k\leq n$. Then there exist a $\ll$-filtered family
$\mathcal{F}_{k}\subseteq\mathcal{O}(X_{k})$ and an open set $O_{k}$ in
$X_{k}$ such that $\bigcap\mathcal{F}_{k}\subseteq O_{k}$, but $U\nsubseteq
O_{k}$ for any $U\in\mathcal{F}_{k}$. Define
$\mathcal{F}=\left\\{\prod_{1\leq k\leq n}U_{k}:\forall
k,U_{k}\in\mathcal{F}_{k}\right\\}.$
By Lemma 3.20, one can deduce that $\mathcal{F}$ is a $\ll$-filtered family of
$\mathcal{O}(X)$ such that $\bigcap\mathcal{F}\subseteq\prod_{1\leq k\leq
n}O_{k}\in\mathcal{O}(X)$. Since $X$ is open well-filtered, there exists
$\prod_{1\leq k\leq n}U_{k}\in\mathcal{F}$ (i.e., $U_{k}\in\mathcal{F}_{k}$
for $1\leq k\leq n$) such that $\prod_{1\leq k\leq
n}U_{k}\subseteq\prod_{1\leq k\leq n}O_{k}$. Note that each $U_{k}$ ($1\leq
k\leq n$) is not empty, which follows that $U_{k}\subseteq O_{k}$ for $1\leq
k\leq n$, a contradiction.
## 4 Upper spaces and open well-filteredness
In this section, we prove that if a space $X$ is open well-filtered then its
upper space (or the Smyth power space) is also open well-filtered. The proof
here makes use of a technique employed in [5].
For any topological space $X$, we use $\mathcal{D}(X)$ to denote the set of
all nonempty compact saturated subsets of $X$. The _upper Vietoris topology_
on $\mathcal{D}(X)$ is the topology that has $\\{\Box U:U\in\mathcal{O}(X)\\}$
as a base, where $\Box U=\\{K\in\mathcal{D}(X):K\subseteq U\\}$. The set
$\mathcal{D}(X)$ equipped with the upper Vietoris topology is called the
_upper space_ or _Smyth power space_ of $X$. Note that
$\\{\diamondsuit\,F:X\setminus F\in\mathcal{O}(X)\\}$ is a base of the co-
upper Vietoris topology, where $\diamondsuit\,F=\\{K\in\mathcal{D}(X):K\cap
F\not=\emptyset\\}$.
###### Remark 4.1.
Let $X$ be a $T_{0}$ space, and $U,U_{1},U_{2}\in\mathcal{O}(X)$.
1. (1)
$\Box U_{1}\subseteq\Box U_{2}$ if and only if $U_{1}\subseteq U_{2}$.
2. (2)
For any $x\notin U$, $\diamondsuit{\rm cl}(\\{x\\})\cap\Box U=\emptyset$.
3. (3)
If $\Box U\subseteq\Box U_{1}\cup\Box U_{2}$, then $\Box U\subseteq\Box U_{1}$
or $\Box U\subseteq\Box U_{2}$ [5].
We now state and prove the main result in this section.
###### Theorem 4.2.
For any open well-filtered space $X$, the upper space $\mathcal{D}(X)$ is open
well-filtered.
###### Proof 4.3.
Assume that $X$ is an open well-filtered space.
Let $\\{\mathcal{U}_{i}:i\in I\\}$ be a $\ll$-filtered family of open sets
$\mathcal{U}_{i}$ in $\mathcal{D}(X)$ and
$\bigcap\\{\mathcal{U}_{i}:i\in I\\}\subseteq\mathcal{U}$
for an open set $\mathcal{U}$ in $\mathcal{D}(X)$.
For each $i\in I$, let $\mathcal{U}_{i}=\bigcup\\{\Box U_{i,t}:t\in T_{i}\\}$,
where $U_{i,t}\in\mathcal{O}(X)$.
Assume that for each $i\in I$, $\mathcal{U}_{i}\not\subseteq\mathcal{U}$. Let
$\mathcal{C}=\mathcal{D}(X)\setminus\mathcal{U}$. Then $\mathcal{C}$ is a
closed set in $\mathcal{D}(X)$ such that
$\mathcal{C}\cap\mathcal{U}_{i}\neq\emptyset$ for all $i\in I$. By Lemma 2.11,
there is a minimal closed set $\mathcal{C}_{0}\subseteq\mathcal{C}$ which also
has a nonempty intersection with every $\mathcal{U}_{i}$.
For each $i\in I$, let
$\hat{\mathcal{U}}_{i}=\bigcup\\{\Box U_{i,t}:\Box
U_{i,t}\cap\mathcal{C}_{0}\not=\emptyset,t\in T_{i}\\}.$
_Fact 1:_ If
$\mathcal{U}_{i_{1}}\ll\mathcal{U}_{i_{2}}\ll\mathcal{U}_{i_{3}}$, then
$\hat{\mathcal{U}}_{i_{1}}\ll\hat{\mathcal{U}}_{i_{3}}$. Therefore,
$\\{\hat{\mathcal{U}}_{i}:i\in I\\}$ is $\ll$-filtered.
We just need to verify the first statement. To see this, let
$\\{\mathcal{V}_{l}:l\in L\\}$ be a directed open cover of
$\hat{\mathcal{U}}_{i_{3}}$. Then
$\\{\mathcal{V}_{l}\cup(\mathcal{D}(X)\setminus\mathcal{C}_{0}):l\in L\\}$ is
a directed open cover of $\mathcal{U}_{i_{3}}$. By the assumption, there is a
$l_{0}\in L$ such that
$\mathcal{U}_{i_{2}}\subseteq\mathcal{V}_{l_{0}}\cup(\mathcal{D}(X)\setminus\mathcal{C}_{0})$.
By $\mathcal{U}_{i_{1}}\ll\mathcal{U}_{i_{2}}$ and the structure of the upper
Vietoris topology, there exist
$\Box W_{1},\Box W_{2},\cdots,\Box W_{n}\mbox{ contained in
}\mathcal{V}_{l_{0}},$ $\Box G_{1},\Box G_{2},\cdots,\Box G_{m}\mbox{
contained in }\mathcal{D}(X)\setminus\mathcal{C}_{0}$
such that
$\mathcal{U}_{i_{1}}\subseteq\bigcup\\{\Box W_{k}:1\leq k\leq
n\\}\cup\bigcup\\{\Box G_{h}:1\leq h\leq m\\}.$
Note that $\hat{\mathcal{U}}_{i_{1}}\subseteq\mathcal{U}_{i_{1}}$. For each
$t\in T_{i_{1}}$ such that $\Box U_{i_{1},t}\cap\mathcal{C}_{0}\neq\emptyset$,
by Remark 4.1, $\Box U_{i_{1},t}\subseteq\Box W_{k}$ for some $1\leq k\leq n$,
or $\Box U_{i_{1},t}\subseteq\Box G_{h}$ for some $1\leq h\leq m$. But $\Box
G_{h}\cap\mathcal{C}_{0}=\emptyset$, hence $\Box U_{i_{1},t}\subseteq\Box
W_{k}\subseteq\mathcal{V}_{l_{0}}$ for some $k$. It then follows that
$\hat{\mathcal{U}}_{i_{1}}\subseteq\mathcal{V}_{l_{0}}.$
Therefore $\hat{\mathcal{U}}_{i_{1}}\ll\hat{\mathcal{U}}_{i_{3}}$ holds.
For each $i\in I$, let
$U_{i}=\bigcup\\{U_{i,t}:\Box U_{i,t}\cap\mathcal{C}_{0}\not=\emptyset\\}.$
_Fact 2:_ If $\hat{\mathcal{U}}_{i_{1}}\ll\hat{\mathcal{U}}_{i_{2}}$, then
$U_{i_{1}}\ll U_{i_{2}}.$ Hence $\\{U_{i}:i\in I\\}$ is a $\ll$-filtered
family of open sets in $X$.
As a matter of fact, it is easy to see that if $\\{W_{j}:j\in
J\\}\subseteq\mathcal{O}(X)$ is a directed open cover of $U_{i_{2}}$, then
$\\{\Box W_{j}:j\in J\\}$ is a directed open cover of
$\hat{\mathcal{U}}_{i_{2}}$, hence there is $j_{0}\in J$ such that
$\hat{\mathcal{U}}_{i_{1}}\subseteq\Box W_{j_{0}}$, thus $U_{i_{1}}\subseteq
W_{j_{0}}$.
Let
$K=\bigcap\\{U_{i}:i\in I\\}.$
By Proposition 2.4, $K$ is a nonempty saturated compact set, that is
$K\in\mathcal{D}(X)$.
_Fact 3:_ $K\not\in\mathcal{U}$.
Indeed, if $K\in\mathcal{U}$, then there is an open set $E$ of $X$ such that
$K\in\Box E\subseteq\mathcal{U}$. Then $K=\bigcap\\{U_{i}:i\in I\\}\subseteq
E$, and since $X$ is open well-filtered, there is $i_{0}$ such that
$U_{i_{0}}\subseteq E$.
Choose one $U_{i_{0},t_{0}}$ such that $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\not=\emptyset$. Then
$U_{i_{0},t_{0}}\subseteq U_{i_{0}}\subseteq E$, and it follows that
$\emptyset\neq\Box U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\subseteq\Box
E\cap\mathcal{C}\subseteq\mathcal{U}\cap\mathcal{C}=\emptyset,$
a contradiction.
_Fact 4:_ $K\in\bigcap\\{\hat{\mathcal{U}}_{i}:i\in I\\}$.
As a matter of fact, if the statement is not true, then there is $i_{0}\in I$
such that $K\not\in\hat{\mathcal{U}}_{i_{0}}$. By the definition of
$\hat{\mathcal{U}}_{i_{0}}$. Take any $U_{i_{0},t_{0}}$ with $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\not=\emptyset$. Then $K\not\subseteq
U_{i_{0},t_{0}}$. Take an $e\in K\setminus U_{i_{0},t_{0}}$ and let
$\mathcal{F}=\diamondsuit{\rm cl}(\\{e\\})$. Then by Remark 4.1
$\mathcal{F}\cap\Box U_{i_{0},t_{0}}=\emptyset$.
We show that $\mathcal{C}_{0}\cap\mathcal{F}\cap\mathcal{U}_{i}\not=\emptyset$
for all $i\in I$. For this, it is enough to show that for any $i\in I$, there
is a $t_{i}\in T_{i}$ such that
$\Box U_{i,t_{i}}\cap\mathcal{F}\cap\mathcal{C}_{0}\not=\emptyset.$
If not, there exists $i_{1}\in I$ such that
$\Box
U_{i_{1},t}\subseteq(\mathcal{D}(X)\setminus\mathcal{F})\cup(\mathcal{D}(X)\setminus\mathcal{C}_{0})$
for all $t\in T_{i_{1}}$. Choose $i_{2}$ such that
$\hat{\mathcal{U}}_{i_{2}}\ll\hat{\mathcal{U}}_{i_{1}}$. Then there are
$\Box V^{1}_{1},\Box V^{1}_{2},\cdots,\Box V^{1}_{m}\mbox{ contained in
}\mathcal{D}(X)\setminus\mathcal{C}_{0},$
and
$\Box V^{2}_{1},\Box V^{2}_{2},\cdots,\Box V^{2}_{n}\mbox{ contained in
}\mathcal{D}(X)\setminus\mathcal{F}$
such that
$\hat{\mathcal{U}}_{i_{2}}\subseteq\bigcup\\{\Box V^{1}_{k}:1\leq k\leq
m\\}\cup\bigcup\\{\Box V^{2}_{l}:1\leq l\leq n\\}.$
By the definition of $K$, we have that $e\in K\subseteq U_{i_{2}}$, and then
there is $t^{\prime}\in T_{i_{2}}$ such that $e\in U_{i_{2},t^{\prime}}$ where
$\Box U_{i_{2},t^{\prime}}\cap\mathcal{C}_{0}\not=\emptyset$. Thus
$Sat_{X}(\\{e\\})\in\mathcal{F}\cap\Box U_{i_{2},t^{\prime}}\neq\emptyset$.
Now $\Box
U_{i_{2},t^{\prime}}\subseteq\hat{\mathcal{U}}_{i_{2}}\subseteq\bigcup\\{\Box
V^{1}_{k}:1\leq k\leq m\\}\cup\bigcup\\{\Box V^{2}_{l}:1\leq l\leq n\\}$. By
Remark 2.1, we have that
$\Box U_{i_{2},t^{\prime}}\subseteq\Box V^{1}_{k}\mbox{ for some }1\leq k\leq
m,\mbox{ or }\Box U_{i_{2},t^{\prime}}\subseteq\Box V^{2}_{l}\mbox{ for some
}1\leq l\leq n.$
But for any $k$, since $\Box
U_{i_{2},t^{\prime}}\cap\mathcal{C}_{0}\neq\emptyset$ and $\Box
V^{1}_{k}\cap\mathcal{C}_{0}=\emptyset$, it follows that $\Box
U_{i_{2},t^{\prime}}\nsubseteq\Box V^{1}_{k}$, and for any $l$, since $\Box
U_{i_{2},t^{\prime}}\cap\mathcal{F}\neq\emptyset$ and $\Box
V^{2}_{l}\cap\mathcal{F}=\emptyset$, it follows that $\Box
U_{i_{2},t^{\prime}}\nsubseteq\Box V^{2}_{l}$, a contradiction. This shows
that $\mathcal{C}_{0}\cap\mathcal{F}\cap\mathcal{U}_{i}\not=\emptyset$ for all
$i\in I$.
By the minimality of $\mathcal{C}_{0}$, we have that
$\mathcal{C}_{0}\subseteq\mathcal{F}$. Then $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\subseteq\Box
U_{i_{0},t_{0}}\cap\mathcal{F}$. But, as we pointed out earlier, $\Box
U_{i_{0},t_{0}}\cap\mathcal{F}=\emptyset$, thus $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}=\emptyset$. This contradicts that $\Box
U_{i_{0},t_{0}}\cap\mathcal{C}_{0}\not=\emptyset$.
All these together show that $K\in\bigcap\\{\hat{\mathcal{U}}_{i}:i\in I\\}$.
Now, Fact 3 and Fact 4 contradict the assumption
$\bigcap\\{\mathcal{U}_{i\in I}:i\in I\\}\subseteq\mathcal{U}.$
The proof is thus completed.
## 5 Summary
In this paper, we mainly considered the preservation of the open well-
filteredness by some standardly constructed spaces from an open well-filtered
space. The table below summarizes the main results, where “sp.” denotes
“subspaces”.
open sp. | closed sp. | saturated sp. | retract | upper space | product
---|---|---|---|---|---
$\checkmark$ | $\times$ | $\times$ | $\times$ | $\checkmark$ | $?$
Note that in many other cases, the ground spaces usually inherit the
corresponding properties of their upper spaces as they can be embedded into
the upper spaces under the principle filter mappings. At the moment, the
following problem is still open.
###### Problem 5.1.
Is it true that a space is open well-filtered if its upper space is open well-
filtered?
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|
# Joint modelling of longitudinal and time-to-event data applied to group
sequential clinical trials
Abigail J. Burdon1,∗, Lisa V. Hampson2 and Christopher Jennison3
1MRC Biostatistics Unit, University of Cambridge, Robinson Way, Cambridge, CB2
0SR, U.K
2Statistical Methodology, Novartis Pharma AG, Basel BA2 7AY, Swizerland
3Department of Mathematical Sciences, University of Bath, Claverton Down, Bath
BA2 7AY, U.K
∗Email<EMAIL_ADDRESS>
###### Abstract
Often in Phase 3 clinical trials measuring a long-term time-to-event endpoint,
such as overall survival or progression-free survival, investigators also
collect repeated measures on biomarkers which may be predictive of the primary
endpoint. Although these data may not be leveraged directly to support early
stopping decisions, can we make greater use of these data to increase
efficiency and improve interim decision making? We present a joint model for
longitudinal and time-to-event data and a method which establishes the
distribution of successive estimates of parameters in the joint model across
interim analyses. With this in place, we can use the estimates to define both
efficacy and futility stopping rules. Using simulation, we evaluate the
benefits of incorporating biomarker information and the affects on interim
decision making.
_K_ eywords
Efficient designs, group sequential, joint modelling, longitudinal data, time-
to-event data.
## 1 Introduction
Interest in joint modelling is motivated by clinical trials where the
biomarker is predictive of a time-to-event (TTE) outcome. Overall survival and
progression-free survival are examples of TTE endpoints. For example, Goldman
et al. (1996) use CD4 lymphocyte cell counts as a surrogate endpoint for
survival in a clinical trial comparing the efficacy and safety of two
antiviral drugs for HIV-infected patients. Taylor et al. (2013) use joint
models to predict survival times of patients with prostate cancer based on
prostate specific antigen (PSA) levels measured by blood tests at multiple
hospital visits.
To date, most group sequential designs for survival trials focus on a single
primary endpoint. Designs which leverage repeated measurements on continuous
or binary endpoints have been reported such as those by Galbraith and
Marschner (2003) and this work extends this idea to survival trials with
repeated measurements on a predictive biomarker. We shall focus on monitoring
randomised control trials so that patients are randomised at baseline to
receive either a novel treatment or control and the objective is to test for
superioirity of the treatment versus control. Hence, let $\eta$ denote the
treatment effect in a statistical model and let $\eta_{0}$ be the true value
of $\eta$, then we shall test the null hypothesis
$H_{0}:\eta_{0}\leq 0\hskip 14.22636pt\text{vs}\hskip
14.22636ptH_{A}:\eta_{0}>0.$ (1)
In the group sequential setting with a total of $K$ analyses, we shall test
$H_{0}$ at analys $k$ by defining treatment effect estimates
$\hat{\theta}^{(k)},$ information levels $\mathcal{I}_{k}$ and standardised
$Z-$statistics given by $-\hat{\eta}^{(k)}\sqrt{\mathcal{I}_{k}}$ for
$k=1,\dots,K.$ The aim will then be to determine the upper boudary constants
$b_{1},\dots,b_{K}$ and lower boundary constants $a_{1},\dots,a_{K}$ for the
group sequential test as in Figure 1. These boundaries are calculated to
ensure such that the Type 1 error rate is $\alpha$ and power is $1-\beta$ when
$\eta=-0.5$. In particular, we shall use the error spending test, also known
as Gordon Lan and DeMets (1983), to calculate the boundary points.
Figure 1: Group sequential trial with $k=5$ analyses. Blue points represent
the upper boundary constants $b_{1},\dots,b_{K}$ and red points give the lower
boundary constants $a_{1},\dots,a_{K}.$.
We shall develop methods for designing and analysing a group sequential trial
based on a joint model for longitudinal and TTE data. We may believe that a
trend in the trajectory of the biomarker is predictive of the TTE outcome, and
we would like to know whether these additional longitudinal data can be used
to inform early stopping decisions. Suppose that biomarker observations are
available but have not been used in the analysis. We shall assess the change
in efficiency of the trial when these observations are included in the
analysis. We shall focus on efficiency measured in terms of the number of
patients that need be recruited to achieve a certain power, and we and show
that, in some scenarios, the trial using the longitudinal data is up to 1.7
times as efficient as the trial which discards the longitudinal data.
## 2 Joint modelling
### 2.1 Joint model
The joint model that we consider is given in Equation (2) by Tsiatis and
Davidian (2001). We shall refer to the authors as TD for short. There are two
processes in this model which represent the survival and longitudinal parts
separately, and these processes are linked through the hazard rate of the
survival process. First we consider the longitudinal data. Suppose that
$X_{i}(t)$ is the true value of the biomarker at time $t$ for subject $i$ and
that $W_{i}(t)$ is the observed value of the biomarker at time $t$ for patient
$i$. Then the longitudinal model takes the form
$\displaystyle X_{i}(t)$ $\displaystyle=b_{0i}+b_{1i}t$ (2) $\displaystyle
W_{i}(t)$ $\displaystyle=X_{i}(t)+\epsilon_{i}(t)$ (3)
where $\mathbf{b}_{i}=(b_{i0},b_{i1})$ is a vector of patient specific random
effects and $\epsilon_{i}(t)$ is the measurement error. In general, the vector
$\mathbf{b}_{i}$ can have dimension $p$ and the function $X_{i}(t)$ need not
be constrained to a linear function of $t.$ We consider a random effects model
where each $\mathbf{b}_{i}$ is a random variable with density function
$f(\cdot).$ The measurement errors are assumed to be independent and if the
biomarker for patient $i$ is measured at times $t_{i1},\dots t_{im_{i}}$, then
$\epsilon_{i}(t_{ij})|\mathbf{b}_{i}\sim N(0,\sigma^{2})\text{ for
}j=1,\dots,m_{i}$ and $\epsilon(t)$ and $\epsilon(t^{\prime})$ are independent
for $t\neq t^{\prime}.$ For now, we shall assume that $\sigma^{2}$ is known
and we later describe how $\sigma^{2}$ can be estimated.
The model for the survival endpoint is a Cox proportional hazards model in
which the underlying trajectory $X_{i}(t)$ acts as a time-varying covariate
with coefficient $\gamma$. Let $Z_{i}$ be an indicator function that patient
$i$ receives the experimental treatment and let $\eta$ be the corresponding
treatment effect. Then the hazard function is given by
$h_{i}(t)=h_{0}(t)\exp\\{\gamma X_{i}(t)+\eta Z_{i}\\},$ (4)
where $h_{0}(\cdot)$ is the baseline hazard function. In general, $Z_{i}$ may
be a $p\times 1$ column vector of coefficients and $\eta$ is the corresponding
coefficient vector of length $p$. In this model, $\gamma$ is the parameter
that determines the correlation between the longitudinal data and the TTE
endpoint. We show in Section 4 that if $\gamma=0$ then the investigator may
pay a small penalty for having tried to leverage the biomarker data. However,
if $\gamma>0$, then there is a large benefit from fitting this joint model to
the data. Together, Equations (2), (3) and (4) define the joint model.
### 2.2 Conditional Score
In the fixed sample setting, TD present the “conditional score" method for
fitting the joint model to the data. The method adapts the general theoretical
work by Stefanski and Carroll (1987) who find unbiased score functions by
conditioning on certain parameter-depedent sufficient statistics. This is a
desireable method because the analysis is semi-parametric so that there are no
distributional assumptions required for the random effects. Further, the
authors show that the parameter estimates are normally distributed and
unbiased. We shall now extend the fixed sample theory for the joint model to
group sequential trials. To perform a group sequential trial with $K$
analyses, we need to know the joint distribution of the sequence of treatment
effect estimates that will be obtained at analyses $k=1,\dots,K.$ To determine
this distribution, we shall define group sequential versions of all objects
included in the single-stage conditional score, which are calculated using
data obtained at that analysis. Equivalent definitions of TD’s single-stage
conditional score, and associated functions, can be found by setting $K=k=1$
The conditional score methodology builds upon the theory of counting
processes. In the general definition, a counting process is a step-function
increasing in integer increments and the survival counting process is a step
function jumping from 0 to 1 at the failure time for an uncensored
observation.
The censoring mechanism is used to keep patients in the at-risk set who have
yet to experience an event. For patient $i$ with time-to-failure random
variable $F_{i}$, let $C_{i}(k)$ be the time-to-censoring random variable at
analysis $k$. This censoring event includes “end of study" censoring for the
total follow-up time of patient $i$ at analysis $k$, then at analysis $k$ the
event time random variable is $T_{i}(k)=\min\\{F_{i},C_{i}(k)\\}.$ The
observed event time at analysis $k$ is $t_{i}(k)$ and the observed censoring
indicator is $\delta_{i}(k)=\mathbb{I}\\{F_{i}\leq C_{i}(k)\\}.$ In the
conditional score approach, to be included in the at-risk set at time $t$ the
patient must have at least two longitudinal observations to fit the
longitudinal regression model. The at-risk process at analysis $k$ is an
indicator for not yet observing the event, not yet censored, or not having
enough longitudinal observations. Let $v_{i2}$ be the time of the second
longitudinal observation for patient $i$, then at analysis $k$, the at-risk
process and counting process for the joint model are
$\displaystyle Y_{i}(k,t)$ $\displaystyle=\mathbb{I}\\{t_{i}(k)\geq
t,v_{i2}\leq t\\}$ $\displaystyle N_{i}(k,t)$
$\displaystyle=\mathbb{I}\\{t_{i}(k)\leq t,\delta_{i}(k)=1,v_{i2}\leq t\\}.$
The function
$\displaystyle dN_{i}(k,t)$ $\displaystyle=N_{i}(k,t+dt)-N_{i}(k,t^{-})$
$\displaystyle=\mathbb{I}\\{t\leq t_{i}(k)<t+dt,\delta_{i}(k)=1,v_{i2}\leq
t\\}$
presents us with useful notation: for any function or stochastic process
$f(\cdot),$ the stochastic integral
$\int_{0}^{\infty}f(u)dN_{i}(k,u)=f(t_{i})$ (5)
is $f$ evaluated at the place where $N_{i}(k,t)$ jumps from 0 to 1 if
$\delta_{i}(k)=1$ and $v_{i2}\leq t$, and 0 otherwise.
By analogy to survival analysis, we seek a compensated counting process with
expectation zero. This property leads us to define an estimating equation from
which we can obtain treatment effect estimates that are asymptotically
normally distributed by Section 5.3 by Van der Vaart (2000). In the usual
survival analysis setting, we can calculate the compensated counting process
by subtracting the intensity process from the counting process itself. In the
joint modelling setting, this is not as simple because the randomness of the
nuisance parameters $\mathbf{b}$ mean that the intensity process is
unpredictable. To overcome this, TD introduce a “conditional intensity
process" which is conditional on a certain “sufficient statistic". The origins
of the conditional intensity process and sufficient statistic are not crucial
for our purpose. What we actually use are the definitions and properties that
are derived from these. In what follows, we shall present TD’s definitions of
the single-stage conditional intensity process, sufficient statistic and
compensated counting process and we extend these to the group sequential
versions. We shall then show that the group sequential compensated counting
process has expectation zero.
For patient $i$, let $v_{i}(u)$ be set of all time points for measurements of
the biomarker, up to and including time $u$. Let $\hat{X}_{i}(u)$ be the
ordinary least squares estimate of $X_{i}(u)$ for patient $i$ based on the set
of measurements taken at times $v_{i}(u)$. That is, calculate
$\hat{b}_{0i}(u)$ and $\hat{b}_{1i}(u)$ based on measurements taken at times
$v_{i}(u)$, then $\hat{X}_{i}(u)=\hat{b}_{0i}(u)+\hat{b}_{1i}(u)u$. As we pass
time $v_{ij},$ a new observation $W_{ij}$ is generated and the formula for
$\hat{X}_{i}(u)$ is updated for larger values of $u$. This seems strange since
at an early time point, $s$ where $s<u$, we use data $v_{i}(s)$ in the
calculation of $\hat{X}_{i}(s)$ even though there may be more data available
at time points $v_{i}(u)$. However, this is necessary for the martingale
property to hold in later results. Suppose that $\sigma^{2}\theta_{i}(u)$ is
the variance of the estimator $\hat{X}_{i}(u)$ at time $u$. TD define the
sufficient statistic to be the function
$\displaystyle S_{i}(k,t,\gamma,\sigma^{2})$
$\displaystyle=\hat{X}_{i}(t)+\gamma\sigma^{2}\theta_{i}(t)dN_{i}(k,t)$
$\displaystyle=\hat{b}_{0i}(t)+\hat{b}_{1i}(t)t+\gamma\sigma^{2}\theta_{i}(t)dN_{i}(k,t)$
which is defined for all $t\in(v_{i2},t_{i})$ for patient $i$. The
corresponding conditional intensity process and compensated counting process
are defined by
$\displaystyle\lambda^{C}_{i}(k,t)$ $\displaystyle=lim_{dt\downarrow
0}\frac{\mathbb{P}(dN_{i}(k,t)=1|S_{i}(k,t,\gamma,\sigma^{2}),t_{i}(t),Z_{i},Y_{i}(k,t))}{dt}$
(6) $\displaystyle=h_{0}(t)\exp\\{\gamma
S_{i}(k,t,\gamma,\sigma^{2})-\gamma^{2}\sigma^{2}\theta_{i}(t)/2+\eta^{T}Z_{i}\\}Y_{i}(k,t)$
(7) $\displaystyle=h_{0}(t)E_{0i}(t,\gamma,\eta,\sigma^{2})Y_{i}(k,t)$
$\displaystyle M_{i}(k,t)$
$\displaystyle=N_{i}(k,t)-\int_{0}^{t}\lambda^{C}_{i}(k,t)du$ $\displaystyle
dM_{i}(k,t)$ $\displaystyle=dN_{i}(k,t)-\lambda^{C}_{i}(k,t).$
We show below that this compensated counting process has expectation zero
conditional on $S_{i}(k,t,\gamma,\sigma^{2})$ and use this result to obtain
the asymptotic distribution of some parameter estimates in the joint model.
Specifically, we will determine the distribution of the estimates
$\hat{\gamma}^{(k)},\hat{\eta}^{(k)}$ and $\hat{\sigma}^{(k)2}$ which are
unbiased estimates, at analysis $k$, of $\gamma,\eta$ and $\sigma^{2}$
respectively.
###### Lemma 1.
The function $dM_{i}(k,t)$ has expectation zero conditional on the sufficient
statistic, that is
$\mathbb{E}(dM_{i}(k,t)|S_{i}(k,t,\gamma,\sigma^{2}),t_{i}(t),Z_{i},Y_{i}(k,t))=0.$
###### Proof.
See Tsiatis and Davidian (2001). ∎
TD present the fixed sample conditional score in their Equation (6). We
present some functions that are needed to define the group sequential
conditional score at analysis $k$ and the derivate of such an object with
respect to parameters $\gamma$ and $\eta$. Let
$J_{i}(k,t,\gamma,\sigma^{2})=\\{S_{i}(k,t,\gamma,\sigma^{2})-\hat{X}_{i}(t)-\gamma\sigma^{2}\theta_{i}(t),0,\dots,0\\}^{T},\hskip
28.45274pt\Gamma_{i}(k,t,\sigma^{2})=\begin{bmatrix}\sigma^{2}\theta_{i}(t)&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}\end{bmatrix}$
be a $(p+1)\times 1$ vector and $(p+1)\times(p+1)$ matrix respectively. Now,
for the following functions, we drop the dependency of all functions on the
parameters $k,t,\gamma,\eta$ and $\sigma^{2}.$ Then the functions of interest
are
$\displaystyle S_{c}^{(0)}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}Y_{i}E_{0i}$ $\displaystyle C^{(1)}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}J_{i}Y_{i}E_{0i}$ $\displaystyle
S_{c}^{(1)}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left\\{\begin{array}[]{c}S_{i}\\\
Z_{i}\end{array}\right\\}Y_{i}E_{0i}$ $\displaystyle C^{(2)}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left\\{\begin{array}[]{c}S_{i}\\\
Z_{i}\end{array}\right\\}J_{i}^{T}Y_{i}E_{0i}$ (12) $\displaystyle
S_{c}^{(2)}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left\\{\begin{array}[]{c}S_{i}\\\
Z_{i}\end{array}\right\\}\left\\{\begin{array}[]{c}S_{i}\\\
Z_{i}\end{array}\right\\}^{T}Y_{i}E_{0i}$ $\displaystyle C^{(3)}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\Gamma_{i}(u)dN_{i}(k,u)Y_{i}E_{0i}$
(17) $\displaystyle E_{c}$ $\displaystyle=\frac{S_{c}^{(1)}}{S_{c}^{(0)}}$
$\displaystyle V^{(2)}_{c}$
$\displaystyle=\frac{C^{(2)}}{S_{c}^{(0)}}-\frac{S_{c}^{(1)}C^{(1)^{T}}}{[S_{c}^{(0)}]^{2}}$
$\displaystyle V^{(1)}_{c}$
$\displaystyle=\frac{S_{c}^{(2)}}{S_{c}^{(0)}}-\frac{S_{c}^{(1)}S_{c}^{(1)^{T}}}{[S_{c}^{(0)}]^{2}}$
$\displaystyle V^{(3)}_{c}$ $\displaystyle=\frac{C^{(3)}}{S_{c}^{(0)}}.$
These functions are analogous to to those in Jennison and Turnbull (1997) for
survival data. In the simple survival model, we have that $S^{(1)}=\partial
S^{(0)}/\partial\boldsymbol{\theta}$ and $S^{(2)}=\partial
S^{(1)}/\partial\boldsymbol{\theta}$ where $\boldsymbol{\theta}$ is the vector
of parameters in the hazard function of the simple survival model. In a
similar manner, the superscripts $(1),(2)$ and $(3)$ refer to the order of the
differentiation for certain functions which we explain further in Section 2.3.
The function $E_{c}(k,t,\gamma,\eta,\sigma^{2})$ has the interpretation of the
expectation of the vector $\\{S_{i}(k,t,\gamma,\sigma^{2}),Z_{i}^{T}\\}^{T}$
at analysis $k$ weighted by the conditional intensity process. Let $\tau_{k}$
be the maximum follow-up time at analysis $k$, then the conditional score for
analysis $k$ is
$U_{c}(k,\gamma,\eta,\sigma^{2})=\int_{0}^{\tau_{k}}\sum_{i=1}^{n}\left(\\{S_{i}(k,u,\gamma,\sigma^{2}),Z_{i}^{T}\\}^{T}-E_{c}(k,u,\gamma,\eta,\sigma^{2})\right)dN_{i}(k,u).$
(18)
### 2.3 Differentiation of the conditional score
Another object of importance is the first derivative of the conditional score
function, Equation (18), with respect to parameters $\gamma$ and $\eta.$ This
matrix plays a key role in the definition of the covariance matrix for the
estimates $\hat{\gamma}$ and $\hat{\eta}$ and has a likeness to the Fisher
information matrix which is the derivative of the score statistic for general
statistical models. TD comment that the variance matrix can be found, however
they do not present an equation for such an object. Details for the following
calculation can be found in the supplementary materials. The first derivative
the conditional score of Equation (18) with respect to $\gamma$ and $\eta$ is
$\displaystyle\frac{\partial}{\partial(\gamma,\eta)^{T}}U_{c}(k,\gamma,\eta,\sigma^{2})=\sum_{i=1}^{n}\int_{0}^{\infty}\bigg{[}$
$\displaystyle\Gamma_{i}(k,u,\sigma^{2})-V_{c}^{(1)}(k,u,\gamma,\eta,\sigma^{2})$
$\displaystyle+$ $\displaystyle
V_{c}^{(2)}(k,u,\gamma,\eta,\sigma^{2})+V_{c}^{(3)}(k,u,\gamma,\eta,\sigma^{2})\bigg{]}dN_{i}(k,u).$
(19)
### 2.4 Asymptotic theory for parameter estimates $\hat{\gamma},\hat{\eta}$
and $\hat{\sigma}^{2}$ in the joint model
We shall now prove that the estimates
$\hat{\gamma}^{(1)},\hat{\eta}^{(1)},\dots,\hat{\gamma}^{(K)},\hat{\eta}^{(K)}$
are asymptotically multivariate normally distributed and we shall derive an
explicit form for the covariance matrix of this vector of parameters. In what
follows, we shall determine the limiting distribution as $n\rightarrow\infty.$
For clarity, we denote the objects
$\hat{\gamma}_{n}^{(k)},\hat{\eta}_{n}^{(k)}$ and
$U_{c}^{(n)}(k,\gamma,\eta,\sigma)$ as dependent on $n$.
###### Theorem 1.
Suppose that $\gamma_{0},\eta_{0}$ and $\sigma^{2}_{0}$ are the true values of
the parameters $\gamma,\eta$ and $\sigma^{2}$ respectively. For each
$k=1,\dots,K$, let $\hat{\gamma}_{n}^{(k)}$ and $\hat{\eta}_{n}^{(k)}$ be the
values of $\gamma$ and $\eta$ which are the solution to the equation
$(U_{c}^{(n)}(k,\gamma,\eta,\sigma_{0})=0$ and suppose that Conditions 2 hold.
Then the vector
$(\hat{\gamma}_{n}^{(1)},\hat{\eta}_{n}^{(1)},\dots,\hat{\gamma}_{n}^{(K)},\hat{\eta}_{n}^{(K)})^{T}$
converges in distribution to a multivariate Gaussian random variable,
specifically
$n^{\frac{1}{2}}\left(\begin{array}[]{c}\hat{\gamma}_{n}^{(1)}-\gamma_{0}\\\
\hat{\eta}_{n}^{(1)}-\eta_{0}\\\ \vdots\\\
\hat{\gamma}_{n}^{(K)}-\gamma_{0}\\\ \hat{\eta}_{n}^{(K)}-\eta_{0}\\\
\end{array}\right)\xrightarrow{d}N\left(\begin{bmatrix}\mathbf{0}\\\
\mathbf{0}\\\ \vdots\\\
\mathbf{0}\end{bmatrix},\Sigma=\begin{bmatrix}\Sigma_{11}&\Sigma_{12}&\cdots&\Sigma_{1K}\\\
\Sigma_{12}&\Sigma_{22}&\cdots&\Sigma_{2K}\\\ \vdots&\vdots&\ddots&\vdots\\\
\Sigma_{1K}&\Sigma_{2K}&\cdots&\Sigma_{KK}\end{bmatrix}\right)$
where
$\Sigma_{k_{1}k_{2}}=(A^{(k_{1})})^{-1}B^{(k_{1})}((A^{(k_{2})})^{-1})^{T}$
(20)
and the matrices $A^{(k)}$ and $B^{(k)}$ are defined by
$\displaystyle A^{(k)}$
$\displaystyle=\int_{0}^{\tau_{k}}\left[\mathbb{E}(\Gamma_{i}(k,u,\sigma^{2}_{0}))-v_{c}^{(1)}(k,u,\gamma_{0},\eta_{0},\sigma_{0}^{2})-v_{c}^{(2)}(k,u,\gamma_{0},\eta_{0},\sigma_{0}^{2})\right]s^{(0)}_{c}(k,u,\gamma_{0},\eta_{0},\sigma_{0}^{2})h_{0}(u)du$
(21) $\displaystyle B^{(k)}$
$\displaystyle=\int_{0}^{\tau_{k}}v_{c}^{(1)}(k,u,\gamma_{0},\eta_{0},\sigma_{0}^{2})s^{(0)}_{c}(k,u,\gamma_{0},\eta_{0},\sigma_{0}^{2})h_{0}(u)du.$
(22)
###### Proof.
We begin this proof by considering the set of $K(p+1)$ stacked equations
$\begin{bmatrix}U_{c}(1,\gamma,\eta,\sigma^{2})\\\ \vdots\\\
U_{c}(K,\gamma,\eta,\sigma^{2})\end{bmatrix}=\begin{bmatrix}\mathbf{0}\\\
\vdots\\\ \mathbf{0}\end{bmatrix}$ (23)
and we shall show that these equations define a multidimensional estimating
equation. That is, for each $k=1,\dots,K$, the function
$U_{c}(k,\gamma,\eta,\sigma^{2})$ has expectation zero. The arguments given by
TD about asymptotic normality in the fixed sample case apply for each
$k=1,\dots,K$. We present some of these arguments which are useful for the
derivation of the covariance matrix in the group sequential setting. We shall
write $U_{c}(k,\gamma,\eta,\sigma^{2})$ as
$\displaystyle\int_{0}^{\tau_{k}}\sum_{i=1}^{n}\left(\\{S_{i}(k,u,\gamma,\sigma^{2}),Z_{i}^{T}\\}^{T}-e_{c}(k,u,\gamma,\eta,\sigma^{2})\right)dM_{i}(k,u)$
(24) $\displaystyle+$
$\displaystyle\int_{0}^{\tau_{k}}\sum_{i=1}^{n}\left(e_{c}(k,u,\gamma,\eta,\sigma^{2})-E_{c}(k,u,\gamma,\eta,\sigma^{2})\right)dM_{i}(k,u).$
(25)
By the same argument as TD, $n^{-1}$ times Expression (25) converges in
probability to zero in a neighbourhood of $(\gamma_{0},\eta_{0})$ and we
deduce that the behaviour of the estimates $\hat{\gamma}^{(k)}$ and
$\hat{\eta}^{(k)}$ will be dictated by Expression (24). Then we have that the
expectation of Expression (24) is
$\mathbb{E}\left(\int_{0}^{\tau_{k}}\sum_{i=1}^{n}\left(\\{S_{i}(k,u,\gamma,\sigma^{2}),Z_{i}^{T}\\}^{T}-e_{c}(k,u,\gamma,\eta,\sigma^{2})\right)dM_{i}(k,u)\right)=0$
(26)
for each $k=1,\dots,K$. Therefore combined with the fact that Expression (25)
converges in probability to zero, we have that Equation (23) defines a
multidimensional estimating equation. Therefore, the estimates
$(\hat{\gamma}^{(k)},\hat{\eta}^{(k)})$ for each $k=1,\dots,K$ are
asymptotically multivariate normal and unbiased for parameters $\gamma$ and
$\eta$ by Section 5.3 by Van der Vaart (2000). It remains to determine the
covariance matrix.
In the general setting for statistical models, the sandwich estimator can be
used to robustly estimate the variance matrix for estimates that are the
solutions to estimating equations as in Section 2.6 by Wakefield (2013). The
sandwich estimator requires deriving $B=Var(U(\theta))$ and $A=\partial
U(\theta)/\partial\theta$ where $U(\cdot)$ is an estimating function for
parameter $\theta.$ The similarity here is with matrices $A^{(k)}$ and
$B^{(k)}$ in Equations (21) and (22) which are the sequential versions of such
objects for the conditional score. Wakefield (2013) prove the asymptotic
normality of estimates obtained using estimating equations and the asymptotic
convergence of the sandwich estimator. The proof by Wakefield (2013) applies
to scalar estimates and one-dimensional estimating functions so we shall
extend this proof to derive the covariance matrix for the vector of parameters
in the joint model as oppose to the scalar sandwich variance.
Following Wakefield (2013), we apply the standard Taylor expansion results to
each row of Equation (23) and aggregate the results. Details of this
calculation are found in the supplementary materials. Let $\bar{\mathbf{A}}$
be the block diagonal matrix whose $k^{th}$ diagonal matrix is the $p\times p$
matrix $\mathbf{A}^{(k)}.$ Then we obtain
$-n^{-\frac{1}{2}}\bar{\mathbf{A}}^{-1}\begin{bmatrix}U_{c}(1,\gamma,\eta,\sigma^{2})\\\
\vdots\\\
U_{c}(K,\gamma,\eta,\sigma^{2})\end{bmatrix}=\bar{\mathbf{A}}^{-1}\begin{bmatrix}n^{-1}\frac{\partial}{\partial(\gamma,\eta)^{T}}U_{c}(1,\gamma,\eta,\sigma^{2})|_{(\gamma^{*(1)}\eta^{*(1)})}\\\
\vdots\\\
n^{-1}\frac{\partial}{\partial(\gamma,\eta)^{T}}U_{c}(K,\gamma,\eta,\sigma^{2})|_{(\gamma^{*(K)}\eta^{*(K)})}\end{bmatrix}\cdot
n^{\frac{1}{2}}\begin{bmatrix}\hat{\gamma}_{n}^{(1)}-\gamma_{0}\\\
\hat{\eta}_{n}^{(1)}-\eta_{0}\\\ \vdots\\\
\hat{\gamma}_{n}^{(K)}-\gamma_{0}\\\ \hat{\eta}_{n}^{(K)}-\eta_{0}\\\
\end{bmatrix}$
where $(\gamma^{*(k)}\eta^{*(k)})$ lies on the line segment between
$(\gamma_{0},\eta_{0})$ and $(\hat{\gamma}^{(k)}_{n},\hat{\eta}^{(k)}_{n})$.
Suppose that we could show that
$\displaystyle
n^{-1}\frac{\partial}{\partial(\gamma,\eta^{T})^{T}}U_{c}^{(n)}(k,\gamma,\eta,\sigma_{0}^{2})\bigg{\rvert}_{\gamma=\gamma^{*}_{n},\eta=\eta^{*}_{n}}\xrightarrow{p}\mathbf{A}^{(k)}\hskip
56.9055pt\text{and}$ (27)
$\displaystyle-n^{-\frac{1}{2}}\begin{bmatrix}U_{c}(1,\gamma,\eta,\sigma^{2})\\\
\vdots\\\
U_{c}(K,\gamma,\eta,\sigma^{2})\end{bmatrix}\xrightarrow{d}N\left(\begin{bmatrix}0\\\
\vdots\\\
0\end{bmatrix},\begin{bmatrix}\mathbf{B}^{(1)}&\cdots&\mathbf{B}^{(1)}\\\
\vdots&\ddots&\vdots\\\
\mathbf{B}^{(1)}&\cdots&\mathbf{B}^{(K)}\end{bmatrix}\right),$ (28)
then by an application of Slutsky’s Theorem, we have the desired result.
A simple application of the triangle inequality proves that condition (27)
holds. The proof of convergence in probability closely follows the standard
results for survival data seen in Theorem VII.2.2 by Andersen and Gill (1982)
and the details of this step are found in the supplementary materials.
To prove condition (28) holds, first note that Expression (24) is a sum over
$n$ independent and identically distributed terms. Then the multivariate
Central Limit Theorem can be applied to the vector
$(U_{c}(1,\gamma,\eta,\sigma^{2}),\dots,U_{c}(K,\gamma,\eta,\sigma^{2}))$ to
establish asymptotic normality. It then remains to show that
$Cov(U_{c}(k_{1},\gamma,\eta,\sigma^{2}),U_{c}(k_{2},\gamma,\eta,\sigma^{2}))\xrightarrow{p}\mathbf{B}^{(k_{1})}$
for $k_{1}\leq k_{2}.$
We now follow a similar structure to the partial likelihood function for
survival data given by Jennison and Turnbull (1997) and we create a new
counting process that allows the conditional score statistic to be written as
the sum of distinct increments. This counting process is defined by
$DN_{i}(k,t)=N_{i}(k,t)-N_{i}(k-1,t)$ for $k=1,\dots,K$.The corresponding
compensated counting process is therefore given by
$DM_{i}(k,t)=DN_{i}(k,t)-\int_{0}^{t}h_{0}(u)E_{0i}(t,\gamma,\eta,\sigma^{2})(Y_{i}(k,u)-Y_{i}(k-1,u))du$
for $k=1,\dots,K$ and $DM_{i}(0,t)=0.$ The event for patient $i$ can only
occur in one interval, and therefore we have
$N_{i}(k,t)=\sum_{l=1}^{k}DN_{i}(l,t)$. For consistency with Andersen and Gill
(1982) and Jennison and Turnbull (1997), let Expression (24) be denoted by
$W^{(n)}_{j}(k,\tau_{k},\gamma,\eta,\sigma^{2})$. Then by the definition of
the difference in counting process, we have
$W^{(n)}_{j}(k,\tau_{k},\gamma,\eta,\sigma^{2})=\int_{0}^{\tau}\sum_{i=1}^{n}\sum_{l=1}^{k}H_{ij}^{(n)}(l,u,\gamma,\eta,\sigma^{2})dDM_{i}(l,u)$
where
$H_{ij}^{(n)}(l,u,\gamma,\eta,\sigma^{2})=n^{-\frac{1}{2}}(\\{S_{i}(k,u,\gamma,\sigma^{2}),Z_{i}\\}^{T}_{j}-e_{c}(l,u,\gamma,\eta,\sigma^{2})_{j})$
and $W^{(n)}_{j}(k,\tau_{k},\gamma,\eta,\sigma^{2})$ has expectation zero by a
simple manipulation of Equation (26).
The main result, with details given in the supplementary materials, is as
follows
$\displaystyle
Cov\left(W^{(n)}_{j_{1}}(k_{1},\tau_{k_{1}},\gamma_{0},\eta_{0},\sigma_{0}^{2}),W^{(n)}_{j_{2}}(k_{2},\tau_{k_{2}},\gamma_{0},\eta_{0},\sigma_{0}^{2})\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(\sum_{i=1}^{n}\int_{0}^{\tau_{k_{1}}}H_{ij_{1}}^{(n)}(k_{1},u,\gamma_{0},\eta_{0},\sigma_{0}^{2})H_{ij_{2}}^{(n)}(k_{1},u,\gamma_{0},\eta_{0},\sigma_{0}^{2})h_{0}(u)E_{0i}(u,\gamma_{0},\eta_{0},\sigma_{0}^{2})Y_{i}(k_{1},u)du\right).$
(29)
To complete the proof, it remains to show that Equation (29) converges in
probability to $\mathbf{B}^{(k_{1})}$ and we shall do so by using the
definitions of the objects $s_{c}^{(j)}$ and $c^{(j)}$ for $j=1,0,1,2$ and
$e_{c},v^{(1)}_{c}$ and $v^{(2)}$ given in Appendix 1. We are assuming that
these limits exist by Conditions 2 and we shall exploit the relationships
between these terms. This working is exactly the same as for standard survival
data given by Andersen and Gill (1982), with the details given in the
supplementary materials, and we see that
$\displaystyle
Cov\left(W^{(n)}_{j_{1}}(k_{1},\tau_{k_{1}},\gamma_{0},\eta_{0},\sigma_{0}^{2}),W^{(n)}_{j_{2}}(k_{2},\tau_{k_{2}},\gamma_{0},\eta_{0},\sigma_{0}^{2})\right)$
$\displaystyle\xrightarrow{p}$
$\displaystyle\left(\int_{0}^{\infty}v^{(1)}_{c}(k_{1},u,\gamma_{0},\eta_{0},\sigma^{2}_{0})s^{(0)}_{c}(k_{1},u,\gamma_{0},\eta_{0},\sigma^{2}_{0})h_{0}(u)du\right)_{j_{1}j_{2}}=B^{(k_{1})}_{j_{1}j_{2}}.$
By the argument that the behaviour of the estimates $\hat{\gamma}^{(k)}$ and
$\hat{\eta}^{(k)}$ is dictated by Expression (24), we have
$Cov\left(U_{c}(k_{1},\gamma_{0},\eta_{0},\sigma^{2}_{0}),U_{c}(k_{2},\gamma_{0},\eta_{0},\sigma^{2}_{0})\right)\xrightarrow{p}\mathbf{B}^{(k_{1})}\hskip
28.45274pt\text{ for }k_{1}\leq k_{2}$
which is the result. ∎
Similar to the fixed sample case, we have assumed that $\sigma^{2}_{0}$ is
known in the derivation of the distribution of $\hat{\gamma}^{(k)}_{n}$ and
$\hat{\eta}^{(k)}_{n}.$ This is not generally the case but by arguments in
Carroll et al. (2006) Section A.3.3, we can find a consistent estimate to
replace $\sigma^{2}_{0}$ with in the group sequential conditional score
function. At analysis $k$ this estimate is given by
$\hat{\sigma}^{(k)2}=\frac{\sum_{i=1}^{n}\mathbb{I}\\{m_{i}(k)>2\\}R_{i}(k)}{\sum_{i=1}^{n}\mathbb{I}\\{m_{i}(k)>2\\}(m_{i}(k)-2)},$
(30)
where $R_{i}(k)$ is the residual sum of squares for the least squares fit to
all $m_{i}(k)$ observations for patient $i$ available at analysis $k$.
### 2.5 A group sequential trial design based on the conditional score
analysis
We shall use Theorem 1 to create a group sequential test based on the joint
model. Let $\gamma_{0},\eta_{0}$ and $\sigma^{2}_{0}$ be the true values of
the parameters $\gamma,\eta$ and $\sigma^{2}$ respectively. Using the group
sequential conditional score method, let $\hat{\gamma}^{(k)},\hat{\eta}^{(k)}$
be the values of the parameters $\gamma$ and $\eta$ such that
$U_{c}(k,\gamma,\eta,\sigma^{2})=0$ where the conditional score function is
calculated using Equation (18). Further, let $\hat{\sigma}^{(k)2}$ be the
estimate for $\sigma^{2}$ given in Equation (30). By Theorem 1, for each
$k=1,\dots,K$, the marginal distribution of the parameter $\hat{\eta}^{(k)}$
is
$\sqrt{n}(\hat{\eta}^{(k)}-\eta_{0})\xrightarrow{d}N(0,\Sigma^{(k)}_{22})$
where
$\Sigma^{(k)}=(A^{(k)})^{-1}B^{(k)}((A^{(k)})^{-1})^{T}$
and the matrices $A^{(k)}$ and $B^{(k)}$ are defined by Equations (21)–(22).
Note that the subscript notation in the covariance matrix represents that the
parameter $\eta$ is the second parameter in the vector $(\gamma,\eta)$. In the
more general case where $Z_{i}$ is a vector of dimension $p$ including other
covariates appart from treatment indicator, then the information would be
indexed by the corresponding dimension of the vector $(\gamma,\eta)^{T}$ which
relates to the treatment effect. The matrices $A^{(k)}$ and $B^{(k)}$ are
estimated using
$\displaystyle\hat{A}^{(k)}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\infty}\left[\Gamma_{i}(k,u,\hat{\sigma}^{(k)2})-V_{c}^{(1)}(k,u,\hat{\gamma}^{(k)},\hat{\eta}^{(k)},\hat{\sigma}^{(k)2})+V_{c}^{(2)}(k,u,\hat{\gamma}^{(k)},\hat{\eta}^{(k)},\hat{\sigma}^{(k)2})\right]dN_{i}(k,u)$
(31) $\displaystyle\hat{B}^{(k)}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\infty}\left[V_{c}^{(1)}(k,u,\hat{\gamma}^{(k)},\hat{\eta}^{(k)},\hat{\sigma}^{(k)2})\right]dN_{i}(k,u).$
(32)
The information matrix at analysis $k$ of the group sequential trial is given
by
$\mathcal{I}_{k}=\frac{1}{n}\left[(\hat{A}^{(k)})^{-1}\hat{B}^{(k)}((\hat{A}^{(k)})^{-1})^{T}\right]^{-1}_{22}.$
Further, a standardised test statistic at analysis $k$ is given by
$Z_{k}=\hat{\eta}^{(k)}\sqrt{\mathcal{I}_{k}}.$
## 3 Designing group sequential trials when the canonical joint distribution
does not hold
### 3.1 Simulation evidence that the trial is conservative with respect to
type 1 error rate
We discuss the implications for a group sequential test when the sequence of
test statistics does not follow the usual canonical joint distribution and we
shall show that in some scenarios, it is appropriate to make this assumption
anyway and proceed with the trial as though the canonical joint distribution
holds anyway. Let $\hat{\theta}^{(1)},\dots,\hat{\theta}^{(K)}$ be the
sequence of treatment effect estimates in a group sequential trial from data
available at analyses $1,\dots,K$ respectively and
$\mathcal{I}_{1},\dots,\mathcal{I}_{K}$ are the associated observed
information levels. The “canonical joint distribution" for the sequence of
estimates $\hat{\theta}^{(1)},\dots,\hat{\theta}^{(K)}$ is such that
1. 1.
$(\hat{\theta}^{(1)},\dots,\hat{\theta}^{(K)})$ is multivariate normal
2. 2.
$\hat{\theta}^{(k)}\sim N(\theta,\mathcal{I}_{k}^{-1}),\hskip 28.45274pt1\leq
k\leq K$
3. 3.
$Cov(\hat{\theta}^{(k_{1})},\hat{\theta}^{(k_{2})})=\mathcal{I}_{k_{2}}^{-1},\hskip
28.45274pt1\leq k_{1}\leq k_{2}\leq K.$
Supposing that the canonical joint distribution holds for the sequence of
treatment effect estimates in the joint model, then the library of standard
group sequential designs, such as Pocock (1977) and O’Brien and Fleming
(1979), could be directly applied in our setting
In Section 2 we saw that asymptotically the sequence of treatment effect
estimates in a group sequential trial obtained from the joint model is
multivariate normally distributed. Further, each of these estimates is
asymptotically unbiased. The first two conditions for the canonical
distribution of the sequence of test statistics are satisfied. However, for
the joint model, we have shown that
$Var(\hat{\eta}^{(k)})=\left[(A^{(k)})^{-1}B^{(k)}(A^{(k)})^{-1}\right]_{22}\text{
for }k=1,\dots,K$ (33)
and
$Cov(\hat{\eta}^{(k_{1})},\hat{\eta}^{(k_{2})})=\left[A^{(k_{1})})^{-1}B^{(k_{1})}(A^{(k_{2})})^{-1}\right]_{22}\text{
for }k_{1}<k_{2}.$ (34)
This implies that the third condition for the canonical joint distribution,
which is the Markov property, is not satisfied. In this section, we discuss
the implications of performing a group sequential trial when the assumption of
a canonical joint distribution fails. We first show that there are some small
differences between the matrices $(A^{(k)})^{-1}B^{(k)}$ and the identity
matrix and describe why this difference is important. Then, we discuss some
alternative methods which aim to correct for this violation of the Markov
property. In method 1, the trial is performed acting as though the canonical
joint distribution holds, and we present some theory that this method controls
the type 1 error rate conservatively.
The theory presented is for the case when a non-binding futility boundary is
used which is where stopping for futility at an interim analysis is not
mandatory as described in GUIDANCE (2018). FDA guidance recommends using non-
binding futility rules because if binding rules are employed and not followed,
then type 1 error rates are inflated. The calculation of the type 1 error rate
therefore only depends on the upper boundary $b_{1},b_{2}$ which is depicted
in Figure 1.
The problem of not obtaining the canonical joint distribution is not unique to
the conditional score method and the proof that this assumption holds is not
always trivial. Slud and Wei (1982) design a group sequential test which uses
the modified-Wilcoxon statistic for two-sample survival data. The authors show
that the increments in test statistics are correlated and their alternative
proposed method has a similar structure to our method 2. Then, Gu and Ying
(1993) present a score process for the analysis of regression data under
general right censorship and they implement the repeated significance method
of Slud and Wei (1982) for the sequential analysis of such data. Further, Cook
and Lawless (1996) discuss a modification of the error spending approach (also
known as Gordon Lan and DeMets (1983)) for sequences of treatment effect
estimates that do not have the independent increments structure.
If the relationship $A^{(k)}=B^{(k)}$ holds for each $k=1,\dots,K,$ then
$Cov(\hat{\eta}^{(k_{1})},\hat{\eta}^{(k_{2})})=\left[(A^{(k_{2})})^{-1}\right]_{p+1,p+1}=Var(\hat{\eta}^{(k_{2})})$
and the third condition holds. Therefore, we shall measure the magnitude of
the violation by considering the matrix $(A^{(k)})^{-1}B^{(k)}$ and to what
extent it differs from the identity matrix. By definition, the variance of the
estimate $\hat{\eta}^{(k)}$ is found in the bottom right entry of the matrix
$(\hat{A}^{(k)})^{-1}\hat{B}^{(k)}(\hat{A}^{(k)})^{-1}$ and hence, the bottom
row of $(\hat{A}^{(k)})^{-1}\hat{B}^{(k)}$ is of interest here.
We can find estimates $\hat{A}^{(k)}$ and $\hat{B}^{(k)}$ from simulated data.
We have done this using a large sample size of n=4800 patients to reduce noise
in these estimates. This calculation is computationally expensive and takes
roughly two hours to compute for this value of $n$. This is appropriate
because, although both matrices depend on the sample size $n$, they can each
be written in the form $\hat{A}^{(k)}=(1/n)\sum_{i=1}^{n}X_{i}(k,\gamma,\eta)$
and $\hat{B}^{(k)}=(1/n)\sum_{i=1}^{n}Y_{i}(k,\gamma,\eta)$ for some functions
$X_{i}(k,\gamma,\eta)$ and $Y_{i}(k,\gamma,\eta).$ Therefore, in the formula
$(\hat{A}^{(k)})^{-1}\hat{B}^{(k)}$, the value of $n$ cancels out, and we are
left with a function that converges in distribution to $(A^{(k)})^{-1}B^{(k)}$
as $n\rightarrow\infty.$ Further, to reduce simulation error, the true values
of $\gamma$ and $\eta$ are used in this calculation, which is appropriate
because of consistency of the estimates $\hat{\gamma}$ and $\hat{\eta}$.
Table 1 shows the matrix $(\hat{A}^{(1)})^{-1}\hat{B}^{(1)}$ for $\eta=0$ and
different values of $\gamma$ and $\sigma^{2}.$ We have chosen to investigate
the properties of this matrix at the first analysis because we see empirically
that the majority of problems occur at early interim analyses. We simulated a
data set with parameter values $\gamma=0,0.03,0.06,0.09,\sigma^{2}=0,1,10,100$
and $\eta=0.$
| $\boldsymbol{\gamma}$
---|---
$\mathbf{\sigma^{2}}$ | $\mathbf{0}$ | $\mathbf{0.03}$ | $\mathbf{0.06}$ | $\mathbf{0.09}$
$\mathbf{0}$ | $\left(\begin{array}[]{cc}1.00&0.00\\\ 0.00&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.00&0.00\\\ 0.00&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.00&0.00\\\ 0.00&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.00&0.00\\\ 0.00&1.00\end{array}\right)$
$\mathbf{1}$ | $\left(\begin{array}[]{cc}1.00&0.00\\\ 0.00&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.01&0.00\\\ 0.00&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.01&0.00\\\ 0.00&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.02&0.00\\\ 0.00&1.00\end{array}\right)$
$\mathbf{10}$ | $\left(\begin{array}[]{cc}1.03&0.00\\\ 0.01&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.06&0.00\\\ 0.00&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.12&0.00\\\ -0.02&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.22&0.00\\\ 0.00&1.00\end{array}\right)$
$\mathbf{100}$ | $\left(\begin{array}[]{cc}1.28&0.00\\\ -0.10&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}1.63&0.00\\\ -0.47&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}2.32&0.00\\\ -0.01&1.00\end{array}\right)$ | $\left(\begin{array}[]{cc}3.49&0.00\\\ 1.00&1.00\end{array}\right)$
* •
Parameter values $\gamma=0,0.03,0.06,0.09$ and $\sigma^{2}=0,1,10,100$ in the
joint model simulated with 4800 patients.
Table 1: Matrix $(\hat{A}^{(1)})^{-1}\hat{B}^{(1)}$ for the null hypothesis
$\eta=0$.
The matrices $A^{(k)},\hat{A}^{(k)},B^{(k)}$ and $\hat{B}^{(k)}$ are each of
dimension $2\times 2.$ The function $V^{(2)}(k,u,\gamma,\eta)$ is such that
$\left[V^{(2)}(k,u,\gamma,\eta)\right]_{12}=\left[V^{(2)}(k,u,\gamma,\eta)\right]_{22}=0$,
and hence by Equations (31) and (32) it can be shown that
$[\hat{A}^{(k)}]_{12}=[\hat{B}^{(k)}]_{12}$ and
$[\hat{A}^{(k)}]_{22}=[\hat{B}^{(k)}]_{22}.$ Further simple algebraic
manipulation gives $[(\hat{A}^{(k)})^{-1}\hat{B}^{(k)}]_{12}=0$ and
$[(\hat{A}^{(k)})^{-1}\hat{B}^{(k)}]_{22}=1$ exactly, which is shown in Table
1. The fact that $[(\hat{A}^{(1)})^{-1}\hat{B}^{(1)}]_{11}$ is a long way from
$1$ for $\sigma^{2}=10$ and $\sigma^{2}=100$ is therefore not a problem. As
$\sigma^{2}$ increases, the absolute value of
$[(\hat{A}^{(1)})^{-1}\hat{B}^{(1)}]_{21}$ increases, but the value of
$\gamma$ has a small impact on the value of
$[(\hat{A}^{(1)})^{-1}\hat{B}^{(1)}]_{21}.$ Thus, we may expect large values
of $\sigma^{2}$ to affect the achieved type 1 error rate.
### 3.2 Method 1 - canonical joint distribution assumed
We consider alternative methods for creating a group sequential trial when the
canonical joint distribution does not hold. In the first method, we construct
the group sequential test by estimating $Var(\hat{\eta}^{(k)}),k=1,\dots,K$
from the data and supposing $Cov(\hat{\eta}^{(k_{1})},\hat{\eta}^{(k_{2})})$
for $k_{1}<k_{2}$ are as specified in the canonical joint distribution. We
shall prove that we have type 1 error rate less than $\alpha$ and we also
show, through simulation, that this method performs satisfactorily in practice
with error rates diverging minimally from planned significance and power.
For this proof, we consider the case where $K=2$ and the futility boundary is
non-binding. We present a sketch proof in the supplementary materials for the
case $K=3$ and we believe that the results generalise for cases $K>3.$ To
prove that the type 1 error rates are conservative, we shall compare the
probabilities of crossing the boundaries of a group sequential trial for two
sequences of treatment effect estimates; one where the canonical joint
distribution does not hold and one where this assumption does hold. For a
group sequential trial with $K=2$, suppose that
$\hat{\eta}_{1},\hat{\eta}_{2}$ are the sequence of treatment effect estimates
that are calculated using the conditional score method. Let the true variance-
covariance matrix for this sequence of estimates be $\Sigma$. Proceeding using
method 1, we let the information levels be calculated as
$\mathcal{I}_{k}=(\Sigma_{kk})^{-1}$ and the $Z-$statistic is given by
$Z_{k}=\hat{\eta}_{k}\sqrt{\mathcal{I}_{k}}$ for $k=1,2.$ Under $H_{0}$ we
have
$\begin{bmatrix}\hat{\eta}_{1}\\\ \hat{\eta}_{2}\end{bmatrix}\sim
N_{K}\left[\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left(\begin{array}[]{cc}\Sigma_{11}&\Sigma_{12}\\\
\Sigma_{12}&\Sigma_{22}\end{array}\right)\right],\hskip
28.45274pt\begin{bmatrix}Z_{1}\\\ Z_{2}\end{bmatrix}\sim
N_{K}\left[\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left(\begin{array}[]{cc}1&\rho\\\
\rho&1\end{array}\right)\right]$ (35)
where the correlation parameter $\rho$ is given by
$\rho=Cov(Z_{1},Z_{2})=\frac{\Sigma_{12}}{\sqrt{\Sigma_{11}\Sigma_{22}}}.$
(36)
Suppose instead, that we have a different sequence of treatment effect
estimates $\hat{\eta}^{*}_{1},\hat{\eta}^{*}_{2}$ with distribution given
below. The values of $\Sigma_{11}$ and $\Sigma_{22}$ are the same as in (35)
and using the same information levels given by
$\mathcal{I}_{k}=(\Sigma_{kk})^{-1}$, we define the standardised statistics
$Z^{*}_{k}=\hat{\eta}^{*}_{k}\sqrt{\mathcal{I}_{k}}$ for $k=1,\dots,K.$ The
joint distribution of the sequence $\hat{\eta}^{*}_{1},\hat{\eta}^{*}_{2}$ and
the distribution of the $Z-$statistics are given by
$\begin{bmatrix}\hat{\eta}_{1}^{*}\\\ \hat{\eta}_{2}^{*}\end{bmatrix}\sim
N_{K}\left[\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left(\begin{array}[]{cc}\Sigma_{11}&\Sigma_{22}\\\
\Sigma_{22}&\Sigma_{22}\end{array}\right)\right],\hskip
28.45274pt\begin{bmatrix}Z_{1}^{*}\\\ Z_{2}^{*}\end{bmatrix}\sim
N_{K}\left[\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left(\begin{array}[]{cc}1&\rho^{*}\\\
\rho^{*}&1\end{array}\right)\right]$ (37)
with correlation parameter
$\rho_{12}^{*}=Cov(Z_{1}^{*},Z_{1}^{*})=\sqrt{\frac{\Sigma_{22}}{{\Sigma_{11}}}}.$
(38)
This sequence of treatment effect estimates
$\hat{\eta}^{*}_{1},\hat{\eta}^{*}_{2}$ therefore has the canonical joint
distribution and $Z^{*}_{1},Z^{*}_{2}$ has the canonical joint distribution
for a sequence of $Z$-statistics, with information levels
$\mathcal{I}_{k}=1/\Sigma_{kk}$ for $k=1,\dots,K$.
The upper boundary points $b_{1},b_{2}$ are calculated under the assumption
that the canonical joint distribution holds so as to give a group sequential
test with the correct type 1 error rate $\alpha$. Hence, we have that
$\mathbb{P}_{\eta=0}(Z^{*}_{1}>b_{1}\cup Z^{*}_{2}>b_{2})=\alpha.$
We consider the probability of rejecting $H_{0}$ when we apply this boundary
to the sequence $Z_{1},Z_{2}.$ We aim to prove that
$\mathbb{P}_{\eta=0}(Z_{1}>b_{1}\cup Z_{2}>b_{2})\leq\alpha.$ (39)
The following conditions are needed for the proof that type 1 error rate is
conservative.
###### Conditions 1.
1. 1.
The upper boundary of a group sequential trial, on the $Z-$scale, is such that
$b_{1}\geq b_{2}\geq 0.$
2. 2.
We have $\Sigma_{12}\geq\Sigma_{22}.$
Condition 1 of Conditions 1 holds under common error spending functions with
increasing information sequences, which are most often used in practice.
Condition 2 of Conditions 1 should be checked by simulation before proceeding
with the analysis. To do so, the investigator would choose sensible values for
all the parameters in the joint model, simulate a large dataset of $4800$
patients using these parameter values, and calculate an estimate for the
variance-covariance matrix $\Sigma$ for the sequence of estimates
$\hat{\eta}^{(1)},\dots,\hat{\eta}^{(K)}.$ We have found that calculations for
various examples has always lead to condition 2 being satisfied. Further, the
scenarios that we have checked span a 3-dimensional grid of $\eta,\gamma$ and
$\sigma^{2}$ values each ranging from small to large and hence, we believe
that the scenarios we have checked span a suitable range of the parameter
values. In the rare event that this condition does not hold, a solution is to
employ method 2, which will be described later. It can also be seen by simple
algebraic manipulation that condition 2 implies $\rho\geq\rho^{*}.$
The following theorem shows that Equation (39) holds.
###### Theorem 2.
Let $Z_{1},Z_{2}$ be the standardised statistics of a group sequential trial
with distribution given by (35) and let $Z^{*}_{1},Z^{*}_{2}$ be the
statistics with distribution given by (37). Let $\alpha$ be the planned type 1
error rate and suppose that $b_{1},b_{2}$ are the upper boundary points on the
$Z$-scale such that
$\mathbb{P}_{\eta=0}(Z^{*}_{1}>b_{1}\cup Z^{*}_{2}>b_{2})=\alpha.$
Suppose that Conditions 1 hold. Then the Type 1 error rate when applying the
boundary for $Z^{*}_{1},\dots,Z^{*}_{K}$ to $Z_{1},\dots,Z_{K}$ is
$\mathbb{P}_{\eta=0}(Z_{1}>b_{1}\cup Z_{2}>b_{2})\leq\alpha.$
###### Proof.
The problem is equivalent to proving that
$\mathbb{P}_{\eta=0}(Z_{1}>b_{1}\cup
Z_{2}>b_{2})\leq\mathbb{P}_{\eta=0}(Z^{*}_{1}>b_{1}\cup Z^{*}_{2}>).$
and by another representation for the above probabilities, we aim to show that
$\displaystyle\mathbb{P}_{\eta=0}(Z_{1}>b_{1})+\mathbb{P}_{\eta=0}(Z_{1}>b_{2})-\mathbb{P}_{\eta=0}(Z_{1}>b_{1}\cap
Z_{2}>b_{2})\leq\mathbb{P}_{\eta=0}(Z_{1}>b_{1}\cap Z_{2}>b_{2})$
$\displaystyle\leq$
$\displaystyle\mathbb{P}_{\eta=0}(Z^{*}_{1}>b_{1})+\mathbb{P}_{\eta=0}(Z^{*}_{1}>b_{2})-\mathbb{P}_{\eta=0}(Z^{*}_{1}>b_{1}\cap
Z^{*}_{2}>b_{2})\leq\mathbb{P}_{\eta=0}(Z_{1}>b_{1}\cap Z_{2}>b_{2}).$
Under $\eta=0$, the marginal distributions of $Z_{1},Z_{2},Z^{*}_{1}$ and
$Z^{*}_{2}$ are equivalent and are all $N(0,1)$ random variables and hence the
probabilities are such that
$\mathbb{P}_{\eta=0}(Z_{k}>b_{k})=\mathbb{P}_{\eta=0}(Z^{*}_{k}>b_{k})$ for
each $k=1,2.$ Therefore, the problem is reduced to showing that
$\mathbb{P}_{\eta=0}(Z^{*}_{1}>b_{1}\cap
Z^{*}_{2}>b_{2})\leq\mathbb{P}_{\eta=0}(Z_{1}>b_{1}\cap Z_{2}>b_{2}).$ (40)
when $\rho^{*}\leq\rho.$
In the below calculations, we appeal to the fact that for two random variables
which are bivariate normally distributed, the conditional distribution of one
normal random variable on the other normal random variable is also normal.
Specifically, we have that $Z_{2}|Z_{1}=z_{1}\sim N(\rho z_{1},1-\rho^{2})$.
Let $\phi(\cdot)$ and $\Phi(\cdot)$ denote the probability density function
and cumulative distribution function of a standard normal random variable
respectively, then the probability on the left hand side in Equation (40) is
$\mathbb{P}_{\eta=0}(Z_{1}>b_{1}\cap
Z_{2}>b_{2})=\int_{b_{1}}^{\infty}\left[1-\Phi\left(\frac{b_{2}-\rho
z_{1}}{\sqrt{1-\rho^{2}}}\right)\right]\phi(z_{1})dz_{1}.$
The corresponding calculation where $Z^{*}_{1}$ and $Z^{*}_{2}$ replace
$Z_{1}$ and $Z_{2}$ yields the following
$\mathbb{P}_{\eta=0}(Z^{*}_{1}>b_{1}\cap
Z^{*}_{2}>b_{2})=\int_{b_{1}}^{\infty}\left[1-\Phi\left(\frac{b_{2}-\rho^{*}z_{1}}{\sqrt{1-{\rho^{*}}^{2}}}\right)\right]\phi(z_{1})dz_{1}$
and since $\Phi(\cdot)$ is strictly increasing, it suffices to show that
whenever $z_{1}>b_{1}$, then
$\frac{b_{2}-\rho
z_{1}}{\sqrt{1-{\rho}^{2}}}\leq\frac{b_{2}-\rho^{*}z_{1}}{\sqrt{1-{\rho^{*}}^{2}}}.$
(41)
We have by assumption that $\rho^{*}\leq\rho.$ Further note that by definition
$0\leq\rho\leq 1$ and $0\leq\rho^{*}\leq 1.$ The following shows a simple
algebraic manipulation of this inequality, which gives
$\rho^{*}\leq\rho\iff\frac{\sqrt{1-{\rho^{*}}^{2}}-\sqrt{1-\rho^{2}}}{\rho\sqrt{1-{\rho^{*}}^{2}}-\rho^{*}\sqrt{1-\rho^{2}}}\leq
1.$
Finally, using the above inequality and Conditions 1 that $0\leq b_{2}\leq
b_{1}$, we have for $z_{1}\geq b_{1}$ that
$b_{2}\frac{\sqrt{1-{\rho^{*}}^{2}}-\sqrt{1-\rho^{2}}}{\rho\sqrt{1-{\rho^{*}}^{2}}-\rho^{*}\sqrt{1-\rho^{2}}}\leq
b_{2}\leq b_{1}\leq z_{1}$
and a simple rearrangement shows that Equation (41) is satisfied. ∎
Table 2 show estimates of type 1 error rate. This was by simulating $10^{4}$
data sets, each with a sample size $n=365.$ This sample size was chosen
because it gives power 0.9 for $\gamma=0.06,\sigma^{2}=1,\phi=2.5$ and for
other parameter values, the power was close to 0.9. For each data set, we
calculate the estimates $\hat{\eta}^{(1)},\dots,\hat{\eta}^{(K)}$ and
estimates of the covariance matrices $\Sigma^{(1)},\dots,\Sigma^{(K)}$ using
estimates
$\hat{A}^{(1)},\dots,\hat{A}^{(K)},\hat{B}^{(1)},\dots,\hat{B}^{(K)}$ given by
Equations (31) and (32). The boundary points $a_{1},\dots,a_{K}$ and
$b_{1},\dots,b_{K}$ are then calculated under the assumption that the
canonical joint distribution holds and the type 1 error rate is calculated as
the proportion of replicates that reject the null hypothesis. This simulation
study is computationally expensive and with $10^{4}$ replicates, there is
noise in the simulation results. Taking this into account, the simulation
results support the relevance of asymptotic theory since all empirical type 1
error rates are within 2 standard deviations of 0.025.
Method | $\boldsymbol{\phi}$ | $\boldsymbol{\sigma^{2}}$ | Type 1 error | Percentage of time method fails
---|---|---|---|---
| | | $\boldsymbol{\gamma}\mathbf{=0}$ | $\boldsymbol{\gamma}\mathbf{=0.03}$ | $\boldsymbol{\gamma}\mathbf{=0.06}$ | $\boldsymbol{\gamma}\mathbf{=0.09}$ | $\boldsymbol{\gamma}\mathbf{=0}$ | $\boldsymbol{\gamma}\mathbf{=0.03}$ | $\boldsymbol{\gamma}\mathbf{=0.06}$ | $\boldsymbol{\gamma}\mathbf{=0.09}$
1 | $\mathbf{2.5}$ | $\mathbf{0}$ | 0.022 | 0.022 | 0.022 | 0.023 | 0.000 | 0.000 | 0.000 | 0.000
1 | $\mathbf{2.5}$ | $\mathbf{1}$ | 0.024 | 0.025 | 0.026 | 0.023 | 0.000 | 0.000 | 0.000 | 0.000
1 | $\mathbf{2.5}$ | $\mathbf{10}$ | 0.023 | 0.024 | 0.023 | 0.025 | 0.000 | 0.000 | 0.000 | 0.000
1 | $\mathbf{2.5}$ | $\mathbf{100}$ | 0.023 | 0.028 | 0.026 | 0.024 | 0.000 | 0.000 | 0.000 | 0.000
2 | $\mathbf{2.5}$ | $\mathbf{0}$ | 0.022 | 0.022 | 0.022 | 0.023 | 0.000 | 0.000 | 0.000 | 0.000
2 | $\mathbf{2.5}$ | $\mathbf{1}$ | 0.024 | 0.025 | 0.026 | 0.023 | 0.000 | 0.000 | 0.000 | 0.000
2 | $\mathbf{2.5}$ | $\mathbf{10}$ | 0.023 | 0.024 | 0.023 | 0.025 | 0.040 | 0.100 | 0.120 | 0.060
2 | $\mathbf{2.5}$ | $\mathbf{100}$ | 0.022 | 0.028 | 0.026 | 0.029 | 49.140 | 37.360 | 35.950 | 40.420
3 | $\mathbf{2.5}$ | $\mathbf{0}$ | 0.028 | 0.024 | 0.023 | 0.023 | 0.000 | 0.000 | 0.000 | 0.000
3 | $\mathbf{2.5}$ | $\mathbf{1}$ | 0.028 | 0.026 | 0.027 | 0.023 | 0.000 | 0.000 | 0.000 | 0.000
3 | $\mathbf{2.5}$ | $\mathbf{10}$ | 0.029 | 0.025 | 0.023 | 0.026 | 0.140 | 0.050 | 0.060 | 0.310
3 | $\mathbf{2.5}$ | $\mathbf{100}$ | 0.024 | 0.028 | 0.027 | 0.029 | 1.930 | 1.950 | 1.840 | 1.850
* •
Calculated using a simulation study with 365 patients and $10^{4}$ replicates.
* •
Standard error is 0.0016 when the method does not fail.
Table 2: Type 1 error rates and percentage of times that each method fails.
### 3.3 Method 2 - Use the complete structure of the covariance matrix
The second method for dealing with estimates from the joint model does not
rely on the canonical joint distribution assumption. Instead, the group
sequential boundaries are calculated using the complete structure of the
variance-covariance matrix for the sequence of treatment effect estimates
across analyses. This differs from when the canonical joint distribution is
assumed because in such a case, only the variances are required and the
covariances are ignored. To calculate the boundary points of the group
sequential test, we are required to calculate a $K-$dimensional integral over
the joint probability distribution of the sequence of test statistics. This
integration calculation can be performed numerically using the R package
mvtnorm by Genz et al. (2020).
However, this method poses some practical difficulties. For example, during
the conditional score method, the variance-covariance matrix $\Sigma$ in
Equation (20) is estimated with error which can sometimes result in a non
positive-definite estimate $\hat{\Sigma}.$ In such a case, the boundary
calculations cannot be performed. Slud and Wei (1982) allude to a similar
problem in their sequential analysis of modified-Wilcoxon scores for two-
sample survival data. Table 2 shows the percentage of times that this problem
occurs. We see that for extremely noisey longitudinal data with
$\sigma^{2}=100$ this problem occurs roughly 40% of the time and for
$\sigma^{2}=10$ we have the problem occurring infrequently. This is not a
problem for small $\sigma^{2}$ and in such a case, the type 1 error rates are
as expected. In summary, this method makes no assumption about the covariance
structure of the sequence of treatment effect estimates and therefore the type
1 error rate is preserved.
### 3.4 Method 3 - Create an asymptotically efficient estimate
For the final method, we follow the approach of Van Lancker et al. (2022)
where a new estimator is created which reaches asymptotic efficiency as it is
designed in such a way to minimise the variance. The efficient estimate at
analysis $k$ is a linear combination of the original estimates at analyses up
to and including $k$. We choose the weights of the linear combination using a
Lagrange multiplier method in such a way that the variance is minimised. Ou
rmotivation for this method follows the simple result by Jennison and Turnbull
(1997), that all asymptotically efficient estimators have the canonical joint
distribution. Van Lancker et al. (2022) show that this new estimator has the
correct canonical distribution, and hence the group sequential methods can be
used without hesitation.
Similarly to method 2, there are some limitations to this method as it relies
too heavily on accurately estimating the covariance matrix of the sequence of
treatment effect estimates. Van Lancker et al. (2022) do not observe these
issues since their results focus on simulations with sample sizes much larger
than our chosen $n=365$. In some cases, because the variance-covariance matrix
is estimated with error, it is possible to choose the weights of the Lagrange
multiplier in such a way that the new estimate has negative variance. Table 2
shows a similar pattern to method 2, that this numerical problem occurs when
the variance of the longitudinal data is extreme so that $\sigma^{2}=100$,
occurs infrequently for $\sigma^{2}=10$ and no problems occur for small
$\sigma^{2}.$
Given that $Cov(\hat{\eta}^{(k_{1})},\hat{\eta}^{(k_{2})})$ is very close to
$Var(\hat{\eta}^{(k_{2})})$ for $k_{1}<k_{2}$, there is not a lot to be gained
by the more complex methods 2 and 3. The calculations are sensitive to errors
in estimates of $Cov(\hat{\eta}^{(k_{1})},\hat{\eta}^{(k_{2})})$ and
$Var(\hat{\theta}^{(k_{2})})$. In some cases, the errors in these calculations
lead to these methods simply not working. This raises doubts about how well
they work in less extreme cases. Despite these problems, methods 2 and 3
perform adequately in simulation studies. However, this does not change our
view that method 1 is preferable.
## 4 Results
We aim to assess the efficiency gain when longitudinal data are included in
the analysis compared to when this longitudinal data is available, yet
ignored. In this case, we believe that our joint model is correctly specified
and therefore, we shall simulate clinical trial data from the true joint model
and analyse it in two separate ways. The first way is to fit the data to the
joint model using the conditional score method to find a treatment effect
estimate and the second way is to ignore the longitudinal data, fitting a Cox
model to the survival data and find the maximum partial likelihood estimate of
the treatment effect. We are interested in comparing the sample sizes required
in each method to achieve the same power. A comparison of these sample sizes
reflects the efficiency of the test incorporating the longitudinal data.
We shall simulate using the joint model in Equation (4). To fit the joint
model to the data, we do not need to assume a distribution for the random
effects, however we must specify this for simulation purposes and we shall
simulate using the following
$\begin{bmatrix}b_{i0}\\\ b_{i1}\end{bmatrix}\sim
N\left(\begin{bmatrix}\mu_{0}\\\
\mu_{1}\end{bmatrix},\begin{bmatrix}\phi_{0}^{2}&0\\\
0&\phi_{1}^{2}\end{bmatrix}\right).$
Unless otherwise stated, throughout the simulation studies, we shall use
parameter values
$(\mu_{0},\mu_{1})=(6,3),\phi_{0}^{2}=12.25,\phi_{1}^{2}=6.25,\sigma^{2}=10,h_{0}(t)=5.5,\gamma=0.03,\lambda=0.022\text{
and }\eta=-0.5.$ (42)
These parameter values are based on the aids dataset in the R package JM by
Rizopoulos (2010).
We shall test the one-sided hypothesis
$H^{(J)}_{0}:\eta_{J}\geq 0,\hskip 28.45274ptH^{(J)}_{A}:\eta_{J}<0.$
Here the subscript notation $J$ represents that the parameter $\theta_{J}$ is
from the "joint" model. We fit the joint model using the conditional score
method to find a treatment effect estimate $\hat{\eta}_{J}$ in order to
perform this hypothesis test.
Our aim is to find the sample size, $n_{J}$, required using the conditional
score method to achieve Type 1 error rate $\alpha=0.025$ when the true
treatment effect is $\eta_{J}=0$ and power $1-\beta=0.9$ when $\eta_{J}=-0.5$.
An estimate of the sample size will be calculated by simulation. The trial is
designed with 2 years recruitment and 3 years follow-up and the group
sequential trial has analyses at $20,30,40,50$ and $60$ months. When
increasing the sample size, we do so by increasing the rate of recruitment so
accrual and follow-up periods in the the trial design stay fixed. This is to
ensure that differences in power are purely due to the sample size and not
changes in the trial design as sample size increases.
The trial uses an error spending design given by Gordon Lan and DeMets (1983).
That is, the boundary constants $b_{1},\dots,b_{K}$ are chosen to satisfy
$\displaystyle\mathbb{P}_{\eta=0}(Z_{1}>b_{1})$
$\displaystyle=\min\\{\alpha(\mathcal{I}_{1}/\mathcal{I}_{max})^{2},\alpha\\}$
$\displaystyle\mathbb{P}_{\eta=0}(Z_{1}<b_{1},\dots,Z_{k-1}<b_{k-1},Z_{k}>b_{k})$
$\displaystyle=\min\\{\alpha(\mathcal{I}_{k}/\mathcal{I}_{max})^{2},\alpha\\}-\min\\{\alpha(\mathcal{I}_{k-1}/\mathcal{I}_{max})^{2},\alpha\\}\text{
for }k=2,\dots,K$
where the value of $\mathcal{I}_{max}$ is calculated to ensure that the trial
has power $1-\beta$ when $\eta=-0.5$ as described by Jennison and Turnbull
(2000).
Maximum sample sizes $n_{J}$ are given in Table 3. The first analysis, at 20
months, occurs just before the end of the recruitment period which is 2 years.
Trials that terminate at the first interim analysis may recruit less than
$n_{J}$ patients, however this occurs with very small probability. Hence, the
expected sample size will be very close to the maximum sample size for each
model and therefore the maximum sample size is a useful measure to compare
methods. The sample size increases as $\sigma^{2}$ increases. This reflects
that noisy longitudinal data is associated with high variance or small
information levels. Sample sizes are particularly high in each case where
$\sigma^{2}=100,$ which has been chosen as an extreme value. Further, sample
sizes appear to increase slightly with $\gamma$ and decrease slightly with
$\phi^{2}_{1}$.
We now consider the analysis when the longitudinal data is ignored. We believe
the joint model to be true and correct, however we shall fit the data to a Cox
model. To do so, we shall simulate data from the joint model and then fit this
data to a misspecified Cox proportional hazards model. The Cox model is given
by:
$\lambda_{i}(t)=\tilde{\lambda}_{0}(t)\exp\\{\eta_{C}Z_{i}\\}.$ (43)
For this clinical trial, we test the hypothesis
$H_{0}^{(C)}:\eta_{C}\geq 0,\hskip 28.45274ptH_{A}^{(C)}:\eta_{C}<0$ (44)
and we find a treatment effect estimate $\hat{\eta}_{C}$ using the maximum
partial likelihood method as in Jennison and Turnbull (1997).
Although this model is misspecified, type 1 error is not affected. This is
because, under $H_{0}^{(J)}$ we have $\eta_{J}=0$ and there is no difference
between treatment groups in overall survival. When fitting the Cox model to
the data, the longitudinal data trajectory is reflected in the function
$\tilde{\lambda}_{0}(t)$ so that we also have that $\eta_{C}=0.$ Hence,
$H_{0}^{(C)}$ is also true.
Let $n_{C}$ be the sample size such that we achieve type 1 error
$\alpha=0.025$ when $\eta_{J}=0$ and power $1-\beta=0.9$ when $\eta_{J}=-0.5$
when we perform the hypothesis test in (44). Values of $n_{C}$ and $n_{J}$ are
given in Table 3. The values of $\gamma$ and $\phi_{1}^{2}$ for simulation are
varied. Notice that $n_{C}$ does not change with $\sigma^{2}$ since this plays
no role in simulating survival times, and the longitudinal data, which is
affected by $\sigma^{2}$, is ignored. As the value of $\gamma$ increases, the
sample size $n_{C}$ increases. This represents that as the longitudinal data
has more weight in the survival hazard rate, ignoring the longitudinal data
results in an increasingly inefficient clinical trial. When $\gamma=0,$ this
represents the case where longitudinal data is available yet has no influence
on the survival function. In this case, $n_{C}<n_{J}$ and it is more efficient
to fit the data to the simple Cox model. We see that the value of $n_{C}$
increasess with $\phi_{1}^{2}$. This is the variance amongst patients of the
slopes of the longitudinal trajectories. This indicates that the simple Cox
model is unable to account for large differences between individual patients.
To compare the sample sizes obtained using the joint model and the
misspecified Cox model, we define “relative efficiency" to be
$RE=\frac{n_{C}}{n_{J}}.$ Using this definition, when $RE>1$ we interpret this
as the joint model analysis being the more efficient model to use and
similarly when $RE<1,$ the Cox model analysis is the more efficient analysis
method.
Table 3 shows the relative efficiency results. We see that RE increases with
$\gamma$, increases with $\phi_{1}^{2}$ and remains constant with $\sigma^{2}$
apart from the case where $\sigma^{2}=100$ which reflects extremely noisy
data. Also, we see that $RE=0.97$ when $\gamma=0$ and $\sigma^{2}=100$ which
indicates that when the longitudinal data is not correlated with the survival
endpoint and the longitudinal data is noisy, the simple Cox model is a
slightly more efficient method for estimating the treatment effect. Apart from
the case where $\gamma=0$, it is always more efficient to analyse the data
using the joint modelling approach. Even when $\gamma=0,$ fitting the data to
the simple Cox model for survival data is only marginally more efficient than
fitting the data to the joint model. In the extreme case, 2.63 times as many
patients are required to analyse the data using the Cox model as when the
joint modelling framework is used.
$\boldsymbol{\phi}$ | $\boldsymbol{\sigma^{2}}$ | $\boldsymbol{\gamma}\mathbf{=0}$ | $\boldsymbol{\gamma}\mathbf{=0.03}$ | $\boldsymbol{\gamma}\mathbf{=0.06}$ | $\boldsymbol{\gamma}\mathbf{=0.09}$
---|---|---|---|---|---
| | $n_{C}$ | $n_{J}$ | RE | $n_{C}$ | $n_{J}$ | RE | $n_{C}$ | $n_{J}$ | RE | $n_{C}$ | $n_{J}$ | RE
$\mathbf{2.5}$ | $\mathbf{0}$ | 363 | 363 | 1.00 | 421 | 365 | 1.16 | 528 | 364 | 1.45 | 607 | 365 | 1.67
$\mathbf{2.5}$ | $\mathbf{1}$ | 363 | 364 | 1.00 | 421 | 365 | 1.16 | 528 | 365 | 1.45 | 607 | 369 | 1.65
$\mathbf{2.5}$ | $\mathbf{10}$ | 363 | 364 | 1.00 | 421 | 364 | 1.16 | 528 | 365 | 1.45 | 607 | 375 | 1.62
$\mathbf{2.5}$ | $\mathbf{100}$ | 363 | 373 | 0.97 | 421 | 374 | 1.13 | 528 | 420 | 1.26 | 607 | 522 | 1.16
$\mathbf{0}$ | $\mathbf{1}$ | 351 | 362 | 0.97 | 371 | 363 | 1.02 | 401 | 367 | 1.09 | 439 | 357 | 1.23
$\mathbf{2.5}$ | $\mathbf{1}$ | 363 | 364 | 1.00 | 421 | 365 | 1.16 | 528 | 365 | 1.45 | 607 | 369 | 1.65
$\mathbf{5}$ | $\mathbf{1}$ | 366 | 365 | 1.00 | 462 | 378 | 1.22 | 720 | 391 | 1.84 | 990 | 421 | 2.35
$\mathbf{7.5}$ | $\mathbf{1}$ | 380 | 372 | 1.02 | 501 | 404 | 1.24 | 804 | 416 | 1.93 | 1185 | 450 | 2.63
* •
$RE=n_{C}/n_{J}$, $\gamma$ is the correlation parameter between the
longitudinal and survival endpoints,$\sigma^{2}$ is the measurement error of
the longitudinal data and $\phi_{2}$ is the variance of the random effects
$b_{1i},\dots,b_{1n}$.
Table 3: Maximum sample sizes required for power 0.9 when fitting the joint
model and the misspecified Cox model to the data.
## 5 Conclusions
The conditional score method is used to find a treatment effect estimate in
the joint model of longitudinal and time to event data and we have displayed
new theoretical results for the distribution of the sequence of treatment
effect estimates $\hat{\eta}_{1},\dots,\hat{\eta}_{K}$ found using the
conditional score method in a group sequential trial. Although the canonical
joint distribution for the sequence $\hat{\eta}_{1},\dots,\hat{\eta}_{K}$ does
not hold, we show that it is sensible and practical to proceed assuming that
the canonical joint distribution holds anyway. In particular, we have proven
that by assuming the canonical joint distribution holds, and using a non-
binding futility boundary, the trial is conservative with respect to type 1
error rates. Finally, using simulation studies we have seen that the
deviations from planned type 1 error $\alpha$ are minimal. Other benefits of
using the conditional score method are that no distributional assumptions are
required for the random effects of the longitudinal data and the analysis is
semi-parametric so that it is not necessary to estimate the baseline hazard
function.
We have shown that by including the longitudinal data, compared to the case
where the longitudinal data is observed but left out of the analysis, we can
greatly improve the efficiency of the trial with respect to sample size. In
some cases, 2.63 times as many patients are required to achieve the same power
in the analysis where the longitudinal data is left out.
## Appendix 1
### Conditions for the asymptotic theory of parameter estimates
To ensure the existence of the asymptotic covariance matrix $\Sigma$, we
require the probabilistic limits of $S_{c}^{(0)}(k),\dots,S_{c}^{(2)}(k)$ and
$C^{(1)}(k),\dots,C^{(3)}(k)$ given by (17) to exist. The limits are defined
through the following conditions.
###### Conditions 2.
1. 1.
There exist neighbourhoods $\Gamma$ of $\gamma_{0}$ and $N$ of $\eta_{0}$ and
for each $k=1,\dots,K$ there are functions
$s_{c}^{(0)}(k,t,\gamma,\eta,\sigma^{2})$,
$s_{c}^{(1)}(k,t,\gamma,\eta,\sigma^{2})$,
$s_{c}^{(2)}(k,t,\gamma,\eta,\sigma^{2})$,
$c^{(1)}(k,t,\gamma,\eta,\sigma^{2})$ and
$c^{(2)}(k,t,\gamma,\eta,\sigma^{2})$ defined on $[0,\infty)\times\Gamma\times
N$ such that
$\displaystyle\sup_{t\in[0,\infty),\gamma\in\Gamma,\eta\in
N}\left\|S_{c}^{(j)}(k,t,\gamma,\eta,\sigma^{2})-s_{c}^{(j)}(k,t,\gamma,\eta,\sigma^{2})\right\|$
$\displaystyle\xrightarrow{p}0\text{ for }j=0,1,2$
$\displaystyle\sup_{t\in[0,\infty),\gamma\in\Gamma,\eta\in
N}\left\|C^{(j)}(k,t,\gamma,\eta,\sigma^{2})-c^{(j)}(k,t,\gamma,\eta,\sigma^{2})\right\|$
$\displaystyle\xrightarrow{p}0\text{ for }j=1,2.$
2. 2.
Each $s_{c}^{(j)}(k,t,\gamma,\eta,\sigma^{2})$ and
$c^{(j)}(k,t,\gamma,\eta,\sigma^{2})$ is a continuous function of
$\gamma\in\Gamma$ and $\eta\in N$ uniformly in $t\in[0,\infty),$ and bounded
on $[0,\infty)\times\Gamma\times N$.
3. 3.
For each $k=1,\dots,K$ $s_{c}^{(0)}$ and $c^{(0)}$ are bounded away from zero
on $[0,\infty)\times\Gamma\times N$.
It is clear that the probabilistic limits $e_{c}(k,t,\gamma,\eta,\sigma^{2})$
of $E_{c}(k,t,\gamma,\eta,\sigma^{2})$,
$v^{(1)}_{c}(k,t,\gamma,\eta,\sigma^{2})$ of
$V^{(1)}_{c}(k,t,\gamma,\eta,\sigma^{2})$ and
$v^{(2)}_{c}(k,t,\gamma,\eta,\sigma^{2})$ of
$V^{(2)}_{c}(k,t,\gamma,\eta,\sigma^{2})$ exist and can expressed in terms of
$s_{c}^{(j)}(k,t,\gamma,\eta,\sigma^{2})$ and
$c^{(j)}(k,t,\gamma,\eta,\sigma^{2})$ for $j=0,1,2$ and these are
$e_{c}=\frac{s_{c}^{(1)}}{s_{c}^{(0)}},\hskip
28.45274ptv^{(1)}_{c}=\frac{s_{c}^{(2)}}{s_{c}^{(0)}}-\frac{s_{c}^{(1)}s_{c}^{(1)^{T}}}{[s_{c}^{(0)}]^{2}},\hskip
28.45274ptv^{(2)}_{c}=\frac{c^{(2)}}{s_{c}^{(0)}}-\frac{s_{c}^{(1)}c^{(1)^{T}}}{[s_{c}^{(0)}]^{2}}.$
## Software
All statistical computing and analyses were performed using the software
environment R version 4.0.2. Programming code for sample size calculations, is
available at https://github.com/abigailburdon/Conditional-score-GST.
## Acknowledgements
This research was funded by the Engineering and Physical Sciences Research
Council.
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|
# The Quantum Mechanical Problem of a Particle on a Ring with Delta Well
Raphael J.F. Berger Division of Chemistry and Physics of Materials, Paris-
Lodron University Salzburg, Salzburg, A-5020, Austria
<EMAIL_ADDRESS>
(Date: August 28, 2024)
###### Abstract.
The problem of a spin-free electron with mass $m$, charge $e$ confined onto a
ring of radius $R_{0}$ and with an attractive Dirac delta potentential with
scaling factor (depth) $\kappa$ in non-relativistic theory has closed form
analytical solutions. The single bound state function is of the form of a
hyperbolic cosine that however contains a paramter $d>0$ which is the single
positive real solution of the transcentdental equation $\coth(d)=\lambda d$
for non zero real $\lambda=\frac{2}{\pi\kappa}$. The energy eigenvalue of the
bound state
$\varepsilon=-\frac{d^{2}}{2\pi^{2}}\approx\frac{qemR_{0}}{2\hbar^{2}}$. In
addition a discrete inifinty of unbounded solutions exists, formally they are
obtained from the terms for the bound solution by substituting $d\to id$
yielding $\cot(d)=\lambda d$ as characteristic equation with the correspondig
set of solutions $d_{k},k\in\mathbb{N}$, the respective state functions obtain
via $\cosh(x)\overset{x\to ix}{\longrightarrow}\cos(x)$ of the form of cosine
functions.
RJFB acknowledges funding from DFG (German Research Foundation) within the
Priority Program SPP1807 “Control of LD in molecular chemistry”, grant
BE4632/2-2, project no. 271386299)
## 1\. Introduction
There is only roughly a dozen of quantum mechanical (QM) systems with an
analytical solution.[Wikipedia contributors(2022)] QM problems with analytical
solutions are not only of great of didactical use for showin how these
problems can be solved but often serve also as physical toy model systems for
otherwise unsovable problems which however share the principal
characteristics.
One example herefore is the particle in the Dirac delta potential[Wikipedia
contributors(2022)] which not ony serves with its bound state as an one-
dimensional analog of the hydrogen atom but can also be interpreted as the
simplest model for electron scattering when one regards the unbound states and
allows for simple calculation of reflection and transmissoin rates on step-
and related potentials, which is for example of relevance for the theory of
scanning tunnel microscopy.
In the context of our research on symmetry breaking in rotationally invariant
systems[Berger and Viel(2020)] we came across the analogous problem where the
particle is but confined to an atomic scale ring. As it turned out that this
QM model system has not yet been described in the literature (to the best of
our knowledge) we report on our results in the following.
## 2\. Solution
### 2.1. Schrödinger equation
A spin-free electron (i.e. a particle with charge $-e$ and mass $m_{e}$) on in
a ring shaped space with radius $R_{0}$ and a $\delta$ function well of
amplitude $-qe$ corresponding to an attractive potential with an integrated
total charge of $+qe$ is regarded in non-relativistic quantum mechanic theory.
As is well known from textbooks the Hamiltonian for the particle on a ring of
radius $R_{0}$ in 2D polar coordinates
($r\in\mathbb{R}^{+},\vartheta\in[0,2\pi)$) in SI units is given by
(1) $\displaystyle\hat{H^{\prime}}$
$\displaystyle=-\frac{\hbar^{2}}{2mR_{0}^{2}}\frac{\partial^{2}}{\partial\vartheta^{2}}$
The ring shall contain an attractive potential $\hat{V}$ with respect to the
electron in the form of a Dirac-$\delta$ function111The dimension of the
argument of the $\delta$ function has to be coosen such as $\int\delta\;dx=1$
is dimensionless and the potential shall integrate over the whole space
($R_{0}\times[0,2\pi)$) to the product of electron and potential charge of
$-eq<0$:
(2) $\displaystyle\hat{V}$
$\displaystyle=-qe\delta((\vartheta-\vartheta_{0})l),$
were $\vartheta$ is given in units of $\mathrm{[}rad]$, thus formally
$l=1\mathrm{rad}^{-1}=1\frac{1m}{1m}=1$ and hence can be dropped in the
following. Without loss of generality we will set $\vartheta_{0}=0$ such that
the Schrödinger equation for the problem becomes
$\displaystyle\hat{H}\psi(\vartheta)$ $\displaystyle=E\psi(\vartheta)$
$\displaystyle(\hat{H^{\prime}}+\hat{V})\psi(\vartheta)$
$\displaystyle=E\psi(\vartheta)$ (3)
$\displaystyle\left(-\frac{\hbar^{2}}{2mR_{0}^{2}}\frac{\partial^{2}}{\partial\vartheta^{2}}-qe\delta(\vartheta)\right)\psi(\vartheta)$
$\displaystyle=E\psi(\vartheta)$
$\displaystyle\left(\frac{\partial^{2}}{\partial\vartheta^{2}}+\frac{2qemR_{0}^{2}}{\hbar^{2}}\delta(\vartheta)+E\frac{2mR_{0}^{2}}{\hbar^{2}}\right)\psi(\vartheta)$
$\displaystyle=0$
For simplicity we combine the constants in
(4) $\displaystyle\kappa=\frac{2qemR_{0}^{2}}{\hbar^{2}}$
and
(5) $\displaystyle\epsilon=\frac{2mR_{0}^{2}}{\hbar^{2}}E$
such that we obtain
(6) $\displaystyle\psi^{\prime\prime}+(\epsilon+\kappa\delta)\psi$
$\displaystyle=0$
### 2.2. Bound state ($E<0$)
The assumption of $(E-V)<0$ or (ignoring the singularity at the origin) $E<0$,
or $\epsilon<0$, respectively thus leads to the bound state solutions. For the
symmetry of the problem and the form of the differential equation (6) we chose
the Ansatz
(7)
$\displaystyle\psi(\vartheta;d)=N^{\prime}(e^{-d\vartheta}+e^{d(\vartheta-2\pi)})=N\cosh(d(x-\pi))$
Yielding the normalization constant
(8) $\displaystyle N=\sqrt{\frac{\sinh(2\pi d)}{2d}+\pi}^{-1}$
To determine the exponent $d$ one in principle has to insert (7) into (6) and
attempt to match $d$ to the boundary conditions. One boundary condition, the
symmtry of the system, was already accounted for choosing the same exponent
$d$ for both functions but with different sign. Since a $\delta$-function is
appearing one has to chose an appropriate strategy to evaluate the result of
combining (7) and (6). The strategy is to integrate the Schrödinger equation
in an $\epsilon$ ball around the origin of the $\delta$-function and to
perform the limit of $\epsilon\to 0$, this results in
$\displaystyle\lim_{\epsilon\to
0}\int_{0-\epsilon}^{0+\epsilon}\left(\frac{\partial^{2}}{\partial\vartheta^{2}}\psi(\vartheta)\right)+\kappa\delta(\vartheta)\psi(\vartheta)-E^{\prime}\psi(\vartheta)\;d\vartheta$
$\displaystyle=\lim_{\epsilon\to 0}\int_{0-\epsilon}^{0+\epsilon}0d\vartheta$
(9) $\displaystyle\psi^{\prime}(0^{+})-\psi^{\prime}(0^{-})+\kappa\psi(0)$
$\displaystyle=0$
Inserting (7) in (9) yields
$\displaystyle-2d+2de^{-2\pi d}+\kappa(1+e^{-2\pi d})$ $\displaystyle=0$ (10)
$\displaystyle\coth(\pi d)$ $\displaystyle=\frac{2}{\kappa}d$
using $d^{\prime}=\pi d$ and $\lambda=\frac{2}{\pi\kappa}$ we obtain
(11) $\displaystyle\coth(d^{\prime})$ $\displaystyle=\lambda d^{\prime}$
here for $\lambda>0$ and real $d^{\prime}$ exactly one pair of solutions
$d^{\prime}_{+}=-d^{\prime}_{-}$ exists and yielding the same function
$\psi(\vartheta;d)$ due to the axial symmetry of $\cosh$, hence we can drop
the $\pm$ indices in the following and only regard the positive solution
$d^{\prime}$, which is it the same time the only solution to (6).
Equation (11) has no symbolic closed form solution, but
(12) $\coth(d^{\prime})\approx 1$
holds with accuracy increasing in $d^{\prime}$ (see Fig. 1).
Figure 1. Realtive error in % obtained from using (12) in dependence on
$d^{\prime}$
Using (12) the solution of (10) can be approximated as
$\displaystyle d\approx d^{0}=\frac{\kappa}{2}=\frac{qemR_{0}^{2}}{\hbar^{2}}$
with deviations decreasing with charge and ring size. In the Table 1 some
exemplary values for $R_{0}$ set to 1 (all atomic units) are shown. At
$R_{0}=1$ bohr and $q=1e$ we have $\epsilon=-d^{2}=-\frac{d^{\prime
2}}{\pi^{2}}$ and $E=-\frac{d^{\prime
2}\hbar^{2}}{2m\pi^{2}R_{0}^{2}}\approx-0.0145$ Hartree (for comparision in
the approximation (12) it yields
$E^{0}=-\frac{{d^{0}}^{2}}{\pi^{2}}\approx-0.0127$ Hartree (further values are
listed in Table 1). In addition we note that with increasing radius $R_{0}$
the approximative solutions will be of increasing quality.
$q$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
${d^{0}}^{\prime}$ | 0.5 | 1.0 | 1.5 | 2.0 | 2.5
$d^{\prime}$ | 0.53575 | 1.00366 | 1.50024 | 2.00001 | 2.5
${E^{0}}^{\prime}$ | -0.01267 | -0.05066 | -0.11399 | -0.202642 | -0.31663
$E^{0}$ | -0.01454 | -0.05103 | -0.11402 | -0.202642 | -0.31663
Table 1. Numerically exact ($d^{\prime}$) and approximated solutions
(${d^{0}}^{\prime}$) of (10) at different total charges $q$ of the Dirac delta
potential and corresponding approximate (${E^{0}}^{\prime}$) and numerically
exact energies (${E^{0}}^{\prime}$) at $1$ Bohr radius.
A graphical display of the wave function of the bound state is shown in Fig.
2.
Figure 2. Stereographic projection of $\psi(\vartheta)$ for the bound state of
the partice in the ring with $\delta$ well.
### 2.3. Unbound states ($E>0$)
In the spirit of (7) the unbound states can be obtained from the Ansatz
(13)
$\displaystyle\psi(\vartheta;di)=C(e^{-di\vartheta}+e^{di(\vartheta-2\pi)})$
Yielding in analogy to (10)
$\displaystyle-2id+2ide^{-i2\pi d}+\kappa(1+e^{-i2\pi d})$ $\displaystyle=0$
(14) $\displaystyle\cot(\pi d)$ $\displaystyle=\frac{2}{\kappa}d$
In contrast to (10) this yields for finite positive $\kappa$ an infinite set
of solutions. As in (10) the system is strictly not analytically solvable in
terms of a finite closed expression. However, for sufficiently large $d$ or
large values of $\kappa$ the solutions are approaching
(15) $d_{n_{+/-}}\approx d^{(0)}_{n_{+/-}}=\pm n\frac{\kappa}{2}$
The first 5 unbound solutions for $d$ and the corresponding energies of the
sytem with charge $+1e$ and $R=1$ bohr are
$n$ | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
$d^{(0)}_{n_{+}}$ | 0.5 | 1.0 | 2.0 | 3.0 | 4.0
$d_{n_{+}}$ | 0.34278 | 1.15979 | 2.09395 | 3.06518 | 4.04963
$E_{n}$ | 0.05875 | 0.67256 | 2.19231 | 4.69766 | 8.19976
Since $\psi(\vartheta;di)$ are not purely real functions, we first decompose
them into real and imaginary part
(16) $\displaystyle\Re{[\psi(\vartheta;di)]}$
$\displaystyle=C[\cos(d\vartheta)+\cos(d(2\pi-\vartheta))]$ (17)
$\displaystyle=\left(\frac{\sin(2\pi
d)}{2d}+\pi\right)^{-\frac{1}{2}}\cos[d(\pi-\vartheta^{\prime})]$ (18)
$\displaystyle\Im{[\psi(\vartheta;di)]}$
$\displaystyle=C[\sin(d\vartheta)+\sin(d(2\pi-\vartheta))]$
and we note that (18) in general is dicontinuous at the $\delta$ well, thus
must be rejected. (16) can be rewritten as a single cosine function
originating at the position opposing the origin $\vartheta^{\prime}_{o}=\pi$
$\displaystyle\Re{[\psi(\vartheta;di)]}$
$\displaystyle=2C\cos(d\pi)[\cos(d(\pi-\vartheta))]$ (19)
$\displaystyle=C^{\prime}\cos[d(\pi-\vartheta^{\prime})]$
with $-\pi\leq\vartheta^{\prime}<\pi$ and the normalisation constant
(20) $\displaystyle C^{\prime}=\left(\frac{\sin(2\pi
d)}{2d}+\pi\right)^{-\frac{1}{2}}$
In summary this yields the unbounded state functions
(21) $\displaystyle\psi_{n}$ $\displaystyle=\sqrt{\frac{\sin(4\pi
d_{n})}{4d_{n}}+\pi}^{-1}\cos[d_{n}(\pi-\vartheta)]$
for $-\pi\leq\vartheta<\pi$ and $n>0$ with energies
(22) $\displaystyle E_{n}=-\frac{\hbar^{2}}{2mR_{0}^{2}}d_{n}^{2}$
where $d_{n}$ are corresponding to the positive solutions of
(23) $\displaystyle\cot(\pi d_{n})$
$\displaystyle=\frac{2\pi\hbar^{2}}{qemR_{0}}d_{n}.$
which are approximated for large $q$, $R_{0}$ or $n$ by
(24) $d_{n}\approx d^{0}_{n}=n\frac{qR_{0}em}{2\pi\hbar^{2}}$
where we have removed the symmetry equivalent negative solutions, since $\cos$
is an even function, thus droped the sign indices, as compared to (15). Hereby
we note that in comparison to the particle in the ring (without additional
well potential) we have lost the two-fold degeneracy of the higher (non-
ground) states. Which is an obvious consequence of the symmetry breaking due
to the potential.
### 2.4. $\kappa$ dependence
The $\kappa$ dependence of the real solutions of (28) is illustrated in the
graph below. The approximative solutions (24) we have used are based on the
asymptotic approximation of the $\tan$ branches to $x=2n+1$ for
$n\in\mathbb{R}$.
Figure 3. Graphical representation of the solutions of the characteristic
equation (23) for the problem with $\kappa=2$
## 3\. Smooth wave function for the bound state
Independently of the physical significance, one can ask for conditions under
which the wave function becomes smooth at the position of the Dirac delta
function potential. Smoothness at $\vartheta=0$ requires $d=1/2$ in the
bounded state (7). Hence we obtain from (10), (4) and using atomic units
(25) $\displaystyle R_{0}=\sqrt{\frac{\tanh{\pi/2}}{2q}}$
yielding, e.g. $R_{0}=0.677183$ Bohr for $q=1$ at $E_{0}=-0.01267$ Hartree.
## 4\. Summary
Due to the relations $\cos(ix)=\cosh(x)$ and $\sin(ix)=\sinh(x)$ we can
combine the bound (7) and the unbound (21) solutions in one expression
(26) $\displaystyle\psi_{n}$ $\displaystyle=\sqrt{\frac{\sin(4\pi
a_{n})}{4a_{n}}+\pi}^{-1}\cos[a_{n}(\pi-\vartheta)]$
with $\kappa=qR_{0}\frac{em}{\pi\hbar^{2}}$, and the corresponding energies
(27) $\displaystyle E_{n}=\frac{\hbar^{2}}{2mR_{0}^{2}}a_{n}^{2},\;\;\forall
n\in\mathbb{N}_{0}.$
and where $a_{0}$ is the (single) purely imaginary solution and $a_{n}$ with
$n>0$ are the purely real solutions of the equation
(28) $\cot(\pi a)=\frac{2}{\kappa}a$
## References
* [Wikipedia contributors(2022)] Wikipedia contributors, _List of quantum-mechanical systems with analytical solutions — Wikipedia, The Free Encyclopedia_ , https://en.wikipedia.org/w/index.php?title=List_of_quantum-mechanical_systems_with_analytical_solutions&oldid=1121884573, 2022, [Online; accessed 28-November-2022].
* [Wikipedia contributors(2022)] Wikipedia contributors, _Delta potential — Wikipedia, The Free Encyclopedia_ , https://en.wikipedia.org/w/index.php?title=Delta_potential&oldid=1117753099, 2022, [Online; accessed 28-November-2022].
* [Berger and Viel(2020)] R. J. F. Berger and A. Viel, _Zeitschrift für Naturforschung B_ , 2020, 75, 327–339.
|
# Dirac Kondo effect under magnetic catalysis
Koichi Hattori ID<EMAIL_ADDRESS>Zhejiang Institute of Modern
Physics, Department of Physics, Zhejiang University, Hangzhou, 310027, China
Research Center for Nuclear Physics (RCNP), Osaka University, Osaka 567-0047,
Japan Daiki Suenaga ID<EMAIL_ADDRESS>Strangeness Nuclear Physics
Laboratory, RIKEN Nishina Center, Wako 351-0198, Japan Research Center for
Nuclear Physics (RCNP), Osaka University, Osaka 567-0047, Japan Kei Suzuki ID
<EMAIL_ADDRESS>Advanced Science Research Center, Japan
Atomic Energy Agency (JAEA), Tokai 319-1195, Japan Shigehiro Yasui ID
<EMAIL_ADDRESS>Research and Education Center for Natural Sciences, Keio
University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan
###### Abstract
We develop a mean-field theory of a novel Kondo effect emerging in systems
without a Fermi surface, which instead emerges under strong magnetic fields.
We determine the magnitude of the Kondo condensate which is a particle pairing
composed of conducting Dirac fermions and localized impurities. We focus on
the competition between the Kondo effect and the energy gap formation that
stems from the pairing among the Dirac fermions leading to the dynamical
chiral symmetry breaking. We find that this competition induces a quantum
critical point. We also investigate finite-temperature effects. This system at
vanishing fermion density can be studied with Monte Carlo lattice simulations
which do not suffer from the sign problem.
Introduction.— Quantum systems under strong magnetic fields have attracted
much attention over the last century in various physical systems (see, e.g.,
Refs. Grasso and Rubinstein (2001); Dittrich and Gies (2000); Yoshioka (2002);
Dunne (2004); Harding and Lai (2006); Mourou _et al._ (2006) for reviews). In
the last decade, even more, common interests have been developed with the aid
of advanced engineering of ultrarelativistic heavy-ion collisions Kharzeev
_et al._ (2013); Kharzeev (2014); Kharzeev _et al._ (2016); Hattori and Huang
(2017); Koch _et al._ (2017) and of Dirac/Weyl semimetals Burkov (2015);
Kharzeev (2015); Miransky and Shovkovy (2015); Armitage _et al._ (2018) as
well as the tremendous progress in the numerical lattice simulations D’Elia
(2013); Endrődi (2015) (see also references therein). These situations
motivate us to study phases of matter under strong magnetic fields. As we will
discuss in detail, it has been pointed out that strong magnetic fields
catalyze particle pairing phenomena, inducing nonpertubative modification of
the ground state of the system when the Landau quantization becomes sizable.
In this Letter, we investigate an interacting system composed of relativistic
light fermions and nonrelativistic heavy impurities under a strong magnetic
field at vanishing fermion density. Such relativistic fermions, governed by
the Dirac equation, appear not only as elementary particles in high-energy
physics but also in Dirac semimetals in condensed matter physics. The
conventional Kondo effect is absent at vanishing fermion density, whereas our
system exhibits the Kondo effect and the chiral symmetry breaking induced by
strong external magnetic fields. We call the Kondo effect appearing in Dirac
fermion systems the Dirac Kondo effect. These phenomena are induced by two
pairing patterns: the Kondo condensate, pairing between the light fermion and
the heavy impurity, and the chiral condensate, pairing between the light
fermion and its antiparticle. In fact, each condensate is induced even with
infinitesimal attractive interactions between the pair. That is, the growth of
each condensate is an inevitable fate of the system when a magnetic field
increases and temperature decreases. The common concept behind this statement
is the magnetic catalysis as first pointed out in the formation of the chiral
condensate Gusynin _et al._ (1994, 1995a, 1995b, 1996) and refined in the
formation of the Kondo condensate Ozaki _et al._ (2016).111 While the
magnetic catalysis is conventionally meant for the chiral condensate, the key
concept, the dimensional reduction in the phase space stemming from the Landau
degeneracy, is found to be more general and can lead to the realization of
various pairing patterns in analogy with the Fermi surface effect Polchinski
(1992); Shankar (1994).
This system provides an interesting question arising from the competition
between the chiral and Kondo condensates, which can drastically change the
fate of the system. We will find that one condensate interrupts the growth of
the other destructively, and there occurs a transition from the vacuum
completely dominated by the chiral condensate to that accompanied by the Kondo
condensate. At zero temperature, these two phases are divided by a quantum
critical point at a certain critical magnetic-field strength where the Kondo
condensate starts to grow. The chiral condensate no longer grows above the
critical point and saturates at a constant value.
Our findings have an impact on the heart of many-body quantum physics. It is a
general and central issue to determine the ground state of a system with
condensates. The ground states can exhibit nonperturbative quantum phenomena
such as superconductivity Bardeen _et al._ (1957); Polchinski (1992); Shankar
(1994) and the Kondo effect Kondo (1964); Abrikosov (1965); Anderson (1970);
Wilson (1975); Hewson (1993); Yosida (1996); Yamada (2004); Coleman (2015).
Relativistic counterparts are the spontaneous chiral symmetry breaking Nambu
and Jona-Lasinio (1961a, b); Klevansky (1992); Hatsuda and Kunihiro (1994),
the color superconductivity Iwasaki and Iwado (1995); Alford _et al._ (1998);
Rapp _et al._ (1998); Alford _et al._ (2008); Fukushima and Hatsuda (2011),
and the newly proposed QCD Kondo effect Yasui and Sudoh (2013); Hattori _et
al._ (2015) (see also Refs. Kanazawa and Uchino (2016); Kimura and Ozaki
(2017); Yasui _et al._ (2019); Yasui (2017); Yasui _et al._ (2017); Suzuki
_et al._ (2017); Yasui and Ozaki (2017); Kimura and Ozaki (2019); Fariello
_et al._ (2019); Hattori _et al._ (2019); Suenaga _et al._ (2020a, b);
Kanazawa (2020); Araki _et al._ (2021a, b); Kimura (2021); Suenaga _et al._
(2021); Ishikawa _et al._ (2021); Suenaga _et al._ (2022) for the
developments in the QCD Kondo effect). Relativistic systems can be also
realized with graphene and Dirac/Weyl semimetals that provide useful platforms
in the study of the magnetic catalysis Semenoff _et al._ (1999, 1998);
Khveshchenko (2001); Gorbar _et al._ (2002); Gusynin _et al._ (2006); Herbut
(2007); Herbut and Roy (2008); Gusynin _et al._ (2009); Roy and Herbut
(2011); Roy _et al._ (2014); Roy and Sau (2015); Zhang and Shindou (2017);
Song _et al._ (2017); DeTar _et al._ (2016); Boyda _et al._ (2014); DeTar
_et al._ (2017); Otsuka _et al._ (2016); Tada (2020) and the Kondo effect
Principi _et al._ (2015); Yanagisawa (2015a, b); Mitchell and Fritz (2015);
Sun _et al._ (2015); Feng _et al._ ; Kanazawa and Uchino (2016); Lai _et
al._ (2018); Ok _et al._ ; Ma _et al._ (2018); Li _et al._ (2018); Dzsaber
_et al._ (2021); Lü _et al._ (2019); Kim and Han (2019); Grefe _et al._
(2020, ); Pedrosa _et al._ (2021) (see Ref. Miransky and Shovkovy (2015) and
Ref. Fritz and Vojta (2013) for reviews).
Despite the many foregoing studies, it should be stressed that the competition
of the Kondo condensate with other condensates is yet elusive; The interplay
appears both in destructive and constructive manners. There is a destructive
competition between the Kondo effect and superconductivity Soda _et al._
(1967); Fowler and Maki (1967); *fowler1970conditions; Zittartz and Müller-
Hartmann ; *muller1970theory; *muller1971kondo; Sakurai (1970); Matsuura and
Nagaoka (1976); *matsuura1977effects; *matsuura1977theory; Jarrell _et al._
(1990); Satori _et al._ (1992); *shiba1993numerical; *sakai1993numerical;
Salkola _et al._ (1997); Bauer _et al._ (2013); Kiršanskas _et al._ (2015);
Kanazawa and Uchino (2016); Lee _et al._ (2017); Moca _et al._ (2021) (see
also, e.g., Refs. Maple _et al._ (1972); Yazdani _et al._ (1997); Franke
_et al._ (2011); Hatter _et al._ (2015); Balatsky _et al._ (2006) for
experiments). Two of the present authors have also discussed the competition
between the chiral symmetry breaking and the QCD Kondo effect at finite baryon
density Suzuki _et al._ (2017) (see also Refs. Kanazawa (2020); Ishikawa _et
al._ (2021)). On the other hand, constructive interplay can induce “heavy
fermion” superconductivity Hewson (1993); Balatsky _et al._ (2006); Coleman
(2006); Pfleiderer (2009); Stewart (2017). It is an interesting question how
the Kondo condensate could affect the critical point from the interplay
between the chiral condensate and the color superconductivity Hatsuda _et
al._ (2006); Yamamoto _et al._ (2007); Fukushima and Hatsuda (2011).
Formulation.— Nonperturbative phenomena, such as the formation of condensates,
can be studied by the mean-field method (See Refs. Read and Newns (1983);
Coleman (1983) for early works on the Kondo effect). In Refs. Yasui _et al._
(2019, 2017), the QCD Kondo effect was investigated with the mean-field
approach applied to a four-point interaction model analogous to the
Nambu–Jona-Lasinio (NJL) model (for applications, see Refs. Suzuki _et al._
(2017); Yasui and Ozaki (2017); Fariello _et al._ (2019); Suenaga _et al._
(2020a, b); Kanazawa (2020); Araki _et al._ (2021a); Suenaga _et al._
(2021); Ishikawa _et al._ (2021); Suenaga _et al._ (2022)).
Referring to those studies, we extract the essence of the chiral symmetry
breaking and the Kondo effect in the Dirac fermion systems in strong magnetic
fields. For this purpose, we use the following Lagrangian (see Supplemental
material)
$\displaystyle{\cal L}=$
$\displaystyle\bar{\psi}(i\not{\partial}_{\parallel}-m_{l})\psi+\frac{G_{ll}}{2N}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\psi)^{2}\right]$
(1)
$\displaystyle+\sum_{c=\pm}\Big{[}\,c\bar{\Psi}_{v}^{c}i\partial_{0}\Psi_{v}^{c}$
$\displaystyle+\frac{G_{hl}}{N}\left\\{(\bar{\psi}\Psi_{v}^{c})(\bar{\Psi}_{v}^{c}\psi)+(\bar{\psi}i\gamma_{5}\Psi_{v}^{c})(\bar{\Psi}_{v}^{c}i\gamma_{5}\psi)\right\\}\,\Big{]},$
where $N$ stands for the number of degrees of freedom for a non-Abelian
interaction ($N=3$ for quarks in QCD). This Lagrangian contains four
eigenmodes of $\psi$ for the spin-polarized light fermion and its antifermion
in the lowest Landau level (LLL), where $m_{l}$ is the light-fermion mass. We
assume that a magnetic field is applied in the third ($z$) direction. Then,
$\psi$ has the kinetic term
$\not{\partial}_{\parallel}=\gamma^{0}\partial_{0}+\gamma^{3}\partial_{3}$ and
is an eigenstate of the spin along the magnetic field due to the Zeeman
effect, i.e., $i\gamma^{1}\gamma^{2}\psi={\rm sgn}(q_{l}B)\psi$ with an
electric charge $q_{l}$.
The Lagrangian (1) also contains $\Psi_{v}^{+}$ and $\Psi_{v}^{-}$ for the
heavy-fermion and its antifermion fields introduced as impurities. They stem
from the original Dirac field $\Psi$ as $\Psi_{v}^{\pm}\equiv e^{\pm
im_{h}v\cdot x}{\mathcal{Q}}_{\pm}\Psi$ after the factorization of the plane-
wave components at the position $x^{\mu}$ and the projection by
${\mathcal{Q}}_{\pm}=\frac{1}{2}(1\pm v^{\mu}\gamma_{\mu})$. Here, the four
momentum is given as $m_{h}v^{\mu}$ with $m_{h}$ and $v^{\mu}$ being the
heavy-fermion mass and the four velocity, respectively. In the rest flame of
heavy fermions, $v^{\mu}=(1,0,0,0)$,
${\mathcal{Q}}_{\pm}=\frac{1}{2}(1\pm\gamma_{0})$ is the projection operator
to the particle and antiparticle components.222 This treatment is similar to
but slightly different from the conventional framework of the heavy-quark
effective theory (HQET) Eichten and Hill (1990); Georgi (1990); Neubert
(1994); Manohar and Wise (2000) in high-enerygy physics, which is an effective
field theory for heavy quarks in QCD. In the conventional HQET, one retains
either $\Psi_{v}^{+}$ or $\Psi_{v}^{-}$. On the other hand, in this work we
retain both of $\Psi_{v}^{+}$ and $\Psi_{v}^{-}$ to include the heavy fermion
and antifermion in a charge-conjugation invariant manner Körner and Thompson
(1991); Balk _et al._ (1994). $G_{hl}>0$ and $G_{ll}>0$ are the four-point
coupling constants for the interactions between the heavy and light fermions
and between the light fermions, respectively.
We analyze the Lagrangian (1) with the mean-field approximation. Analogously
to Ref. Araki _et al._ (2021a) in the (3+1) dimensional spacetime, the chiral
and Kondo condensates are assumed to be
$\displaystyle\langle\bar{\psi}\psi\rangle_{\mathrm{LLL}}\equiv-\frac{N}{G_{ll}}M,\quad\langle\bar{\psi}\Psi_{v}^{\pm}\rangle_{\mathrm{LLL}}\equiv\frac{N}{G_{hl}}\Delta,$
(2)
with the gaps $M$ and $\Delta$. By explicitly diagonalizing the mean-field
Lagrangian, we obtain the energy-momentum dispersion relations of the four
eigenmodes
$\displaystyle E_{\pm}(p_{z})$
$\displaystyle\equiv\pm\frac{1}{2}\left(\sqrt{E_{p_{z}}^{2}+|2\Delta|^{2}}\pm
E_{p_{z}}\right),$ (3a) $\displaystyle\tilde{E}_{\pm}(p_{z})$
$\displaystyle\equiv\pm\frac{1}{2}\left(\sqrt{E_{p_{z}}^{2}+|2\Delta|^{2}}\mp
E_{p_{z}}\right),$ (3b)
where $E_{p_{z}}\equiv\sqrt{p_{z}^{2}+(m_{l}+M)^{2}}$. Straightforwardly, we
obtain the thermodynamic potential $\Omega(M,\Delta)\equiv-T\ln Z$ with
temperature $T=1/\beta$ and the partition function $Z$ at the one-loop level
Kapusta and Gale (2011):
$\displaystyle\Omega(M,\Delta)$
$\displaystyle=N\left[\,\frac{M^{2}}{2G_{ll}}+\frac{|2\Delta|^{2}}{2G_{hl}}+\Omega_{\rm
vac}+\Omega_{T}\,\right],$ $\displaystyle\Omega_{\rm vac}(M,\Delta)$
$\displaystyle=-\rho_{B}\sum_{{\cal
E}_{i}}\int_{-\Lambda}^{\Lambda}\frac{dp_{z}}{2\pi}\frac{1}{2}|{\cal E}_{i}|,$
(4) $\displaystyle\Omega_{T}(M,\Delta)$ $\displaystyle=-\rho_{B}\sum_{{\cal
E}_{i}}\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}\frac{1}{\beta}\ln(1+e^{-\beta|{\cal
E}_{i}|}),$
where $\Omega_{{\rm vac}/T}$ are the contributions from the vacuum bubbles and
the thermal excitation, respectively, and $\rho_{B}=|q_{l}B|/(2\pi)$ is the
Landau degeneracy factor. The sum runs over the four eigenmodes ${\cal
E}_{i}=\\{E_{+},E_{-},\tilde{E}_{+},\tilde{E}_{-}\\}$. We are left with
evaluating the one-dimensional momentum integral whose interval is limited
within the ultraviolet cutoff $\lambda$. All the dimensionful quantities can
be scaled by $\Lambda$ because of the dimensional reduction.
The thermodynamic potential (4) is inevitably unstable at the origin
$M=0=\Delta$ due to the dimensional reduction at zero temperature, which
implies that the system always favors nonzero condensates. This can be
confirmed with an analytic expression of $\Omega_{\rm vac}$ as follows. We
take the massless limit ($m_{l}=0$) for a demonstration. One can simply
perform the $p_{z}$ integral and expand the result as
$\displaystyle\Omega_{\rm
vac}(\Phi)=-\frac{\rho_{B}}{2\pi}\biggl{(}\Lambda^{2}+\Phi^{2}\biggl{(}\frac{1}{2}+\ln\frac{2\Lambda}{\Phi}\biggr{)}+{\mathcal{O}}(\Phi^{4})\biggr{)},$
(5)
where $\Phi^{2}=M^{2}+|2\Delta|^{2}$. Stability at the origin is determined by
the sign of the quadratic terms in $\Phi$. The quadratic term enhanced by a
logarithmic factor $\ln\Lambda/\Phi$, diverging as $\Phi\to 0$, makes the
potential convex upward at the origin for any coupling strengths Gusynin _et
al._ (1994, 1995a, 1995b, 1996); Fukushima and Pawlowski (2012). At finite
temperature, the origin can be a stable point since the thermal contribution
$\Omega_{T}$ yields a term that can cancel the above-mentioned logarithmic
term.
Zero-temperature results.— We first investigate the ground state at zero
temperature. We consider several cases classified by the relative magnitude
between $G_{hl}$ and $G_{ll}$ in Eq. (1). When $G_{hl}=G_{ll}$, the
thermodynamic potential (4) is invariant under a rotation in the $M$-$\Delta$
plane in the chiral limit, i.e., the vanishing light-fermion mass limit
($m_{l}=0$), and has a degenerate minimum on a circle at $\Phi^{2}={\rm
constant}$. A finite $m_{l}$ breaks the rotational invariance. As a result,
the potential is tilted toward the $M$ axis, and has a minimum at a nonzero
$M$ and vanishing $\Delta$. This is an analog of the chiral condensate in the
effective models of QCD with the nonzero current quark mass (see, e.g., Ref.
Hatsuda and Kunihiro (1994)). When $G_{hl}\neq G_{ll}$, there is no longer the
degeneracy on the circle. When $G_{hl}<G_{ll}$, the tendency that $M$ is
favored over $\Delta$ is enhanced due to the suppression of the potential
energy along the $M$ axis for any $m_{l}$. However, when $G_{hl}>G_{ll}$,
competition appears between the suppression of the potential energy and the
tilting effect. This is a nontrivial case as we investigate in detail below.
First, we consider a special case $G_{hl}>G_{ll}=0$ and elucidate the effects
of the light-fermion mass $m_{l}$ on the Kondo effect. In the chiral limit, we
find that there is no phase transition characterized by a singular behavior of
the $\Delta$, as shown by the dotted line in Fig. 1. On the other hand, when a
nonzero $m_{l}$ is switched on, the monotonic increase changes to a second-
order phase transition, as shown by the blue dashed line, at a critical
magnetic-field strength $B_{cK}$. Thus, the light-fermion mass plays an
important role in the Kondo effect.
Next, we investigate the competition between the chiral and Kondo condensates
with nonzero $G_{ll}$ and $G_{hl}$. One can find analytic solutions of the
stationary conditions $\partial\Omega/\partial
M=0=\partial\Omega/\partial\Delta$ as follows (See Supplemental material for
details). One immediately notices a trivial solution $\Delta=0$ accompanied by
a nontrivial solution $M\neq 0$. This set is denoted as
$(M_{1},\Delta_{1}=0)$. One can find another set of solutions
$(M_{2},\Delta_{2})$ as
$\displaystyle M_{2}$ $\displaystyle=\frac{m_{l}G_{ll}}{G_{hl}-G_{ll}},$ (6a)
$\displaystyle\Delta_{2}$
$\displaystyle=\frac{1}{2}\sqrt{\frac{\Lambda^{2}}{\sinh^{2}\bigl{(}\frac{2\pi^{2}}{\rho_{B}G_{hl}}\bigr{)}}-\frac{m_{l}^{2}G_{hl}^{2}}{(G_{hl}-G_{ll})^{2}}}.$
(6b)
Both the two sets, $(M_{1},\Delta_{1}=0)$ and $(M_{2},\Delta_{2})$, satisfy
the stationary conditions for any values of the parameters as long as
$G_{hl}>G_{ll}$.
Figure 1: Magnetic-field dependences of Kondo condensate $\Delta$ (blue) and
chiral condensate $M$ (red). The solid lines show the full results with fixed
parameters, $m_{l}/\Lambda=0.01$, $G_{ll}\Lambda^{2}=1.0$, and
$G_{hl}\Lambda^{2}=3.0$. The dotted and dashed lines show $\Delta$ with
$m_{l}=G_{ll}=0$ and with $G_{ll}=0$, respectively.
Figure 2: Temperature dependences of Kondo condensate $\Delta$ and chiral
condensate $M$ at a fixed magnetic-field strength $|q_{l}B|/\Lambda^{2}=1.5$.
The legends are the same as in Fig. 1. The red dashed line shows $M$ at
$G_{hl}=0$. Figure 3: Magnetic-field and temperature dependences of chiral
condensate $M$ (left) and Kondo condensate $\Delta$ (right).
We numerically investigated which, or any other set of solutions, serves as
the global minimum of the thermodynamic potential ${\Omega}(M,\Delta)$. The
numerical results are shown with the solid lines in Fig. 1. We find that,
below a certain critical magnetic-field strength $B_{c}$, the first solutions,
$(M_{1},\Delta_{1}=0)$, are realized: The Kondo effect is prohibited by the
existence of the chiral condensate. By solving $\Delta_{2}=0$ in Eq. (6b), we
obtain the critical strength of magnetic field:
$\displaystyle q_{l}B_{c}=\frac{4\pi^{3}}{G_{hl}\,{\rm
acrsinh}\Bigl{(}\Lambda\frac{|G_{hl}-G_{ll}|}{m_{l}G_{hl}}\Bigr{)}}\sim\frac{4\pi^{3}m_{l}}{\Lambda|G_{hl}-G_{ll}|},$
(7)
where the rightmost side holds with a small value of $|G_{hl}-G_{ll}|$. The
true vacuum switches over from $(M_{1},\Delta_{1}=0)$ to $(M_{2},\Delta_{2})$
at $B_{c}$, which is thus identified as a quantum critical point. The analytic
solution (6) shows a nontrivial behavior of the condensates above $B_{c}$: The
Kondo condensate grows up as the magnetic field increases, while the chiral
condensate is forced to be a constant value ($M_{2}$). This is an anomalous
saturation of the chiral condensate due to the formation of the Kondo
condensate. As in Fig. 1, a “plateau” of the chiral condensate above $B_{c}$
serves as a clear signal of the appearance of the Kondo condensate at zero
temperature.
Finite-temperature results.— Finally, we investigate the phase diagram at
finite temperature. In Fig. 2, we show the numerical results for the
temperature dependence of the condensates, where the magnetic-field strength
and the heavy-light coupling are fixed at $|q_{l}B|/\Lambda^{2}=1.5$ and
$G_{hl}\Lambda^{2}=3.0$, respectively. The Kondo condensate at $G_{ll}=0$ and
$m_{l}=0$ is shown by the blue dotted line, where we find a phase transition
at the critical temperature $T_{cK0}$. When we switch on a light-fermion mass
$m_{l}/\Lambda=0.01$, as shown by the blue dashed line, we observe that the
magnitude of condensate is slightly suppressed, and the critical temperature
decreases to $T_{cK}$. This suppression effect by $m_{l}$ is consistent with
our result at zero temperature.
Before discussing the competition, we check the spontaneous chiral symmetry
breaking with nonzero $G_{ll}$ at $G_{hl}=0$. The chiral condensate $M$ is
shown by the red dashed line in Fig. 2. The chiral condensate hardly changes
within the plot range of Fig. 2 but decreases at higher temperature. The
chiral condensate at nonzero $m_{l}$ shows a crossover transition, and its
pseudocritical temperature is located in the higher-temperature region, not
shown in the figure.
The numerical results including both nonzero $G_{ll}$ and $G_{hl}$ are shown
by solid lines in Fig. 2. We find that the chiral and Kondo condensates
coexist at low temperature below $T_{c}$. Here, the critical temperature
$T_{c}$ for the Kondo condensate decreases from $T_{cK}$ due to the
competition. As we increase temperature across $T_{c}$, the Kondo condensate
melts away and the chiral condensate starts to increase abruptly at $T_{c}$
since the chiral condensate is released from the competition with the Kondo
condensate. Such a steep increase in the chiral condensate can be indirect
evidence of the Kondo condensate.
Figure 4: Schematic phase diagram extracted from Fig. 3.
In Fig. 3, we show the condensate values on the $T$-$B$ plane. The plateau,
which we have observed in the chiral condensate $M$ at zero temperature, also
extends to nonzero temperature. In addition, one can read off the orders of
phase transitions: While both condensates indicate second-order phase
transitions at zero temperature, switching on a nonzero temperature leads to
the first-order phase transitions. In Fig. 4, we show a schematic phase
diagram, where the first-order phase transition line at finite temperature is
connected to the quantum critical point at zero temperature.
Conclusion.— We have investigated the novel phase diagram of Dirac fermions,
which is characterized by the chiral and Kondo condensates. We analytically
found the critical magnetic field $B_{c}$ where the new quantum critical point
emerges from the competition between the two condensates. This competition
also gives rise to a saturation behavior of the chiral condensate above
$B_{c}$ at low temperature. At higher temperature, the Kondo condensate melts
away. This is the end of the competition, which is signaled by a steep
increase of the chiral condensate.
One of the relevant questions is the excitation spectrum when the ground state
varies with an increasing magnetic field. It is interesting to investigate an
extension of the so-called Yu-Shiba-Rusinov state in superconductor Yu (1965);
Shiba (1968); Rusinov (1969a, b) (see also Refs. Jarrell _et al._ (1990);
Satori _et al._ (1992); *shiba1993numerical; *sakai1993numerical; Salkola
_et al._ (1997); Bauer _et al._ (2013); Kiršanskas _et al._ (2015); Lee _et
al._ (2017); Moca _et al._ (2021) and Ref. Kanazawa and Uchino (2016) for an
analog in QCD), the Nambu-Goldstone bosons, and non-Fermi liquid behavior near
quantum critical points Hertz (1976); Doniach (1977); Millis (1993) pursued
with the experimental progress (see, e.g., Refs. Vojta (2003); Löhneysen _et
al._ (2007); Gegenwart _et al._ (2008); Si and Steglich (2010)). It is
important to investigate the critical exponents with possible excitations. One
can also study an extension to the multi-channel Kondo effect Nozières and
Blandin (1980), where a non-Fermi liquid system can appear depending on the
number of channels (see, e.g., Refs. Nozières and Blandin (1980); Affleck
(1995); Kanazawa and Uchino (2016); Kimura and Ozaki (2017); Kimura (2021)).
Numerical simulations on the lattice have provided useful approaches to
nonperturbative many-body phenomena. The intertwined dynamics of the chiral
symmetry breaking and the Kondo effect may be simulated in strong magnetic
fields. Lattice QCD simulations elucidated the magnetic catalysis in QCD with
high precision (see, e.g., Refs. Bali _et al._ (2012a, b); Endrődi (2015);
D’Elia _et al._ (2018); Endrődi _et al._ (2019); Ding _et al._ (2021);
D’Elia _et al._ (2022)), and they can be also applied to the counterpart in
solid-state physics. The magnetically induced Kondo effect deserves further
study beyond the mean-field model calculation and should be confirmed by
direct measurements of the Kondo condensate constructed from the correlation
function of the heavy and light fermions or by indirect ones through special
behaviors of the chiral condensate pointed out in this work.
Acknowledgements.— The authors thank Kazunori Itakura for helpful discussions.
This work was supported by Japan Society for the Promotion of Science (JSPS)
KAKENHI under grant Nos. JP17K14277, JP20K14476, 20K03948 and 22H02316, and by
the RIKEN special postdoctoral researcher program.
### Supplementary material for “Dirac Kondo effect under magnetic catalysis”
## I Mixing term
We examine a sequence of the terms generated from an interaction Lagrangian
that has the Lorentz and $\text{SU}(N)$ symmetries Ebert _et al._ (1995);
Yasui and Sudoh (2013); Yasui _et al._ (2019, 2017):
$\displaystyle{\cal
L}_{\text{int}}=-G(\bar{\psi}\gamma^{\mu}T^{a}\psi)(\bar{\Psi}\gamma_{\mu}T^{a}\Psi),$
(S1)
where $\gamma^{\mu}$ ($\mu=0,1,2,3$) are the Dirac matrices, and $T^{a}$
($a=1,2,\dots,N^{2}-1$) are the generator matrices of the $\text{SU}(N)$
group. The light and heavy fermions, $\psi$ and $\Psi$, belong to the
fundamental representation of the $\text{SU}(N)$ symmetry. The sums over $\mu$
and $a$ are implicitly taken. Here, $G>0$ is the coupling constant. This is a
simple analog of the NJL interaction Nambu and Jona-Lasinio (1961a, b);
Klevansky (1992); Hatsuda and Kunihiro (1994), where the inter-fermion
interactions are understood as the mimic of the gluon exchange interactions
between quarks in QCD.
The matrix structures in Eq. (S1) can be decomposed by utilizing the Fierz
transformations, i.e.,
$\displaystyle(\gamma^{\mu})_{\alpha\beta}(\gamma_{\mu})_{\gamma\delta}$
$\displaystyle=$
$\displaystyle\delta_{\alpha\delta}\delta_{\gamma\beta}+(i\gamma_{5})_{\alpha\delta}(i\gamma_{5})_{\gamma\beta}$
$\displaystyle-\frac{1}{2}(\gamma^{\mu})_{\alpha\delta}(\gamma_{\mu})_{\gamma\beta}-\frac{1}{2}(\gamma^{\mu}\gamma_{5})_{\alpha\delta}(\gamma_{\mu}\gamma_{5})_{\gamma\beta},$
for the Dirac matrices and
$\displaystyle(T^{a})_{ij}(T^{a})_{kl}=\frac{N^{2}-1}{2N^{2}}\delta_{il}\delta_{kj}-\frac{1}{N}(T^{a})_{il}(T^{a})_{kj},$
(S3)
for the $\text{SU}(N)$ generators. As we briefly discuss below (cf. Refs.
Yasui _et al._ (2019); Yasui (2017); Yasui _et al._ (2017)), we identify the
dominant terms, that are maintained in Eq. (1), as the singlet channel
$\delta_{il}\delta_{kj}$ in the $\text{SU}(N)$ generators. Then, we have a
relation between the coupling strengths, $G_{hl}$ in Eq. (1) and $G$ in Eq.
(S1), as $G_{hl}/N=G(N^{2}-1)/(2N^{2})$. As for the Dirac matrices, we focus
on the scalar $\delta_{\alpha\delta}\delta_{\gamma\beta}$ and pseudoscalar
$(i\gamma_{5})_{\alpha\delta}(i\gamma_{5})_{\gamma\beta}$ terms, and potential
realization of the vector and axial-vector condensates in magnetic fields are
not considered in Eq. (2).
We note that the above singlet channel is the most dominant one in the large
$N$ limit Yasui _et al._ (2019); Yasui (2017); Yasui _et al._ (2017), where
the mean-field approximation can be justified.333See, e.g., Refs. Read and
Newns (1983); Coleman (1983) for early works on the large $N$ limit applied to
the Kondo effect and Ref. Coleman (2015) for generalities. The other channel,
that is, the adjoint representation from the second term on the right-hand
side of Eq. (S3), is suppressed in comparison to the singlet channel in the
large $N$ limit.
In addition to the “particle-antiparticle” channel
($\psi\Gamma^{\prime}\bar{\Psi}$ and $\bar{\Psi}\bar{\Gamma}^{\prime}\psi$
types) in Eq. (1), one can get the “particle-particle” channels
($\psi^{t}\Gamma\Psi$ and $\Psi^{t}\Gamma^{t}\psi$ types) from Eq. (S1) as
interactions between the light and heavy fermions (see, e.g., Ref. Buballa
(2005) for similar discussions in case of light-light fermion interactions).
Here $\Gamma$, $\Gamma^{t}$ (the transpose of $\Gamma$), $\Gamma^{\prime}$,
and $\bar{\Gamma}^{\prime}=\gamma_{0}\Gamma^{\prime{\dagger}}\gamma_{0}$ (the
complex conjugate of $\Gamma^{\prime}$) are appropriate combinations of the
Dirac matrices resulting from the Fierz transformations. These terms are also
suppressed in the large $N$ limit.
## II Solutions for gap equations
From Eq. (4), the stationary conditions $\partial\Omega/\partial
M=0=\partial\Omega/\partial\Delta$ lead to the gap equations
$\displaystyle\frac{M}{G_{ll}}+\frac{M+m_{l}}{2\pi}\rho_{B}\ln\Xi=0,\,\,\frac{\Delta}{G_{hl}}+\frac{\Delta}{2\pi}\rho_{B}\ln\Xi=0,$
(S4)
where
$\displaystyle\Xi=\frac{(M+m_{l})^{2}+4|\Delta|^{2}}{(\Lambda+\sqrt{(M+m_{l})^{2}+4|\Delta|^{2}+\Lambda^{2}})^{2}}.$
(S5)
By solving these gap equations, we can determine $M$ and $\Delta$.
### II.1 Without competition
We briefly investigate special cases where $G_{hl}=0$ or $G_{ll}=0$. According
to the discussion around Eq. (5), we find nonzero solutions for the gap
equations at any coupling strengths. When $G_{hl}=0$ in the Lagrangian (1),
the gap equation becomes the first form in Eq. (S4) with $\Delta=0$. In the
chiral limit $m_{l}=0$, we find an analytic solution ($M>0$):
$\displaystyle
M=\frac{\Lambda}{\sinh\bigl{(}\frac{\pi}{\rho_{B}G_{ll}}\bigr{)}}\to 2\Lambda
e^{-\frac{\pi}{\rho_{B}G_{ll}}},$ (S6)
which approaches the rightmost side for a small value of $G_{ll}$. The chiral
symmetry is always broken under a strong magnetic field and a nonzero coupling
constant by the mechanism of magnetic catalysis Gusynin _et al._ (1994,
1995a, 1995b, 1996).
On the other hand, when $G_{ll}=0$ in the Lagrangian (1), the gap equation
becomes the second form of Eq. (S4) with $M=0$. In the chiral limit, we find
an analytic solution ($\Delta>0$):
$\displaystyle\Delta=\frac{\Lambda}{2\sinh\bigl{(}\frac{\pi}{\rho_{B}G_{hl}}\bigr{)}}\to\Lambda
e^{-\frac{\pi}{\rho_{B}G_{hl}}},$ (S7)
which approaches the rightmost side for a small value of $G_{hl}$. This
solution clearly indicates the occurrence of the Kondo effect by a strong
magnetic field. The novel magnetically induced Kondo effect was previously
suggested also by a renormalization-group (RG) analysis Ozaki _et al._ (2016)
and further analyzed by a conformal-field theory approach Kimura and Ozaki
(2019). Our mean-field solution (S7) is consistent with the emergent scale in
the previous RG analysis Ozaki _et al._ (2016) and determines the magnitude
of the Kondo condensate for the first time. The magnetic-field dependence of
$\Delta$ at $G_{ll}\Lambda^{2}=1.0$ is shown with the blue dotted line in Fig.
1.
### II.2 With competition
We also obtain analytic solutions in the presence of the competition between
the Kondo effect and the magnetic catalysis with nonzero $G_{hl}$ and
$G_{ll}$. We immediately notice that the gap equations (S4) have a trivial
solution $\Delta=0$. In this case, $M$ takes a nontrivial solution which is an
extension of Eq. (S6) with a nonzero $m_{l}$. We call these solutions
$(M_{1},\Delta_{1}=0)$. By eliminating $\ln\Xi$ in Eq. (S4), one can find
another set of solutions that we call $(M_{2},\Delta_{2})$ and show in Eq.
(6).
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|
# Variable selection and covariance structure identification using sparse
loadings
Jan O. Bauer<EMAIL_ADDRESS>Faculty of Business and Economics, University
of Basel
Peter Merian-Weg 6, 4002 Basel, Switzerland. Baden-Wuerttemberg Cooperative
State University Mannheim, Coblitzallee 1-9, 68163 Mannheim, Germany
###### Abstract
We provide sparse principal loading analysis which is a new concept that
reduces dimensionality of cross sectional data and identifies the underlying
covariance structure. Sparse principal loading analysis selects a subset of
existing variables for dimensionality reduction while variables that have a
small distorting effect on the covariance matrix are discarded. Therefore, we
show how to detect these variables and provide methods to assess their
magnitude of distortion. Sparse principal loading analysis is twofold and can
also identify the underlying block diagonal covariance structure using sparse
loadings. This is a new approach in this context and we provide a required
criterion to evaluate if the found block-structure fits the sample. The method
uses sparse loadings rather than eigenvectors to decompose the covariance
matrix which can result in a large loss of information if the loadings of
choice are too sparse. However, we show that this is no concern in our new
concept because sparseness is controlled by the aforementioned evaluation
criterion. Further, we show the advantages of sparse principal loading
analysis both in the context of variable selection and covariance structure
detection, and illustrate the performance of the method with simulations and
on real datasets. Supplementary material for this article is available online.
_Keywords:_ Covariance Structure, Dimensionality Reduction, Principal Loading
Analysis, Sparse Principal Component Analysis, Sparsity, Variable Selection
_2020 Mathematics Subject Classification:_ 62H25, 62F07, 65F15, 65F50.
## 1 Introduction
Principal loading analysis (PLA) is a method developed by Bauer and Drabant
(2021) to reduce dimensionality using variable selection. The method chooses a
subset of variables based on the impact of linked eigenvectors on the
covariance matrix. PLA relies on hard-thresholding to link the eigenvectors to
variables. Bauer and Drabant (2021) and Bauer (2021) provide recommendations
for the threshold based on simulations in their work. However, since simulated
results have their limitations, it is natural to search for an extension that
does not rely on simulations. One approach by Bauer and Drabant (2023) is to
analyze PLA from a regression point of view to obtain threshold values in line
with coefficient significant tests of multivariate linear regression.
In this work, we provide the new concept of sparse principal loading analysis
(SPLA) which gives two major contributions. First, SPLA does not rely on
thresholding and therefore overcomes the concerns discussed above. The
proposed concept is based on sparse loadings rather than eigenvectors to make
the usage of a threshold redundant. Transforming a random vector using sparse
loadings yields variables that are potentially correlated. Therefore, we
propose the usage of a measure based on linear regression to correct
calculation of the explained variance. Second, SPLA identifies the block
covariance structure of the underlying random variables using sparse loadings.
This is a new approach in this context. We contribute a criterion to evaluate
the detected structure which is necessary to find the one that fits the
underlying block-structure.
Calculation of sparse loadings is not new and the respective literature is
broad. Jolliffe et al. (2003) proposed a method that constrains the loadings
using lasso regularization (see Tibshirani (1996)). Zou et al. (2006)
formulated the calculation of the eigenvectors as a regression problem with
quadratic penalization. The eigenvectors are then sparsed using lasso
regularization as well. Both regularizations together yield the elastic net
(see Zou and Hastie (2005)). Qi et al. (2013) consider regularization by an
elasticnet-type norm. On the other hand, Shen and Huang (2008) utilized the
connection between the eigendecomposition of the covariance matrix and the
singular value decomposition of the sample matrix to derive sparse loadings.
Their approach is based on a regularized singular value decomposition which
has also been used by Witten et al. (2009) for their method.
Our work is twofold since SPLA also identifies the underlying block-diagonal-
structure of the covariance matrix. There are recent parametric (see, e.g.,
Srivastava and Reid (2012) and Yamada et al. (2017)) and non-parametric works
(see, e.g., Pavlenko et al. (2012) and Devijver and Gallopin (2018))
especially in the high-dimensional context that have the same goal. The reason
is that various techniques exist to estimate covariance matrices that follow a
sparse structure (see, e.g., Ledoit and Wolf (2004), Karoui (2008), Bickel and
Levina (2008), and many more). Further, detection of the block-diagonal
structure prior to network inferences for Gaussian graphical models can
improve performance (Tan et al., 2015).
This article is organized as follows: Section 2 provides notation needed for
the remainder of this work. In Section 3, we motivate the new concept of SPLA.
We discuss how SPLA is used to detect an underlying block-diagonal covariance
structure in Section 4. In Section 5, we provide calculations of the explained
total variance of SPLA which is required for application. Sparseness of the
loadings to identify an underlying block-structure is essential for SPLA. We
provide a criterion to evaluate the found block covariance structure in
Section 6. Further, we contribute the algorithm to conduct SPLA in practice.
Although a sparser model is easier to interpret, it is problematic to work
with a model that is sparser than the underlying covariance matrix since this
results in too little explanatory power. Therefore, we also discuss this
variance-interpretability-trade-off and show respective solutions. A
simulation study to evaluate the performance for block-diagonal covariance
detection is given in Section 7. In Section 8, we provide real data examples
for variable selection and block covariance structure detection, and we give
an example for block covariance structure detection for synthetic data. SPLA
will also be compared to PLA. We take a resume and suggest extensions in
Section 9. Proofs and more detailed results are deferred to the supplementary
material.
## 2 Setup
We first state some notation in this section. Afterwards, we give three
assumptions used throughout this work and elaborate their respective
objective.
We use the $\mathcal{L}_{2}$ norm $\|\bm{v}\|_{2}$ and the
$\mathcal{L}_{\infty}$ norm $\|\bm{v}\|_{\infty}$ for column vectors. For any
matrix $\bm{A}$, $\|\bm{A}\|$ denotes the matrix norm and we will use the
spectral norm $\|\bm{A}\|_{2}$ and the infinity norm $\|\bm{A}\|_{\infty}$ as
induced norms, and the Frobenius norm $\|\bm{A}\|_{F}$. Further, ${\rm
tr}(\bm{A})$ denotes the trace and $\bm{A}^{+}$ denotes the Moore–Penrose
inverse of a matrix $\bm{A}$. $\mathcal{O}_{p}(\cdot)$ denotes stochastic
boundedness.
We consider a random vector
$\bm{X}=(X_{1},\ldots,X_{M})\equiv(\bm{X}_{\mathcal{K}},\bm{X}_{\mathcal{D}})\in\mathbb{R}^{M}$
consisting of two random vectors
$\bm{X}_{\mathcal{K}}^{\top}\in\mathbb{R}^{K}$ and
$\bm{X}_{\mathcal{D}}^{\top}\in\mathbb{R}^{D}$ such that $K+D=M$. The
covariance matrix of $\bm{X}$ is given by
$\tilde{\bm{\Sigma}}=(\tilde{\sigma}_{i,j})$ for $i,j\in\\{1,\ldots,M\\}$, and
we consider a perturbation matrix $\bm{\mathrm{E}}$ such that
$\tilde{\bm{\Sigma}}\equiv\begin{pmatrix}\tilde{\bm{\Sigma}}_{1}&\tilde{\bm{\Sigma}}_{12}\\\
\tilde{\bm{\Sigma}}_{12}^{\top}&\tilde{\bm{\Sigma}}_{2}\end{pmatrix}\equiv\begin{pmatrix}\bm{\Sigma}_{1}&\bm{0}\\\
\bm{0}&\bm{\Sigma}_{2}\end{pmatrix}+\begin{pmatrix}\bm{0}&\bm{\mathrm{E}}_{12}\\\
\bm{\mathrm{E}}_{12}^{\top}&\bm{0}\end{pmatrix}\equiv\bm{\Sigma}+\bm{\mathrm{E}}\;,$
(1)
can be written as a sum of matrices. The covariance matrices of
$\bm{X}_{\mathcal{K}}$ and $\bm{X}_{\mathcal{D}}$ are given by
$\tilde{\bm{\Sigma}}_{1}\equiv\bm{\Sigma}_{1}$ and
$\tilde{\bm{\Sigma}}_{2}\equiv\bm{\Sigma}_{2}$ respectively. $\bm{\Sigma}$ as
well as $\bm{\mathrm{E}}$ are technical constructions to facilitate the
distinction of the covariance matrices of the two random vectors
$\bm{X}_{\mathcal{K}}$ and $\bm{X}_{\mathcal{D}}$ from the covariances
$\bm{\mathrm{E}}$ between the two random vectors. The covariance matrix
follows the eigendecomposition
$\tilde{\bm{\Sigma}}=\bm{V}\bm{\Lambda}\bm{V}^{\top}$. $\bm{\Lambda}$ is
hereby a diagonal matrix containing the eigenvalues
$\lambda_{1}>\cdots>\lambda_{M}$ of $\tilde{\bm{\Sigma}}$ in descending order,
and $\bm{V}=(\bm{v}_{1},\ldots,\bm{v}_{M})$ is an orthonormal matrix
containing the respective eigenvectors where
$\bm{v}_{i}^{\top}=(v_{i}^{(1)},\ldots,v_{i}^{(M)})$ for all
$i\in\\{1,\ldots,M\\}$. Further, we will introduce sparse representations of
the eigenvectors throughout this work where
$\bm{U}=(\bm{u}_{1},\ldots,\bm{u}_{M})$ denotes the matrix containing these
sparse loadings.
The independent and identically distributed (IID) sample
$\bm{x}=(\bm{x}_{1},\ldots,\bm{x}_{M})\equiv(\bm{x}_{\mathcal{K}},\bm{x}_{\mathcal{D}})$
$\in\mathbb{R}^{N\times M}$ contains $N$ observations of the random vector
$\bm{X}$. In general, the sample counterparts are indicated using a hat. For
example,
$\hat{\tilde{\bm{\Sigma}}}\equiv\hat{\bm{\Sigma}}+\hat{\bm{\mathrm{E}}}$ is
the sample covariance matrix of $\bm{x}$.
Variables or blocks of variables are linked to sparse loadings in SPLA. If we
want to reduce dimensionality, we look for a set of variables $\mathcal{D}$ to
discard while keeping the remaining variables $\mathcal{K}$. If we want to
detect the underlying block covariance structure, our interest is to identify
if two disjoint blocks of variables $\mathcal{D}$ and $\mathcal{K}$ do exist.
Hence, we define $\mathcal{D}\subset\\{1,\ldots,M\\}$ with ${\rm
card}(\mathcal{D})=D$ such that $1\leq D<M$. $\mathcal{D}$ will contain the
indices of a variable-block $\\{X_{d}\\}\equiv\\{X_{d}\\}_{d\in\mathcal{D}}$,
and we introduce the quasi-complement
$\mathcal{K}\equiv\\{1,\ldots,M\\}\backslash\mathcal{D}$ with ${\rm
card}(\mathcal{K})=K$ to label the remaining block of variables
$\\{X_{k}\\}\equiv\\{X_{k}\\}_{k\in\mathcal{K}}$. In an analogous manner,
$\mathit{\Delta}$ with ${\rm card}(\mathit{\Delta})=D$ will be used to index
eigenvectors with respective eigenvalues linked to the $\\{X_{d}\\}$.
For convenience purposes, we consider only the two blocks $\\{X_{d}\\}$ and
$\\{X_{k}\\}$ throughout this work except for the examples in Section 8.
Further, since $\bm{X}\equiv(\bm{X}_{\mathcal{K}},\bm{X}_{\mathcal{D}})$, the
index sets are given by $\mathcal{D}=\\{K+1,\ldots,M\\}$ and
$\mathcal{K}=\\{1,\ldots,K\\}$ respectively. However, our results remain true
for the trivial extension to any number of blocks and for the general case if
$\mathcal{D}\subset\\{1,\ldots,M\\}$. All matrices following an analogous
block-structure as $\tilde{\bm{\Sigma}}$ given in (1) are then to be replaced
by their general form such as
$\tilde{\bm{\Sigma}}_{1}=\tilde{\bm{\Sigma}}_{[\mathcal{K},\mathcal{K}]}$,
$\tilde{\bm{\Sigma}}_{2}=\tilde{\bm{\Sigma}}_{[\mathcal{D},\mathcal{D}]}$,
$\tilde{\bm{\Sigma}}_{12}=\tilde{\bm{\Sigma}}_{[\mathcal{K},\mathcal{D}]}$,
and
$\tilde{\bm{\Sigma}}_{12}^{\top}=\tilde{\bm{\Sigma}}_{[\mathcal{D},\mathcal{K}]}$.
For our purposes, this matrix-notation is only needed for the proofs to which
we refer for a more elaborate explanation.
Given the notation above, the following assumptions are made:
###### Assumption 1.
$\mathbb{E}(\bm{X})=\bm{0}$.
We assume the random variables to have mean zero for convenience purposes to
simplify the covariance matrix to
$\mathbb{E}(\bm{X}^{\top}\bm{X})=\tilde{\bm{\Sigma}}$.
###### Assumption 2.
$\tilde{\bm{\Sigma}}$ is positive definite.
We want to extract the information contained in the $\\{X_{d}\\}$ that can be
explained by the $\\{X_{k}\\}$. This is done by regression projection where
$\tilde{\bm{\Sigma}}$ is required to be invertible. This corresponds to the
full rank assumption of $\bm{x}$ in the sample case. Since
$\tilde{\bm{\Sigma}}=\bm{\Sigma}$ if $\bm{\mathrm{E}}=\bm{0}$ by construction,
we implicitly assume that $\bm{\Sigma}$ is positive definite as well.
###### Assumption 3.
$\bm{U}$ is a block-diagonal matrix with $\bm{U}^{\top}\bm{U}=\bm{I}$.
Orthogonality of the sparse loadings is needed to obtain results regarding the
explained total variance in this work. For convenience purposes, we therefore
assume orthogonality throughout the whole work which holds without loss of
generality: if $\bm{U}$ was not orthogonal, we would orthogonalize the
loadings by using the singular value decomposition of $\bm{U}$ with singular
values replaced by ones as it is done in a Procrustes problem (see, e.g.,
Mardia et al. (1979)). This can be done due to the block-diagonal shape of
$\bm{U}$ which will be introduced in (3), and because the exact values of the
non-zero components of $\bm{U}$ are not of concern. Both will be discussed
elaborately in Section 3.
## 3 Methodology and Motivation
Firstly, we recap PLA in this section. The method relies on choosing a
threshold value which, however, can be difficult in practice. Hence, we
motivate the usage of SPLA based on sparse loadings which does not require
hard-thresholding anymore.
PLA is a concept for dimensionality reduction where a subset of existing
variables is selected while the other variables are discarded. We refer to
Bauer and Drabant (2021) for an elaborate explanation. However, the intuition
is that blocks of variables are discarded which distort the covariance matrix
only slightly. We scan for a block $\bm{X}_{\mathcal{D}}$ by checking if all
$K$ elements $k\in\mathcal{K}$ of $D$ eigenvectors
$\\{v_{\delta}^{(k)}\\}_{\delta\in\mathit{\Delta}}$ of the covariance matrix
$\tilde{\bm{\Sigma}}$ are smaller in absolute terms than a certain cut-off
value $\tau$. That is to say, we check if $|v_{\delta}^{(k)}|\leq\tau$ for all
$(k,\delta)\in\mathcal{K}\times\mathit{\Delta}\equiv\\{(k,\delta):k\in\mathcal{K},\delta\in\mathit{\Delta}\\}$.
For $\bm{X}=(\bm{X}_{\mathcal{K}},\bm{X}_{\mathcal{D}})$, we therefore scan if
the eigenvectors $\bm{V}$ are of shape
$\displaystyle\begin{split}\bm{v}_{\delta^{c}}&=\begin{pmatrix}\bm{\ast}_{K}\\\
\bm{\varepsilon}_{D}\end{pmatrix}\;\text{ for
}\delta^{c}\in\mathit{\Delta}^{c}\equiv\\{1,\ldots,M\\}\backslash\mathit{\Delta}\;\text{
with }|\mathit{\Delta}^{c}|=K\\\
\bm{v}_{\delta}&=\begin{pmatrix}\bm{\varepsilon}_{K}\\\
\bm{\ast}_{D}\end{pmatrix}\;\text{ for }\delta\in\mathit{\Delta}\;\text{ with
}|\mathit{\Delta}|=D\end{split}$ (2)
and hence represent the blocks $\bm{x}_{\mathcal{K}}$ and
$\bm{x}_{\mathcal{D}}$ respectively. $\bm{\ast}_{K}\in\mathbb{R}^{K}$,
$\bm{\ast}_{D}\in\mathbb{R}^{D}$ are hereby vectors containing components
larger than $\tau$ in absolute terms, and
$\bm{\varepsilon}_{K}\in\mathbb{R}^{K}$,
$\bm{\varepsilon}_{D}\in\mathbb{R}^{D}$ are vectors containing components that
are all smaller than $\tau$ in absolute terms respectively. The intuition
comes from the special case when $\tau=0$, i.e. when $\bm{\mathrm{E}}=\bm{0}$,
since then the $\\{\bm{v}_{\delta^{c}}\\}_{\delta^{c}\in\mathit{\Delta}^{c}}$
and $\\{\bm{v}_{\delta}\\}_{\delta\in\mathit{\Delta}}$ represent the exclusive
distortion of the covariance matrix by the $\\{X_{k}\\}$ and the $\\{X_{d}\\}$
respectively.
###### Lemma 1.
The eigenvectors $\bm{V}\equiv(\bm{v}_{1},\ldots,\bm{v}_{M})$ of the
covariance matrix $\tilde{\bm{\Sigma}}\equiv\bm{\Sigma}+\bm{\mathrm{E}}$ from
(1) are of shape $\bm{v}_{\delta^{c}}=(\bm{\ast}_{K},\bm{0}_{D})^{\top}$ and
$\bm{v}_{\delta}=(\bm{0}_{K},\bm{\ast}_{D})^{\top}$ if and only if
$\bm{\mathrm{E}}=\bm{0}$.
In practice, we are more likely to face the situation that
$\bm{\mathrm{E}}\neq\bm{0}$. However, if the perturbation $\bm{\mathrm{E}}$ is
small, then the components of $\bm{\varepsilon}_{K}$ and
$\bm{\varepsilon}_{D}$ are small as well because they are bounded by
$\bm{\mathrm{E}}$.
###### Corollary 1.
For $\bm{\varepsilon}_{K}\in\mathbb{R}^{K}$ and
$\bm{\varepsilon}_{D}\in\mathbb{R}^{D}$ from (2) it holds that
$\|\bm{\varepsilon}_{K}\|_{\infty}=\mathcal{O}_{p}(\|\bm{\mathrm{E}}\|_{l})$
and
$\|\bm{\varepsilon}_{D}\|_{\infty}=\mathcal{O}_{p}(\|\bm{\mathrm{E}}\|_{l})$
for $l\in\\{2,\infty\\}$.
Therefore, if the eigenvectors are of shape as in (2) with
$\bm{\varepsilon}_{K}$, $\bm{\varepsilon}_{D}$ having all components smaller
than $\tau$ in absolute terms, then most distortion of the variables
$\\{X_{d}\\}$ is geometrically represented by the eigenvectors
$\\{\bm{v}_{\delta}\\}_{\delta\in\mathit{\Delta}}$. The explained variance of
the $\\{X_{d}\\}$ can then be evaluated either by
$\sum_{d\in\mathcal{D}}\tilde{\sigma}_{d,d}$ or approximated by
$(\sum_{m}\lambda_{m})^{-1}(\sum_{\delta\in\mathit{\Delta}}\lambda_{\delta})$
where $\lambda_{i}$ is the eigenvalue corresponding to the eigenvector
$\bm{v}_{i}$. If the explained variance is sufficiently small for the
underlying purpose of application, the variables $\\{X_{d}\\}$ are discarded.
The same holds for the block $\\{X_{k}\\}$ in an analogous manner.
The concern in PLA is the proper choice of the threshold $\tau$ which is
elementary for the detection of the blocks $\\{X_{d}\\}$ and $\\{X_{k}\\}$
respectively. There are no theoretical derivations for the choice of the
threshold yet. Bauer and Drabant (2021) provided simulation results and Bauer
and Drabant (2023) built a connection between $\tau$ and hypothesis testing in
multivariate linear regression. However, if we use sparse loadings instead of
eigenvectors, we do not rely on a threshold anymore. This is the intuition of
the new concept of SPLA: Instead of checking if the eigenvectors follow a
shape according to (2), we rather construct sparse loadings of shape
$\displaystyle\begin{split}\bm{u}_{\delta^{c}}&=\begin{pmatrix}\bm{\ast}_{K}\\\
\bm{0}_{D}\end{pmatrix}\;\text{ for
}\delta^{c}\in\mathit{\Delta}^{c}\equiv\\{1,\ldots,M\\}\backslash\mathit{\Delta}\;\text{
with }|\mathit{\Delta}^{c}|=K\\\ \bm{u}_{\delta}&=\begin{pmatrix}\bm{0}_{K}\\\
\bm{\ast}_{D}\end{pmatrix}\;\text{ for }\delta\in\mathit{\Delta}\;\text{ with
}|\mathit{\Delta}|=D\end{split}$
which represent the underlying covariance matrix. While eigenvectors follow a
natural order due to the size of their respective eigenvalues, for our
purposes such an order is not required for the sparse loadings. Instead, we
will always consider $\bm{U}$ to be a matrix of block diagonal shape
$\bm{U}\equiv\begin{pmatrix}\bm{\ast}_{K\times K}&\bm{0}_{K\times D}\\\
\bm{0}_{D\times K}&\bm{\ast}_{D\times
D}\end{pmatrix}\equiv\begin{pmatrix}\bm{U}_{1}&\bm{0}\\\
\bm{0}&\bm{U}_{2}\end{pmatrix}\;,$ (3)
which reduces $\mathit{\Delta}^{c}=\\{1,\ldots,K\\}$ and
$\mathit{\Delta}=\\{K+1,\ldots,M\\}$. If the sparse loadings do not follow
this shape by construction, we will transform $\bm{U}$ to a matrix of block-
diagonal shape $\bm{P}_{1}\bm{U}\bm{P}_{2}$ using permutation matrices
$\bm{P}_{1}$ and $\bm{P}_{2}$ for row and column permutations respectively. We
note that the exact component-values in $\bm{\ast}_{K\times K}$ and
$\bm{\ast}_{D\times D}$ are not of concern, since we are only interested if an
underlying structure representing the blocks $\\{X_{k}\\}$ and $\\{X_{d}\\}$
does exist. Therefore, we can always orthogonalize $\bm{U}$ as described in
Assumption 3 despite of its changes of the component-values.
## 4 Block Covariance Structure Identification
In this section, we discuss how SPLA identifies if the random vector $\bm{X}$
follows a block covariance structure.
According to Lemma 1, it is reasonable to consider the eigenvectors for
identification of the shape of the covariance matrix. Further, it holds that
the covariance matrix follows a block-diagonal structure if all eigenvectors
are either close to $(\bm{\ast}_{K},\bm{0}_{D})^{\top}$ or close to
$(\bm{0}_{K},\bm{\ast}_{D})^{\top}$.
###### Lemma 2.
For $\bm{\varepsilon}_{K}\in\mathbb{R}^{K}$ and
$\bm{\varepsilon}_{D}\in\mathbb{R}^{D}$ from (2) it holds that
$\|\bm{\mathrm{E}}\|_{l}=\mathcal{O}_{p}(\|(\bm{\varepsilon}_{K},\bm{\varepsilon}_{D})^{\top}\|_{\infty})$
for $l\in\\{2,F,\infty\\}$.
Instead of checking if the eigenvectors are close to the aforementioned shape
in the $\mathcal{L}_{1}$ norm, we rather suggest to check if sparse loadings
exist that are not only of shape
$\bm{u}_{\delta^{c}}=(\bm{\ast}_{K},\bm{0}_{D})^{\top}$ and
$\bm{u}_{\delta}=(\bm{0}_{K},\bm{\ast}_{D})^{\top}$ but also represent the
underlying covariance matrix adequately. A criterion to evaluate if the sparse
loadings are representative for the covariance matrix and therefore for the
random vector are provided in Section 6.
Note that using sparse loadings instead of eigenvectors loses the bounds on
the perturbation. When using hard-thresholding on the eigenvectors in PLA, the
magnitude of $\bm{\mathrm{E}}$ relates to the identification as a block-
diagonal covariance matrix as follows:
###### Corollary 2.
Let $\tau$ be the threshold for PLA and $\lambda_{1}>\cdots>\lambda_{M}$ be
the eigenvalues of $\tilde{\bm{\Sigma}}$ with $\lambda_{0}\equiv\infty$ and
$\lambda_{M+1}\equiv-\infty$. If the covariance matrix does not follow a
block-structure in PLA-sense, then
$\|\bm{\mathrm{E}}\|_{l}>\tau\cdot
2^{-3/2}\min\limits_{j\in\\{1,\ldots,M+1\\}}(\lambda_{j-1}-\lambda_{j})\;,$
for $l\in\\{2,\infty\\}$.
This bound on $\bm{\mathrm{E}}$ does not hold for the concept of SPLA when
using sparse loadings however. The reason is that SPLA considers block
covariance structure identification based on the covariance matrix and on the
precision within a block, and not exclusively on the covariances between
blocks. Hence, SPLA takes into account the relation between $\bm{\mathrm{E}}$
and $\bm{\Sigma}_{1}^{-1}$. This relation is important to evaluate if the
found covariance structure is reasonable and will be considered more
elaborately in Section 6.
## 5 Explained variance
Transforming the random vector using sparse loadings yields variables that are
potentially correlated. This is a concern when calculating the explained
variance for each loading or for each block of variables. Solutions to that
issue will be addressed in this section.
Since $\mathbb{E}[(\bm{X}\bm{v}_{i})^{\top}\bm{X}\bm{v}_{j}]=0$ and
$\mathbb{E}[(\bm{X}\bm{v}_{i})^{\top}\bm{X}\bm{v}_{i}]=\lambda_{j}$ for
$i,j\in\\{1,\ldots,M\\}$ and $i\neq j$, it is well known that the eigenvalues
contain the magnitude of dispersion of $\bm{X}$ along the eigenvectors. Hence,
the overall distortion of the covariance matrix is given by
$\sum_{i}\lambda_{i}$. In the sparse case, however, the projected variables
$\\{\bm{X}\bm{u}_{i}\\}\equiv\\{\bm{X}\bm{u}_{i}\\}_{i\in\\{1,\ldots,M\\}}$
might be correlated such that
$\mathbb{E}[(\bm{X}\bm{u}_{i})^{\top}\bm{X}\bm{u}_{j}]\neq 0$ for
$i,j\in\\{1,\ldots,M\\}$. Therefore, quasi-eigenvalues
$\mathbb{E}[(\bm{X}\bm{u}_{i})^{\top}\bm{X}\bm{u}_{i}]=\bm{u}_{i}^{\top}\tilde{\bm{\Sigma}}\bm{u}_{i}$
overvalue the explained total variance because the $\\{\bm{X}\bm{u}_{i}\\}$
contain similar contributions.
To find a different measure for the explained variance, we recap that the
partial covariance matrix $\tilde{\bm{\Sigma}}_{2\cdot 1}$ contains the
information left in $\bm{X}_{\mathcal{D}}$ after eliminating the effects of
$\bm{X}_{\mathcal{K}}$ by regression projection.
###### Remark 1.
Let $\bm{X}\equiv(\bm{X}_{\mathcal{K}},\bm{X}_{\mathcal{D}})$ with
$\mathbb{E}[\bm{X}^{\top}\bm{X}]=\tilde{\bm{\Sigma}}$. It holds that
$\mathbb{E}[(\bm{X}_{\mathcal{D}}-\bm{X}_{\mathcal{K}}\bm{\beta})^{\top}(\bm{X}_{\mathcal{D}}-\bm{X}_{\mathcal{K}}\bm{\beta})]=\bm{\Sigma}_{2}-\bm{\mathrm{E}}_{12}^{\top}\bm{\Sigma}_{1}^{-1}\bm{\mathrm{E}}_{12}\equiv\tilde{\bm{\Sigma}}_{2\cdot
1}\;,$ (4)
where $\bm{\beta}$ contains the respective regression coefficients.
Hence, an initial proposal is to use a measure for the explained variance that
is based on the partial covariance matrix $\tilde{\bm{\Sigma}}_{2\cdot 1}$ to
overcome the concern of correlation among the $\\{\bm{X}\bm{u}_{i}\\}$. McCabe
(1984) discussed several measures and we will use one of them for our examples
in Section 8. However, in order to find the variables we want to discard as
well as the number of variables we want to discard, each partial covariance
matrix for any possible combination of variables $\sum_{m=1}^{M-1}{M\choose
m}=2^{M}-2$ has to be calculated which is not feasible in practice. Further,
calculating the partial covariance matrix is computationally demanding.
Therefore, we suggest to firstly use a different evaluation of the explained
variance to find possible blocks of variables we consider to discard.
Zou et al. (2006) proposed to correct the explained variance using regression
projection as well. For calculating the explained variance of
$\bm{X}\bm{u}_{i}$, they suggest to correct upwardly for
$\\{\bm{X}\bm{u}_{j}\\}_{j\in\\{1,\ldots,i-1\\}}$ and therefore to use
$\mathbb{E}[(\bm{X}\bm{u}_{i}-\bm{X}\bm{U}_{i-1}\bm{\beta})^{\top}(\bm{X}\bm{u}_{i}-\bm{X}\bm{U}_{i-1}\bm{\beta})]\equiv
r_{i,i}^{2}\;,$ (5)
where $\bm{U}_{i-1}\equiv(\bm{u}_{1},\ldots,\bm{u}_{i-1})$, and $\bm{\beta}$
contains the respective regression coefficients. The advantage is the
following computationally more efficient measure for application in the sample
case using a QR decomposition.
###### Remark 2.
Let $\bm{x}\hat{\bm{U}}=\hat{\bm{Q}}\hat{\bm{R}}$ with
$\hat{\bm{R}}=(\hat{r}_{i,j})$ be the QR decomposition of the sample
$\bm{x}\hat{\bm{U}}$. It holds that
$\|\bm{x}\hat{\bm{u}}_{i}-\bm{x}\hat{\bm{U}}_{i-1}\hat{\bm{\beta}}\|_{2}^{2}=\hat{r}_{i,i}^{2}\;,$
(6)
where $\hat{\bm{U}}_{i-1}\equiv(\hat{\bm{u}}_{1},\ldots,\hat{\bm{u}}_{i-1})$,
and $\hat{\bm{\beta}}$ contains the respective regression coefficients.
Further, $(N-1)^{-1}\hat{r}_{i,i}^{2}\overset{p}{\longrightarrow}r_{i,i}^{2}$
is the sample counterpart of $r_{i,i}^{2}$ for $i\in\\{1,\ldots,M\\}$.
Hence, in the sample case the explained variance of $\bm{X}\bm{u}_{i}$
corrected by $\\{\bm{X}\bm{u}_{j}\\}_{j\in\\{1,\ldots,i-1\\}}$ can be
estimated by $(N-1)^{-1}\hat{r}_{i,i}^{2}$, and the explained total variance
can be estimated by $(N-1)^{-1}\sum_{i}\hat{r}_{i,i}^{2}$ respectively.
However, while (6) gives a computationally efficient measure, there are also
two downsides: Firstly, the evaluation of blocks of variables is not taken
into account since the measure considers each sparse loading as an $1\times 1$
block. Secondly, the measure is biased because we do not correct for all
sparse loadings. Since we correct for more loadings when $i$ increases, it
holds that this bias decreases the larger $i$. However, (6) still provides a
good first intuition regarding the explained variance which benefits from the
efficient calculation. Therefore, we propose that SPLA is based on (6) for a
first analysis to reduce the number of block-combinations that have to be
evaluated as well as on the partial covariance matrix (4) for the final
evaluation.
## 6 Identifying the Underlying Block-Structure
When calculating sparse loadings, we might create sparse loadings even if the
underlying eigenvector structure is not sparse-ish, or we might create
loadings that are sparser than the underlying eigenvector structure. In
Section 6.1, we therefore discuss how to evaluate if the obtained sparse
loadings are reasonable or not. Further, we recap methods to calculate sparse
loadings. Afterwards, we have all pieces together to provide the algorithm for
SPLA in Section 6.2. Additionally, we show in Section 6.3 that the explained
total variance of the sparse model is controlled by the introduced evaluation
in Section 6.1.
### 6.1 Sparseness Evaluation Criterion
The sparser the loadings, the easier it is to interpret the underlying
structure of the sample. Of course, it is only reasonable to enforce sparse
loadings in application if the underlying population eigenvectors are sparse-
ish. However, since we have only access to the sample eigenvectors in
practice, we need a criterion to decide if the found sparseness is reasonable
or not.
Let therefore $\bm{u}_{\delta^{\ast}}$ be the sparse loading associated with
the block $\\{X_{d}\\}$ with the smallest index. As a sparseness criterion, we
propose to evaluate the ratio between the corrected explained variance
$r_{{\delta^{\ast}},{\delta^{\ast}}}$ of $\bm{u}_{\delta^{\ast}}$ and its
uncorrected variance
$\mathbb{E}[(\bm{X}\bm{u}_{\delta^{\ast}})^{2}]=\bm{u}_{{\delta^{\ast}}}^{\top}\tilde{\bm{\Sigma}}\bm{u}_{{\delta^{\ast}}}$.
We use $\bm{u}_{\delta^{\ast}}$ because if we used another $\bm{u}_{d}$ with
$d<{\delta^{\ast}}$, then this sparse loading would be corrected by
$\bm{u}_{\delta^{\ast}}$ i.e. it would be corrected by variables included in
the same block. It holds the following result for the corrected explained
variance given by the largest sparse loading $\bm{u}_{\delta^{\ast}}$:
###### Theorem 1.
Let $1\notin\mathit{\Delta}$ and let
$\\{\bm{u}_{\delta}\\}_{\delta\in\mathit{\Delta}}$ be the sparse loadings
associated with the block $\\{X_{d}\\}$. It holds for
$r_{{\delta^{\ast}},{\delta^{\ast}}}^{2}\equiv\mathbb{E}[(\bm{X}\bm{u}_{\delta^{\ast}}-\bm{X}\bm{U}_{{\delta^{\ast}}-1}\bm{\beta})^{\top}(\bm{X}\bm{u}_{\delta^{\ast}}-\bm{X}\bm{U}_{{\delta^{\ast}}-1}\bm{\beta})]$
that
$r_{{\delta^{\ast}},{\delta^{\ast}}}^{2}=\bm{u}_{{\delta^{\ast}}}^{\top}\tilde{\bm{\Sigma}}\bm{u}_{{\delta^{\ast}}}-\mathcal{O}_{p}(\|\bm{\mathrm{E}}\bm{u}_{{\delta^{\ast}}}\|_{2}^{2}\|\bm{\Sigma}_{1}^{-1}\|_{2})\;,$
where ${\delta^{\ast}}\equiv\min\mathit{\Delta}$ is the smallest index of the
sparse loadings associated with the $\\{X_{d}\\}$,
$\bm{U}_{{\delta^{\ast}}-1}\equiv(\bm{u}_{1},\ldots,\bm{u}_{{\delta^{\ast}}-1})$,
and $\bm{\beta}$ contains the respective regression coefficients.
The bound depends on the covariances between the $\\{X_{d}\\}$ and the
$\\{X_{k}\\}$, and on the precision matrix of the $\\{X_{k}\\}$. The latter
one follows the indication by Bauer and Drabant (2021) that blocks become more
distinguishable the stronger the relation is within the blocks. In case the
underlying covariance matrix follows a perfect block-diagonal structure, i.e.
if $\bm{\mathrm{E}}=\bm{0}$, we can conclude that
$r_{{\delta^{\ast}},{\delta^{\ast}}}^{2}=\bm{u}_{{\delta^{\ast}}}^{\top}\tilde{\bm{\Sigma}}\bm{u}_{{\delta^{\ast}}}$
reduces to the largest sparse quasi-eigenvalue. We note, however, that the
bound depends not on the plain covariances between the blocks but rather on a
weighted covariance $\bm{\mathrm{E}}\bm{u}_{{\delta^{\ast}}}$ with weights
$\bm{u}_{{\delta^{\ast}}}$. This is problematic because different loadings of
same shape yield different evaluation outcomes. Additionally, variables with
zero-components in the corresponding loading are not considered in the
criterion. However, we can fix this issue by giving equal weights to all
variables. Therefore, the closer the ratio of the following criterion is to
one, the more likely we detected the correct underlying sparse structure.
###### Remark 3.
Let $1\notin\mathit{\Delta}$ and let
${\delta^{\ast}}\equiv\min\mathit{\Delta}$ be the smallest index of the sparse
loadings associated with the $\\{X_{d}\\}$. Let further
$\bm{w}_{{\delta^{\ast}}}\equiv(\bm{0}_{K},\bm{1}_{D})^{\top}$. We replace the
${\delta^{\ast}}$th loading by
$\bm{u}_{{\delta^{\ast}}}^{\top}\equiv\bm{w}_{{\delta^{\ast}}}/\|\bm{w}_{{\delta^{\ast}}}\|_{2}=D^{-1/2}\bm{w}_{{\delta^{\ast}}}$
and we replace $\\{\bm{u}_{i}\\}_{i\neq{\delta^{\ast}}}$ such that Assumption
3 is satisfied. The block evaluation criterion (EC) for $\\{X_{d}\\}$ is given
by
${\rm
EC}\equiv\frac{r_{{\delta^{\ast}},{\delta^{\ast}}}^{2}}{\bm{u}_{{\delta^{\ast}}}^{\top}\tilde{\bm{\Sigma}}\bm{u}_{{\delta^{\ast}}}}\in(0,1]$
(7)
and it holds that
${\rm
EC}=\frac{\bm{u}_{{\delta^{\ast}}}^{\top}\tilde{\bm{\Sigma}}\bm{u}_{{\delta^{\ast}}}-\mathcal{O}_{p}(\|D^{-1/2}\bm{\mathrm{E}}_{12}\bm{1}_{D}\|_{2}^{2}\|\bm{\Sigma}_{1}^{-1}\|_{2})}{\bm{u}_{{\delta^{\ast}}}^{\top}\tilde{\bm{\Sigma}}\bm{u}_{{\delta^{\ast}}}}$
due to Theorem 1.
Clearly, (7) equals one for the extreme case when $\bm{\mathrm{E}}=\bm{0}$ and
therefore when the two random vectors $\bm{X}_{\mathcal{K}}$ and
$\bm{X}_{\mathcal{D}}$ are perfectly uncorrelated. Further, the criterion is
positive by construction because it is a scaled variance term with
$\tilde{\bm{\Sigma}}$ being positive definite (Assumption 2).
The vector $\bm{w}_{{\delta^{\ast}}}$ weighs all covariances between
$\\{X_{d}\\}$ and $\\{X_{k}\\}$ equally making the EC a corrected criterion.
When replacing a loading $\bm{u}_{{\delta^{\ast}}}$, we have to change all
remaining loadings $\\{\bm{u}_{i}\\}_{i\neq{\delta^{\ast}}}$ to maintain
orthogonality of $\bm{U}$. However, such a change is quickly implemented and a
respective procedure is given in Supplementary Material 1.
For completion, we shall mention that we obtain similar bounds to Theorem 1
also for the case when the block-diagonal shape of the loadings given in (3)
is not satisfied. However, in this case it is not feasible to correct for all
$K$ loadings associated with the block $\\{X_{k}\\}$ since we would correct
for variables contained in the same block otherwise.
###### Lemma 3.
Let $1\notin\mathit{\Delta}$ and let
$\\{\bm{u}_{\delta}\\}_{\delta\in\mathit{\Delta}}$ be the sparse loadings
associated with the block $\\{X_{d}\\}$. If the block-diagonal condition of
$\bm{U}$ in Assumption 3 is violated, it holds for
$r_{{\delta^{\ast}},{\delta^{\ast}}}^{2}\equiv\mathbb{E}[(\bm{X}\bm{u}_{\delta^{\ast}}-\bm{X}\bm{U}_{{\delta^{\ast}}-1}\bm{\beta})^{\top}(\bm{X}\bm{u}_{\delta^{\ast}}-\bm{X}\bm{U}_{{\delta^{\ast}}-1}\bm{\beta})]$
that
$r_{{\delta^{\ast}},{\delta^{\ast}}}^{2}=\bm{u}_{\delta^{\ast}}^{\top}\tilde{\bm{\Sigma}}\bm{u}_{\delta^{\ast}}-\mathcal{O}_{p}(\|\bm{\mathrm{E}}\bm{u}_{\delta^{\ast}}\|_{2}^{2}\|\bm{\Sigma}_{1}^{-1}\|_{2})\;,$
where ${\delta^{\ast}}\equiv\min\mathit{\Delta}$ is the smallest index of the
sparse loadings associated with the $\\{X_{d}\\}$,
$\bm{U}_{{\delta^{\ast}}-1}\equiv(\bm{u}_{\kappa_{1}},\bm{u}_{\kappa_{2}},\ldots)$
with $\kappa_{1},\kappa_{2},\ldots<{\delta^{\ast}}$ and ${\rm
card}(\\{\kappa_{1},\kappa_{2},\ldots\\})<K$, and $\bm{\beta}$ contains the
respective regression coefficients.
We evaluate the sparseness only in an upward manner in the EC due to the
nature of the measure $r_{i,i}^{2}$ from $(\ref{eq:expvarzouPopulation})$ as
discussed in Section 5 in relation to Remark 2. Further, following Theorem 1
we evaluate a block with respect to its largest associated sparse loading. If
we used a different loading $\bm{u}_{\delta}$ with
$\delta\neq{\delta^{\ast}}\equiv\min\mathit{\Delta}$, the bound on the quasi-
eigenvalue would not only depend on $\bm{\mathrm{E}}_{12}$ and
$\bm{\Sigma}_{1}^{-1}$, but also on $\bm{\Sigma}_{2}$.
Note that ${\delta^{\ast}}=1$, i.e. using the first loading $\bm{u}_{1}$, is
not feasible for the evaluation criterion EC. We have to keep this in mind for
the general case with more than two blocks. Despite ${\delta^{\ast}}=1$ is
mathematically possible in (6), no additional information is provided since
loadings are corrected only in an upward direction. Therefore, the first
loading is not corrected for the effect of other loadings and therefore the EC
always equals one in this case.
###### Corollary 3.
Let $\\{\bm{u}_{\delta}\\}_{\delta\in\mathit{\Delta}}$ be the sparse loadings
associated with the block $\\{X_{d}\\}$. If
${\delta^{\ast}}\equiv\min\mathit{\Delta}=1$, then the block-structure
evaluation criterion EC from (7) reduces to one.
If this situation arises, however, one evaluates the block $\\{X_{k}\\}$
instead of $\\{X_{d}\\}$ by using its largest loading $\bm{u}_{\kappa^{\ast}}$
with $\kappa^{\ast}=\min\\{1,\ldots,M\\}\backslash\mathit{\Delta}$. Clearly,
verifying the block $\\{X_{k}\\}$ implies verification of $\\{X_{d}\\}$ as a
block. The condition that $1\notin\mathit{\Delta}$ in Theorem 1 is fulfilled
by construction in this work since we assumed for convenience that
$\mathcal{D}=\mathit{\Delta}=\\{K+1,\ldots,M\\}$ and therefore that
${\delta^{\ast}}\equiv\min\mathit{\Delta}=K+1$. However, in a more general
setting $1\in\mathit{\Delta}$ is possible as can be seen in the examples in
Section 8.
We note that other evaluation criteria based on the partial covariance matrix
can also be used.
###### Lemma 4.
Let $\tilde{\bm{\Sigma}}_{2\cdot
1}\equiv\bm{\Sigma}_{2}-\bm{\mathrm{E}}_{12}^{\top}\bm{\Sigma}_{1}^{-1}\bm{\mathrm{E}}_{12}$
be the partial covariance matrix. It holds that
$\frac{\|\tilde{\bm{\Sigma}}_{2\cdot
1}\|}{\|\bm{\Sigma}_{2}\|}=\frac{\|\bm{\Sigma}_{2}\|-\mathcal{O}_{p}(\|\bm{\mathrm{E}}_{12}\|^{2}\|\bm{\Sigma}_{1}^{-1}\|)}{\|\bm{\Sigma}_{2}\|}\in(0,1]\;.$
As mentioned in Section 5, however, calculation of the partial covariance
matrix is computationally costly.
### 6.2 Sparse Principal Loading Analysis
In Section 5, we proposed that a measure for the explained variance in SPLA
should be based on the QR decomposition (6) for a first analysis as well as on
the partial covariance matrix (4) for the final evaluation. To decide if the
found block-structure is reasonable, we additionally check the evaluation
criterion.
The missing piece is the calculation of the sparse loadings. Guerra-Urzola et
al. (2021) compared the performance of several methods and assessed that the
one by Shen and Huang (2008) performs best. Since it is a special case of the
approach by Witten et al. (2009), we use their method for the majority of this
work. For some of our examples, however, we applied the procedure of Zou et
al. (2006). The latter one allows sparsening of each loading individually by
penalization which allows to create any composition of sparse loadings and has
been used for the examples in Section 8.1 and Section 8.2 respectively. Bot
methods are recapped in Supplementary Material 1, however, we refer to the
original works for elaborate explanations.
We can now contribute an algorithm for SPLA. Discard $\\{X_{d}\\}$ according
to SPLA proceeds as follows:
Algorithm 1 Sparse principal loading analysis
1:Calculate the sparse loadings $\\{\bm{u}_{i}\\}$.
2:Assess if the detected sparseness is meaningful by checking the EC from (7).
Repeat step 1-2 as long as the sparseness is reasonable.
3:Evaluate the detected blocks of variables using their explained variances
from (6) and select potential blocks for discarding accordingly.
4:Check the choice in step 3 using the partial covariance matrices from (4).
Discard the blocks if selection is verified.
In case we are only interested in finding the underlying block covariance
structure, the first three steps 1 to 3 of Algorithm 1 are sufficient.
Further, in step 2 we decide that the detected sparseness is not reasonable
when the evaluation criterion falls below a fixed parameter $c_{\rm EC}$.
### 6.3 Variance-Interpretability-Trade-Off
In this section, we discuss the trade-off between the explained total variance
and interpretability, and elaborate the importance of a large explained total
variance. We will show that the variance-interpretability-trade-off is
addressed appropriately when the evaluation of the underlying sparseness
discussed in Section 6.1 is considered.
For applications, we have to be aware of the trade-off between the explained
total variance and interpretability of SPLA. We discussed in Section 6.1 that
the more sparse the loadings, the easier it is to interpret the underlying
sample. However, this comes at the price of a small explained total variance
which, on the other hand, might make interpretation not reasonable anymore
because the sparse loadings simply provide too little information.
In application, the explained total variance of the sample sparse loadings
according to (6) relative to the total variance of $\bm{x}$ is given by
$(N-1)^{-1}\sum_{i}\hat{r}_{i,i}^{2}/{\rm tr}(\hat{\tilde{\bm{\Sigma}}})$. Due
to the perturbation
$\hat{\bm{\mathrm{E}}}\equiv\hat{\tilde{\bm{\Sigma}}}-\hat{\bm{\Sigma}}$, we
obtain the following result:
###### Theorem 2.
Let $\varepsilon(\bm{x})$ be a perturbation of the sample such that
$\check{\bm{x}}\equiv\bm{x}-\varepsilon(\bm{x})$ has covariance matrix
$\hat{\bm{\Sigma}}$. It holds that
$\|\varepsilon(\bm{x})\|=\mathcal{O}_{p}(\|\hat{\bm{\mathrm{E}}}\|)\;.$
If further
$(N-1)^{-1}\|\hat{\bm{\Sigma}}^{-1}\|_{2}^{1/2}\|\varepsilon(\bm{x})\|_{2}<1$,
it holds that
$(N-1)^{-1}\sum\limits_{i}\hat{r}_{i,i}^{2}={\rm
tr}(\hat{\tilde{\bm{\Sigma}}})-(N-1)^{-1}\mathcal{O}_{p}(\|\hat{\bm{\mathrm{E}}}\|_{F}^{2})\;.$
Theorem 2 contains two implications. Firstly, since
$\|\varepsilon(\bm{x})\|=\mathcal{O}_{p}(\|\hat{\bm{\mathrm{E}}}\|)$, the
condition that
$(N-1)^{-1}\|\hat{\bm{\Sigma}}^{-1}\|_{2}^{1/2}\|\varepsilon(\bm{x})\|_{2}<1$
evaluates the relation within blocks in comparison to the relation between
blocks as discussed in Section 6.1. Secondly, the explained total variance
drops in case we erroneously assign a block. In turn, the explained total
variance is large in case we correctly detect a block. However, recap that we
already control for correct specification of blocks in SPLA due to the EC from
(7). We can conclude that the variance-interpretability-trade-off is of no
concern when we follow Algorithm 1 and therefore evaluate the sparseness
according to the evaluation criterion.
## 7 Simulation Study
In this section, we deepen the understanding of SPLA based on simulations. In
Section 7.1, we analyse the behaviour of the EC from Section 6 for block-
diagonal covariance detection when the correlation among blocks varies.
Afterwards, we evaluate the percentage of identified block-diagonal covariance
matrices (identification rate) of SPLA in Section 7.2. We further discuss
considerations regarding our simulation design in Section 7.3. Code to
replicate all simulation results is provided in Supplementary Material 2.
### 7.1 Corrected Evaluation Criterion Analysis
In this section, we simulate a sample following a block-diagonal correlation
matrix of six blocks to demonstrate the evaluation using the EC.
Figure 1: Sample correlation matrices of $\bm{x}^{[0.1]}$, $\bm{x}^{[0.2]}$,
$\bm{x}^{[0.4]}$, and $\bm{x}^{[0.5]}$ respectively.
Let therefore $Y$ and $Z_{i}$ for $i\in\\{1,\ldots,5\\}$ be IID $N(0,10)$
distributed, and let $W$ be $N(0,1)$ distributed. We consider a random vector
$\bm{X}^{[\rho]}=(X_{1},\ldots,X_{14})$ with
$X_{j}=\sqrt{1-\rho}Z_{i}+\sqrt{\rho}Y+W$ for
$(j,i)\in\\{(1,1),(2,1),(3,2),(4,2),\ldots,(13,6),(14,6)\\}$ such that
$\bm{X}^{[\rho]}$ has a covariance matrix with six $2\times 2$ blocks on the
diagonal. Hereby, $\rho$ reflects the approximate correlation between the
blocks. We simulate a sample $\bm{x}^{[\rho]}$ by firstly drawing $N=100$
observations from $\bm{X}^{[\rho]}$ and secondly standardizing the sample.
Therefore, the sample covariance matrix equals the sample correlation matrix
which is illustrated in Figure 1 for $\rho\in\\{0.1,0.2,0.4,0.5\\}$.
Afterwards, we redraw $s\in\\{1,\ldots,100\\}$ samples $\bm{x}^{[\rho|s]}$
with $N=100$ observations and calculate the respective EC according to
$(\ref{eq:CEC})$ for block two, four, and six. The results are illustrated as
boxplots in Figure 2.
Figure 2: EC for block two, four, and six for each of the
$s\in\\{1,\ldots,100\\}$ samples $\bm{x}^{[\rho|s]}$ drawn from
$\bm{X}^{[\rho]}$ for $\rho\in\\{0.1,0.2,0.3,0.4,0.5\\}$ respectively.
The EC decreases with an increase of the correlation among the blocks and,
therefore, with an increase of $\rho$. Further, the EC decreases when
evaluating more blocks. This is due to the construction of the corrected
measure for the explained variance in $(\ref{eq:expvarzou})$ because we
control in an upward direction.
According to the EC for the sixth block in Figure 2, $c_{\rm EC}=0.6$ seems
appropriate because it confirms the block for $\rho<0.3$ and rejects the block
for $\rho>0.5$. This is meaningful, since strong correlation among variables
tells us that they are not separated into different blocks. We therefore use
$c_{\rm EC}=0.6$ for the remainder of this work. For completion, we shall
emphasize that choosing the appropriate sparseness depends on application.
Choosing a too small $c_{\rm EC}$, however, is no concern because erroneously
detected blocks are corrected in step 4 of Algorithm 1 using the partial
covariance matrix.
### 7.2 Performance
In this section, we compare the performance of block-diagonal covariance
structure detection. Recently, Devijver and Gallopin (2018) derived
nonasymptotic approaches that select the number of blocks either using a slope
heuristic robust regression (SHRR) or using a slope heuristic dimension jump
(SHDJ). We refer to their work for an elaborate explanation. These approaches
outperform existing methods when the number of blocks is unknown a priori and
we therefore compare SPLA to them.
Similar to Section 7.1, we simulate $s\in\\{1,\ldots,100\\}$ samples
$\bm{x}^{[\rho|s]}$ for $\rho\in\\{0,0.1,\ldots,0.9\\}$ with number of
observations $N\in\\{1000,500,100,50,20\\}$. We compare the performance of the
methods according to the identification rate and the results are illustrated
in Figure 3.
Figure 3: Performance of the block-diagonal covariance structure detection by
SHDJ, SHRR, and SPLA with $c_{\rm EC}=0.6$ measured by the identification rate
(the percentage of samples where all blocks where identified) on
$s\in\\{1,\ldots,100\\}$ samples $\bm{x}^{[\rho|s]}$ for
$\rho\in\\{0,0.1,\ldots,0.7\\}$ with number of observations
$N\in\\{1000,500,100,50,20\\}$. Further, the EC for each block is illustrated
for the case when $N=100$.
SPLA performs best for large sample sizes. The identification rate decreases
for larger $\rho$ since the covariance matrix can no longer be considered to
be a block-diagonal matrix due to the increasing correlation among variables.
This is illustrated by the decreasing EC. Still, we could increase
identification rate by assuming a small EC to be reasonable as well. This is
illustrated in Figure 4, where the identification rate continuous to be high
for an increasing $\rho$. However, this demonstration of adapting $c_{\rm EC}$
is only given for completion since it might be useful for a different context.
One should be aware that a small EC might not reflect a reasonable block-
structure as discussed in Section 6.
Figure 4: Performance of the block-diagonal covariance structure detection by
SPLA for $c_{\rm EC}\in\\{0.1,0.2,0.3,0.4,0.5,0.6\\}$ measured by the
identification rate (the percentage of samples where all blocks where
identified) on $s\in\\{1,\ldots,100\\}$ samples $\bm{x}^{[\rho|s]}$ for
$\rho\in\\{0,0.1,\ldots,0.9\\}$ with number of observations
$N\in\\{500,100,20\\}$. Identification rates for SHDJ are drawn to make the
graphics comparable to Figure 3.
SHDJ and SHRR increase performance the smaller the sample size because they
are nonasymptotic approaches. Further, they can also be applied for the case
when $M>N$ which is not discussed for SPLA in this work.
### 7.3 Considerations about the Simulation Design
Simulations always face limitations due to their artificialness. However,
there are further considerations due to the nature of SPLA. In the previous
sections, we constructed a spiked covariance model (Johnstone, 2001) similar
to Zou et al. (2006) and Shen and Huang (2008) respectively to cause strong
correlations within each block. The reason is that SPLA might rightfully split
an intended block into several blocks if the within-block-correlation is
small, making it difficult to analyse the obtained results. As an
illustration, we simulated $1000$ times the matrix
$\bm{A}\in\mathbb{R}^{3\times 3}$ by drawing each element $a_{i,j}$ from a
uniform $U(0,1)$ distribution. The covariance matrix
$\hat{\tilde{\bm{\Sigma}}}=\bm{A}^{\top}\bm{A}$ was constructed accordingly
and we consider the respective correlation matrix. By construction, the
correlation matrix reflects three correlated variables. However, in Figure 5
we see that SPLA detected one, two, or three blocks respectively for different
realisations. The EC is small when SPLA detected one block (i.e. no split of
the variables) and large for two or three blocks respectively which indicates
that the found block-structure was reasonable.
Figure 5: Number of detected blocks by SPLA in relation to the EC for
simulated samples with correlation matrices obtained from the simulated
covariance matrix $\hat{\tilde{\bm{\Sigma}}}=\bm{A}^{\top}\bm{A}$.
Therefore, our simulations follow a spiked covariance model having a large
within-block-correlation to ensure that an intended block cannot be
interpreted as several blocks from a SPLA perspective.
## 8 Examples
We provide two examples for SPLA on real data and one example on synthetic
data in this section. The first example in Section 8.1 mainly serves to
illustrate detection of the underlying block covariance structure and results
are compared to the ones obtained by PLA. In the second example in Section
8.2, the principal focus lies on variable selection by SPLA. Afterwards, we
discuss a synthetic example from the sparse principal component analysis
context in Section 8.3. Code to replicate all examples is provided in
Supplementary Material 3.
### 8.1 OECD Sample
For the first example, we use data from Mankiw et al. (1992) that consist of
$N=22$ OECD countries with population larger than one million. The sample
contains the real GDP in 1960 (Y60) and in 1985 (Y85) per person of working
age respectively. Further, it consists of the average annual ratios of real
investment to GDP (I/Y), the percentage of working-age population that is in
secondary school (SCH), the annual population growth from 1960 to 1985 (POP),
and the average of annual ratios of gross domestic expenditure on research and
development to nominal GDP (RD). Data are available in the CRAN contributed
package AER by Kleiber and Zeileis (2008). We standardized the sample i.e. we
consider the correlation matrix rather than the covariance matrix because the
variables have different scales.
The detected blocks by both PLA and SPLA are summarized in Table 1. A large
number of blocks $K$ eases interpretation while a large EC indicates that the
block-structure represents the underlying sample. Therefore, splitting the
OECD sample into the four blocks {I/Y}, {SCH}, {POP}, and {RD, Y85, Y60} with
EC = 0.84 is an appropriate choice since this provides a reasonable as well as
more detailed representation of the underlying sample. The corresponding
sparse loadings which we obtain when performing SPLA according to Algorithm 1
are given in Table 2. In general, we continuously increase the sparseness of
the loadings to find the underlying structure until the EC indicates that
sparseness is too large. This procedure is indicated in Table 1, where we stop
with four blocks since five blocks result in a small EC.
Table 1: Number of blocks $K$ of the OECD sample detected by PLA and SPLA. For PLA, we either provide the threshold $\tau$ or we denote $--$ if the structure cannot be identified by any threshold. For SPLA, we give the smallest EC of all detected blocks to evaluate if the respective structure is reasonable. $K$ | Blocks | PLA | SPLA
---|---|---|---
2 | {I/Y, POP}, {SCH, RD, Y85, Y60} | $--$ | $0.96$
2 | {I/Y, POP, SCH}, {RD, Y85, Y60} | $0.40$ | $0.84$
3 | {I/Y}, {POP}, {SCH, RD, Y85, Y60} | $--$ | $0.96$
3 | {RD}, {Y85, Y60}, {I/Y, SCH, POP} | $0.50$ | $0.53$
4 | {I/Y}, {POP}, {SCH}, {RD, Y85, Y60} | $--$ | $0.84$
5 | {I/Y}, {POP}, {SCH}, {RD}, {Y85, Y60} | $--$ | $0.45$
Note that PLA does not detect as many combinations as SPLA due to its nature
of hard-thresholding. Further, PLA does not provide information to decide
which number of blocks to choose. This is a concern because the detected
blocks with $\tau=0.5$ could be chosen which are a poor representation of the
underlying sample according to the EC.
Table 2: Sample sparse loadings $\hat{\bm{u}}_{i}$ with $i\in\\{1,\ldots,6\\}$ of the covariance matrix of the standardized OECD sample. All loading components equal to zero are left blank. ${\rm SV}\equiv\hat{r}_{i,i}^{2}/{\rm tr}(\hat{\tilde{\bm{\Sigma}}})$ denotes the share of explained total variance (in percent), CV denotes the cumulative share of explained total variance for each block (in percent). $\hat{\bm{u}}_{i}$ | $\hat{\bm{u}}_{3}$ | $\hat{\bm{u}}_{4}$ | $\hat{\bm{u}}_{2}$ | $\hat{\bm{u}}_{1}$ | $\hat{\bm{u}}_{5}$ | $\hat{\bm{u}}_{6}$
---|---|---|---|---|---|---
I/Y | 1.00 | | | | |
SCH | | 1.00 | | | |
RD | | | 1.00 | | |
POP | | | | 0.50 | -0.87 | 0.00
Y85 | | | | 0.64 | 0.36 | 0.68
Y60 | | | | 0.59 | 0.34 | -0.73
EC | | 0.96 | 0.93 | 0.84
SV | 16.67 | 16.04 | 15.57 | 40.26
CV | 16.67 | 32.71 | 48.28 | 88.54
Variable selection appears not to be feasible in this case, because no block
has a small cumulative share of explained total variance and, therefore, there
is no block that distorts the covariance matrix only a little. As discussed in
Section 5 however, we have to be aware that the explained variance given by
$\bm{R}$ is biased. Following Algorithm 1, we therefore check the results
using the partial covariance matrices. Instead of calculating the partial
covariance matrix for all possible combinations $\sum_{m=1}^{6-1}{6\choose
m}=62$, we are now left with calculations for the three respective blocks. The
partial covariance does not indicate that any block should be discarded either
and the respective results can be found in Table 5 for completion.
### 8.2 EXAM Sample
The second example deals with the EXAM sample from Mardia et al. (1979), where
we have access to grades of $N=88$ students in a five part exam containing the
subjects Mechanics (mec), Vectors (vec), Algebra (alg), Analysis (ana), and
Statistics (stat). Each subject has been scored separately. The sample was
taken from the CRAN contributed package bootstrap by Tibshirani (2019).
Following the procedure in Algorithm 1, we obtain the results given in Table
3.
Table 3: Sparse loadings $\bm{u}_{i}$ with $i\in\\{1,\ldots,5\\}$ of the covariance matrix of the EXAM sample. All loading components equal to zero are left blank. ${\rm SV}\equiv r_{i,i}^{2}/{\rm tr}(\tilde{\bm{\Sigma}})$ denotes the share of explained total variance (in percent), CV denotes the cumulative share of explained total variance for each block (in percent). $\bm{u}_{i}$ | $\bm{u}_{1}$ | $\bm{u}_{2}$ | $\bm{u}_{3}$ | $\bm{u}_{4}$ | $\bm{u}_{5}$
---|---|---|---|---|---
vec | 1.00 | | | |
mec | | 1.00 | | |
alg | | | -0.09 | -0.99 | 0.00
ana | | | -0.65 | 0.06 | -0.76
sta | | | -0.75 | 0.06 | 0.66
EC | | 0.74 | 0.72
SV | 13.21 | 19.28 | 38.98
CV | 13.21 | 32.49 | 71.47
Due to the evaluation criteria of $0.74$ and $0.72$, we can conclude that the
block-structure consisting of the three blocks {vec}, {mec}, and {alg, ana,
sta} reflects the EXAM sample well. Further, the shared explained total
variance of {vec} equals 13.21%. Since a single variable out of five should on
average explain 20% and since the 13.27% are biased upwardly as discussed in
Section 5, we consider to discard this variable. We verify or falsify this
selection by checking the partial covariance matrix. Accordingly, the block
{vec} explains only 7.45% (Table 5) and discarding is verified.
### 8.3 Synthetic Example
We use an example based on Zou et al. (2006) and Shen and Huang (2008) to
illustrate the block covariance structure detection of a synthetic sample.
First, we consider a random vector $\bm{X}=(X_{1},\ldots,X_{8})$ with
$X_{j}=Z_{1}+\theta_{j}$ for $j\in\\{1,\ldots,4\\}$ and
$X_{j}=Z_{2}+\theta_{j}$ for $j\in\\{5,\ldots,8\\}$. $Z_{1}$, $Z_{2}$, and the
$\theta_{j}$ are independent and we let $Z_{1}\sim N(0,290)$, $Z_{2}\sim
N(0,300)$ and $\theta_{j}\sim N(0,1)$ for $j\in\\{1,\ldots,10\\}$. A synthetic
sample $\bm{x}$ consisting of $N=5000$ observations is generated accordingly.
By construction, the sample consists of the two blocks
$\\{X_{1},\ldots,X_{4}\\}$ and $\\{X_{5},\ldots,X_{8}\\}$. SPLA detects this
underlying block-structure which is reflected by a EC of $0.9998$ (Table 4).
We extend the synthetic example and add the two random variables
$X_{j}=-0.3\cdot Z_{1}+0.925\cdot Z_{2}+\theta_{j}$ for $j\in\\{9,10\\}$ to
the random vector $\bm{X}=(X_{1},\ldots,X_{10})$. Both Zou et al. (2006) and
Shen and Huang (2008) discuss the appropriate sparseness of the loadings.
Hereby, Zou et al. (2006) use the knowledge about the synthetic sample while
the approach of Shen and Huang (2008) is based on the explained variance of
the most important loadings in a variance-sense. Expressing their results in
SPLA-sense, they discuss if the loadings are of shape
$\begin{pmatrix}\bm{\ast}_{4\times 4}&\bm{0}&\bm{0}\\\
\bm{0}&\bm{\ast}_{4\times 4}&\bm{0}\\\ \bm{0}&\bm{0}&\bm{\ast}_{2\times
2}\end{pmatrix}\;\text{ or }\;\begin{pmatrix}\bm{\ast}_{4\times 4}&\bm{0}\\\
\bm{0}&\bm{\ast}_{6\times 6}\end{pmatrix}$
respectively and therefore if the three blocks $\\{X_{1},\ldots,X_{4}\\}$,
$\\{X_{5},\ldots,X_{8}\\}$ and $\\{X_{9},X_{10}\\}$, or if the two blocks
$\\{X_{1},\ldots,X_{4}\\}$ and $\\{X_{5},\ldots,X_{10}\\}$ represent the
random vector. From a SPLA-perspective, we clearly confirm the two-block-
structure with an EC of $0.9910$ and results can be found in Table 4. The
block $\\{X_{9},X_{10}\\}$ in the three-block-scenario has an EC of $0.0009$
and we can therefore conclude that it is not adequate to split
$\\{X_{1},\ldots,X_{10}\\}$ into $\\{X_{5},\ldots,X_{8}\\}$ and
$\\{X_{9},X_{10}\\}$.
Table 4: Block indices $j$ for the $\\{X_{j}\\}$ of the synthetic examples given by $\bm{X}=(X_{1},\ldots,X_{8})$ and $\bm{X}=(X_{1},\ldots,X_{10})$ respectively. EC is the respective criterion for the blocks according to Remark 3. $\bm{X}$ | $\bm{X}=(X_{1},\ldots,X_{8})$ | $\bm{X}=(X_{1},\ldots,X_{10})$ | $\bm{X}=(X_{1},\ldots,X_{10})$
---|---|---|---
$j$ | $\\{1,\ldots,4\\}$ | $\\{5,\ldots,8\\}$ | $\\{1,\ldots,4\\}$ | $\\{5,\ldots,10\\}$ | $\\{1,\ldots,4\\}$ | $\\{5,\ldots,8\\}$ | $\\{9,10\\}$
EC | | 0.9998 | | 0.9910 | | 0.9998 | 0.0009
## 9 Concluding Remarks
We introduced SPLA as a new concept for variable selection and block
covariance structure identification based on sparse loadings. We proposed how
the sparse loadings can be used to identify the underlying block-structure,
and gave a criterion to evaluate if the detected structure fits the covariance
matrix. Regarding dimensionality reduction, calculation of the explained total
variance is a concern and we provided an eligible measure to control the loss
of explanatory power. Algorithm 1 can be used to apply SPLA for variable
selection and block covariance structure identification in practice. The
algorithm is available in the CRAN contributed package prinvars by Bauer and
Holzapfel (2023).
Some improvements have to be addressed in future research. From an applied
perspective, the choice of the method to calculate the sparse loadings needs
to be discussed. This should address the computational speed by comparing
existing methods that calculate sparse loadings. However, the methods need
also be evaluated with respect to the created block-structure. From a
theoretical perspective, considering how SPLA is related to the optimal
variable selection based on the partial covariance matrix is of interest.
While SPLA yields blocks that possibly explain only little of the overall
variance, it is not clear yet how optimal these blocks are in a variable
selection-sense.
There are methods to calculate sparse loadings that are stable in the high-
dimensional setting. This motivates block covariance structure identification
in the generalized spiked covariance model (Bai and Yao, 2012) or the spiked
population model (Johnstone, 2001) as a special case where the $M\times M$
population covariance matrix is assumed to be of form
$\begin{pmatrix}{\rm Cov}(\bm{X})&\bm{0}\\\
\bm{0}&\bm{V}_{M^{\prime}}\end{pmatrix}$
where $M^{\prime}=M-P$ and $M=M(N)$ such that $M/N\to\theta>0$ ($N\to\infty$).
A natural extension of SPLA would be the identification of this block
covariance structure in this high dimensional scenario.
Further, SPLA in time series analysis and functional data analysis is of
interest. Let therefore $\\{X_{t}\\}_{t\in\mathcal{T}}$ be a stochastic
process where $\mathcal{T}$ is discrete. In time series analysis, the
autocovariance function $\gamma_{X}(s,t)\equiv{\rm Cov}(X_{s},X_{t})$ gives
the covariance with itself at pairs of time points. Further, weak stationarity
of time series requires that $\gamma_{X}(h)\equiv{\rm Cov}(X_{t},X_{t+h})$ for
$t,h\in\mathbb{Z}$ does not depend on time and therefore assumes that the
autocovariance matrix consists of one block in a SPLA-sense. Hence, if the
autocovariance matrix consists of several blocks, SPLA detects structural
breaks where the weak stationary assumption is violated. On the other hand,
SPLA can be used for functional (time) data analysis to detect time intervals
that are less important from a variance perspective. This can be done by
extending the concept of SPLA to the eigenfunctions of the covariance function
of the underlying stochastic process.
## Appendix A Complementary Results
This section provides complementary results about the partial covariance
matrix for both examples discussed in Section 8.1 and Section 8.2
respectively.
Table 5 contains the trace of the sample partial covariance matrix for each of
the identified blocks from Section 8. Considering the trace is in line with
one of the measures given by McCabe (1984). Further, scaling the matrices by
the overall explained variance ${\rm tr}(\hat{\tilde{\bm{\Sigma}}})$ gives us
the contribution of each block corrected by all remaining blocks in percent.
Table 5: Share of the total variance explained by the sample partial covariance matrix $\hat{\tilde{\bm{\Sigma}}}_{2\cdot 1}$ of the variables $\\{X_{d}\\}$ eliminating the effects of the remaining variables $\\{X_{k}\\}$ such that $\\{X_{d}\\}\cup\\{X_{k}\\}=\\{X_{j}\\}_{j\in\\{1,\ldots,M\\}}$ both for the OECD and for the EXAM sample respectively. $\mathcal{D}$ | ${\rm tr}(\hat{\tilde{\bm{\Sigma}}}_{2\cdot 1})/{\rm tr}(\hat{\tilde{\bm{\Sigma}}})\cdot 100$ | $\mathcal{D}$ | ${\rm tr}(\hat{\tilde{\bm{\Sigma}}}_{2\cdot 1})/{\rm tr}(\hat{\tilde{\bm{\Sigma}}})\cdot 100$
---|---|---|---
{I/Y} | 10.23 | {vec} | 7.45
{SCH} | 12.41 | {mec} | 17.97
{POP} | 12.94 | {alg, ana, sta} | 46.49
{RD, Y85, Y60} | 41.73 | |
SUPPLEMENTARY MATERIAL
1. 1.
Methods to calculate sparse loadings and weight matrix construction for Remark
3.
2. 2.
Simulation results
3. 3.
Examples
4. 4.
Proofs
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|
capbtabboxtable[][]
11institutetext: CentraleSupélec, Université Paris-Saclay, 91190 Gif-sur-
Yvette, France 11email<EMAIL_ADDRESS>22institutetext: MICS
Laboratory, CentraleSupélec, Université Paris-Saclay, 91190 Gif-sur-Yvette,
France 22email<EMAIL_ADDRESS>
# Artifact Removal in Histopathology Images
Cameron Dahan 11 Stergios Christodoulidis 22 Maria Vakalopoulou 22 Joseph
Boyd 22
###### Abstract
In the clinical setting of histopathology, whole-slide image (WSI) artifacts
frequently arise, distorting regions of interest, and having a pernicious
impact on WSI analysis. Image-to-image translation networks such as CycleGANs
are in principle capable of learning an artifact removal function from
unpaired data. However, we identify a surjection problem with artifact
removal, and propose a weakly-supervised extension to CycleGAN to address
this. We assemble a pan-cancer dataset comprising artifact and clean tiles
from the TCGA database. Promising results highlight the soundness of our
method.
###### Keywords:
C
ycleGANs; Weakly-supervised; Histopathology; Artifacts
## 1 Introduction
Due to their handling and large size, histopathology slides are susceptible to
contaminants and other artifacts, leading to image artifacts in whole slide
images (WSI). Common artifacts include pen marker, ink, blur, air bubbles,
tissue folds, dust and filaments [12]. Such artifacts create noise and
outliers in WSI datasets, potentially undermining statistical analysis. An
automatic system capable of localising and removing artifacts is therefore of
great interest.
CycleGANs [13] are unpaired image-to-image translation models with several
existing applications in histopathology, notably in stain normalisation [10,
3, 2], stain transfer [4], and cell segmentation [7]. CycleGANs have been
applied to histopathology artifacts before [1], but this was restricted to the
relatively straightforward case of pen marker.
In Section 2 we propose an extension to the CycleGAN framework aimed at
accounting for the surjective nature of artifact removal. In Section 3 we
demonstrate the improvement of our proposed method over baselines.
## 2 Methods
Our artifact removal system is based on the CycleGAN framework, for which we
denote artifact and clean tiles as image domains $A$ and $B$. Accordingly, our
baseline objective function is,
$\displaystyle\mathcal{L}_{base}$
$\displaystyle=\mathcal{L}_{GAN}(G_{AB},D_{B})+\lambda_{ABA}\cdot\mathcal{L}_{CYC}(G_{BA}\circ
G_{AB})+\lambda_{A}\cdot\mathcal{L}_{ID}(G_{BA})$ (1)
$\displaystyle{}+\mathcal{L}_{GAN}(G_{BA},D_{A})+\lambda_{BAB}\cdot\mathcal{L}_{CYC}(G_{AB}\circ
G_{BA})+\lambda_{B}\cdot\mathcal{L}_{ID}(G_{AB}),$
where generators $G_{AB}$ and $G_{BA}$ translate between the image domains,
and $D_{B}$ and $D_{A}$ are their respective discriminators. The loss
$\mathcal{L}_{GAN}$ represents a least squares GAN loss [8];
$\mathcal{L}_{CYC}$ a cycle consistency loss; and $\mathcal{L}_{ID}$ an
identity function loss. The system is optimised through a minmax optimisation
process.
The key observation of our method is that artifact removal is a _surjective_
operation; many artifact images may correspond with the same clean image. If
truly clean, the image should not retain information about the artifact. This
creates a difficult task for generator $G_{BA}$, which is required to
reconstruct artifacts in $\lambda_{ABA}\cdot\mathcal{L}_{CYC}$ (Equation 1).
This creates strong pressure for $G_{AB}$ to only partially clean or otherwise
encode extraneous information within artifact images. This is clearly at odds
with minimising the adversarial loss $\mathcal{L}_{GAN}(G_{AB},D_{B})$
(Equation 1). We refer to this as the surjection problem and hypothesise that
this leads the CycleGAN to a poor local minimum. An initial idea is to simply
set $\lambda_{ABA}=0$, and to rely on the other half of the cycle to enforce
consistency.
### 2.1 Weakly supervised CycleGAN
[width=0.8]tikz/model
Figure 1: Weakly-supervised CycleGAN for artifact$\to$clean$\to$artifact
consistency.
To better address the problem of surjective artifact removal, we leverage the
label of artifact tiles in two alternative models. In the first model, the
label is used as a conditioning variable for both the $G_{BA}$ generator and
the $D_{A}$ discriminator. For each, the condition is encoded as a one-hot
tensor with $7$ class channels, where a channel is set to ones to indicate the
class label. This conditioning is concatenated with the input image channel-
wise. Thus, $G_{BA}$ can learn what class of artifact has been removed. We
refer to this model as $\mathcal{M}_{cond}$.
In the second model we introduce an attention network, $\alpha$. This is a
two-layer MLP applied to each pixel of the penultimate layer, $g_{AB}$, of
generator $G_{AB}$, followed by a sigmoid activation function. The $A\to B\to
A$ cycle-consistency is thus redefined to be,
$G_{BA}\circ
G_{AB}:=G_{BA}\Big{(}concat\big{(}\alpha(g_{AB}),proj_{RGB}(g_{AB})\big{)}\Big{)}$
(2)
where $proj_{RGB}$ is a linear projection followed by a _tanh_ activation, and
$concat$ is channel-wise concatenation. As such, $G_{BA}$ has an additional
channel of input. Since the attention map is only available when computing the
cycle-consistency loss, at all other times we substitute a dummy channel of
Gaussian noise. This model is referred to as $\mathcal{M}_{attn}$.
To enhance $\mathcal{M}_{attn}$ we employ a similar trick as in [11], where
the attentions are combined with the image inputs by element-wise product, and
then fed into an auxiliary convolutional network $C$, which classifies the
artifact under a cross-entropy loss, $\mathcal{L}_{cls}$. The mask is
regularised with smoothness $\mathcal{L}_{smooth}$ and sparsity
$\mathcal{L}_{sparse}$ losses. This compels the attention model to produce
masks that highlight the artifact regions only. Our hypothesis is that this
will allow $G_{AB}$ to decouple the tissue from the artifact, and to offload
all extraneous information onto the attention map. This model is referred to
as $\mathcal{M}_{ws}$, and is depicted in Figure 1. With $C$ guiding the
attention map, the model becomes weakly-supervised, and the objective function
becomes,
$\mathcal{L}_{ws}=\mathcal{L}_{base}+\lambda_{cls}\cdot\mathcal{L}_{cls}+\lambda_{smooth}\cdot\mathcal{L}_{smooth}+\lambda_{sparse}\cdot\mathcal{L}_{sparse}$
(3)
### 2.2 Dataset
Our dataset consists of $28$ WSIs randomly selected from the TCGA database
111https://www.cancer.gov/tcga, and thus spans a wide range of tissue types.
$6556$ non-overlapping tiles of size $300\times 300$px were manually extracted
capturing typical artifact classes at both $40$x and $10$x magnification. For
each artifact tile, a clean equivalent was extracted in close proximity, as in
Supp. Figure 1, to control for sampling bias. Each artifact tile was labelled
into one of seven classes: pen marker, ink, blur, air bubble, tissue fold,
dust and filament, as shown in Supp. Figure 2.
### 2.3 Model training and evaluation
We train $\mathcal{L}_{base}$ with UNet and attention UNet [9] generators,
which we denote $\mathcal{M}_{base}$ and $\mathcal{M}_{dpa}$ respectively.
Apart from $\mathcal{M}_{\lambda_{ABA}=0}$, the hyperparameters
$\lambda_{ABA}=5$, $\lambda_{BAB}=5$, $\lambda_{A}=5$, and $\lambda_{B}=5$ are
fixed for all models. For model $\mathcal{M}_{ws}$ we set $\lambda_{cls}=1$,
$\lambda_{smooth}=1$, and $\lambda_{sparse}=0.1$.
We train all networks with the Adam optimiser [6], with $lr=0.001$,
$\beta_{1}=0.5$, $\beta_{2}=0.999$, and an exponential learning rate decay of
$\gamma=0.9975$. All models were trained for 30 epochs with batch size 16. To
regularise learning, we augment the data with random horizontal and vertical
flips, followed by taking a random $256\times 256$px crop and resizing to
$128\times 128$px. At test time the random crop is replaced by a center crop.
We apply label smoothing to the discriminators and a weight decay of
$1\text{\times}{10}^{-5}$ for all networks.
There is an inherent difficulty in evaluating unpaired datasets, as the
targets are unknown. We therefore compare the distributional output of each
model with the negative samples using Fréchet Inception Distance (FID) on both
train (2620 samples) and test (655 samples) sets to evaluate the results (i.e.
an $80$-$20$ split). Note that FID is conventionally evaluated over $50,000$
samples [5], but this is not possible in our dataset.
## 3 Results
As shown in Table 1, the modified CycleGANs generally perform better, in
particular $\mathcal{M}_{ws}$. The train FID scores are systematically lower
than the test scores, but this does not seem to be a sign of overfitting,
rather the effect of sample size. Controlling for sample size, our best model
achieves a FID of $60.24$ on $655$ training samples and $59.36$ on the test
data. We also note the baseline is slightly improved upon with dot product
attention.
Figure 2 shows examples of artifact tiles from the test set, cleaned tile
model outputs and corresponding attention maps. We note the localisation of
artifact regions in the attention maps, which has been achieved from weak
labels only. We include additional examples in Supp. Figure 3, as well as
failure cases in Supp. Figure 4. Currently, the model struggles with opaque
pen marker and underrepresented classes such as filament artifacts.
| $\mathcal{M}_{base}$ | $\mathcal{M}_{dpa}$ | $\mathcal{M}_{cond}$ | $\mathcal{M}_{\lambda_{ABA}=0}$ | $\mathcal{M}_{attn}$ | $\mathcal{M}_{ws}$
---|---|---|---|---|---|---
Tr. FID | $45.09$ | 44.95 | 42.62 | 42.38 | 45.39 | $\mathbf{34.43}$
Te. FID | $70.71$ | 70.29 | 67.93 | 67.72 | 75.33 | $\mathbf{59.36}$
Table 1: FID scores on train and test data. Best results in bold. Figure 2:
Model inputs, outputs, and attention (top to bottom) for unseen test images.
## 4 Conclusions
In this paper we have identified a limitation of CycleGANs for the surjective
task of artifact removal in histopathology images. We have presented a
mechanism for incorporating weakly-supervised data into a CycleGAN, allowing
it to decouple tissue from artifact, and improving over baselines in its
ability to remove artifacts. An approximate artifact segmentation is a
byproduct of the removal process. Future work could aim to expand the dataset,
as currently some artifact classes are underrepresented.
## References
* [1] Ali, S., Alham, N.K., Verrill, C., Rittscher, J.: Ink removal from histopathology whole slide images by combining classification, detection and image generation models. In: 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019). pp. 928–932. IEEE (2019)
* [2] de Bel, T., Hermsen, M., Kers, J., van der Laak, J., Litjens, G.: Stain-transforming cycle-consistent generative adversarial networks for improved segmentation of renal histopathology. In: International Conference on Medical Imaging with Deep Learning–Full Paper Track (2018)
* [3] BenTaieb, A., Hamarneh, G.: Adversarial stain transfer for histopathology image analysis. IEEE transactions on medical imaging 37(3), 792–802 (2017)
* [4] Boyd, J., Villa, I., Mathieu, M.C., Deutsch, E., Paragios, N., Vakalopoulou, M., Christodoulidis, S.: Region-guided cyclegans for stain transfer in whole slide images. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 356–365. Springer (2022)
* [5] Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., Hochreiter, S.: Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems 30 (2017)
* [6] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)
* [7] Mahmood, F., Borders, D., Chen, R.J., McKay, G.N., Salimian, K.J., Baras, A., Durr, N.J.: Deep adversarial training for multi-organ nuclei segmentation in histopathology images. IEEE transactions on medical imaging 39(11), 3257–3267 (2019)
* [8] Mao, X., Li, Q., Xie, H., Lau, R.Y., Wang, Z., Paul Smolley, S.: Least squares generative adversarial networks. In: Proceedings of the IEEE international conference on computer vision. pp. 2794–2802 (2017)
* [9] Oktay, O., Schlemper, J., Folgoc, L.L., Lee, M., Heinrich, M., Misawa, K., Mori, K., McDonagh, S., Hammerla, N.Y., Kainz, B., et al.: Attention u-net: Learning where to look for the pancreas. arXiv preprint arXiv:1804.03999 (2018)
* [10] Rana, A., Yauney, G., Lowe, A., Shah, P.: Computational histological staining and destaining of prostate core biopsy rgb images with generative adversarial neural networks. In: 2018 17th IEEE International Conference on Machine Learning and Applications (ICMLA). pp. 828–834. IEEE (2018)
* [11] Sahasrabudhe, M., Christodoulidis, S., Salgado, R., Michiels, S., Loi, S., André, F., Paragios, N., Vakalopoulou, M.: Self-supervised nuclei segmentation in histopathological images using attention. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 393–402. Springer (2020)
* [12] Smit, G., Ciompi, F., Cigéhn, M., Bodén, A., van der Laak, J., Mercan, C.: Quality control of whole-slide images through multi-class semantic segmentation of artifacts (2021)
* [13] Zhu, J.Y., Park, T., Isola, P., Efros, A.A.: Unpaired image-to-image translation using cycle-consistent adversarial networks. In: Proceedings of the IEEE international conference on computer vision. pp. 2223–2232 (2017)
## Appendix 0.A Supplementary Figures
(a) Air Bubble
(b) Dust
(c) Filament
(d) Out of focus
(e) Tissue fold
(f) Ink
(g) Pen marker
Figure 3: The seven categories of image artifact studied. Figure 4: Annotated
tissue region comprising positive and negative tiles at two scales. Figure 5:
Artifact removal samples. Figure 6: Artifact removal failure cases.
|
# Few-shot Query-Focused Summarization with Prefix-Merging
Ruifeng Yuan
The Hong Kong Polytechnic University
<EMAIL_ADDRESS>Zili Wang
Xidian University
<EMAIL_ADDRESS>Ziqiang Cao 111Corresponding author
Institute of Artificial Intelligence,
Soochow University, China
<EMAIL_ADDRESS>Wenjie Li
The Hong Kong Polytechnic University
<EMAIL_ADDRESS>
###### Abstract
Query-focused summarization has been considered as an important extension for
text summarization. It aims to generate a concise highlight for a given query.
Different from text summarization, query-focused summarization has long been
plagued by the problem of lacking high-quality large-scale datasets. In this
paper, we investigate the idea that whether we can integrate and transfer the
knowledge of text summarization and question answering to assist the few-shot
learning in query-focused summarization. Here, we propose prefix-merging, a
prefix-based pretraining strategy for few-shot learning in query-focused
summarization. Drawn inspiration from prefix-tuning, we are allowed to
integrate the task knowledge from text summarization and question answering
into a properly designed prefix and apply the merged prefix to query-focused
summarization. With only a small amount of trainable parameters, prefix-
merging outperforms fine-tuning on query-focused summarization. We further
discuss the influence of different prefix designs and propose a visualized
explanation for how prefix-merging works.
## 1 Introduction
Text summarization aims to compress the source document(s) into a shorter
version that contains its important information. As a classic sub-topic for
text summarization, query-focused summarization meets that situation that only
a specific aspect of information is needed to be summarized. In other words,
it aims to generate a summary based on the source content related to a given
query. Hence, this task requires not only to locate relevant content in a
passage as question answering (QA) but also to summarize and generate a
highlight as text summarization. Although text summarization has been widely
studied in recent years, there are fewer attempts on exploring query-focused
summarization Deng et al. (2020); Su et al. (2020); Xu and Lapata (2020a); Su
et al. (2021) after the age of neural model. One main reason is the lack of
generalized large-scale datasets. Compared with the easily accessible nature
reference summaries such as titles or headlines in text summarization, it is
hard to collect large-scale data for query-focused summarization. Meanwhile,
human-written reference summaries have always been costly.
The rapidly developed few-shot learning techniques provides potential cues to
alleviate the problem of lacking large-scale dataset for query-focused
summarization, and knowledge transferring is one of them. In fact, when facing
unseen tasks, it is natural for human beings to integrate and transfer the
knowledge of known tasks to relevant new tasks. Inspired by this, we
innovatively propose to decouple the query-focused summarization to two basic
tasks, i.e. text summarization and question answering, and transfer the
knowledge from these two tasks to query-focused summarization. However, in
parameter-based knowledge learning, previous work are usually one-to-one (pre-
train then fine-tune Yosinski et al. (2014)) or one-to-many (domain/task
adaption Houlsby et al. (2019); Lin et al. (2020)), and seldom of them focus
on many-to-one (integrate basic tasks to a complex one). In this case, the
previous methods may not work well in this task.
In this paper, we propose a pre-trained strategy, prefix-merging, for few-shot
learning in query-focused summarization. In recent prompt-based language
models, the prompt/prefix is considered as containing the knowledge of the
given task, which provides us an explicit way to control the task-specific
knowledge previously dispersed in the language model (LM). For example,
prefix-tuning Li and Liang (2021) achieved a similar result with fine-tuning
by training only the task-specific prefix, a sequence of continuous vectors
that prepend to the input. Following the framework proposed by prefix-tuning,
prefix-merging aims to integrate the task knowledge from text summarization
and question answering into a properly designed prefix and apply the merged
prefix to the more complex task, query-focused summarization.
Generally, there are two straightforward ideas for merging knowledge from
multiple tasks into a prefix: concatenate the separated prefix for different
tasks as a whole or adopt a shared prefix for all the tasks. Considering there
exist both similarities and differences across the tasks, a more flexible
prefix design composed of both task-specific part and shared part is used in
further investigation. Moreover, we propose a self-adaptive prefix merging
that allows the basic tasks themselves to decide the prefix design. Drawn the
inspiration from Xu et al. (2021), we adopt Fisher Information to calculate
the importance scores of the prefix embeddings (basic units for the prefix)
for each basic task. For one task, only the prefix embeddings with top scores
are activated in the following training. Hence, different tasks can adapt to
different parts of prefix automatically. After finishing training the merged
prefix, it is transferred to a downstream task for few-shot learning. In the
experiment, we explore prefix merging in the context of query-focused
summarization, taking PubMedQA Jin et al. (2019) and DUC Dang (2006) as the
evaluation dataset.
Prefix-merging provides a potential solution for the few-shot learning in
complex tasks that can be integrated by the basic tasks. Benefited by the
universality of the prompt-based approach, prefix-merging is not limited by
the model architecture and can be used in both autoregressive LM and encoder-
decoder based LM. We believe this shows a possible direction to the
application of prompt-based approaches. Our contribution can be summarized as
follow:
* •
We provide a new solution for few-shot query-focused summarization by
decoupling it to two basic tasks with large-scale training data, text
summarization and question answering.
* •
We propose prefix-merging that integrates the task-specific knowledge from
basic tasks to assist the learning of a more complex task, which provides a
new solution to many-to-one parameter-transfer learning.
* •
We further expand the application of prompt-based approaches by applying the
prefix to multi-task situation, exploring the interaction between different
task knowledge through prefix.
## 2 Related Work
### 2.1 Query-focused Summarization
Query-focused summarization aims to generate a concise highlight from the
source document(s) according to a specific topic or query, which is considered
as a more complex extension of text summarization. Early works Lin et al.
(2010); Shen and Li (2011) focus on extracting query-related sentences as
summaries, while further works Wang et al. (2016); Li and Li (2014) improve it
by rewriting the extracted sentences with sentence compression. Nema et al.
(2017); Hasselqvist et al. (2017) propose neural-abstractive models with an
additional query attention mechanism to generate the summaries with respect to
the given query. Deng et al. (2020) consider the relation among the query and
source sentences as a multi-hop inference process and generate the summaries
by integrating information from different inference steps. Meanwhile,
researchers also utilized QA models to find the possible query-related
evidence in query-focused summarization. Xu and Lapata (2020a, b) adopts QA
models for sentence-level or paragraph-level answer evidence ranking. Su et
al. (2021) incorporate answer relevance scores generated by QA model as
explicit fine-grained query relevance to a transformer-based abstractive
summarization model. Therefore, we believe the text summarization and QA are
the foundation for query-focused summarization and choose them as the
auxiliary tasks in this work.
### 2.2 Prompt-based Approaches
Prompting originally refers to adding instructions and several examples to a
task input and generating the output from the LM. A fundamental idea for
prompt-based approaches is that let the tasks adapt to the LM. Some
researchers tend to utilize the idea to improve the performance of the model
by making the form of the task closer to the LM. A series of works Petroni et
al. (2019); Jiang et al. (2020); Shin et al. (2020) explore the prompt
engineering and prompt ensemble in natural language understanding tasks. For
instance, instead of manually designing prompt, AutoPrompt Shin et al. (2020)
automatically search for a sequence of discrete words as prompt to extract
knowledge from pre-trained LMs. Other works choose to optimize the prompt in a
continuous space. Qin and Eisner (2021); Liu et al. (2021) adopt hand-designed
prompt as initialization and add learnable perturbation on the prompt. Other
researchers choose to find a parameter-efficient adaption from LM to a
specific task. GPT-3 Brown et al. (2020) adopts manually designed task-
specific prompts to adapt the LM for different generation tasks. Prefix-tuning
proposes “prefix tuning” for language generation task: learning a sequence of
continuous prefixes that are inserted to every transformer layer. Lester et
al. (2021) provides a simplified version of “prefix tuning” with fewer
parameters and more robust prompt initialization on the SuperGLUE tasks. Zhao
et al. (2022) has recently proposed a prefix-based model that utilize domain
words to achieve zero-shot domain adaption on dialogue summarization. In this
work, following the framework of prefix-tuning, we aim to integrate basic
tasks to a more complex one by merging the task knowledge through the prefix.
## 3 Method
Figure 1: Focusing on the encoder layer of BART, the figure shows annotated
examples and comparison between the prefix-merging (top, mid) on the two
auxiliary tasks and applying the merged prefix on the target task with prefix-
tuning (bottom).
### 3.1 Problem Statement
In this work, we aim to transfer the task-specific knowledge from text
summarization and question answering (auxiliary tasks) to query-focused
summarization (target task) to assist its learning. In this case, the query-
focused summarization model can obtain a fair performance even with limited
data. There are mainly two stages to accomplish this. In the first stage, a
model is trained on the large-scale data from two auxiliary tasks to obtain
the potentially useful knowledge for query-focused summarization. Here, we
propose prefix-merging that merges task knowledge from auxiliary tasks into a
particularly designed prefix. In the second stage, we train the model with
data from query-focused summarization but with the assistance of the trained
parameters from the first stage. For prefix-merging, the merged prefix is used
to transfer the knowledge from the first stage to the second stage.
Our prefix-merging is considered as an extension of prefix-tuning, so we have
a brief introduction about it in the section 3.2 as the background of our
method. Then, we introduce our own method from section 3.3 to 3.5, and how we
apply the merged prefix on query-focused summarization in 3.6.
### 3.2 Prefix-tuning
Consider there is a transformer-based encoder-decoder LM $p(y|x)$ such as
BartLewis et al. (2019) and it is parametrized by $\phi$. Taking the encoder
layer in transformer as an example, let $z=[x]$ denote its input sequence. We
use $h_{i}$ to represent the concatenation of all activation from all layers
at the index $i$, and each activation consists of a key-value pair. The
$h_{i}$ for all $i\in x$ in encoder layer is a function of $z_{i}$ and the
other activations in the context based on the LM, as follows:
$h_{i}=LM_{\phi}(z_{i},h_{\neq i})$ (1)
Prefix-tuning prepends a prefix for the encoder layer to obtain
$z=[prefix;x]$, or prepends prefixes for cross-attention layer or self-
attention layer in the decoder to obtain $z=[prefix;x;y]$ or $z=[prefix;y]$.
Here, we use $P_{idx}$ to represent the sequence of prefix embedding indices,
and $|P_{idx}|$ is used to represent the length of the prefix. A trainable
matrix $P_{\theta}\in|P_{idx}|\times dim(h_{i})$ is initialized to store the
prefix parameters. Following the recurrence relation in equation (1), $h_{i}$
is calculated as below in prefix-tuning.
$h_{i}=\left\\{\begin{array}[]{ll}P_{\theta}[i,:],&if\;i\in P_{idx}\\\
LM_{\phi}(z_{i},h_{\neq i}),&otherwise\end{array}\right.$ (2)
Hence, $h_{i}$ becomes a function of the trainable $P_{\theta}$ and it allows
the prefix parameters to control the model by affecting the activations in
every layer of the transformer. During the training in prefix-tuning, the
objective maintains the same as normal task, but only the prefix parameters
$\theta$ are trainable and the parameters of the LM $\phi$ are fixed. In this
case, the prefix parameters contain all the task-specific knowledge learned
from the training.
### 3.3 Intuition for Prefix-merging
Intuitively, to merge the knowledge from different tasks into the prefix, the
simplest way is to concatenate the individual prefix from these tasks. Another
way is to use a shared prefix that is updated by all the tasks. Instead of
using either of the two ways, we choose a more flexible prefix design for
further investigation of the problem. For each task, its prefix consists of a
shared sub-prefix (prefix embeddings shared by all tasks) and a task-specific
sub-prefix (prefix embeddings used for a specific task) whose lengths are
controlled by two hyperparameters. We believe the shared sub-prefix tends to
represent the similarities between all merged tasks, while the task-specific
sub-prefix refers to the uniqueness of each task. Meanwhile, the two mentioned
intuitive methods can also be restored when any of the two hyperparameters is
set to 0.
### 3.4 Prefix-merging
Similar to prefix-tuning, a trainable matrix $P_{\theta}$ is used to store the
prefix parameters. The difference is that there are n different tasks denoted
as $[task_{1},task_{2},..,task_{n}]$ that share or partly share the whole
matrix. For each single task, it corresponds to several prefix embeddings in
the prefix metrix, and we separate them into task-specific unique sub-prefix
with a length of $l_{u}$ and a shared sub-prefix with a length of $l_{s}$.
Figure 1 shows an example of training two auxiliary tasks, text summarization
and question answering, for prefix-merging. Here, both the shared sub-prefix
length and unique sub-prefix length are set to 2. The prefix embedding indices
for text summarization is [1,2,3,4], and it changes to [1,2,5,6] for QA.
In this way, the $P_{\theta}$ has the dimension of $(l_{s}+l_{u}*n)\times
dim(h_{i})$. We use $P_{idx}^{n}$ to represent the the sequence of prefix
embedding indices of $task_{n}$ and its length $|P_{idx}^{n}|$ is equal to
$l_{s}+l_{u}$. As follow, the $h_{i}$ for $task_{n}$ is calculated based on
the following equation:
$h_{i}=\left\\{\begin{array}[]{ll}P_{\theta}[P_{idx}^{n}[i],:],&if\;i\leq|P_{idx}^{n}|\\\
LM_{\phi}(z_{i},h_{\neq i}),&otherwise\end{array}\right.$ (3)
To distinguish the different tasks during the training, we add a task-specific
prompt before the original input tokens following T5 Raffel et al. (2019). As
shown in Figure 1, the prompt is “summarize” for the text summarization and
the prompt is “answer the question” for question answering. During the
training, we adopt a mixed-task training strategy where instances from
different tasks equally exist in the same training batch.
### 3.5 Self-adaptive Prefix-merging
Considering that manual design does not always lead to the best results, we
further propose a self-adaptive prefix-merging. Instead of presetting the
lengths of shared sub-prefix and unique sub-prefix, we aim to let the
auxiliary tasks decide the prefix design. The idea is based on Fisher
Information, a evaluation metric that reflects how much the model output
changes when its parameters change. It can be considered as the importance of
a parameter for the model on a certain set of data Xu et al. (2021). In this
way, we can find the most important sub-prefix for each auxiliary task based
on Fisher Information with the following equation:
$F_{i}=\frac{1}{pq}\sum_{j=1}^{p}\sum_{k=1}^{q}(\frac{\partial
log(p(y_{k}|x_{k};\theta))}{\partial\theta_{j}})^{2}$ (4)
where $F_{(}i)$ refers to the average Fisher information of the $i$-th prefix
embedding, $p$ denotes the number of parameters in the embedding and $q$
represents the number of data. $x$ and $y$ refer to the input and output data
in one auxiliary task.
During the training, we first initialize the prefix as a shared prefix trained
by all auxiliary task for one epoch. Taking $task_{n}$ as an example, we then
conduct a complete forward propagation and back propagation (one epoch) for
all data in $task_{n}$, and calculate the Fisher Information for each prefix
embedding. Only the top-$n$ prefix embeddings will be used in the later
training for $task_{n}$ and others will be masked. In other words, the
$P_{idx}^{n}$ is the indices of the top-$n$ prefix embeddings. After obtaining
the important sub-prefix for each task, naturally, some prefix embeddings are
shared by different tasks while others are task-specific. At last, we continue
the training of the prefix on the auxiliary tasks with the selected sub-
prefix.
### 3.6 Applying the Merged Prefix to the Target Task
After training on the auxiliary tasks, we obtain the prefix parameters that
contain task knowledge from text summarization and question answering. We
apply the knowledge to the target task, query-focused summarization, by using
the merged prefix as initialization and continue prefix-tuning on it, but with
a few differences. As shown in Figure 1, all the prefix parameters are used
for the target task including the shared sub-prefix and all the unique sub-
prefixes. For self-adaptive prefix-merging, only the prefix embedding that is
used for at least one auxiliary task is applied for the target task, otherwise
it will be masked. We also adopt a new prompt that suggests the relation
between the target task and auxiliary tasks. More specifically, we concatenate
the prompt of text summarization and question answering as “summarize and
answer the question” for query-focused summarization.
Data Size | | 50 | | | 150 | | | 300 |
---|---|---|---|---|---|---|---|---|---
Model | R-1 | R-2 | R-L | R-1 | R-2 | R-L | R-1 | R-2 | R-L
Random | 30.33 | 9.96 | 28.00 | 32.08 | 11.67 | 28.97 | 32.79 | 11.92 | 29.51
Unq(30) | 30.81 | 10.97 | 26.52 | 32.13 | 11.73 | 28.23 | 32.37 | 11.86 | 27.81
Unq(20)+Sha(10) | 32.36 | 11.40 | 28.30 | 33.14 | 12.12 | 29.10 | 33.68 | 12.39 | 29.81
Unq(10)+Sha(20) | 32.64 | 11.84 | 28.60 | 33.46 | 12.34 | 29.46 | 33.90 | 12.59 | 30.12
Sha(30) | 32.44 | 11.48 | 28.17 | 33.28 | 12.04 | 29.11 | 33.87 | 12.41 | 29.83
Self-adaptive | 33.18 | 12.01 | 28.45 | 33.66 | 12.40 | 28.98 | 34.19 | 12.65 | 29.53
BART(tar) | 30.95 | 10.54 | 26.87 | 32.28 | 11.46 | 28.23 | 32.52 | 11.63 | 28.33
BART(aux+tar) | 31.65 | 10.75 | 28.18 | 32.23 | 11.27 | 28.57 | 32.66 | 11.62 | 29.16
BART_base(full) | 37.49 | 14.11 | 34.45 | 37.49 | 14.11 | 34.45 | 37.49 | 14.11 | 34.45
Table 1: Evaluation result for query-focused summarization on PubMedQA. We compare the result on three different training data size: 50, 150, 300. Here, we also provide result of BART-base on the full-size training for better comparison. Model | R-1 | R-2 | R-L
---|---|---|---
Random | 33.96 | 6.37 | 23.46
Unq(30) | 34.56 | 6.80 | 24.40
Unq(20)+Sha(10) | 34.54 | 6.64 | 23.53
Unq(10)+Sha(20) | 34.87 | 7.23 | 24.70
Sha(30) | 34.53 | 6.89 | 23.98
Self-adaptive | 34.99 | 7.47 | 24.74
BART(tar) | 33.83 | 6.52 | 22.71
BART(aux+tar) | 12.85 | 4.42 | 15.17
Ext_Oracle | 35.62 | 9.20 | 24.00
Table 2: Evaluation result for query-focused summarization on DUC. Ext_Oracle
refers to oracle extractive result taking the query-related sentences as
input, which can be seen as the upper bound of this experiment.
## 4 Experiment
### 4.1 Datasets
To evaluate the idea of prefix-merging, we take query-focused summary as the
target task, text summarization and question answering as two auxiliary tasks.
We focus on commonly used datasets for query-focused summarization: PubMedQA
and DUC. We also test our model on Debatepedia Nema et al. (2017) and have a
discussion about it in the appendix. In terms of the PubMedQA, it requires the
model to generate a summary containing 1-3 sentences as an answer to a
question based on a medical related document. Since we train the target task
under a few-shot situation, only part of the training set is used in the
experiment and we test the model on the full testing set containing more than
20000 data samples. In terms of the DUC, it is a multi-document query-focused
summarization dataset with hundreds of data samples. Hence, we adopt a
extract-generate framework to conduct the experiment. We first adopt BM25 to
extract a set of query-related sentences from the source documents and use the
concatenation of the query and extracted sentences as the input of our model.
The DUC 2006 is used for training and DUC 2007 is used for testing. In terms
of the two auxiliary tasks, we adopt the XSum dataset Narayan et al. (2018), a
highly abstractive single-document summarization dataset, for the text
summarization, and we use the classic machine reading comprehension dataset
SQUAD 1.1 Rajpurkar et al. (2016) for question answering.
### 4.2 Experiment Setting
Our implementation is based on the BART-large model from HuggingFace and all
the input is truncated to 800 tokens. For the prefix-tuning based method, a
default setting is a learning rate of $5\times 10^{-5}$ and a prefix length of
30. The batch size is set to 48 when conducting prefix-merging, and for few-
shot prefix-tuning, it changes with the size of the training data. In the
experiment, we also use fine-tune based method as a comparison, and the
default setting for it is a learning rate of $2\times 10^{-5}$ and a batch
size of 48. At training time, we adopt the AdamW optimizer with default
hyperparameters. At inference time, we use beam search with a beam size of 2.
The output length limitation is set from 30 to 75 tokens for PubMedQA and 250
to 300 for DUC. Since few-shot learning is sensitive to the training data, we
train the models with three sets of training data and report the average
result on PubMedQA.
As for evaluation metric, following previous works, we apply ROUGE Lin (2004)
including Rouge-1 (R-1), Rouge-2 (R-2) and Rouge-L (R-L) for the query-focused
summarization. We adopt a full Python implementation of the ROUGE-1.5.5, to
conduct the experiment.
### 4.3 Result
We first evaluate the different prefix designs within three different few-shot
learning data sizes (50, 150, 300) for the target task in Table 1. "Unq(n)"
stands for the total number of the prefix embeddings in all unique sub-prefix,
while "Sha(n)" refer to the shared sub-prefix. For example, "Unq(10)+Sha(20)"
represent the merged prefix consists of unique sub-prefix with length 10 (5
for each task) and the shared sub-prefix with length 20. In terms of the self-
adaptive prefix-merging, we initialize the prefix length as 40 and select the
top-25 prefix embeddings for each tasks. In this case, self-adaptive prefix-
merging is more likely to have a comparable parameter numbers with the other
prefix designs, which makes a fair comparison. We also add a baseline
“random”: randomly initialize the prefix and conduct few-shot prefix-tuning on
the query-focused summarization dataset. We further compare our model with
BART in three training settings:(1) BART(tar) refers to fine-tuning the BART
only use the limited data from query-focused summarization; (2) BART(aux+tar)
refers to first fine-tuning on the auxiliary tasks then fine-tuning on query-
focused summarization, which is similar to some previous approaches Yu et al.
(2021) and Fabbri et al. (2020); (3) BART(full) refers to fine-tuning on the
large-scale data from query-focused summarization.
Model | R-1 | R-2 | R-L
---|---|---|---
Fine+Fine | 31.65 | 10.75 | 28.18
Fine+Prefix | 31.64 | 10.79 | 27.57
Prefix+Fine | 32.03 | 11.30 | 28.12
Prefix+Prefix | 33.18 | 12.01 | 28.45
Table 3: The comparison between prefix-merging and fine-tuning with a training
data size of 50.
In Table 3, we compare the prefix-merging with fine-tuning. Since it is a two-
stage training process (training on auxiliary tasks then applying on the
target task), each stage can adopt prefix-based training (only the prefix
parameters are trained and the LM parameters are frozen) or fine-tuning (all
parameters are trained). Therefore, we report four variants in total: (1)
fine-tuning + fine-tuning (Fine+Fine), which is the same as BART(aux+tar); (2)
fine-tuning + prefix-tuning (Fine+Prefix); (3) prefix-merging + fine-tuning
(Prefix+Fine); (4) prefix-merging + prefix-tuning (Prefix+Prefix), which is
our proposed approach in Section 3. Despite the variant (1), we add a prefix
of length 30 to the model. Taking variant (2) as an example, firstly, both the
prefix and the LM are updated by the training data from auxiliary tasks and
then only the prefix parameter is trained on the target task.
Table 4 displays the result of using different auxiliary tasks for query-
focused summarization. “Sum+QA” refers to the best result when using both text
summarization and QA; “Only Sum” and “only QA” are designed for ablation study
where only one of the two tasks is used in stage one. Moreover, we also import
a baseline “Unrelated Task” that takes sentence copying as the auxiliary task,
which contains no useful task knowledge for query-focused summarization. We
use prefix-tuning to train the model when there is only one auxiliary task.
We summarize the experiment result with the following conclusions.
Model | R-1 | R-2 | R-L
---|---|---|---
Unrelated Task | 31.34 | 10.77 | 27.08
Only Sum | 32.38 | 11.56 | 27.75
Only QA | 31.78 | 11.39 | 28.43
Sum and QA | 33.18 | 12.01 | 28.45
Table 4: The comparison between using different auxiliary tasks with a
training data size of 50. Figure 2: The attention score for query-focused
summarization in both encoder and decoder of model “Unq(20)+Sha(10)”.
The self-adaptive prefix-merging achieves a comparable result with the best
manually prefix design. It is not a surprise that self-adaptive prefix-merging
outperforms most of the prefix designs and achieves the best result in both
datasets. One thing that is worth noticing is that the effective length for
self-adaptive prefix-merging is also around 30 (initialized as 40 and 10 are
masked by all tasks), which means the number of parameter maintains equal with
other prefix design. Meanwhile, its proportion of shared sub-prefix and unique
sub-prefix is similar to the best manual design Unq(10)+Sha(20). This suggests
that self-adaptive prefix-merging has the ability to find the best prefix
design automatically. Compared with BART, self-adaptive prefix-merging
outperforms both BART(tar) and BART(aux+tar), which indicates the
effectiveness of prefix-merging. In the experiment on DUC, we notice that
BART(aux+tar) drops a lot compared with other results. We believe this is
because the difference between DUC and datasets used in auxiliary tasks is
relatively huge and the generalization ability of BART is lost after training
on the auxiliary tasks.
Prefix-merging is better than fine-tuning for integrating and transferring
task knowledge to the downstream task. In Table 2, prefix-merging outperforms
fine-tuning with both downstream training approaches. On the one hand, this is
because the generalization ability of the LM is preserved when its parameters
are frozen. On the other hand, we believe using prefix as the container of new
task knowledge is more similar to the natural form of LM. We believe this
shows the potential of prefix-merging in many-to-one knowledge transferring.
The merged prefix contains effective task knowledge from both auxiliary tasks.
The initialization of prefix is believed to have a huge effect on the prefix-
tuning based approaches. Here, “unrelated task” stands for the performance
when the prefix is well-initialized while containing no knowledge for the
target task. Compared to it, using one auxiliary task, either text
summarization or QA, achieve a better result. This suggests that the two tasks
contribute useful knowledge to query-focused summarization. More importantly,
prefix-merging gets the best performance. And this can be achieved only when
the prefix-merging allows the prefix to integrate effective task knowledge
from both tasks.
Model | R-1 | R-2 | R-L
---|---|---|---
–Prefix | 26.56 | 8.19 | 22.16
–Prompt | 32.48 | 11.63 | 28.57
Unq(10)+Sha(20) | 32.64 | 11.84 | 28.60
Sha(40) | 32.60 | 11.74 | 28.54
Self-adaptive | 33.18 | 12.01 | 28.45
Table 5: The experiment result for ablation study with a training data size of
50.
### 4.4 Ablation Study
For more detailed analysis, we design an experiment to explore how different
components contribute to our approach. We remove the prefix (-prefix) and the
prompt (-prompt) from during the training of the query-focused summarization.
The prefix design used here is Unq(10)+Sha(20). We can observe that removing
the prompt has a small negative influence on the result. We believe this is
because the input form of text summarization and QA is different and the model
can distinguish the two tasks even without the given prompt. We also find that
the performance drops a lot once the prefix is removed. This indicates that
the prompt only plays as guidance, while the prefix is the one containing the
task-specific knowledge. For self-adaptive prefix-merging, we compare it with
its base prefix design without self-adaption, Sha(40). Even with more
trainable parameters, self-adaptive prefix-merging still outperforms it. The
result shows that prefix embeddings selected by Fisher Information are crucial
for the tasks.
### 4.5 Prefix Visualization
To have a more direct observation, we visualize the attention on the prefix
during the inference for query-focused summarization in Figure 2. We adopt the
attention weights passing through the Softmax layer and further normalize the
attention weights only on the prefix embeddings. The final attention score is
obtained by averaging attentions from all heads in all layers from 100 random
samples. In Figure 2, the x axis refers to the indices of the prefix embedding
and y axis is the normalized attention score. The straight lines with colors
stand for the position of the three types of sub-prefix, shared sub-prefix
(0-9), unique sub-prefix originated from QA (10-19) and unique sub-prefix
originated from Summarization (20-29), and their heights refer to the average
attention score, which can be considered as the prefix’s contribution to the
query-focused summarization. In this case, it explains how the merged prefix
works for query-focused summarization.
For the decoder, we display the attention in the cross-attention layer. In
terms of the encoder, since the model needs to understand the query, we
believe it is reasonable that the sub-prefix originated from QA plays the most
important role. In terms of the decoder, the sub-prefix originated from QA has
little effect on the model, while the shared sub-prefix and sub-prefix
originated from summarization dominate. This is because generating the query-
focused summaries relies more on generation ability and summarization ability.
These findings suggest that the knowledge from QA and summarization is
properly used for query-focused summarization through the merged prefix.
## 5 Conclusion
In this paper, we show that prefix-merging is an effective approach for
transferring and integrating task knowledge from multiple auxiliary tasks to a
target task with limited data. In the context of query-focused summarization,
integrating text summarization and QA, our approach outperforms the
traditional approach fine-tuning. We further discuss the influence of
different prefix designs and propose a self-adaptive prefix-merging. We also
provide a visualize explanation for how the merged prefix works. Although this
paper focuses on query-focused summarization, we believe these findings
suggest a new application for prompt-based approaches in multi-task situation,
providing guidance for future progress in this field.
## 6 Limitations
Prefix-merging is based on a seq2seq pretrained model, Bart, so it is hard for
our model to deal with long input that exceed the input limitation of the
pretrained model. Hence, we mainly focus on single-document query-focused
summarization. In terms of the experiment, unfortunately, there is seldom few-
shot single-document query-focused summarization model. Although there exist
some multi-document query-focused summarization models with weak supervisionXu
and Lapata (2020b); Laskar et al. (2020); Xu and Lapata (2020a), these models
all follow a coarse-to-fine framework, which make them hard to directly
compared with our model. Hence, we mainly use BART with different training
settings as comparison and focus more on the longitudinal comparison.
Moreover, we believe that prefix-merging has the potential to be used for
other complex tasks that can be integrated from basic tasks. However, we only
finish the research in the context of query-focused summarization, which
leaves future direction for our work.
## Acknowledgements
The work described in this paper was supported by Research Grants Council of
Hong Kong (PolyU/15203617 and PolyU/5210919), National Natural Science
Foundation of China (61672445, 62106165).
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## Appendix A Appendix
### A.1 Experiment on Debatepedia
Debatepedia Nema et al. (2017) is one of the commonly used query-focused
summarization dataset. However, during our experiment, we find a serious but
unintentional data leakage problem between the training set and the testing
set in its standard division. Around 64% of summaries in the testing set
appear or have similar ones in the training set (difference is lower than 2
words). In this case, the model tends to remember the data samples rather
learning to do query-focused summarization.
Data Size | | 50 | | | 150 | | | 300 |
---|---|---|---|---|---|---|---|---|---
Model | R-1 | R-2 | R-L | R-1 | R-2 | R-L | R-1 | R-2 | R-L
Random | 0.20 | 0.30 | 0.45 | 0.30 | 0.22 | 0.36 | 0.08 | 0.11 | 0.45
Unq(30) | 0.45 | 0.23 | 0.40 | 0.41 | 0.36 | 1.02 | 0.08 | 0.08 | 0.11
Unq(20)+Sha(10) | 0.50 | 0.22 | 0.25 | 0.04 | 0.22 | 0.59 | 0.17 | 0.09 | 0.23
Unq(10)+Sha(20) | 0.14 | 0.01 | 0.13 | 0.23 | 0.15 | 0.16 | 0.15 | 0.02 | 0.32
Sha(30) | 0.19 | 0.08 | 0.18 | 0.25 | 0.27 | 0.64 | 0.23 | 0.10 | 0.24
Self-adaptive | 0.38 | 0.16 | 0.48 | 0.10 | 0.20 | 0.71 | 0.12 | 0.08 | 0.35
BART(tar) | 0.53 | 0.52 | 0.51 | 0.21 | 0.15 | 0.25 | 0.32 | 0.22 | 0.14
BART(aux+tar) | 0.47 | 0.32 | 0.43 | 0.24 | 0.17 | 0.21 | 0.18 | 0.14 | 0.16
Table 6: .Standard Deviation of the Results on PubMedQA Model | R-1 | R-2 | R-L
---|---|---|---
Random | 18.57 | 5.50 | 17.50
Unq(30) | 21.53 | 6.77 | 20.07
Unq(20)+Sha(10) | 22.20 | 7.20 | 20.59
Unq(10)+Sha(20) | 22.03 | 7.13 | 20.29
Sha(30) | 21.85 | 6.92 | 20.27
Self-adaptive | 22.29 | 7.39 | 20.53
BART(tar) | 21.60 | 6.81 | 19.89
BART(aux+tar) | 21.36 | 6.24 | 19.00
BART(full) | 57.74 | 43.42 | 57.03
BART(full)_redivided | 24.80 | 8.06 | 22.95
Table 7: Evaluation result on Debatepedia.
This observation is also supported by the experiment. In the upper and middle
part of Table 7, we display the result of few-shot learning with 50 data
samples on standard division of Debatepedia. In the lower part of Table 7, we
show the result of BART training with full-size data but with different data
division. BART(full) represents the standard division and BART(full)_redivided
refers to a new division that do not have the data leakage problem (we achieve
this by redivide all data samples by an alphabetical sort, where similar data
samples tend to gather together rather than scatter in both training and
testing set). We observe a huge gap between the BART(full) and
BART(full)_redivided and it can not be explained by the difference of the
division. Meanwhile, the result of few-shot learning is much lower than the
result of full-size training. Both phenomenon suggest there exist a data
leakage problem. The poor performance of BART on the redivided Debatepedia
also make us question whether Debatepedia is qualified for query-focused
summarization. Hence, we discuss this problem in the appendix and hope more
researchers can notice this.
### A.2 Standard Deviation of the Results on PubMedQA
To have a better understanding of the experiment results on PubMedQA, we
report the standard deviation (std) across multiple runs in the experiment on
PubMedQA in Table 6.
|
# Optical properties of dispersive time-dependent materials
Jamison Sloan1,2<EMAIL_ADDRESS>Nicholas Rivera3,4 John D. Joannopoulos3
Marin Soljačić3 1 Department of Electrical Engineering and Computer Science,
MIT, Cambridge, MA 02139, USA 2 Research Laboratory of Electronics, MIT,
Cambridge, MA 02139, USA 3 Department of Physics, MIT, Cambridge, MA 02139,
USA 4 Department of Physics, Harvard University, Cambridge, MA 02138, USA
###### Abstract
Time-varying optical materials have attracted recent interest for their
potential to enable frequency conversion, nonreciprocal physics, photonic
time-crystals, and more. However, the description of time-varying materials
has been primarily limited to regimes where material resonances (i.e.,
dispersion) can be neglected. In this work, we describe how the optics of
these dispersive time-varying materials emerges from microscopic quantum
mechanical models of time-driven systems. Our results are based on a framework
for describing the optics of dispersive time-varying materials through quantum
mechanical linear response theory. Importantly, we clarify how response
functions for time-varying materials are connected to energy transfer. We
provide three examples of our framework applied to systems which can be used
to model a wide variety of experiments: few level models that can describe
atoms, spins, or superconducting qubits, oscillator models which can describe
the strong response of polar insulators, and strongly driven atom models which
can describe the highly nonperturbative optical response of materials
undergoing high harmonic generation. We anticipate that our results will be
broadly applicable to electromagnetic phenomena in strongly time-varying
systems.
## I Introduction
The propagation of electromagnetic waves through materials represents an
essential component of light-matter interactions, and lies at the heart of
countless physical phenomena and technological applications. In many bulk
materials, the dominant features of electromagnetic wave propagation can be
described by a simple complex refractive index which encodes the speed of wave
propagation, as well as the rate of dissipation. In fact, the ability to
describe a complex many-body system such as a solid with a frequency dependent
refractive index is critical for a practical description of many systems. Over
the last century, a great deal of effort has gone into understanding the
origins and fundamental properties of optical response, leading to important
devices such as detectors, LEDs, solar cells, and lasers. Nowadays, artificial
structures such as photonic crystals, layered 2D materials, and metamaterials
are routinely created to provide further control over optical response,
leading to increased command over the interactions between light and matter.
Many of the basic assumptions about the nature of optical response and wave
propagation rely on considering optical materials as time translation-
invariant — the same at all times. However, a recent surge of interest has
developed in the possibilities that may be enabled by materials which break
this assumption — in other words, materials which vary in time
galiffi2021photonics . In practice, time-varying materials are typically
created by applying strong temporal modulations to stationary materials in the
form of external fields. These time-varying materials may exhibit rich physics
such as frequency conversion zhou2020broadband , scattering from temporal
interfaces pacheco2020temporal ; plansinis2015temporal , nonreciprocity
shaltout2015time ; li2022nonreciprocity , and amplification pendry2020new .
Moreover, a recent interest has sparked in the study of so-called “photonic
time crystals” which have a strong temporally periodic index variation,
enabling new directions in topological physics lustig2018topological and
light-matter interactions dikopoltsev2022light ; lyubarov2022amplified . In
the context of metamaterials, time has recently been identified as additional
degree of freedom which can be added to create “spatiotemporal metamaterials”
caloz2019spacetime ; engheta2021metamaterials . Additionally, the possibility
of strongly time-modulated materials introduces new fundamental questions
about the nature of quantum light-matter interactions in time-modulated
systems, including control over the generation of entangled photon pairs
yablonovitch1989accelerating ; sloan2020casimir ; kort2021space .
To achieve these goals, it will become increasingly important to accurately
describe the optical response of time-varying materials in the most general
setting. Past work on time-varying media has typically assumed that the time
modulations to a material occur at frequencies away from intrinsic resonances
in the material law1994effective ; lustig2018topological ;
zurita2009reflection ; chu1972wave ; harfoush1991scattering ;
fante1971transmission ; holberg1966parametric . In these cases, it is
sufficient to consider a permittivity $\varepsilon(t)$ which is nondispersive,
associated with an instantaneous polarization response. However, there are
many systems, especially those which are varied quickly in time, which are not
adequately accounted for by this framework. For example, the description of
wave propagation on a strongly driven conductor requires the simultaneous
description of driving and plasmonic dispersion. In fact, understanding the
influence of dispersion has recently been identified as a key challenge in the
field of time-varying materials solis2021functional . Some semiclassical
models have been proposed for particular systems ptitcyn2021scattering ;
solis2021functional . Yet, there is still no comprehensive framework for
describing the optical physics of dispersive time dependent materials from
first principles.
It is tempting to take a theoretical model (or data) for the optical response
of an undriven system, and then introduce time dependence. While this is a
valid approach for slow variations, it is not reliable in general. The key
issue is that in a strongly time-modulated system, the optical response
depends on the new microscopic dynamics of the driven system, which will in
general not be adequately captured by this approach. Therefore, the optical
response of time-driven materials should ideally be considered on the basis of
first principles, starting from a microscopic description of the driven
system.
In this work, we present a general framework which describes the optics of
dispersive time-dependent materials based on microscopic quantum mechanical
dynamics. By doing so, we answer a fundamental question about the nature of
energy transfer in time-varying systems, namely the significance (or in some
cases, lack thereof) of the imaginary part of response functions. We also
specialize many of our results to the particularly intriguing case of time-
periodic (i.e. “Floquet”) systems. In this case, many of our results are
simplified by the use of Floquet theory to describe both the quantum and
macroscopic electromagnetic aspects of problems. We provide examples of our
framework across variety of systems: time-modulated superconducting qubits in
the GHz, time-modulated polar insulators with optical phonon resonances, and
strongly driven gasses which exhibit high harmonic generation (HHG). Our
framework, when applied to these systems, enables us to discover a wide range
of physics such as pulse propagation in dispersive photonic time-crystals,
nonperturbative frequency conversion, and energy loss/gain.
These findings point toward a future where time-varying linear response theory
is an important theoretical and experimental tool for studying time-varying
optical materials. In an experimental setting, a great benefit of using linear
response theory in these complex systems is that linear response functions can
be measured, rather than computed. Another advantage of this framework is that
it allows one to characterize nonperturbative nonlinearities, which can be
important in systems which are very strongly driven.
The organization of this work is as follows: In Section II, we give an
overview of our framework which incorporates quantum mechanical linear
response theory, and classical optics to describe wave propagation in
dispersive time-varying materials. This section includes an important
discussion about Kramers-Kronig relations, and how response functions encode
energy transfer in time-varying optical systems. We also summarize some of the
important simplifications when these results are specialized to time-periodic
(Floquet) systems, leading to a compact description of wave propagation in
dispersive photonic time-crystals. After describing the key foundations, we
provide three distinct examples of our framework applied to different types of
microscopic models for driven systems. In Section III.1, we describe wave
propagation and energy transfer in a material whose optical response is
characterized by time-modulated two-level systems. Intriguingly, we find that
in the presence of sufficiently strong modulation, this type of system can
exhibit resonant gain in its ground state. Such a model is relevant for
describing metamaterials which could be formed from networks of
superconducting qubits. In Section III.2, we describe the optical response and
resultant scattering processes in a system described by a time-varying Lorentz
oscillator model which we refer to as a “Lorentz parametric oscillator.” Such
a model is relevant for describing time-modulated polar insulators with an
infrared optical response which is dominated by optical phonon resonances.
Finally, in Section III.3, we describe the highly nonperturbative frequency
conversion which may occur in gases undergoing high harmonic generation (HHG).
This result paves the way toward using strongly time-driven materials to
create artificial optical response at ultraviolet and X-ray frequencies.
Figure 1: General framework for describing the optics of dispersive and
strongly time-dependent systems. (a) Examples of time-dependent quantum
mechanical systems whose optical response requires time-varying linear
response theory. (b) Models of these microscopic dynamics can then be used to
construct macroscopic response functions which may also vary spatially to
account for material structures. For example, a dispersive dielectric
structure $\varepsilon(\mathbf{r},\omega)$ in the absence of time-modulation
can be described in terms of a two-frequency response function
$\varepsilon(\mathbf{r},\omega,\omega^{\prime})$ in the presence of time-
modulation. (c) These response functions can be incorporated into the Maxwell
equations to describe optical features of these systems, such as “free” wave
propagation, scattering, and energy transfer.
## II Theoretical Framework
In this section, we describe our general framework for constructing new time-
dependent optical materials from microscopic quantum mechanical models (Fig.
1). In such models, the dynamics are described by the solution to the
Schrödinger equation with a time-dependent Hamiltonian $H(t)$. Generally,
these microscopic dynamics can depend on many-body effects in a complicated
manner. In this work, we will focus on materials which are well-described by
constructing an effective bulk response from a collection of single particle
dynamics; however, many of our conclusions hold more broadly. Once the
relevant Schrodinger equation has been solved, the dipole response functions
of single particles can be constructed, and then transformed into bulk
macroscopic response functions such as $\varepsilon(\omega,\omega^{\prime})$.
In systems where dissipation mechanisms are important, this process can also
be followed by solving an appropriate master equation which rigorously
incorporates the dissipative dynamics.
With a macroscopic response function in hand, one can then use classical
electrodynamics to describe wave propagation and energy transfer in dispersive
time-varying media. For example, strongly modulated systems which are periodic
in time (photonic “time crystals”) can be associated with a band structure
which indicates the relationship between wavevector and quasi-frequency in the
driven material (see dispersion diagram in Fig. 1c). In systems with strong
light-matter hybridization, this provides a direct way to solve for the
polaritons of the driven system. Another example is the use of time-dependent
response functions to compute frequency-dependent scattering from a structure
such as a thin film of a time-dependent material. Experiments of this type
have been performed on epsilon-near-zero (ENZ) materials alu2007epsilon ;
reshef2019nonlinear which have demonstrated so-called temporal refraction
zhou2020broadband . The linear response formulation that we detail in this
work allows for the prediction of these behaviors in systems where dispersion
is critical.
### II.1 Time-varying linear response theory
The foundation of our approach is time-varying linear response theory, which
describes how some observable of a time-varying quantum mechanical system
evolves due to the presence of a weak probe field kubo1957statistical . For
the context of this discussion, we will focus on the polarizability which
dictates how an electric field probe $\mathbf{E}(t)$ induces a change to the
dipole moment $\Braket{\delta\mathbf{d}(t)}$, where $\Braket{\cdot}$ denotes a
quantum mechanical expectation value. While we will focus on the
polarizability of a single point-like particle, these concepts apply equally
well to other response functions such as susceptibility, permittivity,
conductivity, magnetic permeability, etc.
A dispersive time-driven material must in general be described with response
functions which refer to two times (or two frequencies)
landau2013electrodynamics . This is due to the fact that the system is not
time-translation invariant, and thus memory effects depend on the absolute
time at which a probe interacts with the system of interest. For the
polarizability example, the change to the dipole moment can be written in
terms of the two-time polarizability $\bm{\alpha}(t,t^{\prime})$ tensor and
the probe electric field $\mathbf{E}(t)$ as
$\displaystyle\Braket{\delta\mathbf{d}(t)}$ $\displaystyle=\int
dt^{\prime}\,\bm{\alpha}(t,t^{\prime})\mathbf{E}(t^{\prime})$ (1a)
$\displaystyle\Braket{\delta\mathbf{d}(\omega)}$
$\displaystyle=\int\frac{d\omega^{\prime}}{2\pi}\,\bm{\alpha}(\omega,\omega^{\prime})\mathbf{E}(\omega^{\prime}),$
(1b)
where frequency domain polarizability is defined as:
$\bm{\alpha}(\omega,\omega^{\prime})=\int dt\,dt^{\prime}\,e^{i\omega
t}e^{-i\omega^{\prime}t^{\prime}}\bm{\alpha}(t,t^{\prime}).$ (2)
The phase convention on the Fourier transform is chosen so that in the time-
independent limit,
$\bm{\alpha}(\omega,\omega^{\prime})=2\pi\bm{\alpha}(\omega)\delta(\omega-\omega^{\prime})$;
hence, the usual relation
$\Braket{\delta\mathbf{d}(\omega)}=\bm{\alpha}(\omega)\mathbf{E}(\omega)$ is
recovered.
The response function of any time-dependent system must be characterized by
its temporal microscopic dynamics. In the case of a time-dependent point-like
particle, the polarizability is given by the Kubo formula
$\bm{\alpha}(t,t^{\prime})=\frac{i}{\hbar}\theta(t-t^{\prime})\Braket{[\mathbf{d}(t),\mathbf{d}(t^{\prime})]},$
(3)
where $\mathbf{d}(t)$ is the interaction picture dipole operator, and the
expectation value is taken in the initial state of the system. In solid state
systems where many-body effects are important, the time-varying dielectric
function or conductivity can be appropriately formulated in a similar way
rudner2020floquet ; wackerl2020floquet .
_Causality and K.K. relations:_ In time-independent systems, the requirement
of passivity is tightly linked to the possible forms of a generic response
function $\chi(\omega)$ via the Kramers-Kronig (K.K.) relations
kramers1927diffusion ; kronig1926theory . When time-dependence is introduced,
this passivity requirement dissolves, as the drive provides energy to the
system that can lead to gain, among other effects. Despite this added
complexity, time-varying response functions are still constrained by
causality. Specifically, any time-dependent linear response function
$\chi(t,t^{\prime})$ must obey the relationship
$\chi(t,t^{\prime})=\theta(t-t^{\prime})\chi(t,t^{\prime})$ so that changes to
an observable at time $t$ are only caused by interactions at times
$t^{\prime}<t$. In the frequency domain, we show (Appendix A.2) this
requirement leads to the K.K. relationship:
$\chi(\omega,\omega^{\prime})=\frac{i}{\pi}\mathcal{P}\int
d\omega^{\prime\prime}\frac{\chi(\omega-\omega^{\prime\prime},\omega^{\prime}-\omega^{\prime\prime})}{\omega^{\prime\prime}},$
(4)
where $\mathcal{P}$ denotes the principal value. By taking real and imaginary
parts of Eq. 4, one can obtain a direct relationship between the real and
imaginary parts of $\chi(\omega,\omega^{\prime})$. In the limit that the
material is time translation-invariant, the response function takes the
limiting form $\chi(\omega,\omega^{\prime})\to\chi(\omega)\cdot
2\pi\delta(\omega-\omega^{\prime})$, and the standard K.K. relation is
recovered.
_Energy transfer:_ In a time-independent material, the energy absorbed by the
material from a passing wave at frequency $\omega$ is proportional to
$\operatorname{Im}\chi(\omega)$. These time-dependent K.K. relations raise an
immediate question about energy transfer in time-dependent materials: does
$\operatorname{Im}\chi(\omega,\omega^{\prime})$ still encode energy
dissipation (or gain) for a time-dependent material? We will show here that
this is generally not the case.
To do so, we consider the work done by a probe field on a time-dependent
polarizable particle. The total energy transferred to a point dipole can be
written as $U=\int_{0}^{\infty}d\omega P(\omega)$. In this expression,
$P(\omega)=-\frac{\omega}{\pi}\operatorname{Im}\left[\mathbf{d}(\omega)\cdot\mathbf{E}^{*}(\omega)\right]$
is the energy per unit frequency dissipated, $\mathbf{d}(\omega)$ is the
dipole moment, and $\mathbf{E}(\omega)$ is the probe field. Assuming the point
dipole is described by the polarizability
$\bm{\alpha}(\omega,\omega^{\prime})$, the energy dissipated per frequency is:
$P(\omega)=-\frac{\omega}{\pi}\operatorname{Im}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}\mathbf{E}^{*}(\omega)\bm{\alpha}(\omega,\omega^{\prime})\mathbf{E}(\omega^{\prime}).$
(5)
Unlike the equivalent expression for time-independent media, Eq. 5 does not
posses a clear sign; this is consistent with the general feature of time-
dependent media that a passing wave can lose or gain energy pendry2021gain ;
galiffi2022archimedes (and in fact, we will show cases where $P>0$, in
violation of passivity). Moreover, the amount of energy lost or gained can in
general depend on the phase of passing waves. To see this explicitly, consider
that for a monochromatic probe $E(t)=E_{0}\cos(\omega_{p}t-\phi)$ of a single
polarization incident upon an isotropic particle, the total energy dissipated
is expressed as
$U=\frac{E_{0}^{2}\omega_{p}}{2}\operatorname{Im}\left[\alpha(\omega_{p},\omega_{p})+e^{-2i\phi}\alpha(\omega_{p},-\omega_{p})\right].$
(6)
The form of Eq. 6 explicitly shows a contribution to the energy loss/gain
which depends on the phase $\phi$ of the probe. This is, for example, exactly
the type of physics exemplified by an optical parametric oscillator, where the
signal can either either be exponentially amplified or attenuated depending on
the phase of the input. Later, we give examples of systems where $U$ can take
on either sign, depending on whether the energy of the probe is absorbed or
amplified.
For a monochromatic probe, there are two contributions to the change in
energy, corresponding to the frequency components of the probe at
$\pm\omega_{p}$. The first contribution comes from
$\operatorname{Im}\alpha(\omega_{p},\omega_{p})$, which is independent of the
probe phase. For this term, the positive frequency components of the probe
field induce a dipole moment at that same positive frequency. It is this term
which reduces to the usual relation that energy transfer is proportional to
$\operatorname{Im}\alpha(\omega)$ in a time-independent system. The second
contribution comes from
$\operatorname{Im}[e^{-2i\phi}\alpha(\omega_{p},-\omega_{p})]$, which depends
explicitly on the temporal relationship between the probe field and dynamics
of the driven system through the phase $\phi$. For this term, the negative
frequency component of the probe $-\omega$ is shifted up by $2\omega_{p}$ to
$\omega_{p}$. From this, we can see that dissipated energy can depend on both
the real and imaginary parts of the response function
$\chi(\omega,\omega^{\prime})$.
#### II.1.1 From microscopic to macroscopic
The time-varying response functions discussed here are useful not only for
describing energy transfer into a medium, but also wave propagation. To see
this, we consider the construction of potentially spatially and temporally
varying response functions which are used to describe some time driven
photonic structure. For many systems, the point-like particles described by a
time-dependent polarizability $\alpha(\omega,\omega^{\prime})$ can be used to
describe the optical response of bulk materials, as is routinely considered
for time-independent media. In the simplest possible case, one assumes that
the polarizable particles are packed with a volume density $n$, allowing one
to define a unitless susceptibility $\chi(\omega,\omega^{\prime})$ by
$\chi(\omega,\omega^{\prime})=(n/\varepsilon_{0})\alpha(\omega,\omega^{\prime})$.
Such a scheme neglects local field effects, which can be accounted for using a
Clausius-Mossotti relation, or similar method which is appropriate to the
geometry aspnes1982local . Spatial arguments can also be incorporated in the
case that the material structure varies spatially, or if the time-varying
material possesses some joint spatio-temporal evolution. Such modulations have
been recently considered in the context of constructing “spatiotemporal
metamaterials” caloz2019spacetime and “spatiotemporal photonic crystals”
sharabi2022spatiotemporal .
We can thus write a form of Maxwell’s equations in frequency space which uses
a two-frequency linear response function in its constitutive relation. To do
so, it is helpful to define a permittivity
$\varepsilon(\mathbf{r},\omega,\omega^{\prime})=2\pi\delta(\omega-\omega^{\prime})+\chi(\mathbf{r},\omega,\omega^{\prime})$
which relates the displacement and electric fields as
$\mathbf{D}(\mathbf{r},\omega)=\varepsilon_{0}\int\frac{d\omega}{2\pi}\varepsilon(\mathbf{r},\omega,\omega^{\prime})\mathbf{E}(\mathbf{r},\omega^{\prime})$.
Under this definition, the electric field $\mathbf{E}(\mathbf{r},\omega)$ in
the presence of a current source $\mathbf{J}(\mathbf{r},\omega)$ obeys:
$\begin{split}\nabla\times\nabla\times\mathbf{E}(\mathbf{r},\omega)-\frac{\omega^{2}}{c^{2}}\int\frac{d\omega^{\prime}}{2\pi}&\varepsilon(\mathbf{r},\omega,\omega^{\prime})\mathbf{E}(\mathbf{r},\omega^{\prime})\\\
&=i\omega\mu_{0}\mathbf{J}(\mathbf{r},\omega).\end{split}$ (7)
Due to the time-dependence, this form of the Maxwell equation is nonlocal in
frequency space. In general, FDTD methods may be required in order to simulate
full spatial and temporal dependencies of time-driven systems. However, we
show in the next section that for time-periodic systems, the linear response
functions take a form which enable significant simplifications.
### II.2 Specialization to time-periodic systems
One general class of time-modulated systems which is of particular interest
are those in which the modulation is periodic in time fante1971transmission ;
morgenthaler1958velocity . In certain cases, such systems have been termed
“photonic time crystals” (PTCs). Most PTCs considered so far have been
described in terms of a “nondispersive” permittivity
$\varepsilon(t)=\varepsilon(t+T)$, where $T$ is the period. Furthermore, for
materials to be considered PTCs, it is usually assumed that the relative
temporal variations to the material are substantial ($\Delta\varepsilon\gtrsim
0.1$) so that the nature of wave propagation departs substantially from that
in an unmodulated counterpart. Due to their periodic nature, a spatially
homogeneous PTC can be associated with a band structure which relates
wavenumbers $k$ to _quasifrequencies_ $\Omega$, which lie in a temporal
Brillouin zone (TBZ) set by $\Omega_{0}\equiv 2\pi/T$. This phenomenon is well
studied, and manifests many natural analogies to spatial photonic crystals.
As an important extension of these ideas, we introduce the concept of
dispersive PTCs which result from temporal modulations which are fundamentally
dispersive. In this section, we specialize key results from above to time-
periodic systems. The key result is that in a time-periodic system, a response
function $\chi(\omega,\omega^{\prime})$ can be reduced to an integer series of
response functions $\chi_{k}(\omega)$.
_Form of response functions:_ In time periodic systems, the harmonic nature of
the problem places constraints on the form of the response functions.
Specifically, periodicity imposes the time-domain constraint
$\chi(t,t^{\prime})=\chi(t+T,t^{\prime}+T)$. This immediately dictates that
the frequency response is described by an integer series of response functions
$\chi_{k}(\omega)$, which are defined such that
$\chi(\omega,\omega^{\prime})=\sum_{k=-\infty}^{\infty}\chi_{k}(\omega)\cdot
2\pi\delta(\omega-\omega^{\prime}+k\Omega_{0}).$ (8)
From this form, we see that $\chi_{k}(\omega)$ encodes how an applied field at
frequency $\omega$ induces a response at frequency $\omega+k\Omega_{0}$.
_Kramers-Kronig relations, energy transfer:_ The K.K. relations take a more
familiar form in time-periodic media. By assuming that
$\chi(\omega,\omega^{\prime})$ obeys Eq. 8, we use Eq. 4 to deduce that the
response function for each integer harmonic obeys the usual time-independent
K.K. relation:
$\chi_{k}(\omega)=\frac{i}{\pi}\mathcal{P}\int
d\omega^{\prime}\frac{\chi_{k}(\omega^{\prime})}{\omega-\omega^{\prime}}.$ (9)
For time-periodic media, the integer order response functions allow for a
particularly informative description of loss and gain. In particular,
$\operatorname{Im}\alpha_{0}(\omega)$ encodes loss and gain which is
independent of the probe phase, while $\alpha_{k}(\omega)$ of nonzero order
encode loss and gain which depend on the probe phase. If we send a
monochromatic probe field at such a material, the energy dissipated $U$ (from
Eq. 6) reduces to
$\begin{split}\frac{U}{T}&=\frac{E_{0}^{2}\omega_{p}}{2}\operatorname{Im}\alpha_{0}(\omega_{p})\\\
&+\frac{E_{0}^{2}\omega_{p}}{2}\sum_{k=1}^{\infty}\operatorname{Im}\left[\alpha_{k}(\omega_{p})e^{-2i\phi}\right]\delta_{2\omega_{p}=k\Omega_{0}}.\end{split}$
(10)
This equation reveals several key pieces of information about the nature of
energy transfer in time-periodic systems. We discuss these features by
examining the two terms.
(1) In the first term, the imaginary part of the zeroth harmonic response
function $\operatorname{Im}\alpha_{0}(\omega)$ _carries unambiguous
information about energy transfer which does not depend on the phase of the
probe field._ Physically, $\alpha_{0}(\omega)$ encodes the polarization which
is created at the same frequency as the probe, and is thus most closely
connected to the dispersive response function of the undriven medium.
Moreover, unlike in a ground state time-independent system,
$\operatorname{Im}\alpha_{0}(\omega)$ is not restricted to be positive. This
is analogous to an active medium which is pumped to an excited state, which
can exhibit gain as characterized by a negative imaginary part of some
response function. Instead here, the passivity can be broken by time-
dependence rather than a population inversion.
(2) In the second term, response function of orders $k\geq 1$ can affect the
dissipated energy through phase-dependent effects under a special resonance
condition. This resonance condition is indicated by a Kronecker delta function
which requires that $2\omega_{p}=k\Omega_{0}$ for integer $k$. Physically,
this condition results from negative frequency probe field components
$-\omega_{p}$ which create polarization at frequencies which are shifted over
by an integer number of harmonics: this positive frequency polarization can
then interact with the positive frequency component $\omega_{p}$ of the probe
field to do work (positive or negative). It is worth noting that this is the
same type of mechanism responsible for parametric amplification processes as
described in nonlinear optics, which are known to be sensitive to phase
boyd2019nonlinear .
For a given $\omega_{p}$ and $\Omega$, only one value of $k$, if any, can
satisfy this condition. This potential additional term contains a phase
dependence within the $\operatorname{Im}$ operator. This leads us to conclude
that _the real and imaginary parts of $\alpha_{k}(\omega)$ for $k\neq 0$ have
no unique physical significance as far as energy transfer is concerned._ Thus
the sign of $\operatorname{Im}\left[\alpha_{k}(\omega)e^{-2i\phi}\right]$, and
the sign of the energy transfer, depends on the phase relationship between the
probe field and the underlying microscopic dynamics that govern
$\alpha_{k}(\omega)$.
_Eigenmodes in periodic systems:_ In a time-periodic system, we can seek
solutions to the sourceless Maxwell equation in a bulk medium. Due to the time
periodicity, and spatial translation invariance, the Maxwell-Floquet modes
take the form
$\mathbf{E}(\mathbf{r},t)=e^{i\mathbf{k}\cdot\mathbf{r}}\sum_{n}u_{\Omega,n}e^{-i(\Omega+n\Omega_{0})t}$,
where $\mathbf{k}$ is the wavevector, $\Omega$ is a quasifrequency in the
first temporal Brillouin zone $[-\frac{\pi}{T},\frac{\pi}{T})$, and
$u_{\Omega,n}$ are a sequence of coefficients. In a medium with permittivity
$\varepsilon(\omega,\omega)=2\pi\delta(\omega-\omega^{\prime})\varepsilon_{\text{bg}}(\omega)+\sum_{k}\Delta\chi_{k}(\omega)2\pi\delta(\omega-\omega^{\prime}+k\Omega_{0})$,
the amplitude of the wavevector $|\mathbf{k}|=k_{\Omega}$ and coefficients can
be found for a given quasifrequency by solving the eigenvalue problem:
$\frac{\Omega_{n}^{2}}{c^{2}}\varepsilon_{\text{bg}}(\Omega_{n})u_{n}+\frac{\Omega_{n}^{2}}{c^{2}}\sum_{m}\Delta\chi_{m}(\Omega_{n})u_{n-m}=k_{\Omega}^{2}u_{n},$
(11)
where $\Omega_{n}\equiv\Omega+n\Omega_{0}$ is the quasifrequency shifted by
$n$ harmonics. This relation can be cast into a linear matrix problem which
yields the band structures of dispersive photonic time crystals, examples of
which are shown later in the text. We note that in the presence of very large
loss or gain, an eigenmode expansion may not strictly form a complete basis
for the set of possible solutions. However, in many cases, the eigenmode
expansion may still provide accurate information about the dispersion
relation. In systems where this approximation breaks down, Green’s function
methods can be employed to describe the propagation of waves from sources.
## III Example Systems
### III.1 Two-level system
Figure 2: Wave propagating in a time-driven two-level system from perturbative
to nonperturbative regimes. (a-c) Real and imaginary parts of zeroth order
polarizability $\alpha_{0}(\omega)$ as a function of frequency for three
different drive strengths $\delta\omega/\omega_{0}=\\{0,0.4,1.5\\}$ and
driving frequency $\Omega_{0}=\omega_{0}/2$. Loss parameter is taken to be
$\Gamma=10^{-3}\omega_{0}$ for all peaks. (d-f) Dispersion relations for a
time-periodic bulk medium which is composed of particles described by the
polarizabilities $\alpha_{k}(\omega)$. Dispersion relations are plotted as a
function of the quasifrequency $\Omega$ which lies in the temporal Brillouin
zone $-\Omega_{0}/2<\Omega\leq\Omega_{0}/2$. Sufficiently strong driving
causes a gap to open in the momentum (panel g). (h-j) Propagation of a
Gaussian wavepacket constructed from the modes indicated by a red $\times$ in
corresponding panels (d-g). The line traced out in $(x,t)$ space indicates the
group velocity of the packet. As the strength of the time modulation
increases, new features such as temporal beating due to interference of
harmonics, and wavepacket spreading due to group velocity dispersion.
In this section, we discuss a two-level system (2LS) which has its energy
splitting modulated strongly in time. When the modulation frequencies are
close to the splitting frequency itself, the dispersive framework outlined
above is required to describe linear response correctly. We focus specifically
on a system with a Hamiltonian
$H_{\text{2LS}}(t)=\frac{\omega_{0}}{2}\left(1+\delta\omega\cos(\Omega_{0}t)\right)\sigma_{z}$.
Since the Hamiltonian is periodic in time, we can use the insight of Floquet
theory that the system should behave like a stationary system, but with a
ladder of quasi-energy levels. Physical systems which have realized
Hamiltonians of this type include driven spins jiang2021floquet ,
superconducting qubits deng2015observation , quantum dots stehlik2016double ,
and strongly modulated Rydberg atoms clark2019interacting . A particularly
appealing aspect of microwave schemes is that very strong modulations can be
readily induced, leading to the nonperturbative regime of effects which do not
fall under the purview of perturbative nonlinear optics. While we focus here
on the $\sigma_{z}$-type modulation of a two-level system, many of the
principles discussed here should carry over naturally to other types of two-
level modulations, as well as systems with more discrete levels.
While an undriven 2LS is restricted to make ground-excited transitions at the
bare resonance frequency $\omega_{0}$, the time dependent 2LS we describe here
can make transitions separated by harmonics of the drive. Thus, in a system
which is well described in terms of a few energy levels, the driven system can
exhibit optical response at frequencies different than that of the undriven
system. Using the Floquet states of the Hamiltonian, and the Kubo formula, we
find the single-particle polarizability:
$\alpha_{k}(\omega)=\frac{d^{2}}{\hbar}\Braket{\sigma_{z}}_{0}\sum_{n}\left[\frac{J_{n+k}J_{n}}{\omega-\omega_{n}+i\Gamma_{n}}-\frac{J_{n-k}J_{n}}{\omega+\omega_{n}+i\Gamma_{n}}\right].$
(12)
Here, $\omega_{n}\equiv\omega_{0}+n\Omega_{0}$ is the bare transition
frequency shifted by $n$ harmonics, $J_{n}\equiv
J_{n}(\delta\omega/\Omega_{0})$ is the $n$-th order Bessel function evaluated
at the driving strength parameter $\delta\omega/\Omega$, $\Gamma_{n}$ is the
linewidth associated with each transition between Floquet levels, and
$\Braket{\sigma_{z}}_{0}$ is an expectation value taken in the equilibrium
state. More discussion about this equilibrium as well as the damping rates is
shown in the Appendix.
As the fractional change in the frequency $\delta\omega$ increases, the
optical response of the system moves from a perturbative to nonperturbative
regime (Figs. 2a-c). The plots show the response function of zeroth harmonic
order $\alpha_{0}(\omega)$ for three modulation strengths
$\delta\omega/\omega_{0}=0,0.4,1.5$, and with a modulation frequency
$\Omega_{0}=\omega_{0}/2$. This response function encodes the amplitude with
which a probe at frequency $\omega_{p}$ generates a change in the dipole
moment at that same frequency. With no modulation, this describes the
polarizability of a static 2LS, which is given by a Lorentzian (Fig. 2a). For
a stronger drive ($\delta\omega/\omega_{0}=0.4)$, sidepeaks emerge, which
indicate optical response at $\omega_{0}\pm\Omega_{0}$. At this strength of
modulation, only the first harmonic contributes substantially, although others
are present at levels which are not yet apparent in Fig. 2b. This behavior
shifts as the modulation strength nears or exceeds the static energy splitting
$\omega_{0}$. Fig. 2c shows an example of the nonperturbative regime, in which
multiple harmonic orders are relevant. In this extreme limit, the strongest
optical response actually occurs at $2\omega_{0}$, indicating that absorptions
several steps up the Floquet ladder occur more strongly than the transition at
$\omega_{0}$. We also note that the overall magnitude of the response peaks is
seen to decrease with increasing $\delta\omega$. In this sense, the optical
response of the system is allocated across more frequencies, but with less
response at each frequency. In this particular case, the sum rule
$\sum_{m=-\infty}^{\infty}J_{m}^{2}=1$ fixes the total response across all
frequencies clark2019interacting .
_Bulk wave propagation:_ Each of the three examples of optical response regime
described above comes with its own implications for wave propagation in a bulk
optical system which is characterized by these single-particle models. To
demonstrate this, we consider a bulk optical medium which consists of time-
dependent point particles packed with a number density $n$. The system is
equivalently described by a plasma frequency $\omega_{p}$. It is then possible
to compute the dispersion relation of plane waves which propagate in such a
uniform time-dependent medium. Figs. 2d-f show the dispersion relations of
bulk media with $\omega_{p}=\omega_{0}/2$ for the modulation parameters given
in each corresponding column. Dispersion relations indicate the relationship
between the wavevector $k$ and the quasienergy $\Omega$, which is taken to lie
in the first temporal Brillouin zone (TBZ)
$-\Omega_{0}/2<\Omega\leq\Omega_{0}/2$. For the undriven medium
($\delta\omega=0$), the dispersion relation is the same as that of a time-
independent Lorentz oscillator, but with frequencies folded into the first
TBZ. Features such as the light-like and polariton-like parts of the
dispersion can still be identified. In this regime, wavepackets propagate in
the usual way (Fig. 2g).
Stronger driving ($\delta\omega/\omega_{0}=0.4$) brings new changes to the
band structure. For example, bands near the edges at $\Omega=\Omega_{0}/2$
have moved up and down in pairs (marked by an arrow in Fig. 2e), and some
curvature has been introduced into bands. Additionally, there is an avoided
crossing of the two lowest bands. While the wavepacket at the point marked on
the band structure propagates coherently and with a similar group velocity to
its unmodulated counterpart, a new beating behavior emerges in the amplitude
due to the presence of multiple temporal harmonics in the modes which comprise
the packet. As it propagates, the wavepacket exchanges energy back and forth
with the medium through this behavior which is only possible in the presence
of broken time-translation symmetry.
With driving strength in the extreme nonperturbative regime
($\delta\omega/\omega_{0}=1.5$), the band structure changes substantially.
Most prominently, wavevector gaps are introduced, representing wavelengths
which cannot propagate in the medium. Band gaps in “photonic time crystals”
have been identified previously in non-dispersive settings
biancalana2007dynamics ; reyes2015observation . Additionally, the crossed
bands shown in Figs. 2d,e are seen to hybridize with one another, eliminating
these sharp crossings. The panel below (Fig. 2i) shows that a wavepacket
centered around the marked mode propagates with amplitude oscillations, as
well as dispersion. This dispersion can be attributed to the band curvature
which has developed (red “x” in Fig. 2g), as compared to the linear dispersion
at the corresponding points of Figs. 2d,e.
Figure 3: Linear response representation of ground state gain in a driven two-
level system. (a) Floquet level diagram for a two-level system driven at
frequency $\Omega/\omega_{0}=0.4$ with strength $\delta\omega/\omega_{0}=2.2$.
Arrows indicate absorptive and emissive transitions from the thermodynamic
ground state to the excited state. (b) Energy transfer properties of the
driven system can be visualized through $\operatorname{Im}\alpha_{0}(\omega)$.
Transitions with loss correspond to peaks where
$\operatorname{Im}\alpha_{0}(\omega)>0$, while transitions with gain
correspond to peaks where $\operatorname{Im}\alpha_{0}(\omega)<0$.
Loss and gain: We now discuss loss and gain in these types of systems, as
described in the time-dependent linear response framework. In the absence of
any driving, it is well known that a two-level system in its thermodynamic
ground state can only absorb energy. The single-particle polarizability in
this case is given by a Lorentzian form (Fig. 2a). The quantity
$\operatorname{Im}\alpha_{0}(\omega)\geq 0$ gives the frequency dependent
loss, which peaks at $\omega_{0}$ due to absorptive transitions from the
ground to excited state. In a Floquet system, transitions from the ground to
excited state can also occur due to absorption of a photon at a frequency
$\omega_{0}+k\Omega_{0}$ for some integer $k$. These resonances correspond
exactly to the peaks shown in Figs. 2b,c, and also exhibit the property
$\operatorname{Im}\alpha_{0}(\omega)>0$. As discussed in the theory section,
the quantity $\operatorname{Im}\alpha_{0}(\omega)$ does possess significance
for phase-independent energy transfer. From this we see that the example
parameters used in Fig. 2 give purely absorptive systems.
To complete our discussion of a modulated 2LS, we give an example of how such
a time-dependent two-level system can exhibit both resonant gain and loss in
its thermal equilibrium state. To do so, we consider a modulated two-level
system with $\Omega=0.4\omega_{0}$. When strongly modulated
($\delta\omega/\omega_{0}=2.2$), a substantial contribution emerges from
Floquet sidebands which fall below the level of the original ground state.
This means that the system can make ground to excited state transitions at
frequencies $\omega_{0}+k\Omega<0$ for sufficiently negative integers $k$.
These transitions are schematically shown in Fig. 3a. In the polarizability,
these transitions appear as peaks with $\operatorname{Im}\alpha_{0}(\omega)<0$
around the relevant resonances, corresponding to energy gain in the Floquet
ground state. The gain peaks appear next to other peaks where
$\operatorname{Im}\alpha_{0}(\omega)>0$, which correspond to absorptive
transitions of the form shown in Fig. 2. Thus, a two-level system modulated in
this way can provide either absorption or gain to a probe field, depending on
the frequency.
### III.2 Time-dependent Lorentz oscillator
In this section, we use our framework to describe a medium which behaves as a
harmonic oscillator with a time-varying frequency. One system which can be
modeled this way under certain conditions is a polar insulator which is
strongly driven by an external field. In polar insulators, such as silicon
carbide (SiC) and hexagonal boron nitride (hBN), the optical response over
some frequency ranges is dominated by optical phonon resonances
basov2016polaritons . In undriven polar insulators, these resonances lead to
well-established peaks at the transverse optical (TO) phonon frequency
$\omega_{\text{TO}}$, which are described by a Lorentz oscillator model.
However, in the presence of strong laser pulses, the TO phonon frequency of
such a material may acquire a time dependence cartella2018parametric . If the
frequency of the modulating pulse is on the order of $\omega_{\text{TO}}$
itself, then a dispersive time-dependent framework is needed to capture
optical behaviors around $\omega_{\text{TO}}$.
To do this, we use a model which we refer to as the “Lorentz Parametric
Oscillator” (LPO). In this model, we assume that the response of the polar
insulator can be characterized by that of a collection of point-like
polarizable particles. Each of these particles is a harmonic oscillator with a
time-varying resonance frequency $\omega(t)^{2}=\omega_{0}^{2}(1+f(t))$, so
that the Hamiltonian governing the oscillator is
$H_{\text{LPO}}(t)=\frac{p^{2}}{2m}+\frac{1}{2}m\omega(t)^{2}x^{2}$, where $x$
and $p$ are the position and momentum operators, and $m$ is the effective mass
of the atom in the lattice. For $|f(t)|\ll 1$, the first order perturbative
correction to the polarizability is given as
$\alpha(\omega,\omega^{\prime})=2\pi\alpha^{(0)}(\omega)\delta(\omega-\omega^{\prime})+\alpha^{(1)}(\omega,\omega^{\prime})$
(13)
where
$\alpha^{(0)}(\omega)=\frac{q^{2}}{m}\frac{1}{\omega_{0}^{2}-\omega_{0}^{2}-i\omega\Gamma}$
is the ordinary Lorentz Oscillator contribution from $f(t)=0$. In cases where
the perturbative approximation breaks down, one can equivalently solve the
equation of motion for a harmonic oscillator with time-varying frequency
(sometimes referred to as the “Mathieu equation” ruby1996applications )
numerically and take Fourier transforms in order to obtain
$\alpha(\omega,\omega^{\prime})$ more generally. However, in most cases, the
perturbative regime should apply, and the first order frequency space
correction is given by
$\alpha^{(1)}(\omega,\omega^{\prime})=-\frac{q^{2}}{m}\frac{\omega_{0}^{2}f(\omega-\omega^{\prime})}{(\omega_{0}^{2}-\omega^{2}-i\omega\Gamma)(\omega_{0}^{2}-\omega^{\prime
2}-i\omega^{\prime}\Gamma)},$ (14)
where $f(\omega)$ is the Fourier transform of the modulation. The expression
features two resonant Lorentz oscillator factors in the denominator at
$\omega$ and $\omega^{\prime}$, and is consistent with expressions for
resonant nonlinearities, such as Kerr nonlinearities around an atomic
resonance boyd2019nonlinear .
Figure 4: Optics of a time-modulated harmonic oscillator. (a) Quantum harmonic
oscillator of frequency $\omega_{0}$ subject to a frequency-modulation at
frequency $\Omega_{0}$. Panel below shows the real and imaginary parts of
$\chi_{1}(\omega)$ for parametric resonance when $\Omega_{0}=2\omega_{0}$, at
a modulation strength of $\delta\omega=10^{-3}\Omega_{0}$. (b) Incident probe
field on a slab of material described by
$\varepsilon(\omega,\omega^{\prime})$. The slab of material has thickness $L$.
Since the medium is time-varying, reflected and transmitted waves can be
shifted by integer multiples of the drive frequency $\Omega_{0}$. (c) Band
structure of a nondispersive medium with a time-dependent refractive index
profile. This is the homogeneous medium dispersion relation $\omega=ck/n$
folded into the temporal Brillouin zone
$-\Omega_{0}/2<\Omega\leq\Omega_{0}/2$. (d) Band structure for the dispersive
Lorentz parametric oscillator medium depicted in (a), with
$\Omega_{0}=2\omega_{0}$. (e-f) Transmission spectrum for probe and shifted
probe frequencies for the configuration depicted in (b) for both nondispersive
and dispersive modulations.
To give an example of how this dispersive time-dependent response function can
be used in optics, we consider the scattering of an incident wave from a slab
of material described by $\chi(\omega,\omega^{\prime})$ for a periodic
modulation $f(t)=\delta\omega\cos(\Omega_{0}t)$. The general concept of
scattering from dispersive time-dependent materials was recently explored in
ptitcyn2021scattering . In this example, we will focus specifically on how
dispersive resonance can greatly impact the reflection and transmission of
waves from a material. To demonstrate this, we consider a thin film scattering
problem which consists of a weak probe field at frequency $\omega_{p}$
incident on a material $\varepsilon(\omega,\omega^{\prime})$ of length $L$. If
the material is time dependent with some periodicity $\Omega_{0}$, then in
general, there will be reflected and transmitted waves of shifted frequencies
$\omega_{p}+k\Omega_{0}$, where $k$ is an integer. To elucidate the effect of
dispersive resonances, we compare the scattering problem for two different
time-dependent materials: (1) a nondispersive material which has a
permittivity
$\varepsilon(t)=\varepsilon_{\text{bg}}(t)+\delta\varepsilon\,f(t)$, and (2) a
dispersive material described the LPO model detailed above. For both
materials, we use a periodic modulation profile $f(t)=\cos(\Omega_{0}t)$,
specifically focusing on the parametric resonance case given by
$\Omega_{0}=2\omega_{0}$.
Figs. 4c-d show the photonic time-crystal band structures corresponding to
materials (1) and (2) for $\delta\varepsilon=\delta\omega=10^{-3}$. In the
absence of dispersion, these two descriptions coincide. In the nondispersive
case, the weak interaction means that the band structure simply corresponds to
the undriven material dispersion
$\omega_{k}=ck/\sqrt{\varepsilon_{\text{bg}}}$ folded into the first temporal
Brillouin zone with quasifrequencies $-\Omega_{0}/2<\Omega\leq\Omega_{0}/2$.
Similarly, the dispersive band structure can be understood as that of an
undriven Lorentz Oscillator dispersion folded into the TBZ. In this particular
case we have chosen for parametric resonance ($\Omega_{0}=2\omega_{0}$), the
steep branch of the dispersion due to the resonance at $\omega_{0}$ coincides
with the band edge at $\Omega_{0}/2$. This parametric resonance leads to
pronounced effects on incoming waves.
In Figs. 4e-f, we show results for the transmission amplitude for an incident
field at $\omega$ for shifted frequencies $\omega$, $\omega-\Omega_{0}$, and
$\Omega_{0}$ for a thin film created from the nondispersive and dispersive
materials described above. For the nondispersive modulation, the vast majority
of the transmitted field lies at the incident frequency. In contrast, the
dispersive material exhibits peaks of resonant conversion for the incident
frequency $3\omega_{0}$. This occurs because at the parametric resonance
condition ($\Omega=2\omega_{0}$), the downshifted harmonic lies at
$\omega_{0}$, which is resonant with the oscillator denominators of Eq. 14.
Since the interaction takes place in a thin film, radiation at incident or new
frequencies may continue to re-interact with the material, leading to cascaded
harmonics. This is, for example, the origin of a similar response for incident
field of $5\omega_{0}$. The differences between behavior between the
dispersive and nondispersive models highlight the importance of using a model
which is consistent with underlying microscopic dynamics.
We now comment briefly about the relationship between the time-modulated two-
level and Lorentz oscillator models. In the limit of time-independent systems,
it is well known that both of these models exhibit the same form of dipole
response, given by $\alpha^{(0)}(\omega)$. The intuition behind this is that a
weak field which probes the ground state of a harmonic oscillator can
effectively only “see” the first transition of the oscillator ladder, so the
two-level model is recovered. This close relationship dissolves when time-
modulations are introduced, as we have seen when comparing the two systems. A
parametric drive involves more states of the harmonic oscillator ladder into
the dynamics, so that the parametric oscillator model is fundamentally
different than a modulated two-level system. The departure of these models
here is not dissimilar to what unfolds in the nonlinear response of the static
systems: the two-level system displays resonant nonlinearities, while the
oscillator exhibits no nonlinearity at all. This serves as a clear example,
then, that models for the optical response of strongly driven materials should
be considered carefully on the basis of microscopic dynamics.
Figure 5: High harmonic generation (HHG) as a time-dependent optical medium.
(a) The general setup of HHG consists of a gas sample which is irradiated with
an extremely strong IR laser pulses $E_{\text{drive}}(t)$. In a 1D Coulomb
potential model, the ground state electron is ejected into the ionized
continuum, generating a dipole moment $d(t)$ in the process. This induced
dipole moment contains many harmonics of the drive frequency, as shown by the
plot of $|d(\omega)|^{2}$. The harmonics continue up to a cutoff frequency,
which is related to the ionization energy of the atom. (b) Two-time atomic
susceptibility $\alpha(t,t^{\prime})$ of a single particle which undergoes the
modulation shown in (a). (c) Two-frequency atomic susceptibility
$\alpha(\omega,\omega^{\prime})$ which shows the frequency decomposition of
$\alpha(t,t^{\prime})$. Harmonic stripes can be seen for
$\omega^{\prime}=\omega+k_{\text{even}}\Omega_{0}$ due to the quasi-periodic
nature of the modulation. (d-f) Induced dipole moments by probe fields at
different frequencies and phases. (g-i) Induced dipole moment spectra
corresponding to each of the probes in (d-f). Probe frequencies are marked
with an arrow. Frequencies of the peaks are marked with dashed lines.
### III.3 High harmonic generation
In this section, we show how the time-dependent linear response framework
provides a pathway to describe IR to X-ray frequency optics of systems which
exhibit high harmonic generation (HHG). The most basic configuration for such
a system is a gas cell which is pumped with an extremely intense infrared
laser pulse. For sufficiently strong pumps, the system exhibits highly
nonperturbative effects of strong field physics, and many pump photons can be
converted into single photons of frequencies which are more than one hundred
times higher than that of the pump mcpherson1987studies ; lewenstein1994theory
. These systems serve as valuable sources of UV and X-ray photons which are
difficult to produce by any other means, and also generate harmonic combs
which form attosecond pulses paul2001observation ; krausz2009attosecond . More
recently, HHG has also been observed from solids ghimire2019high .
Although HHG systems have been studied for decades for light generation, there
are untapped opportunities to use them as venues for new optical interactions.
We propose that HHG systems represent an intriguing platform to study the
optics of time-varying materials at high frequencies. From this point of view,
the strongly driven gas can itself be considered a time-varying optical
medium. Due to the strong field strengths and atomic resonances involved in
these systems, dispersion plays an important role.
The general setup of an HHG system is shown in Fig. 5a. In the absence of any
probe field, the driven system acquires a dipole moment which oscillates at
many harmonic multiples of the driving frequency, leading to HHG. If the
emitted photons are considered quantum mechanically, HHG can be equivalently
characterized as spontaneous emission from transitions between the Floquet
quasi-energy levels of the driven system gorlach2020quantum .
Using a 1D model of an atomic potential, we numerically solved the time-
dependent Schrodinger equation for
$H(t)=\frac{p^{2}}{2m}+V(x)-E_{0}x\sin(\Omega_{0}t)g(t)$, where
$V(x)=-1/\sqrt{x^{2}+a^{2}}$ is a “soft Coulomb” potential regulated by the
parameter $a$, and $g(t)$ is an envelope function which turns the drive on and
off. Parameters were chosen so that the ionization energy of the potential
matches that of Neon. Using numerical evolution of this Hamiltonian, we
directly computed the atomic polarizability $\alpha(t,t^{\prime})$ using the
Kubo formula (Eq. 3) 111In a Hermitian system, the polarizability
$\alpha(t,t^{\prime})$ can be computed by evolving all eigenstates of the
Hamiltonian in time, and then taking expectation values of the appropriate
operators to implement the time-domain Kubo formula directly. However, the
numerical models used for HHG typically rely on absorbing boundary conditions,
which break the Hermiticity, and thus the completeness of eigenstates.
Therefore, we compute the time-domain response function via the Liouvillian
evolution in a manner consistent with the quantum regression theorem. We have
verified that the response function $\alpha(t,t^{\prime})$ obtained through
this method generates dipole responses to probe fields which match those
obtained by directly incorporating the probe into the Hamiltonian.. The
results of this calculation are shown in Fig. 5b. As dictated by the causality
constraint, the dipole response is nonzero only for times $t>t^{\prime}$. Even
though this system is driven for a relatively small number of periods, some
features of the Floquet regime emerge. From the theoretical discussion around
Eq. 8, we know that for a perfectly time periodic system, the frequency
polarizability $\alpha(\omega,\omega^{\prime})$ converges to a series of delta
functions spaced at integer multiples of the drive. By numerically
transforming the time-domain polarizability, we show that this holds
approximately true. The squared magnitude of the frequency-domain
polarizability $|\alpha(\omega,\omega^{\prime})|^{2}$ is shown in Fig. 5c over
a range of harmonics. The clear diagonal stripes adhere to lines for which
$\omega^{\prime}=\omega+k\Omega_{0}$. For a general system, $k$ can be any
integer. However, the inversion symmetry of the potential and driving field we
have chosen dictates that $k$ may only take on even values, as is consistent
with the general framework for selection rules in HHG neufeld2019floquet . In
the limit of weak driving, where only a small number of harmonics can be
produced, this constraint reduces to the well-known fact that centrosymmetric
materials have no $\chi^{(2)}$ nonlinearity.
We now explore the consequences of these response functions for weak probes
which interact with the driven system. Figs. 5d-f, show the change to the
time-dependent dipole moment $\Delta d(t)$ induced by a probe field
$E(t)=E_{p}\cos(\omega_{p}t-\phi)e^{-(t-t_{0})^{2}/2\tau^{2}}$ for different
probe frequencies and phases. Generically, the dipole moment peaks can appear
at $\pm\omega_{p}+k\Omega_{0}$, where $k$ is an even integer. Different
behaviors emerge depending on the frequency $\omega_{p}$ and phase $\phi$ of
the probe, which are visualized through the frequency spectrum $|\Delta
d(\omega)|^{2}$ for different probe parameters.
For an odd-harmonic probe ($\omega_{p}=11\Omega_{0}$), the dipole responds at
other odd frequencies (Fig. 5g). A black arrow marks harmonic 11, which
oscillates at an amplitude higher than that of the surrounding peaks.
Nevertheless, notable contributions come from many peaks, extending up through
around harmonic 50. Additionally, we note that the phase of the probe with
respect to the pump can contribute substantially to the resulting output. In
particular, we show examples of the phases $\phi=\pi/2,\pi$. For some of the
harmonics produced by the probe, the amplitude can vary by an order of
magnitude or more depending on the probe phase. This type of behavior has
actually already been observed in the context of so-called “two-color high
harmonic generation” in which some harmonic of the drive (usually the third),
is sent into the sample along with the drive itself kim2005highly . In some
sense, the pump-probe schemes discussed here are similar in nature. The main
development here is that while experiments so far have considered only a few
harmonics of the drive as a probe, our insights from Floquet linear response
indicate that such HHG systems should also exhibit response at many harmonics
of the drive, providing a way to engineer optical response at UV and X-ray
frequencies.
So far, we have shown that an odd harmonic probe can essentially modify the
normal HHG spectrum (which also consists only of odd harmonics in this
example). However, we now show that weak probe fields sent at the time-driven
system can also be used to generate dipole moments at frequencies which are
not produced in absence of the probe. For example, sending in a probe
frequency at some even harmonic $m$ induces a comb of dipole moments at
frequencies $k_{\text{even}}\Omega_{0}$. This change to the dipole moment will
radiate at even harmonics, which are not produced by the system in absence of
the probe. Such a configuration is shown in Figs. 5e,h, where the probe is
sent at harmonic 10. The result is an even comb of induced dipole moments,
which will then radiate into even harmonics. Similarly to the odd-probe, the
probe harmonic stands out in strength above the others, and the probe phase
can strongly influence the output.
Finally, we show that by probing at a non-harmonic frequency results in a pair
of interleaved combs (Fig. 5f,i). In particular, sending in a weak probe of
harmonic $\omega_{p}=22.5\omega_{0}$ produces dipole moment peaks at
$\pm\omega_{p}+k_{\text{even}}\Omega_{0}$, leading to induced dipole moments
at non-harmonic frequencies, but which are separated by even multiples of the
drive. Moreover, this example indicates that the driven system will respond
optically at tens of harmonics, which for a near-IR pump corresponds to
optical response at wavelengths of 10’s of nanometers or below. Thus, these
results show promise for the potential to use existing HHG systems as a
platform to study the optical response of strongly driven systems, potentially
leading to controllable optical materials which can respond and convert
frequencies in the UV and X-ray regime. While we have focused for clarity on a
single-particle polarizability model, the two-frequency linear response
framework naturally lends itself to the inclusion of spatial aspects of HHG
problems, which can enable studies of phase-matching and wave propagation
l1991higher ; salieres1995coherence .
We additionally note that the time-dependent linear response is particularly
appealing for the study of probe-response in HHG, because once a function such
as $\alpha(t,t^{\prime})$ is computed through potentially time-consuming
quantum mechanical simulations (i.e. results of Fig. 5b), the response to any
probe can be computed as a simple convolution integral. In fact, through
energy dissipation/gain measurements, it may be possible in some cases to
determine $\alpha(\omega,\omega^{\prime})$ experimentally, enabling inferences
about how the system will response to other probes. Such an approach may be
particularly appealing in order to construct probe fields which will
selectively enhance or suppress the generation of certain harmonics, which
could relate to optimization of 2-color HHG processes kim2005highly , or more
complex analogs. This framework could also be used to describe the high
frequency parametric gain which has been observed in HHG systems
seres2010laser , allowing for the creation of more efficient systems which
amplify high frequency radiation.
## IV Conclusion and outlook
In summary, we have presented a framework for describing electromagnetic
response and wave propagation in dispersive time-varying quantum systems. We
have established fundamental properties of time-varying response functions,
with a special focus on developing forms for use in time-periodic (Floquet)
systems. We have addressed fundamental questions about the nature of energy
transfer (gain and loss) in these systems. In fact, the relationship between
K.K. relations and energy transfer in time-varying materials that we have
established can enable the use of absorption/gain measurements to construct
the full complex response functions of time-varying systems. Additionally, we
have shown selected examples of this framework to address a diverse set of
problems which raise implications for superconducting microwave circuits,
polar insulators in the IR, and UV/X-ray optics of HHG systems. We hope that
this unifying approach will reveal further similarities among fields which
would normally be considered disparate.
One important future direction is the development of microscopic models to
describe time-varying linear response in more complex systems. For example,
many recent works have focused on the electronic states that can be created in
Floquet-driven matter (with a particular focus on topological electronic
properties) bao2022light . However, there is still much work to be done to use
these descriptions of strongly driven solids to infer the optical properties
of such materials. In some cases, free electron or few-band models may be
sufficient to capture the key physics. In more complicated situations, time-
dependent density functional theory (TDDFT) may serve as an essential tool for
computing time-varying response functions (for example,
$\varepsilon(\omega,\omega^{\prime})$ for a strongly driven semiconductor).
Once the optical response of a strongly driven material is appropriately
characterized, it can be incorporated into either classical or quantum
descriptions of electromagnetic phenomena.
In the classical domain, time-varying materials can lead to the propagation of
new types of excitations in materials, and new mechanisms for gain. It is well
known that the propagation of waves in a medium with a simple periodic time-
varying permittivity can in theory lead to PTCs with momentum bandgaps. In
more complex settings where dispersion is important, strong temporal driving
may lead to the generation of new “floquet polaritons” in either bulk or
structured media. Such floquet polaritons on 2D materials could open up a new
set of directions for the broad field of polaritonics.
In the quantum domain, the appropriate use of response functions to describe
time-varying materials may enable a general description of quantum light-
matter interactions in time-varying materials. This can eventually lead to an
accurate quantum picture of “photonic quasiparticles” rivera2020light in
time-varying materials which may interact with matter. The study of
fundamental quantum light-matter interaction processes in time-driven
materials is the first step toward answering questions about how they can be
used to construct new devices such as amplifiers or lasers with new output
properties, or at frequencies which have been historically difficult to
achieve.
###### Acknowledgements.
We thank Prof. Ido Kaminer for useful discussions. J.S. acknowledges support a
Mathworks fellowship, as well as earlier support from a National Defense
Science and Engineering Graduate Fellowship of the Department of Defense
(F-1730184536). N.R. acknowledges the support of a Junior Fellowship from the
Harvard Society of Fellows, as well as earlier support from a Computational
Science Graduate Fellowship of the Department of Energy (DE-FG02-97ER25308),
and a Dean’s Fellowship from the MIT School of Science. This material is based
on work supported in part by the Defense Advanced Research Projects Agency
(DARPA) under agreement no. HR00112090081. This material is based upon work
supported in part by the Air Force Office of Scientific Research under the
award number FA9550-20-1-0115; the work is also supported in part by the U. S.
Army Research Office through the Institute for Soldier Nanotechnologies at
MIT, under Collaborative Agreement Number W911NF-18-2-0048.
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## Appendix A Derivations of general properties
### A.1 Kubo formula for Floquet systems
Here, we derive an expression for the two-frequency atomic polarizability
$\alpha(\omega,\omega^{\prime})$ of a generic single-electron system which is
varied in time. We assume that the unperturbed system is described by a
Hamiltonian $H_{0}(t)$, with the corresponding time evolution operator
$U_{0}(t)$ which satisfies $i\hbar\partial_{t}U_{0}(t)=H_{0}(t)U_{0}(t)$ We
can send an electric field probe which couples to the dipole operator $d$ of
the unperturbed system as $V(t)=-dE(t)$. We have written our expressions in
terms of a single coordinate, but all that follows can be easily generalized
to include vector directions for fields and dipole moments. Then, by linear
response theory, the change in the dipole moment is given as
$\delta\Braket{d(t)}=\int_{-\infty}^{\infty}dt^{\prime}\,\alpha(t,t^{\prime})E(t^{\prime}),$
(15)
where the Kubo formula for the two-time polarizability is given as
$\alpha(t,t^{\prime})=\frac{i}{\hbar}\theta(t-t^{\prime})\Braket{\psi_{0}}{[d_{I}(t),d_{I}(t^{\prime})]}{\psi_{0}}.$
(16)
Here, $d_{I}(t)=U_{0}^{\dagger}(t)dU_{0}(t)$ is the dipole moment operator in
the interaction picture, and $\ket{\psi_{0}}$ is the state of the system
before the probe is applied. In the frequency domain, we can write
$\delta\Braket{d(\omega)}=\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}\alpha(\omega,\omega^{\prime})E(\omega^{\prime}),$
(17)
where $\alpha(\omega,\omega^{\prime})\equiv\int dt\,dt^{\prime}\,e^{i(\omega
t-\omega^{\prime}t^{\prime})}\alpha(t,t^{\prime})$. In general,
$\alpha(\omega,\omega^{\prime})$ can be a function of the two continuous
frequencies $\omega,\omega^{\prime}$.
We may also specialize this result to systems which are time-modulated
periodically. This will allow us to make key simplifications. Namely, Floquet
theory will be used to decompose the problem into harmonics. In this case, we
assume that the time dependent Hamiltonian $H_{0}(t)$ has a period $T$, so
that $H_{0}(t+T)=H_{0}(t)$. The frequency associated with this period is
$\Omega_{0}\equiv 2\pi/T$. In this case, solutions to the time-dependent
Schrodinger equation
$i\hbar\partial_{t}\ket{\psi_{\alpha}(t)}=H_{0}(t)\ket{\psi_{\alpha}(t)}$ can
be written in terms of Floquet states
$\ket{\psi_{\alpha}(t)}=e^{-i\epsilon_{\alpha}t}\ket{\phi_{\alpha}(t)}$ (18)
where $\epsilon_{\alpha}$ is the Floquet quasi-energy which lies in the
Brillouin zone, and $\ket{\phi_{\alpha}(t)}$ is a periodic function (known as
a Floquet mode) which can be decomposed in terms of harmonics as
$\ket{\phi_{\alpha}(t)}=\sum_{n}e^{in\Omega_{0}t}\ket{\phi_{\alpha}^{n}}$
By assuming periodicity, we can write a form for $\alpha(t,t^{\prime})$ in
terms of the Floquet modes. Inserting a complete set of Floquet states
$\ket{\psi_{\alpha}(0)}$, we can write
$\displaystyle\Braket{\psi_{0}}{d_{I}(t)d_{I}(t^{\prime})}{\psi_{0}}$
$\displaystyle=\sum_{\alpha}\Braket{\psi_{0}}{U_{0}^{\dagger}(t)dU_{0}(t)}{\psi_{\alpha}}\Braket{\psi_{\alpha}}{U_{0}^{\dagger}(t^{\prime})dU_{0}(t^{\prime})}{\psi_{0}}$
(19)
$\displaystyle=\sum_{\alpha}\Braket{\psi_{0}(t)}{d}{\psi_{\alpha}(t)}\Braket{\psi_{\alpha}(t^{\prime})}{d}{\psi_{0}(t^{\prime})}$
(20)
Substituting these states into the above expression for dipole expectation
value, we find
$\Braket{\psi_{0}}{d_{I}(t)d_{I}(t^{\prime})}{\psi_{0}}=\sum_{\alpha}\sum_{k,l,m,n}e^{-i(\epsilon_{\alpha}-\epsilon_{0})t+i(n-k)\Omega_{0}t}e^{i(\epsilon_{\alpha}-\epsilon_{0})t^{\prime}-i(m-l)\Omega_{0}t^{\prime}}\Braket{\phi_{0}^{k}}{d}{\phi_{\alpha}^{n}}\Braket{\phi_{\alpha}^{m}}{d}{\phi_{0}^{l}}$
(21)
Next we use
$\theta(t-t^{\prime})=i\int\frac{d\omega^{\prime\prime}}{2\pi}\frac{e^{-i\omega^{\prime\prime}(t-t^{\prime})}}{\omega^{\prime\prime}+i\eta}$
to write
$\begin{split}\frac{i}{\hbar}\int dt\,dt^{\prime}\,e^{i\omega
t-i\omega^{\prime}t^{\prime}}\theta(t-t^{\prime})&\Braket{\psi_{0}}{d_{I}(t)d_{I}(t^{\prime})}{\psi_{0}}=-\frac{1}{\hbar}\int\frac{d\omega^{\prime\prime}}{2\pi}\frac{1}{\omega^{\prime\prime}+i\eta}\int
dt\,dt^{\prime}\,e^{i\omega
t-i\omega^{\prime}t^{\prime}}e^{-i\omega^{\prime\prime}(t-t^{\prime})}\\\
&\times\left(\sum_{\alpha}\sum_{k,l,m,n}e^{-i(\epsilon_{\alpha}-\epsilon_{0})t+i(n-k)\Omega_{0}t}e^{i(\epsilon_{\alpha}-\epsilon_{0})t^{\prime}-i(m-l)\Omega_{0}t^{\prime}}\Braket{\phi_{0}^{k}}{d}{\phi_{\alpha}^{n}}\Braket{\phi_{\alpha}^{m}}{d}{\phi_{0}^{l}}\right)\end{split}$
(22)
After simplifying this expression, one obtains
$\begin{split}\alpha(\omega,\omega^{\prime})=-\frac{2\pi}{\hbar}\sum_{\alpha}\sum_{k,l,m,n}&\left(\frac{\Braket{\phi_{0}^{k}}{d}{\phi_{\alpha}^{n}}\Braket{\phi_{\alpha}^{m}}{d}{\phi_{0}^{l}}}{\omega+(n-k)\Omega_{0}-(\epsilon_{\alpha}-\epsilon_{0})+i\eta}-\frac{\Braket{\phi_{0}^{k}}{d}{\phi_{\alpha}^{n}}\Braket{\phi_{\alpha}^{m}}{d}{\phi_{0}^{l}}}{\omega-(m-l)\Omega_{0}+(\epsilon_{\alpha}-\epsilon_{0})+i\eta}\right)\\\
&\hskip
56.9055pt\times\delta\left[\omega-\omega^{\prime}-(m+k-n-l)\Omega_{0}\right]\end{split}$
(23)
We immediately note that $\alpha(\omega,\omega^{\prime})$ takes the form of a
sum over delta functions of the form
$\delta(\omega-\omega^{\prime}-k\Omega_{0})$, where $k$ is an integer. In
other words, an incident field at frequency $\omega^{\prime}$ can induce a
dipole moment at frequencies $\omega^{\prime}+k\Omega_{0}$. This property
emerges solely from assuming that the system is periodic. As such, for
periodic systems, we may replace $\alpha(\omega,\omega^{\prime})$ with a
series of single-argument functions $\alpha_{k}(\omega)$ defined by
$\alpha(\omega,\omega^{\prime})=\sum_{k}\alpha_{k}(\omega)2\pi\delta(\omega-\omega^{\prime}-k\Omega_{0}).$
(24)
### A.2 Derivation of generalized K.K. Relations
In this section, we derive Kramers Kronig (K.K.) relations for linear response
functions $\chi(\omega,\omega^{\prime})$ which are not time-translation
invariant. The conventional proof of Kramer’s Kronig relations uses complex
analysis to show that optical passivity of a linear response function
$\chi(\omega)$ implies to the relation
$\chi(\omega)=\frac{i}{\pi}\mathcal{P}\int
d\omega^{\prime\prime}\frac{\chi(\omega^{\prime\prime})}{\omega-\omega^{\prime\prime}},$
(25)
where $\mathcal{P}$ denotes the principle value of the integral. Then,
splitting this equation into complex components yields a direct relationship
between the real and imaginary parts of $\chi(\omega)$. In a passive system,
$\operatorname{Im}\chi(\omega)$ encodes the dissipation of the material. A
basic consideration of energy conservation requires that
$\operatorname{Im}\chi(\omega)\geq 0$ for all $\omega>0$ to ensure that inputs
are attenuated over time, rather than amplified. While the conventional K.K.
relation is usually explained in terms of optical passivity (and complex poles
in the upper half plane), Eq. 25 can also be derived as an immediate
consequence of _causality_ : the fact that a system can only respond to an
impulse after its application.
While linear response functions in time-varying systems are not necessarily
passive, they must respect causality. We will use this constraint to derive
K.K. relations for linear response functions in time-dependent systems. For a
time-dependent response function $\chi(t,t^{\prime})$, the causality
constraint can be expressed in terms of the heaviside function $\theta(t)$ as
$\chi(t,t^{\prime})=\theta(t-t^{\prime})\chi(t,t^{\prime}).$ (26)
To derive the K.K. relation, we take the two-time Fourier transform of both
sides, using the convention $\chi(\omega,\omega^{\prime})=\int
dt\,dt^{\prime}\,e^{i(\omega t-\omega^{\prime}t^{\prime})}\chi(t,t^{\prime})$.
This allows us to write
$\chi(\omega,\omega^{\prime})=\int dt\,dt^{\prime}\,e^{i(\omega
t-\omega^{\prime}t^{\prime})}\theta(t-t^{\prime})\chi(t,t^{\prime}).$ (27)
The right hand side can be evaluated using the convolution theorem. To perform
this, we note that the double Fourier transform of the heaviside function is
given by
$\theta(\omega,\omega^{\prime})=\int dt\,dt^{\prime}\,e^{i(\omega
t-\omega^{\prime}t^{\prime})}\theta(t-t^{\prime})=2\pi\delta(\omega-\omega^{\prime})\left[2\pi\delta(\omega+\omega^{\prime})+\frac{2i}{\omega+\omega^{\prime}}\right].$
(28)
The convolution integral thus gives
$\begin{split}\int dt\,dt^{\prime}\,e^{i(\omega
t-\omega^{\prime}t^{\prime})}\theta(t-t^{\prime})\chi(t,t^{\prime})&=\int\frac{d\nu\,d\nu^{\prime}}{(2\pi)^{2}}\chi(\omega-\nu,\omega^{\prime}-\nu^{\prime})\theta(\nu,\nu^{\prime})\\\
&=\frac{1}{2\pi}\int
d\nu\chi(\omega-\nu,\omega^{\prime}-\nu)\left[2\pi(\nu+\nu^{\prime})+\frac{i}{\nu}\right]\\\
&=\frac{1}{2}\chi(\omega,\omega^{\prime})+\frac{i}{2\pi}\int
d\nu\,\frac{\chi(\omega-\nu,\omega^{\prime}-\nu)}{\nu}\end{split}$ (29)
Plugging this back into Eq. 27, we can solve for
$\chi(\omega,\omega^{\prime})$ to give the K.K. relation
$\chi(\omega,\omega^{\prime})=\frac{i}{\pi}\mathcal{P}\int
d\omega^{\prime\prime}\frac{\chi(\omega-\omega^{\prime\prime},\omega^{\prime}-\omega^{\prime\prime})}{\omega^{\prime\prime}}.$
(30)
To see how this general relation relates to the usual time-translation-
invariant case, we take
$\chi(\omega,\omega^{\prime})=2\pi\delta(\omega-\omega^{\prime})\chi(\omega)$.
Substituting this form into the new relation gives
$\chi(\omega)=\frac{i}{\pi}\int
d\omega^{\prime\prime}\frac{\chi(\omega-\omega^{\prime\prime})}{\omega^{\prime\prime}}$,
which after changing the integration variable matches the usual form noted in
Eq. 25. The new K.K. given in Eq. 30 importantly indicates that even without
passivity, causality still requires a strict relation between the real and
imaginary parts of $\chi(\omega,\omega^{\prime}$. Specifically, these are:
$\displaystyle\operatorname{Re}\chi(\omega,\omega^{\prime})$
$\displaystyle=-\frac{1}{\pi}\mathcal{P}\int
d\omega^{\prime\prime}\frac{\operatorname{Im}\chi(\omega-\omega^{\prime\prime},\omega^{\prime}-\omega^{\prime\prime})}{\omega^{\prime\prime}}$
(31) $\displaystyle\operatorname{Im}\chi(\omega,\omega^{\prime})$
$\displaystyle=\frac{1}{\pi}\mathcal{P}\int
d\omega^{\prime\prime}\frac{\operatorname{Re}\chi(\omega-\omega^{\prime\prime},\omega^{\prime}-\omega^{\prime\prime})}{\omega^{\prime\prime}}.$
(32)
_Specialization to the Floquet case:_ While the K.K. relation Eq. 30 is valid
for any linear time-varying system, we can make simplifications in the case
where the time-varying system is periodic with frequency $\Omega_{0}$. In this
case, we have seen that the response function necessarily takes the form of a
sum over harmonics
$\chi(\omega,\omega^{\prime})=\sum_{k}\chi_{k}(\omega)2\pi\delta(\omega-\omega^{\prime}-k\Omega_{0})$.
Plugging this assumption into Eq. 30 shows that the harmonics behave
independently from one another. Specifically, each harmonic component
$\chi_{k}(\omega)$ actually satisfies the time-invariant K.K. relation
$\chi_{k}(\omega)=\frac{i}{\pi}\mathcal{P}\int
d\omega^{\prime\prime}\frac{\chi_{k}(\omega^{\prime\prime})}{\omega-\omega^{\prime\prime}}.$
(33)
## Appendix B Two level system derivations
In this section, we provide details of the derivation of the polarizability
for the time-modulated two-level system (2LS) discussed in the main text.
There, we assumed that the Hamiltonian of the driven two-level system takes
the form
$H_{0}(t)=\frac{\sigma_{z}}{2}\left(\omega_{0}+\delta\omega\cos\Omega_{0}t\right).$
(34)
Our goal here is to compute the atomic polarizability, which in time domain is
given via the Kubo formalism as
$\alpha(t,t^{\prime})=\frac{i}{\hbar}\theta(t-t^{\prime})\braket{\psi_{0}}{[d_{I}(t),d_{I}(t^{\prime})]}{\psi_{0}},$
(35)
where $d_{I}(t)$ is the dipole moment operator in the interaction picture of
the time dependent Hamiltonian in absence of the probe field. Computing this
dipole operator is done with the aid of the unitary time evolution operator
$U_{0}(t)$ which corresponds to $H_{0}(t)$. This Hamiltonian commutes with
itself at all times, which means that the unitary evolution operator can be
evaluated directly without any concerns of time ordering as
$U_{0}(t)=\exp\left[-i\int^{t}dt^{\prime}\,H_{0}(t^{\prime})\right]=e^{-i\omega_{0}\sigma_{z}t/2}e^{-i(\delta\omega/2\Omega_{0})\sigma_{z}\sin\Omega_{0}t}$
(36)
We note that the second can be expanded as a Floquet series using the Jacobi-
Anger expansion $e^{iz\sin\theta}=\sum_{n}J_{n}(z)e^{in\theta}$. This means
that we can write expressions for the time-dependent Floquet states
$\displaystyle\ket{g(t)}$
$\displaystyle=e^{-i\omega_{0}t/2}\sum_{m}e^{im\Omega_{0}t}J_{m}\left(-\frac{\delta\omega}{2\Omega_{0}}\right)\ket{g}$
(37) $\displaystyle\ket{e(t)}$
$\displaystyle=e^{i\omega_{0}t/2}\sum_{m}e^{im\Omega_{0}t}J_{m}\left(\frac{\delta\omega}{2\Omega_{0}}\right)\ket{e}$
(38)
The states clearly evolve in the form of Eq. A4. We see that the two Floquet
states consist of the original eigenstates $\ket{g}$ and $\ket{e}$ with a
time-dependent phase attached. This is a feature of this particularly simple
example, as the Hamiltonian (Eq. 34) has only a $\sigma_{z}$ term, so the
states necessarily evolve independently. Thus in this case, the Floquet levels
can be written as $\ket{g_{m}}=J_{m}(-\delta\omega/2\Omega_{0})\ket{g}$ and
$\ket{e_{m}}=J_{m}(\delta\omega/2\Omega_{0})\ket{e}$.
### B.1 Derivation of the polarizability
Here, we calculate the interaction picture dipole operator $d_{I}(t)$ and
evaluate $\alpha(t,t^{\prime})$ directly from the commutator form of the Kubo
formula (Eq. 35). The dipole operator is proportional to
$\sigma_{x}=\sigma_{+}+\sigma_{-}$, where $\sigma_{\pm}$ are the standard
state raising and lowering operators. Using the unitary time evolution
operator (Eq. 36), we compute the interaction picture operators
$\displaystyle\sigma_{-}(t)$
$\displaystyle=\sigma_{-}\sum_{n}J_{n}e^{-i\omega_{n}t}$ (39)
$\displaystyle\sigma_{+}(t)$
$\displaystyle=\sigma_{+}\sum_{n}J_{n}e^{i\omega_{n}t},$ (40)
where $\omega_{n}\equiv\omega_{0}+n\Omega_{0}$ is the bare transition
frequency shifted by $n$ harmonics of the drive, and we have introduced the
notation $J_{n}\equiv J_{n}(\delta\omega/\Omega_{0})$ for convenience. The
relevant commutator for use in the Kubo formula is then easily computed as
$[\sigma_{x}(t),\sigma_{x}(t^{\prime})]=\sigma_{z}\sum_{m,n}J_{m}J_{n}\left(e^{-i\omega_{n}t}e^{-i\omega_{m}t}-\mathrm{c.c}\right),$
(41)
so that the time domain response function is
$\alpha(t,t^{\prime})=i\frac{d^{2}}{\hbar}\theta(t-t^{\prime})\Braket{\sigma_{z}}_{0}\sum_{m,n}\left(e^{-i\omega_{n}t}e^{-i\omega_{m}t}-\mathrm{c.c}\right).$
(42)
Here, $\Braket{\sigma_{z}}_{0}$ is the expectation value of $\sigma_{z}$ taken
in the steady state of the driven system. The exact ground and excited state
probabilities will depend on the damping environment of the time-dependent
particle, as will be discussed in the next section. Taking two Fourier
transforms as defined by Eq. 2 yields the frequency domain result
$\alpha(\omega,\omega^{\prime})=\frac{2\pi
d^{2}}{\hbar}\Braket{\sigma_{z}}_{0}\sum_{m,n}J_{m}J_{n}\left[\frac{\delta(\omega-\omega^{\prime}+(n-m)\Omega_{0})}{\omega^{\prime}+\omega_{m}+i\Gamma_{m}}-\frac{\delta(\omega-\omega^{\prime}-(n-m)\Omega_{0})}{\omega^{\prime}-\omega_{m}+i\Gamma_{m}}\right].$
(43)
As required by the periodic nature of the problem, the two-frequency
polarizability can be written as a sum over delta functions (Eq. A10). In
particular, the integer-order polarizability functions can be written as
$\alpha_{k}(\omega)=\frac{d^{2}}{\hbar}\Braket{\sigma_{z}}_{0}\sum_{n}\left[\frac{J_{n+k}J_{n}}{\omega-\omega_{n}+i\Gamma_{n}}-\frac{J_{n-k}J_{n}}{\omega+\omega_{n}+i\Gamma_{n}}\right]$
(44)
This result shows that the polarizability of our example time-driven two-level
model essentially amounts to the polarizability of many two-level systems at
frequencies spaced out from $\omega_{0}$ by integer shifts of the driving
frequency $\Omega$. This result is entirely consistent with the wisdom of
Floquet theory which states that a periodically driven quantum system should
behave in many ways like a time-independent system with many more quasi-energy
levels corresponding to harmonics. As we will detail in the next section, the
damping coefficients are given by $\Gamma_{n}=2\pi
g^{2}J_{n}^{2}\rho(\omega_{n})$, where $\rho(\omega)$ is the density of
electromagnetic states at frequency $\omega$, and $g$ is the weak coupling
associated with the undriven system which gives rise to a damping rate
$\Gamma_{0}\equiv 2\pi g^{2}\rho(\omega_{0})$.
### B.2 The effects of damping
In the section above, we defined the linear response functions
$\alpha_{k}(\omega)$ for a time-modulated two-level system. In doing so, we
introduced damping in a somewhat ad-hoc way. This approach is well-established
for static systems. In this section, we place these decay coefficients on more
rigorous footing by applying standard density matrix damping theory to the
Floquet system. We begin by assuming that the driven two-level system,
described by $H_{0}(t)$, is coupled to a continuous bath of harmonic
oscillator modes:
$H/\hbar=H_{0}(t)+\sum_{k}\nu_{k}b_{k}^{\dagger}b_{k}+\sum_{k}g_{k}\sigma_{x}(b_{k}+b_{k}^{\dagger}).$
(45)
Here, $b_{k}$ is the annihilation operator for the bath mode of frequency
$\nu_{k}$, which is coupled to the dipole of the driven 2LS with a coefficient
$g_{k}$. Furthermore, we assume that the bath modes are in thermal equilibrium
such that the reservoir average is
$\Braket{b_{k}^{\dagger}b_{k^{\prime}}}=n_{\text{th}}\delta_{kk^{\prime}}$,
where $n_{\text{th}}$ is the photon number of the bath at thermal equilibrium.
Following the usual sequence of manipulations scully1999quantum , one can find
an equation for the reduced density matrix (corresponding to the 2LS only),
which is given as:
$\dot{\rho}_{S}(t)=-\mathrm{Tr}_{R}\int_{0}^{t}dt^{\prime}\,\left[V(t),[V(t^{\prime}),\rho_{S}(t^{\prime})\otimes\rho_{R}(0)]\right].$
(46)
Here, $V(t)$ is the interaction picture Hamiltonian obtained by transforming
the dipole-bath interaction term according to the unitary evolution operator
of the non-interaction systems. Note that we have neglected a term which is
linear in the bath operators, since such a term will vanish when reservoir
expectation values are taken. Additionally, we have made the usual assumption
that the density matrix of the reservoir is unchanged by weak interactions
with the system, so that the quantity $\rho_{R}(0)$ can be used at all times.
For the specific 2LS model that we consider, the interaction picture
Hamiltonian is found to be:
$V(t)=\sum_{k}g_{k}\sum_{n}J_{n}\left(\sigma_{+}e^{i\omega_{n}t}+\sigma_{-}e^{-i\omega_{n}t}\right)\left(b_{k}e^{-i\nu_{k}t}+b_{k}^{\dagger}e^{i\nu_{k}t}\right)$
(47)
We can now implement a type of rotating wave approximation (RWA) for each of
the transitions between Floquet levels. For a two level system as considered
here, these transitions can be divided into two types, which we treat
separately:
1. 1.
Transitions in which the excited state relaxes to the ground state by emission
of a photon at frequency $\omega_{n}=\nu_{k}$. In this case, we have the
interaction Hamiltonian
$V_{n}(t)=J_{n}\sum_{k}g_{k}\left(\sigma_{-}b_{k}^{\dagger}e^{-i(\omega_{n}-\nu_{k})t}+\sigma_{+}b_{k}e^{i(\omega_{n}-\nu_{k})t}\right).$
(48)
By using this Hamiltonian in Eq. 46, we find
$\dot{\rho}(t)=-n_{\text{th}}\frac{\Gamma_{n}}{2}\left(\rho\sigma_{-}\sigma_{+}-2\sigma_{+}\rho\sigma_{-}+\sigma_{-}\sigma_{+}\rho\right)-(n_{\text{th}}+1)\frac{\Gamma_{n}}{2}(\sigma_{+}\sigma_{-}\rho-2\sigma_{-}\rho\sigma_{+}+\rho\sigma_{+}\sigma_{-}),$
(49)
where $\Gamma_{n}\equiv 2\pi g(\omega_{n})^{2}\rho(\omega_{n})$ is a modified
relaxation rate defined in terms of the coupling coefficient $g$ and density
of states $\rho$, both evaluated at the shifted frequency $\omega_{n}$. For
excited to ground processes, a total decay rate can be defined as a sum of
decay rates for all harmonics which allow emission (i.e. which satisfy
$\omega_{n}>0$). Specifically, this is
$\Gamma_{e}\equiv\sum_{\omega_{n}>0}\Gamma_{n}$ (50)
We also note that the decay rates $\Gamma_{n}$ referenced here are identical
to those that would be calculated by use of the Floquet Fermi Golden rule for
transitions between Floquet states rudner2020floquet .
2. 2.
Transitions in which the ground state relaxes to the excited state by emission
of a photon at $-\omega_{m}=\nu_{k}$. In the absence of time dependence, such
a process cannot be energy conserving. However, harmonic shifts permit the
existence of integers $m<0$ such that $-\omega_{m}=-\omega_{0}-m\Omega>0$.
Under the RWA, these transitions are accounted for by Hamiltonian terms which
are usually discarded as counter-rotating terms:
$V_{m}(t)=J_{m}\sum_{k}g_{k}\left(\sigma_{-}b_{k}e^{-i(\omega_{m}+\nu_{k})t}+\sigma_{+}b_{k}^{\dagger}e^{i(\omega_{n}+\nu_{k})t}\right)$
(51)
Following a similar procedure, we find
$\dot{\rho}(t)=-n_{\text{th}}\frac{\Gamma_{m}}{2}(\sigma_{+}\sigma_{-}\rho-2\sigma_{-}\rho\sigma_{+}+\rho\sigma_{+}\sigma_{-})-(n_{\text{th}}+1)\frac{\Gamma_{m}}{2}\left(\rho\sigma_{-}\sigma_{+}-2\sigma_{+}\rho\sigma_{-}+\sigma_{-}\sigma_{+}\rho\right),$
(52)
The only difference between these Lindblad terms and the ones in Eq. B16 is
that the $n_{\text{th}}$ and $n_{\text{th}}$ terms are switched. Also
similarly to case (1), we can define a total rate of ground state decay
$\Gamma_{g}\equiv\sum_{\omega_{m}<0}\Gamma_{m}$ (53)
By taking the above manipulations to hold for all relevant values of $m$ and
$n$, a Lindblad operator which incorporates all decays of the system can be
constructed:
$\begin{split}\dot{\rho}(t)=&-n_{\text{th}}\frac{\Gamma_{e}}{2}\left(\rho\sigma_{-}\sigma_{+}-2\sigma_{+}\rho\sigma_{-}+\sigma_{-}\sigma_{+}\rho\right)\\\
&-(n_{\text{th}}+1)\frac{\Gamma_{e}}{2}(\sigma_{+}\sigma_{-}\rho-2\sigma_{-}\rho\sigma_{+}+\rho\sigma_{+}\sigma_{-})\\\
&-n_{\text{th}}\frac{\Gamma_{g}}{2}(\sigma_{+}\sigma_{-}\rho-2\sigma_{-}\rho\sigma_{+}+\rho\sigma_{+}\sigma_{-})\\\
&-(n_{\text{th}}+1)\frac{\Gamma_{g}}{2}\left(\rho\sigma_{-}\sigma_{+}-2\sigma_{+}\rho\sigma_{-}+\sigma_{-}\sigma_{+}\rho\right)\end{split}$
(54)
Assuming no thermal background contribution ($n_{\text{th}}=0$), the equations
of motion for the diagonal density matrix take a particularly simple form
which enable us to calculate the steady state populations of the driven
system:
$\displaystyle\dot{\rho}_{gg}$
$\displaystyle=-\Gamma_{g}\rho_{gg}+\Gamma_{e}\rho_{ee}$ (55)
$\displaystyle\dot{\rho}_{ee}$
$\displaystyle=-\Gamma_{e}\rho_{ee}+\Gamma_{g}\rho_{gg}$ (56)
By setting derivatives to zero, the population equation is easily solved:
$\rho_{gg}=\Gamma_{e}/(\Gamma_{e}+\Gamma_{g})$,
$\rho_{ee}=\Gamma_{g}/(\Gamma_{e}+\Gamma_{g})$. Thus the total inversion level
referenced in the linear response coefficients is
$\Braket{\sigma_{z}}_{0}=\rho_{ee}-\rho_{gg}=(\Gamma_{g}-\Gamma_{e})/(\Gamma_{g}+\Gamma_{e})$.
_Additional remarks on damping:_ The above give an outline of how damping can
be accounted for in Floquet systems, which a special emphasis on the
implications for their optical response. We have focused here on a two-level
model which is $\sigma_{z}$-modulated for simplicity. That said, the concepts
explained here should apply more generally to multi-level systems which are
modulated in more complex ways, such as the types of systems explored in
electromagnetically induced transparency (EIT). It is also worth noting that
issues of degeneracies (or quasi-degeneracies) can introduce substantial
complications into the analysis of Floquet systems. Such cases should be
treated carefully, with guidance from works on this topic such as
hone2009statistical .
## Appendix C Lorentz parametric oscillator derivations
### C.1 Semiclassical approach
The Lorentz oscillator is an important and commonly used model for dispersive
dielectric functions, for example in the presence of resonances that result
from optical phonons in polar insulators. In this section, we outline the
semiclassical approach taken to derive the optical response of a parametric
oscillator. To do so, we can write the equation of motion of a forced harmonic
oscillator with a resonance frequency which is perturbed by some amount
$f(t)$:
$\ddot{x}(t)+\Gamma\dot{x}(t)+\omega_{0}^{2}(1+f(t))x(t)=F(t).$ (57)
In the absence of any parametric driving, this equation of course has the
Fourier domain solution
$x(\omega)=F(\omega)/(\omega_{0}^{2}-\omega^{2}-i\omega\Gamma).$
To proceed in the parametric case, we can solve for the Green’s function. The
time domain Green’s function $K(t,t^{\prime})$ is defined to satisfy:
$\left[\partial_{t}^{2}+\Gamma\partial_{t}+\omega_{0}^{2}(1+f(t))\right]K(t,t^{\prime})=\delta(t-t^{\prime}).$
(58)
Rather than solving for $K(t,t^{\prime})$ in the time domain and the
transforming into frequency space, we will find it more useful to transform
into frequency space and then consider perturbative solutions for
$K(\omega,\omega^{\prime})$ directly. Using the Fourier transform convention
$f(\omega,\omega^{\prime})=\int dt\,dt^{\prime}\,e^{i\omega
t}f(t,t^{\prime})e^{-i\omega^{\prime}t^{\prime}}$, we find that:
$\left[\omega_{0}^{2}-\omega^{2}-i\omega\Gamma\right]K(\omega,\omega^{\prime})+\omega_{0}^{2}\int\frac{d\omega^{\prime\prime}}{2\pi}\tilde{f}(\omega-\omega^{\prime\prime})K(\omega^{\prime\prime},\omega^{\prime})=2\pi\delta(\omega-\omega^{\prime}).$
(59)
By solving perturbatively, we obtain the correction:
$K(\omega,\omega^{\prime})=\frac{2\pi\delta(\omega-\omega^{\prime})}{\omega_{0}^{2}-\omega^{2}-i\omega\Gamma}-\frac{\omega_{0}^{2}\tilde{f}(\omega-\omega^{\prime})}{(\omega_{0}^{2}-\omega^{2}-i\omega\Gamma)(\omega_{0}^{2}-\omega^{\prime
2}-i\omega^{\prime}\Gamma)}+\mathcal{O}(f^{3}).$ (60)
For our purposes, this form is suitable. Evaluation in the time domain for
parametric oscillator:
A particularly important case of this is the case of parametric resonance in
which the driving frequency is twice the resonance frequency. To aid this, we
define
$\chi_{\pm}^{(1)}(t,t^{\prime})=-\delta
f\omega_{0}^{2}\int\frac{d\omega\,d\omega^{\prime}}{(2\pi)^{2}}e^{-i\omega
t}\frac{\pi\delta(\omega-\omega^{\prime}\pm\Omega)}{(\omega_{0}^{2}-\omega^{2}-i\omega\Gamma)(\omega_{0}^{2}-\omega^{\prime
2}-i\omega^{\prime}\Gamma)}e^{i\omega^{\prime}t^{\prime}}.$ (61)
We begin by completing the $\omega^{\prime}$ delta function integral which
sets $\omega^{\prime}\to\omega\pm\Omega$. This gives
$\displaystyle\chi_{\pm}^{(1)}(t,t^{\prime})$
$\displaystyle=-\frac{1}{2}\delta f\omega_{0}^{2}e^{\pm i\Omega
t^{\prime}}\int\frac{d\omega}{2\pi}e^{-i\omega(t-t^{\prime})}\frac{1}{(\omega_{0}^{2}-\omega^{2}-i\omega\Gamma)}\frac{1}{(\omega_{0}^{2}-(\omega\pm\Omega)^{2}-i(\omega\pm\Omega)\Gamma)}$
(62) $\displaystyle=-\frac{1}{2}\delta f\omega_{0}^{2}e^{\pm i\Omega
t^{\prime}}\int d\tau\,e^{\pm
i\Omega\tau}\chi^{(0)}(t-t^{\prime}-\tau)\chi^{(0)}(\tau)$ (63)
$\displaystyle=-\frac{\delta f\omega_{0}^{2}e^{\pm i\Omega
t^{\prime}}e^{-\frac{\Gamma}{2}(t-t^{\prime})}}{2\omega_{r}^{2}}\int_{0}^{\infty}d\tau\,\theta(t-t^{\prime}-\tau)e^{\pm
i\Omega\tau}\sin[\omega_{r}(t-t^{\prime}-\tau)]\sin[\omega_{r}\tau],$ (64)
where we have used $\omega_{r}=\frac{1}{2}\sqrt{4\omega_{0}^{2}-\Gamma^{2}}$,
and taken advantage of convolutions, shifting frequency arguments, and the
unperturbed harmonic oscillator response function $\chi^{(0)}(t-t^{\prime})$.
Performing the integration, and then adding the positive and negative
frequency contribution as
$\chi^{(1)}(t,t^{\prime})=\chi_{+}^{(1)}(t,t^{\prime})+\chi_{-}^{(1)}(t,t^{\prime})$
gives the final result
$\displaystyle\chi^{(1)}(t,t^{\prime})$ $\displaystyle=\frac{2\delta
f\omega_{0}^{2}}{\Omega\omega_{r}\left(4\omega_{r}^{2}-\Omega^{2}\right)}\theta(t-t^{\prime})e^{-\frac{1}{2}\Gamma(t-t^{\prime})}\cos\left(\frac{\Omega}{2}(t+t^{\prime})\right)$
(65)
$\displaystyle\times\left[2\omega_{r}\sin\left(\frac{\Omega}{2}(t-t^{\prime})\right)\cos\left(\omega_{r}(t-t^{\prime})\right)-\Omega\cos\left(\frac{\Omega}{2}(t-t^{\prime})\right)\sin\left(\omega_{r}(t-t^{\prime})\right)\right].$
(66)
The expression has several encouraging features. First, the expression is
manifestly causal with the $\theta(t-t^{\prime})$ dependence. Secondly, all
terms have an explicit $(t-t^{\prime})$ dependence, _except for the
$\cos(\Omega(t+t^{\prime})/2)$ term_, which comes explicitly from the broken
time translation invariance of the system. Additionally, the expression
contains explicit exponential decay set by the loss parameter $\Gamma$, just
as $\chi^{(0)}(t-t^{\prime})$ does.
### C.2 Quantum mechanical approach
We now provide a quantum derivation of the Lorentz parametric oscillator
susceptibility. We do this by calculating the dipole susceptibility
$\alpha(t,t^{\prime})$ for a quantum mechanical parametric oscillator, and
showing that it matches the semiclassical approach at first order in
perturbation theory. We start with a Hamiltonian of a one dimensional quantum
harmonic oscillator with a resonant frequency $\omega_{0}$ which oscillates at
frequency $\Omega$.
$H(t)=\frac{p^{2}}{2m}+\frac{1}{2}m\omega_{0}^{2}\left(1+\delta\omega\cos\Omega
t\right)x^{2}.$ (67)
In the basis of the unperturbed Harmonic oscillator, we can write this in
terms of creation and annihilation operators as
$H(t)=\hbar\omega_{0}a^{\dagger}a+\frac{\hbar\omega_{0}}{4}\delta\epsilon\cos\Omega
t\left(a+a^{\dagger}\right)^{2}.$ (68)
The Floquet states, to first order in $\delta\epsilon$, are given as
$\ket{\psi_{n}^{(1)}(t)}=e^{-i\omega_{n}t}\left[(1+S_{n})\ket{n}+P_{n-1}\ket{n-2}+-P_{n+1}^{*}\ket{n+2}\right]$
(69)
where we defined $S_{n}(t)\equiv-i\delta\epsilon(2n+1)\eta_{0}(t)$ and
$P_{n}(t)\equiv-i\delta\epsilon\sqrt{n(n+1)}\eta_{1}(t)$, where
$\eta_{0}(t)=\frac{\omega_{0}\sin\Omega t}{4\Omega},\hskip
28.45274pt\eta_{1}(t)=\frac{\omega_{0}}{4}\frac{\Omega\sin\Omega
t-2i\omega_{0}\cos\Omega t}{\Omega^{2}-4\omega_{0}^{2}}$ (70)
Using this, we can write the unitary time evolution order to first order as
$U(t)=\sum_{n}e^{-i\omega_{n}t}\left[(1+S_{n})\ket{n}\bra{n}+P_{n-1}\ket{n-2}\bra{n}-P_{n+1}^{*}\ket{n+2}\bra{n}\right].$
(71)
Then we need the dipole moment $d(a+a^{\dagger})$ in the interaction picture
(i.e. transformed by $U(t)$). Doing this, and discarding terms greater than
first order in $\delta\epsilon$, we find
$U^{\dagger}(t)(a+a^{\dagger})U(t)=\sum_{n}\sum_{k=\pm 1,\pm
3}e^{-i(\omega_{n}-\omega_{n+k})t}A_{k}\ket{n+k}\bra{n}$ (72)
where
$\displaystyle A_{1}$
$\displaystyle=\left(1+S_{n+1}^{*}\right)\left[\sqrt{n+1}\left(1+S_{n}\right)-\sqrt{n+2}P_{n+1}^{*}\right]+P_{n}^{*}\left[\sqrt{n}\left(1+S_{n}\right)+\sqrt{n-1}P_{n-1}\right]$
(73) $\displaystyle A_{-1}$
$\displaystyle=\left(1+S_{n-1}^{*}\right)\left[\sqrt{n}\left(1+S_{n}\right)+\sqrt{n-1}P_{n-1}\right]-P_{n}\left[\sqrt{n+1}\left(1+S_{n}\right)-\sqrt{n+2}P_{n+1}^{*}\right]$
(74) $\displaystyle A_{3}$
$\displaystyle=-\sqrt{n+3}P_{n+1}^{*}+P_{n+2}^{*}\left[\sqrt{n+1}(1+S_{n})-\sqrt{n+2}P_{n+1}^{*}\right]$
(75) $\displaystyle A_{-3}$
$\displaystyle=\sqrt{n-2}P_{n-1}-P_{n-2}\left[\sqrt{n}(1+S_{n})+\sqrt{n-1}P_{n-1}\right]$
(76)
Then we find that
$\Braket{0}{d_{I}(t)d_{I}(t^{\prime})}{0}=d^{2}e^{-i\omega_{0}(t-t^{\prime})}\left(1+S_{0}^{*}(t)+S_{1}(t)-\sqrt{2}P_{1}(t)\right)\left(1+S_{0}(t^{\prime})+S_{1}^{*}(t^{\prime})-\sqrt{2}P_{1}^{*}(t^{\prime})\right)$
(77)
Several of these terms are second order in $\delta\epsilon$ and can be
discarded. Doing this, the commutator for linear response is
$\Braket{0}{\left[d_{I}(t,)d_{I}(t^{\prime})\right]}{0}=\left(e^{-i\omega_{0}(t-t^{\prime})}-\text{c.c}\right)-2i\delta\epsilon
d^{2}e^{-i\omega_{0}(t-t^{\prime})}\left[\eta_{0}(t)-\eta_{0}(t^{\prime})-\left(\eta_{1}(t)-\eta_{1}^{*}(t^{\prime})\right)\right]-\text{c.c}$
(78)
Then the polarizability to first order can be expressed as
$\alpha(t,t^{\prime})=\alpha^{(0)}(t-t^{\prime})+\alpha^{(1)}(t,t^{\prime})+\mathcal{O}(\delta\epsilon^{2})$
(79)
where
$\alpha^{(0)}(t-t^{\prime})=\frac{2d^{2}}{\hbar}\theta(t-t^{\prime})\sin\left(\omega_{0}(t-t^{\prime})\right)$
(80)
and
$\begin{split}\alpha^{(1)}(t,t^{\prime})&=-\frac{4\delta\omega
d^{2}\omega_{0}^{2}}{\hbar\Omega\left(\Omega^{2}-4\omega_{0}^{2}\right)}\theta(t-t^{\prime})\cos\left(\frac{\Omega}{2}(t+t^{\prime})\right)\\\
&\times\left[2\omega_{0}\sin\left(\frac{\Omega}{2}(t-t^{\prime})\right)\cos\left(\omega_{0}(t-t^{\prime})\right)-\Omega\cos\left(\frac{\Omega}{2}(t-t^{\prime})\right)\sin\left(\Omega_{0}(t-t^{\prime})\right)\right]\end{split}$
(81)
In the limit of no dissipation, this is equivalent to the Lorentz parametric
oscillator model that was derived semiclassically.
## Appendix D Electrodynamics with time-varying materials
### D.1 Maxwell’s Equations in Time Dependent Materials
In this section, we will formulate Maxwell’s equations in media which are
time-periodic and dispersive. By making the assumption of a spatially
homogeneous medium, we will be able to cast Maxwell’s equations into a Floquet
eigenvalue problem for the wavevectors and quasifrequencies which can
propagate. By solving this problem numerically, we can compute band
structures. In the absence of external sources, the Maxwell equation for the
electric field can be written as
$\nabla\times\nabla\times\mathbf{E}(\mathbf{r},\omega)-\frac{\omega^{2}}{c^{2}}\int\frac{d\omega^{\prime}}{2\pi}\varepsilon(\mathbf{r},\omega,\omega^{\prime})\mathbf{E}(\mathbf{r},\omega^{\prime})=0.$
(82)
If we write epsilon as time-independent background
$\varepsilon_{\text{bg}}(\omega)$ plus a perturbation
$\Delta\chi(\omega,\omega^{\prime})$ which results from a time dependence, we
can write
$\varepsilon(\mathbf{r},\omega,\omega^{\prime})=\varepsilon_{\text{bg}}(\mathbf{r},\omega)[2\pi\delta(\omega-\omega^{\prime})]+\Delta\chi(\mathbf{r},\omega,\omega^{\prime})$.
Substituting this into the Maxwell equation, we obtain the form
$\nabla\times\nabla\times\mathbf{E}(\mathbf{r},\omega)-\frac{\omega^{2}}{c^{2}}\varepsilon_{\text{bg}}(\mathbf{r},\omega)\mathbf{E}(\mathbf{r},\omega)=\frac{\omega^{2}}{c^{2}}\int\frac{d\omega^{\prime}}{2\pi}\Delta\chi(\mathbf{r},\omega,\omega^{\prime})\mathbf{E}(\mathbf{r},\omega^{\prime}).$
(83)
If the driving is periodic, and the medium is assumed be uniform in space,
then the response function can be cast into the form
$\Delta\chi(\omega,\omega^{\prime})=\sum_{k}\Delta\chi_{k}(\omega)2\pi\delta(\omega-\omega^{\prime}-k\Omega_{0}).$
(84)
Substituting this into the Maxwell equation gives
$\nabla\times\nabla\times\mathbf{E}(\omega)-\frac{\omega^{2}}{c^{2}}\varepsilon_{\text{bg}}(\omega)\mathbf{E}(\omega)=\frac{\omega^{2}}{c^{2}}\sum_{k}\Delta\chi_{k}(\omega)\mathbf{E}(\omega-k\Omega_{0}).$
(85)
Assuming a bulk medium which can be spatially decomposed into plane waves, and
invoking the Bloch-Floquet requirement for the time-dependent portion of the
mode functions, we can write
$\mathbf{E}(\mathbf{r},t)=e^{i\mathbf{k}\cdot\mathbf{r}}\sum_{n}u_{\Omega
n}e^{-i(\Omega+n\Omega_{0})t}\implies\mathbf{E}(\mathbf{r},\omega)=e^{i\mathbf{k}\cdot\mathbf{r}}2\pi\sum_{n}u_{\Omega
n}\delta(\omega-\Omega-n\Omega_{0}).$ (86)
For convenience, we will define $\Omega_{n}\equiv\Omega+n\Omega_{0}$.
Substituting this into the Maxwell equation, integrating both sides in
$\omega$ to isolate frequency components, and relabeling sums gives the final
central equation result
$\left[k^{2}-\frac{\Omega_{n}^{2}}{c^{2}}\varepsilon_{\text{bg}}(\Omega_{n})\right]u_{n}=\frac{\Omega_{n}^{2}}{c^{2}}\sum_{m}\Delta\chi_{m}(\Omega_{n})u_{n-m}.$
(87)
Rewriting this as an eigenvalue problem for $k$ in terms of $\Omega$, we have
$\frac{\Omega_{n}^{2}}{c^{2}}\varepsilon_{\text{bg}}(\Omega_{n})u_{n}+\frac{\Omega_{n}^{2}}{c^{2}}\sum_{m}\Delta\chi_{m}(\Omega_{n})u_{n-m}=k_{\Omega}^{2}u_{n},$
(88)
which casts the dispersion as a relatively simple central equation eigenvalue
problem. Eq. 11 can be easily implemented as a matrix eigenvalue problem,
yielding the wavevector $k_{\Omega}$ and Floquet mode amplitudes $u_{\Omega
n}$ at each quasifrequency $\Omega\in[-\Omega_{0}/2,\Omega_{0}/2)$.
### D.2 Reflection and Transmission
Once the Maxwell Floquet solutions have been obtained, as described in the
previous section, they can be used to solve classical optics problems. In this
section, we show the illustrative example of a reflection-transmission problem
of a monochromatic field incident on a slab of material characterized by a
time-dependent linear response function.
We write the Floquet modes as
$u_{b,\tilde{\Omega}}(t)=\sum_{m}u_{b,\tilde{\Omega}}^{(m)}e^{-i\Omega_{0}mt},$
(89)
where $b$ is a band index, and $\sum_{m}$ is a sum over harmonics of the
driving frequency $\Omega_{0}$. Next we can consider an interface between two
materials (1) and (2), where (1) for now is air, and (2) is a material
described with this kind of formalism, and we assume we have solved for the
bands. Furthermore, we assume that the electric field is polarized in the
plane of the interface (s-polarized).
In this analysis, it will be helpful to express the incident frequency as
$\omega_{0}=\tilde{\omega}_{0}+s\Omega_{0}$, where $\tilde{\omega}_{0}$ is
confined to lie in the first Brillouin zone
$-\frac{\Omega_{0}}{2}<\tilde{\omega}_{0}<\frac{\Omega_{0}}{2}$. In this case,
$s$ is easily interpreted as a number of harmonics by which the true incoming
frequency is offset from the BZ frequency that is seen in the time-dependent
medium. Thus we can write the incident field as
$E^{\text{inc}}(x,t)=e^{ik_{0}x}e^{-i\omega_{0}t},$ (90)
where $\omega_{0}$ is the frequency of the incident plane wave, and
$k_{0}=k(\omega_{0})=\omega_{0}/c$ is the vacuum dispersion. For now, we
assume that the reflected field can have any frequency, so we can write very
generally
$E^{(\text{r})}(x,t)=\sum_{\omega>0}r(\omega)e^{-ik(\omega)x}e^{-i\omega t}.$
(91)
Finally, the transmitted field is
$\displaystyle E^{(\text{t})}(x,t)$
$\displaystyle=\sum_{b}t_{b}e^{iq_{b}(\tilde{\omega}_{0}x}e^{-i\tilde{\omega}_{0}t}u_{b,\tilde{\omega}_{0}}(t)$
(92)
$\displaystyle=\sum_{b,m}t_{b}u_{b,\tilde{\omega}_{0}}^{(m)}e^{iq_{b}(\tilde{\omega}_{0})x}e^{-i(\tilde{\omega}_{0}+m\Omega_{0})t}.$
(93)
The boundary conditions at the interface are given by
$E^{\text{inc}}(0,t)+E^{\text{r}}(x,t)=E^{\text{t}}(0,t)$ and
$\partial_{x}E^{\text{inc}}(x,t)|_{x=0}+\partial_{x}E^{\text{r}}(x,t)|_{x=0}=\partial_{x}E^{\text{t}}(x,t)|_{x=0}$.
We can write a matrix equation down for the boundary conditions. Now we have
boundary conditions at $x=\pm L/2$. The first two rows are for continuity of
the field at $\pm L/2$ respectively. The last two rows are for continuity of
the derivative. Doing this, we find the matrix equations
$\begin{pmatrix}-I*e^{ik_{0}(-L/2)}&\mathbf{u}_{b}e^{ik_{b}(\tilde{\omega}_{0})(-L/2)}&\mathbf{u}_{b}e^{-ik_{b}(\tilde{\omega}_{0})(-L/2)}&\mathbf{0}\\\
\mathbf{0}&\mathbf{u}_{b}e^{ik_{b}(\tilde{\omega}_{0})(L/2)}&\mathbf{u}_{b}e^{-ik_{b}(\tilde{\omega}_{0})(L/2)}&-I*e^{ik_{0}(L/2)}\\\
(\Omega_{m}/c)e^{i(\Omega_{m}/c)(-L/2)}&k_{b}(\tilde{\omega_{0}})\mathbf{u}_{b}e^{ik_{b}(\tilde{\omega}_{0})(-L/2)}&-k_{b}(\tilde{\omega_{0}})\mathbf{u}_{b}e^{-ik_{b}(\tilde{\omega}_{0})(-L/2)}&\mathbf{0}\\\
\mathbf{0}&k_{b}(\tilde{\omega_{0}})\mathbf{u}_{b}e^{ik_{b}(\tilde{\omega}_{0})(L/2)}&-k_{b}(\tilde{\omega_{0}})\mathbf{u}_{b}e^{-ik_{b}(\tilde{\omega}_{0})(L/2)}&(\Omega_{m}/c)e^{i(\Omega_{m}/c)(L/2)}\end{pmatrix}\begin{pmatrix}\mathbf{r}\\\
\mathbf{a}\\\ \mathbf{b}\\\
\mathbf{t}\end{pmatrix}=\begin{pmatrix}\mathbf{v}_{1}\\\ \mathbf{v}_{2}\\\
\mathbf{v}_{3}\\\ \mathbf{v}_{4}\end{pmatrix}$ (94)
Finally, we have $\mathbf{v}_{2}=\mathbf{v}_{4}=0$. Then we have
$(\mathbf{v}_{1})_{s}=e^{ik_{0}(-L/2)}$ and
$(\mathbf{v}_{3})_{s}=(\Omega_{s}/c)e^{i(\Omega_{s}/c)(-L/2)}$, where the $s$
index refers to the index that matches $s$ (so actually $M+1+s$). By
performing matrix inversion, one can find the reflected, transmitted, and
internal fields for the scattering problem which is described in Fig. 4 of the
main text.
|
# Robust boundary integral equations for the solution of elastic scattering
problems via Helmholtz decompositions
Víctor Domínguez Dep. Ingeniería Matemática e Informática, Universidad Pública
de Navarra. Campus de Tudela 31500 - Tudela, Spain, e-mail:
<EMAIL_ADDRESS>Catalin Turc Department of Mathematical
Sciences, New Jersey Institute of Technology, Univ. Heights. 323 Dr. M. L.
King Jr. Blvd, Newark, NJ 07102, USA, e-mail<EMAIL_ADDRESS>
(November 28, 2022)
###### Abstract
Helmholtz decompositions of the elastic fields open up new avenues for the
solution of linear elastic scattering problems via boundary integral equations
(BIE) [19]. The main appeal of this approach is that the ensuing systems of
BIE feature only integral operators associated with the Helmholtz equation.
However, these BIE involve non standard boundary integral operators that do
not result after the application of either the Dirichlet or the Neumann trace
to Helmholtz single and double layer potentials. Rather, the Helmholtz
decomposition approach leads to BIE formulations of elastic scattering
problems with Neumann boundary conditions that involve boundary traces of the
Hessians of Helmholtz layer potential. As a consequence, the classical
combined field approach applied in the framework of the Helmholtz
decompositions leads to BIE formulations which, although robust, are not of
the second kind. Following the regularizing methodology introduced in [3] we
design and analyze novel robust Helmholtz decomposition BIE for the solution
of elastic scattering that are of the second kind in the case of smooth
scatterers in two dimensions. We present a variety of numerical results based
on Nyström discretizations that illustrate the good performance of the second
kind regularized formulations in connections to iterative solvers.
Keywords: Time-harmonic Navier scattering problems, Helmholtz decomposition,
boundary integral equations, pseudodifferential calculus, Nyström
discretizations, preconditioners.
AMS subject classifications: 65N38, 35J05, 65T40, 65F08
## 1 Introduction
The extension of Boundary Integral Equation (BIE) based discretizations for
the numerical solution of acoustic scattering problems (i.e. Helmholtz
equations) to their elastic scattering problems (i.e. Navier equations)
counterpart is thought to be more or less straightforward. Typically, BIE for
elastic scattering problems are based on the Navier Green’s function, which
albeit more complicated than the Helmholtz Green’s function (not in the least
because it involves two wave-numbers), exhibits the same singularity type as
the latter. Therefore, discretization methods that rely on kernel independent
quadratures, that is these quadratures resolve in a black box manner the
singularities of Green’s functions together with their derivatives, can be
extended from the Helmholtz to the Navier case without much fuss. Amongst
these types of discretizations we mention the $\delta$-BEM methods [14, 12,
13, 15] and the Density Interpolation Methods [21]. On the other hand, other
discretizations strategy such as the Kussmaul-Martensen singularity splitting
Nyström methods can be extended from the Helmholtz to the Navier setting but
require cumbersome modifications [10, 17].
Recently a new BIE approach to elastic scattering problems shortcuts the need
to use Navier Green’s functions and relies entirely on Helmholtz layer
potentials [19, 28]. This approach is based on Helmholtz decompositions of
elastic waves into compressional and shear waves, a manner which reduces the
elastic scattering problem, at least in two dimensions, to the solution of two
Helmholtz equations coupled by their boundary values on the scatterers. In
three dimensions this approach leads to coupling the solution of a Helmholtz
scattering problem to that of a Maxwell scattering problem [20]. These recent
contributions consider only single layer potential representations of
Helmholtz fields and elastodynamic fields with Dirichlet boundary conditions
on the boundary of the scatterers, and, therefore, the ensuing BIE are not
robust for all frequencies. We extended the Helmholtz decomposition approach
in two dimensions to Neumann boundary conditions as well as for combined field
representations in [18]. The extension of the Helmholtz decomposition approach
to the Neumann case is nontrivial since it gives rise to non standard Boundary
Integral Operators (BIOs) which arise from applications of boundary traces to
Hessians of single and double layer Helmholtz potentials. More importantly,
the BIE Helmholtz decomposition route is a viable approach to the solution of
elastic scattering problems in as much as robust formulations are used. The
goal of this paper is to derive and analyze such robust formulations,
including some which are of the second kind in the case of smooth scatterers.
The most widely used strategy to deliver robust BIE formulations for the
solution of time harmonic scattering problems is the combined field (CFIE)
strategy [4, 6]. The analysis of such formulations relies on the Fredholm
theory and the well posedness of Robin boundary value problems in
bounded/interior domains. The analysis of the Fredholm property of the BIOs
that arise in the combined field strategy in the Helmholtz decomposition
framework for elastic scattering problems is a bit more delicate, as it
requires the use of lower order terms in the asymptotic expansion in the
pseudodifferential sense of the constitutive BIOs. The reason for this more
involved analysis is the fact that the principal symbols of these operators
(unlike the Helmholtz case) are defective, which was already remarked in [19].
However, the ensuing CFIE feature pseudodifferential operators of order one
(Dirichlet) and respectively two (Neumann) and a such are not ideal for
iterative solutions.
In order to design integral formulations of the second kind for the solution
of elastic scattering problems via the Helmholtz decomposition approach we
employ the general methodology in [3]. Using coercive approximations of
Dirichlet to Neumann (DtN) operators, we construct certain regularizing
operators that are approximations of the operators that map the boundary
conditions in the Helmholtz decomposition approach to the Helmholtz Cauchy
data on the boundary of the scatterer. The regularizing operators we consider,
whose construction relies on the pseudodifferential calculus, are based on
either square root Fourier multipliers or Helmholtz BIOs, and are
straightforward to implement in existing discretizations methodologies for
Helmholtz BIOs. More importantly, we prove that in the case of smooth
scatterers the regularized formulations are robust and of the second kind, and
we provide numerical evidence about the superior performance of these
formulations over the combined field formulations with respect to iterative
solvers. Thus, we show in this paper that it is possible to construct BIE with
good spectral properties for the solution of the elastic scattering problems
in two dimensions via Helmholtz decompositions, making this approach a viable
alternative to the recently introduced BIE based on the Navier Green’s
function [8, 9, 17, 5]. The extension of this approach to three dimensional
elastic scattering problem is currently ongoing.
The paper is organized as follows: in Section 2 we introduce the Navier
equations in two dimensions and we present the Helmholtz decomposition
approach; in Section 3 we present the trace for the gradient and Hessian of
the boundary layer operators for Helmholtz equation. We also review certain
properties of Helmholtz BIOs with asymptotic expansions of in the
pseudodifferential sense. The proof of these result is postponed to the self-
contained Appendix and presented here just for the sake of completeness; in
Section 4 we analyze the nonstandard BIOs that arise in the Helmholtz
decomposition approach for Dirichlet (subsection 4.1) and Neumann (subsection
4.2) boundary conditions. Finally, we present in Section 5 a variety of
numerical results illustrating the iterative behavior of solvers based on
Nyström discretizations of the various BIE formulations of the Helmholtz
decomposition approach in the high frequency regime.
## 2 Navier equations
For any vector function ${\bf
u}=(u_{1},u_{2})^{\top}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ (vectors in this
paper will be always regarded as column vectors) the strain tensor in a linear
isotropic and homogeneous elastic medium with Lamé constants $\lambda$ and
$\mu$ is defined as
$\bm{\epsilon}({\bf u}):=\frac{1}{2}(\nabla{\bf u}+(\nabla{\bf
u})^{\top})=\begin{bmatrix}\partial_{x_{1}}u_{1}&\tfrac{1}{2}\left(\partial_{x_{1}}u_{2}+\partial_{x_{2}}u_{1}\right)\\\
\tfrac{1}{2}\left(\partial_{x_{1}}u_{2}+\partial_{x_{2}}u_{1}\right)&\partial_{x_{2}}u_{2}\end{bmatrix}.$
The stress tensor is then given by
$\bm{\sigma}({\bf u}):=2\mu\bm{\epsilon}({\bf
u})+\lambda(\operatorname{div}{\bf u})I_{2}$
where $I_{2}$ is the identity matrix of order 2 and the Lamé coefficients
$\lambda$ are assumed to satisfy $\lambda,\ \lambda+2\mu>0$. The time-harmonic
elastic wave (Navier) equation is
$\operatorname{div}\bm{\sigma}({\bf u})+\omega^{2}{\bf u}=\mu\Delta{\bf
u}+(\lambda+\mu)\nabla(\operatorname{div}{\bf u})+\omega^{2}{\bf u}=0$
where the frequency $\omega\in\mathbb{R}^{+}$ and the divergence operator
${\rm div}$ is applied row-wise.
Considering a bounded domain $\Omega$ in $\mathbb{R}^{2}$ whose boundary
$\Gamma$ is a closed smooth curve, we are interested in solving the
impenetrable elastic scattering problem in the exterior of $\Omega$, denoted
from now on as $\Omega^{+}$. That is, we look for solutions of the time-
harmonic Navier equation
$\operatorname{div}\bm{\sigma}({\bf u})+\omega^{2}{\bf u}=0\quad{\rm in}\
\mathbb{R}^{2}\setminus\Omega$ (2.1)
that satisfy the Kupradze radiation condition at infinity [1, 27]: if
$\mathbf{u}_{p}:=-\frac{1}{k_{p}^{2}}\nabla\operatorname{div}\mathbf{u},\quad\mathbf{u}_{s}:=\mathbf{u}-\mathbf{u}_{p}=\operatorname{\overrightarrow{\mathrm{curl}}}{\rm{curl}\
{\bf u}}$ (2.2)
($\operatorname{\overrightarrow{\mathrm{curl}}}\varphi:=(\partial_{x_{2}}\varphi,-\partial_{x_{1}}\varphi)$,
$\operatorname{curl}{\bf u}:=\partial_{x_{2}}u_{1}-\partial_{x_{1}}u_{2}$ are
respectively the vector and scalar curl, or rotational, operator) with
$k_{p}^{2}:=\frac{\omega^{2}}{\lambda+2\mu},\quad
k_{s}^{2}:=\frac{\omega^{2}}{\mu}$ (2.3)
the associated the pressure and stress wave-numbers wave-numbers, then
$\frac{\partial\mathbf{u}_{p}}{\partial\widehat{\bm{x}}}(\bm{x})-ik_{p}\mathbf{u}_{p}(\bm{x}),\quad\frac{\partial\mathbf{u}_{s}}{\partial\widehat{\bm{x}}}(\bm{x})-ik_{s}\mathbf{u}_{s}(\bm{x})=o\left(|\bm{x}|^{-1/2}\right),\quad\widehat{\bm{x}}:=\frac{1}{|\bm{x}|}\bm{x}.$
On the boundary $\Gamma$ the solution ${\bf u}$ of (2.1) satisfies either the
Dirichlet boundary condition
${\bf u}={\bf f}\quad{\rm on}\ \Gamma$
or the Neumann boundary condition
$T{\bf u}:=\bm{\sigma}({\bf u})\bm{n}=\lambda\operatorname{div}{\bf
u}-2\mu(\nabla{\bf u})\bm{t}={\bf g}\quad{\rm on}\ \Gamma.$
Here $\bm{n}$ is unit normal vector pointing outward and
$\bm{t}:=-\mathrm{Q}\bm{n},\quad\mathrm{Q}:=\begin{bmatrix}&1\\\
-1&\end{bmatrix}$
the unit tangent field positively (counterclockwise if $\Gamma$ is simply
connected) oriented.
In view of (2.2) we can look for the fields ${\bf u}$ in the form
${\bf u}=\nabla u_{p}+\operatorname{\overrightarrow{\mathrm{curl}}}{u_{s}}$
(2.4)
where $u_{p}$ and $u_{s}$ are respectively solutions of the Helmholtz
equations in $\Omega^{+}$ with wave-numbers $k_{p}$ and $k_{s}$ fulfilling the
radioactive, or Sommerfeld, condition at infinity (as consequence of the
Kupradze radiation condition) .
In the case of Dirichlet boundary conditions, by taking the scalar product of
the decomposition (2.4) with the orthogonal frame $(\bm{t},\bm{n})$ on
$\Gamma$, it is straightforward to see that $u_{p}$ and $u_{s}$ must satisfy
the following coupled boundary conditions
$\left|\begin{array}[]{rcll}\partial_{\bm{n}}u_{p}+\partial_{\bm{t}}u_{s}&=&-{\bf
u}^{\rm inc}\cdot\bm{n},&{\rm on}\ \Gamma\\\
\partial_{\bm{t}}u_{p}-\partial_{\bm{n}}u_{s}&=&-{\bf u}^{\rm
inc}\cdot\bm{t},&{\rm on}\ \Gamma.\end{array}\right.$ (2.5)
In the case of Neumann boundary conditions, the same approach described above
combined with the identities
$\displaystyle T[\nabla u_{p}]$ $\displaystyle=$ $\displaystyle
2\mu\operatorname{Hes}u_{p}+\lambda\Delta
u_{p}=2\mu\operatorname{Hes}u_{p}-\lambda k_{p}^{2}u_{p}$ $\displaystyle
T[\overrightarrow{{\rm curl}}_{p}u_{s}]$ $\displaystyle=$ $\displaystyle{\rm
Q}(2\mu\operatorname{Hes}u_{s}-\mu\Delta u_{s})={\rm
Q}(2\mu\operatorname{Hes}u_{s}+\mu k^{2}_{s}u_{s})$
where $\operatorname{Hes}:=\nabla\nabla^{\top}$ is the Hessian matrix
operator, imply that $u_{p}$ and $u_{s}$ must satisfy the following coupled
boundary conditions
$\left|\begin{array}[]{rclrcl}2\mu\bm{n}^{\top}\operatorname{Hes}u_{p}{\bm{n}}-\lambda
k_{p}^{2}u_{p}&+&2\mu\bm{t}^{\top}\operatorname{Hes}u_{s}{\bm{n}}&=&-T{\bf
u}^{\rm inc}\cdot{\bm{n}},&{\rm on}\ \Gamma\\\
2\mu\bm{t}^{\top}\operatorname{Hes}u_{p}{\bm{n}}&-&2\mu\bm{n}^{\top}\operatorname{Hes}u_{s}{\bm{n}}-\mu
k_{s}^{2}u_{s}&=&-T{\bf u}^{\rm inc}\cdot{\bm{t}},&{\rm on}\ \Gamma.\\\
\end{array}\right.$ (2.6)
The goal of this paper is to develop robust BIE formulations for the solution
of elastic scattering problems based on the Helmholtz decomposition approach
in connection with the systems of boundary conditions (2.5) and respectively
(2.6). The BIE we develop use Helmholtz potentials, for which reason we review
certain properties of those next section.
###### Remark 2.1
Throughout this article, except in the Appendix, we will assume that $\Gamma$
is of length $2\pi$. This simplifies some of the following expressions. The
modifications needed to cover the case of arbitrary length curves essentially
consist of replacing the wave-number(s) $k$ (as well as $k_{p}$, $k_{s}$ and
its complexifications $\widetilde{k}_{p}$ and $\widetilde{k}_{s}$) by
$Lk/(2\pi)$, its characteristic length. We can see this either by adapting the
analysis or by a simple scaling argument.
## 3 Helmholtz BIOs, gradient and Hessian calculations
For a given wave-number $k$ and a functional density $\varphi$ on the boundary
$\Gamma$ we define the Helmholtz single and double layer potentials in the
form
$\operatorname{SL}_{k}[\varphi]({\bm{x}}):=\int_{\Gamma}\phi_{k}({\bm{x}}-{\bm{y}})\varphi({\bm{y}}){{\rm
d}{\bm{y}}},\quad\operatorname{DL}_{k}[\varphi]({\bm{x}}):=\int_{\Gamma}\frac{\partial\phi_{k}({\bm{x}}-{\bm{y}})}{\partial\bm{n}({\bm{y}})}\varphi({\bm{y}}){{\rm
d}{\bm{y}}},\ {\bm{x}}\in\mathbb{R}^{2}\setminus\Gamma.$
where $k>0$ and
$\phi_{k}({\bm{x}})=\frac{i}{4}H_{0}^{(1)}(k|{\bm{x}}|)$
is the fundamental solution of the Helmholtz equation.
The four BIOs of the Calderón’s calculus associated with the Helmholtz
equation are defined by applying the exterior/interior Dirichlet and Neumann
traces on $\Gamma$ (denoted in what follows by $\gamma^{+}/\gamma^{-}$ and
$\partial_{\bm{n}}^{+}/\partial_{\bm{n}}^{-}$ respectively) to the Helmholtz
single and double layer potentials defined above via the classical relations
[22, 29, 31]
$\displaystyle\gamma^{\pm}\operatorname{SL}_{k}\varphi$
$\displaystyle=\operatorname{V}_{k}\varphi,\quad$
$\displaystyle\partial_{\bm{n}}^{\pm}\operatorname{SL}_{k}\varphi$
$\displaystyle=\mp\frac{1}{2}\varphi+\operatorname{K}_{k}^{\top}\varphi,$
(3.1) $\displaystyle\gamma^{\pm}\operatorname{DL}_{k}\varphi$
$\displaystyle=\pm\frac{1}{2}\varphi+\operatorname{K}_{k}\varphi,\quad$
$\displaystyle\partial_{\bm{n}}^{\pm}\operatorname{DL}_{k}\varphi$
$\displaystyle=\operatorname{W}_{k}\varphi$
where, for ${\bm{x}}\in\Gamma$,
$\displaystyle(\operatorname{V}_{k}\varphi)({\bm{x}})$ $\displaystyle:=$
$\displaystyle\int_{\Gamma}\phi_{k}({\bm{x}}-{\bm{y}})\varphi({\bm{y}}){{\rm
d}{\bm{y}}},$ $\displaystyle(\operatorname{K}_{k}\varphi)({\bm{x}})$
$\displaystyle:=$
$\displaystyle\int_{\Gamma}\frac{\partial\phi_{k}({\bm{x}}-{\bm{y}})}{\partial\bm{n}({\bm{y}})}\varphi({\bm{y}}){{\rm
d}{\bm{y}}},$ $\displaystyle(\operatorname{K}_{k}^{\top}\varphi)({\bm{x}})$
$\displaystyle:=$
$\displaystyle\int_{\Gamma}\frac{\partial\phi_{k}({\bm{x}}-{\bm{y}})}{\partial\bm{n}({\bm{x}})}\varphi({\bm{y}}){{\rm
d}{\bm{y}}},$ $\displaystyle(\operatorname{W}_{k}\varphi)({\bm{x}})$
$\displaystyle:=$
$\displaystyle\mathrm{f.p.}\int_{\Gamma}\frac{\partial^{2}\phi_{k}({\bm{x}}-{\bm{y}})}{\partial\bm{n}({\bm{x}})\,\partial\bm{n}({\bm{y}})}\varphi({\bm{y}}){{\rm
d}{\bm{y}}},\ $
(“f.p.” stands for finite part since the kernel of the operator is strongly
singular) are respectively the single layer, double layer, adjoint double and
hypersingular operator. The well-know jump relations, which can be extended to
merely Lipschitz curves, are then simple byproducts:
$\displaystyle=\gamma^{+}\operatorname{SL}_{k}\varphi-\gamma^{-}\operatorname{SL}_{k}\varphi=0\quad$
$\displaystyle[\partial_{\bm{n}}\operatorname{SL}_{k}\varphi]$
$\displaystyle=\partial_{\bm{n}}^{+}\operatorname{SL}_{k}\varphi-\partial_{\bm{n}}^{-}\operatorname{SL}_{k}\varphi=\varphi$
(3.2) $\displaystyle[\gamma\operatorname{DL}_{k}\varphi]$
$\displaystyle=\gamma^{+}\operatorname{DL}_{k}\varphi-\gamma^{-}\operatorname{DL}_{k}\varphi=-\varphi,\quad$
$\displaystyle[\partial_{\bm{n}}\operatorname{DL}_{k}\varphi]$
$\displaystyle=\partial_{\bm{n}}^{+}\operatorname{DL}_{k}\varphi-\partial_{\bm{n}}^{-}\operatorname{DL}_{k}\varphi=0.$
Let $H^{s}(\Gamma)$ be the Sobolev space on $\Gamma$ of order $s$. It is well
known then that, for smooth $\Gamma$,
$\operatorname{K}_{k},\ \operatorname{K}_{k}^{\top}:H^{s}(\Gamma)\to
H^{s+3}(\Gamma),\quad\operatorname{V}_{k}:H^{s}(\Gamma)\to
H^{s+1}(\Gamma),\quad\operatorname{W}_{k}:H^{s+1}(\Gamma)\to H^{s+1}(\Gamma).$
That is, we have pseudodifferential operators of order $-3$, $-1$ and $1$
respectively. We will then write
$\operatorname{K}_{k},\
\operatorname{K}_{k}^{\top}\in\mathrm{OPS}(-3),\quad\operatorname{V}_{k}\in\mathrm{OPS}(-1),\quad\operatorname{W}_{k}\in\mathrm{OPS}(1).$
Having reviewed properties of Helmholtz BIOs that will be useful in what
follows, we turn our attention to connecting the systems of boundary
conditions (2.5) and (2.6) to the four Helmholtz BIOs of the Calderón
calculus. In particular, since the boundary conditions (2.6) feature the
Hessian operators, more involved calculations are necessary to express them
via the classical four BIOs. We detail in what follows these derivations which
are largely based on integration by parts techniques akin to those leading to
Maue’s identity (3.12).
We will follow from now on the convention that for a vector function
$\bm{\varphi}=(\varphi_{1},\varphi_{2})^{\top}$ and a pseudodifferential
operator $\mathrm{R}$,
$\mathrm{R}\bm{\varphi}:=(\mathrm{R}{\varphi}_{1},\mathrm{R}{\varphi}_{2})^{\top}.$
(3.3)
The following identities will be used in the proofs of the next results
$\bm{t}\bm{t}^{\top}+\bm{n}\bm{n}^{\top}=I_{2},\qquad\bm{t}\bm{t}^{\top}-\bm{n}\bm{n}^{\top}=I_{2}-2\bm{n}\bm{n}^{\top}.$
(3.4)
(Notice in pass that the second one is just a reflection, a Householder
matrix, about the $\bm{t}$ direction.) Finally, the signed curvature $\kappa$
given
$\kappa=\bm{n}\cdot\partial_{\bm{t}}\bm{t}$ (3.5)
will appear profusely in the expressions below.
We will study now the exterior trace for the gradient of the single and double
layer potentials. Although these results (Propositions 3.1 and 3.3) can be
certainly found in the literature, we cite for example [26, 23, 24], we will
prefer to present their proofs here for the sake of completeness and to
prepare both the statements and the proofs themselves of the corresponding
results for the Hessian matrix of the layer boundary potentials for which the
authors were not able to find detailed proofs. Clearly very similar results
can be derived for the interior traces with the same techniques but since our
aim is the study of exterior problems, we omit the study of this case, leaving
it as (a simple) exercise for the reader.
###### Proposition 3.1
For any smooth closed curve sufficiently smooth,
$\gamma^{+}\nabla{\rm
SL}_{k}\varphi=-\frac{1}{2}\varphi\,\bm{n}+\nabla\operatorname{V}_{k}\varphi$
(3.6)
with
$\displaystyle(\nabla\operatorname{V}_{k})\varphi$ $\displaystyle:=$
$\displaystyle-\operatorname{K}_{k}(\varphi\,\bm{n})+\operatorname{V}_{k}(\partial_{\bm{t}}(\varphi\,\bm{t}))$
(3.7a) $\displaystyle=$
$\displaystyle-\operatorname{K}_{k}(\varphi\,\bm{n})+\operatorname{V}_{k}(\partial_{\bm{t}}\varphi\,\bm{t})-\operatorname{V}_{k}(\kappa\varphi\,\bm{n})$
(3.7b) $\displaystyle=$
$\displaystyle(\operatorname{K}_{k}^{\top}\varphi)\bm{n}+(\partial_{\bm{t}}\operatorname{V}_{k}\varphi)\bm{t}$
$\displaystyle=$ $\displaystyle{\rm
p.v.}\int_{\Gamma}(\nabla\phi_{k})(\cdot-{\bm{y}})\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}$
where p.v. stands for “principal value” of the integral.
Proof. For ${\bm{x}}\in\Omega^{+}$ sufficiently close to $\Gamma$ extend
$\bm{n}({\bm{x}})=\bm{n}(\widehat{{\bm{x}}}),\quad\bm{t}({\bm{x}})=\bm{t}(\widehat{{\bm{x}}}),\qquad{\bm{x}}=\widehat{{\bm{x}}}+\varepsilon\bm{n}(\widehat{{\bm{x}}}),\quad\widehat{{\bm{x}}}\in\Gamma$
with $\varepsilon>0$ sufficiently small.
First, to prove (3.7) we note that by (3.4)
$\displaystyle(\nabla{\rm SL}_{k}\varphi)({\bm{x}})$ $\displaystyle=$
$\displaystyle\int_{\Gamma}\left(\nabla_{{\bm{x}}}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\cdot\bm{n}({\bm{x}})\right)\bm{n}({\bm{x}})\varphi({\bm{x}})\,{\rm
d}{\bm{y}}+\int_{\Gamma}\left(\nabla_{{\bm{x}}}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\cdot\bm{t}({\bm{x}})\right)\bm{t}({\bm{x}})\varphi({\bm{y}})\,{\rm
d}{\bm{y}}$ $\displaystyle=$
$\displaystyle[\partial_{\bm{n}}\operatorname{SL}_{k}\varphi]({\bm{x}})\bm{n}({\bm{x}})+(\nabla_{\bm{x}}[\operatorname{SL}_{k}\varphi]({\bm{x}})\cdot{\bm{t}}({\bm{x}})){\bm{t}}({\bm{x}})$
and applied then (3.1).
Alternatively, by the first identity in (3.4)
$\displaystyle(\nabla{\rm SL}_{k}\varphi)({\bm{x}})$ $\displaystyle=$
$\displaystyle\int_{\Gamma}\nabla_{{\bm{x}}}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\varphi({\bm{y}})\,{\rm
d}{\bm{y}}=-\int_{\Gamma}\nabla_{{\bm{y}}}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\varphi({\bm{y}})\,{\rm
d}{\bm{y}}$ $\displaystyle=$
$\displaystyle-\int_{\Gamma}\left(\nabla_{{\bm{y}}}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\cdot\bm{t}({\bm{y}})\right)\bm{t}({\bm{y}})\varphi({\bm{y}})\,{\rm
d}{\bm{y}}-\int_{\Gamma}\left(\nabla_{{\bm{y}}}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\cdot\bm{n}({\bm{y}})\right)\bm{n}({\bm{y}})\varphi({\bm{y}})\,{\rm
d}{\bm{y}}$ $\displaystyle=$
$\displaystyle-\int_{\Gamma}\partial_{\bm{t}({\bm{y}})}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\left(\bm{t}({\bm{y}})\varphi({\bm{y}})\right)\,{\rm
d}{\bm{y}}-\int_{\Gamma}\left(\nabla_{{\bm{y}}}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\cdot\bm{n}({\bm{y}})\right)\bm{n}({\bm{y}})\varphi({\bm{y}})\,{\rm
d}{\bm{y}}$ $\displaystyle=$
$\displaystyle\int_{\Gamma}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\partial_{\bm{t}({\bm{y}})}\left(\bm{t}({\bm{y}})\varphi({\bm{y}})\right)\,{\rm
d}{\bm{y}}-\int_{\Gamma}\left(\nabla_{{\bm{y}}}\phi_{k}({{\bm{x}}}-{{\bm{y}}})\cdot\bm{n}({\bm{y}})\right)\bm{n}({\bm{y}})\varphi({\bm{y}})\,{\rm
d}{\bm{y}}$ $\displaystyle=$ $\displaystyle\left({\rm
SL}_{k}\partial_{\bm{t}}(\varphi\,\bm{t})\right)({\bm{x}})-\left({\rm
DL}_{k}(\varphi\,\bm{n})\right)({\bm{x}})$ $\displaystyle=$
$\displaystyle\left({\rm
SL}_{k}(\partial_{\bm{t}}\varphi\,\bm{t})\right)({\bm{x}})-\left({\rm
SL}_{k}(\kappa\varphi\,\bm{n})\right)({\bm{x}})-\left({\rm
DL}_{k}(\varphi\,\bm{n})\right)({\bm{x}}).$
(Notice that integration by parts has been applied in the last part of the
argument). Clearly, (3.7a) and (3.7b) follows again from the jump relations
(3.1). $\Box$
Recall the relations
$\overrightarrow{{\rm curl}}\,{u}\cdot\bm{n}=\nabla
u\cdot\bm{t}=\partial_{\bm{t}}u,\quad\overrightarrow{{\rm
curl}}\,{u}\cdot\bm{t}=-\nabla
u\cdot\bm{n}=-\partial_{\bm{n}}u,\quad\nabla(\operatorname{div}{\bf
u})=\Delta{\bf u}+\overrightarrow{{\rm curl}}\,\operatorname{curl}{\bf u}.$
(3.8)
($\Delta{\bf u}$ is just, according to the convention (3.3), the vector
laplacian) and
$\operatorname{curl}({u{\bf v}})=\nabla u\cdot\mathrm{Q}{\bf
v}+u\operatorname{curl}{\bf v}.$ (3.9)
###### Lemma 3.2
It holds
$\nabla{\rm DL}_{k}\varphi=k^{2}{\rm
SL}_{k}(\varphi\,\bm{n})+\overrightarrow{{\rm curl}}\,{\rm
SL}_{k}\partial_{\bm{t}}\varphi$ (3.10)
Proof. Note first that
$\left(\nabla{\rm
DL}_{k}\varphi\right)({\bm{x}})=\int_{\Gamma}\nabla_{{\bm{x}}}(\nabla_{{\bm{y}}}\phi_{k}({\bm{x}}-{\bm{y}})\cdot\bm{n}({\bm{y}}))\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}=-\int_{\Gamma}\nabla_{{\bm{x}}}(\operatorname{div}_{{\bm{x}}}(\phi_{k}({\bm{x}}-{\bm{y}})\bm{n}({\bm{y}}))\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}.$
The third identity in (3.8) implies then
$\displaystyle\left(\nabla{\rm DL}_{k}\varphi\right)({\bm{x}})$
$\displaystyle=$
$\displaystyle-\int_{\Gamma}\Delta_{{\bm{x}}}\phi_{k}({\bm{x}}-{\bm{y}})\bm{n}({\bm{y}}))\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}-\int_{\Gamma}\overrightarrow{{\rm
curl}}\,_{{\bm{x}}}(\operatorname{curl}_{{\bm{x}}}\phi_{k}({\bm{x}}-{\bm{y}})\bm{n}({\bm{y}}))\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}$
$\displaystyle=$ $\displaystyle
k^{2}\int_{\Gamma}\phi_{k}({\bm{x}}-{\bm{y}})\bm{n}({\bm{y}}))\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}-\int_{\Gamma}\overrightarrow{{\rm
curl}}\,_{{\bm{x}}}(\nabla_{{\bm{y}}}\phi_{k}({\bm{x}}-{\bm{y}})\cdot\bm{t}({\bm{y}}))\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}$
$\displaystyle=$ $\displaystyle
k^{2}[\operatorname{SL}_{k}(\varphi\bm{n})]({\bm{x}})-\int_{\Gamma}\overrightarrow{{\rm
curl}}\,_{{\bm{x}}}\partial_{\bm{t}({\bm{y}})}\phi_{k}({\bm{x}}-{\bm{y}})\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}$
where we have made use of the first identity in (3.8) and that $\phi_{k}$ is
the fundamental solution of the Helmholtz equation. The result follows now by
integration by parts in the last integral.
$\Box$
We are ready to prove the counterpart of Proposition 3.1 for the double layer
operator
###### Proposition 3.3
It holds
$\gamma^{+}\nabla{\rm
DL}_{k}\varphi=\frac{1}{2}\partial_{\bm{t}}\varphi\,\bm{t}+\nabla\operatorname{K}_{k}\varphi$
where
$\displaystyle(\nabla\operatorname{K}_{k}\varphi)({\bm{x}})$ $\displaystyle:=$
$\displaystyle{\rm
p.v.}\int_{\Gamma}\left(\nabla_{{\bm{x}}}\partial_{\bm{n}({\bm{y}})}\phi_{k}(\cdot-{\bm{y}})\right)\\!({\bm{x}})\,\varphi({\bm{y}})\,\mathrm{d}{\bm{y}}$
(3.11a) $\displaystyle=$ $\displaystyle
k^{2}\operatorname{V}_{k}(\varphi\bm{n})({\bm{x}})+\left(\partial_{\bm{t}}\operatorname{V}_{k}\partial_{\bm{t}}\varphi\right)\\!({\bm{x}})\,\bm{n}({\bm{x}})-\big{(}\operatorname{K}_{k}^{\top}\partial_{\bm{t}}\varphi\big{)}({\bm{x}})\,{\bm{t}}({\bm{x}})$
$\displaystyle=$
$\displaystyle\big{[}\operatorname{W}_{k}\varphi\big{]}({\bm{x}})\,\bm{n}({\bm{x}})+\big{[}k^{2}\operatorname{V}_{k}(\varphi\bm{n})\cdot\bm{t}-\operatorname{K}_{k}^{\top}\partial_{\bm{t}}\varphi\big{]}({\bm{x}})\,\bm{t}({\bm{x}}).$
(3.11b)
Proof. From Lemma 3.2 we derive
$\displaystyle\gamma^{+}(\bm{t}\cdot\nabla{\rm DL}_{k}\varphi)$
$\displaystyle=$ $\displaystyle\gamma^{+}(k^{2}\bm{t}\cdot{\rm
SL}_{k}(\varphi\,\bm{n})+\bm{t}\cdot\overrightarrow{{\rm curl}}\,{\rm
SL}_{k}\partial_{\bm{t}}\varphi)$ $\displaystyle=$ $\displaystyle
k^{2}\gamma^{+}(\bm{t}\cdot{\rm SL}_{k}(\varphi\,\bm{n})-\bm{n}\cdot\nabla{\rm
SL}_{k}\partial_{\bm{t}}\varphi)$ $\displaystyle=$ $\displaystyle
k^{2}\bm{t}\cdot\operatorname{V}_{k}(\varphi\,\bm{n})-\operatorname{K}_{k}^{\top}\partial_{\bm{t}}\varphi+\frac{1}{2}\partial_{\bm{t}}u_{s}$
$\displaystyle\gamma^{+}(\bm{n}\cdot\nabla{\rm DL}_{k}\varphi)$
$\displaystyle=$ $\displaystyle\gamma^{+}(k^{2}\bm{n}\cdot{\rm
SL}_{k}(\varphi\,\bm{n})+\bm{t}\nabla{\rm SL}_{k}\partial_{\bm{t}}\varphi)$
$\displaystyle=$ $\displaystyle k^{2}\bm{n}\cdot{\rm
V}_{k}(\varphi\,\bm{n})+\partial_{\bm{t}}\operatorname{V}_{k}\partial_{\bm{t}}\varphi.$
Alternatively,
$\gamma^{+}(\bm{n}\cdot\nabla{\rm DL}_{k}\varphi)=\partial_{\bm{n}}{\rm
DL}_{k}\varphi=\operatorname{W}_{k}\varphi.$
Using (3.4) the proof is finished. $\Box$
Notice that as byproduct the well-known Maue identity (cf. [29, Ch. 9], [26,
Ch. 7] or [31, Ch. 2]) has been derived for the hypersingular operator:
$\operatorname{W}_{k}\varphi=\partial_{\bm{t}}\operatorname{V}_{k}\partial_{\bm{t}}\varphi+\bm{n}\cdot\operatorname{V}_{k}(\bm{n}\varphi)$
(3.12)
We next move to consider the trace for the Hessian for the Helmholtz single
and double layer operator.
###### Theorem 3.4
It holds
$\displaystyle\gamma^{+}(\operatorname{Hes}{\rm SL}_{k}\varphi)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{\bm{t}}\varphi\,(\bm{n}\bm{t}^{\top}+\bm{t}\bm{n}^{\top})-\frac{1}{2}\kappa\varphi\,(I_{2}-2\bm{n}\bm{n}^{\top})$
(3.13)
$\displaystyle+\nabla\operatorname{V}_{k}(\partial_{\bm{t}}\varphi\,\bm{t}^{\top})-\nabla\operatorname{V}_{k}(\kappa\varphi\,\bm{n}^{\top})-\nabla\operatorname{K}_{k}(\varphi\,\bm{n}^{\top})$
$\displaystyle\gamma^{+}(\operatorname{Hes}{\rm DL}_{k}\varphi)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\bm{t}}^{2}\varphi\,(I_{2}-2\bm{n}\bm{n}^{\top})-\frac{1}{2}\kappa\partial_{\bm{t}}\varphi\,(\bm{n}\bm{t}^{\top}+\bm{t}\bm{n}^{\top})-\frac{1}{2}k^{2}\varphi\,\bm{n}\bm{n}^{\top}$
(3.14)
$\displaystyle+k^{2}\nabla\operatorname{V}_{k}(\varphi\,\bm{n}^{\top})+\nabla\operatorname{V}_{k}(\partial^{2}_{\bm{t}}\varphi\,\bm{n}^{\top})+\nabla\operatorname{V}_{k}(\kappa\partial_{\bm{t}}\varphi\,\bm{t}^{\top})+\nabla\operatorname{K}_{k}(\partial_{\bm{t}}\varphi\,\bm{t}^{\top}).$
Proof. We first consider (3) in Proposition 3.1:
$\displaystyle(\nabla^{\top}{\rm SL}_{k}\varphi)({\bm{x}})$ $\displaystyle=$
$\displaystyle\big{(}{\rm
SL}_{k}(\partial_{\bm{t}}\varphi\,\bm{t}^{\top})\big{)}({\bm{x}})-\big{(}{\rm
SL}_{k}(\kappa\varphi\,\bm{n}^{\top})\big{)}({\bm{x}})-\big{(}{\rm
DL}_{k}(\varphi\,\bm{n}^{\top})\big{)}({\bm{x}}).$
Therefore (note that $\operatorname{Hes}u=\nabla(\nabla^{\top}u)$), we can
apply Proposition 3.1 (twice) for the single layer terms and Proposition 3.3
for the double layer to obtain
$\displaystyle\gamma^{+}\operatorname{Hes}{\rm SL}_{k}\varphi$
$\displaystyle=$
$\displaystyle\nabla\operatorname{V}_{k}(\partial_{\bm{t}}\varphi\,\bm{t}^{\top})-\nabla\operatorname{V}_{k}(\kappa\varphi\,\bm{n}^{\top})-\nabla\operatorname{K}_{k}(\varphi\,\bm{n}^{\top})$
$\displaystyle-\frac{1}{2}\partial_{\bm{t}}\varphi\,\bm{n}\bm{t}^{\top}+\frac{1}{2}\kappa\varphi\,(\bm{n}\bm{n}^{\top})-\bm{t}\,\frac{1}{2}\partial_{\bm{t}}(\varphi\,\bm{n}^{\top})$
$\displaystyle=$
$\displaystyle\nabla\operatorname{V}_{k}(\partial_{\bm{t}}\varphi\,\bm{t}^{\top})-\nabla\operatorname{V}_{k}(\kappa\varphi\,\bm{n}^{\top})-\nabla\operatorname{K}_{k}(\varphi\,\bm{n}^{\top})$
$\displaystyle-\frac{1}{2}\partial_{\bm{t}}\varphi\,(\bm{n}\bm{t}^{\top}+\bm{t}\bm{n}^{\top})-\frac{1}{2}\kappa\varphi\,(\bm{t}\bm{t}^{\top}-\bm{n}\bm{n}^{\top})$
Then (3.13) follows now from (3.4).
To prove (3.14) we start from Lemma 3.2 which implies, using that
$\overrightarrow{{\rm curl}}\,=\mathrm{Q}\nabla$,
$\displaystyle\operatorname{Hes}{\rm DL}_{k}\varphi$ $\displaystyle=$
$\displaystyle
k^{2}\nabla\mathrm{SL}_{k}(\varphi\bm{n}^{\top})+\nabla\nabla^{\top}\mathrm{SL}_{k}(\partial_{\bm{t}}\varphi)\mathrm{Q}^{\top}.$
Proposition 3.1 for the first term and (3.13) for the second one imply
$\displaystyle\operatorname{Hes}{\rm DL}_{k}\varphi$ $\displaystyle=$
$\displaystyle-\frac{1}{2}k^{2}\varphi\bm{n}\bm{n}^{\top}+k^{2}\nabla\operatorname{V}_{k}(\varphi\bm{n}^{\top})\bm{n}$
$\displaystyle+\Big{(}-\frac{1}{2}\partial_{\bm{t}}^{2}\varphi\,(\bm{n}\bm{t}^{\top}+\bm{t}\bm{n}^{\top})-\frac{1}{2}\kappa\partial_{\bm{t}}\varphi\,(I_{2}-2\bm{n}\bm{n}^{\top})$
$\displaystyle+\nabla\operatorname{V}_{k}(\partial^{2}_{\bm{t}}\varphi\,\bm{t}^{\top})-\nabla\operatorname{V}_{k}(\kappa\partial_{\bm{t}}\varphi\,\bm{n}^{\top})-\nabla\operatorname{K}_{k}(\partial_{\bm{t}}\varphi\,\bm{n}^{\top})\Big{)}\mathrm{Q}^{\top}.$
The result follows from the relations $\mathrm{Q}\bm{n}=-\bm{t},\
\mathrm{Q}\bm{t}=\bm{n}.$
$\Box$
Let us write explicitly (3.13)-(3.14) in a way it will be used later. Hence,
using (3.7) for $\nabla\operatorname{V}_{k}$ and (3.11b)
$\nabla\operatorname{K}_{k}$ we can write
$\displaystyle\gamma^{+}(\operatorname{Hes}{\rm SL}_{k}\varphi)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{\bm{t}}\varphi({\bm{n}}{\bm{t}}^{\top}+{\bm{t}}{\bm{n}}^{\top})-\frac{1}{2}\kappa\varphi(I_{2}-2\bm{n}\bm{n}^{\top})$
$\displaystyle+\Big{[}\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\,\bm{t})-\operatorname{K}_{k}^{\top}(\kappa\varphi\,\bm{n})-\operatorname{W}_{k}(\varphi\,\bm{n})\Big{]}{\bm{n}}^{\top}$
$\displaystyle+\Big{[}\partial_{\bm{t}}\operatorname{V}_{k}(\partial_{\bm{t}}\varphi\,\bm{t})-\partial_{\bm{t}}\operatorname{V}_{k}(\kappa\varphi\bm{n})-k^{2}\operatorname{V}_{k}(\varphi\,\bm{n}\bm{n}^{\top})\bm{t}+\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\,\bm{n})\Big{]}{\bm{t}}^{\top}$
and
$\displaystyle\gamma^{+}(\operatorname{Hes}{\rm DL}_{k}\varphi)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\bm{t}}^{2}\varphi(I_{2}-2\bm{n}\bm{n}^{\top})-\frac{1}{2}\kappa\partial_{\bm{t}}\varphi({\bm{n}}{\bm{t}}^{\top}+{\bm{t}}{\bm{n}}^{\top})-\frac{1}{2}\kappa^{2}\varphi\,\bm{n}\bm{n}^{\top}$
$\displaystyle+\Big{[}k^{2}\operatorname{K}_{k}^{\top}(\varphi\,\bm{n})+\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}^{2}\varphi\,\bm{n})+\operatorname{K}_{k}^{\top}(\kappa\partial_{\bm{t}}\varphi\bm{t})+\operatorname{W}_{k}(\partial_{\bm{t}}\varphi\,\bm{t})\Big{]}{\bm{n}}^{\top}$
$\displaystyle+\Big{[}k^{2}\partial_{\bm{t}}\operatorname{V}_{k}(\varphi\,\bm{n})+\partial_{\bm{t}}\operatorname{V}_{k}(\partial_{\bm{t}}^{2}\varphi\bm{n})+\partial_{\bm{t}}\operatorname{V}_{k}(\kappa\varphi\bm{t})+k^{2}\operatorname{V}_{k}(\varphi\,\bm{t}\bm{n}^{\top})\bm{t}-\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\,\bm{t})\Big{]}{\bm{t}}^{\top}.$
We will summarize the results proven in this section in a way that will be
used in both the analysis and implementation of the numerical algorithms:
###### Theorem 3.5
It holds
$\displaystyle\partial_{\bm{n}}{\rm SL}_{k}\varphi=\gamma^{+}(\nabla{\rm
SL}_{k}\varphi)\cdot\bm{n}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\varphi+\operatorname{K}^{\top}\varphi$
$\displaystyle\partial_{\bm{t}}{\rm SL}_{k}\varphi=\gamma^{+}(\nabla{\rm
SL}_{k}\varphi)\cdot\bm{t}$ $\displaystyle=$
$\displaystyle\partial_{\bm{t}}\operatorname{V}_{k}\varphi$
and
$\displaystyle\partial_{\bm{n}}{\rm DL}_{k}\varphi=\gamma^{+}(\nabla{\rm
DL}_{k}\varphi)\cdot\bm{n}$ $\displaystyle=$
$\displaystyle\operatorname{W}_{k}\varphi=\partial_{\bm{t}}\operatorname{V}_{k}\partial_{\bm{t}}\varphi+k^{2}\bm{n}\cdot{\rm
V}_{k}(\varphi\,\bm{n})$ $\displaystyle\partial_{\bm{t}}{\rm
DL}_{k}\varphi=\gamma^{+}(\nabla{\rm DL}_{k}\varphi)\cdot\bm{t}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\bm{t}}\varphi+k^{2}\operatorname{V}_{k}(\varphi\bm{n})\cdot\bm{t}-\operatorname{K}_{k}^{\top}\partial_{\bm{t}}\varphi.$
###### Theorem 3.6
For the trace of the Hessian matrix of the single layer boundary potential it
holds
$\displaystyle\gamma^{+}\,\bm{n}^{\top}(\operatorname{Hes}{\rm
SL}_{k}\varphi)\bm{n}$ $\displaystyle=$
$\displaystyle-\gamma^{+}\,\bm{t}^{\top}(\operatorname{Hes}{\rm
SL}_{k}\varphi)\bm{t}-k^{2}\operatorname{V}_{k}\varphi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\kappa\varphi+\bm{n}\cdot\Big{[}\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\,\bm{t})-\operatorname{K}_{k}^{\top}(\kappa\varphi\,\bm{n})-\operatorname{W}_{k}(\varphi\,\bm{n})\Big{]}$
$\displaystyle\gamma^{+}\,\bm{t}^{\top}(\operatorname{Hes}{\rm
SL}_{k}\varphi)\bm{t}$ $\displaystyle=$
$\displaystyle-\gamma^{+}\,\bm{n}^{\top}(\operatorname{Hes}{\rm
SL}_{k}\varphi)\bm{n}-k^{2}\operatorname{V}_{k}\varphi$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\kappa\varphi+\bm{t}\cdot\Big{[}\partial_{\bm{t}}\operatorname{V}_{k}(\partial_{\bm{t}}\varphi\,\bm{t})-\partial_{\bm{t}}\operatorname{V}_{k}(\kappa\varphi\bm{n})+k^{2}\operatorname{V}_{k}(\varphi\,\bm{n}\bm{n}^{\top})\bm{t}+\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\,\bm{n})\Big{]}$
$\displaystyle\gamma^{+}\,\bm{n}^{\top}(\operatorname{Hes}{\rm
SL}_{k}\varphi)\bm{t}$ $\displaystyle=$
$\displaystyle\gamma^{+}\,\bm{t}^{\top}(\operatorname{Hes}{\rm
SL}_{k}\varphi)\bm{n}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{\bm{t}}\varphi+{\bm{n}}\cdot\Big{[}\partial_{\bm{t}}\operatorname{V}_{k}(\varphi\,\bm{t})-\partial_{\bm{t}}\operatorname{V}_{k}(\kappa\varphi\bm{n})+k^{2}\operatorname{V}_{k}(\varphi\,\bm{n}\bm{n}^{\top})\bm{t}+\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\,\bm{n})\Big{]}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{\bm{t}}\varphi+{\bm{t}}\cdot\Big{[}\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\,\bm{t})-\operatorname{K}_{k}^{\top}(\kappa\varphi\,\bm{n})-\operatorname{W}_{k}(\varphi\,\bm{n})\Big{]}.$
Similarly, for the trace of the Hessian matrix of the double layer potential
we have
$\displaystyle\gamma^{+}\,\bm{n}^{\top}(\operatorname{Hes}{\rm
DL}_{k}\varphi)\bm{n}$ $\displaystyle=$
$\displaystyle-\gamma^{+}\,\bm{t}^{\top}(\operatorname{Hes}{\rm
DL}_{k}\varphi)\bm{t}-{k^{2}\left(\frac{1}{2}\varphi+\operatorname{K}_{k}\varphi\right)}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{\bm{t}}^{2}\varphi-\frac{1}{2}k^{2}\varphi\,+\bm{n}\cdot\Big{[}k^{2}\operatorname{K}_{k}^{\top}(\varphi\,\bm{n})+\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}^{2}\varphi\,\bm{n})+\operatorname{K}_{k}^{\top}(\kappa\partial_{\bm{t}}\varphi\bm{t})+\operatorname{W}_{k}(\partial_{\bm{t}}\varphi\,\bm{t})\Big{]}$
$\displaystyle\gamma^{+}\,\bm{t}^{\top}(\operatorname{Hes}{\rm
DL}_{k}\varphi)\bm{t}$ $\displaystyle=$
$\displaystyle-\gamma^{+}\,\bm{n}^{\top}(\operatorname{Hes}{\rm
DL}_{k}\varphi)\bm{n}-{k^{2}\left(\frac{1}{2}\varphi+\operatorname{K}_{k}\varphi\right)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\bm{t}}^{2}\varphi+\bm{t}\cdot\Big{[}k^{2}\partial_{\bm{t}}\operatorname{V}_{k}(\varphi\,\bm{n})+\partial_{\bm{t}}\operatorname{V}_{k}(\partial_{\bm{t}}^{2}\varphi\bm{n})+\partial_{\bm{t}}\operatorname{V}_{k}(\kappa\varphi\bm{t})+k^{2}\operatorname{V}_{k}(\varphi\,\bm{t}\bm{n}^{\top})\bm{t}$
$\displaystyle-\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\bm{t})\Big{]}$
$\displaystyle\gamma^{+}\,\bm{n}^{\top}(\operatorname{Hes}{\rm
DL}_{k}\varphi)\bm{t}$ $\displaystyle=$
$\displaystyle\gamma^{+}\,\bm{t}^{\top}(\operatorname{Hes}{\rm
DL}_{k}\varphi)\bm{n}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\kappa\partial_{\bm{t}}\varphi+{\bm{n}}\cdot\Big{[}k^{2}\partial_{\bm{t}}\operatorname{V}_{k}(\varphi\,\bm{n})+\partial_{\bm{t}}\operatorname{V}_{k}(\partial_{\bm{t}}^{2}\varphi\bm{n})+\partial_{\bm{t}}\operatorname{V}_{k}(\kappa{\partial_{\bm{t}}}\varphi\bm{t})+k^{2}\operatorname{V}_{k}({\partial_{\bm{t}}}\varphi\,\bm{t}\bm{n}^{\top})\bm{t}$
$\displaystyle-\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}\varphi\bm{t})\Big{]}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\kappa\partial_{\bm{t}}\varphi+{\bm{t}}\cdot\Big{[}k^{2}\operatorname{K}_{k}^{\top}(\varphi\,\bm{n})+\operatorname{K}_{k}^{\top}(\partial_{\bm{t}}^{2}\varphi\,\bm{n})+\operatorname{K}_{k}^{\top}(\kappa\partial_{\bm{t}}\varphi\bm{t})+\operatorname{W}_{k}(\partial_{\bm{t}}\varphi\,\bm{t})\Big{]}.$
To conclude this section we will write the principal part of the normal and
tangential derivatives for the layer potentials in terms of two basic
pseudodifferential operator: powers of the tangential derivative and the
Hilbert transform. Hence, let us denote
$\mathrm{D}=\partial_{\bm{t}}:H^{s}(\Gamma)\to
H^{s-1}(\Gamma),\quad\mathrm{D}_{-1}:=\mathrm{D}^{\dagger}:H^{s}(\Gamma)\to
H^{s+1}(\Gamma)$
the tangential derivative and its pseudoinverse. We will keep this double
notation, ${\mathrm{D}}$ and $\partial_{\bm{t}}$, for the tangential
derivative, favouring the former when dealing with expansions of the boundary
operators.
We will define the integer powers of $\mathrm{D}$ in the natural way: with
$r\in\mathbb{N}$, set
$\mathrm{D}_{r}:=\mathrm{D}^{r},\quad\mathrm{D}_{-r}:=(\mathrm{D}_{-1})^{-r},\quad\mathrm{D}_{0}:=\mathrm{D}^{r}\mathrm{D}^{-r}.$
Clearly,
$\mathrm{D}_{r}\mathrm{D}_{s}=\mathrm{D}_{r+s},\qquad\mathrm{D}_{0}\varphi=\varphi-\mathrm{J}\varphi,\text{
with }\mathrm{J}\varphi:=\frac{1}{L}\int_{\Gamma}\varphi.$
On the other hand, let $\operatorname{H}:H^{s}(\Gamma)\to H^{s}(\Gamma)$ be
the Hilbert transform or Hilbert singular operator (cf. [31, Ch. 5], [26, Ch.
7]) given by
$(\operatorname{H}\varphi)({\bf
x}(t))=-\frac{1}{2\pi}\mathrm{p.v.}\int_{0}^{2\pi}\cot\left(\frac{\cdot-\tau}{2}\right)\varphi({\bf
x}(\tau))\,{\rm
d}\tau+i\mathrm{J}\varphi=i\bigg{[}\widehat{\varphi}(0)+\sum_{n\neq
0}\mathop{\rm\rm sign}(n)\widehat{\varphi}(n)e_{n}(t)\bigg{]},$
where ${\bf x}:\mathbb{R}\to\Gamma$ is (a) $2\pi-$periodic arc-length
parameterization (recall Remark 2.1) positively oriented of $\Gamma$ and
$\widehat{\varphi}(n):=\frac{1}{{2\pi}}\int_{0}^{2\pi}\varphi({\bf
x}(\tau))e_{-n}(\tau)\,{\rm d}\,\tau,\quad e_{n}(\tau):=\exp(i\,n\tau),\quad
n\in\mathbb{Z}$ (3.17)
the Fourier coefficients of $\varphi\circ{\bf x}$. Clearly
$\operatorname{H}^{2}=-\operatorname{I}.$
The operators $\mathrm{D}^{r}$ can be also written in the Fourier basis:
$(\mathrm{D}_{r}\varphi)({\bf x}(t))=\sum_{n\neq
0}\left(in\right)^{r}\widehat{\varphi}(n)e_{n}(t)$
which as byproduct shows that
$\operatorname{H}\mathrm{D}_{r}=\mathrm{D}_{r}\operatorname{H}.$
Furthermore, since for any smooth function $a:\Gamma\to\mathbb{C}$,
$\operatorname{H}a-a\operatorname{H}$ can be written as an integral operator
with smooth kernel we can conclude that $a\operatorname{H}$ and
$\operatorname{H}a$ differ in a smoothing operator. The notation, which will
be extensively used from now on,
${\mathrm{A}}_{1}={\mathrm{A}}_{2}+\mathrm{OPS}(r)\ \Leftrightarrow\
{\mathrm{A}}_{1}-{\mathrm{A}}_{2}\in\mathrm{OPS}(r)$
allows us to simply write
$a\operatorname{H}=\operatorname{H}a+\mathrm{OPS}(-\infty).$
Similarly
$a\mathrm{D}_{r}=\mathrm{D}_{r}a+\mathrm{OPS}(r-1).$
(It is trivial to prove for positive integers $r$ and a simple exercise for
negative integers; see the Appendix).
The last two results of this section detail how the normal and tangential
derivatives of the boundary layer operators can be expanded in terms of these
basic operators. The proof is a consequence of Theorem 3.5-3.6 and Theorem A.5
in the Appendix:
###### Proposition 3.7
It holds
$\displaystyle\gamma^{+}\,\nabla{\rm SL}_{k}[\,\cdot\,]\cdot\bm{n}$
$\displaystyle\ =\ -\frac{1}{2}\operatorname{I}-\frac{1}{2}\kappa
k^{2}\mathrm{D}_{-3}\operatorname{H}+\mathrm{OPS}(-4),$ (3.18)
$\displaystyle\gamma^{+}\,\nabla{\rm SL}_{k}[\,\cdot\,]\cdot\bm{t}$
$\displaystyle\
=\frac{1}{2}\operatorname{H}-\frac{1}{4}k^{2}\mathrm{D}_{-2}\operatorname{H}+\mathrm{OPS}(-4)$
and
$\displaystyle\gamma^{+}\,\nabla{\rm DL}_{k}[\,\cdot\,]\cdot\bm{n}$
$\displaystyle\ =\
\frac{1}{2}\operatorname{H}\mathrm{D}+\frac{1}{4}k^{2}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-3),$
(3.19) $\displaystyle\gamma^{+}\,\nabla{\rm DL}_{k}[\,\cdot\,]\cdot\bm{t}$
$\displaystyle\ =\
\frac{1}{2}\mathrm{D}+k^{2}\kappa\mathrm{D}_{-2}\operatorname{H}+\mathrm{OPS}(-3).$
###### Proposition 3.8
It holds
$\displaystyle\gamma^{+}\,\bm{n}^{\top}\operatorname{Hes}{\rm
SL}_{k}[\,\cdot\,]\bm{n}$ $\displaystyle\ =\
-\frac{1}{2}\mathrm{D}\operatorname{H}+\frac{1}{2}\kappa\operatorname{I}-\frac{k^{2}}{4}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-3),$
(3.20) $\displaystyle\gamma^{+}\,\bm{t}^{\top}\operatorname{Hes}{\rm
SL}_{k}[\,\cdot\,]\bm{t}$ $\displaystyle\ =\
\frac{1}{2}\mathrm{D}\operatorname{H}-\frac{1}{2}\kappa\operatorname{I}-\frac{k^{2}}{4}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-3),$
$\displaystyle\gamma^{+}\,\bm{t}^{\top}\operatorname{Hes}{\rm
SL}_{k}[\,\cdot\,]\bm{n}$ $\displaystyle\ =\
\gamma^{+}\,\bm{n}^{\top}\operatorname{Hes}{\rm SL}_{k}[\,\cdot\,]\bm{t}$
$\displaystyle\ =\
-\frac{1}{2}\mathrm{D}\varphi-\frac{1}{2}\kappa\operatorname{H}-\frac{k^{2}\kappa}{4}\kappa\mathrm{D}_{-2}\operatorname{H}+\mathrm{OPS}(-3),$
and
$\displaystyle\gamma^{+}\,\bm{n}^{\top}\operatorname{Hes}{\rm
DL}_{k}[\,\cdot\,]\bm{n}$ $\displaystyle\ =\
-\frac{1}{2}\mathrm{D}_{2}-\frac{1}{2}\kappa\operatorname{H}\mathrm{D}-\frac{1}{2}k^{2}\operatorname{I}-\frac{k^{2}}{4}\kappa\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-2),$
(3.21) $\displaystyle\gamma^{+}\,\bm{t}^{\top}\operatorname{Hes}{\rm
DL}_{k}[\,\cdot\,]\bm{t}$ $\displaystyle\ =\
\frac{1}{2}\mathrm{D}_{2}+\frac{1}{2}\kappa\operatorname{H}\mathrm{D}+\frac{k^{2}}{4}\kappa\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-2),$
$\displaystyle\gamma^{+}\,\bm{t}^{\top}\operatorname{Hes}{\rm
DL}_{k}[\,\cdot\,]\bm{n}$ $\displaystyle\ =\
\gamma^{+}\,\bm{n}^{\top}\operatorname{Hes}{\rm
DL}_{k}[\,\cdot\,]\bm{t}=\frac{1}{2}\operatorname{H}\mathrm{D}_{2}-\frac{1}{2}\kappa\mathrm{D}+\frac{k^{2}}{4}\operatorname{H}+\mathrm{OPS}(-2).$
## 4 Boundary integral formulations
From now on, and to alleviate the notation, we will use “$p$” and “$s$” in
subscripts to refer to $k_{p}$ and $k_{s}$, the pressure and stress wave-
numbers so that
$\operatorname{SL}_{p}=\operatorname{SL}_{k_{p}},\
\operatorname{SL}_{s}=\operatorname{SL}_{k_{s}},\
\operatorname{DL}_{p}=\operatorname{DL}_{k_{p}},\
\operatorname{DL}_{s}=\operatorname{DL}_{k_{s}},$
and similarly for the associated BIOs.
In what follows the operator matrix
${\cal H}_{0}:=\begin{bmatrix}\operatorname{I}&-\operatorname{H}\\\
-\operatorname{H}&-\operatorname{I}\end{bmatrix}\in\mathrm{OPS}(0)$
will play an essential role. Notice that
${\cal H}_{0}^{2}=0.$
That is, ${\cal H}_{0}$ is nilpotent of index 2.
We note also that the matrix operator
$\begin{bmatrix}a_{11}\operatorname{I}&a_{12}\operatorname{H}\\\
a_{12}\operatorname{H}&a_{22}\operatorname{I}\end{bmatrix}\in\mathrm{OPS}(0)$
(4.1a) with $a_{ij}\in\mathbb{C}$ is invertible if and only if
$\det\begin{bmatrix}a_{11}&a_{21}i\\\
a_{12}i&a_{22}\end{bmatrix}=a_{11}a_{22}+a_{12}a_{22}\neq 0$ (4.1b) and that
the inverse is given by
$\frac{1}{a_{11}a_{22}+a_{12}a_{21}}\begin{bmatrix}a_{22}\operatorname{I}&-a_{12}\operatorname{H}\\\
-a_{21}\operatorname{H}&a_{11}\operatorname{I}\end{bmatrix}.$ (4.1c)
In particular, this also implies that $\mathcal{H}_{0}$ is not invertible.
For any matrix ${\cal A}$ and scalar operator $\mathrm{B}$ with
${\cal A}=\begin{bmatrix}\mathrm{A}_{11}&\mathrm{A}_{12}\\\
\mathrm{A}_{21}&\mathrm{A}_{22}\\\ \end{bmatrix},\quad\mathrm{B}$
we will denote
${\cal A}\,\mathrm{B}={\cal
A}\otimes\mathrm{B}=\begin{bmatrix}\mathrm{A}_{11}\mathrm{B}&\mathrm{A}_{12}\mathrm{B}\\\
\mathrm{A}_{21}\mathrm{B}&\mathrm{A}_{22}\mathrm{B}\end{bmatrix},\qquad\mathrm{B}\,{\cal
A}=\mathrm{B}\otimes{\cal
A}=\begin{bmatrix}\mathrm{B}\mathrm{A}_{11}&\mathrm{B}\mathrm{A}_{12}\\\
\mathrm{B}\mathrm{A}_{21}&\mathrm{B}\mathrm{A}_{22}\end{bmatrix}.$
Clearly, if $a_{ij}\in\mathbb{C}$,
$\begin{bmatrix}a_{11}\operatorname{I}&a_{12}\operatorname{H}\\\
a_{12}\operatorname{H}&a_{22}\operatorname{I}\end{bmatrix}\mathrm{D}_{r}\operatorname{H}=\mathrm{D}_{r}\operatorname{H}\begin{bmatrix}a_{11}\operatorname{I}&a_{12}\operatorname{H}\\\
a_{12}\operatorname{H}&a_{22}\operatorname{I}\end{bmatrix}.$
We will work in this section with Fourier multiplier, or diagonal, scalar
$(\mathrm{A}\varphi)\circ{\bf
x}=\sum_{n\in\mathbb{Z}}\mathrm{A}(n)\widehat{\varphi}(n)e_{n}$
and matrix operator,
$(\mathcal{A}\bm{\varphi})\circ{\bf
x}:=\sum_{n\in\mathbb{Z}}\mathcal{A}(n)\begin{bmatrix}\widehat{\varphi}_{1}(n)\\\
\widehat{\varphi}_{2}(n)\end{bmatrix}e_{n}$
where $\mathrm{A}(n)\in\mathbb{C}$ and the complex matrix $\mathcal{A}(n)$ a
$2\times 2$ are the so-called principal symbol of $\mathrm{A}$ and
$\mathcal{A}$. Besides, $e_{n}$ and ${\bf x}$ are the complex exponential and
the (or an) arc-length parameterization of the curve such as it was introduced
in (3.17).
Therefore we can write
${\cal H}_{0}\bm{\varphi}=\begin{bmatrix}1&-i\\\
-i&-1\end{bmatrix}\begin{bmatrix}\widehat{\varphi}_{1}(0)\\\
\widehat{\varphi}_{2}(0)\end{bmatrix}+\sum_{n\neq
0}\begin{bmatrix}1&-i\operatorname{sign}(n)\\\ -i\operatorname{sign}(n)&-1\\\
\end{bmatrix}\begin{bmatrix}\widehat{\varphi}_{1}(n)\\\
\widehat{\varphi}_{2}(n)\end{bmatrix}e_{n}.$
It is well known (see also the Appendix of this work) that
$\mathrm{A}:H^{s}(\Gamma)\to:H^{s-m}(\Gamma),\quad\mathcal{A}:H^{s}(\Gamma)\times
H^{s}(\Gamma)\to H^{s-m}(\Gamma)\times H^{s-m}(\Gamma)$
are continuous provided that, respectively,
$|\mathrm{A}(n)|\leq C(1+|n|)^{m},\quad\|\mathcal{A}(n)\|_{2}\leq
C(1+|n|)^{m}$
with $C$ independent of $n$. Injectivity for matrix, respectively scalar,
operator is equivalent to $\mathcal{A}(n)-$invertibility, resp. $|A(n)|\neq
0$. Finally, $\mathrm{A}^{-1}(n)$ and $\mathcal{A}^{-1}(n)$ are the principal
symbols of $\mathrm{A}^{-1}$ and $\mathcal{A}^{-1}$ whenever these inverses
exist.
### 4.1 Dirichlet problem
#### 4.1.1 Combined formulations
If we look for combined field representations of the Helmholtz fields $u_{p}$
and $u_{s}$ defined in equation (2.4) satisfying the system of boundary
conditions (2.5) in the form
$u_{p}=\operatorname{DL}_{p}\varphi_{p}-ik\operatorname{SL}_{p}\varphi_{p}\qquad
u_{s}=\operatorname{DL}_{s}\varphi_{s}-ik\operatorname{SL}_{s}\varphi_{s}\quad
k>0$ (4.2)
the results in Theorem 3.5 led to the CFIE formulation
$\left({\cal A}_{\rm DL}-ik{\cal A}_{\rm
SL}\right)\begin{bmatrix}\varphi_{p}\\\
\varphi_{s}\end{bmatrix}=-\begin{bmatrix}{\bf u}^{\rm inc}\cdot\bm{n}\\\ {\bf
u}^{\rm inc}\cdot\bm{t}\end{bmatrix}$
in terms of the matrix BIO
$\displaystyle{\cal A}_{\rm DL}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\operatorname{W}_{p}&\frac{1}{2}\partial_{\bm{t}}+k_{s}^{2}\bm{t}\cdot\operatorname{V}_{s}[\bm{n}\,\cdot\,]-\operatorname{K}_{s}^{\top}\partial_{\bm{t}}\\\
\frac{1}{2}\partial_{\bm{t}}+k_{p}^{2}\bm{t}\cdot\operatorname{V}_{p}[\bm{n}\,\cdot\,]-\operatorname{K}_{p}^{\top}\partial_{\bm{t}}&-\operatorname{W}_{s}\end{bmatrix}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathcal{H}_{0}\mathrm{D}+\frac{1}{4}\begin{bmatrix}k_{p}^{2}&\\\
&-k_{s}^{2}\end{bmatrix}\operatorname{H}\mathrm{D}+\mathrm{OPS}(-2)$
$\displaystyle{\cal A}_{\rm SL}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}-\frac{1}{2}\operatorname{I}+\operatorname{K}_{p}^{\top}&\partial_{\bm{t}}\operatorname{V}_{s}\\\
\partial_{\bm{t}}\operatorname{V}_{p}&\frac{1}{2}\operatorname{I}-\operatorname{K}_{s}^{\top}\end{bmatrix}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\mathcal{H}_{0}-\frac{1}{4}\begin{bmatrix}&k_{s}^{2}\\\
k_{p}^{2}\end{bmatrix}\operatorname{H}\mathrm{D}_{-2}+\mathrm{OPS}(-3).$
We then have
$\displaystyle{\cal A}_{\rm CFIE}$ $\displaystyle:={\cal A}_{\rm DL}-ik{\cal
A}_{\rm SL}$ (4.5)
$\displaystyle=\frac{1}{2}\left[\operatorname{H}\mathrm{D}+ik\right]\mathcal{H}_{0}+\frac{1}{4}\begin{bmatrix}k_{p}^{2}&\\\
&-k_{s}^{2}\end{bmatrix}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-2).$
###### Theorem 4.1
The ${\cal A}_{\rm CFIE}$ defined in equation (4.5) enjoys the property ${\cal
A}_{\rm CFIE}^{2}\in\mathrm{OPS}(0)$ and is invertible.
Proof. Clearly
$\displaystyle{\cal A}_{\rm CFIE}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\underbrace{\left[\left(\operatorname{H}\mathrm{D}+ik\right)\mathcal{H}_{0}\right]^{2}}_{=0}-\frac{1}{8}\begin{bmatrix}k_{p}^{2}&\\\
&-k_{s}^{2}\end{bmatrix}\mathcal{H}_{0}-\frac{1}{8}\mathcal{H}_{0}\begin{bmatrix}k_{p}^{2}&\\\
&-k_{s}^{2}\end{bmatrix}+\mathrm{OPS}(-1)$ $\displaystyle=$
$\displaystyle-\frac{1}{8}\begin{bmatrix}2k_{p}^{2}\operatorname{I}&-(k_{p}^{2}+k_{s}^{2})\operatorname{H}\\\
-(k_{p}^{2}+k_{s}^{2})\operatorname{H}&-2k_{s}^{2}\operatorname{I}\end{bmatrix}+\mathrm{OPS}(-1).$
The principal part is clearly invertible with
$\begin{bmatrix}2k_{p}^{2}\operatorname{I}&-(k_{p}^{2}+k_{s}^{2})\operatorname{H}\\\
-(k_{p}^{2}+k_{s}^{2})\operatorname{H}&-2k_{s}^{2}\operatorname{I}\end{bmatrix}^{-1}=\frac{1}{(k_{p}-k_{s})^{2}}\begin{bmatrix}-2k_{s}^{2}\operatorname{I}&(k_{p}^{2}+k_{s}^{2})\operatorname{H}\\\
(k_{p}^{2}+k_{s}^{2})\operatorname{H}&2k_{p}^{2}\operatorname{I}\end{bmatrix}$
Therefore, the invertibility of the operator ${\cal A}_{\rm CFIE}^{2}$ is
equivalent to the injectivity of the operator ${\cal A}_{\rm CFIE}$. The
latter, in turn, can be established in a straightforward manner via classical
arguments related to the well posedness of Helmholtz CFIE formulations.
Indeed, assuming $(\varphi_{p},\varphi_{s})^{\top}\in\mathop{\rm Ker}({\cal
A}_{\rm CFIE})$, we define the associated elastic field via the Helmholtz
combined field representations (4.2) with these densities
${\bf u}=\nabla u_{p}+\operatorname{\overrightarrow{\mathrm{curl}}}{u_{s}}$
and we see that ${\bf u}$ is a radiative solution of the Navier equations in
$\Omega^{+}$ satisfying ${\bf u}\cdot\bm{t}=0$ and ${\bf u}\cdot\bm{n}=0$ on
$\Gamma$ and therefore ${\bf u}=0$ on $\Gamma$. We can invoke the uniqueness
result for solutions of exterior Navier problems with Dirichlet boundary
conditions to get that ${\bf u}=0$ in $\Omega^{+}$. Therefore we obtain
$\operatorname{div}{\bf u}=0$ in $\Omega^{+}$ which amounts to $\Delta
u_{p}=0$ in $\Omega^{+}$. Given that $u_{p}$ is a solution of the Helmholtz
equation with wave-number $k_{p}$ in $\Omega^{+}$ we derive $u_{p}=0$ in
$\Omega^{+}$. Then, we have
$\operatorname{\overrightarrow{\mathrm{curl}}}{u_{s}}=0$ and hence $\Delta
u_{s}=0$ in $\Omega^{+}$ which, since $u_{s}$ is a solution of the Helmholtz
equation with wave-number $k_{s}$, implies that $u_{s}=0$ in $\Omega^{+}$.
Finally, using the jump conditions of Helmholtz layer potentials (3.2), we see
that $u_{p}$ and $u_{s}$ are solutions of Helmholtz equations in $\Omega$ with
wave-numbers $k_{p}$ and $k_{s}$ satisfying the Robin conditions
$\partial_{\bm{n}}u_{p}-iku_{p}=0\qquad\partial_{\bm{n}}u_{s}-iku_{s}=0\qquad{\rm
on}\ \Gamma.$
Therefore, $u_{p}=0$ and $u_{s}=0$ in $\Omega$ as well, and hence
$\varphi_{p}=0$ and $\varphi_{s}=0$ on $\Gamma$. $\Box$
Following the Helmholtz paradigm, it would appear more natural to look for
combined field representations of the fields $u_{p}$ and $u_{s}$ in the form
$u_{p}:=\operatorname{DL}_{p}\varphi_{p}-ik_{p}\operatorname{SL}_{p}\varphi_{p}\qquad
u_{p}:=\operatorname{DL}_{s}\varphi_{s}-ik_{s}\operatorname{SL}_{s}\varphi_{s}$
(4.6)
leading to the CFIE formulation
$\widetilde{\cal A}_{\rm CFIE}\begin{bmatrix}\varphi_{p}\\\
\varphi_{s}\end{bmatrix}:=\left({\cal A}_{\rm DL}-\begin{bmatrix}ik_{p}&\\\
&ik_{s}\end{bmatrix}{\cal A}_{\rm SL}\right)\begin{bmatrix}\varphi_{p}\\\
\varphi_{s}\end{bmatrix}=-\begin{bmatrix}{\bf u}^{\rm inc}\cdot\bm{n}\\\ {\bf
u}^{\rm inc}\cdot\bm{t}\end{bmatrix}.$
In this case it can be easily proven that
$\displaystyle\widetilde{\cal A}_{\rm CFIE}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[\left(\operatorname{H}\mathrm{D}+i\begin{bmatrix}k_{p}&\\\
&k_{s}\end{bmatrix}\right)\mathcal{H}_{0}\right]^{2}-\frac{1}{8}\begin{bmatrix}k_{p}^{2}&\\\
&-k_{s}^{2}\end{bmatrix}\mathcal{H}_{0}-\frac{1}{8}\mathcal{H}_{0}\begin{bmatrix}k_{p}^{2}&\\\
&-k_{s}^{2}\end{bmatrix}+\mathrm{OPS}(-1)$ $\displaystyle=$
$\displaystyle\frac{1}{4}i\mathcal{H}_{0}\begin{bmatrix}k_{p}&\\\
&k_{s}\end{bmatrix}\mathcal{H}_{0}\mathrm{D}+\mathrm{OPS}(0)$ $\displaystyle=$
$\displaystyle\frac{i}{4}(k_{p}-k_{s})\mathcal{H}_{0}\operatorname{H}\mathrm{D}+\mathrm{OPS}(0)$
Then $\widetilde{\cal A}_{\rm CFIE}^{2}\in\mathrm{OPS}(1)$ and its principal
part is not invertible.
#### 4.1.2 Regularized formulations
The design of regularized formulations starts with Green’s identities in
$\Omega^{+}$
$u_{p}=\operatorname{DL}_{p}[\gamma
u_{p}]-\operatorname{SL}_{p}[\partial_{\bm{n}}u_{p}]\qquad
u_{s}=\operatorname{DL}_{s}[\gamma
u_{s}]-\operatorname{SL}_{s}[\partial_{\bm{n}}u_{s}].$
The main thrust in regularized formulations [3] is the construction of a
certain regularizing operator ${\cal R}$ that is an approximation of the
operator that maps the left hand side of equation (2.5) to the Dirichlet
Cauchy data $(\gamma u_{p},\gamma u_{s})$ on $\Gamma$ corresponding to the
Helmholtz solutions $u_{p}$ and $u_{s}$ in $\Omega^{+}$. At the core of the
construction of regularizing operators lies the use of coercive approximations
of DtN operators which are typically represented in the form of Fourier square
root multipliers. Once such an operator is constructed, approximations of DtN
operators are used to access the Neumann Cauchy data
$(\partial_{\bm{n}}u_{p},\partial_{\bm{n}}u_{s})$ on $\Gamma$, which in turn
leads via Green’s identities to representations of the fields $(u_{p},u_{s})$
in $\Omega^{+}$. Specifically, using (cf. (A.16) in Theorem A.3)
${\rm
DtN}_{k}=2\operatorname{W}_{k}+\mathrm{OPS}(-2)=\operatorname{H}\mathrm{D}_{1}+\frac{k^{2}}{2}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-2)$
(4.7)
we can easily deduce (cf. (A.12) in Theorem A.3)
$\displaystyle{\rm DtN}_{p}$
$\displaystyle=2\operatorname{W}_{\widetilde{k}_{p}}+\mathrm{OPS}(-1)=\frac{1}{2}\operatorname{H}\mathrm{D}+\mathrm{OPS}(-1)$
(4.8) $\displaystyle{\rm DtN}_{s}$
$\displaystyle=2\operatorname{W}_{\widetilde{k}_{s}}+\mathrm{OPS}(-1)=\frac{1}{2}\operatorname{H}\mathrm{D}+\mathrm{OPS}(-1)$
where
$\widetilde{k}_{p}:=k_{p}+i\varepsilon_{p}\qquad\widetilde{k}_{s}:=k_{s}+i\varepsilon_{s},\qquad\varepsilon_{p}>0,\
\varepsilon_{s}>0.$ (4.9)
We express then the fields $(u_{p},u_{s})$ in the combined field form
$\displaystyle u_{p}$ $\displaystyle=$ $\displaystyle{\rm
DL}_{p}\varphi_{p}-2{\rm
SL}_{p}\operatorname{W}_{\widetilde{k}_{p}}\varphi_{p}$ $\displaystyle u_{s}$
$\displaystyle=$ $\displaystyle{\rm DL}_{s}\varphi_{s}-2{\rm
SL}_{s}\operatorname{W}_{\widetilde{k}_{s}}\varphi_{s}$ (4.10)
through a regularizing operator ${\cal R}$ in terms of boundary functional
densities $(\varphi_{p},\varphi_{s})$
$\begin{bmatrix}\varphi_{p}\\\ \varphi_{s}\end{bmatrix}={\cal
R}\begin{bmatrix}f_{p}\\\ f_{s}\end{bmatrix}$
for appropriate $(f_{p},f_{s})$. The enforcement of boundary conditions (2.5)
on the combined field representation (4.1.2) leads to the CFIER formulation to
be solved for the unknown densities $(f_{p},f_{s})$
${\cal A}^{\rm comb}{\cal R}\begin{bmatrix}f_{p}\\\
f_{s}\end{bmatrix}=\begin{bmatrix}-{\bf u}^{\rm inc}\cdot\bm{n}\\\ -{\bf
u}^{\rm inc}\cdot\bm{t}\end{bmatrix},\quad{\cal A}^{\rm comb}:=\left({\cal
A}_{\rm DL}-2{\cal A}_{\rm
SL}\begin{bmatrix}\operatorname{W}_{\widetilde{k}_{p}}&\\\
&\operatorname{W}_{\widetilde{k}_{s}}\end{bmatrix}\right)$ (4.11)
(See (4.1.1) and (4.1.1)). Provided the regularizing operator ${\cal R}$ is
constructed per the prescriptions above, the CFIER operator in the left hand
side of equation (4.11) is an approximation of the identity operator. Clearly,
the main challenge in this approach is the construction of the regularizing
operator ${\cal R}$. Using the DtN maps ${\rm DtN}_{p}$ and ${\rm DtN}_{s}$,
the _exact_ regularizing operator $\mathcal{R}^{\rm ex}_{\rm D}$ has the
property
$\begin{bmatrix}{\rm DtN}_{p}&\partial_{\bm{t}}\\\ \partial_{\bm{t}}&-{\rm
DtN}_{s}\end{bmatrix}\mathcal{R}^{\rm ex}_{\rm D}=\mathcal{I}.$
and therefore, at least formally,
$\displaystyle\mathcal{R}^{\rm ex}_{\rm D}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}{\rm DtN}_{p}&\partial_{\bm{t}}\\\
\partial_{\bm{t}}&-{\rm DtN}_{s}\end{bmatrix}^{-1}$ $\displaystyle=$
$\displaystyle({\rm DtN}_{p}{\rm
DtN}_{s}+\partial_{\bm{t}}^{2})^{-1}\begin{bmatrix}{\rm
DtN}_{s}&\partial_{\bm{t}}\\\ -\partial_{\bm{t}}&{\rm DtN}_{p}\end{bmatrix}.$
Notice that by (4.7)
${\rm DtN}_{p}{\rm
DtN}_{s}+\partial_{\bm{t}}^{2}=\left(\operatorname{H}\mathrm{D}+\frac{k_{p}^{2}}{4}\operatorname{H}\mathrm{D}_{-1}\right)\left(\operatorname{H}\mathrm{D}+\frac{k_{s}^{2}}{4}\operatorname{H}\mathrm{D}_{-1}\right)+\mathrm{D}_{2}+\mathrm{OPS}(-2)=\frac{k_{p}^{2}+k_{s}^{2}}{4}\operatorname{I}+\mathrm{OPS}(-2)$
which lead us to propose
$\mathcal{R}_{\rm
D}=-\frac{2}{\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}}\mathcal{H}_{0}\operatorname{H}\mathrm{D}+\frac{1}{\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}}\begin{bmatrix}-\widetilde{k}_{s}^{2}\\\
&\widetilde{k}_{p}^{2}\end{bmatrix}\operatorname{H}\mathrm{D}_{-1}+\begin{bmatrix}\mathrm{J}&\\\
&\mathrm{J}\end{bmatrix}$ (4.12)
as an approximation of $\mathcal{R}_{\rm D}^{\rm ex}$. It is easy to see that
$\mathcal{R}_{\rm D}\begin{bmatrix}{\rm DtN}_{p}&\partial_{\bm{t}}\\\
\partial_{\bm{t}}&-{\rm DtN}_{s}\end{bmatrix}=\mathcal{I}+\mathrm{OPS}(-1).$
Moreover ${\cal R}_{\rm D}:H^{s}(\Gamma)\times H^{s}(\Gamma)\to
H^{s-1}(\Gamma)\times H^{s-1}(\Gamma)$, i.e. ${\cal R}_{\rm
D}\in\mathrm{OPS}(1)$, and it is injective since the Fourier matrix principal
symbol is
$\mathcal{R}_{\rm D}(0):=\begin{bmatrix}1&\\\
&1\end{bmatrix},\qquad\mathcal{R}_{\rm
D}(n)=\frac{2}{\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}}\begin{bmatrix}|n|+\frac{\widetilde{k}_{s}^{2}}{2|n|}&-ni\\\
-ni&-|n|-\frac{\widetilde{k}^{2}_{p}}{2|n|}\end{bmatrix}$
which is clearly invertible, but not uniformly invertible, for all $n$.
Alternatively, we can construct an approximation for $\mathcal{R}^{\rm
ex}_{\rm D}$ via the Fourier multipliers
$2\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{p}}]$ and
$2\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{s}}]$ whose principal symbols
are defined (cf. (A.15) in Theorem A.3) as
$2\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{p}}](n)=-(n^{2}-\widetilde{k}_{p}^{2})^{1/2}\qquad
2\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{s}}]=-(n^{2}-\widetilde{k}_{s}^{2})^{1/2}.$
(4.13)
Using formulas (4.13), and taking into account (4.7)–(4.9), we obtain
$\mathrm{OPS}(1)\ni\widetilde{\mathcal{R}}_{\rm D}\approx\mathcal{R}^{\rm
ex}_{\rm D}$, the Fourier multiplier of matrix principal symbol
$\widetilde{\mathcal{R}}_{\rm D}(0)=\begin{bmatrix}1&\\\
&1\end{bmatrix},\quad\widetilde{\mathcal{R}}_{\rm
D}(n):=\Delta(n,\widetilde{k}_{p},\widetilde{k}_{s})\begin{bmatrix}(n^{2}-\widetilde{k}_{s}^{2})^{1/2}&-in\\\
-in&-(n^{2}-\widetilde{k}_{p}^{2})^{1/2}\end{bmatrix},\quad n\neq 0,$ (4.14)
where
$\Delta(n,\widetilde{k}_{p},\widetilde{k}_{s}):=\frac{1}{n^{2}-(n^{2}-\widetilde{k}_{p}^{2})^{1/2}(n^{2}-\widetilde{k}_{s}^{2})^{1/2}}=\frac{n^{2}+(n^{2}-\widetilde{k}_{p}^{2})^{1/2}(n^{2}-\widetilde{k}_{s}^{2})^{1/2}}{(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})n^{2}-\widetilde{k}_{p}^{2}\widetilde{k}_{s}^{2}}.$
(4.15)
Note that $\Delta(n,\widetilde{k}_{p},\widetilde{k}_{s})$ is well defined
since the denominator has positive imaginary part (cf. (4.9)).
Furthermore, since
$\Delta(n,\widetilde{k}_{p},\widetilde{k}_{s})=\frac{2}{\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}}-\frac{\widetilde{k}_{p}^{2}-\widetilde{k}_{s}^{2}}{2(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})n^{2}}+{\cal
O}(n^{-4})$
we easily deduce
$\widetilde{\mathcal{R}}_{\rm D}=\mathcal{R}_{\rm
D}-\frac{\widetilde{k}_{p}^{2}-\widetilde{k}_{s}^{2}}{2(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})}\mathcal{H}_{0}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-2)$
(4.16)
Before entering into the analysis, let us note that
$\displaystyle\mathcal{A}^{\rm comb}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{\rm DL}-2{\cal A}_{\rm
SL}\begin{bmatrix}\operatorname{W}_{\widetilde{k}_{p}}&\\\
&\operatorname{W}_{\widetilde{k}_{s}}\end{bmatrix}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathcal{H}_{0}\operatorname{H}\mathrm{D}+\frac{1}{4}\begin{bmatrix}k_{p}^{2}&\\\
&-k_{s}^{2}\end{bmatrix}\operatorname{H}\mathrm{D}_{-1}$
$\displaystyle+\left(\frac{1}{2}\mathcal{H}_{0}+\frac{1}{4}\begin{bmatrix}&k_{s}^{2}\\\
k_{p}^{2}&\end{bmatrix}\operatorname{H}\mathrm{D}_{-2}\right)\left(\operatorname{H}\mathrm{D}+\frac{1}{2}\begin{bmatrix}\widetilde{k}_{p}^{2}&\\\
&\widetilde{k}_{s}^{2}\end{bmatrix}\operatorname{H}\mathrm{D}_{-1}\right)+\mathrm{OPS}(-2)$
$\displaystyle=$ $\displaystyle\mathcal{A}^{\rm
comb}_{0}+\mathrm{OPS}(-2),\qquad\quad\mathcal{A}^{\rm
comb}_{0}:=\mathcal{H}_{0}\operatorname{H}\mathrm{D}+\frac{1}{4}\begin{bmatrix}(k_{p}^{2}+\widetilde{k}_{p}^{2})\operatorname{I}&(k_{s}^{2}-\widetilde{k}_{s}^{2})\operatorname{H}\\\
(k_{p}^{2}-\widetilde{k}_{p}^{2})\operatorname{H}&-(k_{s}^{2}+\widetilde{k}_{s}^{2})\operatorname{I}\end{bmatrix}\operatorname{H}\mathrm{D}_{-1}$
###### Lemma 4.2
The CFIER operators
$\displaystyle{\cal A}_{\rm CFIER}$ $\displaystyle:={\cal A}^{\rm comb}{\cal
R}_{\rm D},\qquad\widetilde{\cal A}_{\rm CFIER}$ $\displaystyle:={\cal A}^{\rm
comb}\widetilde{\cal R}_{\rm D}.$ (4.17)
are compact perturbations of an invertible pseudodifferential operator of
order 0.
Proof. Clearly (recall that $\mathcal{H}_{0}$ commutes with $\operatorname{H}$
and $\mathrm{D}$ and that $\mathcal{H}_{0}^{2}=0$),
$\displaystyle{\cal A}_{\rm CFIER}$ $\displaystyle=$
$\displaystyle\mathcal{A}^{\rm comb}_{0}\mathcal{R}_{\rm D}+\mathrm{OPS}(-1)$
$\displaystyle=$
$\displaystyle\frac{1}{\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}}\left(\mathcal{H}_{0}\mathrm{D}+\frac{1}{4}\begin{bmatrix}(k_{p}^{2}+\widetilde{k}_{p}^{2})\operatorname{I}&(k_{s}^{2}-\widetilde{k}_{s}^{2})\operatorname{H}\\\
(k_{p}^{2}-\widetilde{k}_{p}^{2})\operatorname{H}&-(k_{s}^{2}+\widetilde{k}_{s}^{2})\operatorname{I}\end{bmatrix}\mathrm{D}_{-1}\right)\left(2\mathcal{H}_{0}\mathrm{D}+\begin{bmatrix}\widetilde{k}_{s}^{2}\\\
&-\widetilde{k}_{p}^{2}\end{bmatrix}\mathrm{D}_{-1}\right)$
$\displaystyle+\mathrm{OPS}(-1)$ $\displaystyle=$
$\displaystyle\frac{1}{2(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})}\begin{bmatrix}(\alpha+\widetilde{\alpha})\operatorname{I}&-(\alpha-\widetilde{\alpha})\operatorname{H}\\\
(\alpha-\widetilde{\alpha})\operatorname{H}&(\alpha+\widetilde{\alpha})\operatorname{I}\end{bmatrix}+\mathrm{OPS}(-1)$
where
$\alpha=k_{p}^{2}+k_{s}^{2},\quad\widetilde{\alpha}=\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}.$
Besides, by (4.16),
$\displaystyle\widetilde{\cal A}_{\rm CFIER}$ $\displaystyle=$
$\displaystyle{\cal A}^{\rm comb}\widetilde{\cal R}_{\rm D}=\mathcal{A}^{\rm
comb}\mathcal{R}_{\rm
D}-\frac{\widetilde{k}_{p}^{2}-\widetilde{k}_{s}^{2}}{2(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})}\mathcal{A}^{\rm
comb}\mathcal{H}_{0}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-1)$
$\displaystyle=$ $\displaystyle{\cal A}_{\rm
CFIER}+\frac{\widetilde{k}_{p}^{2}-\widetilde{k}_{s}^{2}}{2(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})}\underbrace{\left(\mathcal{H}_{0}+\frac{1}{4}\begin{bmatrix}(k_{p}^{2}+\widetilde{k}_{p}^{2})\operatorname{I}&(k_{s}^{2}-\widetilde{k}_{s}^{2})\operatorname{H}\\\
(k_{p}^{2}-\widetilde{k}_{p}^{2})\operatorname{H}&-(k_{s}^{2}+\widetilde{k}_{s}^{2})\operatorname{I}\end{bmatrix}\mathrm{D}_{-2}\right)\mathcal{H}_{0}}_{\in\mathrm{OPS}(-2)}+\mathrm{OPS}(-1).$
In short,
$\widetilde{\cal A}_{\rm CFIER}={\cal A}_{\rm
CFIER}+\mathrm{OPS}(-1)=\frac{1}{2(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})}\begin{bmatrix}(\alpha+\widetilde{\alpha})\operatorname{I}&-(\alpha-\widetilde{\alpha})\operatorname{H}\\\
(\alpha-\widetilde{\alpha})\operatorname{H}&(\alpha+\widetilde{\alpha})\operatorname{I}\end{bmatrix}+\mathrm{OPS}(-1).$
Since
$(\alpha+\widetilde{\alpha})^{2}-(\alpha-\widetilde{\alpha})^{2}=4\alpha\widetilde{\alpha}=4(k_{p}^{2}+k_{s}^{2})(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})\neq
0$
we conclude, cf. (4.1), that the principal part is an invertible operator of
order 0 that finishes the proof. $\Box$
We prove now the main result concerning regularized formulations for the
solution of elastic scattering problems with Dirichlet boundary conditions:
###### Theorem 4.3
The operators ${\cal A}_{\rm CFIER}$ and $\widetilde{\cal A}_{\rm CFIER}$ are
invertible pseudodifferential operators of order zero.
Proof. Since ${\mathcal{R}}_{\rm D}$ and $\widetilde{\mathcal{R}}_{\rm D}$ are
injective, by Fredholm alternative, and in view of Lemma 4.2, it suffices to
show that $\mathcal{A}^{\rm comb}$ cf. (4.11) is injective too. Let
$(\varphi_{p},\varphi_{s})\in\mathop{\rm Ker}({\cal A}_{\rm comb})$ and define
$u_{p},\ u_{s}:\mathbb{R}^{2}\setminus\Gamma\to\mathbb{C}$ given by
$\displaystyle u_{p}$ $\displaystyle:=$ $\displaystyle{\rm
DL}_{p}\varphi_{p}-2{\rm
SL}_{p}\operatorname{W}_{\widetilde{k}_{p}}\varphi_{p},$ $\displaystyle u_{s}$
$\displaystyle:=$ $\displaystyle{\rm DL}_{s}\varphi_{s}-2{\rm
SL}_{s}\operatorname{W}_{\widetilde{k}_{s}}\varphi_{s}.$
From the jump relations cf. (3.2)
$[\partial_{\bm{n}}u_{p}]-2\operatorname{W}_{p}[\gamma
u_{p}]=0,\quad[\partial_{\bm{n}}u_{s}]-2\operatorname{W}_{s}[\gamma u_{s}]=0.$
But on account of the uniqueness of solutions of the elastic scattering
problem with Dirichlet boundary conditions in $\Omega^{+}$, we obtain via the
same arguments as in Theorem 4.1 that $u_{p}=0$ and $u_{s}=0$ in $\Omega^{+}$.
Hence
$\partial_{\bm{n}}^{-}u_{p}-2\operatorname{W}_{\widetilde{k}_{p}}\gamma^{-}u_{p}=0\qquad\partial_{\bm{n}}^{-}u_{s}-2\operatorname{W}_{\widetilde{k}_{s}}\gamma^{-}u_{s}=0\quad{\rm
on}\ \Gamma.$
Given the coercivity property [2]
$\Im\int_{\Gamma}\operatorname{W}_{\widetilde{k}}f\
\overline{f}>0,\qquad\Im\widetilde{k}>0,\qquad f\neq 0.$
But as consequence of the Green Identity
$\Im\int_{\Gamma}\partial_{\bm{n}}^{-}u_{p}\overline{u}_{p}=\Im\int_{\Gamma}\partial_{\bm{n}}^{-}u_{s}\overline{u}_{s}=0$
from where we conclude that $u_{p}=0$ and $u_{s}=0$ in $\Omega$. Consequently,
it follows that $\varphi_{p}=0$ and $\varphi_{s}=0$ on $\Gamma$ and the result
is proven. $\Box$
In an effort to simplify the regularizing operator, we can consider an
alternative regularizer ${\cal R}_{{\rm D},1}\in\mathrm{OPS}(1)$ which is
defined as the Fourier multiplier matrix operator whose symbol is given by the
formula
$\mathcal{R}_{{\rm
D},1}(n):=\frac{2}{\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}}\begin{bmatrix}(n^{2}-\widetilde{k_{s}}^{2})^{1/2}&-in\\\
-in&-(n^{2}-\widetilde{k}_{p}^{2})^{1/2}\end{bmatrix}.$ (4.18)
The regularizing operator defined in equation (4.18) defines also a well posed
CFIER formulation since
$\mathcal{R}_{{\rm D},1}-\mathcal{R}_{{\rm D}}=\mathrm{OPS}(-2)$
(see (4.12)) and therefore (see Lemma 4.2)
${\cal A}^{\rm comb}\mathcal{R}_{{\rm D},1}={\cal A}_{\rm
CFIER}+\mathrm{OPS}(-1).$
Interestingly, it is possible to propose a different regularizing operator
that involves Helmholtz BIOs rather than Fourier multipliers. Indeed, since
the operator
$\mathcal{R}_{{\rm
D},2}:=\frac{1}{\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}}\begin{bmatrix}2\operatorname{W}_{\widetilde{k}_{p}}&\partial_{\bm{t}}\\\
\partial_{\bm{t}}&-2\operatorname{W}_{\widetilde{k}_{s}}\end{bmatrix}.$ (4.19)
has the property (cf. Theorem A.3)
$\mathcal{R}_{{\rm D},2}-\mathcal{R}_{{\rm D},1}\in\mathrm{OPS}(-3)$
we can see that the ensuing CFIER formulations based on the operator
$\mathcal{R}_{{\rm D},2}$ lead again to Fredholm operators of index 0. In
order to establish the invertibility of the CFIER operator in this case we
rely on the following result
###### Lemma 4.4
The operator $\mathcal{R}_{{\rm D},2}$ is an injective pseudodifferential
operator of order $1$.
Proof. Assume $(\varphi_{p},\varphi_{s})\in\mathop{\rm Ker}(\mathcal{R}_{{\rm
D},2})$, that is
$\displaystyle
2\operatorname{W}_{\widetilde{k}_{p}}\varphi_{p}+\partial_{\bm{t}}\varphi_{s}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\partial_{\bm{t}}\varphi_{p}-2\operatorname{W}_{\widetilde{k}_{s}}\varphi_{s}$
$\displaystyle=$ $\displaystyle 0.$
Then
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\int_{\Gamma}\overline{\varphi_{p}}\,{\operatorname{W}_{\widetilde{k}_{p}}\varphi_{p}}+2\int_{\Gamma}\overline{\varphi}_{s}\,\operatorname{W}_{\widetilde{k}_{s}}\varphi_{s}+\int_{\Gamma}\overline{\varphi}_{p}\partial_{\bm{t}}\varphi_{s}-\int_{\Gamma}\overline{\varphi}_{s}\partial_{\bm{t}}\varphi_{p}$
$\displaystyle=$ $\displaystyle
2\int_{\Gamma}\overline{\varphi_{p}}\,{\operatorname{W}_{\widetilde{k}_{p}}\varphi_{p}}+2\int_{\Gamma}\overline{\varphi}_{s}\,\operatorname{W}_{\widetilde{k}_{s}}+\int_{\Gamma}\overline{\varphi_{p}}\partial_{\bm{t}}\varphi_{s}+\int_{\Gamma}\overline{\partial_{\bm{t}}\varphi}_{s}\varphi_{p}.$
(Notice that integration by parts has been applied in the last step). Then,
$\Im\left(\int_{\Gamma}\overline{\varphi}_{p}\,{\operatorname{W}_{\widetilde{k}_{p}}\varphi_{p}}+\int_{\Gamma}\overline{\varphi}_{s}\,\operatorname{W}_{\widetilde{k}_{s}}{\varphi}_{s}\right)=0$
Given that for $\Im\widetilde{k}>0$
$\int_{\Gamma}\operatorname{W}_{\widetilde{k}}\varphi\overline{\varphi}>0,\qquad\text{for
}\varphi\neq 0$
we conclude $\varphi_{p}=\varphi_{s}=0$, which completes the proof. $\Box$
###### Remark 4.5
It is possible to replace the DtN approximations ${\rm DtN}_{p}\approx
2\operatorname{W}_{\widetilde{k}_{p}}$ and ${\rm DtN}_{s}\approx
2\operatorname{W}_{\widetilde{k}_{s}}$ by the Fourier multipliers
$2\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{p}}]$ and respectively
$2\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{s}}]$ defined in equations
(4.13) in the construction of the CFIER operators considered above. In that
case, the CFIER operators are of the form
${\cal A}_{\rm CFIER}^{\rm PS}=\left({\cal A}_{\rm DL}-2{\cal A}_{\rm
SL}\begin{bmatrix}\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{p}}]&\\\
&\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{s}}]\end{bmatrix}\right){\cal
R}_{\rm D}.$ (4.20)
Given that
$\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{p}}]-\operatorname{W}_{\widetilde{k}_{p}}\in\mathrm{OPS}(-3)$
and
$\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{s}}]-\operatorname{W}_{\widetilde{k}_{s}}\in\mathrm{OPS}(-3)$,
and $\Im\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{p}}]>0,\
\Im\mathrm{PS}[\operatorname{W}_{\widetilde{k}_{s}}]>0$ the CFIER formulations
define in equations (4.20) are well posed.
### 4.2 Neumann Problem
#### 4.2.1 Double and single layer formulations
We turn our attention to the Neumann problem where the Helmholtz fields
$u_{p}$ and $u_{s}$ in the Helmholtz decomposition (2.4) satisfy the system of
boundary conditions (2.6). Looking for Helmholtz double layer potential
representations of $u_{p}$ and $u_{s}$ in $\Omega^{+}$ of the form
$u_{p}={\rm DL}_{p}\,\varphi_{p},\quad u_{s}={\rm DL}_{s}\,\varphi_{s}$
the system of boundary conditions (2.6) can be recast as a system of BIE
involving the matrix operator ${\cal B}_{\rm DL}$ defined as:
$\displaystyle{\cal B}_{\rm DL}\begin{bmatrix}\varphi_{p}\\\
\varphi_{s}\end{bmatrix}\ :=$
$\displaystyle\mu\begin{bmatrix}-\partial_{\bm{t}}^{2}\varphi_{p}&2{\bm{t}}\cdot\operatorname{W}_{s}(\partial_{\bm{t}}\varphi_{s}\,\bm{t})\\\
2{\bm{t}}\cdot\operatorname{W}_{p}(\partial_{\bm{t}}\varphi_{p}\,\bm{t})&\partial_{\bm{t}}^{2}\varphi_{s}\end{bmatrix}+\mu\begin{bmatrix}2{\bm{n}}\cdot\operatorname{W}_{p}(\partial_{\bm{t}}\varphi_{p}\,\bm{t})&-\kappa\partial_{\bm{t}}\varphi_{s}\\\
-\kappa\partial_{\bm{t}}\varphi_{p}&-2{\bm{n}}\cdot\operatorname{W}_{s}(\partial_{\bm{t}}\varphi_{s}\,\bm{t})\end{bmatrix}$
(4.21) $\displaystyle-\mu\begin{bmatrix}k_{p}^{2}\varphi_{p}&\\\
&-k_{s}^{2}\varphi_{s}\end{bmatrix}+2\mu\begin{bmatrix}\bm{n}\cdot\operatorname{K}_{p}^{\top}(\partial_{\bm{t}}^{2}\varphi_{p}\,\bm{n})&\\\
&-\bm{n}\cdot\operatorname{K}_{s}^{\top}(\partial_{\bm{t}}^{2}\varphi_{s}\,\bm{n})\end{bmatrix}$
$\displaystyle+2\mu\begin{bmatrix}&{\bm{t}}\cdot\left(\operatorname{K}_{p}^{\top}(\partial_{\bm{t}}^{2}\varphi_{p}\,\bm{n})+\operatorname{K}_{p}^{\top}(\partial_{\bm{t}}\varphi\,{\bm{t}})\right)\\\
{\bm{t}}\cdot\left(\operatorname{K}_{s}^{\top}(\partial_{\bm{t}}^{2}\varphi_{s}\,\bm{n})+\operatorname{K}_{s}^{\top}(\partial_{\bm{t}}\varphi\,{\bm{t}})\right)&\end{bmatrix}$
$\displaystyle+2\mu\begin{bmatrix}\bm{n}\cdot\left(k_{p}^{2}\operatorname{K}_{p}^{\top}(\varphi_{p}\,\bm{n})+\operatorname{K}_{p}^{\top}(\kappa\partial_{\bm{t}}\varphi_{p}\bm{t})\right)&\\\
&-\bm{n}\cdot\left(k_{s}^{2}\operatorname{K}_{s}^{\top}(\varphi_{s}\,\bm{n})+\operatorname{K}_{s}^{\top}(\kappa\partial_{\bm{t}}\varphi_{s}\bm{t})\right)\end{bmatrix}$
$\displaystyle+2\mu\begin{bmatrix}&k_{s}^{2}{\bm{t}}\cdot\operatorname{K}_{s}^{\top}(\varphi_{p}\,\bm{n})\\\
k_{p}^{2}{\bm{t}}\cdot\operatorname{K}_{p}^{\top}(\varphi_{s}\,\bm{n})&\end{bmatrix}-\begin{bmatrix}\lambda
k_{p}^{2}(\frac{1}{2}\varphi_{p}+\operatorname{K}_{p}\varphi_{p})\\\ &\mu
k_{s}^{2}(\frac{1}{2}\varphi_{s}+\operatorname{K}_{s}\varphi_{s})\end{bmatrix}.$
Applying the results in Theorem 3.6 and A.5 (or alternatively, (3.21) in
Proposition 3.8) we can show that
${\cal B}_{\rm DL}:=-\mu{\cal H}_{0}\partial_{\bm{t}}^{2}-\mu{\cal
H}_{0}H\kappa\partial_{\bm{t}}-\frac{1}{2}\mu\begin{bmatrix}2k_{p}^{2}\operatorname{I}&-k_{s}^{2}\mathrm{H}\\\
-k_{p}^{2}\mathrm{H}&-k_{s}^{2}\operatorname{I}\end{bmatrix}-\frac{\lambda}{2}\begin{bmatrix}k_{p}^{2}\operatorname{I}&0\\\
0&0\end{bmatrix}+\mathrm{OPS}(-1).$ (4.22)
On the other hand, if we seek fields $u_{p}$ and $u_{s}$ in the form of
Helmholtz single layer potentials with wave-numbers $k_{p}$ and $k_{s}$, that
is
$u_{p}={\rm SL}_{p}\,\varphi_{p},\quad u_{s}={\rm SL}_{s}\,\varphi_{s},$
the system of boundary conditions (2.6) gives rise to the system of BIE
involving the matrix operator ${\cal B}_{\rm SL}$ defined as
$\begin{aligned} {\cal B}_{\rm SL}\begin{bmatrix}\varphi_{p}\\\
\varphi_{s}\end{bmatrix}\ =\
&\mu\begin{bmatrix}-2\bm{n}\cdot\operatorname{W}_{k}(\varphi_{p}\,\bm{n})\Big{]}&-\partial_{\bm{t}}\varphi_{s}\\\
-\partial_{\bm{t}}\varphi_{p}&2\bm{n}\cdot\operatorname{W}_{s}(\varphi_{s}\,\bm{n})\Big{]}\end{bmatrix}+\mu\begin{bmatrix}\kappa\varphi_{p}&-2{\bm{t}}\cdot\operatorname{W}_{k}(\varphi_{s}\,\bm{n})\\\
-2{\bm{t}}\cdot\operatorname{W}_{k}(\varphi_{p}\,\bm{n})&-\kappa\varphi_{s}\end{bmatrix}\\\
&+2\mu\begin{bmatrix}-\bm{n}\cdot\operatorname{K}_{p}^{\top}(\kappa\varphi_{p}\,\bm{n})&{\bm{t}}\cdot\operatorname{K}_{s}^{\top}(\varphi_{s}\,\bm{t})\\\
\bm{t}\cdot\operatorname{K}_{p}^{\top}(\varphi_{p}\,\bm{t})&{\bm{n}}\cdot\operatorname{K}_{s}^{\top}(\varphi_{s}\,\bm{n})\end{bmatrix}+2\mu\begin{bmatrix}\bm{n}\cdot\operatorname{K}_{p}^{\top}(\varphi_{p}\,\bm{t})&-{\bm{t}}\cdot\operatorname{K}_{s}^{\top}(\kappa\varphi_{s}\,\bm{n})\\\
-{\bm{t}}\cdot\operatorname{K}_{p}^{\top}(\kappa\varphi_{p}\,\bm{n})&-\bm{n}\cdot\operatorname{K}_{s}^{\top}(\varphi_{s}\,\bm{t})\end{bmatrix}\\\
&-\begin{bmatrix}\lambda k_{p}^{2}\operatorname{V}_{p}\varphi_{p}\\\ &\mu
k_{s}^{2}\operatorname{V}_{s}\varphi_{s}\end{bmatrix}\end{aligned}.$ (4.23)
Theorem 3.6 and A.5 (or alternatively, (3.20) in Proposition 3.8) implies
${\cal B}_{\rm SL}=-\mu{\cal H}_{0}\operatorname{H}\mathrm{D}+\mu{\cal
H}_{0}\kappa-\frac{1}{2}(\lambda+\mu)k_{p}^{2}\begin{bmatrix}\operatorname{I}&0\\\
0&0\end{bmatrix}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-2).$ (4.24)
Looking for fields $u_{p}$ and $u_{s}$ in the combined field form (4.2) we are
led to a CFIE formulation for the Neumann problem with the underlying operator
${\cal B}_{\rm CFIE}={\cal B}_{\rm DL}-ik{\cal B}_{\rm SL}.$
In this case, following the same approach as in the Dirichlet case, we obtain
by combining formulas (4.22) and (4.24)
$\displaystyle{\cal B}_{\rm CFIE}^{2}$ $\displaystyle\ =\
\frac{1}{2}\mu\begin{bmatrix}k_{p}^{2}\left(2\lambda+3\mu)-k_{s}^{2}\mu\right)\operatorname{I}&-\left(k_{p}^{2}(\lambda+2\mu)-k_{s}^{2}\mu\right)\operatorname{H}\\\
-\left(k_{p}^{2}(\lambda+2\mu)-k_{s}^{2}\mu\right)\operatorname{H}&\mu\left(k_{s}^{2}-k_{p}^{2}\right)\operatorname{I}\end{bmatrix}\mathrm{D}_{2}+\mathrm{OPS}(1)$
(4.25) $\displaystyle\ =\
\frac{\mu(\mu+\lambda)}{2(\lambda+2\mu)}\omega^{2}\begin{bmatrix}\operatorname{I}&\\\
&\operatorname{I}\end{bmatrix}(\mathrm{D}^{2}-\operatorname{I})+\mathrm{OPS}(1),\quad\text{(by
\eqref{eq:ks:kp})}$
(We have used the definition of the wave-lengths $k_{p}$ and $k_{s}$ given in
(2.3)). Since the leading order operator in formulas (4.25) is invertible,
because so is $\mathrm{D}^{2}-\operatorname{I}$, the invertibility of the CFIE
operator can be established using identical arguments as in the Dirichlet case
(the only notable difference is that the uniqueness of solutions of Navier
equations with Neumann boundary conditions in $\Omega^{+}$ has to be invoked).
We note that both operator ${\cal B}_{\rm CFIE}\in\mathrm{OPS}(2)$ and ${\cal
B}_{\rm CFIE}^{2}\in\mathrm{OPS}(2)$, and as such the CFIE formulations are
not of the second kind. We describe in what follows a method that delivers
robust BIE formulations of the second kind in the Neumann case.
###### Remark 4.6
As in the Dirichlet case we can be tempted to use the combined potential (4.6)
but, as in Dirichlet case, the principal part of the square resulting operator
is not invertible since it is given by
$-i(k_{p}-k_{s})\mathcal{H}_{0}\operatorname{H}\mathrm{D}_{3}.$
#### 4.2.2 Regularized formulations
The main idea in the derivations of regularized BIE formulations for the
boundary value system (2.6) is the construction of suitable approximations to
the operator ${\cal R}_{\rm N}^{\rm ex}$ that maps the left hand side of the
system (2.6) to the Cauchy data of $(u_{p},u_{s})$ on $\Gamma$. Starting with
the formula
$\nabla=\bm{n}\partial_{\bm{n}}+\bm{t}\partial_{\bm{t}}$
we get
$\displaystyle\nabla\nabla^{\top}$ $\displaystyle=$
$\displaystyle(\bm{n}\partial_{\bm{n}}+\bm{t}\partial_{\bm{t}})(\bm{n}^{\top}\partial_{\bm{n}}+\bm{t}^{\top}\partial_{\bm{t}})$
(4.26) $\displaystyle=$
$\displaystyle\bm{n}\bm{n}^{\top}\partial_{\bm{n}}^{2}+\bm{t}\bm{t}^{\top}\partial_{\bm{t}}^{2}+\bm{n}\bm{t}^{\top}\partial_{\bm{n}}\partial_{\bm{t}}+\bm{t}\bm{n}^{\top}\partial_{\bm{t}}\partial_{\bm{n}}-\kappa\bm{n}\bm{n}^{\top}\partial_{\bm{n}}+\kappa\bm{t}\bm{t}^{\top}\partial_{\bm{n}}.$
Again, using DtN operators whenever the normal derivative $\partial_{\bm{n}}$
appears in the last formula above, and neglecting the lower order
contributions that contain the curvature in formulas (4.26), a regularizing
operator ${\cal R}_{\rm N}$ that satisfies the identity
${\cal R}_{\rm N}\approx{\cal R}_{\rm N}^{\rm ex},\quad\text{with }\quad{\cal
R}_{\rm N}^{\rm ex}:=\begin{bmatrix}2\mu{\rm DtN}_{p}^{2}-\lambda
k_{p}^{2}\operatorname{I}&2\mu\partial_{\bm{t}}{\rm DtN}_{s}\\\
2\mu\partial_{\bm{t}}{\rm DtN}_{p}&-2\mu{\rm DtN}_{s}^{2}-\mu
k_{s}^{2}\operatorname{I}\end{bmatrix}^{-1}$ (4.27)
is suggested. Straightforward calculations, with the usual approximation for
operators ${\rm DtN}_{p}$ and ${\rm DtN}_{s}$, shows that, at least formally,
$({\cal R}_{\rm N}^{\rm ex})^{-1}=\begin{bmatrix}2\mu{\rm DtN}_{p}^{2}-\lambda
k_{p}^{2}\operatorname{I}&2\mu\partial_{\bm{t}}{\rm DtN}_{s}\\\
2\mu\partial_{\bm{t}}{\rm DtN}_{p}&-2\mu{\rm DtN}_{s}^{2}-\mu
k_{s}^{2}\operatorname{I}\end{bmatrix}=2\mu\begin{bmatrix}n^{2}-\frac{1}{2}k_{s}^{2}&-in(n^{2}-{k}_{s}^{2})^{1/2}\\\
-in(n^{2}-{k}_{p}^{2})^{1/2}&\frac{1}{2}k_{s}^{2}-n^{2}\end{bmatrix}+\mathrm{OPS}(-1)$
In order to ensure the invertibility of the principal part we consider
complexified $\widetilde{k}_{p}$ and $\widetilde{k}_{s}$ in the off-diagonal
terms. That is, we define ${\cal R}_{\rm N}$ so that
${\cal R}_{\rm
N}(n):=\frac{1}{2\mu}\begin{bmatrix}n^{2}-\frac{1}{2}k_{s}^{2}&-in(n^{2}-\widetilde{k}_{s}^{2})^{1/2}\\\
-in(n^{2}-\widetilde{k}_{p}^{2})^{1/2}&\frac{1}{2}k_{s}^{2}-n^{2}\end{bmatrix}^{-1}=\frac{1}{2\mu}\Delta(n)\begin{bmatrix}-(n^{2}-\frac{1}{2}k_{s}^{2})&in(n^{2}-\widetilde{k}_{s}^{2})^{1/2}\\\
in(n^{2}-\widetilde{k}_{p}^{2})^{1/2}&n^{2}-\frac{1}{2}k_{s}^{2}\end{bmatrix}$
(4.28)
with
$\Delta^{-1}(n)=-\left(n^{2}-\frac{1}{2}k_{s}^{2}\right)^{2}+n^{2}(n^{2}-\widetilde{k}_{p}^{2})^{1/2}(n^{2}-\widetilde{k}_{s}^{2})^{1/2}\neq
0.$
(Notice that $\Delta$ is well defined since $\Im\Delta^{-1}(n)<0$). Then it
can be proved after some direct but tedious calculations that
$\displaystyle{\cal R}_{\rm N}\ =$
$\displaystyle\frac{1}{2\mu}\left(\frac{2}{\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2}}\mathrm{D}_{-2}+\frac{2k_{s}^{4}+(\widetilde{k}_{p}^{2}-\widetilde{k}_{s}^{2})^{2}}{2(k_{p}^{2}+\widetilde{k}_{s}^{2}-2k_{s}^{2})^{2}}\mathrm{D}_{-4}\right)\left(\mathcal{H}_{0}\mathrm{D}_{2}-\frac{1}{2}\begin{bmatrix}k_{s}^{2}\operatorname{I}&-\widetilde{k}_{s}^{2}\operatorname{H}\\\
-\widetilde{k}_{p}^{2}\operatorname{H}&-k_{s}^{2}\operatorname{I}\end{bmatrix}\right)$
(4.29) $\displaystyle\quad+\mathrm{OPS}(-4)$ $\displaystyle=$
$\displaystyle\frac{1}{2\mu}\left(\frac{2}{\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2}}\mathcal{I}+\frac{2k_{s}^{4}+(\widetilde{k}_{p}^{2}-\widetilde{k}_{s}^{2})^{2}}{(\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2})^{2}}\mathrm{D}_{-2}\right)\mathcal{H}_{0}$
$\displaystyle-\frac{1}{2\mu(\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2})}\begin{bmatrix}k_{s}^{2}\operatorname{I}&-\widetilde{k}_{s}^{2}\operatorname{H}\\\
-\widetilde{k}_{p}^{2}\operatorname{H}&-k_{s}^{2}\operatorname{I}\end{bmatrix}\mathrm{D}_{-2}+\mathrm{OPS}(-4)$
###### Remark 4.7
It is possible to used complexified wave-numbers for the definition of the
Fourier multipliers in equations (4.13) when we consider approximations of DtN
operators in equation (4.27) and to consider only the pseudodifferential
operators of order 2 that constitute the leading order contribution to the
expression on the left hand side of equation (2.6). In that case, we are led
to solve the following matrix equation
$2\mu\begin{bmatrix}n^{2}-\widetilde{k}_{p}^{2}&-in(n^{2}-\widetilde{k}_{s}^{2})^{1/2}\\\
-in(n^{2}-\widetilde{k}_{s}^{2})^{1/2}&\widetilde{k}_{s}^{2}-n^{2}\end{bmatrix}{\cal
R}_{\rm N}(n)=\mathcal{I}.$ (4.30)
Although such a choice works equally well in practice, it is more difficult to
establish the unique solvability of equations (4.30). Using the actual wave-
numbers $k_{p}$ and $k_{s}$ when approximating the operators ${\rm
DtN}_{p}^{2}$ and respectively ${\rm DtN}_{s}^{2}$ is also more intuitive if
we write the Laplacian operator $\Delta$ in the $(\bm{n},\bm{t})$ frame and
taking into account the fact that $u_{p}$ and $u_{s}$ are solutions of the
Helmholtz equation with those wave-numbers.
Using the regularizing operator defined in equation (4.28) we employ the same
strategy as in the Dirichlet case and we arrive at the CFIER formulations
${\cal B}^{\rm comb}{\cal R}_{\rm N}\begin{bmatrix}g^{R}_{p}\\\
g^{R}_{s}\end{bmatrix}=-\begin{bmatrix}T{\bf u}^{\rm inc}\cdot{\bm{n}}\\\
T{\bf u}^{\rm inc}\cdot{\bm{t}}\end{bmatrix}$ (4.31)
with, see (4.22), (4.23) and (4.8),
$\displaystyle{\cal B}^{\rm comb}$ $\displaystyle\ :={\cal B}_{\rm DL}-2{\cal
B}_{\rm SL}\begin{bmatrix}\operatorname{W}_{\widetilde{k}_{p}}&\\\
&\operatorname{W}_{\widetilde{k}_{s}}\end{bmatrix}$ (4.32)
$\displaystyle=-2\mu\mathcal{H}_{0}(\mathrm{D}_{2}+\kappa\operatorname{H}\mathrm{D})-\frac{1}{2}\mu\mathcal{H}_{0}\begin{bmatrix}\widetilde{k}_{p}^{2}&\\\
&\widetilde{k}_{s}^{2}\end{bmatrix}+\frac{1}{4}\begin{bmatrix}-3k_{p}^{2}(\lambda+2\mu)\operatorname{I}&2k_{s}^{2}\mu\operatorname{H}\\\
2k_{p}^{2}\mu\operatorname{H}&3k_{s}^{2}\mu\operatorname{I}\\\
\end{bmatrix}+\mathrm{OPS}(-1).$
###### Lemma 4.8
It holds
$\displaystyle{\cal B}^{\rm comb}{\cal R}_{\rm N}$ $\displaystyle\ =\
\begin{bmatrix}\left(2\mu\left(-3\widetilde{k}_{p}^{2}+3k_{s}^{2}+\widetilde{k}_{s}^{2}\right)-3k_{p}^{2}(\lambda+2\mu)\right)\operatorname{I}&\left(3k_{p}^{2}(\lambda+2\mu)+2\mu\left(\widetilde{k}_{p}^{2}+k_{s}^{2}-3\widetilde{k}_{s}^{2}\right)\right)\operatorname{H}\\\
\mu\left(2k_{p}^{2}+6\widetilde{k}_{p}^{2}-7k_{s}^{2}-2\widetilde{k}_{s}^{2}\right)\operatorname{H}&\mu\left(2k_{p}^{2}+2\widetilde{k}_{p}^{2}+k_{s}^{2}-6\widetilde{k}_{s}^{2}\right)\operatorname{I}\end{bmatrix}$
$\displaystyle\ \qquad+\mathrm{OPS}(-1).$
Furthermore, the principal part is an invertible operator of order zero.
Proof. Using (4.29) (recall again that ${\cal H}_{0}^{2}=0$) we conclude
$\begin{aligned} \bigg{(}{\cal B}_{\rm DL}&-2{\cal B}_{\rm
SL}\begin{bmatrix}\operatorname{W}_{\widetilde{k}_{p}}&\\\
&\operatorname{W}_{\widetilde{k}_{s}}\end{bmatrix}\bigg{)}{\cal R}_{\rm N}\\\
&\ =\
\frac{1}{\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2}}\bigg{(}\mathcal{H}_{0}\begin{bmatrix}k_{s}^{2}\operatorname{I}&-\widetilde{k}_{s}^{2}\operatorname{H}\\\
-\widetilde{k}_{p}^{2}\operatorname{H}&-k_{s}^{2}\operatorname{I}\end{bmatrix}-\frac{1}{2}{\cal
H}_{0}\begin{bmatrix}\widetilde{k}_{p}^{2}&\\\
&\widetilde{k}_{s}^{2}\end{bmatrix}{\cal H}_{0}\\\
&\qquad-\frac{1}{4\mu}\begin{bmatrix}-3k_{p}^{2}(\lambda+2\mu)\operatorname{I}&2k_{s}^{2}\mu\operatorname{H}\\\
2k_{p}^{2}\mu\operatorname{H}&3k_{s}^{2}\mu\operatorname{I}\\\
\end{bmatrix}\mathcal{H}_{0}\bigg{)}+\mathrm{OPS}(-1)\end{aligned}.$
from where the first result follows. The principal part is invertible since
the determinant of the principal part (cf. (4.1)) is given (see Lemma below)
by
$\frac{(3\lambda+4\mu)k_{p}^{2}+k_{s}^{2}\mu}{4\mu\left(\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2}\right)}=\frac{(2\lambda+3\mu)k_{p}^{2}}{2(\mu(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2})-2w^{2})}\neq
0.$
$\Box$
###### Lemma 4.9
The matrix
$A=\frac{1}{4\mu(\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2})}\begin{bmatrix}2\mu\left(-3\widetilde{k}_{p}^{2}+3k_{s}^{2}+\widetilde{k}_{s}^{2}\right)-3k_{p}^{2}(\lambda+2\mu)&\left(3k_{p}^{2}(\lambda+2\mu)+2\mu\left(\widetilde{k}_{p}^{2}+k_{s}^{2}-3\widetilde{k}_{s}^{2}\right)\right)i\\\
\mu\left(2k_{p}^{2}+6\widetilde{k}_{p}^{2}-7k_{s}^{2}-2\widetilde{k}_{s}^{2}\right)i&\mu\left(2k_{p}^{2}+2\widetilde{k}_{p}^{2}+k_{s}^{2}-6\widetilde{k}_{s}^{2}\right)\end{bmatrix}$
is invertible with
$\det
A=\frac{k_{p}^{2}(3\lambda+4\mu)+k_{s}^{2}\mu}{4\mu\left(\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2}\right)}\neq
0.$
Proof. Notice that
$A\begin{bmatrix}i\\\ 1\end{bmatrix}=-\begin{bmatrix}i\\\ 1\end{bmatrix}.$
Hence
$\det A=-(1+\mathop{\rm Tr}(A))$
from where the result follows. $\Box$
Just like in the Dirichlet case (see (4.18)), we can construct simpler
regularizing operators starting from equation (4.28) and using asymptotic
arguments. For instance, let us consider the alternative regularizing operator
${\cal R}_{{\rm N},1}\in\mathrm{OPS}(0)$ defined as the Fourier multiplier
whose symbol is given by
${\cal R}_{{\rm
N},1}(n)=\begin{bmatrix}1&-in(n^{2}-\widetilde{k}_{s}^{2})^{-1/2}\\\
-in(n^{2}-\widetilde{k}_{p}^{2})^{-1/2}&-1\end{bmatrix}.$ (4.33)
It is straightforward to see that
${\cal R}_{{\rm
N},1}=\mathcal{H}_{0}-\frac{1}{2}\begin{bmatrix}0&\widetilde{k}_{s}^{2}\operatorname{I}\\\
\widetilde{k}_{p}^{2}\operatorname{I}&0\end{bmatrix}\operatorname{H}\mathrm{D}_{-2}+\mathrm{OPS}(-4)$
(4.34)
which is considerably simpler than ${\cal R}_{\rm N}$ (cf. (4.28)) and
satisfies
${\cal R}_{{\rm
N},1}=\frac{1}{\mu(\widetilde{k}_{p}^{2}-2k_{s}^{2}+\widetilde{k}_{s}^{2})}{\cal
R}_{\rm N}+\mathrm{OPS}(-2).$
Next, we establish
###### Lemma 4.10
It holds
$\displaystyle{\cal B}^{\rm comb}{\cal R}_{{\rm N},1}$ $\displaystyle\ =\
\begin{bmatrix}\left(2\mu\left(\widetilde{k}_{p}^{2}+k_{s}^{2}+\widetilde{k}_{s}^{2}\right)-3k_{p}^{2}(\lambda+2\mu)\right)\operatorname{I}&\left(3k_{p}^{2}(\lambda+2\mu)+2\mu\left(\widetilde{k}_{p}^{2}-k_{s}^{2}+\widetilde{k}_{s}^{2}\right)\right)\operatorname{H}\\\
\mu\left(2k_{p}^{2}-2\widetilde{k}_{p}^{2}-3k_{s}^{2}-2\widetilde{k}_{s}^{2}\right)\operatorname{H}&\mu\left(2k_{p}^{2}+2\widetilde{k}_{p}^{2}-3k_{s}^{2}+2\widetilde{k}_{s}^{2}\right)\operatorname{I}\end{bmatrix}$
(4.35) $\displaystyle\ \qquad+\mathrm{OPS}(-1).$
Moreover, the principal part of this operator given above is an invertible
operator of order zero.
Proof. Similarly as before, using (4.34) we obtain
$\displaystyle\left({\cal B}_{\rm DL}-2{\cal B}_{\rm
SL}\begin{bmatrix}\operatorname{W}_{\widetilde{k}_{p}}&\\\
&\operatorname{W}_{\widetilde{k}_{s}}\end{bmatrix}\right){\cal R}_{{\rm N},1}$
$\displaystyle\qquad=\
\frac{1}{4}\begin{bmatrix}\left(2\mu\left(\widetilde{k}_{p}^{2}+k_{s}^{2}+\widetilde{k}_{s}^{2}\right)-3k_{p}^{2}(\lambda+2\mu)\right)\operatorname{I}&\left(3k_{p}^{2}(\lambda+2\mu)+2\mu\left(\widetilde{k}_{p}^{2}-k_{s}^{2}+\widetilde{k}_{s}^{2}\right)\right)\operatorname{H}\\\
\mu\left(2k_{p}^{2}-2\widetilde{k}_{p}^{2}-3k_{s}^{2}-2\widetilde{k}_{s}^{2}\right)\operatorname{H}&\mu\left(2k_{p}^{2}+2\widetilde{k}_{p}^{2}-3k_{s}^{2}+2\widetilde{k}_{s}^{2}\right)\operatorname{I}\end{bmatrix}+\mathrm{OPS}(-1)$
The principal symbol is invertible since its determinant is different from
zero (see Lemma below). $\Box$
###### Lemma 4.11
Let
$A_{2}=\frac{1}{4}\begin{bmatrix}2\mu\left(\widetilde{k}_{p}^{2}+k_{s}^{2}+\widetilde{k}_{s}^{2}\right)-3k_{p}^{2}(\lambda+2\mu)&\left(3k_{p}^{2}(\lambda+2\mu)+2\mu\left(\widetilde{k}_{p}^{2}-k_{s}^{2}+\widetilde{k}_{s}^{2}\right)\right)i\\\
\mu\left(2k_{p}^{2}-2\widetilde{k}_{p}^{2}-3k_{s}^{2}-2\widetilde{k}_{s}^{2}\right)i&\mu\left(2k_{p}^{2}+2\widetilde{k}_{p}^{2}-3k_{s}^{2}+2\widetilde{k}_{s}^{2}\right)\end{bmatrix}.$
Then
$\det
A_{2}=-\frac{1}{4}\mu\left(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}\right)\left(k_{p}^{2}(3\lambda+4\mu)+k_{s}^{2}\mu\right).$
Proof. Clearly
$A_{2}\begin{bmatrix}i\\\
1\end{bmatrix}=\mu\left(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}\right)\begin{bmatrix}i\\\
1\end{bmatrix}$
which implies the result from the relation
$\det
A_{2}=\mu\left(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}\right)\left(\mathop{\rm
Tr}(A_{2})-\mu\left(\widetilde{k}_{p}^{2}+\widetilde{k}_{s}^{2}\right)\right).$
$\Box$
The arguments in the proof of Theorem 4.3 carry over in the Neumann case and
we have
###### Theorem 4.12
The operators
${\cal B}^{\rm comb}{\cal R}_{\rm N},\quad{\cal B}^{\rm comb}{\cal R}_{{\rm
N},1}$
associated to CFIER formulations considered above are invertible
pseudodifferential operators of order zero.
Therefore the CFIER formulations are well posed.
Proof. Indeed, the uniqueness of the solution of the elastic scattering
equation in $\Omega^{+}$ with Neumann boundary conditions on $\Gamma$,
together with the coercivity of the operators
$\operatorname{W}_{\widetilde{k}_{p}},\ \operatorname{W}_{\widetilde{k}_{s}}$
and the invertibility of the operators ${\cal R}_{\rm N}$ and ${\cal R}_{{\rm
N},1}$ are used in the same manner as in the proof of Theorem 4.3 to conclude
the injectivity of the Neumann CFIER operators. Given that we have already
established the Fredholmness of these operators, the proof is complete. $\Box$
Finally, it is also possible to construct regularizing operators that involve
Helmholtz BIOs only. Indeed, it is straightforward to see that the operator
${\cal R}_{{\rm
N},2}:=\begin{bmatrix}\operatorname{I}&2\partial_{\bm{t}}\operatorname{V}_{\widetilde{k}_{s}}\\\
2\partial_{\bm{t}}\operatorname{V}_{\widetilde{k}_{p}}&-\operatorname{I}\end{bmatrix}$
(4.36)
has the property
${\cal R}_{{\rm N},2}-{\cal R}_{{\rm N},1}\in\mathrm{OPS}(-4).$
Therefore, the CFIER formulations based on the newly defined operator ${\cal
R}_{{\rm N},2}$ are Fredholm. Their well posedness is a consequence of the
following
###### Lemma 4.13
The operator ${\cal R}_{{\rm N},2}$ is injective.
Proof. Assume $(f,g)\in\mathop{\rm Ker}(\mathcal{R}_{N,2})$, that is
$\displaystyle f+2\partial_{\bm{t}}\operatorname{V}_{\widetilde{k}_{s}}g$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle
2\partial_{\bm{t}}\operatorname{V}_{\widetilde{k}_{p}}f-g$ $\displaystyle=$
$\displaystyle 0.$
First, we remark that by integrating each of the equations above on $\Gamma$
we obtain $\int_{\Gamma}f=\int_{\Gamma}g=0$. Therefore there exist two
functional densities $F$ and $G$ on $\Gamma$ such that
$\partial_{\bm{t}}F=f\qquad\partial_{\bm{t}}G=g.$
We have then
$2\int_{\Gamma}\overline{F}\partial_{\bm{t}}\operatorname{V}_{\widetilde{k}_{p}}f+2\int_{\Gamma}\overline{G}\partial_{\bm{t}}\operatorname{V}_{\widetilde{k}_{s}}g+\int_{\Gamma}f\overline{G}-\int_{\Gamma}g\overline{F}=0.$
By integration by parts
$-2\int_{\Gamma}f\operatorname{V}_{\widetilde{k}_{p}}f-2\int_{\Gamma}g\operatorname{V}_{\widetilde{k}_{s}}g-\left(\int_{\Gamma}F\overline{g}+\int_{\Gamma}g\overline{F}\right)=0.$
Taking the imaginary part of the last equation we obtain
$\Im\int_{\Gamma}f\operatorname{V}_{\widetilde{k}_{p}}f+\Im\int_{\Gamma}g\operatorname{V}_{\widetilde{k}_{s}}g=0$
but both terms are positive for $f,g\neq 0$. Hence we conclude $f=0$ and
$g=0$, which completes the proof. $\Box$
###### Theorem 4.14
The operator
${\cal B}^{\rm comb}{\cal R}_{{\rm N},2}$
is an invertible pseudodifferential operators of order zero. Therefore the
CFIER formulations is well posed.
## 5 Numerical results
In this brief section we check some of the robust formulations proposed in
this paper. It is rather straightforward to extend the Nyström discretization
based on global trigonometric interpolation and Kussmaul-Martensen singularity
splittings for the four Helmholtz BIOs [25] to Nyström discretizations of the
BIOs introduced in Sections 4.1 and 4.2. These extensions were described
already in our previous contribution [18]. In addition, the Fourier
multipliers required by the various regularizing operators considered in this
paper are easy to discretize using global trigonometric interpolation. We
present in this section numerical results concerning the iterative behavior of
solvers based on the Nyström discretization of CFIE and CFIER Helmholtz
decomposition formulations using GMRES [30] iterative solvers. Specifically,
we report numbers of GMRES iterations required by various formulations to
reach GMRES residuals of $10^{-5}$ for discretizations sizes that give rise to
results accurate to at least four digits in the far field (which was estimated
using reference solutions produced with the high-order Nyström solvers based
on Navier Green functions [17]).
In all the numerical experiments in this section we assumed plane wave
incident fields of the form
${\bf u}^{\rm
inc}({\bm{x}})=\frac{1}{\mu}e^{ik_{s}{\bm{x}}\cdot{\bm{d}}}({\bm{d}}\cdot\mathrm{Q}{\bm{p}}){\bm{d}}+\frac{1}{\lambda+2\mu}e^{ik_{p}{\bm{x}}\cdot{\bm{d}}}({\bm{d}}\cdot{\bm{p}}){\bm{d}}$
(5.1)
where the direction ${\bm{d}}$ has unit length $|{\bm{d}}|=1$. Specifically,
we considered plane waves of direction
${\bm{d}}=\begin{bmatrix}0&-1\end{bmatrix}^{\top}$ and
${\bm{p}}=\begin{bmatrix}1&0\end{bmatrix}^{\top}$ in all of our numerical
experiments, and thus the incident wave is a shear S-wave. We observed that
other choices of the direction ${\bm{d}}$ and of the vector ${\bm{p}}$
(including cases when ${\bm{p}}=\pm{\bm{d}}$—the incident plane is then a
pressure wave or P-wave) lead to virtually identical results. In all the
numerical experiments we considered Lamé constants $\lambda=2$ and $\mu=1$ so
that $k_{s}=2k_{p}$.
We present numerical results for three smooth scatterer shapes. Namely, a unit
circle, the kite parametrized by
${\bf x}_{2}(t)=\frac{1}{L_{1}}(\cos{t}+0.65\cos{2t}-0.65,1.5\sin{t}),\ $
and a smooth cavity whose parametrization is given by
${\bf x}_{3}(t)=\frac{1}{L_{2}}(12\cos t+24\cos 2t,28\sin t+17\sin 2t+18\sin
3t-2\sin 4t)$
The constants $L_{1}\approx 1.17145$ and $L_{2}\approx 56.2295$ are taken so
that the length of the three curves is $2\pi$. In order to be consistent with
the derivation of regularizing operators in Sections 4.1 and 4.2, we used in
our numerical experiments arc length parametrizations of the three shapes. For
the CFIE formulations based on the representations (4.2) we selected the
coupling parameter $k=k_{p}$. for both the Dirichlet and Neumann case; we have
found in practice that other choices of the coupling parameter such as
$k=k_{s}$, or even the classical representations in Remark 4.6 lead to similar
iterative behaviors for CFIE formulations. The CFIE formulations are of the
first kind for both Dirichlet and Neumann cases since the operators ${\cal
A}_{\rm CFIE}\in\mathrm{OPS}(1)$ and ${\cal B}_{\rm CFIE}\in\mathrm{OPS}(2)$,
and thus the numbers of GMRES iterations required by CFIE formulations
increase with the discretization size. The increase of the numbers of GMRES
iterations with respect to discretization size is more dramatic for Neumann
boundary conditions (i.e. for CFIE based on the operator ${\cal B}_{\rm
CFIE}$—see Figure 3 where we can see almost a linear growth with respect to
the discretization size) but quite mild for Dirichlet boundary conditions
(i.e. for CFIE based on the operator ${\cal A}_{\rm CFIE}$). In the Dirichlet
case, and in the light of the result in Theorem 4.1, an obvious preconditioned
CFIE formulation takes advantage of the fact that ${\cal A}_{\rm
CFIE}^{2}\in\mathrm{OPS}(0)$. Unfortunately, this preconditioning strategies
leads only to modest gains in numbers of iterations (i.e. about 20%) despite
the resulting formulation being of the second kind cf. Theorem 4.1.
The CFIER formulations, on the other hand, are second kind integral equations
for both types of boundary conditions. We present in Figure 2 and 3 high
frequency results based on CFIER formulations with regularizing operators
${\cal R}_{\rm D}$ defined in equations (4.14) in the Dirichlet case and
respectively operators ${\cal R}_{\rm N}$ defined in equations (4.28) in the
Neumann case with the choice of complex wave-numbers
$\widetilde{k}_{p,s}=k_{p,s}+0.4\ i\ K^{2/3}k_{p,s}^{1/3},\qquad
K=\max_{\Gamma}|\kappa|.$
$-1$$0$$1$$-1$$0$$1$
$-2$$-1$$0$$1$$-1$$0$$1$
$-0.5$$0$$0.5$$-0.5$$0$$0.5$
$-2/3$0$2/3$$-2/3$0$2/3$
$-2/3$0$2/3$$-2/3$0$2/3$
Figure 1: Geometries for the experiments considered in this section. Top row,
smooth curves: the unit circle, the kite and the cavity curve; bottom row, a
square and the $L-$shaped (Lipschitz) domain; Notice that all the curves are
of length $2\pi$.
The use of regularizing operators ${\cal R}_{{\rm D},1}$ defined in equations
(4.18) and ${\cal R}_{{\rm D},2}$ defined in equations (4.19) in the Dirichlet
CFIER formulations leads to almost identical results and similarly for the use
of regularizing operators ${\cal R}_{{\rm N},1}$ defined in equations (4.33)
and ${\cal R}_{{\rm N},2}$ defined in equations (4.36) in the Neumann CFIER
formulations. Since the additional computational cost incurred by the
implementation of the regularizing operators is negligible with respect to
that of the CFIE operators, we conclude that the use of CFIER formulation
gives rise to significant gains in the high frequency regime. We illustrate in
Figure 4 the eigenvalue distribution corresponding to the CFIER formulations
in the case of the kite geometry for the frequency $\omega=40$ and both
Dirichlet and Neumann boundary conditions. We observe a very strong clustering
of the eigenvalues in the Dirichlet case, justifying the very small numbers of
GMRES iterations needed for convergence (cf. Figure 2). In the Neumann case
the spectrum of the CFIER operator is more widespread, yet the clustering of
eigenvalues around the value $1$ is still observed.
Figure 2: Numbers of GMRES iterations required to reach residuals of $10^{-5}$
for the CFIE and CFIER formulations for the circle (left), kite (middle) and
the smooth cavity (right) in the case of Dirichlet boundary conditions and
frequencies $\omega=10,20,40,80,160$ with Lamé parameters $\lambda=2$ and
$\mu=1$ under plane wave incidence. We used Nyström discretizations
corresponding to 8 points per the shorter wavelength. The numbers of
iterations are independent of the direction and polarization of the plane
wave.
Figure 3: Numbers of GMRES iterations required to reach residuals of $10^{-5}$
for the CFIE and CFIER formulations for the circle (left), kite (middle) and
the smooth cavity (right) in the case of Neumann boundary conditions and
frequencies $\omega=10,20,40,80,160$ with Lamé parameters $\lambda=2$ and
$\mu=1$ under plane wave incidence. We used Nyström discretizations
corresponding to 8 points per the shorter wavelength. In order to illustrate
the effect of discretization size on the iterative behavior of the CFIE
formulations, we report iteration counts ”CFIE refined” corresponding to
discretizations refined by a factor of two. The numbers of iterations are
independent of the direction and polarization of the plane wave.
Figure 4: Eigenvalue distribution of the CFIER operators in the case of the
kite geometry and $\omega=40$ for the Dirichlet (left) and Neumann (right)
cases.
Finally, we illustrate in Figure 5 the iterative behavior of the CFIE and
CFIER formulations in the case of Lipschitz scatterers in the high frequency
regime. Specifically, we considered a square and an L-shaped scatterer of
lengths $2\pi$ with arc length parametrizations, equipped with sigmoidal
graded meshes that accumulate points polynomially (e.g. polynomials of degree
three were used in our numerical experiments) towards the corner points. We
remark that the various well posedness proofs of the formulations considered
in this paper relied heavily on the smoothness of the curve $\Gamma$. Indeed,
a key ingredient in the analysis was the increased regularity of the double
layer operators, that is $\operatorname{K}_{k}^{\top}\in\mathrm{OPS}(-3)$,
which in the case of Lipschitz curves is only
$\operatorname{K}_{k}^{\top}\in\mathrm{OPS}(0)$. As a consequence, the CFIER
formulations are no longer of the second kind. Nevertheless, the CFIER
formulations still outperform the CFIE formulations. For instance, we did not
observe convergence when GMRES solvers were applied to CFIE formulations in
the Neumann case.
Figure 5: Numbers of GMRES iterations required to reach residuals of $10^{-4}$
for the CFIE and CFIER formulations for the square and the L-shaped scatterers
in the high frequency regime for the Dirichlet (left) and Neumann (right)
boundary conditions and the same material parameters as in the previous cases,
In the case of Neumann boundary conditions, the solvers based on CFIE
formulations did not converge.
## 6 Conclusions
We introduced and analyzed CFIER formulations for the solution of two
dimensional elastic scattering problems via Helmholtz decompositions. Despite
featuring non standard BIOs, we showed that these CFIER formulations are well
posed in the case of smooth scatterers. The CFIER formulations, being of the
second kind, possess superior spectral properties to the classical CFIE
formulations for both Dirichlet and Neumann boundary conditions. The extension
of our approach to the case of homogeneous penetrable scatterers is
straightforward and is currently being pursued. The extension of the CFIER
methodology to three dimensional elastic scattering problems via Helmholtz
decompositions, on the other hand, is more challenging as it requires
incorporation of both Helmholtz and Maxwell BIOs.
## Acknowledgments
Catalin Turc gratefully acknowledges support from NSF through contract
DMS-1908602. Víctor Domínguez is partially supported by project “Adquisición
de conocimiento y minería de datos, funciones especiales y métodos numéricos
avanzados” from Universidad Pública de Navarra.
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## Appendix A Helmholtz BIOs and pseudodifferential operator calculus
For a given wave-number $k$ and a functional density $\varphi$ on the boundary
$\Gamma$ denote in this section the Helmholtz single and double layer
potentials in the form
$\operatorname{SL}_{k,\Gamma}[\varphi]({\bm{x}}):=\int_{\Gamma}H_{0}^{(1)}(k|{\bm{x}}-{\bm{y}}|)\varphi({\bm{y}}){{\rm
d}{\bm{y}}},\quad{\operatorname{DL}_{k,\Gamma}}[\varphi]({\bm{x}}):=\int_{\Gamma}\frac{\partial
H_{0}^{(1)}(k|{\bm{x}}-{\bm{y}}|)}{\partial\bm{n}({\bm{y}})}\varphi({\bm{y}}){{\rm
d}{\bm{y}}},\ {\bm{x}}\in\mathbb{R}^{2}\setminus\Gamma.$
Any exterior, radiating, solution of the Helmholtz equation can be written
$u({\bm{x}})=\operatorname{DL}_{k,\Gamma}[\gamma_{\Gamma}^{+}u]({\bm{x}})-\operatorname{SL}_{k,\Gamma}[\partial_{\bm{n}}^{+}u]({\bm{x}}),\quad{\bm{x}}\in\Omega^{+}.$
(A.1)
The associated layer operators (read by rows in the matrix operator: double
layer, single layer, hypersingular and adjoing double layer) are defined as
$\begin{bmatrix}\gamma^{+}\\\
\partial_{\bm{n}}^{+}\end{bmatrix}\begin{bmatrix}\mathrm{DL}_{k,\Gamma}&-\mathrm{SL}_{k,\Gamma}\end{bmatrix}=\begin{bmatrix}\operatorname{K}_{k,\Gamma}&-\operatorname{V}_{k,\Gamma}\\\
\operatorname{W}_{k,\Gamma}&-\operatorname{K}^{\top}_{k,\Gamma}\end{bmatrix}=:{\cal
C}_{k,\Gamma}:H^{s}(\Gamma)\times H^{s-1}(\Gamma)\to H^{s}(\Gamma)\times
H^{s-1}(\Gamma).$
The matrix operator ${\cal C}_{k,\Gamma}$ is then continuous for any $s$ if
$\Gamma$ is smooth. Calderón identities can be derived from the fact that (see
for instance [31, Ch.2 ], [22, Ch. 1] or [29, Chapter 6-7]) that
$\left(\frac{1}{2}\mathcal{I}+{\cal
C}_{k,\Gamma}\right)^{2}=\frac{1}{2}\mathcal{I}+{\cal C}_{k,\Gamma},$
which are equivalent to
$\displaystyle\operatorname{V}_{k,\Gamma}\operatorname{W}_{k,\Gamma}$
$\displaystyle=-\frac{1}{4}\operatorname{I}+\operatorname{K}_{k,\Gamma}^{2},$
$\displaystyle\quad\operatorname{W}_{k,\Gamma}\operatorname{V}_{k,\Gamma}$
$\displaystyle=-\frac{1}{4}\operatorname{I}+(\operatorname{K}^{\top}_{k,\Gamma})^{2}$
(A.2)
$\displaystyle\operatorname{V}_{k,\Gamma}\operatorname{K}_{k,\Gamma}^{\top}$
$\displaystyle=\operatorname{K}_{k,\Gamma}\operatorname{V}_{k,\Gamma},\quad$
$\displaystyle\operatorname{W}_{k,\Gamma}\operatorname{K}_{k,\Gamma}$
$\displaystyle=\operatorname{K}^{\top}_{k,\Gamma}\operatorname{W}_{k,\Gamma}.$
It is convenient to present the analysis of robust BIE formulations of
Helmholtz decompositions of Navier equations in the framework of periodic
pseudodifferential operators. Consider then a regular positive oriented arc-
length parameterization of $\Gamma$, ${\bf x}:[0,L]\to\Gamma$ where $L$ is the
length of the curve. For any function, or distribution in the general case,
$\varphi_{\Gamma}:\Gamma\to\mathbb{C}$ we will denote by
$\varphi(\tau)=\varphi_{\Gamma}({\bf x}(\tau))$
its parameterized counterpart. We will extend this convention to the
operators. For instance,
$(\operatorname{V}_{k}\varphi)(\tau)=\frac{i}{4}\int_{0}^{L}H_{0}^{(1)}(k|{\bf
x}(t)-{\bf x}(\tau)|)\varphi_{\Gamma}({\bf x}(\tau))\,{\rm d}\tau.$
is the parameterized version of $\operatorname{V}_{k,\Gamma}$.
The unit tangent and normal parameterized vector to $\Gamma$ (at ${\bf
x}(\tau)$) are then given by
$\bm{t}(\tau)={\bf x}^{\prime}(\tau),\quad\bm{n}(\tau)=\mathrm{Q}{\bf
x}^{\prime}(\tau),\quad\mathrm{Q}=\begin{bmatrix}&1\\\ -1&\end{bmatrix}$
so that
$(\partial_{\bm{t}}\varphi_{\Gamma})\circ{\bf
x}=\varphi^{\prime}=:\mathrm{D}\varphi.$
Besides, the (parameterized) signed quadrature can be then expressed as
$\kappa=-\bm{t}\cdot\mathrm{D}\bm{n}.$ (A.3)
It is a well-established result, see for instance [26, Ch. 8], that Sobolev
spaces on $\Gamma$ $H^{s}(\Gamma)$ can be then identified with the
$L-$periodic Sobolev spaces
$H^{s}=\Big{\\{}\varphi\in{\cal D}^{\prime}(\mathbb{R})\ :\
\varphi(\cdot+L)=\varphi,\quad\|\varphi\|_{s}<\infty\Big{\\}}$
where, with
$\quad\widehat{\varphi}(n)=\frac{1}{L}\int_{0}^{L}\varphi(\tau)e_{-n}(\tau)\,{\rm
d}\tau,\qquad e_{n}(\tau)=\exp\Big{(}\frac{2\pi i}{L}n\tau\Big{)}$
the Fourier coefficients, the Sobolev norm is given by
$\|\varphi\|_{s}^{2}=|\widehat{\varphi}(0)|^{2}+\sum_{n\neq
0}|n|^{2s}|\widehat{\varphi}(n)|^{2}.$
Clearly,
$\varphi=\widehat{\varphi}(0)+\sum_{n\neq 0}\widehat{\varphi}(n)e_{n}$
with convergence in $H^{s}$. The set $\\{H^{s}\\}_{s\in\mathbb{R}}$ is a
Hilbert scale, meaning that $H^{s}$ is actually a Hilbert space, that
$H^{t}\subset H^{s}$ for any $t>s$ with compact and dense injection and that
$\bigcap_{s}H^{s}={\cal D}:=\\{\varphi\in{\cal C}^{\infty}(\mathbb{R})\ :\
\varphi=\varphi(\cdot+L)\\},\quad\bigcup_{s}H^{s}={\cal D}^{\prime}.$
We will denote by $\mathrm{OPS}(m)$ the class of periodic pseudodifferential
operators of order $m\in\mathbb{Z}$ on $\Gamma$. That is,
$\mathrm{A}\in\mathrm{OPS}(m)$ if $\mathrm{A}:H^{s}\to H^{s-m}$ is continuous
for any $s$. For convenience, we will write that
$\mathrm{A}=\mathrm{B}+\mathrm{OPS}(m-1),$
if $\mathrm{A}-\mathrm{B}\in\mathrm{OPS}(m-1)$
Trivially, $\mathrm{A}\in\mathrm{OPS}(m)$ implies that
$\mathrm{A}\in\mathrm{OPS}(m+1)$ and as an operator in ${\mathrm{OPS}}(m+1)$
$\mathrm{A}$ is compact. We also set
${\mathrm{OPS}}(-\infty):=\bigcap_{n\in\mathbb{N}}\mathrm{OPS}(n)$
the class of smoothing operators which in turn can be identified with integral
operators with $L-$periodic smooth kernel.
In connection to periodic pseudodifferential operators, Fourier multipliers
will play a central role in what follows:
$\mathrm{A}\varphi=\sum_{m=-\infty}^{\infty}\mathrm{A}(n)\widehat{\varphi}(n)e_{n}\qquad$
(A.4)
where $\mathrm{A}(n)\in\mathbb{C}$ are referred to as the symbol of
$\mathrm{A}$. Clearly, if $\mathrm{A}$ is a Fourier multiplier defined as in
(A.4), if there exists $c>0$ and $r$ such that $|\mathrm{A}(n)|\leq c|n|^{r}$
for all $n\in\mathbb{Z}$ (which for simplicity we shall denote in what follows
as $\mathrm{A}(n)=\mathcal{O}(n^{r})$) then $\mathrm{A}\in\mathrm{OPS}(r)$.
Equivalently, if
$\sum_{n\in\mathbb{Z}}|\mathrm{A}(n)|<\infty,$
then with the function
$a(t)=\frac{1}{L}\sum_{n\in\mathbb{Z}}\mathrm{A}(n)e_{n}(t)\in
L^{\infty}(\mathbb{R}),\quad\text{that
is,\quad}\widehat{a}(n)=\frac{1}{L}\mathrm{A}(n),$
$\mathrm{A}$ is just a convolution operator:
$\mathrm{A}f=\int_{0}^{L}a(\cdot-\tau)\varphi(\tau)\,{\rm d}\tau.$
Note that the tangential derivative becomes a Fourier multiplier:
$\mathrm{D}\varphi=\sum_{n\neq 0}\left(\frac{2\pi
im}{L}\right)\widehat{\varphi}(m)e_{m}=\varphi^{\prime}$
and that for any nonnegative integer $r$,
$\mathrm{D}^{r}=\mathrm{D}_{r},\quad\mathrm{D}_{r}\varphi:=\sum_{n\neq
0}\left(\frac{2\pi im}{L}\right)^{r}\widehat{\varphi}(m)e_{m}.$
We will extend this definition to set $\mathrm{D}_{r}$ for negative integer
values of $r$ too.
Three additional Fourier multiplier operators we will required in our
analysis. First, the Bessel operator
$\displaystyle\Lambda\varphi$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int_{0}^{L}\log\left(4e^{-1}\sin^{2}\left(\frac{\pi}{L}(\cdot-\tau)\right)\right)\varphi(\tau)\,{\rm
d}\tau=\frac{L}{2\pi}\left[\widehat{\varphi}(0)+\sum_{n\neq
0}\frac{1}{|n|}\widehat{\varphi}(n)e_{n}\right],$
next, the Hilbert transform, or Hilbert singular operator,
$\operatorname{H}\varphi:=-\frac{1}{L}\mathrm{p.v.}\int_{0}^{L}\cot\left(\frac{\pi}{L}(\cdot-\tau)\right)\varphi(\tau)\,{\rm
d}\tau+i\mathrm{J}\varphi=i\bigg{[}\widehat{\varphi}(0)+\sum_{n\neq
0}\mathop{\rm\rm sign}(n)\widehat{\varphi}(n)e_{n}\bigg{]},$
with
$\mathrm{J}\varphi:=\widehat{\varphi}(0)=\frac{1}{L}\int_{\Gamma}\varphi_{\Gamma}.$
the mean operator. We notice then
$\displaystyle\mathrm{D}\Lambda$
$\displaystyle=\Lambda\mathrm{D}=i\operatorname{H}+\mathrm{J},$
$\displaystyle\quad\Lambda$
$\displaystyle=\operatorname{H}\mathrm{D}_{-1}+\frac{L}{2\pi}\mathrm{J},$
$\displaystyle\mathrm{D}\mathrm{D}_{-1}$
$\displaystyle=\operatorname{I}-\mathrm{J}=-\operatorname{H}^{2}-\mathrm{J}$
$\displaystyle\quad\Lambda^{-1}$
$\displaystyle=-\mathrm{D}\operatorname{H}+\mathrm{J}=-\mathrm{D}_{2}\Lambda+\frac{2\pi}{L}\mathrm{J}.$
By a direct analysis of the resulting kernel we can easily check
$a\operatorname{H}-\operatorname{H}a\in\mathrm{OPS}(-\infty)$ (A.5)
for any smooth function $a$. Similarly.
$a\mathrm{D}_{r}-\mathrm{D}_{r}a\in\mathrm{OPS}(r-1)$ (A.6)
which is trivial to show for positive $r$ and an easy consequence, from
negative values of $r$, of the equality
$\mathrm{D}_{-1}(a\varphi)=a\mathrm{D}_{-1}\varphi-\mathrm{D}_{-1}(a^{\prime}\mathrm{D}_{-1}\varphi)-(\mathrm{D}_{-1}a)\mathrm{J}\varphi.$
which implies, still for $r<0$
$\mathrm{D}_{r}(a\varphi)=\sum_{m=-n}^{r}\binom{r}{-m}(\mathrm{D}_{r-m}a)(\mathrm{D}_{m}\varphi)+\mathrm{OPS}(-n-1).$
(A.7)
We point out that if
$\\{\operatorname{K}_{0},\operatorname{V}_{0},\operatorname{W}_{0},\operatorname{K}^{\top}_{0}\\}$
are the boundary layer operators associated to the Laplace equation ($k=0$),
$\displaystyle\operatorname{V}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda+\mathrm{OPS}(-\infty)=\frac{1}{2}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-\infty),$
$\displaystyle\operatorname{W}_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathrm{\Lambda}^{-1}_{0}+\mathrm{OPS}(-\infty)=\frac{1}{2}\mathrm{D}\operatorname{H}+\mathrm{OPS}(-\infty)$
$\displaystyle\operatorname{K}_{0}$ $\displaystyle=$
$\displaystyle\mathrm{OPS}(-\infty)\ =\operatorname{K}_{0}^{\top}.$
since the functions
$\log\left(\frac{\left|\sin\left(\frac{\pi}{L}(t-\tau)\right)\right|}{|{\bf
x}(t)-{\bf
x}(\tau)|^{2}}\right),\quad\partial_{\tau}\partial_{t}\log\left|\sin\left(\frac{\pi}{L}(t-\tau)\right)\right|-\frac{\bm{n}({\bf
x}(t))\cdot\bm{n}({\bf x}(\tau))}{|{\bf x}(t)-{\bf x}(\tau)|^{2}}$
are smooth.
Our aim is to extend such expansions for the Calderon operators associated to
Helmholtz equation. For such purposes, let us define for non-negative integer
values of $r$
$\alpha_{2r}(\tau)=-\frac{1}{2\pi}\left[2\sin^{2}\left(\frac{\pi}{L}\tau\right)\right]^{r}\log\left(4e^{-1}\sin^{2}\left(\frac{\pi}{L}\tau\right)\right)$
and denote the associated multiplier operator by $\Lambda_{2r}$. Note that
$\alpha_{0}(n)=\alpha$, and therefore $\Lambda_{0}=\Lambda$, and that
$\widehat{\alpha}_{2r}(n)=-\tfrac{1}{2}\widehat{\alpha}_{2r-2}(n-1)+\widehat{\alpha}_{2r-2}(n)-\tfrac{1}{2}\widehat{\alpha}_{2r-2}(n+1)$
which implies
$\widehat{\alpha}_{2r}(n)=\frac{(-1)^{r}(2r)!L}{2^{r}(2\pi)}\frac{1}{|n|^{2r+1}}+{\cal
O}(n^{-2r-3})=\frac{(-1)^{r}(2r)!2^{r}\pi^{2r}}{L^{2r}}|\widehat{\alpha}_{0}(n)|^{2r+1}+{\cal
O}(n^{-2r-3})$
That is, $\Lambda_{2r}\in\mathrm{OPS}(-2r-1)$ with
$\Lambda_{2r}=\frac{(2r)!2^{r}\pi^{2r}}{L^{2r}}(-1)^{r}\Lambda^{2r+1}+\mathrm{OPS}(-2r-3).$
(Actually, $\Lambda_{2r}$ can be expanded in powers of $\Lambda^{2\ell+1}$
with $\ell\geq r$ is needed). In particular,
$\Lambda_{2}=-\frac{(2\pi)^{2}}{L^{2}}\Lambda^{3}+\mathrm{OPS}(-5).$
Similarly, we can set
$\alpha_{2r+1}(\tau)=(e_{1}(\tau)-1)\alpha_{2r}(\tau),$
show next that
$\widehat{\alpha}_{2r+1}(n)=\frac{(-1)^{r}(2r+1)!L}{2^{r}(2\pi)}\frac{1}{n^{2r+1}}+{\cal
O}(n^{-2r-4}).$
and conclude that the associated Fourier multiplier operator satisfies
$\Lambda_{2r+1}=\frac{(2r+1)!2^{r+1}\pi^{2r+1}}{L^{2r+1}}(-1)^{r}\Lambda^{2r+2}+\mathrm{OPS}(-2r-3).$
Finally, it is a well established result that periodic integral operators
$\mathrm{L}_{r}\varphi=\int_{0}^{L}D(\cdot,\tau)\alpha_{r}(\cdot-\tau)\varphi(\tau)\,{\rm
d}\tau,$
with $D$ a $L-$periodic smooth function, belongs to $\mathrm{OPS}(-r-1)$ (see
[31]).
The well posedness of the various BIE formulations considered in this paper
relies heavily on the following result which provides decompositions of the
Helmholtz operators $\operatorname{V}_{k}$ and $\operatorname{W}_{k}$ in sums
of Fourier multiplier pseudodifferential operators of orders $-1$ and $-3$ for
the first operator and $1$ and $-1$ for the second plus remainders that are
smoother. Specifically, we establish
###### Proposition A.1
It holds
$\displaystyle\operatorname{V}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda+\frac{k^{2}}{4}\Lambda^{3}+\mathrm{OPS}(-5),$
(A.8) $\displaystyle\operatorname{W}_{k}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Lambda^{-1}+\frac{k^{2}}{4}\Lambda+\mathrm{OPS}(-3),$
(A.9) $\displaystyle\operatorname{K}_{k}$ $\displaystyle=$
$\displaystyle\frac{k^{2}}{2}\kappa\Lambda^{3}+\mathrm{OPS}(-4)=\operatorname{K}_{k}^{\top},$
(A.10)
Proof. From the decomposition of the Bessel functions (see for instance [26,
Ch. 12] or [11, §10]) we get the decomposition:
$\frac{i}{4}H_{0}^{(i)}(t)=-\frac{1}{4\pi}\log t^{2}+\frac{1}{16\pi}t^{2}\log
t^{2}+\underbrace{\frac{1}{4\pi}\frac{1}{t^{4}}(1-\frac{1}{4}t^{2}-J_{0}(t))}_{B_{0}(t)}t^{4}\log
t^{2}+C_{0}(t),$
with $B_{0}$ and $C_{0}$ smooth. Since
$|{\bf x}(t)-{\bf
x}(\tau)|^{2}=\frac{L^{2}}{\pi^{2}}\sin^{2}\left(\frac{\pi}{L}(t-\tau)\right)+D_{1}(t,\tau)\sin^{4}\left(\frac{\pi}{L}(t-\tau)\right),$
$D_{1}$ being smooth, the following decomposition holds
$\displaystyle\frac{i}{4}H_{0}^{(i)}(k|{\bf x}(t)-{\bf x}(\tau)|)$
$\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\log\left(4e^{-1}\sin^{2}\left(\frac{\pi}{L}(t-\tau)\right)\right)$
$\displaystyle+\frac{(Lk)^{2}}{4}\frac{1}{(2\pi)^{3}}\left(2\sin^{2}\frac{\pi}{L}(t-\tau)\right)\log\left(4e^{-1}\sin^{2}\left(\frac{\pi}{L}(t-\tau)\right)\right)$
$\displaystyle+D_{2}(t,\tau)\sin^{4}\left(\frac{\pi}{L}(t-\tau)\right)\log\sin^{2}(4e^{-1}\pi(t-\tau))+D_{3}(t,\tau),$
where $D_{1},\ D_{2}$ smooth bi-periodic functions. Therefore,
$\displaystyle\operatorname{V}_{k}\varphi$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda\varphi-\frac{(Lk)^{2}}{4}\frac{1}{(2\pi)^{2}}\int_{0}^{L}\alpha_{2}(\,\cdot\,-\tau)\,\varphi(\tau)\,{\rm
d}\tau$
$\displaystyle+\int_{0}^{L}D_{2}(t,\tau)\alpha_{4}(\,\cdot\,-\tau)\,\varphi(\tau)\,{\rm
d}\tau+\int_{0}^{L}B(\,\cdot\,,\tau)\,\varphi(\tau)\,{\rm d}\tau$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda\varphi-\frac{k^{2}}{4}\frac{1}{(2\pi)^{2}}\Lambda_{2}\varphi+\mathrm{OPS}(-5)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda\varphi+\frac{k^{2}}{4}\Lambda^{3}\varphi+\mathrm{OPS}(-5).$
The analysis of $\mathrm{K}_{k}$ is very similar. Indeed, the kernel is given
by
$\frac{i}{4}H_{1}^{(1)}(k|{\bm{x}}-{\bm{y}}|)k|{\bm{x}}-{\bm{y}}|\frac{({\bm{x}}-{\bm{y}})\cdot\bm{n}({\bm{y}})}{|{\bm{x}}-{\bm{y}}|^{2}},$
which with the decomposition
$\frac{i}{4}H_{1}^{(1)}(t)t=-\frac{1}{8\pi}t^{2}\log
t^{2}+\underbrace{\frac{1}{4\pi
t^{2}}\left(\frac{1}{2}-\frac{1}{t}J_{1}(t)\right)}_{E_{1}(t)}t^{4}\log
t^{2}+C_{1}(t).$
and the fact that
$\frac{({\bm{x}}-{\bm{y}})\cdot\bm{n}({\bm{y}})}{|{\bm{x}}-{\bm{y}}|^{2}}=-\frac{1}{2}\kappa({\bm{x}})+E_{2}({\bm{x}},{\bm{y}})|{\bm{x}}-{\bm{y}}|$
($E_{1},\ E_{2}$ are again smooth functions) allow us to write
$\displaystyle\operatorname{K}_{k}\varphi$ $\displaystyle=$
$\displaystyle-\frac{(Lk)^{2}}{2(2\pi)^{2}}\kappa\Lambda_{2}\varphi+\int_{0}^{L}E_{2}(t,\tau)\alpha_{3}(\,\cdot\,-\tau)\,\varphi(\tau)\,{\rm
d}\tau+\int_{0}^{L}E_{3}(\,\cdot\,,\tau)\,\varphi(\tau)\,{\rm d}\tau$
$\displaystyle=$
$\displaystyle\frac{k^{2}}{2}\kappa\Lambda^{3}\varphi+\mathrm{OPS}(-4).$
The case $\operatorname{K}^{\top}$ is consequence of (A.5)-(A.6) since
$\Lambda(\kappa\cdot)=\kappa\Lambda+\mathrm{OPS}(-2)$
which with the fact $\Lambda^{\top}=\Lambda$.
Finally, for the hypersingular operator, we note that with the identity
$\operatorname{W}_{k}\operatorname{V}_{k}=-\frac{1}{4}\operatorname{I}+(\operatorname{K}_{k}^{\top})^{2}$
cf. (A.2),
$\displaystyle\operatorname{W}_{k}$ $\displaystyle=$
$\displaystyle\operatorname{W}_{k}\left(\operatorname{I}-\frac{k^{4}}{4}\Lambda^{4}\right)+\mathrm{OPS}(-3)$
$\displaystyle=$
$\displaystyle\operatorname{W}_{k}\left(\frac{1}{2}\Lambda+\frac{k^{2}}{4}\Lambda^{3}\right)\left(2\Lambda^{-1}-k^{2}\Lambda\right)+\mathrm{OPS}(-3)$
$\displaystyle=$
$\displaystyle\operatorname{W}_{k}\operatorname{V}_{k}\left(2\Lambda^{-1}-k^{2}\Lambda\right)+\underbrace{\operatorname{W}_{k}\left(\frac{1}{2}\Lambda+\frac{k^{2}}{4}\Lambda^{3}-\operatorname{V}_{k}\right)\left(2\Lambda^{-1}-k^{2}\Lambda\right)}_{\in\mathrm{OPS}(-3)}+\mathrm{OPS}(-3)$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\left(2\Lambda^{-1}-k^{2}\Lambda\right)+\underbrace{(\operatorname{K}_{k}^{\top})^{2}\left(2\Lambda^{-1}-k^{2}\Lambda\right)}_{\in\mathrm{OPS}(-5)}+\mathrm{OPS}(-3)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\Lambda^{-1}+\frac{k^{2}}{4}\Lambda+\mathrm{OPS}(-3)$
and the result is proven. $\Box$
###### Remark A.2
This result appears in a slightly different form [22] (equations (10.4.5)).
Indeed, several terms in the asymptotic expansion of the principal symbol of
the periodic pseudodifferential operator $\operatorname{V}_{k}$ are provided
in equations (10.4.5) in [22], and they coincide with the symbols of the
operators in the decomposition we provide in Proposition A.1. Indeed,
It is also evident that the expansions in the four operators for Helmholtz
equation can be continued in powers of $\Lambda_{0}$ (or equivalently)
$\mathrm{D}\operatorname{H}$), all of them being negative except the first
term for $\operatorname{W}_{k}$. Such as expansions have appeared previously
in the literature (see for instance [31] and, with applications to the study
and design of numerical methods, in [16, 7]).
###### Theorem A.3
It holds
$\displaystyle\operatorname{V}_{k}$ $\displaystyle=$
$\displaystyle\operatorname{V}_{0}+2k^{2}\operatorname{V}_{0}^{3}+\mathrm{OPS}(-5)=\frac{1}{2}\operatorname{H}\mathrm{D}_{-1}-\frac{k^{2}}{4}\operatorname{H}\mathrm{D}_{-3}+\mathrm{OPS}(-5),$
(A.11) $\displaystyle\operatorname{W}_{k}$ $\displaystyle=$
$\displaystyle\operatorname{W}_{0}+\frac{k^{2}}{2}\operatorname{V}_{0}+\mathrm{OPS}(-3)=\frac{1}{2}\operatorname{H}\mathrm{D}_{1}+\frac{k^{2}}{4}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-3),$
(A.12) $\displaystyle\operatorname{K}_{k}$ $\displaystyle=$
$\displaystyle\operatorname{K}_{k}^{\top}=4k^{2}\kappa\operatorname{V}_{0}^{3}+\mathrm{OPS}(-4)=-\frac{1}{2}\kappa
k^{2}\operatorname{H}\mathrm{D}_{-3}+\mathrm{OPS}(-4)$ (A.13)
Equivalently,
$\displaystyle\operatorname{V}_{k}\varphi$ $\displaystyle=$
$\displaystyle\frac{L}{4\pi}\sum_{n\neq
k}\left(n^{2}-\left(\frac{kL}{2\pi}\right)^{2}\right)^{-1/2}\widehat{\varphi}(n)e_{n}+\mathrm{OPS}(-5),$
(A.14) $\displaystyle\operatorname{W}_{k}\varphi$ $\displaystyle=$
$\displaystyle-\frac{\pi}{L}\sum_{n}\left(n^{2}-\left(\frac{kL}{2\pi}\right)^{2}\right)^{1/2}\widehat{\varphi}(n)e_{n}+\mathrm{OPS}(-3).$
(A.15)
Finally, the Dirichlet-to-Neumann operator satisfies
$\displaystyle\mathrm{DtN}_{k}\varphi$
$\displaystyle=\operatorname{H}\mathrm{D}+\frac{k^{2}}{2}\operatorname{H}\mathrm{D}_{-1}-\kappa
k^{2}\mathrm{D}_{-2}+\mathrm{OPS}(-4)$ (A.16) $\displaystyle\ =\
\frac{1}{L}\sum_{n}\left(n^{2}-\left(\frac{kL}{2\pi}\right)^{2}\right)^{1/2}\widehat{\varphi}(n)e_{n}+\mathrm{OPS}(-2)=2\operatorname{W}_{k}+\mathrm{OPS}(-2).$
Proof. Expansions A.11-(A.13) follows from Theorem A.1 whereas Properties
(A.14)-(A.15) follows from
$(n^{2}-k^{2})^{-1/2}=\frac{\operatorname{sign}(n)i}{in}-\frac{\operatorname{sign}(n)i}{(in)^{3}}+{\cal
O}(n^{-5}),\quad-(n^{2}-k^{2})^{1/2}=(\operatorname{sign}(n)i)(in)+\frac{\operatorname{sign}(n)i}{in}+{\cal
O}(n^{-3}).$
Finally, if $\operatorname{V}_{k}$ is invertible, using (A.2)
$\displaystyle\mathrm{DtN}_{k}$ $\displaystyle=$
$\displaystyle\operatorname{V}_{k}^{-1}(-\frac{1}{2}\operatorname{I}+\operatorname{K}_{k})=-4\operatorname{W}_{k}(-\frac{1}{2}\operatorname{I}+\operatorname{K}_{k})+\mathrm{OPS}(-5)$
$\displaystyle=$ $\displaystyle
2\operatorname{W}_{k}-4\operatorname{W}_{k}\operatorname{K}_{k}+\mathrm{OPS}(-5).$
Now (A.16) is straightforward to derive. If $\operatorname{V}_{k}$ fails to be
invertible we can use the alternative expression for the Dirichlet-to-Neumann
operator
$\mathrm{DtN}_{k}=\left(\frac{1}{2}\operatorname{I}+\operatorname{K}_{k}^{\top}\right)^{-1}\operatorname{W}_{k}$
(notice that at least one of $\operatorname{V}_{k}$ or
$(\frac{1}{2}\operatorname{I}+\operatorname{K}_{k}^{\top})$ must be
invertible) and proceed similarly. $\Box$
###### Lemma A.4
It holds
$\displaystyle\bm{n}\cdot\operatorname{W}_{k}(\bm{n}\cdot)=\bm{t}\cdot\operatorname{W}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\Lambda^{-1}+\frac{k^{2}}{4}\Lambda+\mathrm{OPS}(-3),$
(A.17a)
$\displaystyle\bm{t}\cdot\operatorname{W}_{k}(\bm{n}\cdot)=-\bm{n}\cdot\operatorname{W}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\kappa\mathrm{D}\Lambda+\frac{k^{2}}{4}\kappa\mathrm{D}\Lambda^{3}+\mathrm{OPS}(-3),$
(A.17b)
$\displaystyle\bm{n}\cdot\operatorname{V}_{k}(\bm{n}\cdot)=\bm{t}\cdot\operatorname{V}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda+\frac{1}{4}(2\kappa+k^{2})\Lambda^{3}+\mathrm{OPS}(-4),\
$ (A.17c)
$\displaystyle\bm{t}\cdot\operatorname{V}_{k}(\bm{n}\cdot)=\bm{n}\cdot\operatorname{V}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\kappa\mathrm{D}\Lambda^{3}+\mathrm{OPS}(-4),\ $
(A.17d)
$\displaystyle\bm{n}\cdot\mathrm{D}\operatorname{V}_{k}(\bm{n}\cdot)=\bm{t}\cdot\mathrm{D}\operatorname{V}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{D}\Lambda+\frac{k^{2}}{4}\mathrm{D}\Lambda^{3}+\mathrm{OPS}(-4),\
$ (A.17e)
$\displaystyle\bm{t}\cdot\mathrm{D}\operatorname{V}_{k}(\bm{n}\cdot)=-\bm{n}\cdot\mathrm{D}\operatorname{V}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle-\frac{k^{2}}{2}\kappa\Lambda^{3}+\mathrm{OPS}(-4).$ (A.17f)
Proof. We start from the identities
$\displaystyle{\bm{n}}\cdot\mathrm{D}_{2}\bm{n}={\bm{t}}\cdot\mathrm{D}_{2}{\bm{t}}$
$\displaystyle=\ \mathrm{D}_{2}-\kappa^{2}\operatorname{I},$ (A.18)
$\displaystyle{\bm{n}}\cdot\mathrm{D}_{2}{\bm{t}}=-{\bm{t}}\cdot\mathrm{D}_{2}\bm{n}$
$\displaystyle=\ -2\kappa\mathrm{D}+\kappa^{\prime}\,\operatorname{I},\ $
$\displaystyle{\bm{n}}\cdot\mathrm{D}\bm{n}=-{\bm{t}}\cdot\mathrm{D}{\bm{t}}$
$\displaystyle=\ \mathrm{D},\ $
$\displaystyle{\bm{t}}\cdot\mathrm{D}\bm{n}=-{\bm{n}}\cdot\mathrm{D}{\bm{t}}$
$\displaystyle=\ \kappa\operatorname{I},\ $
$\displaystyle{\bm{n}}\cdot\mathrm{D}_{-1}\bm{n}={\bm{t}}\cdot\mathrm{D}_{-1}{\bm{t}}$
$\displaystyle=\
\mathrm{D}_{-1}-\kappa\mathrm{D}_{-3}+(1+\kappa)(\mathrm{D}\kappa)\mathrm{D}_{-4}+\mathrm{OPS}(-5),\
$
$\displaystyle{\bm{t}}\cdot\mathrm{D}_{-1}\bm{n}=-{\bm{n}}\cdot\mathrm{D}_{-1}{\bm{t}}$
$\displaystyle=\
-\kappa\mathrm{D}_{-2}+\kappa^{\prime}\,\mathrm{D}_{-3}+(\mathrm{D}^{2}\kappa-\kappa^{2})\mathrm{D}_{-4}+\mathrm{OPS}(-5),\
$
$\displaystyle{\bm{n}}\cdot\mathrm{D}_{-2}\bm{n}={\bm{t}}\cdot\mathrm{D}_{-2}{\bm{t}}$
$\displaystyle=\ \mathrm{D}_{-2}+3\kappa^{2}\mathrm{D}_{-4}+\mathrm{OPS}(-5),\
$
$\displaystyle{\bm{t}}\cdot\mathrm{D}_{-2}\bm{n}=-{\bm{n}}\cdot\mathrm{D}_{-2}{\bm{t}}$
$\displaystyle=\
-2\kappa\mathrm{D}_{-3}-3\kappa^{\prime}\,\mathrm{D}_{-4}+\mathrm{OPS}(-4)$
which can be easily proven from (A.3). Hence the result follows from
Proposition A.1 and the commutation properties for $\operatorname{H}$ cf.
(A.5), the Leibnitz rule for the derivatives and its extension cf. (A.7) for
the negative order derivatives $\mathrm{D}_{-r}$. $\Box$
###### Theorem A.5
It holds
$\displaystyle\bm{n}\cdot\operatorname{W}_{k}(\bm{n}\cdot)=\bm{t}\cdot\operatorname{W}_{k}(\bm{t}\cdot)$
$\displaystyle=$ $\displaystyle\operatorname{W}_{k}+\mathrm{OPS}(-3)$ (A.19a)
$\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{D}\operatorname{H}+\frac{k^{2}}{4}\operatorname{H}\mathrm{D}_{-1}+\mathrm{OPS}(-3)$
$\displaystyle\bm{t}\cdot\operatorname{W}_{k}(\bm{n}\cdot)=-\bm{n}\cdot\operatorname{W}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle\kappa\mathrm{D}\operatorname{V}_{k}+\mathrm{OPS}(-3)$ (A.19b)
$\displaystyle=$
$\displaystyle\frac{1}{2}\kappa\operatorname{H}-\frac{k^{2}}{4}\kappa\mathrm{D}_{-2}\operatorname{H}+\mathrm{OPS}(-4)$
$\displaystyle\bm{n}\cdot\operatorname{V}_{k}(\bm{n}\cdot)=\bm{t}\cdot\operatorname{V}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle\operatorname{V}_{k}+4\kappa\operatorname{V}_{k}^{3}+\mathrm{OPS}(-4)$
(A.19c) $\displaystyle=$
$\displaystyle\frac{1}{2}\operatorname{H}\mathrm{D}_{-1}-\frac{1}{4}(k^{2}+2\kappa)\mathrm{D}_{-3}\operatorname{H}+\mathrm{OPS}(-3)$
$\displaystyle\bm{t}\cdot\operatorname{V}_{k}(\bm{n}\cdot)=-\bm{n}\cdot\operatorname{V}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle-4\kappa\operatorname{V}_{k}^{3}+\mathrm{OPS}(-4)$ (A.19d)
$\displaystyle=$
$\displaystyle\frac{1}{2}\kappa\mathrm{D}_{-3}\operatorname{H}+\mathrm{OPS}(-4)\
$
$\displaystyle\bm{n}\cdot\mathrm{D}\operatorname{V}_{k}(\bm{n}\cdot)=\bm{t}\cdot\mathrm{D}\operatorname{V}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle\mathrm{D}\operatorname{V}_{k}+\mathrm{OPS}(-4)$ (A.19e)
$\displaystyle=$
$\displaystyle\frac{1}{2}\operatorname{H}-\frac{k^{2}}{4}\mathrm{D}_{-2}\operatorname{H}+\mathrm{OPS}(-4)\
$
$\displaystyle\bm{t}\cdot\mathrm{D}\operatorname{V}_{k}(\bm{n}\cdot)=-\bm{n}\cdot\mathrm{D}\operatorname{V}_{k}(\bm{t}\cdot)$
$\displaystyle=$
$\displaystyle-4k^{2}\kappa\mathrm{D}\operatorname{V}_{k}^{3}+\mathrm{OPS}(-4)$
(A.19f) $\displaystyle=$
$\displaystyle\frac{1}{2}k^{2}\kappa\operatorname{H}\mathrm{D}_{-2}+\mathrm{OPS}(-4)$
$\displaystyle\bm{n}\cdot\operatorname{K}_{k}(\bm{n}\cdot)=\bm{t}\cdot\operatorname{K}_{k}(\bm{t}\cdot)$
$\displaystyle=$ $\displaystyle
4k^{2}\kappa\operatorname{V}_{k}^{3}+\mathrm{OPS}(-4)$ (A.19g)
$\displaystyle=$
$\displaystyle-\frac{1}{2}k^{2}\kappa\operatorname{H}\mathrm{D}_{-3}+\mathrm{OPS}(-4)\
$
$\displaystyle\bm{t}\cdot\operatorname{K}_{k}(\bm{n}\cdot)=-\bm{n}\cdot\operatorname{K}_{k}(\bm{t}\cdot)$
$\displaystyle=$ $\displaystyle\mathrm{OPS}(-4).$ (A.19h)
Proof. It is consequence of Lemma A.4 and Proposition A.1. $\Box$
|
November, 2022
Revised version: December, 2022
Conformal $(p,q)$ supergeometries in two dimensions
Sergei M. Kuzenko and Emmanouil S. N. Raptakis
Department of Physics M013, The University of Western Australia
35 Stirling Highway, Perth W.A. 6009, Australia
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
We propose a superspace formulation for conformal $(p,q)$ supergravity in two
dimensions as a gauge theory of the superconformal group
$\mathsf{OSp}_{0}(p|2;{\mathbb{R}})\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$
with a flat connection. Upon degauging of certain local symmetries, this
conformal superspace is shown to reduce to a conformally flat
$\mathsf{SO}(p)\times\mathsf{SO}(q)$ superspace with the following properties:
(i) its structure group is a direct product of the Lorentz group and
$\mathsf{SO}(p)\times\mathsf{SO}(q)$; and (ii) the residual local scale
symmetry is realised by super-Weyl transformations with an unconstrained real
parameter. As an application of the formalism, we describe ${\cal N}$-extended
AdS superspace as a maximally symmetric supergeometry in the $p=q\equiv\cal N$
case. If at least one of the parameters $p$ or $q$ is even, alternative
superconformal groups and, thus, conformal superspaces exist. In particular,
if $p=2n$, a possible choice of the superconformal group is
$\mathsf{SU}(1,1|n)\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$, for $n\neq 2$,
and $\mathsf{PSU}(1,1|2)\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$, when $n=2$.
In general, a conformal superspace formulation is associated with a supergroup
$G=G_{L}\times G_{R}$, where the simple supergroups $G_{L}$ and $G_{R}$ can be
any of the extended superconformal groups, which were classified by Günaydin,
Sierra and Townsend. Degauging the corresponding conformal superspace leads to
a conformally flat $H_{L}\times H_{R}$ superspace, where $H_{L}$ ($H_{R}$) is
the $R$-symmetry subgroup of $G_{L}$ ($G_{R}$). Additionally, for the $p,q\leq
2$ cases we propose composite primary multiplets which generate the Gauss-
Bonnet invariant and supersymmetric extensions of the Fradkin-Tseytlin term.
###### Contents
1. 1 Introduction
2. 2 The conformal Killing supervector fields of $\mathbb{M}^{(2|p,q)}$
3. 3 Conformal geometry in two dimensions
1. 3.1 Gauging the conformal algebra
2. 3.2 Degauging to Lorentzian geometry
4. 4 Conformal $(p,q)$ superspace
5. 5 The superspace geometry of $(p,q)$ supergravity
1. 5.1 $p,q>1$ case
2. 5.2 $p>1,~{}q=1$ case
3. 5.3 $p>1,~{}q=0$ case
4. 5.4 $p=q=1$ case
5. 5.5 $p=1,~{}q=0$ case
6. 6 Generalisations and future prospects
7. A Conformal geometry in $d\geq 3$ dimensions
1. A.1 Gauging the conformal algebra in $d\geq 3$ dimensions
2. A.2 Degauging to Lorentzian geometry
8. B Conformal $(1,0)$ superspace with non-vanishing curvature
9. C Compactified Minkowski superspace
## 1 Introduction
Local superconformal symmetry has played a major role in string theory and
supergravity. The ${\cal N}=1$ spinning string can be formulated as a two-
dimensional (2D) linear sigma model coupled either to ${\cal N}=1$ Poincaré
supergravity [1, 2] with super-Weyl invariance [3] or to ${\cal N}=1$
conformal supergravity [4]. Similarly, the ${\cal N}=2$ spinning string can be
realised as a Weyl invariant matter-coupled ${\cal N}=2$ supergravity theory
[5] or as ${\cal N}=2$ conformal supergravity coupled to a linear sigma model
[4].
In the component setting, conformal $(p,q)$ supergravity in two dimensions was
described as a gauge theory of the superconformal algebra
$\mathfrak{osp}(p|2;{\mathbb{R}})\times\mathfrak{osp}(q|2;{\mathbb{R}})$, for
$p,q\leq 2$, in the mid 1980s [4, 6, 7, 8, 9, 10, 11, 12]. Pernici and van
Nieuwenhuizen [13] constructed ${\cal N}=4\equiv(4,4)$ conformal supergravity
as a gauge theory of the superconformal algebra
$\mathfrak{psu}(1,1|2)\times\mathfrak{psu}(1,1|2)$. They coupled ${\cal N}=4$
conformal supergravity to an arbitrary number of ${\cal N}=4$ scalar
multiplets. An alternative approach was put forward by Schoutens [14] who
formulated $d=2$ conformal supergravity with ${\cal N}=0,1,2$ and 4 as gauge
theories corresponding to infinite-dimensional superalgebras.
In this paper, as a generalisation of the component results, we propose
superspace formulations for conformal $(p,q)$ supergravity theories in two
dimensions for arbitrary $p$ and $q$. We mostly concentrate on constructing
conformal $(p,q)$ supergravity as a gauge theory of the superconformal group
$\mathsf{OSp}_{0}(p|2;{\mathbb{R}})\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$
with a flat connection.111We point out that
$\mathsf{OSp}_{0}(p|2;{\mathbb{R}})\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$
is the connected superconformal group of compactified $(p,q)$ Minkowski
superspace in two dimensions, eq. (C.1), see [15] for the technical details.
Strictly speaking, if $p$ is even, $\mathsf{OSp}_{0}(p|2;{\mathbb{R}})$ should
be replaced with $\mathsf{OSp}_{0}(p|2;{\mathbb{R}})/{\mathbb{Z}}_{2}$, and
similar for $q$. However, we will not pay attention to such technical details.
However, our approach allows one to construct a conformal superspace
formulation that is associated with a supergroup $G=G_{L}\times G_{R}$, where
the simple supergroups $G_{L}$ and $G_{R}$ can be any of the extended
superconformal groups, which were classified by Günaydin, Sierra and Townsend
[16] (see also [17]). Our $d=2$ construction is a natural extension of the
conformal superspace approaches in higher dimensions $3\leq d\leq 6$ [18, 19,
20, 21, 22]. From the conceptual point of view, these approaches are
superspace analogues of the formulation for conformal gravity as the gauge
theory of the conformal group in four dimensions [23].
It is appropriate to give a few comments about conformal gravity in $d$
dimensions following the discussions in [24, 20, 22]. Beyond three dimensions,
$d>3$, the algebra of conformal covariant derivatives is
$\displaystyle[\nabla_{a},\nabla_{b}]=-\frac{1}{2}C_{abcd}M^{cd}-\frac{1}{2(d-3)}\nabla^{d}C_{abcd}K^{c}~{},\qquad
d>3~{},$ (1.1)
with $M^{cd}$ and $K^{c}$ being the Lorentz and special conformal generators,
respectively. It is determined by the Weyl tensor $C_{abcd}$, see appendix A.1
for its properties. For $d=3$ the algebra of conformal covariant derivatives
looks simpler
$\displaystyle[\nabla_{a},\nabla_{b}]=-\frac{1}{2}\varepsilon_{abc}W^{cd}K_{d}~{},\qquad
d=3~{},$ (1.2)
where $W_{ab}$ is the Cotton tensor, see appendix A.1 for its properties.
Finally, in the $d=2$ case the conformal connection is flat,
$\displaystyle[\nabla_{a},\nabla_{b}]=0~{},\qquad d=2~{}.$ (1.3)
Actually, the $d=2$ case is somewhat subtle. If one gauges the $d=2$ conformal
group $\mathsf{SL}(2,{\mathbb{R}})\times\mathsf{SL}(2,{\mathbb{R}})$ and
imposes the same constraints as in higher dimensions [23] (see also [24, 20,
22] for a review), then the resulting algebra of conformal covariant
derivatives turns out to be
$\displaystyle[\nabla_{a},\nabla_{b}]=\varepsilon_{ab}W^{c}K_{c}~{},$ (1.4)
where the special conformal curvature $W^{c}$ is a primary field, as discussed
in section 3. However, with $W^{c}\neq 0$ the theory is not equivalent to
conformal gravity, since there is an additional gauge field along with the
gravitational field. That is why one is forced to impose the conformal
flatness condition (1.3). As a result, the special conformal connection
$\mathfrak{f}_{ab}$ (which corresponds to the special conformal generator)
becomes a non-local function of the vielbein (in a gauge where the dilatation
connection $b_{a}$ is gauged away). This is in contrast to the situation in
$d>2$ spacetime dimensions, reviewed in appendix A, where $\mathfrak{f}_{ab}$
is proportional to the Schouten tensor, eq. (A.13).
The lessons from $d=2$ conformal gravity lead us to define $d=2$ conformal
$(p,q)$ supergravity as a gauge theory of the superconformal group
$\mathsf{OSp}_{0}(p|2;{\mathbb{R}})\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$
with a flat connection. Upon degauging of certain local symmetries, this
conformal superspace is shown to reduce to a conformally flat superspace with
its structure group being a direct product of the Lorentz group and
$\mathsf{SO}(p)\times\mathsf{SO}(q)$. The conformal flatness of the resulting
$\mathsf{SO}(p)\times\mathsf{SO}(q)$ superspace means that its supergeometry
is describable locally by a single prepotential modulo purely gauge degrees of
freedom.
An important comment is in order. In the $d=2$ case one can try to follow the
philosophy of [18, 19, 20, 21, 22] to construct conformal superspace
formulations in higher dimensions $3\leq d\leq 6$, which is: the curvature
structure of conformal superspace should resemble that of the super Yang-Mills
one. Then one will end up with a conformal $(p,q)$ superspace with non-
vanishing curvature as an extension of the non-supersymmetric geometry (1.4).
This idea will be elaborated in appendix B for $(1,0)$ supersymmetry.
This paper is organised as follows. Section 2 is devoted to a derivation of
the infinite-dimensional superconformal algebra of Minkowski superspace
$\mathbb{M}^{(2|p,q)}$ by employing its conformal Killing supervector fields.
We then describe the additional constraints of the conformal Killing
supervector fields which single out the finite-dimensional superconformal
algebra $\mathfrak{osp}(p|2;\mathbb{R})\oplus\mathfrak{osp}(q|2;\mathbb{R})$.
In section 3 we review the formulation of conformal gravity in two dimensions
as the gauge theory of the $d=2$ conformal group
$\mathsf{SL}(2,{\mathbb{R}})\times\mathsf{SL}(2,{\mathbb{R}})$. Building on
the construction of conformal gravity, in section 4 we formulate conformal
$(p,q)$ supergravity in two dimensions as a gauge theory of the superconformal
group
$\mathsf{OSp}_{0}(p|2;{\mathbb{R}})\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$
with a flat connection. The procedure of ‘degauging’ from this superconformal
formulation to the $\mathsf{SO}(p)\times\mathsf{SO}(q)$ superspaces is
described in section 5. Section 6 is mostly devoted to generalisations of the
results derived in section 5. Such generalisations become possible in the case
that at least one of the parameters $p$ or $q$ is even, and then alternative
superconformal groups and, thus, conformal superspaces exist. The main body of
this paper is accompanied by three technical appendices. Appendix A reviews
conformal geometry in $d\geq 3$ dimensions following [20, 22]. In appendix B
we construct $\mathcal{N}=(1,0)$ conformal superspace with non-vanishing
curvature. Appendix C is devoted to the supertwistor realisation of
compactified Minkowski superspace $\overline{\mathbb{M}}^{(2|2n,q)}$ as a
homogeneous space of the superconformal group
$\mathsf{SU}(1,1|n)\times{\mathsf{OSp}}_{0}(q|2;{\mathbb{R}})$.
Throughout this paper we will make use of two types of notation, $(p,q)$ and
${\cal N}=(p,q)$, to denote superspaces with $p$ left and $q$ right real
spinor coordinates. Additionally, the special case $p=q$ is also referred to
as ${\cal N}=p$.
## 2 The conformal Killing supervector fields of $\mathbb{M}^{(2|p,q)}$
In the case of a $d$-dimensional superconformal field theory in Minkowski
superspace ${\mathbb{M}}^{d|\delta}$, its symmetries are formulated in terms
of the conformal Killing supervector fields of ${\mathbb{M}}^{d|\delta}$. This
section is devoted the description of the conformal Killing supervector fields
of $\mathbb{M}^{(2|p,q)}$, the $(p,q)$ Minkowski superspace in two dimensions
[25].
Minkowski superspace $\mathbb{M}^{(2|p,q)}$ is parametrised by the real
coordinates $z^{A}=(x^{a},\theta^{+{\overline{I}}},\theta^{-\underline{I}})$,
where $x^{a}=(x^{++},x^{--})=\frac{1}{\sqrt{2}}(x^{0}+x^{1},x^{0}-x^{1})$,
${\overline{I}}=\overline{1},\dots,\overline{p}$ and
$\underline{I}={\underline{1}},\dots,\underline{q}$. Its covariant derivatives
$D_{A}=(\partial_{a},D_{+}^{{\overline{I}}},D_{-}^{\underline{I}})$ take the
form
$\displaystyle\partial_{a}:=\frac{\partial}{\partial
x^{a}}=(\partial_{++},\partial_{--})~{},\quad
D_{+}^{{\overline{I}}}:=\frac{\partial}{\partial\theta^{+{\overline{I}}}}+{\rm
i}\theta^{+{\overline{I}}}\partial_{++}~{},\quad
D_{-}^{\underline{I}}:=\frac{\partial}{\partial\theta^{-\underline{I}}}+{\rm
i}\theta^{-\underline{I}}\partial_{--}~{},$ (2.1)
and satisfy the algebra:
$\displaystyle\\{D_{+}^{{\overline{I}}},D_{+}^{\overline{J}}\\}=2{\rm
i}\delta^{{\overline{I}}\overline{J}}\partial_{++}~{},\qquad\\{D_{-}^{\underline{I}},D_{-}^{\underline{J}}\\}=2{\rm
i}\delta^{\underline{I}\underline{J}}\partial_{--}~{}.$ (2.2)
We emphasise that for $p=0$ the left spinor covariant derivative
$D_{+}^{{\overline{I}}}$ does not appear, similarly $D_{-}^{\underline{I}}$ is
not present for $q=0$.
The conformal Killing supervector fields of $\mathbb{M}^{(2|p,q)}$,
$\xi=\xi^{a}\partial_{a}+\xi^{+{\overline{I}}}D_{+}^{{\overline{I}}}+\xi^{-\underline{I}}D_{-}^{\underline{I}}=\xi^{++}\partial_{++}+\xi^{--}\partial_{--}+\xi^{+{\overline{I}}}D_{+}^{{\overline{I}}}+\xi^{-\underline{I}}D_{-}^{\underline{I}}=\bar{\xi}~{},$
(2.3)
may be defined to satisfy the constraints
$[\xi,D_{+}^{{\overline{I}}}]=-(D_{+}^{{\overline{I}}}\xi^{+\overline{J}})D_{+}^{\overline{J}}~{},\qquad[\xi,D_{-}^{\underline{I}}]=-(D_{-}^{\underline{I}}\xi^{-\underline{J}})D_{-}^{\underline{J}}~{}.$
(2.4)
We note that for vanishing $p$ ($q$), the spinor $\xi^{+{\overline{I}}}$
($\xi^{-{\underline{I}}}$) must be turned off. From $\eqref{2.4}$ we obtain
the fundamental equations
$\displaystyle D_{+}^{{\overline{I}}}\xi^{--}$ $\displaystyle=$ $\displaystyle
0\quad\implies\quad\partial_{++}\xi^{--}=0~{},$ (2.5a) $\displaystyle
D_{-}^{\underline{I}}\xi^{++}$ $\displaystyle=$ $\displaystyle
0\quad\implies\quad\partial_{--}\xi^{++}=0~{},$ (2.5b)
and expressions for the spinor parameters
$\displaystyle\xi^{+{\overline{I}}}=-\frac{{\rm
i}}{2}D_{+}^{{\overline{I}}}\xi^{++}~{},\qquad\xi^{-\underline{I}}=-\frac{{\rm
i}}{2}D_{-}^{\underline{I}}\xi^{--}~{}.$ (2.6)
Hence, we see that every conformal Killing supervector field (2.3) is
completely determined by its vector parameter $\xi^{a}$. Additionally, the
equations (2.5) tell us that
$\displaystyle\xi^{++}=\xi^{++}(x^{++},\theta^{+})\equiv\xi^{++}(\zeta_{L})~{},\qquad\xi^{--}=\xi^{--}(x^{--},\theta^{-})\equiv\xi^{--}(\zeta_{R})~{}.$
(2.7)
Here $\xi^{++}(\zeta_{L})$ and $\xi^{--}(\zeta_{R})$ are arbitrary functions
of $\zeta_{L}$ and $\zeta_{R}$, respectively.
Taking (2.5) into account, the equations (2.4) can be rewritten in the form
$\displaystyle[\xi,D_{+}^{{\overline{I}}}]$
$\displaystyle=-\frac{1}{2}(\sigma[\xi]+K[\xi])D_{+}^{{\overline{I}}}-\rho[\xi]^{{\overline{I}}\overline{J}}D_{+}^{\overline{J}}~{},$
(2.8a) $\displaystyle[\xi,D_{-}^{\underline{I}}]$
$\displaystyle=-\frac{1}{2}(\sigma[\xi]-K[\xi])D_{-}^{\underline{I}}-\rho[\xi]^{\underline{I}\underline{J}}D_{-}^{\underline{J}}~{},$
(2.8b)
where we have defined the following parameters:
$\displaystyle\sigma[\xi]$
$\displaystyle:=\frac{1}{2}\big{(}\partial_{++}\xi^{++}+\partial_{--}\xi^{--}\big{)}~{},$
(2.9a) $\displaystyle K[\xi]$
$\displaystyle:=\frac{1}{2}\big{(}\partial_{++}\xi^{++}-\partial_{--}\xi^{--}\big{)}~{},$
(2.9b) $\displaystyle\rho[\xi]^{{\overline{I}}\overline{J}}$
$\displaystyle:=-\frac{{\rm
i}}{4}\big{[}D_{+}^{{\overline{I}}},D_{+}^{\overline{J}}\big{]}\xi^{++}~{},$
(2.9c) $\displaystyle\rho[\xi]^{\underline{I}\underline{J}}$
$\displaystyle:=-\frac{{\rm
i}}{4}\big{[}D_{-}^{\underline{I}},D_{-}^{\underline{J}}\big{]}\xi^{--}~{}.$
(2.9d)
Their $z$-independent components generate scale, Lorentz, $\mathfrak{so}(p)$
and $\mathfrak{so}(q)$ transformations, respectively. We point out that the
$\mathfrak{so}(p)$ parameter $\rho[\xi]^{{\overline{I}}\overline{J}}$ is a
function of the left variables $\zeta_{L}=(x^{++},\theta^{+})$, while the
$\mathfrak{so}(q)$ parameter $\rho[\xi]^{\underline{I}\underline{J}}$ is a
function of the right variables $\zeta_{R}=(x^{--},\theta^{-})$. Further, the
former (latter) identically vanishes when $p<2$ ($q<2$).
Given two conformal Killing supervectors $\xi_{1}$ and $\xi_{2}$, their
commutator is another conformal Killing supervector $\xi_{3}$
$\displaystyle[\xi_{1},\xi_{2}]=\xi_{3}^{a}\partial_{a}+\xi_{3}^{+{\overline{I}}}D_{+}^{{\overline{I}}}+\xi_{3}^{-{\underline{I}}}D_{-}^{{\underline{I}}}=\xi_{3}~{},$
(2.10a) with the definitions $\displaystyle\xi_{3}^{++}$
$\displaystyle=\xi_{1}^{++}\partial_{++}\xi_{2}^{++}-\xi_{2}^{++}\partial_{++}\xi_{1}^{++}+2{\rm
i}\xi_{1}^{+{\overline{I}}}\xi_{2}^{+{\overline{I}}}~{}\implies~{}D_{-}^{{\underline{I}}}\xi_{3}^{++}=0~{},$
(2.10b) $\displaystyle\xi_{3}^{--}$
$\displaystyle=\xi_{1}^{--}\partial_{--}\xi_{2}^{--}-\xi_{2}^{--}\partial_{--}\xi_{1}^{--}+2{\rm
i}\xi_{1}^{-{\underline{I}}}\xi_{2}^{-{\underline{I}}}~{}\implies~{}D_{+}^{{\overline{I}}}\xi_{3}^{--}=0~{},$
(2.10c)
and the spinor parameters are determined by eq. (2.6). Equations (2.5) imply
that the algebra of conformal Killing supervector fields of
$\mathbb{M}^{(2|p,q)}$ is infinite dimensional. It may be referred to as a
$(p,q)$ super Virasoro algebra. Such superalgebras for the $p=q$ case were
studied in [26, 27, 28].
The superconformal transformation law of a primary tensor superfield $U$ (with
suppressed Lorentz, $\mathsf{SO}(p)$ and $\mathsf{SO}(q)$ indices) is
$\displaystyle\delta_{\xi}U$ $\displaystyle=$
$\displaystyle\Big{\\{}\xi+\lambda_{U}K[\xi]+\Delta_{U}\sigma[\xi]+\frac{1}{2}\rho[\xi]^{{\overline{I}}\overline{J}}\mathfrak{L}^{{\overline{I}}{\overline{J}}}+\frac{1}{2}\rho[\xi]^{\underline{I}\underline{J}}\mathfrak{R}^{{\underline{I}}{\underline{J}}}\Big{\\}}U~{},$
(2.11)
where $\mathfrak{L}^{{\overline{I}}{\overline{J}}}$ and
$\mathfrak{R}^{{\underline{I}}{\underline{J}}}$ are the generators of the
groups $\mathsf{SO}(p)$ and $\mathsf{SO}(q)$, respectively.222As usual, we
adopt the convention where a factor of $1/2$ is inserted when performing a
summations over pairs of antisymmetric indices. The parameters $\lambda_{U}$
and $\Delta_{U}$ are called the Lorentz weight and the dimension (or Weyl
weight) of $U$, respectively. These weights are related if $U$ depends only on
$\zeta_{L}$ or $\zeta_{R}$,
$\displaystyle U$ $\displaystyle=$ $\displaystyle
U(\zeta_{L})\quad\implies\quad\lambda_{U}=\Delta_{U}~{},\qquad\mathfrak{R}^{{\underline{I}}{\underline{J}}}U=0~{},$
(2.12a) $\displaystyle U$ $\displaystyle=$ $\displaystyle
U(\zeta_{R})\quad\implies\quad\lambda_{U}=-\Delta_{U}~{},\qquad\mathfrak{L}^{{\overline{I}}{\overline{J}}}U=0~{}.$
(2.12b)
As pointed out above, the algebra of conformal Killing supervector fields of
$\mathbb{M}^{(2|p,q)}$ is infinite dimensional. It contains a finite
dimensional subalgebra which is singled out by the constraints
$\displaystyle p=0:$
$\displaystyle\quad\partial_{++}\partial_{++}\partial_{++}\xi^{++}=0~{},$
(2.13a) $\displaystyle p=1:$
$\displaystyle\quad\partial_{++}\partial_{++}D_{+}\xi^{++}=0~{},$ (2.13b)
$\displaystyle p=2:$
$\displaystyle\quad\partial_{++}D_{+}^{[{\overline{I}}}D_{+}^{{\overline{J}}]}\xi^{++}=0~{},$
(2.13c) $\displaystyle p>2:$ $\displaystyle\quad
D_{+}^{[{\overline{I}}}D_{+}^{{\overline{J}}}D_{+}^{{\overline{K}}]}\xi^{++}=0~{},$
(2.13d)
and their counterparts in the right sector. Physically, these conditions mean
that $\xi$ generates those infinitesimal superconformal transformations that
belong to the superconformal algebra
$\mathfrak{osp}(p|2;\mathbb{R})\oplus\mathfrak{osp}(q|2;\mathbb{R})$. This is
the Lie algebra of the superconformal group
${\mathsf{OSp}}_{0}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}_{0}(q|2;{\mathbb{R}})$,
which acts on the compactified Minkowski superspace (C.1). It is instructive
to check that (2.10) preserves the conditions (2.13). We will assume (2.13) in
what follows.
Employing (2.5), it is possible to show that the parameters (2.9a) and (2.9b)
satisfy the constraints
$\displaystyle D_{+}^{{\overline{I}}}\sigma[\xi]$
$\displaystyle=D_{+}^{{\overline{I}}}K[\xi]\quad\implies\quad\partial_{++}\sigma[\xi]=\partial_{++}K[\xi]~{},$
(2.14a) $\displaystyle D_{-}^{\underline{I}}\sigma[\xi]$
$\displaystyle=-D_{-}^{\underline{I}}K[\xi]\quad\implies\quad\partial_{--}\sigma[\xi]=-\partial_{--}K[\xi]~{}.$
(2.14b) Next, by using (2.5) in conjunction with (2.13), one obtains the
following constraints on the $R$-symmetry parameters: $\displaystyle
D_{+}^{{\overline{I}}}\rho[\xi]^{\overline{J}{\overline{K}}}$
$\displaystyle=2\delta^{{\overline{I}}[\overline{J}}D_{+}^{{\overline{K}}]}\sigma[\xi]\quad\implies\quad\partial_{++}\rho^{{\overline{I}}{\overline{J}}}[\xi]=0~{},$
(2.14c) $\displaystyle
D_{-}^{\underline{I}}\rho[\xi]^{\overline{J}{\overline{K}}}$
$\displaystyle=0\quad\implies\quad\partial_{--}\rho^{{\overline{I}}{\overline{J}}}[\xi]=0~{},$
(2.14d) $\displaystyle
D_{+}^{{\overline{I}}}\rho[\xi]^{{\underline{J}}{\underline{K}}}$
$\displaystyle=0\quad\implies\quad\partial_{++}\rho^{{\underline{I}}{\underline{J}}}[\xi]=0~{},$
(2.14e) $\displaystyle
D_{-}^{{\underline{I}}}\rho[\xi]^{{\underline{J}}{\underline{K}}}$
$\displaystyle=2\delta^{{\underline{I}}[{\underline{J}}}D_{-}^{{\underline{K}}]}\sigma[\xi]\quad\implies\quad\partial_{--}\rho^{{\underline{I}}{\underline{J}}}[\xi]=0~{},$
(2.14f)
and on the scaling parameter:
$\displaystyle D_{+}^{{\overline{I}}}D_{+}^{{\overline{J}}}\sigma[\xi]$
$\displaystyle=\frac{{\rm
i}}{p}\delta^{{\overline{I}}{\overline{J}}}\partial_{++}\sigma[\xi]\quad\implies\quad\partial_{++}D_{+}^{\overline{I}}\sigma[\xi]=0~{},$
(2.15a) $\displaystyle
D_{-}^{{\underline{I}}}D_{-}^{{\underline{J}}}\sigma[\xi]$
$\displaystyle=\frac{{\rm
i}}{q}\delta^{{\underline{I}}{\underline{J}}}\partial_{--}\sigma[\xi]\quad\implies\quad\partial_{--}D_{-}^{\underline{I}}\sigma[\xi]=0~{}.$
(2.15b)
The above results mean that we may parametrise the conformal Killing
supervector fields obeying (2.13) as
$\xi\equiv\xi(\lambda(P)^{a},\lambda(Q)^{+{\overline{I}}},\lambda(Q)^{-{\underline{I}}},\lambda(M),\lambda(\mathbb{D}),\lambda(\mathfrak{L})^{{\overline{I}}{\overline{J}}},\lambda(\mathfrak{R})^{{\underline{I}}{\underline{J}}},\lambda(K)_{a},\lambda(S)_{+}^{{\overline{I}}},\lambda(S)_{-}^{{\underline{I}}})~{},$
(2.16)
where we have defined the parameters
$\displaystyle\lambda(P)^{a}$
$\displaystyle:=\xi^{a}|_{z=0}~{},\qquad\lambda(Q)^{+{\overline{I}}}:=\xi^{+{\overline{I}}}|_{z=0}~{},\qquad\lambda(Q)^{-{\underline{I}}}:=\xi^{-{\underline{I}}}|_{z=0}~{},$
(2.17a)
$\displaystyle\qquad\quad~{}\,\lambda(M):=K[\xi]|_{z=0}~{},\qquad\quad\,\lambda({\mathbb{D}}):=\sigma[\xi]|_{z=0}~{},$
(2.17b)
$\displaystyle\qquad\quad\lambda(\mathfrak{L})^{{\overline{I}}{\overline{J}}}:=\rho[\xi]^{{\overline{I}}{\overline{J}}}|_{z=0}~{},\;\,\,\quad\lambda(\mathfrak{R})^{{\underline{I}}{\underline{J}}}:=\rho[\xi]^{{\underline{I}}{\underline{J}}}|_{z=0}$
(2.17c) $\displaystyle\lambda(K)_{a}$
$\displaystyle:=\frac{1}{2}\partial_{a}\sigma[\xi]|_{z=0}~{},\quad\lambda(S)_{+}^{{\overline{I}}}:=\frac{1}{2}D_{+}^{\overline{I}}\sigma[\xi]|_{z=0}~{},\quad\lambda(S)_{-}^{{\underline{I}}}:=\frac{1}{2}D_{-}^{\underline{I}}\sigma[\xi]|_{z=0}~{}.$
(2.17d)
In particular, $\xi$ may be represented as
$\xi=\lambda(X)^{\tilde{A}}X_{\tilde{A}}~{},$ (2.18)
where we have introduced a condensed notation for the superconformal
parameters
$\displaystyle\lambda(X)^{\tilde{A}}$
$\displaystyle=(\lambda(P)^{A},\lambda(M),\lambda(\mathbb{D}),\lambda(\mathfrak{L})^{{\overline{I}}{\overline{J}}},\lambda(\mathfrak{R})^{{\underline{I}}{\underline{J}}},\lambda(K)_{A})~{},$
(2.19a) $\displaystyle\lambda(P)^{A}$
$\displaystyle=(\lambda(P)^{a},\lambda(Q)^{+{\overline{I}}},\lambda(Q)^{-{\underline{I}}}),\qquad\lambda(K)_{A}=(\lambda(K)_{a},\lambda(S)_{+}^{{\overline{I}}},\lambda(S)_{-}^{{\underline{I}}})~{},$
(2.19b)
and for generators of the superconformal algebra
$\displaystyle X_{\tilde{A}}$
$\displaystyle=(P_{A},M,\mathbb{D},\mathfrak{L}^{{\overline{I}}{\overline{J}}},\mathfrak{R}^{{\underline{I}}{\underline{J}}},K^{A})~{},$
(2.20a) $\displaystyle P_{A}$
$\displaystyle=(P_{a},Q^{{\overline{I}}}_{+},Q^{{\underline{I}}}_{-})~{},\qquad
K^{A}=(K^{a},S_{+}^{{\overline{I}}},S_{-}^{{\underline{I}}})~{}.$ (2.20b)
Making use of the above results allows us to derive the graded commutation
relations for the superconformal algebra, keeping in mind the relation
$\displaystyle[\xi_{1},\xi_{2}]=-\lambda(X)_{2}^{\tilde{B}}\lambda(X)_{1}^{\tilde{A}}\big{[}X_{\tilde{A}},X_{\tilde{B}}\big{\\}}~{}.$
(2.21)
The commutation relations for the conformal algebra are as follows:
$\displaystyle[M,P_{\pm\pm}]$ $\displaystyle=\pm
P_{\pm\pm}~{},\qquad~{}\,[\mathbb{D},P_{\pm\pm}]=P_{\pm\pm}~{},$ (2.22a)
$\displaystyle[M,K^{\pm\pm}]$ $\displaystyle=\mp
K^{\pm\pm}~{},\qquad[\mathbb{D},K^{\pm\pm}]=-K^{\pm\pm}~{},$ (2.22b)
$\displaystyle\;[K^{\pm\pm},P_{\pm\pm}]=2(\mathbb{D}\pm M)~{}.$ (2.22c) The
$R$-symmetry generators $\mathfrak{L}^{{\overline{I}}{\overline{J}}}$ and
$\mathfrak{R}^{{\underline{I}}{\underline{J}}}$ commute with all the
generators of the conformal group. Amongst themselves, they obey the algebra
$\displaystyle[\mathfrak{L}^{{\overline{I}}{\overline{J}}},\mathfrak{L}^{{\overline{K}}{\overline{L}}}]$
$\displaystyle=2\delta^{{\overline{K}}[{\overline{I}}}\mathfrak{L}^{{\overline{J}}]{\overline{L}}}-2\delta^{{\overline{L}}[{\overline{I}}}\mathfrak{L}^{{\overline{J}}]{\overline{K}}}~{},$
(2.22d)
$\displaystyle[\mathfrak{R}^{{\underline{I}}{\underline{J}}},\mathfrak{R}^{{\underline{K}}{\underline{L}}}]$
$\displaystyle=2\delta^{{\underline{K}}[{\underline{I}}}\mathfrak{R}^{{\underline{J}}]{\underline{L}}}-2\delta^{{\underline{L}}[{\underline{I}}}\mathfrak{R}^{{\underline{J}}]{\underline{K}}}~{}.$
(2.22e) The superconformal algebra is then obtained by extending the
translation generator $P_{a}$ to $P_{A}$ and the special conformal generator
$K^{a}$ to $K^{A}$. The commutation relations involving the $Q$-supersymmetry
generators with the bosonic ones are:
$\displaystyle\big{[}M,Q_{+}^{\overline{I}}\big{]}$
$\displaystyle=\frac{1}{2}Q_{+}^{\overline{I}}~{},\qquad\qquad\quad\big{[}M,Q_{-}^{\underline{I}}\big{]}=-\frac{1}{2}Q_{-}^{\underline{I}}~{},$
(2.22f) $\displaystyle\big{[}\mathbb{D},Q_{+}^{\overline{I}}\big{]}$
$\displaystyle=\frac{1}{2}Q_{+}^{\overline{I}}~{},\qquad\qquad\quad\,\,\big{[}\mathbb{D},Q^{\underline{I}}_{-}\big{]}=\frac{1}{2}Q^{\underline{I}}_{-}~{},$
(2.22g)
$\displaystyle\big{[}\mathfrak{L}^{{\overline{I}}{\overline{J}}},Q_{+}^{\overline{K}}\big{]}$
$\displaystyle=2\delta^{{\overline{K}}[{\overline{I}}}Q_{+}^{{\overline{J}}]}~{},\qquad\;\;\big{[}\mathfrak{R}^{{\underline{I}}{\underline{J}}},Q_{-}^{\underline{K}}\big{]}=2\delta^{{\underline{K}}[{\underline{I}}}Q_{-}^{{\underline{J}}]}~{},$
(2.22h) $\displaystyle\big{[}K^{++},Q_{+}^{\overline{I}}\big{]}$
$\displaystyle={\rm
i}S^{+{\overline{I}}}~{},\qquad\quad\quad\,\big{[}K^{--},Q_{-}^{\underline{I}}\big{]}={\rm
i}S^{-{\underline{I}}}~{}.$ (2.22i) The commutation relations involving the
$S$-supersymmetry generators with the bosonic operators are:
$\displaystyle\big{[}M,S^{+{\overline{I}}}\big{]}$
$\displaystyle=-\frac{1}{2}S^{+{\overline{I}}}~{},\qquad\qquad\quad\big{[}M,S^{-{\underline{I}}}\big{]}=\frac{1}{2}S^{-{\underline{I}}}~{},$
(2.22j) $\displaystyle\big{[}\mathbb{D},S^{+{\overline{I}}}\big{]}$
$\displaystyle=-\frac{1}{2}S^{+{\overline{I}}}~{},\qquad\qquad\quad\,\,\big{[}\mathbb{D},S^{-{\underline{I}}}\big{]}=-\frac{1}{2}S^{-{\underline{I}}}~{},$
(2.22k)
$\displaystyle\big{[}\mathfrak{L}^{{\overline{I}}{\overline{J}}},S^{+{\overline{K}}}\big{]}$
$\displaystyle=2\delta^{{\overline{K}}[{\overline{I}}}S^{+{\overline{J}}]}~{},\qquad\;\;\;\big{[}\mathfrak{R}^{{\underline{I}}{\underline{J}}},S^{-{\underline{K}}}\big{]}=2\delta^{{\underline{K}}[{\underline{I}}}S^{-{\underline{J}}]}~{},$
(2.22l) $\displaystyle\big{[}S^{+{\overline{I}}},P_{++}\big{]}$
$\displaystyle=-2{\rm
i}Q_{+}^{{\overline{I}}}~{},\qquad\quad\quad\;\big{[}S^{-{\underline{I}}},P_{--}\big{]}=-2{\rm
i}Q_{-}^{{\underline{I}}}~{}.$ (2.22m) Finally, the anti-commutation relations
of the fermionic generators are:
$\displaystyle\\{Q_{+}^{{\overline{I}}},Q_{+}^{{\overline{J}}}\\}$
$\displaystyle=2{\rm
i}\delta^{{\overline{I}}{\overline{J}}}P_{++}~{},\qquad\quad\;\;\,\\{Q_{-}^{{\underline{I}}},Q_{-}^{{\underline{J}}}\\}=2{\rm
i}\delta^{{\underline{I}}{\underline{J}}}P_{--}~{},$ (2.22n)
$\displaystyle\\{S^{+{\overline{I}}},S^{+{\overline{J}}}\\}$
$\displaystyle=-4{\rm
i}\delta^{{\overline{I}}{\overline{J}}}K^{++}~{},\qquad\\{S^{-{\underline{I}}},S^{-{\underline{J}}}\\}=-4{\rm
i}\delta^{{\underline{I}}{\underline{J}}}K^{--}~{},$ (2.22o)
$\displaystyle\;\;\\{S^{+{\overline{I}}},Q_{+}^{{\overline{J}}}\\}=2\delta^{{\overline{I}}{\overline{J}}}(\mathbb{D}+M)-2\mathfrak{L}^{{\overline{I}}{\overline{J}}}~{},$
(2.22p)
$\displaystyle\;\;\\{S^{-{\underline{I}}},Q_{-}^{{\underline{J}}}\\}=2\delta^{{\underline{I}}{\underline{J}}}(\mathbb{D}-M)-2\mathfrak{R}^{{\underline{I}}{\underline{J}}}~{}.$
(2.22q)
Note that all remaining (anti-)commutators not listed above vanish
identically.
## 3 Conformal geometry in two dimensions
Before turning to the superconformal case, it is instructive to first consider
conformal gravity as the gauge theory of the $d=2$ conformal group
$\mathsf{SL}(2,{\mathbb{R}})\times\mathsf{SL}(2,{\mathbb{R}})$. Such a
formulation can be extracted from those for the $(1,0)$, ${\cal N}=1$ and
${\cal N}=2$ conformal supergravity theories [4, 6, 7, 8, 9, 10, 11, 14].
However, our discussion below has some specific features, since it is targeted
at formulating $(p,q)$ conformal supergravity in the next section. We will
also emphasise those aspects of conformal gravity which are unique to two
dimensions as compared with the $d>2$ case reviewed in appendix A.
We recall from the previous section that the conformal algebra of
$\mathbb{M}^{2}$ is spanned by the operators $X_{\tilde{a}}$ comprising the
translation ($P_{a}$), Lorentz ($M$), dilatation ($\mathbb{D}$) and special
conformal generators ($K^{a}$), which can be grouped into the two disjoint
subalgebras spanned by $P_{a}$ and $X_{\underline{a}}$:
$\displaystyle X_{\tilde{a}}=(P_{a},X_{\underline{a}})~{},\qquad
X_{\underline{a}}=(M,\mathbb{D},K^{a})~{}.$ (3.1)
Then, the commutation relations (2.22a)-(2.22c) may be rewritten as follows
$\displaystyle[X_{\underline{a}},X_{\underline{b}}]$
$\displaystyle=-f_{\underline{a}\underline{b}}{}^{\underline{c}}X_{\underline{c}}\
,$ (3.2a) $\displaystyle[X_{\underline{a}},P_{{b}}]$
$\displaystyle=-f_{\underline{a}{{b}}}{}^{\underline{c}}X_{\underline{c}}-f_{\underline{a}{{b}}}{}^{{c}}P_{{c}}~{},$
(3.2b)
where $f_{\underline{a}\underline{b}}{}^{\underline{c}}$,
$f_{\underline{a}{{b}}}{}^{\underline{c}}$ and
$f_{\underline{a}{{b}}}{}^{{c}}$ denote the structure coefficients of the
conformal algebra.
### 3.1 Gauging the conformal algebra
Let $\mathcal{M}^{2}$ be a two-dimensional curved spacetime parametrised by
local coordinates $x^{m}$. To gauge the conformal algebra we associate each
non-translational generator $X_{\underline{a}}$ with a connection one-form
$\omega^{\underline{a}}=(\omega,b,\mathfrak{f}_{a})={\rm
d}x^{m}\omega_{m}{}^{\underline{a}}$ and with $P_{a}$ a vielbein one-form
$e^{a}={\rm d}x^{m}e_{m}{}^{a}$, where it is assumed that $e:={\rm
det}(e_{m}{}^{a})\neq 0$, hence there exists a unique inverse vielbein
$e_{a}{}^{m}$
$\displaystyle e_{a}{}^{m}e_{m}{}^{b}=\delta_{a}{}^{b}~{},\qquad
e_{m}{}^{a}e_{a}{}^{n}=\delta_{m}{}^{n}~{}.$ (3.3)
The latter may be used to construct the vector fields
$e_{a}=e_{a}{}^{m}\partial_{m}$, with $\partial_{m}=\partial/\partial x^{m}$,
which constitute a basis for the tangent space at each point of
$\mathcal{M}^{2}$. It may then be used to express the connection in the
vielbein basis as $\omega^{\underline{a}}=e^{b}\omega_{b}{}^{\underline{a}}$,
where $\omega_{b}{}^{\underline{a}}=e_{b}{}^{m}\omega_{m}{}^{\underline{a}}$.
The covariant derivatives $\nabla_{a}$ then take the form
$\displaystyle\nabla_{a}$ $\displaystyle=$ $\displaystyle
e_{a}-\omega_{a}{}^{\underline{b}}X_{\underline{b}}=e_{a}-\omega_{a}M-b_{a}\mathbb{D}-\mathfrak{f}_{ab}K^{b}~{}.$
(3.4)
It should be noted that the translation generators $P_{a}$ do not appear in
the above expression. Instead, we assume that they are replaced by the
covariant derivatives $\nabla_{a}$ and obey the graded commutation relations
(c.f. (3.2b))
$[X_{\underline{a}},\nabla_{b}\\}=-f_{\underline{a}b}{}^{\underline{c}}X_{\underline{c}}-f_{\underline{a}b}{}^{c}\nabla_{c}~{}.$
(3.5)
By definition, the gauge group of conformal gravity is generated by local
transformations of the form
$\displaystyle\delta_{\mathscr{K}}\nabla_{a}$ $\displaystyle=$
$\displaystyle[\mathscr{K},\nabla_{a}]\ ,$ (3.6a) $\displaystyle\mathscr{K}$
$\displaystyle=$
$\displaystyle\xi^{b}\nabla_{b}+\Lambda^{\underline{b}}X_{\underline{b}}=\xi^{b}\nabla_{b}+KM+\sigma\mathbb{D}+\Lambda_{b}K^{b}~{},$
(3.6b)
where the gauge parameters satisfy natural reality conditions. These gauge
transformations act on a conformal tensor field $\mathcal{U}$ (with its
indices suppressed) as
$\displaystyle\delta_{\mathscr{K}}\mathcal{U}={\mathscr{K}}\mathcal{U}~{}.$
(3.7)
Further, we will say that $\mathcal{U}$ is primary if (i) it is annihilated by
the special conformal generators, $K^{a}\mathcal{U}=0$; and (ii) it is an
eigenvector of $\mathbb{D}$. It will be said to have dimension $\Delta$ and
Lorentz weight $\lambda$ if
$\displaystyle\mathbb{D}{\cal U}=\Delta{\cal U}~{},\qquad
M\mathcal{U}=\lambda\mathcal{U}~{}.$ (3.8)
The covariant derivatives (3.4) obey the commutation relations
$\displaystyle\big{[}\nabla_{++},\nabla_{--}\big{]}$
$\displaystyle=-\mathcal{T}^{a}\nabla_{a}-\mathcal{R}(X)^{\underline{a}}X_{\underline{a}}=-{\cal
T}^{++}\nabla_{++}-{\cal
T}^{--}\nabla_{--}-\mathcal{R}(X)^{\underline{a}}X_{\underline{a}}~{},$ (3.9)
where the torsion and curvatures take the form
$\displaystyle\mathcal{T}^{++}$
$\displaystyle=-\mathscr{C}^{++}+\omega^{++}+b^{++}~{},$ (3.10a)
$\displaystyle\mathcal{T}^{--}$
$\displaystyle=-\mathscr{C}^{--}+\omega^{--}-b^{--}~{},$ (3.10b)
$\displaystyle\mathcal{R}(M)$
$\displaystyle=-\frac{1}{2}\mathcal{R}-2(\mathfrak{f}_{++,--}+\mathfrak{f}_{--,++})~{},$
(3.10c) $\displaystyle\mathcal{R}(\mathbb{D})$
$\displaystyle=-\mathscr{C}^{a}b_{a}+e_{++}b_{--}-e_{--}b_{++}+2(\mathfrak{f}_{++,--}-\mathfrak{f}_{--,++})~{},$
(3.10d) $\displaystyle\mathcal{R}(K)_{++}$
$\displaystyle=-\mathscr{C}^{a}\mathfrak{f}_{a,++}+e_{++}\mathfrak{f}_{--,++}-e_{--}f_{++,++}-\omega_{++}f_{--,++}$
$\displaystyle\phantom{=}~{}-b_{++}f_{--,++}+\omega_{--}\mathfrak{f}_{++,++}+b_{--}\mathfrak{f}_{++,++}~{},$
(3.10e) $\displaystyle\mathcal{R}(K)_{--}$
$\displaystyle=-\mathscr{C}^{a}\mathfrak{f}_{a,--}+e_{++}\mathfrak{f}_{--,--}-e_{--}f_{++,--}$
$\displaystyle\phantom{=}~{}-b_{++}f_{--,--}-\omega_{--}\mathfrak{f}_{++,--}+b_{--}\mathfrak{f}_{++,--}+\omega_{++}f_{--,--}~{},$
(3.10f) $\displaystyle\mathcal{R}$
$\displaystyle=2\mathscr{C}^{a}\omega_{a}-2e_{++}\omega_{--}+e_{--}\omega_{++}~{}.$
(3.10g)
Here ${\mathcal{R}}$ is the scalar curvature constructed from the Lorentz
connection $\omega_{a}$ and we have introduced the anholonomy coefficients
$\mathscr{C}^{a}$
$\displaystyle[e_{++},e_{--}]=\mathscr{C}^{a}e_{a}=\mathscr{C}^{++}e_{++}+\mathscr{C}^{--}e_{--}~{}.$
(3.11)
In order for this geometry to describe conformal gravity, it is necessary to
impose certain covariant constraints. Specifically, we require that the
torsion and both Lorentz and dilatation curvatures vanish
$\displaystyle{\cal T}^{a}=0~{},\qquad{\cal R}(M)=0~{},\qquad{\cal
R}(\mathbb{D})=0~{}.$ (3.12)
The constraint ${\cal T}^{a}=0$ determines the Lorentz connection in terms of
the vielbein and dilatation connection
$\displaystyle\omega_{\pm\pm}=\mathscr{C}_{\pm\pm}\pm b_{\pm\pm}~{},$ (3.13)
while ${\cal R}(M)={\cal R}(\mathbb{D})=0$ fixes several components of the
special conformal connection
$\displaystyle\mathfrak{f}_{++,--}$
$\displaystyle=-\frac{1}{8}\big{(}\mathcal{R}-2\mathscr{C}^{a}b_{a}+2e_{++}b_{--}-2e_{--}b_{++}\big{)}~{},$
(3.14a) $\displaystyle\mathfrak{f}_{--,++}$
$\displaystyle=-\frac{1}{8}\big{(}\mathcal{R}+2\mathscr{C}^{a}b_{a}-2e_{++}b_{--}+2e_{--}b_{++}\big{)}~{}.$
(3.14b)
As a result, the algebra of conformal covariant derivatives takes the
form333The ${\cal N}=(1,0)$ superconformal extension of this geometry is
described in appendix B.
$\displaystyle\big{[}\nabla_{++},\nabla_{--}\big{]}$
$\displaystyle=W_{++}K^{++}+W_{--}K^{--}~{},\qquad
W_{a}\equiv-\mathcal{R}(K)_{a}~{},$ (3.15)
where $W_{++}$ and $W_{--}$ have the following conformal properties
$\displaystyle K^{a}W_{++}=0~{},\qquad\mathbb{D}W_{++}=3W_{++}~{},$ (3.16a)
$\displaystyle K^{a}W_{--}=0~{},\qquad\mathbb{D}W_{--}=3W_{--}~{}.$ (3.16b)
Strictly speaking, it is necessary to impose the constraint $W_{a}=0$ for this
geometry to describe conformal gravity.444This is in contrast to the situation
in $d>3$ dimensions, see appendix A. Specifically, in the absence of this
constraint, there are extra degrees of freedom, in addition to the vielbein,
which correspond to the special conformal connections $\mathfrak{f}_{++,++}$
and $\mathfrak{f}_{--,--}$. This will be addressed in further detail below.
### 3.2 Degauging to Lorentzian geometry
According to (3.6), under an infinitesimal special superconformal gauge
transformation $\mathscr{K}=\Lambda_{b}K^{b}$, the dilatation connection
transforms algebraically
$\displaystyle\delta_{\mathscr{K}}b_{a}=-2\Lambda_{a}~{}.$ (3.17)
As a result, we may enforce the gauge $b_{a}=0$, which completely fixes the
freedom to perform special superconformal transformations with unconstrained
$\Lambda_{b}$. Hence, the connection $\mathfrak{f}_{ab}$ is not required for
the covariance of $\nabla_{a}$ and it may be separated
$\displaystyle\nabla_{a}$ $\displaystyle=$ $\displaystyle{\cal
D}_{a}-\mathfrak{f}_{ab}K^{b}~{},$ (3.18)
where the operator ${\cal D}_{a}$ takes the form
$\displaystyle{\cal D}_{a}=e_{a}-\omega_{a}M~{}.$ (3.19)
An important feature of this gauge, which follows from (3.14), is
$\displaystyle\mathfrak{f}_{++,--}=\mathfrak{f}_{--,++}=-\frac{1}{8}\mathcal{R}~{},$
(3.20)
which, keeping in mind the relation
$\displaystyle[{\cal D}_{++},{\cal D}_{--}]$ $\displaystyle=$
$\displaystyle[\nabla_{++},\nabla_{--}]+\big{(}{\cal
D}_{++}\mathfrak{f}_{--,a}-{\cal D}_{--}\mathfrak{f}_{++,a}\big{)}K^{a}$
(3.21)
$\displaystyle+\mathfrak{f}_{++,a}[K^{a},\nabla_{--}]-\mathfrak{f}_{--,a}[K^{a},\nabla_{++}]~{},$
allows one to determine $[{\cal D}_{++},{\cal D}_{--}]$ by a routine
computation. One finds
$\displaystyle[{\cal D}_{++},{\cal D}_{--}]=\frac{1}{2}{\cal R}M~{}.$ (3.22)
Additionally, by analysing the special conformal sector of (3.21), we obtain
the relations
$\displaystyle W_{++}=-\frac{1}{8}{\cal D}_{++}{\cal R}-{\cal
D}_{--}\mathfrak{f}_{++,++}~{},\qquad W_{--}=\frac{1}{8}{\cal D}_{--}{\cal
R}+{\cal D}_{++}\mathfrak{f}_{--,--}~{}.$ (3.23)
In particular, for vanishing $W_{++}$ ($W_{--}$), we see that
$\mathfrak{f}_{++,++}$ ($\mathfrak{f}_{--,--}$) is a non-local function of the
vielbein, which is in contrast to the situation in $d>2$ spacetime dimensions
(see appendix A for a review).
If no constraint is imposed on the conformal curvature tensors $W_{++}$ and
$W_{--}$, then the components $\mathfrak{f}_{++,++}$ and
$\mathfrak{f}_{--,--}$ of the special conformal connection remain independent
fields in addition to the vielbein. Therefore we are forced to impose the
constraints
$\displaystyle W_{++}=0~{},\qquad W_{--}=0~{},$ (3.24)
in order for the vielbein to be the only independent field. Then it follows
from (3.23) that $\mathfrak{f}_{++,++}$ and $\mathfrak{f}_{--,--}$ become non-
local functions of the gravitational field.
Next, it is important to describe the gauge freedom of this geometry, which
corresponds to the residual gauge transformations of (3.6) in the gauge
$b_{a}=0$. These include local $\mathcal{K}$-transformations of the form
$\displaystyle\delta_{\mathcal{K}}{\cal D}_{A}=[\mathcal{K},{\cal
D}_{A}]~{},\qquad\mathcal{K}=\xi^{b}{\cal D}_{b}+KM~{},$ (3.25a) which act on
tensor fields $\mathcal{U}$ (with indices suppressed) as
$\displaystyle\delta_{\cal K}{\cal U}={\cal K}{\cal U}~{}.$ (3.25b)
The gauge transformations (3.25) are not the most general conformal gravity
gauge transformations preserving the gauge $b_{a}=0$. Specifically, it may be
shown that the following transformation also enjoys this property
$\displaystyle\mathscr{K}(\sigma)=\sigma\mathbb{D}+\frac{1}{2}\nabla_{b}\sigma
K^{b}\quad\Longrightarrow\quad\delta_{\mathscr{K}(\sigma)}b_{a}=0~{},$ (3.26)
where $\sigma$ is real but otherwise unconstrained. As a result, it is
necessary to consider how this transformation manifests in the degauged
geometry
$\displaystyle\delta_{\mathscr{K}(\sigma)}\nabla_{A}\equiv\delta_{\sigma}\nabla_{a}=\delta_{\sigma}{\cal
D}_{a}-\delta_{\sigma}(\mathfrak{f}_{ab}K^{b})~{}.$ (3.27)
Employing this relation, we arrive at the transformation laws
$\displaystyle\delta_{\sigma}{\cal D}_{++}$ $\displaystyle=\sigma{\cal
D}_{++}+{\cal D}_{++}\sigma M~{},$ (3.28a) $\displaystyle\delta_{\sigma}{\cal
D}_{--}$ $\displaystyle=\sigma{\cal D}_{--}-{\cal D}_{--}\sigma M~{},$ (3.28b)
$\displaystyle\delta_{\sigma}{\cal R}$ $\displaystyle=2\sigma{\cal R}-4{\cal
D}_{++}{\cal D}_{--}\sigma~{},$ (3.28c)
which are exactly the Weyl transformations of spacetime.
## 4 Conformal $(p,q)$ superspace
In section 2, we have realised the superconformal algebra
$\mathfrak{osp}(p|2;\mathbb{R})\oplus\mathfrak{osp}(q|2;\mathbb{R})$ as the
maximal finite-dimensional subalgebra of the $(p,q)$ super Virasoro algebra.
Now we turn to formulating the corresponding gauge theory. This is known as
conformal superspace and is identified with a pair $({\cal
M}^{(2|p,q)},\nabla)$, where $\mathcal{M}^{(2|p,q)}$ denotes a supermanifold
parametrised by local coordinates $z^{M}$, and $\nabla$ is a covariant
derivative associated with the superconformal algebra. The latter is generated
by the operators $X_{\tilde{A}}$, eq. (2.20), which can be grouped into the
two disjoint subsets $P_{A}$ and $X_{\underline{A}}$,
$\displaystyle X_{\tilde{A}}=(P_{A},X_{\underline{A}})~{},\qquad
X_{\underline{A}}=(M,{\mathbb{D}},\mathfrak{L}^{{\overline{I}}{\overline{J}}},\mathfrak{R}^{{\underline{I}}{\underline{J}}},K^{A})~{},$
(4.1)
each of which constitutes a superalgebra:
$\displaystyle[P_{{A}},P_{{B}}\\}$
$\displaystyle=-f_{{{A}}{{B}}}{}^{{{C}}}P_{{C}}\ ,$ (4.2a)
$\displaystyle[X_{\underline{A}},X_{\underline{B}}\\}$
$\displaystyle=-f_{\underline{A}\underline{B}}{}^{\underline{C}}X_{\underline{C}}\
,$ (4.2b) $\displaystyle[X_{\underline{A}},P_{{B}}\\}$
$\displaystyle=-f_{\underline{A}{{B}}}{}^{\underline{C}}X_{\underline{C}}-f_{\underline{A}{{B}}}{}^{{C}}P_{{C}}\
.$ (4.2c)
The structure constants above may be determined by comparing with equations
(2.22).
In order to define covariant derivatives, it is necessary to associate with
each non-translational generator $X_{\underline{A}}$ a connection one-form
$\Omega^{\underline{A}}=(\Omega,B,\Phi^{{\overline{I}}{\overline{J}}},\Phi^{{\underline{I}}{\underline{J}}},\mathfrak{F}_{A})={\rm
d}z^{M}\Omega_{M}{}^{\underline{A}}$, and with $P_{{A}}$ a supervielbein one-
form $E^{{A}}=(E^{a},E^{+{\overline{I}}},E^{-{\underline{I}}})={\rm
d}z^{{M}}E_{M}{}^{A}$. It is assumed that the supermatrix $E_{M}{}^{A}$ is
nonsingular, $E:={\rm Ber}(E_{M}{}^{A})\neq 0$, hence there exists a unique
inverse supervielbein $E_{A}{}^{M}$
$\displaystyle E_{A}{}^{M}E_{M}{}^{B}=\delta_{A}{}^{B}~{},\qquad
E_{M}{}^{A}E_{A}{}^{N}=\delta_{M}{}^{N}~{}.$ (4.3)
The latter may be used to construct the supervector fields
$E_{A}=E_{A}{}^{M}\partial_{M}$, with $\partial_{M}=\partial/\partial z^{M}$,
which constitute a basis for the tangent space at each point of
$\mathcal{M}^{(2|p,q)}$. The connection may then be expressed in the
supervielbein basis as
$\Omega^{\underline{A}}=E^{B}\Omega_{B}{}^{\underline{A}}$, where
$\Omega_{B}{}^{\underline{A}}=E_{B}{}^{M}\Omega_{M}{}^{\underline{A}}$. The
covariant derivatives
$\nabla_{A}=(\nabla_{a},\nabla_{+}^{{\overline{I}}},\nabla_{-}^{{\underline{I}}})$
then take the form
$\displaystyle\nabla_{A}$ $\displaystyle=$ $\displaystyle
E_{A}-\Omega_{A}{}^{\underline{B}}X_{\underline{B}}=E_{A}-\Omega_{A}M-B_{A}\mathbb{D}-\frac{1}{2}\Phi_{A}^{{\overline{I}}{\overline{J}}}\mathfrak{L}^{{\overline{I}}{\overline{J}}}-\frac{1}{2}\Phi_{A}^{{\underline{I}}{\underline{J}}}\mathfrak{R}^{{\underline{I}}{\underline{J}}}-\mathfrak{F}_{AB}K^{B}~{}.$
(4.4)
It should be noted that the translation generators $P_{A}$ do not appear in
the above expression. Instead, we assume that they are replaced by the
covariant derivatives $\nabla_{A}$ and obey the graded commutation relations
$[X_{\underline{B}},\nabla_{A}\\}=-f_{\underline{B}A}{}^{C}\nabla_{C}-f_{\underline{B}A}{}^{\underline{C}}X_{\underline{C}}~{},$
(4.5)
where the relevant structure constants were defined in equation (4.2c).
By definition, the gauge group of conformal supergravity is generated by local
transformations of the form
$\displaystyle\delta_{\mathscr{K}}\nabla_{A}$ $\displaystyle=$
$\displaystyle[\mathscr{K},\nabla_{A}]\ ,$ (4.6a) $\displaystyle\mathscr{K}$
$\displaystyle=$
$\displaystyle\xi^{B}\nabla_{B}+\Lambda^{\underline{B}}X_{\underline{B}}=\xi^{B}\nabla_{B}+KM+\sigma\mathbb{D}+\frac{1}{2}\rho^{{\overline{I}}{\overline{J}}}\mathfrak{L}^{{\overline{I}}{\overline{J}}}+\frac{1}{2}\rho^{{\underline{I}}{\underline{J}}}\mathfrak{R}^{{\underline{I}}{\underline{J}}}+\Lambda_{B}K^{B}~{},~{}~{}~{}~{}$
(4.6b)
where the gauge parameters satisfy natural reality conditions. These gauge
transformations act on a conformal tensor superfield $\mathcal{U}$ (with its
indices suppressed) as
$\displaystyle\delta_{\mathscr{K}}\mathcal{U}={\mathscr{K}}\mathcal{U}~{}.$
(4.7)
Further, we will say that $\mathcal{U}$ is primary if (i) it is annihilated by
the special conformal generators, $K^{A}\mathcal{U}=0$; and (ii) it is an
eigenvector of $\mathbb{D}$. The superfield is said to have dimension $\Delta$
and Lorentz weight $\lambda$ if $\mathbb{D}{\cal U}=\Delta{\cal U}$ and
$M\mathcal{U}=\lambda\mathcal{U}$.
The covariant derivatives (4.4) obey the graded commutation relations
$\displaystyle\big{[}\nabla_{A},\nabla_{B}\big{\\}}=-\mathcal{T}_{AB}{}^{C}\nabla_{C}-\mathcal{R}(X)_{AB}{}^{\underline{C}}X_{\underline{C}}~{}.$
(4.8)
In conformal superspace, we impose the requirement that torsion
$\mathcal{T}_{AB}{}^{C}$ and curvature tensors
$\mathcal{R}(X)_{AB}{}^{\underline{C}}$ differ from their flat counterparts
only by terms proportional to the conformal curvatures of ${\cal
M}^{(2|p,q)}$. Here we will assume that all such superfields vanish555See
appendix B for a construction of conformal $(1,0)$ superspace with non-
vanishing curvature. and thus the only non-vanishing graded commutators are:
$\displaystyle\\{\nabla_{+}^{{\overline{I}}},\nabla_{+}^{\overline{J}}\\}=2{\rm
i}\delta^{{\overline{I}}\overline{J}}\nabla_{++}~{},\qquad\\{\nabla_{-}^{\underline{I}},\nabla_{-}^{\underline{J}}\\}=2{\rm
i}\delta^{\underline{I}\underline{J}}\nabla_{--}~{}.$ (4.9)
## 5 The superspace geometry of $(p,q)$ supergravity
According to (4.6), under an infinitesimal special superconformal gauge
transformation $\mathscr{K}=\Lambda_{B}K^{B}$, the dilatation connection
transforms algebraically
$\displaystyle\delta_{\mathscr{K}}B_{A}=-2\Lambda_{A}~{}.$ (5.1)
Hence, we may enforce the gauge $B_{A}=0$, which completely fixes the freedom
to perform special superconformal transformations with unconstrained
$\Lambda_{B}$. As a result, the corresponding connection $\mathfrak{F}_{AB}$
is not required for the covariance of $\nabla_{A}$, and it may be separated
$\displaystyle\nabla_{A}$ $\displaystyle=$ $\displaystyle{\cal
D}_{A}-\mathfrak{F}_{AB}K^{B}~{}.$ (5.2)
Here the degauged covariant derivative ${\cal D}_{A}$ involves only the
Lorentz and $R$-symmetry connections (depending on the choice of $p$ and $q$).
Additionally, the special superconformal connection $\mathfrak{F}_{AB}$ may be
related to the torsion and curvatures of the degauged geometry by analysing
the relation
$\displaystyle[{\cal D}_{A},{\cal D}_{B}\\}$ $\displaystyle=$
$\displaystyle[\nabla_{A},\nabla_{B}\\}+\big{(}{\cal
D}_{A}\mathfrak{F}_{BC}-(-1)^{AB}{\cal
D}_{B}\mathfrak{F}_{AC}\big{)}K^{C}+\mathfrak{F}_{AC}[K^{C},\nabla_{B}\\}$
(5.3)
$\displaystyle-(-1)^{AB}\mathfrak{F}_{BC}[K^{C},\nabla_{A}\\}-(-1)^{BC}\mathfrak{F}_{AC}\mathfrak{F}_{BD}[K^{D},K^{C}\\}~{}.$
We will refer to the superspace geometry described by the covariant
derivatives ${\cal D}_{A}$ as curved $\mathsf{SO}(p)\times\mathsf{SO}(q)$
superspace.
### 5.1 $p,q>1$ case
First, we consider the case where $p,q>1$. By a routine calculation, one finds
that the degauged connection $\mathfrak{F}_{AB}$ takes the form
$\displaystyle\mathfrak{F}_{+,-}^{{\overline{I}}\phantom{,,}{\underline{J}}}$
$\displaystyle=$
$\displaystyle-\mathfrak{F}_{-,+}^{{\underline{J}}\phantom{,,}{\overline{I}}}=S^{{\overline{I}}{\underline{J}}}~{},\quad\mathfrak{F}_{+,+}^{{\overline{I}}\phantom{,,}{\overline{J}}}=-\mathfrak{F}_{+,+}^{{\overline{J}}\phantom{,,}{\overline{I}}}=X_{++}^{{\overline{I}}{\overline{J}}}~{},\quad\mathfrak{F}_{-,-}^{{\underline{I}}\phantom{,,}{\underline{J}}}=-\mathfrak{F}_{-,-}^{{\underline{J}}\phantom{,,}{\underline{I}}}=X_{--}^{{\underline{I}}{\underline{J}}}~{},\quad$
(5.4a) $\displaystyle\mathfrak{F}_{+,--}^{{\overline{I}}}$ $\displaystyle=$
$\displaystyle\mathfrak{F}_{--,+}^{~{}~{}~{}~{}\phantom{,}{\overline{I}}}=\frac{{\rm
i}}{q}{\cal
D}_{-}^{{\underline{J}}}S^{{\overline{I}}{\underline{J}}}~{},\qquad\qquad\;\;\,\,\mathfrak{F}_{-,++}^{{\underline{I}}}=\mathfrak{F}_{++,-}^{~{}~{}~{}~{}\phantom{,}{\underline{I}}}=-\frac{{\rm
i}}{p}{\cal D}_{+}^{{\overline{J}}}S^{{\overline{J}}{\underline{I}}}~{},$
(5.4b) $\displaystyle\mathfrak{F}_{+,++}^{{\overline{I}}}$ $\displaystyle=$
$\displaystyle\mathfrak{F}_{++,+}^{~{}~{}~{}~{}\phantom{,}{\overline{I}}}=-\frac{{\rm
i}}{p-1}{\cal
D}_{+}^{{\overline{J}}}X_{++}^{{\overline{J}}{\overline{I}}}~{},\qquad\mathfrak{F}_{-,--}^{{\underline{I}}}=\mathfrak{F}_{--,-}^{~{}~{}~{}~{}\phantom{,}{\underline{I}}}=-\frac{{\rm
i}}{q-1}{\cal
D}_{-}^{{\underline{J}}}X_{--}^{{\underline{J}}{\underline{I}}}~{},$ (5.4f)
$\displaystyle\mathfrak{F}_{++,--}=\mathfrak{F}_{--,++}=\frac{1}{2pq}\big{[}{\cal
D}_{+}^{\overline{I}},{\cal
D}_{-}^{\underline{J}}\big{]}S^{{\overline{I}}{\underline{J}}}-\frac{p+q}{pq}S^{{\overline{I}}{\underline{J}}}S^{{\overline{I}}{\underline{J}}}~{},$
$\displaystyle\qquad\mathfrak{F}_{++,++}=\frac{1}{p(p-1)}{\cal
D}_{+}^{\overline{I}}{\cal
D}_{+}^{\overline{J}}X_{++}^{{\overline{I}}{\overline{J}}}-\frac{2}{p}X_{++}^{{\overline{I}}{\overline{J}}}X_{++}^{{\overline{I}}{\overline{J}}}~{},$
$\displaystyle\qquad\mathfrak{F}_{--,--}=\frac{1}{q(q-1)}{\cal
D}_{-}^{\underline{I}}{\cal
D}_{-}^{\underline{J}}X_{--}^{{\underline{I}}{\underline{J}}}-\frac{2}{q}X_{--}^{{\underline{I}}{\underline{J}}}X_{--}^{{\underline{I}}{\underline{J}}}~{},$
where we have introduced the imaginary dimension-$1$ torsion tensors
$S^{{\overline{I}}{\underline{J}}}$, $X_{++}^{{\overline{I}}{\overline{J}}}$
and $X_{--}^{{\underline{I}}{\underline{J}}}$. In contrast to conformal
gravity, all components of ${\mathfrak{F}}_{AB}$ are determined in terms of
the supergravity multiplet.
The torsion tensors obey the Bianchi identities
$\displaystyle{\cal D}_{+}^{{\overline{I}}}S^{{\overline{J}}{\underline{K}}}$
$\displaystyle=\frac{1}{p}\delta^{{\overline{I}}{\overline{J}}}{\cal
D}_{+}^{{\overline{L}}}S^{{\overline{L}}{\underline{K}}}+{\cal
D}_{-}^{\overline{K}}X_{++}^{{\overline{I}}{\overline{J}}}~{},\quad{\cal
D}_{-}^{{\underline{I}}}S^{{\overline{J}}{\underline{K}}}=\frac{1}{q}\delta^{{\underline{I}}{\underline{K}}}{\cal
D}_{-}^{{\underline{L}}}S^{{\overline{J}}{\underline{L}}}-{\cal
D}_{+}^{\overline{J}}X_{--}^{{\underline{I}}{\underline{K}}}~{},$ (5.5a)
$\displaystyle{\cal
D}_{+}^{{\overline{I}}}X_{++}^{{\overline{J}}{\overline{K}}}$
$\displaystyle=\frac{2}{p-1}\delta^{{\overline{I}}[{\overline{J}}}{\cal
D}_{+}^{|{\overline{L}}}X_{++}^{{\overline{L}}|{\overline{K}}]}~{},\qquad\quad{\cal
D}_{-}^{{\underline{I}}}X_{--}^{{\underline{J}}{\underline{K}}}=\frac{2}{q-1}\delta^{{\underline{I}}[{\underline{J}}}{\cal
D}_{-}^{|{\underline{L}}}X_{--}^{{\underline{L}}|{\underline{K}}]}~{}.$ (5.5b)
Additionally, it may be shown that the algebra obeyed by ${\cal D}_{A}$ takes
the form
$\displaystyle\\{{\cal D}_{+}^{{\overline{I}}},{\cal
D}_{+}^{{\overline{J}}}\\}$ $\displaystyle=$ $\displaystyle 2{\rm
i}\delta^{{\overline{I}}{\overline{J}}}{\cal
D}_{++}-4X_{++}^{{\overline{K}}({\overline{I}}}\mathfrak{L}^{{\overline{J}}){\overline{K}}}~{},$
(5.6a) $\displaystyle\\{{\cal D}_{+}^{{\overline{I}}},{\cal
D}_{-}^{{\underline{J}}}\\}$ $\displaystyle=$
$\displaystyle-4S^{{\overline{I}}{\underline{J}}}M+2S^{{\overline{K}}{\underline{J}}}\mathfrak{L}^{{\overline{K}}{\overline{I}}}-2S^{{\overline{I}}{\underline{K}}}\mathfrak{R}^{{\underline{K}}{\underline{J}}}~{},$
(5.6b) $\displaystyle\\{{\cal D}_{-}^{{\underline{I}}},{\cal
D}_{-}^{{\underline{J}}}\\}$ $\displaystyle=$ $\displaystyle 2{\rm
i}\delta^{{\underline{I}}{\underline{J}}}{\cal
D}_{--}-4X_{--}^{{\underline{K}}({\underline{I}}}\mathfrak{R}^{{\underline{J}}){\underline{K}}}~{},$
(5.6c) $\displaystyle\big{[}{\cal D}_{+}^{{\overline{I}}},{\cal
D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle-2{\rm
i}S^{{\overline{I}}{\underline{J}}}{\cal D}_{-}^{{\underline{J}}}-\frac{4{\rm
i}}{q}{\cal
D}_{-}^{{\underline{J}}}S^{{\overline{I}}{\underline{J}}}M+\frac{2{\rm
i}}{q}{\cal
D}_{-}^{{\underline{K}}}S^{{\overline{J}}{\underline{K}}}\mathfrak{L}^{{\overline{J}}{\overline{I}}}~{},$
(5.6d) $\displaystyle\big{[}{\cal D}_{-}^{{\underline{I}}},{\cal
D}_{++}\big{]}$ $\displaystyle=$ $\displaystyle 2{\rm
i}S^{{\overline{J}}{\underline{I}}}{\cal D}_{+}^{{\overline{J}}}-\frac{4{\rm
i}}{p}{\cal
D}_{+}^{{\overline{J}}}S^{{\overline{J}}{\underline{I}}}M-\frac{2{\rm
i}}{p}{\cal
D}_{+}^{{\overline{K}}}S^{{\overline{K}}{\underline{J}}}\mathfrak{R}^{{\underline{J}}{\underline{I}}}~{},$
(5.6e) $\displaystyle\big{[}{\cal D}_{+}^{{\overline{I}}},{\cal
D}_{++}\big{]}$ $\displaystyle=$ $\displaystyle-2{\rm
i}X_{++}^{{\overline{I}}{\overline{J}}}{\cal
D}_{+}^{{\overline{J}}}-\frac{2{\rm i}}{p-1}{\cal
D}_{+}^{\overline{J}}X_{++}^{{\overline{J}}{\overline{K}}}\mathfrak{L}^{{\overline{K}}{\overline{I}}}~{},$
(5.6f) $\displaystyle\big{[}{\cal D}_{-}^{{\underline{I}}},{\cal
D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle-2{\rm
i}X_{--}^{{\underline{I}}{\underline{J}}}{\cal
D}_{-}^{{\underline{J}}}-\frac{2{\rm i}}{q-1}{\cal
D}_{-}^{\underline{J}}X_{--}^{{\underline{J}}{\underline{K}}}\mathfrak{R}^{{\underline{K}}{\underline{I}}}~{},$
(5.6g) $\displaystyle\big{[}{\cal D}_{++},{\cal D}_{--}\big{]}$
$\displaystyle=$ $\displaystyle-\frac{2}{p}{\cal
D}_{+}^{{\overline{J}}}S^{{\overline{J}}{\underline{I}}}{\cal
D}_{-}^{{\underline{I}}}-\frac{2}{q}{\cal
D}_{-}^{{\underline{J}}}S^{{\overline{I}}{\underline{J}}}{\cal
D}_{+}^{{\overline{I}}}$ (5.6h) $\displaystyle-\frac{2}{pq}\big{(}\big{[}{\cal
D}_{+}^{{\overline{I}}},{\cal
D}_{-}^{{\underline{J}}}\big{]}-2(p+q)S^{{\overline{I}}{\underline{J}}}\big{)}S^{{\overline{I}}{\underline{J}}}M~{}.$
Next, it is important to describe the supergravity gauge freedom of this
geometry, which corresponds to the residual gauge transformations of (4.6) in
the gauge $B_{A}=0$. These include local $\mathcal{K}$-transformations of the
form
$\displaystyle\delta_{\mathcal{K}}{\cal D}_{A}$ $\displaystyle=$
$\displaystyle[\mathcal{K},{\cal D}_{A}]\ ,$ (5.7a) $\displaystyle\mathcal{K}$
$\displaystyle=$ $\displaystyle\xi^{B}{\cal
D}_{B}+KM+\frac{1}{2}\rho^{{\overline{I}}{\overline{J}}}\mathfrak{L}^{{\overline{I}}{\overline{J}}}+\frac{1}{2}\rho^{{\underline{I}}{\underline{J}}}\mathfrak{R}^{{\underline{I}}{\underline{J}}}~{},$
(5.7b) which act on tensor superfields $\mathcal{U}$ (with indices suppressed)
as $\displaystyle\delta_{\cal K}{\cal U}={\cal K}{\cal U}~{}.$ (5.7c)
The gauge transformations (5.7) prove to not be the most general conformal
supergravity gauge transformations preserving the gauge $B_{A}=0$.
Specifically, it may be shown that the following transformation also enjoys
this property
$\displaystyle\mathscr{K}(\sigma)=\sigma\mathbb{D}+\frac{1}{2}\nabla_{B}\sigma
K^{B}\quad\Longrightarrow\quad\delta_{\mathscr{K}(\sigma)}B_{A}=0~{},$ (5.8)
where $\sigma$ is real but otherwise unconstrained.
As a result, it is necessary to consider how this transformation manifests in
the degauged geometry
$\displaystyle\delta_{\mathscr{K}(\sigma)}\nabla_{A}\equiv\delta_{\sigma}\nabla_{A}=\delta_{\sigma}{\cal
D}_{A}-\delta_{\sigma}(\mathfrak{F}_{AB}K^{B})~{}.$ (5.9)
Employing this relation, we arrive at the transformation laws for ${\cal
D}_{A}$
$\displaystyle\delta_{\sigma}{\cal D}_{+}^{\overline{I}}$
$\displaystyle=\frac{1}{2}\sigma{\cal D}_{+}^{\overline{I}}+{\cal
D}_{+}^{\overline{I}}\sigma M-{\cal
D}_{+}^{\overline{J}}\sigma\mathfrak{L}^{{\overline{J}}{\overline{I}}}~{},$
(5.10a) $\displaystyle\delta_{\sigma}{\cal D}_{-}^{\underline{I}}$
$\displaystyle=\frac{1}{2}\sigma{\cal D}_{-}^{\underline{I}}-{\cal
D}_{-}^{\underline{I}}\sigma M-{\cal
D}_{-}^{\underline{J}}\sigma\mathfrak{R}^{{\underline{J}}{\underline{I}}}~{},$
(5.10b) $\displaystyle\delta_{\sigma}{\cal D}_{++}$ $\displaystyle=\sigma{\cal
D}_{++}-{\rm i}{\cal D}_{+}^{\overline{I}}\sigma{\cal
D}_{+}^{\overline{I}}+{\cal D}_{++}\sigma M~{},$ (5.10c)
$\displaystyle\delta_{\sigma}{\cal D}_{--}$ $\displaystyle=\sigma{\cal
D}_{--}-{\rm i}{\cal D}_{-}^{\underline{I}}\sigma{\cal
D}_{-}^{\underline{I}}-{\cal D}_{--}\sigma M~{},$ (5.10d) and, by making use
of (5.4), it may be shown that the torsions transform as follows
$\displaystyle\delta_{\sigma}S^{{\overline{I}}{\underline{J}}}$
$\displaystyle=\sigma S^{{\overline{I}}{\underline{J}}}+\frac{1}{2}{\cal
D}_{+}^{\overline{I}}{\cal D}_{-}^{\underline{J}}\sigma~{},$ (5.10e)
$\displaystyle\delta_{\sigma}X_{++}^{{\overline{I}}{\overline{J}}}$
$\displaystyle=\sigma
X_{++}^{{\overline{I}}{\overline{J}}}+\frac{1}{4}\big{[}{\cal
D}_{+}^{\overline{I}},{\cal D}_{+}^{\overline{J}}\big{]}\sigma~{},$ (5.10f)
$\displaystyle\delta_{\sigma}X_{--}^{{\underline{I}}{\underline{J}}}$
$\displaystyle=\sigma
X_{--}^{{\underline{I}}{\underline{J}}}+\frac{1}{4}\big{[}{\cal
D}_{-}^{\underline{I}},{\cal D}_{-}^{\underline{J}}\big{]}\sigma~{}.$ (5.10g)
These are the super-Weyl transformations of the degauged geometry.
It should be mentioned that, for the special case ${\cal N}=(2,2)$, equivalent
superspace geometry was formulated in the works [29, 30, 31, 32].666This
follows from the fact that the superconformal groups
$\mathsf{OSp}_{0}(2|2;{\mathbb{R}})\times\mathsf{OSp}_{0}(2|2;{\mathbb{R}})$
and $\mathsf{SU}(1,1|1)\times\mathsf{SU}(1,1|1)$ are isomorphic. To see this,
we first eliminate the superfields $X_{++}^{\overline{I}\overline{J}}$ and
$X_{--}^{\underline{I}\underline{J}}$ in (5.6) by redefining the vector
covariant derivatives
$\displaystyle\hat{{\cal D}}_{++}={\cal D}_{++}-{\rm
i}X_{++}^{\overline{I}\overline{J}}\mathfrak{L}^{\overline{I}\overline{J}}~{},\qquad\hat{{\cal
D}}_{--}={\cal D}_{--}-{\rm
i}X_{--}^{\underline{I}\underline{J}}\mathfrak{R}^{\underline{I}\underline{J}}~{}.$
(5.11)
The resulting algebra is as follows
$\displaystyle\\{{\cal D}_{+}^{{\overline{I}}},{\cal
D}_{+}^{{\overline{J}}}\\}$ $\displaystyle=$ $\displaystyle 2{\rm
i}\delta^{{\overline{I}}{\overline{J}}}\hat{{\cal D}}_{++}~{},\qquad\\{{\cal
D}_{-}^{{\underline{I}}},{\cal D}_{-}^{{\underline{J}}}\\}=2{\rm
i}\delta^{{\underline{I}}{\underline{J}}}\hat{{\cal D}}_{--}~{},$ (5.12a)
$\displaystyle\\{{\cal D}_{+}^{{\overline{I}}},{\cal
D}_{-}^{{\underline{J}}}\\}$ $\displaystyle=$
$\displaystyle-4S^{{\overline{I}}{\underline{J}}}M+2S^{{\overline{K}}{\underline{J}}}\mathfrak{L}^{{\overline{K}}{\overline{I}}}-2S^{{\overline{I}}{\underline{K}}}\mathfrak{R}^{{\underline{K}}{\underline{J}}}~{},$
(5.12b) $\displaystyle\big{[}{\cal D}_{+}^{{\overline{I}}},\hat{{\cal
D}}_{++}\big{]}$ $\displaystyle=$ $\displaystyle 0~{},\qquad\big{[}{\cal
D}_{-}^{{\underline{I}}},\hat{{\cal D}}_{--}\big{]}=0~{},$ (5.12c)
$\displaystyle\big{[}{\cal D}_{+}^{{\overline{I}}},\hat{{\cal D}}_{--}\big{]}$
$\displaystyle=$ $\displaystyle-2{\rm i}S^{{\overline{I}}{\underline{J}}}{\cal
D}_{-}^{{\underline{J}}}-2{\rm i}{\cal
D}_{-}^{{\underline{J}}}S^{{\overline{I}}{\underline{J}}}M+{\rm i}{\cal
D}_{-}^{{\underline{K}}}S^{{\overline{J}}{\underline{K}}}\mathfrak{L}^{{\overline{J}}{\overline{I}}}+{\rm
i}{\cal
D}_{-}^{\underline{J}}S^{{\overline{I}}{\underline{K}}}\mathfrak{R}^{{\underline{J}}{\underline{K}}}~{},$
(5.12d) $\displaystyle\big{[}{\cal D}_{-}^{{\underline{I}}},\hat{{\cal
D}}_{++}\big{]}$ $\displaystyle=$ $\displaystyle 2{\rm
i}S^{{\overline{J}}{\underline{I}}}{\cal D}_{+}^{{\overline{J}}}-2{\rm i}{\cal
D}_{+}^{{\overline{J}}}S^{{\overline{J}}{\underline{I}}}M-{\rm i}{\cal
D}_{+}^{\overline{J}}S^{{\overline{K}}{\underline{I}}}\mathfrak{L}^{{\overline{J}}{\overline{K}}}-{\rm
i}{\cal
D}_{+}^{{\overline{K}}}S^{{\overline{K}}{\underline{J}}}\mathfrak{R}^{{\underline{J}}{\underline{I}}}~{},$
(5.12e) $\displaystyle\big{[}\hat{{\cal D}}_{++},\hat{{\cal D}}_{--}\big{]}$
$\displaystyle=$ $\displaystyle-{\cal
D}_{+}^{{\overline{J}}}S^{{\overline{J}}{\underline{I}}}{\cal
D}_{-}^{{\underline{I}}}-{\cal
D}_{-}^{{\underline{J}}}S^{{\overline{I}}{\underline{J}}}{\cal
D}_{+}^{{\overline{I}}}-\frac{1}{2}\big{(}\big{[}{\cal
D}_{+}^{{\overline{I}}},{\cal
D}_{-}^{{\underline{J}}}\big{]}-8S^{{\overline{I}}{\underline{J}}}\big{)}S^{{\overline{I}}{\underline{J}}}M~{},$
(5.12f) $\displaystyle-\frac{1}{4}{\cal D}_{-}^{\underline{K}}{\cal
D}_{+}^{\overline{I}}S^{{\overline{J}}{\underline{K}}}\mathfrak{L}^{{\overline{I}}{\overline{J}}}-\frac{1}{4}{\cal
D}_{+}^{\overline{K}}{\cal
D}_{-}^{\underline{I}}S^{{\overline{K}}{\underline{J}}}\mathfrak{R}^{{\underline{I}}{\underline{J}}}~{}.$
It should be emphasised that the resulting geometry is described solely in
terms of $S^{{\overline{I}}{\underline{J}}}$. Then, to relate the geometry of
[29, 30, 31, 32] to ours it is necessary to express (5.12) in a complex basis
of spinor covariant derivatives
$\displaystyle{\cal D}_{+}:=\frac{1}{\sqrt{2}}({\cal
D}_{+}^{\overline{1}}-{\rm i}{\cal D}_{+}^{\overline{2}})~{},\qquad\bar{{\cal
D}}_{+}=-\frac{1}{\sqrt{2}}({\cal D}_{+}^{\overline{1}}+{\rm i}{\cal
D}_{+}^{\overline{2}})~{},$ (5.13a) $\displaystyle{\cal
D}_{-}:=\frac{1}{\sqrt{2}}({\cal D}_{-}^{\underline{1}}-{\rm i}{\cal
D}_{-}^{\underline{2}})~{},\qquad\bar{{\cal D}}_{-}=-\frac{1}{\sqrt{2}}({\cal
D}_{-}^{\underline{1}}+{\rm i}{\cal D}_{-}^{\underline{2}})~{}.$ (5.13b)
We omit further technical details regarding this procedure, which will appear
in a future work.
To the best of our knowledge, for $p,q>2$ our superspace geometry described by
the equations (5.5) and (5.6) has not appeared in the literature. In
particular, in the ${\cal N}=(4,4)$ case, the above
$\mathsf{SO}(4)\times\mathsf{SO}(4)$ superspace geometry differs from the one
proposed in [33] by the choice of structure group, see section 6 for the
discussion of this formulation.
As follows from (5.4), some expressions are ill-defined if at least one of the
parameters $p$ and $q$ takes values $0$ or $1$. Each of these cases should be
studied separately, which is done in the remainder of this section.
### 5.2 $p>1,~{}q=1$ case
Next, let us consider the case where $p>1,~{}q=1$. For convenience, we will
unambiguously remove bars over left isovector indices, e.g.
${\overline{I}}\equiv I$. By a routine calculation, one obtains the following
components of the degauged connection $\mathfrak{F}_{AB}$
$\displaystyle\mathfrak{F}_{+,-}^{I\phantom{,,}}$ $\displaystyle=$
$\displaystyle-\mathfrak{F}_{-,+}^{\phantom{,,}I}=S^{I}~{},\quad\mathfrak{F}_{+,+}^{I\phantom{,,}J}=-\mathfrak{F}_{+,+}^{J\phantom{,,}I}=X_{++}^{IJ}~{},\quad\mathfrak{F}_{-,-}=0~{},\quad$
(5.14a) $\displaystyle\mathfrak{F}_{+,--}^{I}$ $\displaystyle=$
$\displaystyle\mathfrak{F}_{--,+}^{~{}~{}~{}~{}\phantom{,}I}={\rm i}{\cal
D}_{-}S^{I}~{},\qquad\mathfrak{F}_{-,++}=\mathfrak{F}_{++,-}=-\frac{{\rm
i}}{p}{\cal D}_{+}^{I}S^{I}~{},$ (5.14b)
$\displaystyle\mathfrak{F}_{+,++}^{I}$ $\displaystyle=$
$\displaystyle\mathfrak{F}_{++,+}^{~{}~{}~{}~{}\phantom{,}I}=-\frac{{\rm
i}}{p-1}{\cal D}_{+}^{J}X_{++}^{JI}~{},$ (5.14c)
$\displaystyle\mathfrak{F}_{++,--}$ $\displaystyle=$
$\displaystyle\mathfrak{F}_{--,++}=\frac{1}{2p}\big{[}{\cal D}_{+}^{I},{\cal
D}_{-}\big{]}S^{I}-\frac{p+1}{p}S^{I}S^{I}~{},$ (5.14d)
$\displaystyle\mathfrak{F}_{++,++}$ $\displaystyle=$
$\displaystyle\frac{1}{p(p-1)}{\cal D}_{+}^{I}{\cal
D}_{+}^{J}X_{++}^{IJ}-\frac{2}{p}X_{++}^{IJ}X_{++}^{IJ}~{},$ (5.14e) where we
have introduced the dimension-$1$ torsions $S^{I}$ and $X_{++}^{IJ}$. The
remaining components of $\mathfrak{F}_{AB}$ do not play a role in the degauged
geometry, though they satisfy
$\displaystyle\mathfrak{F}_{-,--}=\mathfrak{F}_{--,-}~{},\quad{\cal
D}_{+}^{I}\mathfrak{F}_{-,--}={\cal
D}_{--}S^{I}~{},\quad\mathfrak{F}_{--,--}=-{\rm i}{\cal
D}_{-}\mathfrak{F}_{-,--}~{}.$ (5.14f)
These equations imply that $\mathfrak{F}_{-,--}$ is a non-local function of
the supergravity multiplet.
The superfields $S^{I}$ and $X_{++}^{IJ}$ obey the Bianchi identities
$\displaystyle{\cal D}_{+}^{I}S^{J}=\frac{1}{p}\delta^{IJ}{\cal
D}_{+}^{K}S^{K}+{\cal D}_{-}X_{++}^{IJ}~{},\qquad{\cal
D}_{+}^{I}X_{++}^{JK}=\frac{2}{p-1}\delta^{I[J}{\cal
D}_{+}^{|L}X_{++}^{L|K]}~{}.$ (5.15)
Additionally, it may be shown that the algebra obeyed by ${\cal D}_{A}$ takes
the form
$\displaystyle\\{{\cal D}_{+}^{I},{\cal D}_{+}^{J}\\}$ $\displaystyle=$
$\displaystyle 2{\rm i}\delta^{IJ}{\cal
D}_{++}-4X_{++}^{K(I}\mathfrak{L}^{J)K}~{},$ (5.16a) $\displaystyle\\{{\cal
D}_{+}^{I},{\cal D}_{-}\\}$ $\displaystyle=$
$\displaystyle-4S^{I}M+2S^{J}\mathfrak{L}^{JI}~{},$ (5.16b)
$\displaystyle\\{{\cal D}_{-},{\cal D}_{-}\\}$ $\displaystyle=$ $\displaystyle
2{\rm i}{\cal D}_{--}~{},$ (5.16c) $\displaystyle\big{[}{\cal D}_{+}^{I},{\cal
D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle-2{\rm i}S^{I}{\cal
D}_{-}-4{\rm i}{\cal D}_{-}S^{I}M+2{\rm i}{\cal
D}_{-}S^{J}\mathfrak{L}^{JI}~{},$ (5.16d) $\displaystyle\big{[}{\cal
D}_{-},{\cal D}_{++}\big{]}$ $\displaystyle=$ $\displaystyle 2{\rm
i}S^{I}{\cal D}_{+}^{I}-\frac{4{\rm i}}{p}{\cal D}_{+}^{I}S^{I}M~{},$ (5.16e)
$\displaystyle\big{[}{\cal D}_{+}^{I},{\cal D}_{++}\big{]}$ $\displaystyle=$
$\displaystyle-2{\rm i}X_{++}^{IJ}{\cal D}_{+}^{J}-\frac{2{\rm i}}{p-1}{\cal
D}_{+}^{J}X_{++}^{JK}\mathfrak{L}^{KI}~{},$ (5.16f) $\displaystyle\big{[}{\cal
D}_{-},{\cal D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle 0~{},$ (5.16g)
$\displaystyle\big{[}{\cal D}_{++},{\cal D}_{--}\big{]}$ $\displaystyle=$
$\displaystyle-\frac{2}{p}{\cal D}_{+}^{I}S^{I}{\cal D}_{-}-2{\cal
D}_{-}S^{I}{\cal D}_{+}^{I}$ (5.16h)
$\displaystyle-\frac{2}{p}\big{(}\big{[}{\cal D}_{+}^{I},{\cal
D}_{-}\big{]}-2(p+1)S^{I}\big{)}S^{I}M~{}.$
The supergravity gauge freedom of the $(p,1)$ geometry (5.16) corresponds to
the residual gauge transformations of (4.6) in the gauge $B_{A}=0$. In
particular, they include the local $\mathcal{K}$-transformations
$\displaystyle\delta_{\mathcal{K}}{\cal D}_{A}$ $\displaystyle=$
$\displaystyle[\mathcal{K},{\cal D}_{A}]\ ,$ (5.17a)
$\displaystyle\mathcal{K}$ $\displaystyle=$ $\displaystyle\xi^{B}{\cal
D}_{B}+KM+\frac{1}{2}\rho^{IJ}\mathfrak{L}^{IJ}~{},$ (5.17b) which act on
tensor superfields $\mathcal{U}$ (with indices suppressed) as $\delta_{\cal
K}{\cal U}={\cal K}{\cal U}$.
Transformations (5.17) prove to not be the most general conformal supergravity
gauge transformations preserving the gauge $B_{A}=0$. Specifically, the
following transformation also enjoys this property
$\displaystyle\mathscr{K}(\sigma)=\sigma\mathbb{D}+\frac{1}{2}\nabla_{B}\sigma
K^{B}\quad\Longrightarrow\quad\delta_{\mathscr{K}(\sigma)}B_{A}=0~{},$ (5.18)
where $\sigma$ is real but otherwise unconstrained.
As a result, it is necessary to consider how this transformation manifests in
the degauged geometry
$\displaystyle\delta_{\mathscr{K}(\sigma)}\nabla_{A}\equiv\delta_{\sigma}\nabla_{A}=\delta_{\sigma}{\cal
D}_{A}-\delta_{\sigma}(\mathfrak{F}_{AB}K^{B})~{}.$ (5.19)
Employing this relation, we arrive at the transformation laws for ${\cal
D}_{A}$
$\displaystyle\delta_{\sigma}{\cal D}_{+}^{I}$
$\displaystyle=\frac{1}{2}\sigma{\cal D}_{+}^{I}+{\cal D}_{+}^{I}\sigma
M-{\cal D}_{+}^{J}\sigma\mathfrak{L}^{JI}~{},$ (5.20a)
$\displaystyle\delta_{\sigma}{\cal D}_{-}$
$\displaystyle=\frac{1}{2}\sigma{\cal D}_{-}-{\cal D}_{-}\sigma M~{},$ (5.20b)
$\displaystyle\delta_{\sigma}{\cal D}_{++}$ $\displaystyle=\sigma{\cal
D}_{++}-{\rm i}{\cal D}_{+}^{I}\sigma{\cal D}_{+}^{I}+{\cal D}_{++}\sigma
M~{},$ (5.20c) $\displaystyle\delta_{\sigma}{\cal D}_{--}$
$\displaystyle=\sigma{\cal D}_{--}-{\rm i}{\cal D}_{-}\sigma{\cal D}_{-}-{\cal
D}_{--}\sigma M~{},$ (5.20d) and, by making use of (5.4), it may be shown that
the torsions transform as follows $\displaystyle\delta_{\sigma}S^{I}$
$\displaystyle=\sigma S^{I}+\frac{1}{2}{\cal D}_{+}^{I}{\cal D}_{-}\sigma~{},$
(5.20e) $\displaystyle\delta_{\sigma}X_{++}^{IJ}$ $\displaystyle=\sigma
X_{++}^{IJ}+\frac{1}{4}\big{[}{\cal D}_{+}^{I},{\cal
D}_{+}^{J}\big{]}\sigma~{}.$ (5.20f)
These are the super-Weyl transformations of the degauged geometry.
It may be shown that, for the special case $p=2$, the superfield $X_{++}^{IJ}$
can be eliminated by performing the redefinition
$\displaystyle\hat{{\cal D}}_{++}={\cal D}_{++}-{\rm
i}X_{++}^{\overline{I}\overline{J}}\mathfrak{L}^{\overline{I}\overline{J}}~{}.$
(5.21)
The resulting algebra then takes the form
$\displaystyle\\{{\cal D}_{+}^{I},{\cal D}_{+}^{J}\\}$ $\displaystyle=$
$\displaystyle 2{\rm i}\delta^{IJ}\hat{{\cal D}}_{++}~{},\qquad\\{{\cal
D}_{-},{\cal D}_{-}\\}=2{\rm i}{\cal D}_{--}~{},$ (5.22a)
$\displaystyle\\{{\cal D}_{+}^{I},{\cal D}_{-}\\}$ $\displaystyle=$
$\displaystyle-4S^{I}M+2S^{J}\mathfrak{L}^{JI}~{},$ (5.22b)
$\displaystyle\big{[}{\cal D}_{+}^{I},{\cal D}_{--}\big{]}$ $\displaystyle=$
$\displaystyle-2{\rm i}S^{I}{\cal D}_{-}-4{\rm i}{\cal D}_{-}S^{I}M+2{\rm
i}{\cal D}_{-}S^{J}\mathfrak{L}^{JI}~{},$ (5.22c) $\displaystyle\big{[}{\cal
D}_{-},\hat{{\cal D}}_{++}\big{]}$ $\displaystyle=$ $\displaystyle 2{\rm
i}S^{I}{\cal D}_{+}^{I}-2{\rm i}{\cal D}_{+}^{I}S^{I}M~{},$ (5.22d)
$\displaystyle\big{[}{\cal D}_{+}^{I},\hat{{\cal D}}_{++}\big{]}$
$\displaystyle=$ $\displaystyle 0~{},\qquad\big{[}{\cal D}_{-},{\cal
D}_{--}\big{]}=0~{},$ (5.22e) $\displaystyle\big{[}\hat{{\cal D}}_{++},{\cal
D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle-{\cal D}_{+}^{I}S^{I}{\cal
D}_{-}-2{\cal D}_{-}S^{I}{\cal D}_{+}^{I}-\big{(}\big{[}{\cal D}_{+}^{I},{\cal
D}_{-}\big{]}-6S^{I}\big{)}S^{I}M$ (5.22f) $\displaystyle-\frac{1}{4}{\cal
D}_{-}{\cal D}_{+}^{I}S^{J}\mathfrak{L}^{IJ}~{}.$
Hence, the resulting geometry is described solely in terms of $S^{I}$.
### 5.3 $p>1,~{}q=0$ case
Now, we consider the case $p>1,~{}q=0$. As in the previous subsection, we
remove bars over left isovector indices; ${\overline{I}}\equiv I$. By a
routine calculation, we readily obtain the following components of the
degauged connection $\mathfrak{F}_{AB}$
$\displaystyle\mathfrak{F}_{+,+}^{I\phantom{,,}J}$ $\displaystyle=$
$\displaystyle-\mathfrak{F}_{+,+}^{J\phantom{,,}I}=X_{++}^{IJ}~{},$ (5.23a)
$\displaystyle\mathfrak{F}_{+,--}^{I}$ $\displaystyle=$
$\displaystyle\mathfrak{F}_{--,+}^{~{}~{}~{}~{}\phantom{,}I}=G_{-}^{I}~{},\qquad\qquad\phantom{-}\,\mathfrak{F}_{+,++}^{I}=\mathfrak{F}_{++,+}^{~{}~{}~{}~{}\phantom{,}I}=-\frac{{\rm
i}}{p-1}{\cal D}_{+}^{J}X_{++}^{JI}~{},$ (5.23b)
$\displaystyle\mathfrak{F}_{++,--}$ $\displaystyle=$
$\displaystyle\mathfrak{F}_{--,++}=-\frac{{\rm i}}{p}{\cal
D}_{+}^{I}G_{-}^{I}~{},\quad\mathfrak{F}_{++,++}=\frac{1}{p(p-1)}{\cal
D}_{+}^{I}{\cal
D}_{+}^{J}X_{++}^{IJ}-\frac{2}{p}X_{++}^{IJ}X_{++}^{IJ}~{},\qquad$ (5.23c)
while $\mathfrak{F}_{--,--}$ only appears in (5.3) through the differential
constraint $\displaystyle{\cal D}_{+}^{I}\mathfrak{F}_{--,--}={\cal
D}_{--}G_{-}^{I}~{}.$ (5.23d)
In the above equations we have introduced the torsions $X_{++}^{IJ}$ and
$G_{-}^{I}$, which obey the Bianchi identities
$\displaystyle{\cal D}_{+}^{I}X_{++}^{JK}=\frac{2}{p-1}\delta^{I[J}{\cal
D}_{+}^{|L}X_{++}^{L|K]}~{},\qquad{\cal
D}_{+}^{I}G_{-}^{J}=\frac{1}{p}\delta^{IJ}{\cal D}_{+}^{K}G_{-}^{K}+{\cal
D}_{--}X_{++}^{IJ}~{}.$ (5.24)
The latter constraint in (5.24) may be solved by expressing $G_{-}^{I}$ in
terms of the dimension-$1$ superfield $G_{--}$
$\displaystyle G_{-}^{I}={\cal D}_{+}^{I}G_{--}\quad\implies\quad{\cal
D}_{+}^{[I}{\cal D}_{+}^{J]}G_{--}={\cal D}_{--}X_{++}^{IJ}~{}.$ (5.25)
However, we will not make use of this solution in what follows.
Making use of these results, we find that the algebra obeyed by ${\cal D}_{A}$
takes the form
$\displaystyle\\{{\cal D}_{+}^{I},{\cal D}_{+}^{J}\\}$ $\displaystyle=$
$\displaystyle 2{\rm i}\delta^{IJ}{\cal
D}_{++}-4X_{++}^{K(I}\mathfrak{L}^{J)K}~{},$ (5.26a)
$\displaystyle\big{[}{\cal D}_{+}^{I},{\cal D}_{--}\big{]}$ $\displaystyle=$
$\displaystyle-4G_{-}^{I}M+2G_{-}^{J}\mathfrak{L}^{JI}~{},$ (5.26b)
$\displaystyle\big{[}{\cal D}_{+}^{I},{\cal D}_{++}\big{]}$ $\displaystyle=$
$\displaystyle-2{\rm i}X_{++}^{IJ}{\cal D}_{+}^{J}-\frac{2{\rm i}}{p-1}{\cal
D}_{+}^{J}X_{++}^{JK}\mathfrak{L}^{KI}~{},$ (5.26c) $\displaystyle\big{[}{\cal
D}_{++},{\cal D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle 2{\rm
i}G_{-}^{I}{\cal D}_{+}^{I}-\frac{4{\rm i}}{p}{\cal D}_{+}^{I}G_{-}^{I}M~{}.$
(5.26d)
It should be noted that ${\cal N}=(p,0)$ superspace geometries, with $p\geq
2$, were discussed in [34], where the structure was chosen to be the Lorentz
group, $\mathsf{SO}(1,1)$, though the $p=2$ case appeared earlier [35].777The
structure group for $(2,0)$ supergravity was enlarged from $\mathsf{SO}(1,1)$
to $\mathsf{SO}(1,1)\times\mathsf{U}(1)$ in [36]. The emergence of the $(2,0)$
geometry proposed in [35] in our framework will be discussed below.
The supergravity gauge transformations of the $(p,0)$ geometry (5.26) may be
obtained from (4.6) after imposing $B_{A}=0$. They include the local
$\mathcal{K}$-transformations
$\displaystyle\delta_{\mathcal{K}}{\cal D}_{A}$ $\displaystyle=$
$\displaystyle[\mathcal{K},{\cal D}_{A}]\ ,$ (5.27a)
$\displaystyle\mathcal{K}$ $\displaystyle=$ $\displaystyle\xi^{B}{\cal
D}_{B}+KM+\frac{1}{2}\rho^{IJ}\mathfrak{L}^{IJ}~{},$ (5.27b) which act on
tensor superfields $\mathcal{U}$ (with indices suppressed) as $\delta_{\cal
K}{\cal U}={\cal K}{\cal U}$.
In addition to (5.27), the following transformation also preserves the gauge
$B_{A}=0$
$\displaystyle\mathscr{K}(\sigma)=\sigma\mathbb{D}+\frac{1}{2}\nabla_{B}\sigma
K^{B}\quad\Longrightarrow\quad\delta_{\mathscr{K}(\sigma)}B_{A}=0~{},$ (5.28)
where $\sigma$ is real but otherwise unconstrained.
It is then necessary to consider how this transformation manifests in the
degauged geometry
$\displaystyle\delta_{\mathscr{K}(\sigma)}\nabla_{A}\equiv\delta_{\sigma}\nabla_{A}=\delta_{\sigma}{\cal
D}_{A}-\delta_{\sigma}(\mathfrak{F}_{AB}K^{B})~{}.$ (5.29)
Employing this relation, we arrive at the transformation laws for ${\cal
D}_{A}$
$\displaystyle\delta_{\sigma}{\cal D}_{+}^{I}$
$\displaystyle=\frac{1}{2}\sigma{\cal D}_{+}^{I}+{\cal D}_{+}^{I}\sigma
M-{\cal D}_{+}^{J}\sigma\mathfrak{L}^{JI}~{},$ (5.30a)
$\displaystyle\delta_{\sigma}{\cal D}_{++}$ $\displaystyle=\sigma{\cal
D}_{++}-{\rm i}{\cal D}_{+}^{I}\sigma{\cal D}_{+}^{I}+{\cal D}_{++}\sigma
M~{},$ (5.30b) $\displaystyle\delta_{\sigma}{\cal D}_{--}$
$\displaystyle=\sigma{\cal D}_{--}-{\cal D}_{--}\sigma M~{},$ (5.30c) and, by
making use of (5.4), it may be shown that the torsions transform as follows
$\displaystyle\delta_{\sigma}X_{++}^{IJ}$ $\displaystyle=\sigma
X_{++}^{IJ}+\frac{1}{4}\big{[}{\cal D}_{+}^{I},{\cal
D}_{+}^{J}\big{]}\sigma~{},$ (5.30d) $\displaystyle\delta_{\sigma}G_{-}^{I}$
$\displaystyle=\frac{3}{2}\sigma G_{-}^{I}+\frac{1}{2}{\cal D}_{+}^{I}{\cal
D}_{--}\sigma~{}.$ (5.30e)
These are exactly the super-Weyl transformations of the degauged geometry.
We emphasise that in special case $p=2$, the superfield $X_{++}^{IJ}$ can be
eliminated by performing the redefinition (5.21). Additionally, it is useful
to work in a complex basis of spinor covariant derivatives, where ${\cal
D}_{+}^{I}$ is replaced by ${\cal D}_{+}=\frac{1}{\sqrt{2}}({\cal
D}_{+}^{1}-{\rm i}{\cal D}_{+}^{2})$ and its conjugate $\bar{\cal
D}_{+}=-\frac{1}{\sqrt{2}}({\cal D}_{+}^{1}+{\rm i}{\cal D}_{+}^{2})$. The
resulting algebra of covariant derivatives is
$\displaystyle\\{{\cal D}_{+},{\cal D}_{+}\\}$ $\displaystyle=$ $\displaystyle
0~{},\qquad\\{{\cal D}_{+},\bar{{\cal D}}_{+}\\}=2{\rm i}\hat{{\cal
D}}_{++}~{},$ (5.31a) $\displaystyle\big{[}{\cal D}_{+},{\cal D}_{--}\big{]}$
$\displaystyle=$ $\displaystyle-4\bar{G}_{-}M-2{\rm
i}\bar{G}_{-}\mathfrak{L}^{12}~{},$ (5.31b) $\displaystyle\big{[}{\cal
D}_{+},\hat{{\cal D}}_{++}\big{]}$ $\displaystyle=$ $\displaystyle 0~{},$
(5.31c) $\displaystyle\big{[}\hat{{\cal D}}_{++},{\cal D}_{--}\big{]}$
$\displaystyle=$ $\displaystyle 2{\rm i}G_{-}{\cal D}_{+}+2{\rm
i}\bar{G}_{-}\bar{{\cal D}}_{+}+2{\rm i}({\cal D}_{+}{G}_{-}+\bar{{\cal
D}}_{+}\bar{G}_{-})M$ (5.31d) $\displaystyle+({\cal D}_{+}{G}_{-}-\bar{{\cal
D}}_{+}\bar{G}_{-})\mathfrak{L}^{12}~{}.$
We see that this supergeometry is described solely in terms of the complex
superfield
$\displaystyle G_{-}:=-\frac{1}{\sqrt{2}}(G_{-}^{1}+{\rm i}G_{-}^{2})$ (5.32)
and its conjugate $\bar{G}_{-}$ It follows from (5.24) that it satisfies the
chirality constraint
$\displaystyle\bar{{\cal D}}_{+}G_{-}=0~{}.$ (5.33)
The $\mathsf{SO}(2)$ gauge freedom enjoyed by the geometry (5.31) may be used
to gauge away its corresponding spinor $\mathsf{SO}(2)$ connection
$\Phi_{+}^{I,\,JK}=-\Phi_{+}^{I,\,JK}$. This can be seen by coupling the
conformal supergravity multiplet to a conformal primary compensator $\phi$
obeying the chirality condition $\bar{{\cal D}}_{+}\phi=0$. Choosing its
weight to be $\Delta_{\phi}=1$, it transforms under super-Weyl (5.30) and
local $\mathsf{SO}(2)$ transformations (5.28) as follows:
$\displaystyle\delta\phi=(\sigma-{\rm i}\rho^{12})\phi~{},$ (5.34)
hence this freedom may be used to impose the gauge $\phi=1$. Associated with
this gauge are the consistency conditions
$\displaystyle\Phi_{+}^{I,12}=0~{},\qquad{\rm i}\bar{{\cal
D}}_{+}\Gamma_{--}=G_{-}~{},\qquad\Gamma_{--}=\bar{\Gamma}_{--}:=\frac{1}{2}\Phi_{--}^{\phantom{--}12}~{}.$
(5.35)
As the $\mathsf{SO}(2)$ connection is now auxiliary, it should be separated by
performing the redefinition
$\displaystyle\hat{{\cal D}}_{--}={\cal
D}_{--}+2\Gamma_{--}\mathfrak{L}^{12}~{}.$ (5.36)
The resulting algebra of covariant derivatives is
$\displaystyle\\{{\cal D}_{+},{\cal D}_{+}\\}$ $\displaystyle=$ $\displaystyle
0~{},\qquad\\{{\cal D}_{+},\bar{{\cal D}}_{+}\\}=2{\rm i}\hat{{\cal
D}}_{++}~{},$ (5.37a) $\displaystyle\big{[}{\cal D}_{+},\hat{{\cal
D}}_{--}\big{]}$ $\displaystyle=$ $\displaystyle-2{\rm i}\Gamma_{--}{\cal
D}_{+}-4{\rm i}\bar{{\cal D}}_{+}\Gamma_{--}M~{},$ (5.37b)
$\displaystyle\big{[}{\cal D}_{+},\hat{{\cal D}}_{++}\big{]}$ $\displaystyle=$
$\displaystyle 0~{},$ (5.37c) $\displaystyle\big{[}\hat{{\cal
D}}_{++},\hat{{\cal D}}_{--}\big{]}$ $\displaystyle=$
$\displaystyle-2\bar{{\cal D}}_{+}\Gamma_{--}{\cal D}_{+}+2{{\cal
D}}_{+}\Gamma_{--}\bar{{\cal D}}_{+}-2[{\cal D}_{+},\bar{{\cal
D}}_{+}]\Gamma_{--}M~{},$ (5.37d)
which coincides with the one appearing in [35].
### 5.4 $p=q=1$ case
Next, we fix $p=q=1$. By a routine calculation, we obtain the following
components of the degauged special conformal connection
$\displaystyle\mathfrak{F}_{+,+}$ $\displaystyle=$ $\displaystyle
0~{},\qquad\mathfrak{F}_{-,-}=0~{},\qquad\mathfrak{F}_{+,-}=-\mathfrak{F}_{-,+}=S$
(5.38a) $\displaystyle\mathfrak{F}_{+,--}$ $\displaystyle=$
$\displaystyle\mathfrak{F}_{--,+}={\rm i}{\cal
D}_{-}S~{},\qquad\mathfrak{F}_{-,++}=\mathfrak{F}_{++,-}=-{\rm i}{\cal
D}_{+}S~{},$ (5.38b) $\displaystyle\mathfrak{F}_{++,++}$ $\displaystyle=$
$\displaystyle-{\rm i}{\cal
D}_{+}\mathfrak{F}_{+,++}~{},\qquad\mathfrak{F}_{--,--}=-{\rm i}{\cal
D}_{-}\mathfrak{F}_{-,--}~{},$ (5.38c) $\displaystyle\mathfrak{F}_{++,--}$
$\displaystyle=$ $\displaystyle\mathfrak{F}_{--,++}=\frac{1}{2}[{\cal
D}_{+},{\cal D}_{-}]S-2S^{2}~{},$ (5.38d) where we have introduced the real
scalar $S$. The remaining components of $\mathfrak{F}_{AB}$ do not play a role
in the degauged geometry, though they satisfy the constraints
$\displaystyle\mathfrak{F}_{+,++}$
$\displaystyle=\mathfrak{F}_{++,+}~{},\qquad\mathfrak{F}_{-,--}=\mathfrak{F}_{--,-}~{},$
(5.38e) $\displaystyle{\cal D}_{-}\mathfrak{F}_{+,++}$ $\displaystyle=-{\cal
D}_{++}S~{},\qquad{\cal D}_{+}\mathfrak{F}_{-,--}={\cal D}_{--}S~{}.$ (5.38f)
It follows that $\mathfrak{F}_{+,++}$ and $\mathfrak{F}_{-,--}$ are non-local
functions of the supergravity multiplet.
It follows from the above results that the algebra obeyed by ${\cal D}_{A}$
takes the form
$\displaystyle\\{{\cal D}_{+},{\cal D}_{+}\\}$ $\displaystyle=$ $\displaystyle
2{\rm i}{\cal D}_{++}~{},\qquad\\{{\cal D}_{-},{\cal D}_{-}\\}=2{\rm i}{\cal
D}_{--}~{},$ (5.39b) $\displaystyle\\{{\cal D}_{+},{\cal D}_{-}\\}=-4SM~{},$
$\displaystyle\big{[}{\cal D}_{+},{\cal D}_{--}\big{]}$ $\displaystyle=$
$\displaystyle-2{\rm i}S{\cal D}_{-}-4{\rm i}({\cal D}_{-}S)M~{},$ (5.39c)
$\displaystyle\big{[}{\cal D}_{-},{\cal D}_{++}\big{]}$ $\displaystyle=$
$\displaystyle 2{\rm i}S{\cal D}_{+}-4{\rm i}({\cal D}_{+}S)M~{},$ (5.39d)
$\displaystyle\big{[}{\cal D}_{++},{\cal D}_{--}\big{]}$ $\displaystyle=$
$\displaystyle-2({\cal D}_{+}S){\cal D}_{-}-2({\cal D}_{-}S){\cal D}_{+}$
(5.39e) $\displaystyle-2\big{(}[{\cal D}_{+},{\cal D}_{-}]-4S\big{)}SM~{}.$
The supergravity gauge freedom of this geometry may be obtained from the
conformal superspace transformations (4.6) after restricting to the gauge
$B_{A}=0$. These include local $\mathcal{K}$-transformations of the form
$\displaystyle\delta_{\mathcal{K}}{\cal D}_{A}$ $\displaystyle=$
$\displaystyle[\mathcal{K},{\cal D}_{A}]~{},\qquad\mathcal{K}=\xi^{B}{\cal
D}_{B}+KM~{},$ (5.40a) which act on tensor superfields $\mathcal{U}$ (with
indices suppressed) as $\delta_{\cal K}{\cal U}={\cal K}{\cal U}$.
In addition to (5.40), it may be shown that the following also preserves the
gauge $B_{A}=0$
$\displaystyle\mathscr{K}(\sigma)=\sigma\mathbb{D}+\frac{1}{2}\nabla_{B}\sigma
K^{B}\quad\Longrightarrow\quad\delta_{\mathscr{K}(\sigma)}B_{A}=0~{},$ (5.41)
where $\sigma$ is real but otherwise unconstrained.
As a result, it is necessary to consider how this transformation manifests in
the degauged geometry
$\displaystyle\delta_{\mathscr{K}(\sigma)}\nabla_{A}\equiv\delta_{\sigma}\nabla_{A}=\delta_{\sigma}{\cal
D}_{A}-\delta_{\sigma}(\mathfrak{F}_{AB}K^{B})~{}.$ (5.42)
Employing this relation, we arrive at the transformation laws for ${\cal
D}_{A}$ and $S$
$\displaystyle\delta_{\sigma}{\cal D}_{+}$
$\displaystyle=\frac{1}{2}\sigma{\cal D}_{+}+{\cal D}_{+}\sigma M~{},$ (5.43a)
$\displaystyle\delta_{\sigma}{\cal D}_{-}$
$\displaystyle=\frac{1}{2}\sigma{\cal D}_{-}-{\cal D}_{-}\sigma M~{},$ (5.43b)
$\displaystyle\delta_{\sigma}{\cal D}_{++}$ $\displaystyle=\sigma{\cal
D}_{++}-{\rm i}{\cal D}_{+}\sigma{\cal D}_{+}+{\cal D}_{++}\sigma M~{},$
(5.43c) $\displaystyle\delta_{\sigma}{\cal D}_{--}$ $\displaystyle=\sigma{\cal
D}_{--}-{\rm i}{\cal D}_{-}\sigma{\cal D}_{-}-{\cal D}_{--}\sigma M~{},$
(5.43d) $\displaystyle\delta_{\sigma}S$ $\displaystyle=\sigma
S+\frac{1}{2}{\cal D}_{+}{\cal D}_{-}\sigma~{}.$ (5.43e)
These are exactly the super-Weyl transformations of the degauged geometry.
The superspace geometry of ${\cal N}=1$ supergravity described above was
originally constructed in [3, 37], see also [38, 39, 40].
### 5.5 $p=1,~{}q=0$ case
Finally, let us consider the case $p=1,~{}q=0$. A routine computation leads to
the degauged special conformal connections
$\displaystyle\mathfrak{F}_{+,+}$ $\displaystyle=$ $\displaystyle
0~{},\qquad\mathfrak{F}_{+,--}=\mathfrak{F}_{--,+}=G_{-}~{},\qquad\mathfrak{F}_{++,--}=\mathfrak{F}_{--,++}=-{\rm
i}{\cal D}_{+}G_{-}~{}.$ (5.44a) where we have introduced the spinor $G_{-}$.
The remaining components of $\mathfrak{F}_{AB}$ do not play a role in the
degauged geometry, though they satisfy the constraints
$\displaystyle\mathfrak{F}_{+,++}$
$\displaystyle=\mathfrak{F}_{++,+}~{},\qquad\mathfrak{F}_{++,++}=-{\rm i}{\cal
D}_{+}\mathfrak{F}_{+,++}~{},$ (5.44b) $\displaystyle{\cal D}_{++}G_{-}$
$\displaystyle={\cal D}_{--}\mathfrak{F}_{+,++}~{},\qquad{\cal
D}_{--}G_{-}={\cal D}_{+}\mathfrak{F}_{--,--}~{}.$ (5.44c)
It is clear that $\mathfrak{F}_{+,++}$ and $\mathfrak{F}_{--,--}$ are non-
local functions of the supergravity multiplet.
It immediately follows that the algebra obeyed by ${\cal D}_{A}$ takes the
form
$\displaystyle\\{{\cal D}_{+},{\cal D}_{+}\\}$ $\displaystyle=$ $\displaystyle
2{\rm i}{\cal D}_{++}~{},$ (5.45a) $\displaystyle\big{[}{\cal D}_{+},{\cal
D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle-4G_{-}M~{},\quad\big{[}{\cal
D}_{+},{\cal D}_{++}\big{]}=0$ (5.45b) $\displaystyle\big{[}{\cal
D}_{++},{\cal D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle 2{\rm
i}G_{-}{\cal D}_{+}+4{\rm i}({\cal D}_{+}G_{-})M~{}.$ (5.45c)
As described in the previous subsections, the supergravity gauge
transformations of this geometry correspond to (4.6) in the gauge $B_{A}=0$.
They include local $\mathcal{K}$-transformations of the form
$\displaystyle\delta_{\mathcal{K}}{\cal D}_{A}$ $\displaystyle=$
$\displaystyle[\mathcal{K},{\cal D}_{A}]~{},\qquad\mathcal{K}=\xi^{B}{\cal
D}_{B}+\omega M~{},$ (5.46a) which acts on tensor superfields $\mathcal{U}$
(with indices suppressed) as $\delta_{\cal K}{\cal U}={\cal K}{\cal U}$.
One may also show that the following transformation also preserves the
$B_{A}=0$ gauge
$\displaystyle\mathscr{K}(\sigma)=\sigma\mathbb{D}+\frac{1}{2}\nabla_{B}\sigma
K^{B}\quad\Longrightarrow\quad\delta_{\mathscr{K}(\sigma)}B_{A}=0~{},$ (5.47)
where $\sigma$ is real but otherwise unconstrained.
As a result, it is necessary to consider how this transformation manifests in
the degauged geometry
$\displaystyle\delta_{\mathscr{K}(\sigma)}\nabla_{A}\equiv\delta_{\sigma}\nabla_{A}=\delta_{\sigma}{\cal
D}_{A}-\delta_{\sigma}(\mathfrak{F}_{AB}K^{B})~{}.$ (5.48)
Employing this relation, we arrive at the transformation laws for ${\cal
D}_{A}$ and $G_{-}$
$\displaystyle\delta_{\sigma}{\cal D}_{+}$
$\displaystyle=\frac{1}{2}\sigma{\cal D}_{+}+{\cal D}_{+}\sigma M~{},$ (5.49a)
$\displaystyle\delta_{\sigma}{\cal D}_{++}$ $\displaystyle=\sigma{\cal
D}_{++}-{\rm i}{\cal D}_{+}\sigma{\cal D}_{+}+{\cal D}_{++}\sigma M~{},$
(5.49b) $\displaystyle\delta_{\sigma}{\cal D}_{--}$ $\displaystyle=\sigma{\cal
D}_{--}-{\cal D}_{--}\sigma M~{},$ (5.49c) $\displaystyle\delta_{\sigma}G_{-}$
$\displaystyle=\frac{3}{2}\sigma G_{-}+\frac{1}{2}{\cal D}_{+}{\cal
D}_{--}\sigma~{}.$ (5.49d)
These are exactly the super-Weyl transformations of the degauged geometry.
It should be noted that the above curved $(1,0)$ superspace geometry was
originally constructed in [35, 41, 34].
## 6 Generalisations and future prospects
Our approach to rigid $(p,q)$ superconformal symmetry was based on the use of
real coordinates
$z^{A}=(x^{a},\theta^{+{\overline{I}}},\theta^{-\underline{I}})$ to
parametrise Minkowski superspace $\mathbb{M}^{(2|p,q)}$. The conformal Killing
supervector fields were defined to satisfy the equation (2.4), which implied
that the algebra of conformal Killing supervector fields of
$\mathbb{M}^{(2|p,q)}$ is infinite dimensional. Its maximal finite-dimensional
subalgebra is singled out by the conditions (2.13), and is isomorphic to
$\mathfrak{osp}(p|2;\mathbb{R})\oplus\mathfrak{osp}(q|2;\mathbb{R})$. The
latter is the Lie algebra of the supergroup
${\mathsf{OSp}}_{0}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}_{0}(q|2;{\mathbb{R}})$,
which is the superconformal group of the compactified Minkowski superspace
(C.1).
In the case that, say, $p$ is even, $p=2n$, the real Grassmann variables
$\theta^{+{\overline{I}}}$ can be replaced with complex ones,
$\displaystyle\theta^{+{\overline{I}}}\to(\theta^{+i}~{},\bar{\theta}^{+}_{i})~{},\qquad\bar{\theta}^{+}_{i}:=\overline{\theta^{+i}}~{},\quad
i=1,\dots,n~{}.$ (6.1)
At the same time, the real covariant derivatives $D_{+}^{\overline{I}}$ should
be replaced with complex ones
$\displaystyle
D^{{\overline{I}}}_{+}\to(D_{+i}~{},\bar{D}_{+}^{i})~{},\qquad\bar{D}_{+}^{i}:=\overline{D_{+i}}~{},$
(6.2)
which obey the algebra
$\displaystyle\big{\\{}D_{+i},\bar{D}_{+}^{j}\big{\\}}=2{\rm
i}\delta_{i}^{j}\partial_{++}~{}.$ (6.3)
Then, equation (2.4) should be replaced with
$\displaystyle[\xi,D_{+i}]=-(D_{+i}\xi^{+j})D_{+j}~{},\qquad[\xi,D_{-}^{\underline{I}}]=-(D_{-}^{\underline{I}}\xi^{-\underline{J}})D_{-}^{\underline{J}}~{},$
(6.4a) where $\xi$ takes the form
$\displaystyle\xi=\xi^{a}\partial_{a}+\xi^{+i}D_{+i}+\bar{\xi}^{+}_{i}\bar{D}_{+}^{i}+\xi^{-{\underline{I}}}D_{-}^{\underline{I}}=\bar{\xi}~{}.$
(6.4b)
One may then perform a similar analysis to that of section 2 to obtain the
corresponding superconformal algebra as a subalgebra of the $(p,q)$ super
Virasoro algebra. We will not perform a complete analysis here and instead
sketch the salient points.
The defining relations (6.4a) imply the master equations
$\displaystyle D_{+i}\xi^{--}=0~{},\qquad
D_{-}^{{\underline{I}}}\xi^{++}=0~{},\qquad D_{+i}D_{+j}\xi^{++}=0~{},$ (6.5)
and yield the following expressions for the spinor parameters
$\displaystyle\xi^{+i}=-\frac{{\rm
i}}{2}\bar{D}_{+}^{i}\xi^{++}~{},\qquad\xi^{-{\underline{I}}}=-\frac{{\rm
i}}{2}D_{-}^{\underline{I}}\xi^{--}~{}.$ (6.6)
It should be noted that the final relation of (6.5) has the following non-
trivial implications, depending on the value of $n$
$\displaystyle n=2:$
$\displaystyle\quad\partial_{++}[D_{+i},\bar{D}_{+}^{i}]\xi^{++}=0~{},$ (6.7a)
$\displaystyle n>2:$
$\displaystyle\quad\partial_{++}[D_{+i},\bar{D}_{+}^{j}]\xi^{++}=0~{}.$ (6.7b)
In particular, it follows from the latter constraint that
$\displaystyle\partial_{++}\partial_{++}D_{+i}\xi^{++}=0~{},$ (6.8)
and thus, for $n>2$, the vector $\xi^{++}$ encodes finitely many parameters.
This is in contrast to the situation in section 2, where it was necessary to
impose the conditions (2.13). This difference is a consequence of (6.4a) being
more restrictive than (2.4). For $n=1,2$ condition (6.8) must instead be
imposed by hand.
Making use of (6.5), the master equations (6.4a) may be written in the form
$\displaystyle[\xi,D_{+i}]$
$\displaystyle=-\frac{1}{2}(\sigma[\xi]+K[\xi]+2\chi[\xi])D_{+i}-\chi[\xi]_{i}{}^{j}D_{+j}~{},$
(6.9a) $\displaystyle[\xi,D_{-}^{\underline{I}}]$
$\displaystyle=-\frac{1}{2}(\sigma[\xi]-K[\xi])D_{-}^{\underline{I}}-\rho[\xi]^{\underline{I}\underline{J}}D_{-}^{\underline{J}}~{},$
(6.9b)
where $\sigma[\xi]$, $K[\xi]$ and $\rho[\xi]^{{\underline{I}}{\underline{J}}}$
were defined in (2.9) and we have made the definitions
$\displaystyle\chi[\xi]:=-\frac{{\rm
i}}{4n}[D_{+i},\bar{D}_{+}^{i}]\xi^{++}~{},\qquad\chi[\xi]_{i}{}^{j}:=-\frac{{\rm
i}}{4n}\Big{(}[D_{+i},\bar{D}_{+}^{j}]-\frac{1}{n}\delta_{i}^{j}[D_{+k},\bar{D}_{+}^{k}]\Big{)}\xi^{++}~{}.$
(6.10)
The former are constrained to satisfy (2.14), while the new parameters obey
$\displaystyle D_{+i}\chi[\xi]$
$\displaystyle=\frac{n-2}{n}D_{+i}\sigma[\xi]~{},\qquad
D_{-}^{\underline{I}}\chi[\xi]=0~{},$ (6.11a) $\displaystyle
D_{+i}\chi[\xi]_{j}{}^{k}$
$\displaystyle=-2\delta_{i}^{k}D_{+j}\sigma[\xi]+\frac{2}{n}\delta_{j}^{k}D_{+i}\sigma[\xi]~{},\qquad
D_{-}^{\underline{I}}\chi[\xi]_{j}{}^{k}=0~{}.$ (6.11b)
One may then continue this analysis, keeping in mind the philosophy of section
2, to derive the superconformal algebra.
The resulting superconformal group for $n\neq 2$ proves to be
$\mathsf{SU}(1,1|n)\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$, with
$\mathsf{U}(n)\times\mathsf{SO}(q)$ being its $R$-symmetry subgroup. The $n=2$
case is special since the diagonal $\mathsf{U}(1)$ subgroup of
$\mathsf{SU}(1,1|2)$ can be factored out, and the $R$-symmetry subgroup of the
superconformal group becomes $\mathsf{SU}(2)\times\mathsf{SO}(q)$. Now, the
construction of conformal $(2n,q)$ superspace can be carried out by gauging
the superconformal group
$\mathsf{SU}(1,1|n)\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$ for $n\neq 2$, or
$\mathsf{PSU}(1,1|2)\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$ for $n=2$. The
degauged version of this superspace may be denoted
$\mathsf{U}(n)\times\mathsf{SO}(q)$ superspace, for $n\neq 2$, and
$\mathsf{SU}(2)\times\mathsf{SO}(q)$ superspace, for $n=2$. Analogous
considerations apply in the case that $q$ is even. In particular, for $p=q=4$,
one can introduce conformal $(4,4)$ superspace as the gauge theory of
$\mathsf{PSU}(1,1|2)\times\mathsf{PSU}(1,1|2)$. Its degauged version turns out
to be the curved $\mathsf{SU}(2)\times\mathsf{SU}(2)$ superspace geometry
introduced in [33].888As pointed out in [33], the super-Weyl and local
$R$-symmetry transformations can be used to partially fix the gauge freedom
such that the resulting supergravity multiplet turns into the one proposed
many years ago in [42, 43]. It should be pointed out that the formulation for
${\cal N}=(4,4)$ matter-coupled supergravity in
$\mathsf{SU}(2)\times\mathsf{SU}(2)$ harmonic superspace was constructed in
[44]. We believe this formulation provides a solution to the torsion
constraints for $\mathsf{SU}(2)\times\mathsf{SU}(2)$ superspace [33] in terms
of unconstrained prepotentials. It would be interesting to prove this
conjecture.
The $d=2$ superconformal groups were classified by Günaydin, Sierra and
Townsend [16] and have the structure
$\displaystyle G=G_{L}\times G_{R}~{},$ (6.12)
where $G_{L}$ and $G_{R}$ are simple supergroups. The supergroups $G_{L}$ and
$G_{R}$ can be any of the following: (i) $\mathsf{OSp}(m|2;{\mathbb{R}})$;
(ii) $\mathsf{SU}(1,1|m)$, for $m\neq 2$, or $\mathsf{PSU}(1,1|2)$; (iii)
$\mathsf{OSp}(4^{*}|2m)$; (iv) $\mathsf{G}(3)$; (v) $\mathsf{F}(4)$; and (vi)
$\mathsf{D}^{1}(2,1,\alpha)$.
Our paper has been devoted to conformal $(p,q)$ supergravity. Various versions
of extended Poincaré supergravity may be obtained from $H_{L}\times H_{R}$
superspaces via coupling to compensating multiplets, following the universal
approach advocated in [45].
Supersymmetric theories in AdS2 have recently attracted much interest, see
e.g. [46] and references therein. Using our construction of
$\mathsf{SO}(p)\times\mathsf{SO}(q)$ superspace, it is possible to work out
the structure of AdS superspaces in two dimensions. This can be achieved by
analogy with the derivation of $(p,q)$ superspace in three dimensions [47].
Specifically, two-dimensional AdS superspaces correspond to those supergravity
backgrounds which satisfy the following requirements:
(i) the torsion and curvature tensors are Lorentz invariant;
(ii) the torsion and curvature tensors are covariantly constant.
Condition (i) means that
$\displaystyle X_{++}^{{\overline{I}}{\overline{J}}}=0~{},\qquad
X_{--}^{{\underline{I}}{\underline{J}}}=0~{}.$ (6.13)
Condition (ii) is equivalent to the requirement
$\displaystyle{\cal D}_{A}S^{{\overline{I}}{\underline{J}}}=0~{},$ (6.14)
which has nontrivial implications. This requirement and the relation (5.6b)
give
$\displaystyle 0=\\{{\cal D}_{+}^{{\overline{I}}},{\cal
D}_{-}^{{\underline{J}}}\\}S^{{\overline{K}}{\underline{L}}}=\delta^{{\overline{I}}{\overline{K}}}S^{{\overline{M}}{\underline{J}}}S^{{\overline{M}}{\underline{L}}}-\delta^{{\underline{L}}{\underline{J}}}S^{{\overline{I}}{\underline{M}}}S^{{\overline{K}}{\underline{M}}}~{}.$
(6.15)
The simplest solution for $p=q\equiv{\cal N}$ is
$\displaystyle
S^{{\overline{I}}{\underline{J}}}=S\delta^{{\overline{I}}{\underline{J}}}~{},\qquad{\cal
D}_{A}S=0~{}.$ (6.16)
It corresponds to a special frame (or a special gauge condition) in which the
left and right $R$-symmetry connections coincide
$\displaystyle\Phi_{A}^{{\overline{I}}{\overline{K}}}\delta^{{\overline{K}}{\underline{J}}}=\delta^{{\overline{I}}{\underline{K}}}\Phi_{A}^{{\underline{K}}{\underline{J}}}~{}.$
(6.17)
This means that the two types of $R$-symmetry indices turn into a single type,
${\overline{I}}={\underline{I}}\equiv I$, and we stay with the diagonal
subgroup of the $R$-symmetry group $\mathsf{SO}({\cal
N})\times\mathsf{SO}({\cal N})$. The latter is generated by
$J^{IJ}=-J^{JI}=\mathfrak{L}^{IJ}+\mathfrak{R}^{IJ}$, which acts on isovectors
as follows
$\displaystyle J^{IJ}\chi^{K}=2\delta^{K[I}\chi^{J]}~{}.$ (6.18)
To summarise, the algebra of covariant derivatives for ${\cal N}$-extended
anti-de-Sitter superspace is
$\displaystyle\\{{\cal D}_{+}^{I},{\cal D}_{+}^{J}\\}$ $\displaystyle=$
$\displaystyle 2{\rm i}\delta^{IJ}{\cal D}_{++}~{},\quad\\{{\cal
D}_{-}^{I},{\cal D}_{-}^{J}\\}=2{\rm i}\delta^{IJ}{\cal D}_{--}~{},$ (6.19a)
$\displaystyle\\{{\cal D}_{+}^{I},{\cal D}_{-}^{J}\\}$ $\displaystyle=$
$\displaystyle-4\delta^{IJ}SM-2SJ^{IJ}~{},$ (6.19b) $\displaystyle\big{[}{\cal
D}_{+}^{I},{\cal D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle-2{\rm
i}S{\cal D}_{-}^{I}~{},\quad\big{[}{\cal D}_{-}^{I},{\cal D}_{++}\big{]}=2{\rm
i}S{\cal D}_{+}^{I}~{},$ (6.19c) $\displaystyle\big{[}{\cal D}_{++},{\cal
D}_{--}\big{]}$ $\displaystyle=$ $\displaystyle 8S^{2}M~{},$ (6.19d)
where it should be understood that $J^{IJ}$ is not present for ${\cal N}=1$.
The AdS curvature $S$ is related to the scalar curvature by
$\mathcal{R}=16S^{2}<0$. The isometry supergroup of this ${\cal N}$-extended
AdS superspace is $\mathsf{OSp}({\cal N}|2;{\mathbb{R}})$, see [16] for the
complete list of AdS2 supergroups.
The geometric structure of two-dimensional $(p,q)$ supersymmetric nonlinear
$\sigma$-models is remarkably rich, see [25, 48, 49, 50, 51] and references
therein; see also [52] for a recent review. Rigid superconformal
$\sigma$-models can be readily coupled to conformal supergravity. For non-
superconformal $\sigma$-models, their uplift to curved superspace may be
achieved by turning on a conformal compensator.
The formalism developed in this work may also be used to construct
supersymmetric extensions of the Gauss-Bonnet invariant by a generalisation of
four-dimensional logarithm construction of [53]. To this end, we consider a
nowhere vanishing primary scalar (super)field $\varphi$ of non-zero dimension
$\Delta$. From $\varphi$, one may construct the following primary descendants:
$\displaystyle{\cal N}=(0,0):$
$\displaystyle\qquad\nabla_{++}\nabla_{--}\,\text{ln}\,\varphi~{},$ (6.20a)
$\displaystyle{\cal N}=(1,0):$
$\displaystyle\qquad\nabla_{--}\nabla_{+}\,\text{ln}\,\varphi~{},$ (6.20b)
$\displaystyle{\cal N}=(1,1):$
$\displaystyle\qquad\frac{1}{2}[\nabla_{+},\nabla_{-}]\,\text{ln}\,\varphi~{}.$
(6.20c)
They can be used to define the (super)conformal functionals:
$\displaystyle{\cal N}=(0,0):$ $\displaystyle\qquad{\cal
S}_{(0,0)}=-\frac{1}{2\Delta}\int{\rm
d}^{2}x\,e\,\nabla_{++}\nabla_{--}\,\text{ln}\,\varphi=-\frac{1}{8}\int{\rm
d}^{2}x\,e\,{\cal R}~{},$ (6.21a) $\displaystyle{\cal N}=(1,0):$
$\displaystyle\qquad{\cal S}_{(1,0)}=-\frac{1}{2\Delta}\int{\rm
d}^{(2|1,0)}z^{-}\,E\,\nabla_{--}\nabla_{+}\,\text{ln}\,\varphi=\int{\rm
d}^{(2|1,0)}z^{-}\,E\,G_{-}~{},$ (6.21b) $\displaystyle{\cal N}=(1,1):$
$\displaystyle\qquad{\cal S}_{(1,1)}=-\frac{1}{4\Delta}\int{\rm
d}^{(2|1,1)}z\,E\,[\nabla_{+},\nabla_{-}]\,\text{ln}\,\varphi=\int{\rm
d}^{(2|1,1)}z\,E\,S~{}.$ (6.21c)
Where in the latter expressions we have degauged the Lagrangian and then
ignored all $\varphi$-dependent surface terms. Remarkably, these functionals
have proven to be independent of $\varphi$. Additionally, the first is simply
the Gauss-Bonnet invariant, while the latter two are its simplest
supersymmetric extensions.
Using the primary (super)fields (6.20) allows us to introduce manifestly
(super)conformal generalisations of the Fradkin-Tseytlin term in string theory
[54]
$\displaystyle S_{\rm FT}=\frac{1}{4\pi}\int{\rm d}^{2}x\,e\,{\cal
R}{\mbox{\boldmath$\Phi$}}~{},$ (6.22)
where $\Phi$ denotes the dilaton field in the curved spacetime in which the
string propagates. If we symbolically denote by $\Omega(\varphi)$ any of the
primaries in (6.20a) – (6.20c) and by $\int{\rm d}\mu$ the integration
measures in (6.21a) – (6.21c), then $I:=\int{\rm
d}\mu\,\Omega(\varphi){\mbox{\boldmath$\Phi$}}$ is invariant under the gauge
group of conformal supergravity for any dimensionless primary scalar $\Phi$.
Here $\varphi$ plays the role of a conformal compensator. Choosing a
(super-)Weyl gauge $\varphi=1$ leads to standard expressions for the Fradkin-
Tseytlin term and its supersymmetric extensions, modulo an overall numerical
coefficient.
The analysis above may be extended to the ${\cal N}=(2,2)$ case. To this end,
it is necessary to work in the complex basis of spinor covariant derivatives,
which is obtained from (5.13) upon the replacement ${\cal
D}\rightarrow\nabla$. This allows us to define two types of constrained
superfields, namely chiral superfields $\Phi$
$\displaystyle\bar{\nabla}_{+}\Phi=0~{},\qquad\bar{\nabla}_{-}\Phi=0~{},$
(6.23a) and twisted chiral superfields $\chi$
$\displaystyle\bar{\nabla}_{+}\chi=0~{},\qquad\nabla_{-}\chi=0~{},$ (6.23b)
where the Lorentz weights of $\Phi$ and $\chi$ are not indicated. Assuming
that they are primary, their superconformal properties are related as follows:
$\displaystyle q^{L}_{\Phi}$
$\displaystyle=\Delta_{\Phi}+\lambda_{\Phi}~{},\qquad
q^{R}_{\Phi}=\Delta_{\Phi}-\lambda_{\Phi}~{},$ (6.24a) $\displaystyle
q^{L}_{\chi}$ $\displaystyle=\Delta_{\chi}+\lambda_{\chi}~{},\qquad
q^{R}_{\chi}=\lambda_{\chi}-\Delta_{\chi}~{}.$ (6.24b)
Here $q^{L}_{\Phi}$ is defined by ${\rm
i}\mathfrak{L}^{\overline{1}\overline{2}}\Phi=q^{L}_{\Phi}\Phi$ and similarly
for $q^{R}_{\Phi}$.
We now specify to primary Lorentz scalars $\Phi$ and $\chi$ of non-zero
dimensions. From these superfields we may construct the primary descendants
$\displaystyle{\cal
N}=(2,2):\qquad\bar{\nabla}_{+}\bar{\nabla}_{-}\,\text{ln}\,\bar{\Phi}~{},\qquad\bar{\nabla}_{+}{\nabla}_{-}\,\text{ln}\,\bar{\chi}~{},$
(6.25)
which are chiral and twisted chiral, respectively. They may be used to define
the superconformal functionals
$\displaystyle\mathcal{S}^{\rm C}_{(2,2)}$
$\displaystyle=-\frac{1}{\Delta_{\Phi}}\int{\rm d}^{2}x{\rm
d}^{2}\theta\,{\cal
E}\,\bar{\nabla}_{+}\bar{\nabla}_{-}\,\text{ln}\,\bar{\Phi}=\int{\rm
d}^{2}x{\rm d}^{2}\theta\,{\cal E}\,\Xi^{\rm C}+\dots~{},$ (6.26a)
$\displaystyle\mathcal{S}^{\rm TC}_{(2,2)}$
$\displaystyle=\frac{1}{\Delta_{\chi}}\int{\rm d}^{2}x{\rm d}{\theta}^{+}{\rm
d}\bar{\theta}^{-}\,\mathfrak{E}\,\bar{\nabla}_{+}{\nabla}_{-}\,\text{ln}\,\bar{\chi}=\int{\rm
d}^{2}x{\rm d}{\theta}^{+}{\rm d}\bar{\theta}^{-}\,\mathfrak{E}\,\Xi^{\rm
TC}+\dots~{},$ (6.26b)
where ${\cal E}$ and $\mathfrak{E}$ are appropriately defined measures for the
chiral and twisted chiral subspaces, respectively, and we have made the
definitions
$\displaystyle\Xi^{\rm C}$ $\displaystyle:=S^{\overline{1}\underline{1}}+{\rm
i}S^{\overline{1}\underline{2}}+{\rm
i}S^{\overline{2}\underline{1}}-S^{\overline{2}\underline{2}}~{},\quad\quad\quad\bar{{\cal
D}}_{+}\Xi^{\rm C}=0~{},\quad\bar{{\cal D}}_{-}\Xi^{\rm C}=0~{},$ (6.27a)
$\displaystyle\Xi^{\rm TC}$ $\displaystyle:=S^{\overline{1}\underline{1}}-{\rm
i}S^{\overline{1}\underline{2}}+{\rm
i}S^{\overline{2}\underline{1}}+S^{\overline{2}\underline{2}}~{},\quad\quad\quad\bar{{\cal
D}}_{+}\Xi^{\rm TC}=0~{},\quad{{\cal D}}_{-}\Xi^{\rm TC}=0~{}.$ (6.27b)
It is not difficult to to show that the functionals (6.26a) and (6.26b) are
independent of $\bar{\Phi}$ and $\bar{\chi}$, respectively. It may also be
seen that these functionals are topological. Degauging the integrand in
(6.26a) gives
$\displaystyle\bar{\nabla}_{+}\bar{\nabla}_{-}\,\text{ln}\,\bar{\Phi}=-\Delta_{\Phi}\Xi^{\rm
C}+\bar{{\cal D}}_{+}\bar{{\cal
D}}_{-}\,\text{ln}\,\bar{\Phi}\equiv-\Delta_{\Phi}\Xi^{\rm C}+\dots~{},$
(6.28)
where the ellipsis denotes a chiral superfield for which we do not yet have an
explicit expression. Both functionals (6.26a) and (6.26b) define ${\cal
N}=(2,2)$ extensions of the Gauss-Bonnet invariant, eq. (6.21a). Component
analyses of these invariants will be given elsewhere.
Given primary dimensionless chiral $\Psi$ and twisted chiral $\Sigma$ scalars,
the following functionals
$\displaystyle\int{\rm d}^{2}x{\rm d}^{2}\theta\,{\cal
E}\,\Psi\bar{\nabla}_{+}\bar{\nabla}_{-}\,\text{ln}\,\bar{\Phi}~{},\qquad\int{\rm
d}^{2}x{\rm d}{\theta}^{+}{\rm
d}\bar{\theta}^{-}\,\mathfrak{E}\,\Sigma\bar{\nabla}_{+}{\nabla}_{-}\,\text{ln}\,\bar{\chi}$
(6.29)
are superconformal invariants. They may be viewed as ${\cal N}=2$
supersymmetric extensions of the Fradkin-Tseytlin term.
The above ${\cal N}=(2,2)$ constructions may also be generalised to the ${\cal
N}=(2,0)$ and ${\cal N}=(2,1)$ cases. In both cases it is necessary to
consider a scalar superfield $\Phi$ of dimension $\Delta$ which is chiral with
respect to the left coordinates, $\bar{\nabla}_{+}\Phi=0$. Using $\Phi$, we
construct the primary left chiral descendants
$\displaystyle{\cal N}=(2,0):$
$\displaystyle\qquad\bar{\nabla}_{+}\nabla_{--}\,\text{ln}\,\bar{\Phi}~{},$
(6.30) $\displaystyle{\cal N}=(2,1):$
$\displaystyle\qquad\bar{\nabla}_{+}\nabla_{-}\,\text{ln}\,\bar{\Phi}~{}.$
(6.31)
They may be used to define the superconformal functionals
$\displaystyle{\cal N}=(2,0):\quad\mathcal{S}_{(2,0)}$
$\displaystyle=-\frac{1}{2\Delta}\int{\rm d}^{2}x{\rm d}{\theta}^{+}\,{\cal
E}_{L}^{(2,0)}\,\bar{\nabla}_{+}{\nabla}_{--}\,\text{ln}\,\bar{\Phi}$
$\displaystyle=2\int{\rm d}^{2}x{\rm d}{\theta}^{+}\,{\cal
E}_{L}^{(2,0)}\,G_{-}~{},$ (6.32) $\displaystyle{\cal
N}=(2,1):\quad\mathcal{S}_{(2,1)}$
$\displaystyle=\frac{1}{\sqrt{2}\Delta}\int{\rm d}^{2}x{\rm d}{\theta}^{+}{\rm
d}{\theta}^{-}\,{\cal
E}_{L}^{(2,1)}\,\bar{\nabla}_{+}{\nabla}_{-}\,\text{ln}\,\bar{\Phi}$
$\displaystyle=\int{\rm d}^{2}x{\rm d}{\theta}^{+}{\rm d}{\theta}^{-}\,{\cal
E}_{L}^{(2,1)}\,\Xi^{\rm LC}+\dots~{},$ (6.33)
where ${\cal E}_{L}^{(2,0)}$ and ${\cal E}_{L}^{(2,1)}$ are the appropriate
integration measures, $G_{-}$ is defined in (5.32) and we have made the
definition
$\displaystyle\Xi^{\rm LC}$ $\displaystyle:=S^{\overline{1}}+{\rm
i}S^{\overline{2}}~{},\quad\quad\quad\bar{{\cal D}}_{+}\Xi^{\rm LC}=0~{}.$
(6.34)
Making use of the primary chiral descendants (6.33) and (6.31) allows us to
define $(2,0)$ and $(2,1)$ supersymmetric analogues of the Fradkin-Tseytlin
term. In both cases such invariants are associated with a primary
dimensionless chiral scalar $\Psi$ and have the explicit form
$\displaystyle{\cal N}=(2,0):\qquad$ $\displaystyle\int{\rm d}^{2}x{\rm
d}{\theta}^{+}\,{\cal
E}_{L}^{(2,0)}\,\Psi\bar{\nabla}_{+}{\nabla}_{--}\,\text{ln}\,\bar{\Phi}$
(6.35) $\displaystyle{\cal N}=(2,1):\qquad$ $\displaystyle\int{\rm d}^{2}x{\rm
d}{\theta}^{+}{\rm d}{\theta}^{-}\,{\cal
E}_{L}^{(2,1)}\,\Psi\bar{\nabla}_{+}{\nabla}_{-}\,\text{ln}\,\bar{\Phi}~{}.$
(6.36)
Recently, new supertwistor formulations were discovered for conformal
supergravity theories in diverse dimensions $3\leq d\leq 6$ [55]. It would be
interesting to extend this approach to the $d=2$ case.
Acknowledgements:
We are grateful to Stefan Theisen for discussions and for pointing out an
error in an earlier version of the manuscript. We thank Daniel Butter for a
question that has led to the material in appendix B. The work of SK is
supported in part by the Australian Research Council, project No. DP200101944.
The work of ER is supported by the Hackett Postgraduate Scholarship UWA, under
the Australian Government Research Training Program.
## Appendix A Conformal geometry in $d\geq 3$ dimensions
This appendix is devoted to a brief review of conformal gravity in $d\geq 3$
dimensions as the gauge theory of the conformal group $\mathsf{SO}(d,2)$. This
approach was pioneered in four dimensions in [23]. Our discussion is aimed at
elucidating the differences between the $d=2$ and $d>2$ cases. We closely
follow the presentations given in [20, 22].
The former is a gauge theory of the conformal algebra $\mathfrak{so}(d,2)$,
which is spanned by the translation ($P_{a}$), Lorentz ($M_{ab}$), dilatation
($\mathbb{D}$) and special conformal generators ($K^{a}$). Their non-vanishing
commutation relations are
$\displaystyle[M_{ab},M_{cd}]$
$\displaystyle=2\eta_{c[a}M_{b]d}-2\eta_{d[a}M_{b]c}~{},$ (A.1a)
$\displaystyle[M_{ab},P_{c}]$
$\displaystyle=2\eta_{c[a}P_{b]}~{},\quad[\mathbb{D},P_{a}]=P_{a}~{},$ (A.1b)
$\displaystyle[M_{ab},K_{c}]$
$\displaystyle=2\eta_{c[a}K_{b]}~{},\quad[\mathbb{D},K_{a}]=-K_{a}~{},$ (A.1c)
$\displaystyle[K_{a},P_{b}]$ $\displaystyle=2\eta_{ab}\mathbb{D}+2M_{ab}~{}.$
(A.1d)
It is convenient to group these generators into the two disjoint subsets
$P_{a}$ and $X_{\underline{a}}$:
$\displaystyle X_{\tilde{a}}=(P_{a},X_{\underline{a}})~{},\qquad
X_{\underline{a}}=(M_{ab},\mathbb{D},K^{a})~{}.$ (A.2)
Then, the conformal algebra (A.1) may be rewritten as follows
$\displaystyle[X_{\underline{a}},X_{\underline{b}}]$
$\displaystyle=-f_{\underline{a}\underline{b}}{}^{\underline{c}}X_{\underline{c}}\
,$ (A.3a) $\displaystyle[X_{\underline{a}},P_{{b}}]$
$\displaystyle=-f_{\underline{a}{{b}}}{}^{\underline{c}}X_{\underline{c}}-f_{\underline{a}{{b}}}{}^{{c}}P_{{c}}\
.$ (A.3b)
### A.1 Gauging the conformal algebra in $d\geq 3$ dimensions
Let $\mathcal{M}^{d}$ be a $d$-dimensional spacetime, $d\geq 3$, parametrised
by the local coodinates $x^{m}$, $m=0,1,\dots,d-1.$ To gauge the conformal
algebra (A.1) it is necessary to associate each non-translational generator
$X_{\underline{a}}$ with a connection one-form
$\omega^{\underline{a}}=(\omega,b,\mathfrak{f}_{a})={\rm
d}x^{m}\omega_{m}{}^{\underline{a}}$ and with $P_{a}$ a vielbein one-form
$e^{a}={\rm d}x^{m}e_{m}{}^{a}$, where it is assumed that $e:={\rm
det}(e_{m}{}^{a})\neq 0$, hence there exists a unique inverse vielbein
$e_{a}{}^{m}$:
$\displaystyle e_{a}{}^{m}e_{m}{}^{b}=\delta_{a}{}^{b}~{},\qquad
e_{m}{}^{a}e_{a}{}^{n}=\delta_{m}{}^{n}~{}.$ (A.4)
The latter may be used to construct the vector fields
$e_{a}=e_{a}{}^{m}\partial_{m}$, which constitute a basis for the tangent
space at each point of $\mathcal{M}^{d}$. It may then be used to express the
connection in the vielbein basis as
$\omega^{\underline{a}}=e^{b}\omega_{b}{}^{\underline{a}}$, where
$\omega_{b}{}^{\underline{a}}=e_{b}{}^{m}\omega_{m}{}^{\underline{a}}$.
The covariant derivatives have the form999We adopt the convention where a
factor of $1/2$ is inserted when performing a summations over pairs of
antisymmetric indices.
$\nabla_{a}=e_{a}{}^{m}\partial_{m}-\frac{1}{2}\omega_{a}{}^{bc}M_{bc}-b_{a}\mathbb{D}-\mathfrak{f}_{ab}K^{b}~{}.$
(A.5)
We note that the translation generators $P_{a}$ do not appear in the covariant
derivatives. Instead, we assume that they are replaced by $\nabla_{a}$ and
obey the commutation relations:
$\displaystyle[X_{\underline{a}},\nabla_{{b}}]$
$\displaystyle=-f_{\underline{a}{{b}}}{}^{\underline{c}}X_{\underline{c}}-f_{\underline{a}{{b}}}{}^{{c}}\nabla_{{c}}\
.$ (A.6)
By definition, the gauge group of conformal gravity is generated by local
transformations of the form
$\displaystyle\delta_{\mathscr{K}}\nabla_{a}$ $\displaystyle=$
$\displaystyle[\mathscr{K},\nabla_{a}]\ ,$ (A.7a) $\displaystyle\mathscr{K}$
$\displaystyle=$
$\displaystyle\xi^{b}\nabla_{b}+\Lambda^{\underline{b}}X_{\underline{b}}=\xi^{b}\nabla_{b}+\frac{1}{2}K^{bc}M_{bc}+\sigma\mathbb{D}+\Lambda_{b}K^{b}~{},$
(A.7b)
where the gauge parameters satisfy natural reality conditions. These gauge
transformations act on a conformal tensor field $\mathcal{U}$ (with its
indices suppressed) as
$\displaystyle\delta_{\mathscr{K}}\mathcal{U}={\mathscr{K}}\mathcal{U}~{}.$
(A.8)
Further, we will say that $\mathcal{U}$ is primary if (i) it is annihilated by
the special conformal generator, $K^{a}\mathcal{U}=0$; and (ii) it is an
eigenvector of $\mathbb{D}$. It will be said to have dimension $\Delta$ if
$\mathbb{D}{\cal U}=\Delta{\cal U}$.
Amongst themselves, the covariant derivatives satisfy the following
commutation relations
$[\nabla_{a},\nabla_{b}]=-{\cal T}_{ab}{}^{c}\nabla_{c}-\frac{1}{2}{\cal
R}(M)_{ab}{}^{cd}M_{cd}-{\cal R}(\mathbb{D})_{ab}\mathbb{D}-{\cal
R}(K)_{abc}K^{c}\ ,$ (A.9)
where we have made the definitions:
$\displaystyle\mathcal{T}_{ab}{}^{c}$
$\displaystyle=-\mathscr{C}_{ab}{}^{c}+2{\omega}_{[ab]}{}^{c}+2{b}_{[a}\delta_{b]}{}^{c}~{},$
(A.10a) $\displaystyle\mathcal{R}(M)_{ab}{}^{cd}$
$\displaystyle=R_{ab}{}^{cd}+8\mathfrak{f}_{[a}{}^{[c}\delta_{b]}{}^{d]}~{},$
(A.10b) $\displaystyle\mathcal{R}(K)_{abc}$
$\displaystyle=-\mathscr{C}_{ab}{}^{d}\mathfrak{f}_{dc}-2{\omega}_{[a|c|}{}^{d}\mathfrak{f}_{b]d}-2{b}_{[a}\mathfrak{f}_{b]c}+2e_{[a}\mathfrak{f}_{b]c}~{},$
(A.10c) $\displaystyle\mathcal{R}(\mathbb{D})_{ab}$
$\displaystyle=-\mathscr{C}_{ab}{}^{c}{b}_{c}+4\mathfrak{f}_{[ab]}+2e_{[a}{b}_{b]}~{},$
(A.10d) $\displaystyle R_{ab}{}^{cd}$
$\displaystyle=-\mathscr{C}_{ab}{}^{f}{\omega}_{f}{}^{cd}+2e_{[a}{\omega}_{b]}{}^{cd}-2{\omega}_{[a}{}^{cf}{\omega}_{b]f}{}^{d}~{}.$
(A.10e)
Here $R_{ab}{}^{cd}$ is the curvature tensor101010It should be emphasised
that, owing to its dependence on the dilatation connection, our curvature
tensor does not satisfy the Bianchi identity $R_{[abc]d}=0$ unless $b_{a}=0$,
see the following subsection. constructed from the Lorentz connection
$\omega_{a}{}^{bc}$ and we have introduced the anholonomy coefficients
$\mathscr{C}_{ab}{}^{c}$
$\displaystyle[e_{a},e_{b}]=\mathscr{C}_{ab}{}^{c}e_{c}~{}.$ (A.11)
In order for the above geometry to describe conformal gravity, it is necessary
to impose certain covariant constraints such that the only independent
geometric fields are the vielbein and dilatation connection. They are as
follows:
$\displaystyle{\cal T}_{ab}{}^{c}$ $\displaystyle=0~{},$ (A.12a)
$\displaystyle\eta^{bd}{\cal R}(M)_{abcd}$ $\displaystyle=0~{}.$ (A.12b)
The first constraint determines $\omega_{a}{}^{bc}$ in terms of the vielbein
and dilatation connection, while the second determines $\mathfrak{f}_{ab}$ to
be
$\mathfrak{f}_{ab}=-\frac{1}{2}P_{ab}=-\frac{1}{2(d-2)}R_{ab}+\frac{1}{4(d-1)(d-2)}\eta_{ab}R~{},$
(A.13)
where $P_{ab}$ is the Schouten tensor and we have defined
$R_{ac}=\eta^{bd}R_{abcd}\ ,\quad R=\eta^{ab}R_{ab}~{}.$ (A.14)
Inserting (A.13) into (A.10b) leads to the result that ${\cal
R}(M)_{ab}{}^{cd}$ is exactly the Weyl tensor
$R(M)_{ab}{}^{cd}=C_{ab}{}^{cd}=R_{ab}{}^{cd}-4P_{[a}{}^{[c}\delta_{b]}{}^{d]}~{},$
(A.15)
which is a primary field of dimension 2
$\displaystyle K_{e}C_{abcd}=0~{},\qquad\mathbb{D}C_{abcd}=2C_{abcd}~{}.$
(A.16)
It should be emphasised that, since $C_{abcd}$ is primary, it is independent
of the dilatation connection $b_{a}$.
Next, it is necessary to analyse the Bianchi identity
$[\nabla_{a},[\nabla_{b},\nabla_{c}]]+[\nabla_{c},[\nabla_{a},\nabla_{b}]]+[\nabla_{b},[\nabla_{c},\nabla_{a}]]=0~{},$
(A.17)
which leads to the identities
$\displaystyle{\cal R}(\mathbb{D})_{ab}$ $\displaystyle=0~{},$ (A.18a)
$\displaystyle{\cal R}(K)_{[abc]}$ $\displaystyle=0\ ,$ (A.18b) $\displaystyle
C_{[abc]}{}^{d}$ $\displaystyle=0\ ,$ (A.18c) $\displaystyle\nabla_{[a}{\cal
R}(K)_{bc]}{}^{d}$ $\displaystyle=0\ ,$ (A.18d)
$\displaystyle\nabla_{[a}C_{bc]}{}^{de}-4{\cal
R}(K)_{[ab}{}^{[d}\delta^{e]}_{c]}$ $\displaystyle=0\ .$ (A.18e)
In particular, constraint (A.18e) implies the important identity
$\frac{1}{2}\nabla_{c}C_{ab}{}^{ce}+(d-3){\cal R}(K)_{ab}{}^{e}-2{\cal
R}(K)_{c[a}{}^{c}\delta^{e}_{b]}=0\ .$ (A.19)
Thus, to continue our analysis it is necessary to consider the cases $d>3$ and
$d=3$ separately.
For $d>3$, it follows from (A.19) that the special conformal curvature takes
the form
$\displaystyle{\cal R}(K)_{abc}=\frac{1}{2(d-3)}\nabla^{d}C_{abcd}~{},$ (A.20)
which implies that the algebra of covariant derivatives is
$\displaystyle[\nabla_{a},\nabla_{b}]=-\frac{1}{2}C_{abcd}M^{cd}-\frac{1}{2(d-3)}\nabla^{d}C_{abcd}K^{c}~{}.$
(A.21)
To conclude our discussion of the $d>3$ case, we list the algebraic properties
of $C_{abcd}$:
$\displaystyle C_{abcd}=C_{[ab][cd]}=C_{cdab}~{},\qquad
C_{a[bcd]}=0~{},\qquad\eta^{bc}C_{abcd}=0~{}.$ (A.22)
The $d=3$ case is special because the Weyl tensor identically vanishes,
$C_{abcd}=0$. As a result, the algebra of covariant derivatives takes the form
$[\nabla_{a},\nabla_{b}]=-{\cal R}(K)_{abc}K^{c}~{},$ (A.23)
and conformal geometry of spacetime is controlled by the primary field ${\cal
R}(K)_{abc}$. One can show that this field takes the form
$\displaystyle{\cal R}(K)_{abc}=-{\cal
D}_{[a}P_{b]c}=-\frac{1}{2}W_{abc}~{},\qquad{\cal
D}_{a}=e_{a}{}^{m}\partial_{m}-\omega_{a}{}^{bc}M_{bc}~{},$ (A.24)
where we have introduced the Lorentz covariant derivative111111This definition
of ${\cal D}_{a}$ is valid for generic spacetime dimensions. ${\cal D}_{a}$
and the Cotton tensor $W_{abc}$. The latter proves to be a primary field of
dimension $3$,
$\displaystyle K_{d}W_{abc}=0~{},\qquad\mathbb{D}W_{abc}=3W_{abc}~{}.$ (A.25)
It is useful to introduce its dual
$\displaystyle W_{ab}=\frac{1}{2}\varepsilon_{acd}W^{cd}{}_{b}~{},$ (A.26)
which proves to be symmetric and traceless,
$\displaystyle W_{ab}=W_{ba}~{},\qquad\eta^{ab}W_{ab}=0~{},$ (A.27)
and also satisfies the conservation equation
$\displaystyle\nabla^{b}W_{ab}=0~{}.$ (A.28)
### A.2 Degauging to Lorentzian geometry
As mentioned above, the only independent geometric fields in this geometry
(for $d\geq 3$) are the vielbein and dilatation gauge field. Actually, the
latter proves to be a purely gauge degree of freedom. Specifically, it
transforms algebraically under special conformal transformations (A.7)
$\displaystyle\mathscr{K}(\Lambda)=\Lambda_{a}K^{a}\qquad\implies\qquad\delta_{\mathscr{K}(\Lambda)}b_{a}=-2\Lambda_{a}~{}.$
(A.29)
Hence, it is possible impose the gauge condition $b_{a}=0$ at the cost of
breaking special conformal symmetry.121212This process is known as
‘degauging’. As a result, the connection $\mathfrak{f}_{ab}$ is no longer
required for the covariance of $\nabla_{a}$ under the residual gauge freedom
and it may be manually extracted
$\displaystyle\nabla_{a}={\cal D}_{a}-\mathfrak{f}_{ab}K^{b}={\cal
D}_{a}+\frac{1}{2}P_{ab}K^{b}~{}.$ (A.30)
It may then be shown that the Lorentz covariant derivatives ${\cal D}_{a}$
satisfy the algebra
$\displaystyle[{\cal D}_{a},{\cal
D}_{b}]=-\frac{1}{2}\big{(}C_{ab}{}^{cd}+4P_{[a}{}^{c}\delta_{b]}{}^{d}\big{)}M_{cd}~{}.$
(A.31)
Next, it is important to describe the supergravity gauge freedom of this
geometry, which corresponds to the residual gauge transformations of (A.7) in
the gauge $b_{a}=0$. These include local $\mathcal{K}$-transformations of the
form
$\displaystyle\delta_{\mathcal{K}}{\cal D}_{A}$ $\displaystyle=$
$\displaystyle[\mathcal{K},{\cal D}_{A}]\ ,$ (A.32a)
$\displaystyle\mathcal{K}$ $\displaystyle=$ $\displaystyle\xi^{b}{\cal
D}_{b}+\frac{1}{2}K^{bc}M_{bc}~{},$ (A.32b) which acts on tensor superfields
$\mathcal{U}$ (with indices suppressed) as $\displaystyle\delta_{\cal K}{\cal
U}={\cal K}{\cal U}~{}.$ (A.32c)
The gauge transformations (A.32) prove to not be the most general conformal
gravity gauge transformations preserving the gauge $b_{a}=0$. Specifically, it
may be shown that the following transformation also enjoys this property
$\displaystyle\mathscr{K}(\sigma)=\sigma\mathbb{D}+\frac{1}{2}\nabla_{b}\sigma
K^{b}\quad\Longrightarrow\quad\delta_{\mathscr{K}(\sigma)}b_{a}=0~{},$ (A.33)
where $\sigma$ is real but otherwise unconstrained.
As a result, it is necessary to consider how this transformation manifests in
the degauged geometry
$\displaystyle\delta_{\mathscr{K}(\sigma)}\nabla_{a}\equiv\delta_{\sigma}\nabla_{a}=\delta_{\sigma}{\cal
D}_{a}-\delta_{\sigma}(\mathfrak{f}_{ab}K^{b})~{}.$ (A.34)
By a routine computation, we obtain
$\displaystyle\delta_{\sigma}{\cal D}_{a}$ $\displaystyle=\sigma{\cal
D}_{a}+{\cal D}^{b}\sigma M_{ba}~{},$ (A.35a)
$\displaystyle\delta_{\sigma}C_{abcd}$ $\displaystyle=2\sigma C_{abcd}~{},$
(A.35b) $\displaystyle\delta_{\sigma}P_{ab}$ $\displaystyle=2\sigma
P_{ab}-{\cal D}_{a}{\cal D}_{b}\sigma~{},$ (A.35c)
which are the standard Weyl transformations.
## Appendix B Conformal $(1,0)$ superspace with non-vanishing curvature
In section 4 we proposed conformal $(p,q)$ superspaces characterised by the
relations (4.9); all conformal curvatures were set to zero. This appendix is
devoted to deriving conformal $(1,0)$ superspace with non-vanishing curvature
as an extension of the non-supersymmetric geometry (3.15).
Guided by the construction in higher dimensions [18, 19, 20, 21, 22], we
require that the covariant derivatives
$\nabla_{A}=(\nabla_{+},\nabla_{++},\nabla_{--})$ obey constraints that are
similar to the super Yang-Mills theory
$\displaystyle\big{\\{}\nabla_{+},\nabla_{+}\big{\\}}=2{\rm
i}\nabla_{++}~{},\qquad\big{[}\nabla_{+},\nabla_{--}\big{]}={\rm
i}\mathscr{W}_{-}~{}.$ (B.1)
Here $\mathscr{W}_{-}=\mathscr{W}(X)_{-}{}^{\tilde{A}}X_{\tilde{A}}$, and
$X_{\tilde{A}}$ denotes the generators of the superconformal algebra (4.1).
The Bianchi identities yield
$\displaystyle\big{[}\nabla_{+},\nabla_{++}\big{]}=0~{},\qquad\big{[}\nabla_{++},\nabla_{--}\big{]}=\big{\\{}\nabla_{+},\mathscr{W}_{-}\big{\\}}~{}.$
(B.2)
Additionally, requiring consistency of (B.1) with the superconformal algebra
leads to
$\displaystyle\big{[}K^{A},\mathscr{W}_{-}\big{\\}}=0~{},\qquad\big{[}\mathbb{D},\mathscr{W}_{-}\big{]}=\frac{3}{2}\mathscr{W}_{-}~{},\qquad\big{[}M,\mathscr{W}_{-}\big{]}=-\frac{1}{2}\mathscr{W}_{-}~{}.$
(B.3)
Hence, we constrain $\mathscr{W}_{-}$ to be
$\displaystyle\mathscr{W}_{-}=\chi_{+}K^{++}+\psi_{---}K^{--}~{},$ (B.4)
where $\chi_{+}$ and $\psi_{---}$ are primary superfields of dimension $5/2$.
At the component level, they contain the conformal curvatures $W_{++}$ and
$W_{--}$ (3.15)
$\displaystyle W_{++}=\nabla_{+}\chi_{+}|_{\theta^{+}=0}~{},\qquad
W_{--}=\nabla_{+}\psi_{---}|_{\theta^{+}=0}~{}.$ (B.5)
Finally, inserting (B.4) into (B.2), we obtain the commutator of vector
derivatives
$\displaystyle\big{[}\nabla_{++},\nabla_{--}\big{]}={\rm
i}\chi_{+}S^{+}+(\nabla_{+}\chi_{+})K^{++}+(\nabla_{+}\psi_{---})K^{--}~{}.$
(B.6)
We note that, according to (4.6), the dilatation connection transforms
algebraically under infinitesimal special superconformal gauge transformations
(4.1). This allows us to fix the gauge $B_{A}=0$ at the expense of this
symmetry. In this gauge the special conformal connection $\mathfrak{F}_{AB}$
is not required for the covariance of $\nabla_{A}$ and may be separated
$\displaystyle\nabla_{A}={\cal D}_{A}-\mathfrak{F}_{AB}K^{B}~{}.$ (B.7)
Here the degauged covariant derivative ${\cal D}_{A}$ involves only the
Lorentz connection and obeys the algebra (5.45). The connection
$\mathfrak{F}_{AB}$ was described in (5.44a) and (5.44b). The constraints
(5.44c) are now replaced with
$\displaystyle\psi_{---}={\cal D}_{--}G_{-}-{\cal
D}_{+}\mathfrak{F}_{--,--}~{},\qquad\chi_{+}=-{\cal D}_{++}G_{-}+{\cal
D}_{--}\mathfrak{F}_{+,++}~{}.$ (B.8)
These relations determine the curvature tensors $\psi_{---}$ and $\chi_{+}$ in
terms of ${\cal D}_{A}$ and $\mathfrak{F}_{AB}$. Keeping (B.5) in mind, it is
clear that this is a $(1,0)$ extension of (3.23). In particular, for vanishing
$\psi_{---}$ and $\chi_{+}$, the connections $\mathfrak{F}_{--,--}$ and
$\mathfrak{F}_{+,++}$ are non-local functions of the supergravity multiplet.
## Appendix C Compactified Minkowski superspace
Superconformal groups (6.12) do not act on Minkowski superspace, since the
special conformal and $S$-supersymmetry transformations are singular at some
points. However, there exists a well defined action of (6.12) on a
compactified $(p,q)$ Minkowski superspace $\overline{\mathbb{M}}^{(2|p,q)}$,
for some $p,q$. In general, $\overline{\mathbb{M}}^{(2|p,q)}$ has the form
$\displaystyle\overline{\mathbb{M}}^{(2|p,q)}=S^{1|p}_{L}\times
S^{1|q}_{R}~{},$ (C.1)
where the bosonic body of $S^{1|n}$ is a circle $S^{1}$. The left superspace
$S^{1|p}_{L}$ is a homogeneous space of the subgroup $G_{L}$ of (6.12), and
similarly in the right sector.
In a recent paper [15], $\overline{\mathbb{M}}^{(2|p,q)}$ was realised as a
homogeneous space of the superconformal group
${\mathsf{OSp}}_{0}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}_{0}(q|2;{\mathbb{R}})$.
Here we present a different construction for the case that $p$ is even,
$p=2n$. Specifically, we describe $\overline{\mathbb{M}}^{(2|2n,q)}$ as a
homogeneous space of the superconformal group
$\displaystyle G=G_{L}\times
G_{R}=\mathsf{SU}(1,1|n)\times{\mathsf{OSp}}_{0}(q|2;{\mathbb{R}})~{}.$ (C.2)
We begin by describing the action of $\mathsf{SU}(1,1|n)$ on $S^{1|2n}$. The
supergroup $\mathsf{SU}(1,1|n)$ is spanned by supermatrices of the form
$\displaystyle g\in\mathsf{SL}(2|n;{\mathbb{C}})~{},\qquad
g^{\dagger}\,\Omega\,g=\Omega~{},\qquad\Omega=\left(\begin{array}[]{cc|c}1&0{}&0\\\
0&-1{}&0\\\ \hline\cr 0&0&{\mathbbm{1}}_{n}\end{array}\right)~{}.$ (C.6)
This supergroup naturally acts on the space of even supertwistors
${\mathbb{C}}^{2|n}$
$\displaystyle X=\left(\begin{array}[]{c}z\\\ w\\\
\hline\cr\varphi^{i}\end{array}\right)~{},\qquad i=1,\dots,n~{},$ (C.10)
where $z,w$ are complex bosonic variables, and $\varphi^{i}$ complex Grassmann
variables. We identify $S^{1|2n}$ with the space of null lines in
${\mathbb{C}}^{2|n}$. By definition, a null supertwistor $X$ is characterised
by the conditions
$\displaystyle X^{\dagger}\Omega X=0~{},\qquad\left(\begin{array}[]{c}z\\\
w\end{array}\right)\neq 0~{}.$ (C.13)
Two null supertwistors $X$ and $X^{\prime}$ are said to be equivalent if
$\displaystyle
X^{\prime}={\mathfrak{c}}X~{},\qquad{\mathfrak{c}}\in{\mathbb{C}}\setminus\\{0\\}~{}.$
(C.14)
Any equivalence class in the set of null supertwistors is called a null line.
Given a null supertwistor $X$ both bosonic components $z$ and $w$ are non-
zero. Making use of the equivalence relation (C.14) allows us to choose, for
each null line, a representative
$\displaystyle X=\left(\begin{array}[]{c}z\\\ 1\\\
\hline\cr\varphi^{i}\end{array}\right)~{},\qquad|z|^{2}=1-\varphi^{\dagger}\varphi~{},$
(C.18)
which is uniquely defined for the null line under consideration. It is seen
that the quotient space is $S^{1|2n}$.
In order to make contact to ordinary Minkowski superspace, it is useful to
switch to a different parametrisation of $\mathsf{SU}(1,1|n)$ and the
associated supertwistor space. Let us introduce the supermatrix
$\displaystyle\Sigma=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cr|c}{1{}}&-{1{}}&0\\\
1{}&1{}&0\\\ \hline\cr
0&0{}&\sqrt{2}\,{\mathbbm{1}}_{n}\end{array}\right)~{},\qquad\Sigma^{\dagger}\Sigma={\mathbbm{1}}_{2+n}~{},$
(C.22)
and associate with it the following similarity transformation:
$\displaystyle g~{}$ $\displaystyle\to$
$\displaystyle~{}\hat{g}=\Sigma\,g\,\Sigma^{-1}~{},\quad
g\in\mathsf{SU}(1,1|n)~{};\qquad X~{}\to~{}\hat{X}=\Sigma\,T~{},\quad
X\in{\mathbb{C}}^{2|n}~{}.$ (C.23)
The supertwistor metric $\Omega$ turns into
$\displaystyle\hat{\Omega}=\Sigma\,\Omega\,\Sigma^{-1}=\left(\begin{array}[]{cc|c}0&{1{}}&0\\\
{1}&0{}&0\\\ \hline\cr 0&0&{\mathbbm{1}}_{n}\end{array}\right)~{}.$ (C.27)
In the new frame, it is not guaranteed that both bosonic components $\hat{z}$
and $\hat{w}$ of a null supertwistor $\hat{X}$ are non-zero. However, at least
one of $\hat{z}$ and $\hat{w}$ is non-vanishing, and we can introduce an open
subset of $S^{1|p}$ which is parametrised by null supertwistors of the form
$\displaystyle\hat{X}=\left(\begin{array}[]{c}1\\\ -{\rm
i}{\mbox{\boldmath$x$}}^{++}\\\
\hline\cr\sqrt{2}\theta^{+i}\end{array}\right)~{},\qquad{\mbox{\boldmath$x$}}^{++}-\bar{\mbox{\boldmath$x$}}^{++}=2{\rm
i}\bar{\theta}^{+}_{i}\theta^{+i}~{},\quad\bar{\theta}^{+}_{i}:=\overline{\theta^{+i}}~{}.$
(C.31)
The constraint on ${\mbox{\boldmath$x$}}^{++}$ is solved by
$\displaystyle{\mbox{\boldmath$x$}}^{++}={x}^{++}+{\rm
i}\bar{\theta}^{+}_{i}\theta^{+i}~{},\qquad\overline{x^{++}}=x^{++}~{}.$
(C.32)
The variables ${\mbox{\boldmath$x$}}^{++}$ and $\theta^{+i}$ parametrise a
chiral subspace of $\mathbb{M}^{(2|2n,q)}$. To deduce the superconformal
transformations of this subspace it is necessary to act on $\hat{X}$ with a
generic group element $\hat{g}\in\mathsf{SU}(1,1|n)$:
$\displaystyle\hat{g}={\rm
e}^{\Lambda_{L}}~{},\qquad\Lambda_{L}=\left(\begin{array}[]{cc|c}-\frac{1}{2}(\sigma+K)-\frac{{\rm
i}n}{n-2}\chi&{{\rm i}b_{++}{}}&\sqrt{2}\eta_{+j}\\\ {-{\rm
i}a^{++}}&\frac{1}{2}(\sigma+K)-\frac{{\rm
i}n}{n-2}\chi{}&\sqrt{2}\bar{\epsilon}^{+}_{j}\\\
\hline\cr\sqrt{2}\epsilon^{+i}&\sqrt{2}\bar{\eta}^{i}_{+}&\lambda^{i}{}_{j}-\frac{2{\rm
i}n}{n-2}\chi\delta^{i}_{j}\end{array}\right)~{},$ (C.36)
where all scalar and vector parameters are real and
$\displaystyle\lambda^{\dagger}=-\lambda~{},\qquad{\rm tr}\ \lambda=0~{}.$
(C.37)
Taking the parameters to be small, one may show that the most general
infinitesimal superconformal transformations on this subspace are:
$\displaystyle\delta\mbox{\boldmath$x$}^{++}$
$\displaystyle=(\sigma+K)\mbox{\boldmath$x$}^{++}+a^{++}+2{\rm
i}\bar{\epsilon}_{i}^{+}\theta^{+i}-\mbox{\boldmath$x$}^{++}b_{++}\mbox{\boldmath$x$}^{++}-2\mbox{\boldmath$x$}^{++}\eta_{+i}\theta^{+i}~{},$
(C.38a) $\displaystyle\delta\theta^{+i}$
$\displaystyle=\frac{1}{2}(\sigma+K)\theta^{+i}-\frac{{\rm
i}n\chi}{n-2}\theta^{+i}+\epsilon^{+i}+\lambda^{i}{}_{j}\theta^{+j}-\theta^{+i}b_{++}\mbox{\boldmath$x$}^{++}$
$\displaystyle\phantom{=}-{\rm
i}\bar{\eta}_{+}^{i}\mbox{\boldmath$x$}^{++}-2\theta^{+i}\eta_{+j}\theta^{+j}~{}.$
(C.38b)
The constant bosonic parameters in (C.38) correspond to dilatations
$(\sigma)$, Lorentz transformations $(K)$, spacetime translations $(a^{++})$,
special conformal transformations $(b_{++})$, chiral transformations $(\chi)$
and $\mathsf{SU}(n)$ rotations $(\lambda^{i}{}_{j})$. The constant fermionic
parameters correspond to $Q$-supersymmetry $(\epsilon^{+i})$ and
$S$-supersymmetry $(\eta_{+j})$ transformations.
Next, we consider the action of $\mathsf{OSp}_{0}(q|2;\mathbb{R})$ on
$S^{1|q}$, see [15] for more details. This supergroup is spanned by
supermatrices of the form
$\displaystyle h\in\mathsf{SL}(2|q;{\mathbb{R}})~{},\qquad h^{\rm
sT}\,\mathbb{J}\,h=\mathbb{J}~{},\qquad\mathbb{J}=\left(\begin{array}[]{cc|c}0&1{}&0\\\
-1&0{}&0\\\ \hline\cr 0&0&{{\rm i}\mathbbm{1}}_{n}\end{array}\right)~{},$
(C.42)
and naturally acts on the space of even supertwistors ${\mathbb{R}}^{2|q}$
$\displaystyle Y=\left(\begin{array}[]{c}a\\\ b\\\
\hline\cr\sigma^{{\underline{I}}}\end{array}\right)~{},\qquad{\underline{I}}=\underline{1},\dots,\underline{q}~{},$
(C.46)
where $a,b$ denote real bosonic variables which are not both zero, and are
$\sigma^{\underline{I}}$ real Grassmann variables. Two supertwistors $Y$ and
$Y^{\prime}$ are equivalent if $Y^{\prime}=\gamma Y$, where
$\gamma\in\mathbb{R}\setminus\\{0\\}$. Assuming that $a\neq 0$, we can choose
the representative
$\displaystyle Y=\left(\begin{array}[]{c}1\\\ -x^{--}\\\ \hline\cr{\rm
i}\theta^{-{\underline{I}}}\end{array}\right)~{},$ (C.50)
where $(x^{--},\theta^{-{\underline{I}}})$ constitute inhomogeneous
coordinates for $S^{1|q}$. To deduce the superconformal transformations of
this subspace it is necessary to act on $Y$ with a generic group element
$h\in{\mathsf{OSp}}_{0}(q|2;{\mathbb{R}})$
$\displaystyle h={\rm
e}^{\Lambda_{R}}~{},\qquad\Lambda_{R}=\left(\begin{array}[]{cc|c}-\frac{1}{2}(\sigma-K)&{-b_{--}{}}&-\eta_{-}^{{\underline{J}}}\\\
{-a^{--}}&\frac{1}{2}(\sigma-K)&\sqrt{2}{\epsilon}^{-{\underline{J}}}\\\
\hline\cr{\rm i}\epsilon^{-{\underline{I}}}&{\rm
i}{\eta}^{{\underline{I}}}_{-}&\rho^{{\underline{I}}{\underline{J}}}\end{array}\right)~{}.$
(C.54)
Here all parameters are real and
$\rho^{{\underline{I}}{\underline{J}}}=-\rho^{{\underline{J}}{\underline{I}}}$.
Taking the parameters to be small, it may be shown that the most general
infinitesimal superconformal transformations on this subspace are:
$\displaystyle\delta x^{--}$
$\displaystyle=(\sigma-K)x^{--}+a^{--}-x^{--}b_{--}x^{--}+{\rm
i}\epsilon^{-{\underline{I}}}\theta^{-{\underline{I}}}+{\rm
i}x^{--}\eta_{-}^{{\underline{I}}}\theta^{-{\underline{I}}}~{},$ (C.55a)
$\displaystyle\delta\theta^{-{\underline{I}}}$
$\displaystyle=\frac{1}{2}(\sigma-K)\theta^{-{\underline{I}}}-\theta^{-{\underline{I}}}b_{--}x^{--}+\epsilon^{-{\underline{I}}}+\rho^{{\underline{I}}{\underline{J}}}\theta^{-{\underline{J}}}-\eta_{-}^{{\underline{I}}}x^{--}-{\rm
i}\theta^{-{\underline{I}}}\eta_{-}^{{\underline{J}}}\theta^{-{\underline{J}}}~{}.$
(C.55b)
The constant bosonic parameters in (C.55) correspond to dilatations
$(\sigma)$, Lorentz transformations $(K)$, spacetime translations $(a^{--})$,
special conformal transformations $(b_{--})$ and $\mathsf{SO}(q)$ rotations
$(\rho^{{\underline{I}}{\underline{J}}})$. The constant fermionic parameters
correspond to $Q$-supersymmetry $(\epsilon^{-{\underline{I}}})$ and
$S$-supersymmetry $(\eta_{-}^{{\underline{I}}})$ transformations. We emphasise
that the parameters $\sigma$ and $K$ in (C.38) and (C.55) are the same.
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|
∎
11institutetext: Devansh Mehta 22institutetext: Voicedeck Technologies
22email<EMAIL_ADDRESS>33institutetext: Kalika Bali 44institutetext:
Microsoft Research India
44email<EMAIL_ADDRESS>
# Learnings from Technological Interventions in a Low Resource Language
Enhancing Information Access in Gondi
Devansh Mehta∗∗ ** Equal Contribution Harshita Diddee∗11footnotemark: 1 Ananya
Saxena Anurag Shukla Sebastin Santy Ramaravind Kommiya Mothilal Brij Mohan
Lal Srivastava Alok Sharma Vishnu Prasad Venkanna U Kalika Bali
(Received: date / Accepted: date)
###### Abstract
The primary obstacle to developing technologies for low-resource languages is
the lack of representative, usable data. In this paper, we report the
deployment of technology-driven data collection methods for creating a corpus
of more than 60,000 translations from Hindi to Gondi, a low-resource
vulnerable language spoken by around 2.3 million tribal people in south and
central India. During this process, we help expand information access in Gondi
across 2 different dimensions (a) The creation of linguistic resources that
can be used by the community, such as a dictionary, children’s stories, Gondi
translations from multiple sources and an Interactive Voice Response (IVR)
based mass awareness platform; (b) Enabling its use in the digital domain by
developing a Hindi-Gondi machine translation model, which is compressed by
nearly 4 times to enable it’s edge deployment on low-resource edge devices and
in areas of little to no internet connectivity. We also present preliminary
evaluations of utilizing the developed machine translation model to provide
assistance to volunteers who are involved in collecting more data for the
target language. Through these interventions, we not only created a refined
and evaluated corpus of 26,240 Hindi-Gondi translations that was used for
building the translation model but also engaged nearly 850 community members
who can help take Gondi onto the internet.
###### Keywords:
Low-Resource Languages, Deployment, Applications
††journal: Language Resources and Evaluation
## 1 Introduction
In the present era of globalization and integration of technology into almost
every aspect of life, native speakers are turning to dominant languages at a
faster rate than ever before. As seen in Figure1, around 40% of all languages
in the world face the danger of extinction in the near future 31 , while 95%
have lost the capacity to ascend to the digital realm kornai2013digital . When
a language spoken in a particular community dies out, we lose a vital part of
the culture that is necessary to completely understand it. At a bare minimum,
languages need to be integrated with the Internet to give them a fighting
chance at survival.
In this work, we design, deploy and critically assess technological
interventions that aim to create a digital ecosystem of Gondi content, a
South-Central Dravidian tribal language spoken by the Gond tribe in Central
India. The case of Gondi is representative of many languages across the world
and presents a unique case study of how a language can be in danger despite
having all the ingredients of sustainability such as (1) long historical
continuity (2) a population of 3 million people speaking it and (3) widely
spoken in around 6 states of India with various dialects and forms. The
complexities arise as Gondi is a predominantly spoken language with no single
standard variety but a number of dialects, some mutually unintelligible
beine1994sociolinguistic . Our objectives with these interventions is creating
a repository of linguistic resources in Gondi that can be used for: (1)
building language technologies like machine translation or speech to text
systems that are essential for taking Gondi onto the internet; and (2)
expanding the information available to the Gond community in their language.
There needs to be groundwork and identification of the real problems that can
be solved by deployment of language technologies in minority communities
dearden2015ethical , as there are legitimate technological and ethical
concerns surrounding the use of technology in low-resource languages
joshi2019unsung ; 10.1145/3530190.3534792 . Technical systems aren’t simply
ethically neutral exchanges of information and those that work in isolation
from social contexts can be detrimental to the ecosystem of minority language
communities. At the same time, inaction on the part of language technologists
is an ethically dubious proposition in its own right, as minority communities
are currently forced to learn mainstream languages for availing the benefits
of the Internet and to access power structures enabling socio-economic
mobility. We believe it to be fairly non-controversial to maintain that anyone
should be able to surf the internet in the language that they are comfortable
in. Ethical issues only arise on the methods one uses to reach this goal.
Figure 1: UNESCO 2017 World Atlas: Demonstrating the endangered state of the
languages of the world.
Our focus when working with the Gond community is centered around devising
novel approaches for creation, evaluation and dissemination of Gondi content,
unlike well-resourced languages where the focus is more on engineering
language technologies using abundantly available data. As far as possible, we
have ensured that the technology interventions described in this study were
led by the Gond community and non-profit organizations working with them.
The first intervention is creating a 3,500 word Gondi dictionary that is
accessible to the community as an Android app. The second is 230 children’s
books that were translated by the Gond community in a 10 day workshop. The
third is crowdsourcing translations from community volunteers via an Android
app, the WhatsApp Business API and an assistive translation tool. The fourth
intervention is a phone number that community members could call to gain
awareness about a local election in their area, upon completion of which they
earn mobile credits. These interventions respectively resulted in facilitating
80-100 community members to create a 3,500 word dictionary; 20 community
members that translated 230 children’s stories from Hindi to Gondi; 219
community members that translated nearly 55,000 sentences; and 557 native
speakers of Gondi that learned about elections in their own language and who
can be called for future workshops.
This data was used to develop a Hindi-Gondi Machine Translation
Model111https://github.com/microsoft/INMT-lite, compressed for streamlined
operation on low-resource edge devices for use by the community. The model was
evaluated by community members that engaged with us through these
interventions and got an average Direct Assessment score specia-
etal-2020-findings-wmt of 63, indicating the semantic accuracy of our model’s
output. We also run preliminary experiments for evaluating the model’s ability
in assisting community members with translations. Our results show promising
efficacy with more than 50% of the suggestions being accepted.
## 2 Context
According to the 2011 census chandramouli2011census , the total population of
the Gond tribe is approximately 11.3 million. However, the total Gondi
speaking population is only around 2.7 million. That is, only about 25 percent
of the entire tribe now speaks it as a first language. UNESCO’s Atlas of the
Worlds Languages in Danger moseley2010atlas lists Gondi as belonging to the
vulnerable category. There is an added difficulty of creating resources for
Gondi due to the linguistic heterogeneity within the Gond community. As seen
in Figure2, Gondi is spread over 6 states in India. It is heavily influenced
by the dominant language of each state to the point where a Gond Adivasi from
Telangana (a Telugu speaking Southern state), finds it difficult to understand
a Gond Adivasi from Madhya Pradesh (a Central state with Hindi as the dominant
language). Recent scholarship has found the influence of these dominant
languages to be a major factor in language loss in the Gond community
boruah2020language .
Figure 2: Gondi speaking areas in India
As a predominantly oral language, the proportion of Gondi speakers is expected
to go down further as opportunities are shrinking to hear the language spoken
outside of their everyday surroundings. All India Radio, the only radio
station in India allowed to broadcast news, does not have regular Gondi news
bulletins. There is no TV station or channel catering to Gondi speakers. There
is also a severe dearth of online content in Gondi, resulting in members of
the tribe having to learn a mainstream language to enjoy the benefits of
internet connectivity. “The Wire”222Gondi Bulletin:
https://youtu.be/M3q2ycJ_U7g is one of the few Indian news outlet that
publishes a news bulletin in Gondi. However, the Gondi spoken in their
broadcast caters to the Gond Adivasis from the Central states of India and it
is difficult for Gonds from the Southern states to understand the content.
Gondi is also not included in the 8th Schedule of the Indian Constitution,
with the result that education and exams for government jobs cannot be
administered in the language. The deleterious effects of the marginalization
of Gondi on their community are manifold. Gondi is considered the lingua
franca of the local insurgents, who use their knowledge of the language and
the perceived neglect by the government to recruit candidates from the tribe
to join their civil war against the state kumar2019gondi . Further, there are
high dropout rates among monolingual children that speak Gondi as a first
language. A UNESCO study found that children whose mother tongue is not the
medium of instruction at primary school are more likely to fail in early
grades or drop out buhmann2008mother , which in turn increases the chances of
them joining the insurgency. Working with the Gond tribes on reviving their
language is thus important not just for cultural reasons, but may also serve
as an instrument for bringing peace to their society. These factors can be
mitigated provided there is political will and a standardized version of the
language that is agreed upon. While the younger, bilingual generation in the
Gond tribe attach less importance to their mother tongue, it was not difficult
to find community members worried about the extinction of their language and
it proved easy to recruit participants for the workshops we held, even without
any financial incentive.
## 3 Related Work
The first section focuses on how we address the constraints of efficiently
collecting data through traditional channels for low-resource languages. The
second section situates our study within existing language preservation and
revitalization programs and lays out our unique contributions to this prior
literature.
##### Data Collection Channels
Traditional channels of data collection can vary across 3 dimensions (a)
Channel of Data Collection (b) Quality of Collected Data and (c) Quantity of
Collected Data. There are two common channels of data collections: automated
mining and human reliant crowdsourcing. Mining includes a wide spectrum of
methods varying from crawling data off the web to utilizing content after
digitization by systems such as Optical Character Recognition. Unfortunately,
these channels assume the existence of data resources in a specific form: for
example, Automatic or Semi Automatic methods like Web mining expects the data
to be digitally readable, while OCR expects data to be digitally consumable by
an existing service, which might not be the case for many languages,
especially low-resource ones rijhwani2020ocr .
In our study, we rely on crowdsourcing large quantities of data through human
reliant channels. This approach is fraught with logistical challenges around
participant recruitment nangia2021ingredients , the cost of response
validation, and mitigating subjectivity in responses goldman-
etal-2018-strategies , as well as ethical considerations pertaining to
devising fair pay kummerfeld-2021-quantifying shmueli2021beyond and
maintaining response diversity by controlling for human bias in the inputs
provided by a crowdsourced audience liu-etal-2022-toward . Our study adds
novel insights to this prior literature by building a dataset from community
volunteers and staff at partner organizations without the provision of any
fiscal incentive, which is often a given attribute of human reliant data
crowdsourcing channels. We find that that a volunteer led workforce for data
generation can yield higher quality datasets by self-selecting for those
motivated to preserve their language.
Prior work has identified quality constraints for corpora arising out of
human-collection channels 10.1162/tacl_a_00447 caswell-etal-2020-language .
Specifically, noise derivative of code-mixing, non-linguistic characters,
short/meta-data segments, untranslated/copied content has been shown to
considerably harm the output of neural machine translation (NMT) systems
khayrallah2018impact . For high resource languages, the sheer quantity of a
corpus is usually significant enough to allow the selection of a subset of
high-quality sentences from the larger set using intelligent validation
techniques, especially since some percentage of noisy data is shown to improve
generalization robustness of NMT models. Training and evaluation corpora in
low-resource languages may not be as effective due to the paucity of data
10.1162/tacl_a_00447 . We observe this explicitly with our interventions,
since a large proportion of our data (nearly 60K data samples that we
collected) were sifted to a set of 27K samples, greatly reducing the effective
yield of our collection channels. Our work thus presents a case study on
developing alternative methods of data collection and evaluation for low-
resource languages.
##### Language Technologies for Revitalization and Preservation
Preservation programs generally attend to the need to develop data corpora in
that language abraham2020crowdsourcing , article , mus2021toward . However,
the notion that language documentation or artifact creation can independently
bring about revitalization is largely dismissed by the community zariquiey-
etal-2022-cld2 . Community-centric technical interventions that can infuse the
language technology into the community for sustained use are increasingly
coming under focus. One dimension of this focus has been towards developing
fundamental NLP tools for low-resource languages such as Part-Of-Speech
taggers finn-etal-2022-developing , parsers bowers-etal-2017-morphological ,
etc. Another dimension is the development of more end-user oriented language
technologies such as edge-deployments bettinson-bird-2017-developing
hermes2013ojibwe bird2014aikuma and interactive interfaces that can be
consumed rapidly by the community little-2017-connecting adams-etal-2021-user
. A critical point to note here is that access bottlenecks can limit the
consumption of outcomes from revitalization and preservation endeavours. For
instance, Roche highlights the political nature of language endangerment,
pointing out that low-resource language communities are usually under-
resourced in other aspects of life roche2020abandoning . Leibling et al
10.1145/3313831.3376261 discuss these constraints across the adaptation of
translation systems to edge devices across three globally distributed and
divergent demographic groups. We contribute to this identified gap by creating
a software artifact that displays NMT translations on low cost devices like
smartphones, which can be used by other speech communities and researchers
working with marginalized populations.
Due to the well-studied constraints of data collection through traditional
channels elazar-etal-2020-extraordinary millour-fort-2020-text liu-
etal-2022-toward , our work explores newer mechanisms for collecting low-
resource language data that are of wider relevance. For example, our Adivasi
Radio translation app allowed users to bulk download a corpus of sentences for
translation, which they could then complete even from areas without Internet.
Another area our work sheds new light on is reducing the cognitive burden
faced by communities in understanding an entirely novel interface for data
provision, as discussed in prior work like 10.1145/2858036.2858448
10.1145/2556288.2557155 . Our study uses low friction approaches for
collecting translations such as through the WhatsApp Business API, which is
well integrated into many communities. Lastly, our work briefly touches upon
the attempt to integrate assistive translation interfaces to provide sub-
optimal help in data collection efforts 10.1145/3290605.3300461
10.1145/3530190.3534792 .
Overall, this paper is a case study in collaborating with target communities
for creating linguistic resources to enable the development of language
technologies that can then give rise to a digital ecosystem for such
languages. In this regard, we follow the lead set by Kornai and Bhattacharyya
in building online tools for low-resource languages like spellchecks that can
be incrementally refined over time kornai2014indian . Our work distinguishes
itself by empirically studying methods of community engagement that are
cognizant of the constraints faced in low-resource environments.
## 4 Technological Interventions
The larger framework of our interventions is that language resource creation
feeds into building language technologies and enhancing access to information
in that language. We rolled out a series of interventions that created
linguistic resources useful to the community, such as children’s stories,
which in turn fed into building language technologies such as a Hindi-Gondi
translation system. These initiatives helped increase information access in
Gondi while supporting the end goal of taking Gondi onto the Internet.
### 4.1 Gondi Dictionary Development
This section discusses the need and complexity of building a common dictionary
across the various Gondi dialects.
#### 4.1.1 Motivation
Several researchers have found that mutual intelligibility of Gondi decreases
with distance, in part due to the influence of dominant state languages
creeping into the various dialects beine1994sociolinguistic ;
shapiro1981language ; tyler1969koya . Beine (1994) conducted mutual
intelligibility tests across the Gondi speaking areas and found there to be 7
mutually unintelligible dialects of the Gondi language (pp 89). He thus
recommended the creation of dialect centres to cover each Gondi speaking
region, with literacy materials separately developed for each center. These
workshops helped determine whether the community wanted separate efforts for
each dialect or a common effort towards one language.
#### 4.1.2 Intervention
Prior to the involvement of our research team, a citizen journalism platform
called CGNet Swara held 7 workshops beginning from 2014 to develop a Gondi
thesaurus containing all the different words used by native speakers from 6
states. Some words, such as water, had as many as 8 different words for it. At
the 8th workshop in 2018, which saw more than 80 people in attendance, the
thesaurus was developed into a dictionary containing 3,500 words.
The dictionary was made into an Android app, Gondi Manak
Shabdkosh333aka.ms/gondi-dictionary 444aka.ms/indian-express-gondi-dictionary,
depicted in Figure3. It allows users to enter a Hindi or Gondi word and hear
or read its equivalent translation, similar to the Ma! Iwaidja dictionary
app555ma-iwaidja-dictionary.soft112.com/ without the wheel based interface for
conjugation and sentence formation.
#### 4.1.3 Takeaways
CGNet Swara reported an overwhelming consensus by the community to remain as
one language. Gondi is primarily used within its local speech community and
participants recognized the need to develop a common vocabulary and spelling
for its emergence as a standard language. The three techniques commonly used
in standardizing languages are comparative (linguistic reconstruction to build
a mother tongue for all), archaizing (deriving a variety from older, written
texts) and the statistical (combining the different dialects having the widest
usage) haugen1966dialect , with most participants favoring the latter
approach.
CGNet’s initial aim with the dictionary app was studying whether it can allow
some basic communication and learning to take place in primary schools where a
teacher does not know Gondi while their student is a monolingual speaker.
However, our approach of finding a central dialect mediating between the
extremes resulted in a variety of Gondi promoted as everyone’s language that
became nobody’s language. For example, monolingual Gonds may only be aware of
their local word for water, but if it hasn’t been selected in the dictionary
then its use in such scenarios would be limited. In retrospect, a thesaurus
might have been more applicable for this particular use case. The dictionary
provided greater utility in pretraining our translation model. Following Wang
et al wang2022expand we augmented digitally consumable monolingual data to
improve performance of the model, as demonstrated in Table 2.
Figure 3: Gondi Manak Shabdkosh, a dictionary of 3,500 words
### 4.2 Creating children’s books in Gondi through Pratham Books
We narrate our efforts at translating children’s stories into Gondi for use in
schools and as a data source for language technologies.
#### 4.2.1 Motivation
Imparting basic, primary education in a child’s mother tongue inculcates pride
in their language and cultural identity formation among younger generations.
It also targets the most important stakeholder for ensuring continued language
use. More practically, having Gondi as a medium of instruction or a subject at
schools can help increase enrollment among monolingual Gond tribes within the
education system and decrease drop-out rates among indigenous children not
conversant in Hindi. This intervention focused on developing the necessary
linguistic resources in Gondi that could serve as educational material.
#### 4.2.2 Intervention
In 2018, our research team conducted a 10 day translation workshop with CGNet
Swara and Pratham Books, a nonprofit publisher with the motto ”Book in every
child’s hands”. 20 Gond tribals from three states of India came together and
translated 230 children’s books from Hindi to Gondi, some of which introduced
Gondi children to climate change for the first time 666Article:
aka.ms/news18-climatechange. These stories were published on the Storyweaver
platform 777storyweaver.org.in/, an initiative of Pratham Books that hosts
more than 15,000 children’s stories in various languages and dialects. To the
best of our knowledge, this is the first online repository of children’s
stories in Gondi 999Article: aka.ms/hindu-gondi-nextgen888aka.ms/storyweaver-
gondi. They were also printed out and distributed in primary schools in the
Gondi speaking districts of Chhattisgarh, with efforts now ongoing to convince
the state government to include them as part of the school curriculum across
the tribal belts of the state.
#### 4.2.3 Takeaways
Although no payment was provided to workshop participants (besides covering
food, travel and lodging), a total of around 20 volunteers translated 8000
sentences over 10 days. Having a volunteer only workforce helped in self-
selecting individuals that cared about the languages survival among future
generations. We found it most effective to create teams of 4-5 people for
translating a story, with more educated participants taking on the role of
writing the translation on paper or typing the written translation into the
computer interface. Monolingual Gond speakers would help find the right
vocabulary for translations, while bilingual speakers would help them
understand the Hindi text that needed translating. Having groups complete
translations (rather than individuals) resulted in considerable debate around
which Gondi words to use for translating a Hindi sentence. This intervention
yielded the most high quality data compared to the other efforts, as is
evident from our manual evaluation of the data referenced in Table 1.
### 4.3 Crowdsourcing Gondi translations through Adivasi Radio
Unlike the Pratham Books workshop featuring in-person group translations,
Adivasi Radio was geared towards the post pandemic world and allowed
volunteers to provide translations from the comfort of their homes.
#### 4.3.1 Motivation
At the children’s book workshop, we found that many participants wanted to
continue the translation work from home but there was no avenue for them to do
so. Taking inspiration from Aikumabird2014aikuma , we developed Adivasi Radio,
an Android application that presents users with Hindi words or sentences for
which they need to provide the Gondi translation. In addition to the
translation role, Adivasi Radio was designed as the go-to place for native
speakers to find outlets and sites publishing Gondi content.
Figure 4: Adivasi Radio app to collect translations and access Gondi content
#### 4.3.2 Intervention
The app was first deployed between March and June 2020, during the initial
months of the coronavirus induced lockdown. The interface of the app can be
seen in Figure4 while translations over time are shown in Figure5. In 4
months, 17,756 sentences were translated through the app from 164 unique
users.
6 superusers, most of whom were staff at our partner organization CGNet Swara,
accounted for the majority of the translations. Many volunteers gave up after
completing a few translations, indicating that sustaining data collection
efforts is a challenge, especially without fiscal incentives.
We uploaded the children stories produced with Pratham Books, Gondi stories on
CGnet Swara’s citizen journalism platform and The Wire’s Gondi news bulletin
into the app. A Devanagari text-to-speech system read out the written Gondi
text so that monolingual speakers could enjoy the content.
Figure 5: Number of translations collected per day.
#### 4.3.3 Takeaways
After Adivasi Radio’s deployment, we held meetings with the translators to
learn about their experience of using the application. We found that several
users resided in areas having weak internet connectivity, thus prompting us to
update the application so that they could bulk download a large corpus of
sentences that they could translate even when they were offline. Similar to
the feature used in d2014mobile ; mehta2020facilitating for enabling citizen
journalism in areas without internet, translated sentences would get sent to
our server whenever the user managed to connect to the internet. The attention
to providing accurate translations was reflected in their request for both a
’skip’ feature to avoid providing translations they were unsure of and an
option to edit their earlier submissions, as they would sometimes realize
mistakes only after completing a translation. However, we found that the skip
feature was used to avoid translating long sentences or short paras of three
or more sentences. This concern could be alleviated by having them translate
sentences forming a cohesive narrative like a Wikipedia article rather than
disjointed sentences unrelated to one another. Finally, we did not see
significant engagement with the uploaded Gondi content available on the app as
most users were bilingual and accustomed to a richer online environment in
Hindi.
Our evaluation of the quality of submitted translations reiterated the
importance of incorporating rigorous validation checks on data collected
through human-channels. Out of the nearly 18000 sentences collected through
this channel, we were only able to utilize 10105 sentences after weeding out
transliteration, skipped sentences and gibberish inputs with a high density of
special and/or numeric characters.
### 4.4 Crowdsourcing Translations through WhatsApp Business API
A growing body of research advocates integration with existing platforms for
coordinating social mobilization instead of creating custom made ones such as
an app or website lambton2020unplatformed ; saldivar2019online ;
starbird2012crowdwork . The majority of Adivasi Radio users made extensive use
of WhatsApp in their daily life, prompting us to explore how we could use this
medium to solicit translations from the community.
Figure 6: Payments were discontinued for translations received via WhatsApp
after finding users were gaming the system
#### 4.4.1 Motivation
During field testing with Adivasi Radio users, we found that some users did
not have space on their phones to install our app. In such cases, we had to
delete media files and unused apps before we could install the Adivasi Radio
application on their phone. Moreover, training had to be provided on how to
navigate the interface of our app. These factors limited our ability to reach
more members in the community without our physical presence.
We thus applied for - and received - permission to use the WhatsApp Business
API, which would enable users to provide translations over WhatsApp directly
to our server without any human intervention. We accessed the API through the
IMI Connect platform, which has a monthly charge of about USD 350 per month in
addition to variable costs per message.
#### 4.4.2 Intervention
To seed usage of the system, we needed users that had a smartphone with
regular internet connectivity and were conversant in both Hindi and Gondi.
From May 2021 to September 2021, we collected 37,173 sentences from 55
volunteers and staff members at CGNet Swara. 6 superusers accounted for nearly
half of the submitted translations. Many translations were written in the
latin script (as opposed to Devanagari), resulting in a little less than
12,000 sentences that could be fed into the translation model.
#### 4.4.3 Takeaways
To address declining interest in completing translations from home via Adivasi
Radio, we released a probe in February 2021 that for the first time paid users
for completing translations via WhatsApp. We found that payment per sentence
translated led to deteriorating quality by rewarding speed over quality.
Indeed, we saw some of the most devoted community members spend more than 10
minutes on a single sentence, even asking their neighbors for the right word,
so that they could get it right. When these members saw their peers provide
inaccurate translations and earn more than them, they withdrew their support.
We ended up discarding nearly 12,000 sentences collected via this process and
switched back to a volunteer driven model. We also observed frustration when
the sentence that translators were given timed out and they were provided a
new sentence upon logging back in, as they had researched the earlier sentence
but there was no option for them to provide it anymore.
We relaunched the WhatsApp chatbot in June 2021 without the option of earning
any money for translations. The majority of the translations came from staff
members at CGNet Swara. In the absence of strict oversight and punishments for
incorrect translations, we recommend paying a fixed salary to translators
rather than one varying by number of sentences translated or other such
quantitative metrics. It also proved effective to integrate data collection
into the day to day responsibilities of employees at partner organizations.
### 4.5 Data Quality
We requested community members to manually evaluate the data collected and
then compare the quality of data collected via each channel.
#### 4.5.1 Annotation Setup
We recruited 3 Gondi speaking annotators from the Indian states of
Maharashtra, Madhya Pradesh and Chhattisgarh to assist us with the manual
evaluations of our data and the translation model developed from it. This
selection was made bearing in mind the dialectal coverage of our dataset, as
most translations were provided by volunteers from these three states. The
annotation instructions were made visible to all annotators via the interface
we provide to them as shown in Figure7. We also discussed a set of samples and
the mechanism to acquaint them with the scoring system before the evaluation
exercise. The participating annotators performed 3 tasks where they scored a
provided sample with the Direct Assessment score described in specia-
etal-2020-findings-wmt . They were compensated using the following
system101010 All compensations are specified in Indian National Rupee \- per
ranking task, each annotator was given Rs. 5 (Rs. 3 base pay, and Rs. 2 upon
completion). Per assistive translation task, they were provided Rs. 5 and per
translation task, they were provided Rs.10 ( Rs. 7 base pay, Rs. 3 upon
completion ).
Figure 7: The scoring instructions are specified in Hindi on the annotation
screen for quick reference. A score slider is provided for limiting scores to
a valid range.
To analyze the quality of the data collected through Pratham Books, Adivasi
Radio and WhatsApp API, we requested each annotator to score 60 sentences
sampled across any two data sources i.e. 2 annotators score 30 instances per
source. These samples were taken from our training corpus, and we apply a
sequence length filter in the range of [4, 15] tokens to avoid undue weightage
to sentences coming from any one source.
#### 4.5.2 Results
We report average direct assessment scores for each data source in Table 1.
Channel | Average DA Score | Standard Deviation
---|---|---
Pratham Books | 65.4 | 16.1
Adivasi Radio | 58.6 | 19.1
WhatsApp | 36.5 | 21.9
Model(Hindi-Gondi) | 63.2 | 23.2
Table 1: Average Direct Assessment Scores with Standard Deviations for data
collected via all channels and the Hindi-Gondi model trained on the data
The average DA scores on the sample set indicate that the data collected from
Pratham Books is of the best quality, followed by Adivasi Radio and WhatsApp
respectively. This could be due to the setup in which participants were
placed: synchronous and supervised for Pratham Books and asynchronous and
unsupervised for WhatsApp or Adivasi Radio.
As is observed with the standard deviation, the inter-annotator score
agreement was the poorest for WhatsApp. The annotators report that WhatsApp
data inputs had a very high degree of code-switching, sometimes not even with
Hindi but other indic languages. For example, all annotators reported
observing Marathi tokens as a part of the provided translations.
### 4.6 Machine Translation
Using the data collected, we train a machine-translation model between Hindi-
Gondi and evaluate the efficacy of its output.
#### 4.6.1 Motivation
The development of the machine translation model aimed at developing a core-
technology that serves the larger goal of improving information access for the
community by enabling automated translation of web content from Hindi to
Gondi. It also let us evaluate the utility of the data when it is plugged into
a real-world NLP usecase. We envisage the usage of this model in other low-
resource languages like that of Bribri, Wixarica and Mundari
https://doi.org/10.48550/arxiv.2210.15184 to demonstrate the applicability of
our artifact across other data-deficient setups.
#### 4.6.2 Intervention
Using the data we had available from all channels, we finetuned MT5 xue2021mt5
for training our translation model in both the Hindi-Gondi and Gondi-Hindi
direction. The model used in this work was adapted from
https://doi.org/10.48550/arxiv.2210.15184 and we replicate the training setup
specified there. The models were evaluated using sacrebleu (v2.2.0) with the
spm tokenizer. As explored by wang2022expand , we used the Hindi-Gondi
dictionary to generate 200K noisy monolingual data to train our models, which
we further used for continual pretraining. We also generated forward
translated data using the same model to fuse with the lexicon-adapted data.
Additionally, we utilize the quantized model developed by
https://doi.org/10.48550/arxiv.2210.15184 to evaluate the performance of the
same model’s compressed version (400 MB in disk size compared to 2.28 GB), for
the purpose of commenting on the edge deployment feasibility in low-resource
environments.
The annotation setup is same as the one described in section 4.5.1. All
annotators scored 40 inferences generated from the model to analyze its
quality.
Model | BLEU | Size
---|---|---
mt5-small (Hindi-Gondi) | 14.3 | 1.2GB
mt5-small (Gondi-Hindi) | 30.9 | 1.2GB
mt5-base (Hindi-Gondi) | 15.6 | 2.2GB
mt5-small with continued pretraining | 14.9 | 1.2GB
mt5-small with fused data (With 50K forward translated) | 12.6 | 1.2GB
Quantized MT5 | 13.8 | 400 MB
Table 2: Performance of the Hindi-Gondi translation model with model
adaptations like lexicon-adaptation, continual pretraining and quantization.
#### 4.6.3 Takeaways
As shown in Table 1, the annotators report an average DA Score of 63 for the
Hindi-Gondi model which connotes that the model’s output was semantically
appropriate but contained an identifiable degree of grammatical errors and
typos. The annotators mentioned that the model’s output was visibly poor in
terms of producing appropriate punctuation. While the structural integrity of
the sentence seems to have been preserved, spelling differences and code-
switched tokens ( i.e. Hindi tokens on the target side ) were common.
Annotators also observed the occurrence of ambiguous-language tokens that they
were unable to identify as being correct in the given context, which could be
attributed to the dialectal differences that existed within the language.
The BLEU scores of the models are reported in Table 2. The significant drop in
the BLEU score in the Hindi-Gondi direction, in comparison to Gondi-Hindi,
appears to be most significantly associated with the inclusion of Hindi in the
pretraining corpus of MT5. The mC4 training corpus contains 1.21% Hindi data,
which greatly appears to aid the target side reconstruction in the Gondi-Hindi
model xue2020mt5 . Additionally, we posit that the mt5 tokenizer, which has
never seen Gondi, expectedly falters in the reconstruction of Gondi on the
target side ( as is seen in the Hindi-Gondi ) model, so we expect a high
degree of tokenization ambiguities in the Gondi output which would adversely
impact the BLEU evaluation. This is further substantiated by observation of
annotators who claimed that in some cases constructions of a Gondi token
emulated the Hindi construction of the token. We intend to do a deeper
investigation of such spurious correlations that massively-multilingual models
might develop when a dominant attribute of language, its script in this case,
matches other well-represented languages in the pretraining corpus despite
being linguistically quite divergent.
One of the primary learnings from this task was the need to integrate
linguistically-motivated validation structures in a data collection channel to
optimize the yield of the data collection process. In our specific case, the
lack of any validation structures around the dialects against which we
collected our data for led to the collection of a dataset with samples from at
least 3 dialects. This especially harmed the impact of the machine translation
system as it’s responses catered to neither dialect completely.
### 4.7 Assistive Translation
We built an interface showing suggestions from our translation model to see if
it improves the quality or speed of translation efforts.
#### 4.7.1 Motivation
We use the same Hindi-Gondi model to generate candidate words that could be
used when translating Hindi sentences to Gondi. The motivation for this task
was to understand if poor-accuracy models could be used to accelerate the data
collection process by providing assistive recommendations to the participating
translators. This task holds unique practical value because it allows us to
comment if sub-optimal machine translations (which would be an obvious
intermediate artifact in such community interventions) can help enhance the
yield of data-generation pipelines in low-resource communities.
Figure 8: Gisting Interface for providing recommendations to data-providers.
Our motivation to develop this system did not come directly from the Gond
community but through partners at an educational institution and the citizen
journalism platform CGNet Swara. They believed it could enable speedy
translation of books for primary education and media stories into Gondi.
#### 4.7.2 Intervention
We realized that developing such a language technology had to take local
constraints into account, which in this specific case entailed the lack of a
stable internet connection. We thus developed an offline variant of the model,
which was compressed enough to operate on relatively low-resource devices
without requiring an internet connection.
The interface for this task, as shown in Figure8, is a screen that shows users
the model’s output as a set of candidate tokens that they can opt for while
submitting the translation. We sample 150 sentences from our development set
and provided it for translation through the Karya app. Inferences for these
sample sentences were provided to the annotators as a panel of assistive
candidate tokens. The annotators were requested to post-edit these candidate
gists until they were correct using the provided suggestions wherever
possible. During this exercise we monitor the number of suggestions accepted
by the annotators over the total number of suggestions presented.
#### 4.7.3 Takeaways
We find that annotators use 3.66 out of an average of 5.56 options shown to
them in each sentence translation iteration. This is further substantiated by
the observations of the annotators, who reported that nearly half of the
suggestions provided by the model were accurate and useful in generating the
final output. They also commented on the interface design by mentioning that
they preferred to type a token displayed in the panel of candidate tokens
instead of clicking on the token to avoid disrupting their flow while typing
out a translation. This indicates that tracking the number of accepted tokens
via clicks would have provided us a pessimistic view of the model’s usefulness
since annotators were visually aided by tokens (which might have been
completely or partially correct) and our experimental design would not have
logged that as a successful instance of aid. Annotators also pointed out that
some stop words were useful candidates in the Bag Of Words interface as they
preferred opting for them while typing a translation (rather than post-editing
them as a part of a gist). Hence, providing even noisy assistance with the
right interface appears to be a promising direction.
The annotators overarching evaluation of the translation system deemed it to
be semantically accurate. This gives hope that we can setup a virtuous cycle
where generation of linguistic resources enables the development of community-
oriented language technologies, which in turn could help in generating more
linguistic resources that can then further improve language technologies.
Additionally, their feedback on finding the candidate translations being
visually assistive encourages a deeper investigation into if and how
relatively low-accuracy models could be leveraged to improve the experience of
data providers in low-resource communities.
### 4.8 Disseminating Gondi content via Interactive Voice Response
Collecting data and building language technologies is only one part of the
equation; the other is creating avenues for communities to hear their language
broadcast over mass media and other communication channels.
#### 4.8.1 Motivation
There is an established body of work on the role of Interactive Voice Response
(IVR) forums for reaching communities that are too poor to afford smartphones,
too remote to access the Internet or too low-literate to navigate text-driven
environments swaminathan2019learn2earn ; dipanjan2019 ; revisitCG ; designLess
; ivrFarmers ; raza2018baang ; sawaal ; raza2013job ; sangeet . As an example,
Learn2Earn swaminathan2019learn2earn is an IVR based system that was used as
an awareness campaign tool to spread farmers’ land rights in rural India, HIV
literacy and voter awareness mehta2020using . Learn2Earn awards mobile
talktime to users who call a toll-free number and answer all multiple-choice
questions on the message correctly. Starting from an initial set of just 17
users, it was successful in spreading land rights awareness to 17,000 farmers
in 45 days via additional rewards for successful referrals
swaminathan2019learn2earn .
We adapted Learn2Earn for spreading voter awareness among Gondi speakers in
Dantewada (a rural district in the state of Chhattisgarh in India), during the
time that a bypoll election was being held. Prior work has shown that larger
the contexts, identities and communicative functions associated with a
language, the more likely it is to thrive walsh2005will . Voter rights content
in local languages during election season has the potential to encourage
conversations on topics of wider contexts and functions as well as contribute
to establishing representative and effective governance.
#### 4.8.2 Intervention
The pilot obtained users through seeding activities and referrals. Figure9
shows the number of unique calls that were made to our system and the number
of users who passed the quiz over time. Further, the Figureshows that a
majority of quiz passers came to know about our system via direct seeding,
though people knew that additional credits can be earned through referrals.
Usage dropped sharply after the elections in September concluded and seeding
activities were discontinued.
Figure 9: Number of calls to the system (blue line) and the number of users
who answered all questions correctly (bar plot). The blue and orange bars
represent users onboarded via seeding and through referrals
---
Metrics | Value
Unique callers (total, during and after seeding) | (557, 480, 77)
Unique callers per day - during seeding (min, mean, median, max) | (13, 48, 49, 80)
Unique callers per day - after seeding (min, mean, median, max) | (0, 4, 3, 20)
Callers answering all questions correctly | 313
Callers answering all questions correctly in their first call | 104
Calls made by callers (min, mean, median, max) | (1, 3, 2, 64)
Table 3: Summary statistics for our Learn2Earn deployment in Gondi
Table 3 highlights some important statistics in our Learn2Earn pilot. We
spread voter awareness in Gondi to 557 speakers, of which a majority (86%)
were reached during active seeding. Nearly 60% of the users correctly answered
all questions, either in their first attempt or in successive attempts,
indicating the effectiveness of the system for content dissemination. Further,
many users called the system more than once, with one user calling 64 times
(to refer someone, users had to place another call to the system). A follow-up
survey revealed that 22 of 113 users surveyed could not vote as they did not
have a voter ID card, potentially useful information for authorities to
increase voter turnout. One disappointing finding was that only about 8.3
percent of users were women.
#### 4.8.3 Takeaways
We see three clear benefits from conducting Learn2Earn pilots in endangered or
vulnerable languages. First, since it is entirely in the spoken form, only
native speakers of an endangered or vulnerable language can comprehend the
content and earn the reward. It is also easily accessible in low-resource
environments as it requires a missed call and nothing more. Second, the phone
numbers collected are an important dataset of speakers of that language and
can be used for future translation workshops and related programs that help
their economic and linguistic development. Finally, an oral language has a
tendency to die out unless there is an opportunity for that language to be
used outside of everyday surroundings, which periodic Learn2Earn campaigns on
important issues can help achieve.
## 5 Discussion
Steven Bird alerts us to the dangers inherent in a technology driven approach
to language revival, contending that commodifying indigenous languages as data
alienates native speakers. He argues for an approach rooted in self-
determination where outside experts are only engaged with to help implement a
program or strategy that the community has already settled on
bird2020decolonising . This sentiment is echoed by Stebbins et al (2017) who
maintain that ‘the community must be the driver and driving interest of the
research and language projects”stebbins2017living . Lewis and Simons carve out
an additional role for reflective practitioners of language development, that
of ”facilitating opportunities for local language users to interact with each
other” book .
There are also concerns that experts profit off their domain expertise in an
indigenous language with little benefit accruing to the communities
themselves. For example, Bird asks ”where are the full implementations,
deployed in robust software, in active use to capture primary data, leading to
curated language products that are being mobilized in speech communities?” By
his yardstick, even the interventions described in our paper fall short as
none have translated to scale or persistent use by the community. With the
benefit of hindsight, we can now reflect on whether our initiatives have been
exploitative.
Our position is that these interventions avoided an exploitative approach by
providing tangible outputs advancing one of the five FAMED conditions laid out
by Lewis and Simons as an ethical guide to language practitioners. This
acronym expands to having Functions associated with the language that are in
use and recognized by the community; a means for community members to Acquire
proficiency in the language; a Motivation for community members to use the
language, which is frequently but not exclusively economic; a policy
Environment that is conducive to the growth of the language; and a
Differentiated sphere for use of the language in adherence to established
societal norms.
Our machine translation artifact can be built upon by the Gondi community and
other low resource communities around the world to expand the functions
associated with their language, through automated translation of digital
content. The assistive translation app can help in language acquisition and
documentation by improving the speed of translating literary material and
providing vocabulary that translators may themselves be unaware of. Our
Learn2Earn pilot provided incentive payments to those speaking Gondi, thus
increasing (at a small scale) the motivation to use and recognize the
language. Translating children’s stories into Gondi can create a fertile
policy environment that makes possible the use of tribal languages in primary
education. The dictionary development process kickstarted a differentiated
sphere of Gondi usage between tribals of different states, whereas it has
historically been used only within their own region. Our thematic approach of
collecting language data to build technologies while simultaneously creating
resources for the community that advance one of the FAMED conditions can help
avoid the pitfall of alienating local speakers by commodifying their language
as data.
Overall, our work raises interesting questions on the role of the outside
interventionist in building langugage technologies and the relationship they
should have with the community. For example, the dictionary development
workshops primarily comprised bilingual and educated Gond speakers, which may
have contributed to their desire for standardizing the language. However, we
realized that this approach would be discriminatory to monolingual Gond
speakers who would have to accept the language thrown at them by the
intellectuals within their community since the standardization efforts would
not be universally agreed upon or even known to the majority of Gond speakers.
As Bird writes, ”No community speaks with a single voice, and in building
relationships we can unwittingly align ourselves with agendas, clans and gate-
keepers.” This prompts reflection on whether the outside interventionist
should simply fill in ”white man’s paperwork” and let the community lead, or
if there is a deeper role that we need to play, one where we use our perceived
neutrality to arbitrate between the different sections of a community. The
latter role finds expression in Lewis and Simons’ contention that ”it may be
an appropriate role for an outsider to act as an intermediary between the
different speech communities.” While we do not presume to have answers to
these questions, we hope that our work helps stimulate wider debate in the
language development community on defining our relationship with the speech
communities we work with and depend upon.
## 6 Conclusion
Engineering is the primary obstacle to designing technologies for well-
resourced languages. For low-resource ones, more focus needs to be given
towards designing methods for robust data collection and evaluation upon which
the language technology can be built.
To keep community members motivated through the data collection process, our
team strived to achieve 2 simultaneous goals: the collection of data upon
which language technologies such as speech to text or machine translation
could be built, and building a literary resource that community members could
point to as an immediate, demonstrable success. For example, the Learn2Earn
pilot in Gondi not only provided the community with an opportunity to earn
money for answering a quiz in their language and referring others to it, but
it also provided a dataset of native Gondi speakers. Similarly, translating
children’s stories and creating a standardized dictionary resulted in both
data upon which a machine translation tool was built and also tangible
language resources that can be used by the community.
We made use of the data collected to develop a machine translation model and
an assistive translation interface. Community evaluators reported that the
model fared poorly in punctuation and grammar, and roughly half the suggested
words were applicable to the sentence they were translating. Our future goals
are improving the models accuracy by feeding it more data and testing its
performance with data from other low-resource languages.
In 2020, the Chhattisgarh government began allowing primary education in 10
tribal languages, including Gondi
111111https://indianexpress.com/article/governance/chhattisgarh-education-
reforms-tribal-languages-to-be-a-medium-of-education-in-pre-school-6271547/.
At the community level, our focus is on working with the government to
integrate the linguistic resources and technological artifacts we’ve created
into their apparatus. We also plan on integrating the translation tool with
CGNet Swara’s citizen journalism newsroom so that they can regularly put out
Gondi content.
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|
# CUNI Submission in WMT22 General Task
Josef Jon Martin Popel Ondřej Bojar
Charles University
<EMAIL_ADDRESS>
We present the CUNI-Bergamot submission for the WMT22 General translation
task. We compete in English$\rightarrow$Czech direction. Our submission
further explores block backtranslation techniques. Compared to the previous
work, we measure performance in terms of COMET score and named entities
translation accuracy. We evaluate performance of MBR decoding compared to
traditional mixed backtranslation training and we show a possible synergy when
using both of the techniques simultaneously. The results show that both
approaches are effective means of improving translation quality and they yield
even better results when combined.
## 1 Introduction
This work focuses on exploring of two methods used in NMT in order to improve
translation quality: backtranslation and Minimum Bayes Risk decoding using
neural-based evaluation metric as the utility function. The methods used and
related work are presented in Section 1. In Section 2 we describe our
experimental setting and results.
## 2 Methods
We describe methods we used to build our system in this section.
### 2.1 Block backtranslation
The translation quality of NMT depends heavily on the amount of parallel
training data. It has been shown that the authentic bilingual data can be
partially supplemented by synthetically parallel, machine translated
monolingual text Bojar and Tamchyna (2011); Sennrich et al. (2016); Xie et al.
(2018); Edunov et al. (2018). Often, the synthetic and authentic parallel data
are mixed in the training dataset but previous research shows that simply
mixing the two types of text does not yield optimal translation quality. We
use block backtranslation (block-BT) in configuration similar to Popel et al.
(2020). This method creates blocks of parallel and synthetic data and presents
them to the neural network separately, switching between the two types during
the training. Since in last year’s WMT, the submission using block-BT by
Gebauer et al. (2021) did not find any improvements, presumably due to
improperly chosen block size, we decided to verify effectiveness of this
method once again.
#### Averaging type
Previous work on block-BT shows the importance of averaging the checkpoints to
combine information from different blocks of training data in order to obtain
good performance. We compare checkpoint averaging with another method of
combining older sets of model’s parameters with the current one – exponential
smoothing. After each update $u$, the current parameters $\Theta_{u}$ are
averaged (with smoothing factor $\alpha$) with parameters after the previous
update $\Theta_{u-1}$:
$\Theta_{u}=\alpha\Theta_{u}+(1-\alpha)\Theta_{u-1}$
Previous work by Popel (2018) contains experiments with exponential averaging,
but only on the level of already saved checkpoints, not online during the
training after each update as for our work.
#### Minimum Bayes Risk decoding
NMT models predict conditional probability distribution over translation
hypotheses given a source sentence. To select the most probable translation
under the model (mode of the model’s distribution), an approximation of MAP
(maximum-a-posteriori) decoding is used, most commonly the beam search Graves
(2012). However, beam search and MAP decoding in general have many
shortcomings described in recent work Stahlberg and Byrne (2019); Meister et
al. (2020) and other approaches have been proposed to generate a high-quality
hypothesis from the model.
One of them, MBR (Minimum Bayes Risk) decoding Goel and Byrne (2000); Kumar
and Byrne (2004), has been proposed as an alternative to MAP. MBR does not
produce a translation with the highest probability, but a translation with the
best value of utility function. This utility function is usually an automatic
machine translation evaluation metric. However, to optimize towards the best
utility function value, it would necessary to know the ideal selection of
hypothesis. In the case of MT, that would mean a perfect, best possible
translation, which of course is not known during the translation process. For
this reason, an approximation of the ideal translation is used, based on the
model’s probability distribution Bryan and Wilker (2021). This can be
implemented as generating a list of hypotheses (e.g. using sampling or beam
search) and then computing utility function of each hypothesis using all the
other hypotheses as the ideal translation approximation (i.e. as references).
This approximation of MBR decoding can be seen as consensus decoding – the
hypothesis most similar to all the others is chosen. Also, in this
implementation, it is more appropriate to name the process reranking, rather
than decoding, and we will do so from now on.
Even though MBR is able to optimize towards many metrics and increase the
scores, these gains did not translate into better human evaluation of the
final translations, when using traditional metrics based on surface
similarities like BLEU. Recent successes in development of novel metrics for
machine translation has renewed interest in this method. Amrhein and Sennrich
(2022a); Freitag et al. (2021); Müller and Sennrich (2021).
## 3 Experiments
In this section we present our experimental setup and results.
### 3.1 Tools
We tokenize the text into subwords using
FactoredSegmenter111https://github.com/microsoft/factored-segmenter and
SentencePiece Kudo and Richardson (2018). We use MarianNMT Junczys-Dowmunt et
al. (2018) to train the models. BLEU scores are computed using SacreBLEU Post
(2018), for COMET scores Rei et al. (2020) we use the original
implementation.222https://github.com/Unbabel/COMET
### 3.2 Datasets
We train English-Czech NMT models for our experiments. We train our models on
CzEng 2.0 (Kocmi et al., 2020). We use all 3 subsets of CzEng corpus: the
originally parallel part, which we call auth, Czech monolingual data
translated into English using MT (csmono) and English monolingual data
translated into Czech using MT (enmono). We use newstest2020 Barrault et al.
(2020) as our dev set and newstest2021 Akhbardeh et al. (2021) as our test
set.
For experiments concerning translation of named entities (NE), we used a test
set originally designed for Czech NLG in restaurant industry domainDušek and
Jurčíček (2019).333https://github.com/UFAL-DSG/cs_restaurant_dataset It
contains sentences which include names of restaurants and addresses in Czech
and their translations in English. We will call this test set the restaurant
test set.
### 3.3 Models
We train Transformer-base (which we denote base) and Transformer-big (big 6-6)
models with standard parameters Vaswani et al. (2017) as pre-configured in
MarianNMT. For the largest model (big 12-6), we use Transformer-big with 12
encoder layers and depth scaled initialization Junczys-Dowmunt (2019); Zhang
et al. (2019).444Training scripts available at:
https://github.com/cepin19/wmt22_general We also used learning rate of
$1\mathrm{e}{-4}$ for the 12 layer model instead of $3\mathrm{e}{-4}$, which
was used for other models. We trained all models for at least 1.4M updates. We
computed validation BLEU scores every 5k updates and we stopped if the score
did not improve for 30 consecutive validations. ]HW, GPUs, training times,
base vs. big vs deeper We trained the models on a heterogenous grid server,
which includes combinations of Quadro RTX 5000, GeForce GTX 1080 Ti, RTX A4000
and GeForce RTX 3090 cards. Typical training time on 4 1080Ti of the base
models for 1.4M updates was 7 days.
### 3.4 Block-BT settings
For all our experiments, we save a checkpoint every 5k updates and we vary
only the size of the blocks during which the training data stay in the same
type (20k, 40k, 80k and 160k updates). The length of the blocks is the same
for all block types. We circle through the block types in the following order:
auth$\rightarrow$csmono$\rightarrow$auth$\rightarrow$enmono.
For checkpoint averaging, we average consecutive 8 checkpoints. For
exponential smoothing, we use default Marian configuration ($\alpha=0.001$,
but there are some slight modifications based on number of updates since start
of the training and batch size).
We also look at the effects of using only backtranslation, or both back- and
forward-translation.
Figure 1: Comparison of different training regimes for EN$\rightarrow$CS
translation on newstest20 in terms of BLEU (top) and COMET (bottom).
Background colors for block-BT regime show which part of training data was
used for a given part of the training. Green means authentic parallel data,
blue is CS$\rightarrow$EN backtranslation and red is EN$\rightarrow$CS forward
translation.
### 3.5 Block-BT results
Model size | Block size | Avg type | update (k) | BLEU | update (k) | COMET
---|---|---|---|---|---|---
base | mixed | exp | 1340 | 34.7 | 1760 | 0.7337
mixed | exp+avg8 | 1365 | 34.7 | 965 | 0.7326
20k | - | 1360 | 34.6 | 640 | 0.7324
exp | 410 | 34.9 | 725 | 0.7406
avg8 | 660 | 34.8 | 1385 | 0.7349
exp+avg8 | 420 | 34.9 | 735 | 0.7399
40k | - | 610 | 34.8 | 1415 | 0.7363
exp | 1130 | 35.3 | 1290 | 0.7474
avg8 | 780 | 35.5 | 1420 | 0.7462
exp+avg8 | 1150 | 35.5 | 1075 | 0.7466
80k | - | 1250 | 34.9 | 960 | 0.7393
exp | 1210 | 35.2 | 1450 | 0.7447
avg8 | 985 | 35.5 | 665 | 0.7474
exp+avg8 | 585 | 35.3 | 1150 | 0.7455
160k | - | 1130 | 34.9 | 1210 | 0.7387
exp | 1125 | 35.3 | 1285 | 0.7453
avg8 | 1135 | 35.5 | 1305 | 0.7467
exp+avg8 | 1145 | 35.3 | 1310 | 0.7473
big 6-6 | 40k | exp | 445 | 35.4 | 1125 | 0.7546
exp+avg8 | 300 | 35.4 | 1310 | 0.7567
big 12-6 | 40k | exp | 130 | 36.1 | 1210 | 0.7848
Table 1: Best COMET and BLEU scores on EN-CS newstest2020 for all the
combinations of models size, training regime and block size. We report the
best score and an number of updates after which was this score reached.
#### Training regime and averaging method
First, we compare different training regimes: mixed-BT, where all the training
datasets are concatenated and shuffled together and block-BT with 40k updates
long blocks and two possible averaging types – exponential smoothing (exp) or
checkpoint averaging (avg8).
Figure 1 shows the behavior of BLEU and COMET scores on newstest2020 during
the training for these configurations inthe interval between 480k and 1280k
updates. The behaviour is not stable earlier than 480k steps and 1280k is the
nearest lower multiplicative for the largest block size. 40k block curve
represents the model without any averaging, 40k block avg8 is the model
trained without exponential smoothing but each checkpoint was averaged with 7
previous checkpoints for the evaluation, 40k block exp model was trained with
continuous exponential smoothing. Finally, we also experimented with
combination of both – exponential smoothing and averaging after the training.
The combination does not improve over the separate averaging techniques and we
omitted the curve from the figure for clarity.
In both metrics, block-BT with either form of averaging outperforms mixed-BT
training. Without any averaging, the advantage of block-BT over mixed-BT is
smaller. Type of averaging does not seem to play a large role – checkpoint
averaging, exponential smoothing and their combination yield very similar best
scores. The best scores on newstest2020 for each combination of parameters are
presented in Table 1.
The curves for checkpoint averaging and exponential smoothing behave
similarly, with exponential averaging reacting faster to change of the block.
Additionally, the avg8 models have higher peaks in enmono (red) blocks,
especially for BLEU scores. The shape of the curves could be tuned by changing
frequency of saving checkpoints and number of checkpoints to be averaged for
checkpoint averaging method, or by changing the $\alpha$ factor for
exponential smoothing.
There are differences in behaviour between BLEU and COMET score curves. Most
notably, COMET is less sensitive to transition from auth (green) to csmono
(blue) blocks. We hypothesize this is caused by lower sensitivity of COMET
score to wrong translation of NE and rare words Amrhein and Sennrich (2022a).
We present further experiments in this direction later. ]There are also peaks
in forward translation, especially for BLEU and avg8 they seem higher than in
auth regions, investingate in NE part.
Figure 2: Comparison of how the block size affects behavior of BLEU (top) and
COMET (bottom) scores during the training for block-BT with exponential
smoothing of the parameters, without checkpoint averaging, on EN-CS
newstest2020.
Figure 3: Comparison of how the block size affects behavior of BLEU (top) and
COMET (bottom) scores during the training or block-BT with checkpoint
averaging and no exponential smoothing of the parameters, on EN-CS
newstest2020.
#### Block size
We assess the influence of block size for both of the two averaging methods.
We compare block sizes of 20k, 40k, 80k and 160k updates. The behaviour of
COMET and BLEU scores is presented in Figures 2 and 3 for exponential
smoothing and checkpoint averaging, respectively. The best scores are again
shown in Table 1.
We see that 20k block size yields noticeably worse results when using
checkpoint averaging than the other sizes. The negative effect of the small
block size is less pronounced when using exponential smoothing, yet still
present. Other block sizes perform similarly in both metrics. This result is
expected, since for 8-checkpoint averaging with 5k updates checkpointing
interval, it is necessary to have a block size of at least 40k updates to fit
all the 8 checkpoints and thus explore all possible ratios of auth and mono
data. ]So it seems that it is important to be able to fit at least n
checkpoints inside a single block for n-checkpoint averaging
Figure 4: Comparison of different training regimes for CS$\rightarrow$EN translation on newstest2020 in terms of BLEU (top) and COMET (bottom). Background colors for block-BT regime show which part of training data was used for a given part of the training. Green means authentic parallel data, blue is CS$\rightarrow$EN forward translation and red is EN$\rightarrow$CS backtranslation. | | | best BLEU | best COMET
---|---|---|---|---
Model | Block | Avg type | update (k) | BLEU | update (k) | COMET
base | mixed | exp | 1405 | 25.2 | 1220 | 0.4149
exp+avg8 | 1430 | 25.1 | 1220 | 0.4114
40k | - | 580 | 25.3 | 1040 | 0.4086
exp | 755 | 25.3 | 570 | 0.4183
avg8 | 765 | 25.4 | 1060 | 0.4175
exp+avg8 | 1080 | 25.2 | 1230 | 0.4186
Table 2: COMET and BLEU scores for Czech to English directions. The best
checkpoints were chosen based on their performance on newstest2020.
#### Reverse direction
For the reverse direction, Czech to English, we performed less extensive
evaluation. We only compare mixed, block-BT with 40k blocks and either
exponential smoothing or checkpoint averaging. The behavior of the metrics is
shown in Figure 4 and the final best scores on newstest2020 are presented in
Table 2. Block-BT still outperforms mixed training, but by a smaller margin
than in the other direction. We attribute this difference to the fact that the
Czech side of the CzEng dataset is more often translationese (originally
English text translated into Czech) and thus differs more from csmono part,
giving space for the larger gains.
#### Backtranslation direction
| | | BLEU | COMET
---|---|---|---|---
dir | regime | datasets | Dev | Test | Dev | Test
encs | mixed | all | 34.7 | 20.9 | 0.7337 | 0.6206
auth+cs | 31.5 | 19.5 | 0.6904 | 0.5779
auth+en | 34.8 | 20.6 | 0.7258 | 0.6097
block | all | 35.3 | 21.1 | 0.7474 | 0.6245
auth+cs | 33.9 | 19.9 | 0.7232 | 0.5908
auth+en | 35.4 | 20.7 | 0.7497 | 0.6147
csen | mixed | all | 25.2 | - | 0.4149 | -
block | all | 25.3 | - | 0.4183 | -
auth+en | 24.3 | - | 0.3682 | -
Table 3: Results on newstest2020 and newstest2021 for various dataset
combinations on dev (newstest2020) and test (newstest2021) sets,
respectivelly, COMET scores are computed by wmt20-comet-da model.
We also evaluate influence of using only backtranslations (i.e. csmono for
en$\rightarrow$cs) as additional synthetic data (monolingual data in target
language automatically translated to source language) or adding also forward
translations (automatic translations from source language to target; enmono)
and we present the results in Table 3. Interestingly, the results show large
gains in both BLEU and COMET when using forward translation. We hypothesize
this is caused by the good quality of the model used to create the translation
for enmono. In such case, the translation model plays the role of the teacher
in teacher$\rightarrow$student training and might lead to good quality
results.
#### Named entities test sets
Figure 5: Behaviour of BLEU (top), COMET (bottom) on newstest2020 and NE translation accuracy on restaurant test set for Czech to English translation with block-BT using exponential smoothing. Update (k) | COMET | BLEU | Acc
---|---|---|---
570 | 0.4183 | 24.9 | 0.607
755 | 0.4038 | 25.3 | 0.629
590 | 0.4099 | 24.9 | 0.636
Table 4: Best checkpoints of Czech to English model trained with 40k blocks
and exponential smoothing in terms of COMET (first row), BLEU (second row) on
newstest2020 and NE translation accuracy on restaurant test set (third row).
From anecdotal evidence, we have seen that checkpoints with large influence of
backtranslated data perform worse on named entities (NE) translation and COMET
and BLEU scores might not reflect this drop of accuracy. We evaluate the
models in terms of accuraccy of NE translation on the restaurant test set. We
selected Czech to English direction, since the evaluation is easier given
lower morphological richness of the target language. ]I also have results for
en-cs, I should integrate themFigure 5 shows comparison of behavior of NE (NE)
translation accuracy on the restaurant test set and COMET and BLEU scores on
newstest2020 for exponential smoothing and checkpoint averaging. NE accuracy
peaks towards the end of auth regions (green). Both COMET and BLEU scores peak
also during the auth part of the training, but, especially for COMET, the peak
occurs in earlier stages after the switch to auth. Overall, BLEU curve
correlates better with the NE accuracy curve. We hypothesize this might be
related to the fact that COMET was found to be insensitive to NE errors by
Amrhein and Sennrich (2022b).
However, it seems that the shift between the accuracy and the other two
metrics is not too large in our settings and choosing the best performing
model in terms of either COMET or BLEU should not hurt NE translation by a
large amount. We further investigate that in Table 4 – we chose the checkpoint
with the best COMET (first row) and the best BLEU (second row) on the
newstest2020 and the checkpoint with the best NE translation accuracy on the
restaurant test set (third row). We compute all three metrics for these three
models. The best COMET checkpoint obtains accuracy of 60.7% on the restaurant
test set, the best BLEU checkpoint reaches the accuracy of 62.9%, while the
best accuracy reached by any checkpoint is 63.6%.
### 3.6 MBR reranking
i | auth | cs | en | AVG comet20 | MBR comet20 | comet21
---|---|---|---|---|---|---
1 | - | - | - | 0.7322 | 0.7888 | 0.0885
2 | 9 | 2 | 1 | 0.7430 | 0.8082 | 0.0946
3 | 4 | 4 | 4 | 0.7408 | 0.8182 | 0.0972
4 | 12 | 0 | 0 | 0.7425 | 0.8010 | 0.0929
5 | 0 | 12 | 0 | 0.7303 | 0.8104 | 0.0949
6 | 0 | 0 | 12 | 0.7372 | 0.7960 | 0.0918
7 | 1 | 7 | 4 | 0.7370 | 0.8232 | 0.0981
8 | 0 | 7 | 5 | 0.7361 | 0.8232 | 0.0980
9 | 2 | 7 | 3 | 0.7377 | 0.8231 | 0.0981
Table 5: Results of MBR reranking on newstest2020 for different selection of
the hypotheses n-best lists produced by checkpoints from different training
blocks. In total, 12 n-best lists produced by transformer-base models are
concatenated and the first three columns show how many n-best lists are used
from each block (the checkpoints for each block are sorted by COMET (wmt20-da
model), so these are produced by the best performing checkpoints). The AVG
COMET20 shows the average wmt20-da COMET scores for the first hypotheses of
each n-best list that was used, MBR COMET20 shows wmt20-da score of the final
sentences after MBR reranking, COMET21 shows results of the same sentences
from wmt21-da model.
We used MBR reranking to rerank concatenation of n-best lists produced by
various checkpoints. In total, we used 6-best lists from 12 checkpoints, i.e.
72 candidate hypotheses for each sentence. We divided the checkpoints based on
which block of the training data they were saved in and sorted them by COMET
score on newstest2020. Using different strategies we selected the best
performing checkpoints to provide the n-best lists. We present the results in
Table 5. The first row shows results for mixed-BT regime, i.e. we concatenated
n-best lists produced by the 12 best performing mixed-BT checkpoints. In the
second row, the block-BT training checkpoints were used to create n-best
lists, selected only based on their COMET scores, without any regard on the
block type they were saved in. In third row, we combine n-best lists from 4
best checkpoints from each type of block. In rows 4-6, we use best checkpoints
from each type of block separately. In the final three rows, we show the
optimal selections which yielded the best score. The results suggest that
larger diversity across block types of the checkpoints improves MBR results:
the combination of n-best lists produced by checkpoints from diverse block
types provides better hypotheses for MBR, even though the average COMET score
of these checkpoints is lower than for the less diverse selection (see rows 2
and 3).
]Todo: I should use BLEURT for final evaluation, since I used comet as a
utility function and I want to see if other metrics also improve ]I could use
dictionary and constrained model to translate rare words (with all the
possible dict translations) and add these translations into the pool, but
probably in some other paper, it would be too much for a system description,
the problem is that comet is not very sensitive to rare word translation ]I
could use combination of different metrics for mbr, but i need to scale them
properly
### 3.7 Submission
auth | cs | en | AVG comet20 | MBR comet20 | comet21
---|---|---|---|---|---
9 | 2 | 8 | 0.7802 | 0.8566 | 0.1114
Table 6: Our final submission for the EN-CS general translation task, based on outputs of the transformer-big 12-6 model. Meaning of the columns is identical to Table 5. System | COMET-B | COMET-C | ChrF-all
---|---|---|---
Online-W | 97.8 | 79.3 | 70.4
Online-B | 97.5 | 76.6 | 71.3
CUNI-Bergamot * | 96.0 | 79.0 | 65.1
JDExploreAcademy * | 95.3 | 77.8 | 67.2
Lan-Bridge | 94.7 | 73.8 | 70.4
Online-A | 92.2 | 71.1 | 67.5
CUNI-DocTransformer * | 91.7 | 72.2 | 66.0
CUNI-Transformer * | 86.6 | 68.6 | 64.2
Online-Y | 83.7 | 62.3 | 64.5
Online-G | 82.3 | 61.5 | 64.6
Table 7: Results of automatic metrics on WMT22 General Task test set.
Constrained submissions are marked by an asterisk, the best scores among
constrained submissions are bold. COMET-B and COMET-C are COMET scores for the
two different references, ChrF is computed using both references together.
Our primary submission is based on the big 12-6 model and MBR reranking. We
explored all the possible combinations of 18 checkpoints from different blocks
as described in the previous section. The results of the best combination are
shown in Table 6. We present the results of the official evaluation in our
task in Table 7. There were 5 submitted systems (4 constrained) and 5 online
services. Our submission ranks first in COMET score among the constrained
systems and third in ChrF.
## 4 Conclusion
We describe our submission to WMT22 and experiments that have led to
development of our system. We confirm effectiveness of block-BT on the recent
COMET metric. We demonstrate the behavior of the translation quality over the
course of the training and discuss the effects of various settings. We also
show that MBR reranking can benefit from more diverse checkpoints created by
block-BT training.
## 5 Acknowledgements
This work was supported by GAČR EXPRO grant NEUREM3 (19-26934X), Bergamot
project (European Union’s Horizon 2020 research and innovation programme under
grant agreement No 825303) and by the Ministry of Education, Youth and Sports
of the Czech Republic, Project No. LM2018101 LINDAT/CLARIAH-CZ.
]compute bleurt for best single and then for the mbr rescored
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|
# Rao-Burbea centroids applied to the statistical characterisation of time
series and images through ordinal patterns
Diego M. Mateos ID Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET), Argentina. Facultad de Ciencia y Tecnología. Universidad Autónoma
de Entre Ríos (UADER). Oro Verde, Entre Ríos, Argentina. Instituto de
Matemática Aplicada del Litoral (IMAL-CONICET-UNL), CCT CONICET, Santa Fé,
Argentina. Corresponding author: Diego M. Mateos<EMAIL_ADDRESS>Leonardo E. Riveaud Facultad de Ingeniería, Universidad Nacional del Comahue
(FAIN, UNComa) Pedro W. Lamberti Consejo Nacional de Investigaciones
Científicas y Técnicas (CONICET), Argentina. Facultad de Matemática
Astronomía, Física y Computación (FaMAF), Universidad Nacional de Córdoba.
Córdoba, Argentina.
###### Abstract
Divergences or similarity measures between probability distributions have
become a very useful tool for studying different aspects of statistical
objects such as time series, networks and images. Notably not every divergence
provides identical results when applied to the same problem. Therefore it is
convenient to have the widest possible set of divergences to be applied to the
problems under study. Besides this choice an essential step in the analysis of
every statistical object is the mapping of each one of their representing
values into an alphabet of symbols conveniently chosen. In this work we attack
both problems, that is, the choice of a family of divergences and the way to
do the map into a symbolic sequence. For advancing in the first task we work
with the family of divergences known as the Burbea-Rao centroids (BRC) and for
the second one we proceed by mapping the original object into a symbolic
sequence through the use of ordinal patterns. Finally we apply our proposals
to analyse simulated and real time series and to real textured images. The
main conclusion of our work is that the best BRC, at least in the studied
cases, is the Jensen Shannon divergence, besides the fact that it verifies
some interesting formal properties.
Keywords— Divergences, Ordinal patterns, Time series analysis, Textured Images
The notion of distinguishability is of crucial importance in probability
theory and in general in every statistical theory. Due to statistical
fluctuations it becomes difficult to distinguish between close probability
distributions. This problem involves two issues. The first one is the
definition of distance measures between probability distributions and the
second one is the introduction of distinguishability criteria. On the other
hand, many physical phenomena can be represented by means of a time series,
which could be generated by chaotic dynamics or by a random proccess. It is
necessary, in order to be able to study these dynamics, to make a
correspondence between the values of the time series under study with a
symbolic sequence. In several occasions this task is the most difficult to
perform. In this work we address the different problems indicated above.
First, we study the applicability of a family of divergences between
probability distributions, known as Burbea-Rao centroids, to the study of time
series. We then investigate the use of ordinal patterns as a way to assign a
symbolic sequence to a time series. With these two tools we study the
statistical characteristics of simulated and real world time series as well as
to textured 2D images. As the Burbea-Rao centroids depend on a concave
functional we test the behaviour of this divergences when we change it. Among
them one of the choosen funtional leads to the Jensen Shannon divergence. In
addition of demonstrating the robustness of the scheme presented it was
possible to conclude that in all the investigated examples the best results
correspond to the Jensen Shannon divergence.
## 1 Introduction
There exists a profusion of divergences between probability distributions.
Remarkably when some of them are applied to the same statistical problem, in
general they do not lead to indistinguishable results. Therefore, might be
useful to have a large set of divergences. In general the divergences have
different origins. Some are statistical, others originated in information
theory and others borrowed from the realm of pure mathematics. The Fisher’s
metric is a conspicuous example of the first kind [1]; and the Kullback-
Leibler is a well known example originated in information theory [2]. Measures
of similarity (or dissimilarity), between probability distributions have
become of great interest in physics (classical and quantum), biology and many
other areas of science and technology [3, 4, 5, 6].
In the realm of statistical physics particularly notable has been the use of
divergences in the context of out-of-equilibrium systems. The obligatory
reference on this subject is the work of G. Crooks and D. Sivak. There these
authors give a physical interpretation to different measures of divergence
when are applied to the study of conjugate ensembles of non equilibrium
trajectories. The relative entropy is related to the dissipation, the Jeffreys
divergence is the average dissipation of the forward and reverse evolution
(hysteresis), the Jensen – Shannon divergence has been proposed as a magnitude
of the time arrow, and the Chernoff divergence is the work cumulant generating
function [7].
As it was said before having a variety of divergence could be useful for both
pure theoretical studies and in the context of applications. Sometimes it is
possible to introduce families of divergences, labelling each member with a
parameter [8, 9] or by giving a general structure depending on certain
functional with adequate characteristics. An example of this last case are the
Csiszar’s divergences or $f$-divergences.
Let $\Omega=\\{\omega_{1},...,\omega_{N+1}\\}$ be a discrete sample space. The
set of probability distributions on $\Omega$ can be identified with the
simplex $\mathbb{P}^{N}=\\{x^{i}\geq
0,i=1,...,N+1,\sum_{i=1}^{N+1}x^{i}=1\\}$. Given two probability distributions
$P=\\{p_{i}\\}_{i=1}^{N+1}$ and $Q=\\{q_{i}\\}_{i=1}^{N+1}$ belonging to
$\mathbb{P}^{N}$, a Csiszar’s divergence is defined by [10]
$\mathcal{D}_{f}(P,Q)=\sum_{i=1}^{N+1}f\left(\frac{p_{i}}{q_{i}}\right)q_{i}$
where $f(x)$ is a convex function such that $f(1)=0$. This family has been
extensively studied in the context of information geometry [11]. A remarkable
result is that when $p_{i}$ and $q_{i}=p_{i}+\delta p_{i}$ are two close
probability distributions, the divergence $\mathcal{D}_{f}(P,Q)$ is
proportional to the Fisher (Riemannian) metric:
$\mathcal{D}_{f}(P,P+\delta
P)\sim\frac{f^{\prime\prime}(1)}{2}\sum_{i}\frac{\delta p_{i}^{2}}{p_{i}}$ (1)
The above mentioned Kullback-Leibler divergence corresponds to $f(t)=t~{}\log
t$ and it is not symmetric and the Jensen-Shannon divergence (JSD) it is
obtained by taking $f(t)=(t+1)\log(\frac{2}{t+1})+t~{}\log t$, which clearly
is symmetric [12].
For the study of dynamical phenomena, we need to have a sequence of
measurements related to them. These sequences are usually given in the form of
time series which allow extracting information on the underlying physical
system under study. Something similar happen with images, where we have a two-
dimensional matrix which contains the information about each pixel of the
images. The time series or images can be associated with a probability
distribution function (PDF) and, from this to have a way for applying the
divergences. To this aim it is necessary to map the time series or images in a
finite alphabet. There exist many methods to discretize a continuous time
series, such as binarization or the wavelet analysis. However these methods do
not take into account the relation between the value for a given time and the
neighbouring values. In 2002 C. Bandt and B. Pompe introduced the ordinal
patterns for time series discretization [13]. This approach has been
extensively used to different problems with very relevant results. Its main
advantage is that reveals the underlying dynamics in the process that
generates the time series. For a review of these properties we suggest the
work of M. Zanin and F. Olivares in which the ordinal pattern approach is
studied in detail [14].
Here we use the Burbea-Rao centroids (BRC) as the dissimilarity measures and
we build a symbolic alphabet formed by ordinal patterns, in which the original
time series or the 2D images are mapped. Then we apply an segmentation
procedure meaning this a way to detect dynamical changes in the corresponding
symbolic sequence. All this scheme is used to the study of the statistical
properties of simulated time series and to real 2D images.
This manuscript is organised as follows. In section 2 we review some
properties of the BRC. We show that the BRC can be thought as a deformations
of the Euclidean metric and we indicate a particular character of the JSD as a
BRC. In section 3 we give a brief explanation of the ordinal patterns and how
them can be evaluated in some concrete cases. In section 4 we combine the use
of the BRC, really a particular case that we called the $\gamma$-divergences,
with the ordinal patterns to four examples, three related to time series
analysis and one to images analysis. Finally, in section 5 we discussed our
results as well as some other possible frameworks of applicability of our
scheme.
## 2 Burbea-Rao centroids
In 1982 J. Burbea and C.R. Rao introduced a family of divergences now known as
the Burbea-Rao centroids (BRC). They started from a concave $\Phi$ functional.
A generic member of this family is given by:
$\mathcal{J}_{\Phi}(P,Q)=\frac{1}{2}[\Phi(P)+\Phi(Q)]-\Phi\left(\frac{P+Q}{2}\right)$
(2)
where $P$ and $Q$ are two discrete probability distributions belongings to
$\mathbb{P}^{N}$.
Incidentally it is worth mentioning that a centroid can be interpreted as a
deformation of the square of the Euclidean metric. Indeed, let us write the
square of the Euclidean distance between $P$ and $Q$ in the form:
$\mathcal{E}(P,Q)=\mathop{\sum}_{i}(q_{i}-p_{i})^{2}=\mathop{\sum}_{i}\left(2~{}q_{i}^{2}+2~{}p_{i}^{2}-(p_{i}+q_{i})^{2}\right)$
(3)
which can be rewritten as a BRC with $\Phi=\sum_{i}x_{i}g(x_{i})$ and
$g(x)=x$. In this sense we can think that a BRC is a “deformation” of the
Euclidean metric through the map $\sum_{i}x_{i}^{2}\rightarrow\Phi$.
Analogously the Jensen-Shannon divergence (JSD) mentioned above can be written
as a BRC with $\Phi=\sum_{i}x_{i}g(x_{i})$ and $g(x)=\frac{1}{2}\log x$. The
explicit expression of the JSD is:
$\mathcal{D}_{JS}(P,Q)=\sum_{i}\frac{1}{2}p_{i}\log p_{i}+\frac{1}{2}q_{i}\log
q_{i}-\frac{1}{2}(p_{i}+q_{i})\log\left(\frac{p_{i}+q_{i}}{2}\right)$ (4)
An important characteristic of the JSD is that it is the only BRC that is a
Csiszar divergence. This can be easily proved by doing a Taylor expansion of
(2) up to second order. Not every BRC is a metric in the sense that verifies
the triangle inequality. However it has been shown that the square root of the
JSD is a true metric for the simplex $\mathbb{P}^{N}$ as well as the Euclidean
metric (3). In general the metric character of a BRC will depend on the
functional $\Phi$.
One advantage of a BRC that is particularly interesting for us is that by
modifying the function $\Phi$, we can determine the divergence that more
adequately allows to characterise the statistical object under study.
### 2.1 Weighted BRC
In 1991 J. Lin introduced a generalised version of the JSD by assigning
different weights to each probability distribution, $P$ and $Q$. Let $\pi_{P}$
and $\pi_{Q}$ two non negative numbers such that $\pi_{P}+\pi_{Q}=1$. Then the
weighted JSD is given by:
$\mathcal{D}_{JS}(P,Q;\pi_{P},\pi_{Q})=H_{S}(\pi_{P}P+\pi_{Q}Q)-\pi_{P}H_{S}(P)-\pi_{Q}H_{S}(Q)$
(5)
with $H_{S}(P)\equiv-\sum_{i}p_{i}\log(p_{i})$ the Shannon entropy. We can
proceed in the same way with a generic BRC and define the weighted BRC as
follows:
$\mathcal{J}_{\Phi}(P,Q,\pi_{P},\pi_{Q})\equiv\pi_{P}\Phi(P)+\pi_{Q}\Phi(Q)-\Phi(\pi_{P}P+\pi_{Q}Q)$
(6)
Due to the functional $\Phi$ it is assumed to be concave, as a consequence of
the Jensen inequality, the quantity (6) is always non negative.
Another possible generalisation of the BRC is to evaluate it among several
probability distributions each one with different weight. To fix ideas let us
suppose $K$ probability distributions $P^{(1)},...,P^{[K)}$ and weights
$\pi_{1},...\pi_{K}$. We can define a weighted RBC among these probability
distributions in the form:
$\mathcal{J}_{\Phi}(P^{(1)},...,P^{(K)};\pi_{1},...,\pi_{N})\equiv\sum_{k=1}^{K}\pi_{k}\Phi(P^{(k)})-\Phi\left(\sum_{k=1}^{K}\pi_{k}P^{(k)}\right)$
(7)
Again due to the concavity of $\Phi$ and the Jensen inequality this last
quantity is non negative.
## 3 Ordinal Patterns and Burbea Rao centroids
As was remarked in the introduction for studying the statistical properties of
a time series, and with the purpose of assigning probability distributions to
it, it is indispensable to map the original data set, that is, the values of
the time series, into symbolic sequences. There are different methods in the
literature for doing this, such as the discretization, binarization and
wavelet’s histograms, among others. An alternative method proposed by Bandt
and Pompe (BP) [13] consists in transforming the time series, via a non
parametric transformation, into a sequence of patterns and then making
inference over these patterns. With this, the analysis gains robustness and
becomes apt to detect relevant causal information related to the unobserved
variables that control the underlying dynamics of the system. The BP approach
is based on the computation of the Shannon entropy from the histogram of
causal patterns. Later the ideas of BP were extended in different ways, being
one particularly relevant for us the the application to the analysis of two
dimensional images [15, 16].
For a time series $\mathcal{X}(t)$ the ordinal patterns procedure is based on
the relative values of the neighbours belonging to the series, and in
consequence takes into account the time structure or causality of the process
that generated the sequence. To understand this idea, let us consider a real-
valued discrete-time series $\mathcal{X}(t)=\\{x_{t}\in\mathbb{R}\\}$, and the
parameters $d\geq 2$ ( embedding dimension) and $\tau\geq 1$ (the time delay).
From these we construct a $d$-dimensional vector $\mathbf{Y}_{t}$:
$\mathbf{Y}_{t}=(x_{t-(d-1)\tau},\dots,x_{t-\tau},x_{t})~{}\qquad~{}{\mathrm{with}}~{}\qquad~{}t\geq(d-1)\tau\
.$ (8)
The dynamical properties of the overall system is preserved by the vector
$\mathbf{Y}_{t}$. We sorted in ascending order the components of the phase
space trajectory $\mathbf{Y}_{t}$. Then, we can define a permutation vector,
$\mathbf{\Pi}_{t}$, with components given by the original position of the
sorted values. Each one of these vectors represents a pattern (or motif) with
$N_{\Pi}=d!$ possible patterns. For example let us consider the time series
$\mathcal{X}(t)=\\{0.42,~{}2.7,~{}4.2,~{}0.35,~{}1.5\\}$ and take the
parameters $d=3$ and $\tau=1$. We define the embedding vectors
$\mathbf{Y}_{t}$ as $\mathbf{Y}_{1}=(0.42,~{}2.7,~{}4.2)$;
$\mathbf{Y}_{2}=(2.7,~{}4.2,~{}0.35)$; $\mathbf{Y}_{3}=(4.2,~{}0.35,~{}1.5)$,
and the respective permutation vectors as $\mathbf{\Pi}_{1}=(0,~{}1,~{}2)$,
$\mathbf{\Pi}_{2}=(1,~{}2,~{}0)$ and $\mathbf{\Pi}_{3}=(2,~{}0,~{}1)$.
Regarding the selection of the parameters, Bandt and Pompe [13] suggested
working with $3\leq d\leq 6$ and specifically consider an embedding delay
$\tau=1$. Nevertheless, other values of $\tau$ could provide additional
information. It has been recently shown that this parameter is strongly
related to the intrinsic time scales of the system under analysis [17, 18]. In
addition, it is important to take into account that for a reliable estimate of
the ordinal probability distribution we need that, the length of the sequence
$N_{\mathcal{X}}$ must to be greater than the number of possible ordinal
patterns: $N_{\Pi}<<N_{\mathcal{X}}$ [13].
Ribeiro and Zunnino extended the permutation entropy framework to two-
dimensional data [15, 16]. We can consider a two-dimensional data array
$\\{x_{u}^{v}\\}$ with $u=1,...,N_{x}$ and $v=1,...,N_{y}$ whose elements
could be consider as pixels of an image. We define the embedding dimensions of
the horizontal ($d_{x}$) and vertical directions ($d_{y}$), and the
corresponding $\tau_{x}$ and $\tau_{y}$. Using a similar scheme that in the
time series case, we slice the data array in partitions of size $d_{x}\times
d_{y}$ defined by:
$\mathbf{Y}_{i}^{j}=\begin{bmatrix}x_{i}^{j}&x_{i}^{j+\tau_{y}}&\ldots&x_{i}^{j+(d_{y}-1)\tau_{y}}\\\
x_{i+\tau_{x}}^{j}&x_{i+\tau_{x}}^{j+\tau_{y}}&\ldots&x_{i+\tau_{x}}^{j+(d_{y}-1)\tau_{y}}\\\
\vdots&\vdots&\ddots&\\\
x_{i+(d_{x}-1)\tau_{x}}^{j}&x_{i+(d_{x}-1)\tau_{x}}^{j+\tau_{y}}&\ldots&x_{i+(d_{x}-1)\tau_{x}}^{j+(d_{y}-1)\tau_{y}}\end{bmatrix}$
(9)
where $i=1,\ldots,n_{x}$ and $j=1,\ldots,n_{y}$ , with
$n_{x}=N_{x}-(d_{x}-1)\tau_{x}$ and $n_{y}=N_{y}-(d_{y}-1)\tau_{y}$.
To associate a permutation symbol with each two-dimensional partition we
concatenate line by line the partition $y_{i}^{j}$
$\begin{split}\mathbf{Y}_{i}^{j}=&(x_{i}^{j},x_{i}^{j+\tau_{y}},\ldots,x_{i}^{j+(d_{y}-1)\tau_{y}},\\\
&x_{i+\tau_{x}}^{j},x_{i+\tau_{x}}^{j+\tau_{y}},\ldots,x_{i+\tau_{x}}^{j+(d_{y}-1)\tau_{y}}\\\
&x_{i+(d_{x}-1)\tau_{x}}^{j},x_{i+(d_{x}-1)\tau_{x}}^{j+\tau_{y}},\ldots,x_{i+(d_{x}-1)\tau_{x}}^{j+(d_{y}-1)\tau_{y}}.)\end{split}$
(10)
Then we evaluate the permutation symbol associated with each data partition as
in the one-dimensional case to define the symbolic array
${\mathbf{\Pi}_{i}^{j}}$ for $i=1,\ldots,n_{x}$ and $j=1,\ldots,n_{y}$ related
to the data set. In this case the number of possible ordinal patterns are
$N_{\Pi}=(d_{x}d_{y})!$ [15]. For a better comprehension of the method we will
present a simple example. Let us suppose we have the following $3\times 3$
array:
$A=\begin{bmatrix}2&3&7\\\ 4&5&6\\\ 1&7&8\end{bmatrix}$
For this case we take $d_{x}=d_{y}=2$ and $\tau_{x}=\tau_{y}=1$ obtaining four
partitions: $\mathbf{Y}_{1}=\left[\begin{smallmatrix}2&3\\\
4&5\end{smallmatrix}\right]=(2,3,4,5)$ with ordinal pattern
$\mathbf{\Pi}_{1}=(0,1,2,3)$; $\mathbf{Y}_{2}=\left[\begin{smallmatrix}3&7\\\
5&6\end{smallmatrix}\right]=(3,7,5,6)$ with ordinal pattern
$\mathbf{\Pi}_{2}=(0,3,1,2)$; $\mathbf{Y}_{3}=\left[\begin{smallmatrix}4&5\\\
1&7\end{smallmatrix}\right]=(4,5,1,7)$ with ordinal pattern
$\mathbf{\Pi}_{3}=(1,2,0,3)$ and
$\mathbf{Y}_{4}=\left[\begin{smallmatrix}5&6\\\
7&8\end{smallmatrix}\right]=(5,6,7,8)$ with ordinal pattern
$\mathbf{\Pi}_{4}=(0,1,2,3)$.
Once we have all the ordinal patters $\mathbf{\Pi}$ corresponding to a signal
or a image we compute the probability distribution $P_{op}=P(\Pi_{i})$ with
$i=1,\ldots,d!$ for one-dimensional sequence or $i=1,\dots,(d_{x}d_{y})!$ for
two-dimensional arrays.
We will compare two signals or images by evaluating a divergence between the
corresponding probability distributions for the associated ordinal patterns,
$P_{op}$ and $Q_{op}$. We used the BRC, that we name the $\gamma$-divergence,
explicitly given by:
$\mathcal{D}_{\gamma}(P_{op}||Q_{op})=2\mathop{\sum}_{i=1}^{N_{\Pi}}\gamma_{g}(p(\Pi_{i}),q(\Pi_{i}))$
(11)
with
$\gamma_{g}(p(\Pi_{i}),q(\Pi_{i}))=\frac{1}{2}p(\Pi_{i})~{}g(p(\Pi_{i}))+\frac{1}{2}q(\Pi_{i})~{}g(q(\Pi_{i}))-\frac{1}{2}(p(\Pi_{i})+q(\Pi_{i}))~{}g\left(\frac{p(\Pi_{i})+q(\Pi_{i})}{2}\right)$
(12)
The only requirement on the function $g(x)$ is that the sum
$\sum_{i}x_{i}g(x_{i})$ results concave.
## 4 Applications
To investigate the behaviour of the $\gamma$-family we use different
$\gamma$-functions, $xg(x)$, by changing the $g$ function. We apply them to
four cases. In the first example we analyse the coupling of the Henon-Henon
system; in the second one we detect the change in the dynamics between chaotic
and stochastic signals; in the third example we apply our scheme to detect the
changes in the stages of an EEG signal and finally we evaluate the distances
between textured 2D images. In each one of this examples, we work with four
$g(x)$ function; i) $g(x)=e^{x}$; $g(x)=~{}log(x)$; $g(x)=~{}\sqrt[]{x}$ and
$g(x)=~{}sinh(x)$.
### 4.1 Henon-Henon coupled system
The coupled Henon-Henon system is described by the following set of equations
[19, 20]:
$\displaystyle y_{1}(n+1)$ $\displaystyle=1.4-y_{1}^{2}(n)+b~{}y_{2}(n)$
$\displaystyle y_{2}(n+1)$ $\displaystyle=y_{1}(n)$ $\displaystyle x_{1}(n+1)$
$\displaystyle=1.4-(\epsilon~{}y_{1}(n)+(1-\epsilon)~{}x_{1}(n)+b~{}x_{2}(n))$
$\displaystyle x_{2}(n+1)$ $\displaystyle=x_{1}(n)$
where, in our choice $b=0.3$. For the simulation the coupling parameter
$\epsilon$ was varied from zero to $1$ in steps of $0.1$. For each $\epsilon$,
$200$ realisations were computed using random initial conditions. The signals
length were $N=100000$ and the ordinal patterns were evaluated with the
parameters $d=3,4,5$ and $\tau=1$.
The Figure 1 shows the results for the four divergences as a function of the
coupling parameter $\epsilon$. The median value of estimator
$\mathcal{D}_{\gamma}$ is maximum for $\epsilon=0$ in all the cases. This is
expected because there is no information sharing between the two systems. The
$\mathcal{D}_{\gamma}$ values decrease with the coupling parameter up to the
value $\epsilon=0.6$. In contrast, for $\epsilon\geq 0.7$ the
$\mathcal{D}_{\gamma}$ values are zero. For these values of the coupling
parameter, the Henon-Henon system is synchronised in such a way that both
systems are statistically indistinguishable. Analogous results were obtained
using the parameter $d=3,5$.
Similar behaviour has been already observed on other information quantifiers
such as the transfer entropy [19, 20, 21]. It is important to note that for
the function $g(x)=log(x)$ (that is, the JSD) the values between the maximum
and zero is $\sim 0.3$ almost an order of magnitude higher than for the other
functions $g(x)$.
Figure 1: Henon-Henon coupled system. Boxplot $\gamma$-divergence as a
function of the coupling parameter $\epsilon$. $\mathcal{D}_{\gamma}$ was
calculated with function $g(x)$ A) $e^{x}$, B) $log(x)$, C) $\sqrt[]{x}$, D)
$sinh(x)$. The ordinal patterns parameter used were $d=4$ and $\tau=1$.
### 4.2 Dynamical changes detection
In this example we use the $\gamma$-divergence to detect changes in the
dynamics of a signal. For this, we generated a group of mixed signals:
stochastic-chaotic and chaotic-chaotic. For the chaotic signal we use the
logistic map [22] given by the equation
$x_{n+1}=r~{}x_{n}\left(1-x_{n}\right)$ for $r=4$ and the cubic map [23]
$x_{n+1}=A~{}x_{n}\left(1-x_{n}^{2}\right)$ for $A=3$. For stochastic signal
we use a white noise. We generate $N=100$ mixed signals
$S_{s-c}^{i}=S_{s}+S_{c}=2000+2000=4000$ and
$S_{c_{1}-c_{2}}^{i}=S_{c_{1}}+S_{c_{2}}=2000+2000=4000$ with $i=1,...,100$
(Figure 2 top). For each signal we used a pointer method to study the
divergence. This method consists of a sliding pointer $p$ which move point to
point over the signal in the range $2d!\leq i\leq L-2d!$ splitting the signals
in two subsequences $S_{l}[x(1),\ldots,x(p)]$ and $S_{r}[x(p+1),\ldots,x(L)]$.
In this case we apply the weighted divergence define in eq.7 with weight
$w_{l}=L_{S_{l}}/L$ and $w_{r}=L_{S_{r}}/L$. We compute the
$\mathcal{D}_{\gamma}^{w_{l},w_{r}}(S^{l}||S^{r})$ for all $i$-signal and
finally measure the mean value $\mu=<\mathcal{D}_{\gamma}>_{i}$ and the
standard deviation $\sigma$. In figure 2 are shown the results for the two
studied cases: white noise-logistic map signal (A) and cubic-logistic maps
signal (B). The figure shows the $\mu$ (straight line) and $\sigma$ (shadow
bar) for the four above indicated $g(x)$ functions. I both cases the maximum
value ($\mathcal{D}_{\gamma}^{max}$) of the divergence is reached in the
transition between the two signals. Is important to remark that for every
function $g(x)$ the method detects the change in the dynamics but the highest
values of $\mathcal{D}_{\gamma}^{max}$ corresponds to $g(x)=log(x)$.
Figure 2: Dynamical changes detection in mixed signal using
$\gamma$-divergence for different function $g(x)$. A) Mixed signal white noise
- logistic map, B) Mixed signal cubic map - logistic map. The ordinal patterns
parameter used were $d=4$ and $\tau=1$.
### 4.3 Detection of different states of sleep
In this example we use the proposed scheme to identify the transition between
sleep states from an electroencephalogram (EEG) record.
The problems associated with sleep have serious repercussions on people’s
health. That is why their study is of great clinical relevance. [24, 25, 26].
Usually two primary sleep stages are identified: rapid eye movement (REM)
sleep and non-REM sleep (NREM). REM is defined as an active sleeping period
showing an intense brain activity. Brain waves remain fast and desynchronised
similar to those in the waking state. In NREM sleep stage the physiological
activity decrease, the brain waves get slower and have greater amplitude,
breathing, and heart rate slows down and blood pressure drops. The NREM phase
is composed by three stages: N1, N2, and N3. The N1 stage is characterised by
perceived drowsiness or transition from being awake to falling asleep observed
by slowing down the brain waves and muscle activity. Stage N2 is a period of
light sleep during which eye movement stops. Brain waves become slower (Theta
waves (4-7 Hz)) with occasional bursts of rapid waves (12-14 Hz) called sleep
spindles, coupled with spontaneous periods of muscle tone mixed. Lastly, stage
N3 is characterised by the presence of Delta (0.5-4 Hz) slow waves,
interspersed with smaller, faster waves [27]. The N3 stage is a deep sleep
stage, without eye movement, and a decrease of muscle activity, resembling a
coma state. Usually, sleepers pass through these four stages (REM, N1, N2, and
N3) cyclically. A complete sleep cycle takes an average of 90 to 110 minutes,
with each one lasting between 5 to 15 minutes. These cycles must be maintained
for healthy body function in awake state [28]. Developing tools that can
detect the changes in dynamic sleep stages over EEG signal are highly
essential for studying patients with sleep disorders [29]. Here we use our
$\gamma$-divergence to detect changes in the dynamics of the EEG to detect the
transition from one state to another.
The data were taken from the Physionet database: The Sleep-EDF Database
[Expanded] [30, 31], and are freely available at [32]. The EEG were recorded
(Fpz-Cz) thorough bipolar channels and the sampling frequency was $100$ Hz.
Initially, we extracted segments from the original signal belonging to the
five different sleep states: Awake, REM, N1, N2, N3 111We used the notes
provided by the database. Each segments had $6000$ points (corresponding to
$60$ sec of recording) and were joined into a single signal 222This was done
for a better visualisation of the results, because the time of each sleep
state can vary from seconds to several minutes. (Figure 3A.) The signal was
preprocessed with a band-pass filter between $0.5-60$ Hz. The ordinal pattern
were computed using the parameter $d=4$ and $\tau=1$.
Figure 3B shows that our method could detect the transition between different
sleep states for all the used functions $g(x)$. The $\log(x)$ function shows
the highest values and the best differentiation between stages. The transition
between Awake-REM and N2-N3 are more remarkable than in REM-N1 and N1-N2.
There is a significant differentiation between N2-N3 because of N3 is the
deepest sleep stage. In this stage the body becomes more insensitive to
outside stimuli. In this state, the EEG signal is mostly composed by slow
waves (Delta and Theta) causing the brain dynamics to be sharply different
from the other states. Lower $\gamma$-divergences values were found between
N1-N2 states showing that both states share similar characteristics in their
dynamics. Particularly, N1 is characterised by drowsiness slowing down the
brain waves and muscle activity. N1-N2) are very similar stages being the main
difference that in N1 occasional bursts of rapid waves (12-14 Hz), called
sleep spindles appear.
Similar result can be found between REM and N1. REM and N1 also present lower
values of the gamma divergence. This is something to be expected considering
that the EEG during the REM stage contains frequencies present in the “awake”
state and in the lighter stages of sleep N1 [33]. Despite the similarities of
the REM and N1, there are still enough differences between them such that
statistically different values are obtained. The presence of $11-16$ Hz
activity (sleep spindles) in N1, and more abundant alpha activity ($8-13$Hz)
in REM sleep means that these two stages present activity at an overlapping
frequency range, which explains the proximity of the divergence values
obtained. Difficulty in detecting N1 and REM sleep has also been found using
other measures [34, mateos2021using].
Figure 3: Application of the $\gamma$-divergence over a sleep EEG signal
using a running windows. A) The EEG signal is composed by $5$ sub-signal
belonging to different sleep stages (Awake, REM, N1, N2, N3). Each state has
$6000$ point corresponding to $60$ sec recording (dashes horizontal lines). B)
Results corresponding to the running windows method (see text for details).
The signal was quantified with the permutation vectors with parameter $d=4$
and $\tau=1$. For all functions, the maximum values
$\mathcal{D}_{\gamma_{max}}$ were reached in the exact point where a
transition between sleep states exists.
### 4.4 Distance between textured images
As a last application we use the $\gamma$-divergence to measure the distance
between 2D textured images. The images were taken from the Normalized Brodatz
Texture (NBT) album avalible in
https://multibandtexture.recherche.usherbrooke.ca. This normalised database
has 112 texture images an improvement regarding the original Brodatz texture
database since that grayscale background effects have been removed [35].
Because of this, it is impossible to discriminate between textures from this
normalised database using only first order statistics. Six different samples of
the NBT database are illustrated in Figure 4A. The images of the NBT album
have dimensions of $640\times 640$ pixels and 8 bits/pixel, which provides 256
grayscale levels.
For the 2D quantization, the ordinal parameters used were $d_{x}=d_{y}=2$ and
$\tau_{x}=\tau_{y}=1$. Figure 4B shows the divergence matrices calculated for
the different functions $g(x)$. We can see all the divergence values allow us
to distinguish different textures from each other. The divergence is higher
when the texture patterns are more different, for example between D15 and D47,
and lower when the patterns have a similar geometry for example between D1 and
D71. The four divergences show similar relative values between the images
however, the values for $g(x)=log(x)$ have an order of magnitude higher than
the other functions.
Figure 4: A) Six samples of the NBT album with their corresponding labels are
illustrated. B) Location of the 112 texture images from the NBT album in the
CECP. $\gamma$-divergence matrix between textured images showing in A using 2D
ordinal patterns with $d_{x}=d_{y}=2$ and $\tau_{x}=\tau_{y}=1$.
## 5 Discussion
We showed some particular aspects of the JSD when thought of as a BRC. In
particular that is the only BRC that is a Csiszar divergence besides being one
of the few that has a metric character. Then we defined the weighted BRC,
which can be explored in the context of Bayesian inference. We restrict
ourselves to the application of these weighted version to the analysis of the
dynamical behaviour of time series. Afterwards, we showed that the ordinal
patterns method is a natural way to apply divergence in signal and image
analysis. The ordinal patterns allow a natural quantification for the use of
one special kind of BRC that we called $\gamma$-divergence.
We applied the $\gamma$-divergence to four examples, three for signals
analysis and one for image analysis. Among the signal analyses, we studied the
coupling between two signals generated by the Hennon-Hennon system. We
observed that divergence can quantify the coupling of the system depending on
the coupling parameter. In the second example, we use divergence to detect
changes in the dynamics of combined signals. We noted that divergence can
accurately detect the transition point between two chaotic-stochastic and
chaotic-chaotic systems. Later, we analysed EEG signal from sleep patient. We
could detect the points where the signal changes its dynamics because of the
change in the state of sleep. Showing it can be an alternative tool for
detecting different sleep states. Finally, we used this family of divergences
to study distances between textured images. We could realised divergence
values were smaller for similar textures and larger for different textures.
One interesting result is for all the examples studied, the largest divergence
values occur for the generating function $g(x)=log(x)$ which is precisely the
Jensen-Shannon divergence.
It is also important to mention that when using divergences, we had consider
the significance of the value obtained. This significance is what allows us to
say if the distance found is really a real and not just a measure of the
statistical fluctuations. In this work we used the standard deviation
calculated over $N$ realizations as a measure of significance. However, many
times we do not have more than one realization being this method useless.
Therefore, a theoretical study of the field is necessary. Some studies on this
topic have already been addressed by Grosse et al. for the Jensen-Shannon
divergence [36]. Similar studies should be carried out for this family of
$\gamma$-divergence. However, this exceeds this work but will be a addressed
in the near future.
## 6 Aknowledgements
We are grateful to Secretaria de Ciencia y Tecnica de la Universidad Nacional
de Córdoba, Argentina, for financial assistance. This work was partially
supported by Grant PIP 519/2019 from CONICET, Argentina and by the Grant PICT
2019-01750 from Angencia Nacional de Promoción de la Investigación, el
Desarrollo Tecnológico y la Inovación, Argentina.
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|
Tools for estimating fake/non-prompt lepton backgrounds with the ATLAS
detector at the LHC Measurements and searches performed with the ATLAS
detector at the CERN Large Hadron Collider often involve signatures with one
or more prompt leptons. Such analyses are subject to ‘fake/non-prompt’ lepton
backgrounds, where either a hadron or a lepton from a hadron decay or an
electron from a photon conversion satisfies the prompt-lepton selection
criteria. These backgrounds often arise within a hadronic jet because of
particle decays in the showering process, particle misidentification or
particle interactions with the detector material. As it is challenging to
model these processes with high accuracy in simulation, their estimation
typically uses data-driven methods. Three methods for carrying out this
estimation are described, along with their implementation in ATLAS and their
performance. CERN-EP-2022-214 JINST
###### Contents
1. 1 Introduction
2. 2 Methods
1. 2.1 Matrix method
1. 2.1.1 Asymptotic matrix method
2. 2.1.2 Poisson likelihood matrix method
2. 2.2 Fake-factor method
3. 2.3 Generalisation for multi-lepton final states
4. 2.4 Use with weighted events
5. 2.5 Performance studies
3. 3 The ATLAS detector
4. 4 Lepton selection criteria
1. 4.1 Electron reconstruction and identification
2. 4.2 Muon reconstruction and identification
3. 4.3 Lepton isolation
4. 4.4 Removing overlaps between jets and leptons
5. 5 Monte Carlo simulation samples
6. 6 Sources of fake/non-prompt leptons
7. 7 Measurement of real and fake/non-prompt lepton efficiencies
1. 7.1 Real-lepton efficiencies
2. 7.2 Fake/non-prompt lepton efficiencies
8. 8 Systematic uncertainties
1. 8.1 Statistical uncertainties in the measured efficiencies
2. 8.2 Systematic uncertainties in the measured efficiencies
3. 8.3 Uncertainties in the modelling of real-lepton processes
4. 8.4 Uncertainties due to biases in the likelihood matrix method
9. 9 Examples of application in ATLAS analyses
1. 9.1 Measurement of the $t\bar{t}Z$ cross-section in final states with three or four leptons
1. 9.1.1 Real-lepton efficiencies
2. 9.1.2 Fake/non-prompt lepton efficiencies
3. 9.1.3 Results in the fake/non-prompt lepton validation regions
2. 9.2 Model-independent search for new phenomena in multi-lepton final states
1. 9.2.1 Fake/non-prompt lepton selection
2. 9.2.2 Fake factors
3. 9.2.3 Validation regions
10. 10 Conclusions
## 1 Introduction
Many measurements and searches for new physics performed with the ATLAS
detector [1] at the Large Hadron Collider (LHC) at CERN require the presence
of one or more leptons (electrons or muons) to indicate that a high-energy
electroweak process occurred in the collision. Lepton candidates are
reconstructed from signals in the inner tracker, calorimeters, and muon
spectrometer [2, 3]. Identification criteria are then applied to suppress
candidates originating from physical objects other than leptons from the hard-
scattering process of the event. These background candidates fall into two
categories: $i$) ‘non-prompt leptons’ from the semileptonic decay of hadrons,
or from photon ($\gamma$) conversions in detector material, and $ii$) ‘fake
leptons’ where the reconstructed object is not, in fact, due to a lepton. In
contrast to the aforementioned categories, ‘real leptons’ are defined as
electrons or muons produced either directly in the hard-scattering process or
directly in the decay of a short-lived non-hadronic resonance (such as a
$W$/$Z$ boson).
The rates at which fake or non-prompt leptons are selected are difficult to
model accurately from simulation. They can depend strongly on details of the
physics simulation, including in non-perturbative regions where the simulation
would not be expected to be reliable. They also depend on the modelling of the
material composition and response of the detector. In addition, fake leptons
sometimes occur with low probability as a result of misidentifying a hadronic
jet in multi-jet events. The computing resources required to simulate these
processes with a sufficient sample size would be prohibitive. Therefore,
‘data-driven’ approaches are commonly used to estimate these backgrounds.
To simplify the adoption of such methods, and to ensure that they are applied
uniformly, a set of standard tools and prescriptions has been developed for
use in ATLAS physics analyses that are subject to fake/non-prompt lepton
backgrounds. The principles and performance of these tools are described in
this paper. The motivation and mathematical basis of the methods are explained
in Section 2; a description of the relevant features of the ATLAS detector is
given in Section 3; the criteria used to select leptons are given in Section
4; the simulated signal and background processes, as well as the different
processes that can lead to fake/non-prompt leptons, are discussed in Sections
5 and 6; the procedures used to measure the efficiencies for real and
fake/non-prompt leptons are described in Section 7; the systematic
uncertainties associated with the methods are described in Section 8; and
Section 9 provides examples of the application of the fake/non-prompt lepton
estimation methods in two published ATLAS physics analyses, with details that
are not included in the existing publications.
## 2 Methods
The fake/non-prompt lepton estimation methods considered in this paper depend
on defining two tiers of lepton selection criteria, called the ‘baseline’ and
‘tight’ criteria. The tight criteria are used to select the signal leptons in
a physics analysis, while the baseline criteria accept all of the tight lepton
candidates as well as an additional set of candidates with a higher rate of
fake/non-prompt contributions. Candidates that satisfy the baseline criteria
but not the tight criteria are called ‘loose’ leptons. If the two sets of
criteria are chosen well, the fraction of real leptons in the baseline sample
that satisfy the tight criteria will be substantially higher than the
corresponding fraction for fake/non-prompt leptons. These fractions are called
the ‘real efficiency’ ($\varepsilon_{\mathrm{r}}$) and ‘fake efficiency’
($\varepsilon_{\mathrm{f}}$), respectively.
The $\varepsilon_{\mathrm{r}}$ values are generally taken from Monte Carlo
(MC) simulated events that are corrected to account for differences between
data and the simulation, while the $\varepsilon_{\mathrm{f}}$ values are
typically measured in a data sample that is orthogonal to the one that is used
for the data analysis, as detailed in Section 7.
### 2.1 Matrix method
With the efficiencies known, a simple counting of the numbers of lepton
candidates that satisfy the tight and loose criteria provides an estimate for
the number of fake/non-prompt leptons. In the simplest case, where an analysis
selects signal events containing exactly one tight lepton candidate and no
loose lepton candidates, the relationship between the numbers of tight and
loose leptons observed in data and the composition of the sample in terms of
real and fake/non-prompt leptons is:
$\left({\begin{array}[]{*{20}{c}}{{N^{\mathrm{t}}}}\\\ {{N^{\mathrm{l}}}}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}\varepsilon_{\mathrm{r}}&\varepsilon_{\mathrm{f}}\\\
{1-\varepsilon_{\mathrm{r}}}&{1-\varepsilon_{\mathrm{f}}}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{N_{\mathrm{r}}^{\mathrm{b}}}\\\
{N_{\mathrm{f}}^{\mathrm{b}}}\end{array}}\right),$ (1)
where $N^{\mathrm{t}}$ and $N^{\mathrm{l}}$ are the numbers of events with
tight and loose lepton candidates, and $N_{\mathrm{r}}^{\mathrm{b}}$ and
$N_{\mathrm{f}}^{\mathrm{b}}$ are the unknown numbers of real and fake/non-
prompt leptons in the baseline sample. In matrix notation, the relationship is
given by:
${\mathbf{N}}_{\mathrm{tl}}={\mathbf{M}}_{\varepsilon}{\mathbf{N}}_{\mathrm{rf}}^{\mathrm{b}}.$
(2)
The fact that the unknown values ($N_{\mathrm{r}}^{\mathrm{b}}$ and
$N_{\mathrm{f}}^{\mathrm{b}}$) and the observed yields ($N^{\mathrm{t}}$ and
$N^{\mathrm{l}}$) are related via the matrix ${\mathbf{M}}_{\varepsilon}$
gives rise to the name of this method: the ‘matrix method’. Inversion of the
matrix allows $N_{\mathrm{f}}^{\mathrm{b}}$ to be determined:
$N_{\mathrm{f}}^{\mathrm{b}}=\frac{1}{\varepsilon_{\mathrm{r}}-\varepsilon_{\mathrm{f}}}\left[(\varepsilon_{\mathrm{r}}-1)N^{\mathrm{t}}+\varepsilon_{\mathrm{r}}N^{\mathrm{l}}\right].$
(3)
In the typical use case, the quantity of interest is the number of events in
the tight sample where the lepton is fake/non-prompt,
$N_{\mathrm{f}}^{\mathrm{t}}$. This is related to the number of such events in
the baseline sample, $N_{\mathrm{f}}^{\mathrm{b}}$, by:
$N_{\mathrm{f}}^{\mathrm{t}}=\varepsilon_{\mathrm{f}}N_{\mathrm{f}}^{\mathrm{b}}.$
(4)
Similarly, the number of real leptons in the tight sample is
$N_{\mathrm{r}}^{\mathrm{t}}=\varepsilon_{\mathrm{r}}N_{\mathrm{r}}^{\mathrm{b}},$
(5)
and these can be treated as elements of a column matrix
${\mathbf{N}}_{\mathrm{rf}}^{\mathrm{t}}$.
The fact that $N^{\mathrm{t}}$ appears in Eq. (1) means that information about
the content of the analysis signal region is used in the estimate, and
therefore an analysis is not completely blinded when using this approach.
Generally, the values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ depend on the lepton candidate’s momentum,
proximity to other objects, or other factors. Details of how these variations
are accounted for in the estimation are given below.
#### 2.1.1 Asymptotic matrix method
In this method, events in the baseline sample are considered one at a time,
and a ‘fake weight’ $w_{i}$ is defined for each event, corresponding to the
two terms in Eq. (3) via Eq. (4):
$w_{i}=\left\\{{\begin{array}[]{*{20}{c}}{\frac{\varepsilon_{\mathrm{f},i}}{\varepsilon_{\mathrm{r},i}-\varepsilon_{\mathrm{f},i}}(\varepsilon_{\mathrm{r},i}-1)}&{\hbox{if
lepton candidate $i$ is tight (so $N^{\mathrm{t}}=1$ and
$N^{\mathrm{l}}=0$)}}\\\\[5.0pt]
{\frac{\varepsilon_{\mathrm{f},i}}{\varepsilon_{\mathrm{r},i}-\varepsilon_{\mathrm{f},i}}}\varepsilon_{\mathrm{r},i}&{\hbox{if
lepton candidate $i$ is loose (so $N^{\mathrm{t}}=0$ and
$N^{\mathrm{l}}=1$)}}\end{array}},\right.$
where $\varepsilon_{\mathrm{r},i}$ and $\varepsilon_{\mathrm{f},i}$ are the
values of $\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ that are
appropriate for lepton $i$. Since $\varepsilon_{\mathrm{r},i}$ is always less
than one, the weight for an event with a tight lepton candidate is negative.
By extension, the total fake/non-prompt lepton background in the tight sample
is then estimated by
$N_{\mathrm{f}}^{\mathrm{t}}=\sum_{\mathrm{events}}w_{i}.$
This approach is convenient since the $w_{i}$ need only be calculated once and
then can be stored with the event, allowing the distribution of the fake-
lepton yield to be binned in any variable of interest, as well as a simple re-
computation of $N_{\mathrm{f}}^{\mathrm{t}}$ if the event selection is
modified. One drawback is that since the value of $w_{i}$ may be negative,
there is no guarantee that $N_{\mathrm{f}}^{\mathrm{t}}$ will be positive. The
value of $N_{\mathrm{f}}^{\mathrm{t}}$ is also sensitive to fluctuations in
the input $\varepsilon_{\mathrm{r},i}$ and $\varepsilon_{\mathrm{f},i}$
values.
The statistical uncertainty of $N_{\mathrm{f}}^{\mathrm{t}}$ is given by:
$\sigma_{N_{\mathrm{f}}^{\mathrm{t}}}=\sqrt{\sum_{\mathrm{events}}w^{2}_{i}}.$
(6)
The method that makes use of the $w_{i}$ is known as the ‘asymptotic matrix
method’, since Eq. (6) is only valid in the asymptotic limit with a large
number of events.
#### 2.1.2 Poisson likelihood matrix method
In this method [4]111An earlier variant of the likelihood matrix method is
described in Ref. [5]. the elements of
${\mathbf{N}}_{\mathrm{rf}}^{\mathrm{t}}$ are treated as free parameters,
which are varied to maximise the likelihood of the observed
${\mathbf{N}}_{\mathrm{tl}}$ values. By doing so, the Poisson-distributed
nature of the ${\mathbf{N}}_{\mathrm{tl}}$ values is taken into account, so
there is no need to use the asymptotic approximation. In the fit, the
${\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}$ values are converted to
${\mathbf{N}}^{\mathrm{b}}_{\mathrm{rf}}$ using Eqs. (4) and (5), where the
entries in the matrix ${\mathbf{M}}_{\varepsilon}$ are calculated using the
averages of the prompt and fake/non-prompt lepton efficiencies in the baseline
sample:
${\mathbf{M}}_{\varepsilon}=\left({\begin{array}[]{*{20}{c}}\langle\varepsilon_{\mathrm{r}}\rangle&\langle\varepsilon_{\mathrm{f}}\rangle\\\
{1-\langle\varepsilon_{\mathrm{r}}\rangle}&{1-\langle\varepsilon_{\mathrm{f}}\rangle}\\\
\end{array}}\right).$ (7)
The resulting ${\mathbf{N}}^{\mathrm{b}}_{\mathrm{rf}}$ values are used to
obtain the expectation values for ${\mathbf{N}}_{\mathrm{tl}}$ using Eq. (2).
These expectation values are denoted by
${\mathbf{N}}_{\mathrm{tl},\mathrm{exp}}$. The
${\mathbf{N}}_{\mathrm{rf}}^{\mathrm{t}}$ parameters are adjusted (subject to
the constraint that they must be non-negative) to maximise the joint Poisson
likelihood
$L\left({\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}\right)\equiv\prod_{i}P\left[N_{\mathrm{tl}_{i}}|N_{\mathrm{tl},\mathrm{exp}_{i}}\left({\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}\right)\right],$
(8)
where the product is over the elements of ${\mathbf{N}}_{\mathrm{tl}}$;
$N_{\mathrm{tl}_{i}}$ and $N_{\mathrm{tl},\mathrm{exp}_{i}}$ are the $i$th
elements of ${\mathbf{N}}_{\mathrm{tl}}$ and
${\mathbf{N}}_{\mathrm{tl},\mathrm{exp}}$, respectively; and
$P\left[n|\mu\right]$ is the Poisson probability for observing $n$ events when
$\mu$ are expected. The output of the fit consists of an estimate of the
number of fake/non-prompt leptons and the uncertainty in this quantity, which
is obtained by noting the value for which $-\ln L$ exceeds its minimum by
$0.5$.
The primary advantages of the Poisson likelihood approach are that the result
is constrained to be non-negative, and the uncertainty is a better
approximation to the range that gives 68% coverage, particularly in samples
with few events. In addition, in some scenarios it can provide smaller
statistical uncertainties than the asymptotic matrix method or the fake-factor
method (described in Section 2.2). The main drawback is that the estimated
yield must be calculated for the sample as a whole rather than from a sum of
individual event weights, which complicates the process of producing a
distribution of the fake-lepton yield, as required for any differential
measurement (to do so, the likelihood must be applied in every bin of the
distribution).
### 2.2 Fake-factor method
The fact that real-lepton kinematic distributions and efficiencies are
generally modelled well in simulation, and that scale factors can be applied
to account for any differences observed between the values in simulation and
in data control samples, leads to an alternative method that uses simulation,
rather than the data, to measure the real-lepton contribution to the loose
lepton sample.
The number of fake-lepton events in the loose sample is
$N_{\mathrm{f}}^{\mathrm{l}}=N^{\mathrm{l}}-N_{\mathrm{r}}^{\mathrm{l}}=(1-\varepsilon_{\mathrm{f}})N_{\mathrm{f}}^{\mathrm{b}}$
and the number of fake-lepton events in the tight sample is
$N_{\mathrm{f}}^{\mathrm{t}}=\varepsilon_{\mathrm{f}}N_{\mathrm{f}}^{\mathrm{b}}=F(N^{\mathrm{l}}-N_{\mathrm{r}}^{\mathrm{l}}),$
where the ‘fake factor’ $F$ is defined as
$F\equiv\varepsilon_{\mathrm{f}}/(1-\varepsilon_{\mathrm{f}})$. Thus, in the
fake-factor method, the number of tight fake/non-prompt leptons for a given
analysis can be computed using the fake factor $F$, the total number of loose
lepton candidates $N^{\mathrm{l}}$, and the number of real leptons in the
loose lepton sample $N_{\mathrm{r}}^{\mathrm{l}}$, where the latter quantity
can be estimated using MC simulated samples, and its contribution subsequently
subtracted from the quantity $FN^{\mathrm{l}}$ observed in the data. In
practice, the calculation is performed on an event-by-event basis to account
for potential variations in $F$ due to properties of the lepton:
$N_{\mathrm{f}}^{\mathrm{t}}=\sum_{\mathrm{data},i=1}^{N^{\mathrm{l}}}F_{i}-\sum_{\mathrm{MC},j=1}^{N^{\mathrm{l}}_{\mathrm{MC}}}w_{\mathrm{MC},j}F_{j},$
where $F_{i}$ is the fake factor appropriate for lepton $i$, all sources of
prompt leptons are considered in the sum over MC simulated events,
$N^{\mathrm{l}}_{\mathrm{MC}}$ is the number of MC events in the loose sample,
and $w_{\mathrm{MC},j}$ is the weight assigned to simulated event $j$, based
on the cross-section of the simulated process and any corrections to the
selection efficiency that may be needed to reflect the performance on data
events.
The main advantage of the fake-factor method is that this result does not
depend on $N^{\mathrm{t}}$, i.e. the yield in the analysis signal region. This
means that unlike the matrix method, the fake-factor method can be applied
while remaining ‘blind’ to the contents of the signal region. However, the
method does have some of the same drawbacks as the asymptotic matrix method,
namely the possibility of $N_{\mathrm{f}}^{\mathrm{t}}$ being negative, and
sensitivity to fluctuations in the $\varepsilon_{\mathrm{f}}$ values.
### 2.3 Generalisation for multi-lepton final states
The above methods can be generalised to cases where multiple baseline lepton
candidates are considered in each event. For the matrix methods, this is done
by increasing the dimensionality of ${\mathbf{M}_{\varepsilon}}$,
${\mathbf{N}}_{\mathrm{tl}}$, and
${\mathbf{N}}_{\mathrm{rf}}^{\mathrm{(t,b)}}$ to $2^{n_{\mathrm{b}}}$, where
$n_{\mathrm{b}}$ is the number of baseline lepton candidates in each event.
The estimated fake/non-prompt yield depends on the requirements of a
particular analysis in three ways: first, from the requirement placed on the
desired number $n_{\mathrm{t}}$ of tight lepton candidates per event; second,
whether or not events with additional loose lepton candidates are vetoed; and
third, on the minimum number of fake/non-prompt leptons $m_{\mathrm{f}}$
defining the background to be evaluated with one of the data-driven
methods.222An example of a case where $m_{\mathrm{f}}$ is greater than one
would be a dilepton analysis where there are backgrounds from both
$W(\rightarrow\ell\nu)+j$ events where the jet forms a fake/non-prompt lepton,
and dijet events where both jets form fake/non-prompt leptons. The analysers
may choose to estimate the first contribution from MC simulation, and use
data-driven methods for the second contribution, and therefore setting
$m_{\mathrm{f}}=2$ for the data-driven approach is required to avoid double-
counting. The consideration of $m_{\mathrm{f}}$ is reflected in the transition
from the number of fake/non-prompt lepton events in the baseline sample to the
number in the tight sample, in a generalisation of Eq. (4):
$N_{\mathrm{f}}^{\mathrm{t}}=\sum_{i}g_{i}\left(\varepsilon_{\mathrm{r}},\varepsilon_{\mathrm{f}}\right)N_{\mathrm{rf},i}^{\mathrm{b}}.$
Here, the sum is over all combinations of real and fake/non-prompt leptons
that include at least $m_{\mathrm{f}}$ fake/non-prompt leptons, $g_{i}$ is a
function of the real and fake/non-prompt efficiencies that will result in the
required number of tight lepton candidates for a given set of real and
fake/non-prompt leptons, and $N_{\mathrm{rf},i}^{\mathrm{b}}$ is the $i$th
element of $\mathbf{N_{\mathrm{rf}}^{\mathrm{b}}}$
To address the requirements on $n_{\mathrm{t}}$ and the possible presence of
additional loose lepton candidates, the analysis must consider all events in
the baseline sample with lepton multiplicities up to the sum of the allowed
numbers of tight and loose lepton candidates in the signal region.
As an example, consider the case where $n_{\mathrm{b}}=2$. If an analysis
selects signal events containing exactly two tight lepton candidates and no
loose lepton candidates, and the background with $m_{\mathrm{f}}\geq 1$ is
being evaluated, then
$N_{\mathrm{f}}^{\mathrm{t}}=\varepsilon_{\mathrm{r},1}\varepsilon_{\mathrm{f},2}N_{\mathrm{r_{1}f_{2}}}^{\mathrm{b}}+\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{r},2}N_{\mathrm{f_{1}r_{2}}}^{\mathrm{b}}+\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{f},2}N_{\mathrm{f_{1}f_{2}}}^{\mathrm{b}},$
where $N_{\mathrm{r_{1}f_{2}}}^{\mathrm{b}}$ is the number of events in the
baseline sample where the first lepton candidate is real and the second is
fake/non-prompt, and $N_{\mathrm{f_{1}r_{2}}}^{\mathrm{b}}$ and
$N_{\mathrm{f_{1}f_{2}}}^{\mathrm{b}}$ are defined correspondingly. The
ordering of the lepton candidates is typically according to $p_{\text{T}}$,
but the method does not depend on the ordering used.
For an analysis with $m_{\mathrm{f}}\geq 1$ that accepts events with two tight
lepton candidates or one tight and one loose lepton candidate, the expression
would change to
$N_{\mathrm{f}}^{\mathrm{t}}=(\varepsilon_{\mathrm{r},1}+\varepsilon_{\mathrm{f},2}-\varepsilon_{\mathrm{r},1}\varepsilon_{\mathrm{f},2})N_{\mathrm{r_{1}f_{2}}}^{\mathrm{b}}+(\varepsilon_{\mathrm{f},1}+\varepsilon_{\mathrm{r},2}-\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{r},2})N_{\mathrm{f_{1}r_{2}}}^{\mathrm{b}}+(\varepsilon_{\mathrm{f},1}+\varepsilon_{\mathrm{f},2}-\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{f},2})N_{\mathrm{f_{1}f_{2}}}^{\mathrm{b}},$
where the additional terms are needed to account for the additional ways that
fake/non-prompt leptons might satisfy the signal selection (and the terms
involving products of the efficiencies are subtracted to avoid double-
counting). For an analysis that imposes the same lepton candidate requirements
but only the background with $m_{\mathrm{f}}=2$ is being evaluated, the
expression is:
$N_{\mathrm{f}}^{\mathrm{t}}=(\varepsilon_{\mathrm{f},1}+\varepsilon_{\mathrm{f},2}-\varepsilon_{\mathrm{f},1}\varepsilon_{\mathrm{f},2})N_{\mathrm{f_{1}f_{2}}}^{\mathrm{b}}.$
The methods can also be extended to cases where there are more than two levels
of lepton selection criteria, or distinct categories of fake/non-prompt
leptons, as in Ref. [6].
As with the matrix method, the fake-factor method can be generalised to higher
lepton candidate multiplicities. In the dilepton final state, the number of
events with two tight lepton candidates, of which at least one is fake/non-
prompt, is:
$N_{\mathrm{f}}^{\mathrm{t}}=N^{\mathrm{t_{1}t_{2}}}-N_{\mathrm{r_{1}r_{2}}}^{\mathrm{t_{1}t_{2}}}=\\\
F_{1}(N^{\mathrm{l_{1}t_{2}}}-N_{\mathrm{r_{1}r_{1}}}^{\mathrm{l_{1}t_{2}}})+F_{2}(N^{\mathrm{t_{1}l_{2}}}-N_{\mathrm{r_{1}r_{2}}}^{\mathrm{t_{1}l_{2}}})-F_{1}F_{2}(N^{\mathrm{l_{1}l_{2}}}-N_{\mathrm{r_{1}r_{2}}}^{\mathrm{l_{1}l_{2}}}),$
(9)
where $N^{\mathrm{t(l)_{1}t(l)_{2}}}$ is the number of events where the first
lepton candidate is tight (loose) and the second lepton candidate is tight
(loose), $N_{\mathrm{r_{1}r_{2}}}^{\mathrm{t(l)_{1}t(l)_{2}}}$ is the
contribution to $N^{\mathrm{t(l)_{1}t(l)_{2}}}$ from events where both lepton
candidates are real leptons, and $F_{1}$ and $F_{2}$ correspond to the fake
factors associated with the first and second lepton candidate, respectively.
However, the algebraic simplification that leads to Eq. (9), where the result
depends simply on products of the fake factors and the observed tight and
loose lepton candidate yields, with a correction term that depends only on
events where all the leptons are real, does not hold for all possible event
selections nor all values of $m_{\mathrm{f}}$; such a simplification is
restricted to cases where the baseline and tight candidate lepton
multiplicities are the same in all events, and where $m_{\mathrm{f}}=1$.
### 2.4 Use with weighted events
In some cases, such as for self-consistency tests using simulated events, it
may be advantageous to weight the events that are input to the fake/non-prompt
background estimate. This is straightforward for the fake-factor and
asymptotic matrix methods, since the weight returned by the method for each
event can be multiplied by the event weight $w_{\mathrm{evt}_{i}}$. For the
likelihood matrix method, this is handled by using the scaled Poisson
distribution [7], in which the values of $N_{\mathrm{tl},\mathrm{exp}_{i}}$
and $N_{\mathrm{tl},\mathrm{evt}_{i}}$ from Eq. (8) are scaled according to
the event weights $w_{\mathrm{evt}_{i}}$:
$s_{\mathrm{tl}}\equiv{\sum_{\mathrm{tl},\mathrm{obs}}w_{\mathrm{evt}_{i}}\over\sum_{\mathrm{tl},\mathrm{obs}}w_{\mathrm{evt}_{i}}^{2}},$
so that the likelihood becomes
$L\left({\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}\right)=\prod_{i}P\left[N_{\mathrm{tl}_{i}}/s_{\mathrm{tl}}|N_{\mathrm{tl},\mathrm{exp}_{i}}\left({\mathbf{N}}^{\mathrm{t}}_{\mathrm{rf}}\right)/s_{\mathrm{tl}}\right]$
### 2.5 Performance studies
MC simulations of experiments are utilised to assess the statistical
performance of the methods. These simulations consist of pseudoexperiments
that mimic scenarios that might occur in an actual analysis. Pseudoexperiments
with sample sizes of $10$ or $1000$ dilepton events in the baseline sample are
considered. The fraction of fake/non-prompt leptons in the baseline lepton
sample is varied for each pseudoexperiment, with a uniform distribution
between $0$ and 100%. The values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ for each lepton are drawn from Gaussian
distributions with specified means and widths (limits are imposed such that
the values are always between zero and one, and $\varepsilon_{\mathrm{f}}$ is
always at least 10% less than $\varepsilon_{\mathrm{r}}$). Each lepton is
randomly assigned as fake/non-prompt or real, in accord with the fraction of
fake/non-prompt leptons assumed for the pseudoexperiment. Then each lepton is
judged to either meet or fail to meet the tight selection criteria, based on
whether or not it is a real lepton and the values of
$\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ assigned to it. The
set of simulated leptons is input to the data-driven algorithms, and the
estimated fake yield and its statistical uncertainty are determined for each
pseudoexperiment. These values are then compared with the expectation value
for the number of fake/non-prompt lepton events in each pseudoexperiment,
which is determined by the numbers of real and fake/non-prompt baseline
leptons, the values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ for each lepton, and the value of $m_{\mathrm{f}}$
for the simulated analysis. For example, in a dilepton sample with
$m_{\mathrm{f}}\geq 1$, the expectation value is:
$\langle N_{\mathrm{f}}^{\mathrm{t}}\rangle=\sum_{\textrm{rf
events}}\varepsilon_{\mathrm{r}_{1,i}}\varepsilon_{\mathrm{f}_{2,i}}+\sum_{\textrm{fr
events}}\varepsilon_{\mathrm{f}_{1,i}}\varepsilon_{\mathrm{r}_{2,i}}+\sum_{\textrm{ff
events}}\varepsilon_{\mathrm{f}_{1,i}}\varepsilon_{\mathrm{f}_{2,i}}.$
For the likelihood matrix method, the likelihood maximisation is implemented
using the minuit function minimisation package [8] to minimise the negative
log likelihood. The interface to minuit is provided by the TMinuit class in
root [9].
The fake-factor method requires a two-step process, where the contribution
from real-lepton events is subtracted in the second step. In an actual physics
analysis, this subtraction is done using MC simulation of the prompt lepton
contribution. For the simple MC simulation used here, the second step is
modelled by running each pseudoexperiment a second time with the same
parameters but a statistically independent sample of events, and running the
fake-factor method only on the events that do not have fake/non-prompt
leptons. The second sample has ten times more events than the primary sample,
to be consistent with the usual case where the MC simulation sample has a
multiple of the number of events in the data sample. The result of this second
run is then scaled down by a factor of ten and subtracted from the result when
using the initial set of simulated events.
As an initial example, one can investigate the performance for dilepton events
under conditions that are favourable for estimating the fake/non-prompt lepton
background. This means that the samples are large ($1000$ events per
pseudoexperiment), and the values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ are on average well-separated (here,
$\langle\varepsilon_{\mathrm{r}}\rangle=0.90$ and
$\langle\varepsilon_{\mathrm{f}}\rangle=0.10$). The ratios of the estimated to
true fake yields are shown versus $\langle N_{\mathrm{f}}^{\mathrm{t}}\rangle$
in Figure 1(a). The average statistical uncertainties in the estimates for
each method are shown in Figure 1(b), and the fraction of pseudoexperiments in
which the true fake yield lies within the uncertainty reported for each method
is shown in Figure 1(c). The performance of the methods for dilepton analyses
with low statistical precision ($10$ events per pseudoexperiment) are shown in
Figure 2. Finally, to represent a more challenging situation, the case where
there is less separation between the values of $\varepsilon_{\mathrm{r}}$ and
$\varepsilon_{\mathrm{f}}$ (due, for example, to the application of stricter
online lepton selection criteria that might be required when the LHC runs at
higher luminosities) is also explored. Figure 3 shows the results when
$\langle\varepsilon_{\mathrm{r}}\rangle=0.70$ and
$\langle\varepsilon_{\mathrm{f}}\rangle=0.30$, and there are $10$ events per
pseudoexperiment.
((a))
((b))
((c))
Figure 1: Performance of the three methods in pseudoexperiments with dilepton
events where the leptons had an average $\varepsilon_{\mathrm{r}}=0.9$ and
$\varepsilon_{\mathrm{f}}=0.1$, both values were varied according to a
Gaussian distribution of width $0.1$ when simulating each lepton, and there
were $1000$ events per pseudoexperiment. The quantity $\langle
N_{\mathrm{f}}^{\mathrm{\ t}}\rangle$ is the expectation value for the number
of fake/non-prompt lepton events in each pseudoexperiment. Plot (a) shows the
ratio of the estimated to expected fake-lepton yields, (b) shows the absolute
uncertainty estimate for each method (for the likelihood method the average of
the upward and downward uncertainties is taken), and (c) shows the fraction of
pseudoexperiments where the true fake yield lies within the reported one
standard deviation ($1\sigma$) range.
((a))
((b))
((c))
Figure 2: Performance of the three methods in pseudoexperiments with dilepton
events where the leptons had an average $\varepsilon_{\mathrm{r}}=0.9$ and
$\varepsilon_{\mathrm{f}}=0.1$, both values were varied according to a
Gaussian distribution of width $0.1$ when simulating each lepton, and there
were $10$ events per pseudoexperiment. The quantity $\langle
N_{\mathrm{f}}^{\mathrm{t}}\rangle$ is the expectation value for the number of
fake/non-prompt lepton events in each pseudoexperiment. Plot (a) shows the
ratio of the estimated to expected fake-lepton yields, (b) shows the absolute
uncertainty estimate for each method (for the likelihood method the average of
the upward and downward uncertainties is taken), and (c) shows the fraction of
pseudoexperiments where the true fake yield lies within the reported one
standard deviation ($1\sigma$) range.
((a))
((b))
((c))
Figure 3: Performance of the three methods in pseudoexperiments with dilepton
events where the leptons had an average $\varepsilon_{\mathrm{r}}=0.7$ and
$\varepsilon_{\mathrm{f}}=0.3$, both values were varied according to a
Gaussian distribution of width $0.1$ when simulating each lepton, and there
were $10$ events per pseudoexperiment. The quantity $\langle
N_{\mathrm{f}}^{\mathrm{t}}\rangle$ is the expectation value for the number of
fake/non-prompt lepton events in each pseudoexperiment. Plot (a) shows the
ratio of the estimated to expected fake-lepton yields, (b) shows the absolute
uncertainty estimate for each method (for the likelihood method the average of
the upward and downward uncertainties is taken), and (c) shows the fraction of
pseudoexperiments where the true fake yield lies within the reported one
standard deviation ($1\sigma$) range.
These studies show that all three methods give accurate estimates, with nearly
equivalent performance, in high-statistics samples with a large separation
between $\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ (as shown in
Figure 1). One notable feature is the dip in the uncertainty values for all
three methods near $\langle N_{\mathrm{f}}^{\mathrm{t}}\rangle=10$ in Figure
1(b). This occurs because there are two ways for the model to produce an
expectation around $10$: either a very low fake fraction in the baseline to
start with, or a very large fake fraction so that there are few real leptons
and most of the background is from events with two fake/non-prompt leptons,
which gives a minimum value of $\varepsilon_{\mathrm{f}}^{2}\cdot 1000=10$
when $\varepsilon_{\mathrm{f}}=0.10$. These two processes will have very
different uncertainties. There is also a bias in the likelihood matrix method
toward low values when the true number of fakes is large. This bias arises due
to the averaging of efficiencies over the entire baseline sample (see Eq. (7))
when in fact the efficiencies are on average different for real and fake
leptons. Such differences in the averages occur randomly in the ‘toy’ MC
tests, but can occur systematically in a physics analysis if, for example, the
real and fake efficiencies have different kinematic distributions. The biases
can be mitigated by binning the baseline sample in the variables for which the
real-lepton and fake-lepton distributions may differ, and performing the
likelihood fit separately in each bin. As an example of the effect of such a
binning, the pseudoexperiments can be run with the results binned according to
the values of $\varepsilon_{\mathrm{f}}$. The effect of using two such bins in
the value of $\varepsilon_{\mathrm{f}}$ for each lepton is shown in Figure 4.
When the situation becomes more challenging, such as in Figures 2 and 3, the
characteristics of each method become more distinct. For low-statistics
samples, the likelihood approach tends to exhibit a bias toward high values
when the true number of fakes is small, a natural consequence of the fact that
it cannot return negative values. The coverage of the true value by the
estimated uncertainty is, however, still reasonable. A clear distinction
between the precision of the methods also appears in Figures 2 and 3, where
the likelihood approach has the smallest uncertainty, followed by the
asymptotic matrix method and then the fake-factor method. This is because the
likelihood approach considers lepton efficiencies averaged over the entire
sample and is therefore less susceptible to event-by-event fluctuations in the
values of $\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$.
Figure 4: Ratio of the estimated to expected fake-lepton yields for the
likelihood matrix method in pseudoexperiments with dilepton events where the
leptons had an average $\varepsilon_{\mathrm{r}}=0.9$ and
$\varepsilon_{\mathrm{f}}=0.1$, both values were varied according to a
Gaussian distribution of width $0.1$ when simulating each lepton, and there
were $1000$ events per pseudoexperiment. The triangles show the result with a
single bin, while the circles show the result where the input events are
binned according to the values of $\varepsilon_{\mathrm{f}}$ for each lepton.
Two bins are used for each of the two leptons, and the results from all four
bins are summed to give the estimated $N_{\mathrm{fake}}$ value.
Despite the differences between them, none of the approaches is incorrect, as
shown by the statistical coverage plots: except in extreme cases, the
confidence intervals built from the estimates and their statistical
uncertainty do contain the true fake yield in at least 68% of
pseudoexperiments, and the visible overcoverage is due mostly to the fact that
the uncertainties are computed under the assumption that all of the
${\mathbf{N}}_{\mathrm{tl}}$ values are independent, while the
pseudoexperiments were generated with a fixed number of baseline events for
each, which means there was some anticorrelation among the
${\mathbf{N}}_{\mathrm{tl}}$.
When selecting which method to use, the analyser needs to consider the
relative benefits and complexities of implementing the methods, along with the
size of the uncertainty in the fake/non-prompt lepton background yield
relative to other uncertainties in the analysis.
## 3 The ATLAS detector
While the above description of the matrix and fake-factor methods is general,
the remainder of this paper discusses the application of these methods to
ATLAS physics analyses, and therefore a brief description of the experimental
apparatus follows.
The ATLAS detector [1] at the LHC covers nearly the entire solid angle around
the collision point.333 ATLAS uses a right-handed coordinate system with its
origin at the nominal interaction point (IP) in the centre of the detector and
the $z$-axis along the beam pipe. The $x$-axis points from the IP to the
centre of the LHC ring, and the $y$-axis points upwards. Cylindrical
coordinates $(r,\phi)$ are used in the transverse plane, $\phi$ being the
azimuthal angle around the $z$-axis. The pseudorapidity is defined in terms of
the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$. Angular distance is
measured in units of $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$.
It consists of an inner tracking detector surrounded by a thin superconducting
solenoid, electromagnetic (EM) and hadronic calorimeters, and a muon
spectrometer (MS) incorporating three large superconducting toroidal magnets.
The inner-detector system is immersed in a $2\text{\,}\mathrm{T}$ axial
magnetic field and provides charged-particle tracking in the range
$|\eta|<2.5$.
The high-granularity silicon pixel detector covers the vertex region and
typically provides four measurements per track, the first hit normally being
in the insertable B-layer (IBL) installed before Run 2 [10, 11]. It is
followed by the silicon microstrip tracker, which usually provides eight
measurements per track. These silicon detectors are complemented by the
transition radiation tracker (TRT), which enables radially extended track
reconstruction up to $|\eta|=2.0$. The TRT also provides electron
identification information based on the fraction of hits (typically $30$ in
total) above a higher energy-deposit threshold corresponding to transition
radiation.
The calorimeter system covers the pseudorapidity range $|\eta|<4.9$. Within
the region $|\eta|<3.2$, EM calorimetry is provided by barrel and endcap high-
granularity lead/liquid-argon (LAr) calorimeters, with an additional thin LAr
presampler covering $|\eta|<1.8$ to correct for energy loss in material
upstream of the calorimeters. Hadronic calorimetry is provided by the
steel/scintillator-tile calorimeter, segmented into three barrel structures
within $|\eta|<1.7$, and two copper/LAr hadronic endcap calorimeters. The
solid angle coverage is completed with forward copper/LAr and tungsten/LAr
calorimeter modules optimised for EM and hadronic measurements respectively.
The MS comprises separate trigger and high-precision tracking chambers
measuring the deflection of muons in a magnetic field generated by the
superconducting air-core toroids. The field integral of the toroids ranges
between $2.0$ and $6.0\text{\,}\mathrm{T}\mathrm{m}$ across most of the
detector. A set of precision chambers covers the region $|\eta|<2.7$ with
three layers of monitored drift tubes, complemented by cathode-strip chambers
in the forward region, where the background is highest. The muon trigger
system covers the range $|\eta|<2.4$ with resistive-plate chambers in the
barrel, and thin-gap chambers in the endcap regions. Interesting events are
selected to be recorded by the first-level trigger system implemented in
custom hardware, followed by selections made by algorithms implemented in
software in the high-level trigger [12]. The first-level trigger accepts
events from the $40\text{\,}\mathrm{MHz}$ bunch crossings at a rate below
$100\text{\,}\mathrm{kHz}$, which the high-level trigger reduces in order to
record events to disk at about $1\text{\,}\mathrm{kHz}$.
An extensive software suite [13] is used in data simulation, in the
reconstruction and analysis of real and simulated data, in detector
operations, and in the trigger and data acquisition systems of the experiment.
## 4 Lepton selection criteria
Full descriptions of the electron and muon reconstruction algorithms and
available selection criteria used in ATLAS are provided in Refs. [2] and [3],
respectively. Here the features most relevant to the fake/non-prompt lepton
background estimation are summarised briefly.
### 4.1 Electron reconstruction and identification
Electron candidates are reconstructed within $|\eta|<2.47$ as tracks in the
inner detector matched to energy clusters in the EM calorimeter. In order to
separate true electrons from other unwanted reconstructed candidates, electron
identification (ID) algorithms are used. These rely upon a set of variables
that quantify the distribution of energy in the calorimeter, the quality of
the spatial match between the calorimeter deposit and the associated track,
and the transition radiation signal in the TRT (see Table 1 of Ref. [2] for a
complete list). Rather than place individual requirements on these variables,
they are combined into a likelihood discriminant based upon the probability
density functions of the variables measured for prompt electrons in
$Z\rightarrow e^{+}e^{-}$ events and for background candidates reconstructed
in inclusive collision events.
Since different analyses have different requirements for electron selection
efficiency and background rejection, several ‘working points’ (WPs) are
defined by different values of the likelihood. The likelihood threshold values
are varied according to the $p_{\text{T}}$ and $|\eta|$ of the electron
candidate, so that the selection efficiency varies smoothly with the electron
$p_{\text{T}}$. The most commonly used ID WPs (and their average
efficiencies444 Those efficiencies, as well as those quoted for muons in the
next section, do not include requirements for the leptons to be successfully
identified in the context of the trigger decision, which may include small
additional inefficiencies [14, 15]. measured for typical electroweak
processes) are ‘Loose’ (93%), ‘Medium’ (88%) and ‘Tight’ (80%). In addition to
the listed WPs, there is another (‘LooseAndBLayer’) WP that imposes the same
requirement on the likelihood as the ‘Loose’ WP, but also requires that the
electron track have a hit in the IBL to suppress candidates originating from
photon conversions.
Often, additional requirements are imposed on the impact parameter of the
electron’s track: the impact parameter in the transverse plane, $d_{0}$, with
respect to the centre of the beamspot must satisfy $|d_{0}|<5\sigma(d_{0})$,
where $\sigma(d_{0})$ is its estimated uncertainty, while the longitudinal
separation $z_{0}$ between the point where $d_{0}$ is measured and the chosen
primary vertex of the event, multiplied by a moderating factor $\sin(\theta)$
which accounts for reduced $z_{0}$ accuracy for more forward tracks, cannot
exceed $0.5\text{\,}\mathrm{mm}$ in absolute value.
### 4.2 Muon reconstruction and identification
Muon candidates are reconstructed in the region $|\eta|<2.5$ by combining MS
tracks with matching inner-detector tracks. The muon reconstruction efficiency
is approximately 98% per muon in simulated $Z\rightarrow\mu^{+}\mu^{-}$
events. After reconstruction, high-quality muon candidates used for physics
analyses are selected by a set of requirements on the number of hits in the
different inner subdetectors and different MS stations, on the track fit
properties, and on variables that test the compatibility of the individual
measurements in the two detector systems, as detailed in Ref. [3]. These
criteria reduce the background from in-flight decays of light-flavour hadrons,
which often result in kinked tracks. The most commonly used muon ID WPs (and
their efficiencies measured in $t\bar{t}$ MC events) are ‘Medium’ (98%) and
‘HighPt’ (80%), the latter optimised to offer the best momentum resolution for
$p_{\text{T}}>$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. The same impact
parameter requirements as defined for electrons are also often applied to muon
candidates, with a tighter condition in the transverse plane:
$|d_{0}|<3\sigma(d_{0})$.
### 4.3 Lepton isolation
In addition to the ID criteria mentioned above, most analyses place
requirements on the isolation of the lepton from other detector activity. This
is especially helpful in reducing the contribution from leptons produced in
heavy-flavour decays, or muons from $\pi^{\pm}$ or $K^{\pm}$ decays, within
jets555Jets are reconstructed from clusters of topologically connected
calorimeter cells (topo-clusters), as described in Ref. [16]. The anti-$k_{t}$
algorithm [17, 18] is used to form jets from the topo-clusters, with the
radius parameter $R$ usually set to 0.4; when reconstructing jets formed from
the merged decay products of boosted resonances, a larger $R$ value, typically
10, is used. Typically, jets with $p_{\text{T}}>$20\text{\,}\mathrm{GeV}$$ and
$|\eta|<4.5$ are considered in physics analyses. as there are often other
components of the jet that are near the lepton in these cases. In the context
of the matrix and fake-factor methods, the baseline lepton selection usually
does not require isolation, while the tight lepton selection usually does.
However, many of the single-lepton triggers [15, 14] used in ATLAS require
isolation, so analyses that depend on such triggers cannot avoid applying
isolation requirements for baseline leptons.
The calorimeter isolation [2, 3] is calculated from the sum of transverse
energies of calorimeter energy clusters within $\Delta
R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}=\text{XX}/100$ of the lepton
candidate, not including the contribution expected from the candidate itself.
Typical values of XX are $20$, $30$ or $40$. Expected residual contributions
to the isolation from the lepton candidate, as well as expected contributions
from particles produced by additional proton–proton ($pp$) interactions, are
subtracted [19], resulting in the variable $E_{\mathrm{T}}^{\text{coneXX}}$.
The track-based isolation [2, 3], denoted by $p_{\text{T}}^{\text{coneXX}}$,
is based on tracks near the lepton candidate with either $p_{\text{T}}>0.5$ or
$1\text{\,}\mathrm{GeV}$ that satisfy basic track-quality requirements and are
spatially consistent with the primary vertex of the event. The scalar sum of
the transverse momenta of such tracks, excluding tracks associated with the
lepton candidate, is compared with the $p_{\text{T}}$ of the candidate to
assess the isolation. For muon candidates, only the single track associated
with the candidate is excluded; for electron candidates, additional tracks
consistent with pair-production from a bremsstrahlung photon are also
excluded. The track isolation can also be defined with a variable cone size
($p_{\text{T}}^{\text{varconeXX}}$). In this case, the size of the cone around
the lepton candidate within which tracks are considered is varied as a
function of the $p_{\text{T}}$ of the candidate:
$\Delta
R=\min\left(\frac{$10\text{\,}\mathrm{GeV}$}{p_{\text{T}}\,[\text{GeV}]},\,R_{\mathrm{max}}\right),$
where $R_{\mathrm{max}}$ is the maximum cone size (typically $0.2$ or $0.3$).
Combining selections on track-based and calorimeter-based isolation provides
even better fake/non-prompt lepton background rejection, as the two isolation
variables use complementary information. Track-based isolation was found to be
less sensitive to detector noise and pile-up effects than calorimeter-based
isolation, and the inner detector provides a better $p_{\text{T}}$ measurement
than the calorimeters for individual soft hadrons. On the other hand,
calorimeter-based isolation includes neutral particles as well as some
particles below the inner detector’s track-$p_{\text{T}}$ threshold, which are
ignored when computing track isolation. However, track and calorimeter
isolation variables measure hadronic activity in a redundant manner, since
charged particles are measured by both the calorimeters and the inner
detector, and simple selection cuts applied independently to those two
variables may not achieve optimal rejection power. To avoid this, an analysis
can use a ‘particle-flow’ algorithm, which allows removal of overlapping
contributions from the track-based and calorimeter-based isolation, decreasing
the correlation between the two variables. For the time being, particle-flow-
based isolation variables are defined only for muons, and discussed in Ref.
[3].
For analyses where the fake/non-prompt lepton background may be dominated by
non-prompt electrons and muons from the decays of $b$\- and $c$-hadrons [20,
21], isolation WPs using a boosted decision tree (BDT) discriminant based on
isolation and secondary vertex information, referred to as the non-prompt
lepton BDT, are also proposed.
Several isolation WPs based on tracking, a combination of calorimetry and
tracking, particle-flow, or a non-prompt lepton BDT are defined to allow for
consistency across analyses that require different levels of lepton isolation.
They are described in Refs. [2] and [3].
### 4.4 Removing overlaps between jets and leptons
In some cases the same object can result in multiple signatures in the
detector. For example, an electron will deposit energy in the calorimeter, and
that energy will generally also be clustered into a jet. In addition,
sometimes objects will be spatially correlated due to the underlying physics,
as when a muon is produced by heavy-flavour decay within a jet. To avoid
double-counting, and to select only the isolated leptons that are typically of
interest in physics analyses, an overlap removal (OR) procedure is applied to
resolve these ambiguities. The procedure used for most physics analyses is as
follows, where the lepton candidates are those that satisfy the baseline ID
criteria for the analysis:
* •
All electron candidates that are within $\Delta R=0.01$ of a muon candidate
(or share a track with a muon candidate) are removed.
* •
All jets that are within $\Delta R=0.2$ of any remaining electron candidates
are removed.
* •
All electron candidates that are within $\Delta R=0.4$ of any remaining jet
are removed.
* •
Cases where a remaining jet is within $\Delta R=0.4$ of a muon candidate are
examined to determine the number of tracks associated with the jet. If there
are more than two such tracks the muon candidate is removed; otherwise the jet
is removed.
Variations of this procedure are also supported, primarily for analyses that
focus on heavy-flavour jets or that select boosted massive particles using
large-$R$ jets.
## 5 Monte Carlo simulation samples
The studies of fake/non-prompt leptons that are presented in Sections 6 and 7
make use of large MC samples of simulated events. The production of $t\bar{t}$
events at next-to-leading order (NLO) in the quantum chromodynamics (QCD)
coupling constant $\alpha_{\text{s}}$ is described in Ref. [22] and relies on
the Powheg Box v2 event generator [23] interfaced with Pythia 8.230 [24] for
parton showering and subsequent steps, with the A14 set of tuned parameters
[25]. The parton distribution functions used for matrix element calculation
and parton showering are NNPDF3.0nlo [26] and NNPDF2.3lo [27] respectively.
Generated events were filtered such that at least one of the top quarks decays
semileptonically. The EvtGen 1.2.0 program [28] was used to model heavy-
flavour hadron decays.
The production of Drell–Yan $Z/\gamma^{*}\to\ell^{+}\ell^{-}$ events
($\ell=e,\mu,\tau$) at NLO in $\alpha_{\text{s}}$ is described in Ref. [29]
and relies on the Sherpa 2.2 event generator [30] with the dedicated set of
tuned parameters and the NNPDF3.0nnlo [26] parton distribution function.
Events were generated according to a partition of the phase space described in
Ref. [29], resulting in a set of orthogonal samples which were combined with
weights corresponding to the NLO cross-section calculated by the generator.
The $\ell^{+}\ell^{-}$ invariant mass was required to be at least
$40\text{\,}\mathrm{GeV}$.
A full ATLAS detector simulation [31] based on Geant4 [32] was then used to
faithfully reproduce particle interactions with the detector and its response.
Additional $pp$ interactions in the same or neighbouring bunch crossings were
also simulated in order to reproduce the conditions of data-taking during the
LHC Run 2 operation.
The source of a lepton candidate in simulation, discussed in Section 6, is
identified by using preserved generator-level information to match each
reconstructed lepton candidate to the closest particle produced by the event
generator, based primarily on the angular separation between the latter and
the reconstructed track of the former. If the matched particle is a charged
lepton (or, in the case of electron candidates, a photon), its origin is
checked so that the different sources of non-prompt leptons can be
distinguished.
In Section 9, the simulation of several other processes is involved, and is
performed with a similar workflow; complete information about the MC
generators employed, their configurations, and the cross-section calculations
used to normalise the simulated samples, is available in the references
provided in that section for those two analyses.
## 6 Sources of fake/non-prompt leptons
((a)) ((b))
Figure 5: Relative contributions of fake or non-prompt muons from different
sources as a function of $p_{\text{T}}$ in simulated (a) $t\bar{t}$ and (b)
Drell–Yan processes. Punch-through hadrons are charged hadrons reaching the MS
and leading to the reconstruction of a fake muon candidate. Some other minor
sources are not displayed; the sum of their contributions is less than 2% in
every bin of (a), and less than 4% in bins of (b) up to
$p_{\text{T}}=$100\text{\,}\mathrm{GeV}$$. The muons are required to satisfy
minimal track-quality criteria specified in the text. Error bars represent
statistical uncertainties of the simulation.
((a)) ((b))
Figure 6: Relative contributions of fake or non-prompt electrons from
different sources as a function of $p_{\text{T}}$ in simulated (a) $t\bar{t}$
and (b) Drell–Yan processes. Some other minor sources are not displayed; the
sum of their contributions is less than 0.5% in every bin. These do not
include electron candidates corresponding to high-energy muon ionisation
signal in the calorimeters, or from conversions of final-state radiation (FSR)
photons with small angular separation from a genuine electron or muon. The
electrons are required to satisfy loose ID criteria specified in the text and
Ref. [33]. Error bars represent statistical uncertainties of the simulation.
The relative contributions of fake/non-prompt leptons from different sources
to the sample of selected lepton candidates depend on the energy and spatial
location of the candidate in the detector, the identification, impact
parameter and isolation criteria applied, the overlap removal procedure, and
the nature of the selected final state (e.g. the presence or absence of bottom
quarks).
Figures 5 and 6 present for illustration the relative contributions of the
different sources of fake/non-prompt muons and electrons respectively, as a
function of the transverse momentum of the candidate, as measured in MC
simulated events. They are shown for two different processes: $t\bar{t}$
production, leading to final states rich in heavy-flavour hadrons, and
Drell–Yan production of $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pairs; for $t\bar{t}$
events at least one of the top quarks is required to decay leptonically. For
these particular figures, the reconstruction/ID of electron and muons and the
general event selection follow those described in Ref. [33] for baseline
lepton candidates, which correspond to rather loose criteria (in particular,
no isolation nor transverse impact parameter requirements are applied, and no
overlap removal is done). For both processes, events are considered only if
they contain a pair of reconstructed baseline leptons with identical charges,
a signature for which fake/non-prompt leptons usually represent a non-
negligible source of background.
The ‘prompt $\gamma$-conversion’ category only includes electron candidates
where the photon is separated by $\Delta R>0.1$ from any generator-level
high-$p_{\text{T}}$ electron from the hard-scatter interaction (photons
emitted at smaller separation are generally reconstructed as part of the
electron candidates).
Non-prompt leptons are those arising from electroweak decays of hadrons.
Heavy-flavour $b$\- and $c$-hadrons decay close to the interaction point and
the resulting leptons are distinguishable from real leptons mostly by
isolation and impact parameter.
Light-flavour hadrons can also be a major source of fake leptons via decay-in-
flight in the tracker volume. This happens mainly in final states for which
QCD multi-jet production is a significant contributor. A charged hadron
stopping early in the calorimeter, and generating a narrower-than-average
shower, can mimic the experimental signature of an electron. Electron ID
criteria are particularly powerful in rejecting these candidates rather than
those from other sources, based notably on three-dimensional profiles of the
shower, but since many orders of magnitude more hadrons than leptons are
produced in collisions at the LHC, a substantial number of such fake electrons
may be selected in a physics analysis. With regard to muons, the depth of the
ATLAS calorimeter is sufficient to stop most pions and kaons before they can
reach the MS. Muon candidates arising from kaons decaying semileptonically in
flight before reaching the MS are a more important contribution to the
fake/non-prompt lepton background.
Among hadrons faking electrons, one significant class is neutral pion decays
into photons ($\pi^{0}\to\gamma\gamma$); the collimated photons create a
single energy deposit in the EM calorimeter, while the associated track might
be provided by the conversion of one of the photons in the upstream detector
material. Due to the importance of this phenomenon, the electron ID criteria
specifically discriminate against these by attempting to identify a two-peak
structure in the distribution of the cell energies matched to the electron
cluster [2], which is not present for real electrons. Other contributions such
as Dalitz decay of pions [34] also create comparable experimental signatures.
The conversion of photons into electron–positron pairs represents the last
important class of non-prompt electrons. These $\gamma$-conversions must
typically occur early on (e.g. in the beam pipe) and be largely asymmetric in
the splitting of the momentum between the two electrons; otherwise, the
conversion vertex can be reconstructed, or the candidate electron’s track
lacks hits in the first layers of the inner detector, both leading to the
proper classification of the reconstructed object as a photon instead of an
electron [2]. The origin of the photon itself influences the characterisation
of the candidate as non-prompt or real: photons emitted close to a real
electron, either due to bremsstrahlung or as higher-order quantum
electrodynamic (QED) corrections to the production process, are typically
considered part of the electron candidate (the calorimeter energy deposits
tend to overlap to the extent that a single cluster is reconstructed);
furthermore, from the perspective of quantum field theory, well-defined
electrons must include extra radiation (‘dressed leptons’ [35]). The electron
reconstruction procedure [2] accounts for bremsstrahlung, in particular by
allowing kinks in the track consistent with bremsstrahlung emission in dense
material regions. In contrast, photons from other origins, such as initial-
state radiation, QED processes not involving leptons or where photons are
sufficiently separated from leptons, or hadronic jet fragmentation, may be
considered as sources of fake electrons.
It can be seen in Figure 5 that non-prompt muons constitute the only
substantial contribution to the fake/non-prompt muon background, while for
electrons Figure 6 shows more variety: in general, non-prompt electrons are
particularly represented in the lower $p_{\text{T}}$ range, especially in
processes involving the production of heavy-flavour hadrons, while hadron
fakes and converted photons populate the higher $p_{\text{T}}$
range.666However, for the range $p_{\text{T}}<$10\text{\,}\mathrm{GeV}$$, not
shown in the figure, hadron fakes are also the dominant contribution.
((a)) ((b))
Figure 7: Rejection (defined as 1/$\varepsilon_{\mathrm{f}}$) as a function
of $p_{\text{T}}$ for different sources of fake/non-prompt (a) muons and (b)
electrons in simulated $t\bar{t}$ events. The efficiency
$\varepsilon_{\mathrm{f}}$ is the fraction of lepton candidates selected using
the same baseline criteria as in Figures 5 and 6 that also pass the tight
selection criteria of Ref. [33]. Error bars represent statistical
uncertainties of the simulation, and for purposes of clarity, values with
relative uncertainties greater than 30% are not shown.
The different sources of fake/non-prompt leptons have distinct probabilities
to satisfy the tight lepton selection criteria described in Section 4. Figure
7 illustrates those differences for the particular example of signal leptons
definitions used in Ref. [33] (including lepton–jet overlap removal) and
simulated $t\bar{t}$ events. Such variability between sources is unwelcome, as
the fake efficiencies required for the application of the methods described in
Section 2 then depend upon the relative contributions of each source to the
regions of interest, which may not be easy to assess. It is therefore
desirable to measure the efficiencies in regions similar in composition to the
regions where the background estimate is needed, otherwise large extrapolation
uncertainties may apply.
Precise simulation of these various sources of background, including their
relative contributions, is indeed very challenging, as it relies heavily on
the modelling of the soft-QCD regime by event generators, including modelling
of fragmentation and hadronisation processes, hadron decay modelling, and soft
emissions and detector modelling. Another issue is that only a small fraction
of fake or non-prompt lepton candidates survive the ID and isolation
requirements, so the simulation of a very large number of events is needed to
obtain a statistically accurate prediction. For inclusive processes with large
cross-sections (e.g. multi-jet production), this is often impractical.
For these reasons, many of the fake/non-prompt lepton background predictions
used in ATLAS publications are based on methods using the data, such as the
ones described in this paper. They rely on common properties shared to some
extent by the different sources of fake/non-prompt leptons that differentiate
them from real leptons, such as a high likelihood to not meet the combination
of ID and isolation criteria.
## 7 Measurement of real and fake/non-prompt lepton efficiencies
The matrix and fake-factor methods both rely on knowledge of the efficiency
for leptons that pass the baseline selection to also pass the tight selection.
For the fake-factor method, only the efficiency for fake/non-prompt leptons is
used explicitly in the calculation, while for the matrix method the efficiency
for real leptons must also be measured. In many cases these efficiencies
depend on the properties of the lepton (such as its $p_{\text{T}}$ or angular
distance from a jet) or on the event in which it is found (such as the overall
activity of the event, as measured for example by the number of reconstructed
primary vertices).
### 7.1 Real-lepton efficiencies
The efficiencies of specific working points of lepton selections are
calibrated precisely in ATLAS for general purposes, primarily with ‘tag-and-
probe methods’ based e.g. on $Z\rightarrow\ell\ell$ events. By performing
these measurements on both data and MC-simulated events, ‘scale factors’ (SFs)
that account for differences between the efficiencies observed in data and
simulation are derived. These SFs can then be applied to simulated events
using the selection criteria that are relevant to a given analysis to
determine the appropriate real-lepton efficiencies. This MC-based approach is
valid as long as both the baseline and tight lepton selection criteria are
taken from the set for which SFs have been measured; only in extraordinary
cases would an analysis utilise different selection criteria. Since the
efficiencies depend more strongly on the environment than the SFs do, the main
advantage of this approach over purely data-based measurements is to allow
efficiencies to be obtained directly in the desired environment (i.e. the
region in which the fake/non-prompt background estimate is needed), rather
than being extrapolated from a more distant region which would be needed for
reliable measurements in data.
Details of the real-electron and real-muon efficiency measurements can be
found in Refs. [2] and [3], respectively. The real efficiencies are often
parameterised with respect to the $p_{\text{T}}$ and $|\eta|$ of the leptons,
and measured separately for electrons and muons.
### 7.2 Fake/non-prompt lepton efficiencies
The fake/non-prompt lepton efficiencies are specific to each analysis,
primarily since there are several sources for such leptons (see Section 6),
which will contribute with different weights depending on the chosen selection
criteria. In general, though, the first step in the efficiency measurement is
to identify a region that has a large contribution from fake leptons. Two
approaches are commonly used. In the first, events with a pair of leptons with
the same electric charge are selected. Since such lepton pairs are only rarely
produced at the LHC (via processes such as $WZ\text{\,+\,jets}$,
$ZZ\text{\,+\,jets}$, and $t\bar{t}+X$ ($X=W/Z$) production) it is likely that
one of the two leptons is fake/non-prompt. By placing stringent quality
criteria on one lepton in these events, the probability that the remaining
lepton is fake/non-prompt is enhanced. The second approach is to use single-
lepton events, where criteria are imposed to suppress the contribution from
real leptons. Examples of such criteria are requiring the missing transverse
momentum777 $\mathbf{E_{\text{T}}^{\text{miss}}}$ is defined as the negative
vector sum of the $p_{\text{T}}$ of the reconstructed and calibrated objects
in the event, with a correction applied for inner detector tracks that
originate from the primary collision vertex and are not associated with any
other objects, and $E_{\text{T}}^{\text{miss}}$ is defined as the magnitude of
$\mathbf{E_{\text{T}}^{\text{miss}}}$ [36]. $E_{\text{T}}^{\text{miss}}$, or
transverse mass888Here
$m_{\text{T}}\equiv\sqrt{2\,p_{\text{T}}^{\ell}\,E_{\text{T}}^{\text{miss}}(1-\cos\Delta\phi_{\ell,E_{\text{T}}^{\text{miss}}})}$,
where $\Delta\phi_{\ell,E_{\text{T}}^{\text{miss}}}$ is the azimuthal angle
between the lepton and $\mathbf{E_{\text{T}}^{\text{miss}}}$ directions.
$m_{\text{T}}$, to be below specific thresholds, thereby reducing the
contribution from $W\text{+\,jets}$ or $t\bar{t}$ events, or requiring via the
track impact parameters that the lepton originate from a position inconsistent
with the primary event vertex, thereby enhancing the contribution of leptons
from heavy-flavour decay.
In either approach, there will be a residual contribution from events with
only real leptons in the selected sample. This contribution is typically
estimated using MC simulation and subtracted separately from both the tight
and baseline samples before the ratio of these samples is taken to measure the
efficiency.
As with the real-lepton efficiencies, the fake/non-prompt lepton efficiencies
depend on properties of the lepton candidates or of the event in which they
are found. Therefore, it is generally helpful to bin the efficiencies in the
lepton $p_{\text{T}}$ and $|\eta|$, and possibly in terms of other quantities
as well. The optimal binning to be used is chosen in the context of each
physics analysis, considering the given numbers of events and potential
changes in the relative contributions from the different sources of fake/non-
prompt leptons; illustrative examples are provided in Section 9.
((a)) ((b))
Figure 8: Probability for a muon produced in the decay of a $b$-hadron to
satisfy a track-based isolation requirement, in simulated $t\bar{t}$ events,
shown as a function of (a) the muon $p_{\text{T}}$; and (b) the sum
$p_{\text{T}}+E_{\text{T}}^{\text{cone40}}$ of the muon $p_{\text{T}}$ and
nearby calorimeter energy deposits. In each case, the universality of these
probabilities is gauged by changing the kinematic properties of the parent
hadron (markers of various colours), achieved indirectly by selecting events
in different ranges of global transverse energy $H_{\text{T}}$. Precise
definitions of the isolation criteria, the
$p_{\text{T}}+E_{\text{T}}^{\text{cone40}}$ and $H_{\text{T}}$ variables, as
well as descriptions of the muon and jets selections, are given in Section
7.2.
Adopting a parameterisation of the efficiency with respect to variables other
than $p_{\text{T}}$ and $\eta$ can sometimes be beneficial. For example, when
considering solely the probability that fake or non-prompt leptons satisfy the
isolation criteria, another relevant quantity is the momentum of the parent
jet: the fake efficiency corresponds in that case to the probability that most
of the jet’s visible momentum is carried by the lepton. A parameterisation as
a function of lepton $p_{\text{T}}$ thus assumes that for a particular
$p_{\text{T}}$, the distribution of the parent jet’s momentum is similar in
the regions where the efficiencies are measured and the regions where the
background estimates are needed. If this assumption does not hold, it can be
useful to adopt instead a parameterisation as a function of the parent jet’s
momentum. Since this quantity is not easily accessed experimentally (unlike
the lepton $p_{\text{T}}$), proxy observables are used in practice. An example
of successful application is the analysis in Ref. [37], which employed the sum
of the lepton’s $p_{\text{T}}$ and the transverse energy in a cone around the
lepton as a proxy.
The preceding discussion is also illustrated in Figure 8 for non-prompt muons
produced in the decay of $b$-hadrons in simulated $t\bar{t}$ events. The
probabilities for such muons to satisfy a track-based isolation requirement,
as defined in Section 4, are shown for two alternative parameterisations: one
based on the muon $p_{\text{T}}$, and the other on the scalar sum of the muon
$p_{\text{T}}$ and the transverse energy deposited in calorimeter-cell
clusters within a cone of size $\Delta R=0.4$ around the muon (referred to as
$E_{\text{T}}^{\text{cone40}}$ in Section 4). This scalar sum serves as a
proxy for the parent jet’s transverse momentum.
In the second parameterisation, the fake efficiency is the fraction of jets
with a non-prompt muon and visible momentum
$p_{\text{T}}+E_{\text{T}}^{\text{cone40}}$ in which the muon is mostly
isolated, i.e. $p_{\text{T}}\gg E_{\text{T}}^{\text{cone40}}$. While one might
consider jets with arbitrarily soft muons in the denominator of this fraction,
for practical reasons Figure 8(b) only includes events where muons satisfy
$p_{\text{T}}>$10\text{\,}\mathrm{GeV}$$. To study the dependence of these
efficiencies on the momentum distribution of the underlying jet, different
regions of jet momentum are emphasised by imposing different requirements on
the global transverse energy $H_{\text{T}}$ in the event, defined for this
purpose as the scalar $p_{\text{T}}$ sum of all jets with
$p_{\text{T}}>$25\text{\,}\mathrm{GeV}$$ and $|\eta|<2.8$ that are a distance
$\Delta R>0.6$ from the muon. This quantity is indeed partially correlated
with the kinematics of the muon’s parent jet, via the momentum of the top
quarks producing all these jets. The reconstruction, calibration and selection
of jets and muons for this figure are otherwise those detailed in Ref. [33].
It can be observed that for the case of a $p_{\text{T}}$-dependent
parameterisation, the efficiencies vary strongly with $H_{\text{T}}$, although
large differences occur mostly for $p_{\text{T}}>$40\text{\,}\mathrm{GeV}$$.
Since most non-prompt muons are produced at low $p_{\text{T}}$, the overall
impact of this non-captured dependency might be small, unless regions of
interest in the analysis specifically select high-$p_{\text{T}}$ leptons. In
contrast, the parameterisation as a function of
$p_{\text{T}}+E_{\text{T}}^{\text{cone40}}$ is much less influenced by
$H_{\text{T}}$, making the measured efficiencies less dependent on the event
topology. In practice, a compromise has to be found between this observation
and other elements evoked above justifying a $p_{\text{T}}$-dependent
parameterisation, especially for electrons because ID criteria are usually
employed in addition to isolation. The direct dependency of efficiencies on
other variables that are also correlated with the event topology (e.g.
$E_{\text{T}}^{\text{miss}}$) may also be reduced by such a parameterisation.
As detailed in the next section, suitable uncertainties must be assigned to
the use of the fake efficiencies in different regions than those in which they
are measured, in particular to account for potential differences in relative
contributions of the different sources of fake/non-prompt leptons. To minimize
these uncertainties, it has sometimes been found beneficial to use an approach
closer to that of section 7.1, in which efficiencies are evaluated in the
simulation and supplemented by data-driven correction factors that depend on
the source of the fake/non-prompt lepton. These correction factors are derived
using dedicated control regions that are enriched in a particular source of
fake/non-prompt leptons. The main assumptions are then the universality of the
correction factors across different processes, and the ability of the
simulation to adequately predict the relative contributions of each source in
the regions of interest. Such an approach has for example been used in Ref.
[38].
## 8 Systematic uncertainties
Systematic uncertainties in the fake/non-prompt lepton background estimates
from the matrix method and the fake-factor method arise from uncertainties in
the values of $\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$. These
uncertainties can be traced to statistical uncertainties from the samples used
to measure the efficiencies, to potential biases that may cause the
efficiencies in the signal region for a particular analysis to differ from the
values obtained from control samples (such as differences in the origin of
fake/non-prompt leptons between these regions), and to uncertainties in the
modelling of contamination from real-lepton processes in the samples used to
measure the fake efficiency. Details of these uncertainties and their
estimation are provided below. The overall impact of variations in
$\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ depends on the
characteristics of the analysis. For the matrix method, from Eqs. (3) and (4)
one obtains:
$\displaystyle{\partial(N_{\mathrm{f}}^{\mathrm{t}})\over\partial\varepsilon_{\mathrm{r}}}$
$\displaystyle={\varepsilon_{\mathrm{f}}\over\varepsilon_{\mathrm{r}}(\varepsilon_{\mathrm{r}}-\varepsilon_{\mathrm{f}})}N_{\mathrm{r}}^{\mathrm{t}}$
$\displaystyle{\partial(N_{\mathrm{f}}^{\mathrm{t}})\over\partial\varepsilon_{\mathrm{f}}}$
$\displaystyle={\varepsilon_{\mathrm{r}}\over\varepsilon_{\mathrm{f}}(\varepsilon_{\mathrm{r}}-\varepsilon_{\mathrm{f}})}N_{\mathrm{f}}^{\mathrm{t}}.$
Therefore, the relative importance of uncertainties in the real and fake
efficiencies (${\Delta\varepsilon_{\mathrm{r}}}$ and
${\Delta\varepsilon_{\mathrm{f}}}$) depend on the typical values of
$\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$ and on the values of
$N_{\mathrm{r}}^{\mathrm{t}}$ and $N_{\mathrm{f}}^{\mathrm{t}}$. For analyses
with
$N_{\mathrm{r}}^{\mathrm{t}}\times\frac{\Delta\varepsilon_{\mathrm{r}}}{\varepsilon_{\mathrm{r}}^{2}}>N_{\mathrm{f}}^{\mathrm{t}}\times\frac{\Delta\varepsilon_{\mathrm{f}}}{\varepsilon_{\mathrm{f}}^{2}}$
the uncertainty in $\varepsilon_{\mathrm{r}}$ will dominate, and vice versa.
### 8.1 Statistical uncertainties in the measured efficiencies
Statistical uncertainties in the real-lepton and fake-lepton efficiencies can
be accounted for either by analytically propagating the uncertainties through
to the estimated event yields or by varying the efficiencies input to the
nominal yield calculation by their statistical uncertainties and using the
resulting difference in the estimated fake/non-prompt background yield as the
resulting uncertainty. The real- and fake-efficiency uncertainties are
generally uncorrelated since the efficiencies are measured in statistically
independent samples. In the usual case where the efficiencies are measured in
bins of one or more quantities, the bins are uncorrelated, so the variations
in each bin are applied separately. The total statistical uncertainty in the
fake/non-prompt lepton background estimate is given by the sum in quadrature
of the uncertainties from all variations. This source of systematic
uncertainty is usually not dominant.
### 8.2 Systematic uncertainties in the measured efficiencies
Systematic uncertainties in $\varepsilon_{\mathrm{f}}$ are generally larger
and more challenging to assess than for $\varepsilon_{\mathrm{r}}$:
1. 1.
Real-lepton efficiencies have only slight variations due to event environment
effects since real leptons have small contributions to their measurements from
underlying event and jet activity. This also means there are a wide variety of
samples with which they can be calibrated in great detail.
2. 2.
There are several sources of fake/non-prompt leptons, and the efficiencies may
differ between these sources. Therefore, any differences in the fake/non-
prompt lepton composition between the sample used to measure the efficiencies
and the signal region for an analysis may lead to a bias in the efficiencies.
Several methods are used to estimate systematic uncertainties in the fake-
lepton efficiencies. One is simply to vary the selection criteria for events
in the control region used to measure the fake-lepton efficiencies, since the
composition of fake/non-prompt leptons in the standard and alternative control
regions may differ. A more sophisticated approach is to use MC simulation to
estimate the fake/non-prompt lepton compositions in both the control and
analysis regions. This information, combined with the MC-estimated selection
efficiencies for each source of fake/non-prompt leptons, can be used to
provide an estimate of the uncertainty.
### 8.3 Uncertainties in the modelling of real-lepton processes
When measuring the fake-lepton efficiencies, a correction must be applied to
account for contamination from processes with real leptons in the control
sample used for the measurement:
$\varepsilon_{\mathrm{f}}=\frac{N^{\mathrm{t}}-N_{\mathrm{r}}^{\mathrm{t}}}{(N^{\mathrm{t}}+N^{\mathrm{l}})-(N_{\mathrm{r}}^{\mathrm{t}}+N_{\mathrm{r}}^{\mathrm{l}})}$
where $N_{\mathrm{r}}^{\mathrm{t}}$ and $N_{\mathrm{r}}^{\mathrm{l}}$ are the
numbers of real leptons in the selected tight and loose samples, respectively.
MC simulation of real-lepton processes is generally used to estimate
$N_{\mathrm{r}}^{\mathrm{t}}$ and $N_{\mathrm{r}}^{\mathrm{l}}$, with
corrections applied to account for known differences in object selection
efficiencies between simulation and data. Nonetheless, several sources of
systematic uncertainty in the real-lepton contamination remain: uncertainties
in the cross-sections of the real-lepton processes, uncertainties in the
correction factors, and uncertainties in the parameters, e.g. parton
distribution functions (PDFs) and factorisation/renormalisation scales
($\mu_{\mathrm{f}}$, $\mu_{\mathrm{r}}$), used in the simulation.
### 8.4 Uncertainties due to biases in the likelihood matrix method
The biases that occur in some situations for the likelihood matrix method (see
Section 2.5) may also be considered as a source of systematic uncertainty,
especially when the signal region contains many bins with few events in each.
The magnitude of the bias can be estimated either by repeating the analysis
with coarser bins, or by constraining the total fake/non-prompt lepton yield
estimate to be the value returned by applying the likelihood matrix method to
the entire unbinned sample. Any resulting differences in the binned estimates
can be taken as a systematic uncertainty.
## 9 Examples of application in ATLAS analyses
In this section the application of the fake/non-prompt lepton background
estimation methods is described using two example ATLAS analyses. The first is
a measurement of the $t\bar{t}Z$ differential cross-section using events that
contain three or four lepton candidates, and the second is a model-independent
search for ‘beyond the Standard Model’ (BSM) phenomena in events with three or
more lepton candidates. In both cases, the high lepton multiplicity suppresses
the Standard Model (SM) backgrounds, which makes the relative contribution of
the fake/non-prompt lepton background larger.
Data from $\sqrt{s}=$13\text{\,}\mathrm{TeV}$$ $pp$ collisions recorded by the
ATLAS detector between 2015 and 2018 are used to perform these analyses. In
this period, the LHC delivered colliding beams with a peak instantaneous
luminosity of
$L=$2.1\text{\times}{10}^{34}\text{\,}\mathrm{c}\mathrm{m}^{-2}\mathrm{s}^{-1}$$,
achieved in 2018, and an average number of $pp$ interactions per bunch
crossing of $33.7$. After applying beam, detector and data-quality criteria,
the total integrated luminosity of the dataset is
$139\text{\,}{\mathrm{fb}}^{-1}$ [39]. The uncertainty in the combined
2015–2018 integrated luminosity is 1.7% [40], obtained using the LUCID-2
detector [41] for the primary luminosity measurements.
### 9.1 Measurement of the $t\bar{t}Z$ cross-section in final states with
three or four leptons
This analysis measured the differential production cross-section of the
$t\bar{t}Z$ process in final states with a total of three or four electron and
muon candidates. The tools described above are used for the lepton efficiency
measurements and the application of the likelihood matrix method (described in
Section 2.1.2). In addition to the fake/non-prompt lepton estimation in the
signal regions of the analysis, the results are checked in several validation
regions enriched in fake/non-prompt leptons. The likelihood matrix method was
chosen for this analysis since the number of fake/non-prompt leptons in the
signal regions with three or four leptons is expected to be very low, and the
likelihood matrix method provides more stable results for binned estimations,
which are necessary for differential background predictions in these low-
statistics signal regions. The fraction of events with more than one fake/non-
prompt lepton in the signal or validation regions has been checked and found
to be negligible. More details about the analysis and the fake/non-prompt
lepton background estimation can be found in Ref. [42].
#### 9.1.1 Real-lepton efficiencies
((a))
((b))
Figure 9: The two-dimensional real-lepton efficiencies obtained for (a)
electrons and (b) muons, in bins of $p_{\text{T}}$ and $|\eta|$ of the
leptons. The last $p_{\text{T}}$ bin is inclusive. The real-lepton
efficiencies are obtained using MC simulation, but corrected to match the
performance seen in data control samples.
The first step in the application of the matrix method is to measure the
efficiencies for real and fake/non-prompt leptons that satisfy the baseline
criteria to also satisfy the tight criteria. In the $t\bar{t}Z$ cross-section
measurement, baseline electrons are required to satisfy the ‘LooseAndBLayer’
ID WP, whereas tight electrons are required to satisfy the stricter ‘Medium’
ID criteria and to be isolated from nearby tracks and calorimeter energy
deposits. Both the baseline and tight muons are required to satisfy the
‘Medium’ WP, and tight muons are in addition required to be isolated from
nearby tracks.
As discussed in Section 7.1, the real-lepton efficiencies are obtained using
MC simulation, corrected to match the performance seen in data control
samples. Those efficiencies are shown in Figure 9, binned in lepton
$p_{\text{T}}$ and $|\eta|$. To check for potential dependencies on the number
of additional jets in the events used for the measurements, the efficiencies
are derived for different jet multiplicities. No significant differences
between the real-lepton efficiencies are observed.
#### 9.1.2 Fake/non-prompt lepton efficiencies
The fake/non-prompt lepton efficiencies are measured with same-charge
electron–muon ($e\mu$) or muon–muon ($\mu\mu$) data events using a tag-and-
probe method, where one ‘tag lepton’ with very stringent requirements on
momentum and isolation ($p_{\text{T}}>$40\text{\,}\mathrm{GeV}$$,
$\textrm{max}(p_{\text{T}}^{\textrm{cone}20},\,E_{\text{T}}^{\textrm{cone}20})/p_{\text{T}}<0.01$)
is selected and the remaining ‘probe lepton’ is used for the efficiency
measurement. Events with more than two leptons are not considered. Only the
aforementioned baseline electron or muon requirements are used to select the
probe lepton. The samples are dominated by events containing at least one
fake/non-prompt lepton, and are orthogonal to the signal regions, which
require a minimum of three tight lepton candidates. In addition, $t\bar{t}Z$
events constitute negligible fractions of these samples.
Table 1: Definition of the fake-lepton control regions used for the electron ($e$-fakes-CR) and muon ($\mu$-fakes-CR) fake-efficiency measurements. $N_{\ell}$ is the number of leptons ($\ell$), while $N_{\textrm{jets}}$ ($N_{b\textrm{-jets}}$) is the number of jets ($b$-tagged jets, see text), respectively. For additional details of the definitions of physics objects, see Ref. [42]. Region | $N_{\ell}$ $(\ell=e,\mu)$ | $p_{\text{T}}$ $(\ell)$ | $p_{\text{T}}$ (jets) | $N_{\textrm{jets}}$ | $N_{b\textrm{-jets}}$
---|---|---|---|---|---
$e$-fakes-CR | $=2$ ($e^{\pm}\mu^{\pm}$) | $>$7\text{\,}\mathrm{GeV}$$ | $>$25\text{\,}\mathrm{GeV}$$ | $\geq 1$ | $\geq 1$
$\mu$-fakes-CR | $=2$ ($\mu^{\pm}\mu^{\pm}$) | $>$7\text{\,}\mathrm{GeV}$$ | $>$25\text{\,}\mathrm{GeV}$$ | $\geq 1$ | $\geq 1$
The definition of the regions used for the fake-efficiency measurements is
summarised in Table 1. For the electron fake efficiencies an $e\mu$ signature
is used, with the muon being used as the tag lepton. For the muon fake
efficiencies, the $\mu\mu$ region is used,999In $\mu\mu$ events with both
muons satisfying the tag-selection, the one with the higher $p_{\text{T}}$ is
chosen as the tag lepton. since the $e\mu$ region also contains unwanted
events where an electron with an incorrectly measured charge is selected as
the tag lepton and paired with a real muon.
((a))
((b))
Figure 10: The two-dimensional fake/non-prompt lepton efficiencies measured
for (a) electrons and (b) muons, in bins of $p_{\text{T}}$ and $|\eta|$ of the
leptons. The last $p_{\text{T}}$ bin is inclusive. The indicated uncertainties
show only the statistical errors in the given bins. Compared to the real-
lepton efficiencies shown in Figure 9, the fake/non-prompt lepton efficiencies
depend much more on the specifications of the analysis.
Real-lepton background processes leading to the same-charge dilepton signature
are estimated using MC-simulated events and are subtracted from data to obtain
an unbiased efficiency measurement. The contribution from electrons with
misassigned charge in the same-charge $e\mu$ region is also subtracted using
estimates from MC-simulated events. After this subtraction, the dominant
source of fake/non-prompt leptons is found to be heavy-flavour hadron decays.
The fake efficiencies, binned in lepton $p_{\text{T}}$ and $|\eta|$, are shown
in Figure 10.
It is assumed that since the loose and tight lepton selection criteria depend
on quantities related to the lepton itself or to its immediate surroundings,
the chosen parameterisation captures the main variations in the fake
efficiencies, and residual dependencies on the event environment can be
covered by systematic uncertainties (see Section 8). Indeed, in the
simulation, the dependence of the fake efficiencies on the number of light-
flavour jets or $b$-tagged101010Jets containing $b$-hadrons are identified
(tagged) by the MV2c10 $b$-tagging algorithm [43]. The algorithm uses a
multivariate discriminant with quantities such as the impact parameters of
associated tracks, and well-reconstructed secondary vertices. jets is mild.
The uncertainties are evaluated by comparing, in the simulation, the fake
efficiencies in event selections corresponding to either the measurement
regions in Table 1 or the regions of interest in the analysis for which the
background estimates are needed. These differences, evaluated as a function of
$p_{\text{T}}$ and $|\eta|$, are applied as a systematic uncertainty of the
fake-efficiency measurement (as discussed in Section 8), and are of the order
of 10%–20%, except for muons with $p_{\text{T}}>$50\text{\,}\mathrm{GeV}$$,
for which they reach 40%. Furthermore, normalisation uncertainties are
considered for the real-lepton background processes, which are subtracted in
the fake-efficiency measurement. They are evaluated by scaling the real-lepton
background processes upwards and downwards within their cross-section
uncertainties before the subtraction and using the differences between the
modified and nominal efficiencies as uncertainties, which are added in
quadrature to the aforementioned uncertainties.
#### 9.1.3 Results in the fake/non-prompt lepton validation regions
To validate the performance of the method, predictions are obtained and
compared with data in two dedicated validation regions called
‘VR-$3\ell$-$1b3j$’ and ‘VR-$3\ell$-$1b3j$-no$Z$’, which have a larger
proportion of fake/non-prompt leptons than is expected in the signal regions.
The definitions of these two validation regions are summarised in Table 2. No
charge requirements are placed on the reconstructed lepton candidates in these
regions.
Table 2: Definition of the fake/non-prompt lepton enriched validation regions, VR-$3\ell$-$1b3j$ and VR-$3\ell$-$1b3j$-no$Z$. $N_{\ell}$ stands for the number of leptons (${\ell}$), while $N_{\textrm{jets}}$ ($N_{b\textrm{-jets}}$) is the number of jets ($b$-tagged jets), respectively. The leptons are ordered by decreasing $p_{\text{T}}$ ($\ell_{1,2,3}$). The mass ($m$) variables are discussed in the text. For additional details of the object definitions, see Ref. [42]. Region | $N_{\ell}$ $(\ell=e,\mu)$ | $p_{\text{T}}$ $(\ell_{1,2,3})$ | $p_{\text{T}}$ (jets) | $|m_{\ell\ell}^{\textrm{SF}}-m_{Z}|$ | $N_{\textrm{jets}}$ | $N_{b\textrm{-jets}}$
---|---|---|---|---|---|---
VR-$3\ell$-$1b3j$ | $=3$ | $>27,20,$20\text{\,}\mathrm{GeV}$$ | $>$25\text{\,}\mathrm{GeV}$$ | – | $=3$ | $=1$
| (2 tight, 1 loose) | | | | |
VR-$3\ell$-$1b3j$-no$Z$ | $=3$ | $>27,20,$20\text{\,}\mathrm{GeV}$$ | $>$25\text{\,}\mathrm{GeV}$$ | $>$10\text{\,}\mathrm{GeV}$$ | $=3$ | $=1$
| (all tight) | | | | |
The variable $m_{\ell\ell}^{\textrm{SF}}$ refers to the invariant mass of the
same-flavour opposite-charge (SFOC) lepton pair with the invariant mass
closest to the $Z$ boson mass. VR-$3\ell$-$1b3j$ is a region similar to the
actual signal regions defined in Ref. [42], but without a requirement on the
$Z$ mass for the (SFOC) lepton candidate pair. Therefore, it contains a higher
fraction of fake/non-prompt leptons after the selection. To further enhance
the fake/non-prompt lepton contribution, the third-highest-$p_{\text{T}}$
lepton that satisfies the baseline selection criteria must not satisfy the
tight criteria. An additional validation region, VR-$3\ell$-$1b3j$-no$Z$, is
defined by requiring all three leptons to satisfy the tight selection
criteria, but placing a veto on SFOC lepton pairs that have an invariant mass
consistent with the $Z$ boson, thereby enhancing the fake/non-prompt lepton
fraction in this region. Both regions are orthogonal to the analysis signal
regions and intended to validate the predictions of the matrix method for
different levels of fake/non-prompt lepton contamination.
Some example distributions are shown in Figure 11 for VR-$3\ell$-$1b3j$ and
Figure 12 for VR-$3\ell$-$1b3j$-no$Z$. The processes with three real leptons
(modelled with MC simulations) plus the prediction from the matrix method can
be compared with data in these regions.
The hatched bands in Figures 11 and 12 show only the statistical uncertainties
of the MC prediction and the uncertainties associated with the fake/non-prompt
lepton estimates (i.e. no theoretical or detector-related systematic
uncertainties are included). The total uncertainty associated with the
fake/non-prompt lepton estimate itself contains a systematic component, which
is evaluated from variations of the input fake/real efficiencies (described in
the previous section), and the statistical uncertainty of the data sample to
which the likelihood matrix method is applied.111111The total uncertainties of
the fake/non-prompt lepton estimates may be different from the uncertainties
reported in Ref. [42], as the number of loose leptons in these validation
regions is larger than in the signal region of the $t\bar{t}Z$ analysis and,
therefore, the statistical uncertainties are smaller.
There is generally good agreement between the data and the total background
estimate, except that the background is overestimated at low $\Delta
R(\ell_{1},\ell_{2})$ and low $E_{\text{T}}^{\text{miss}}$ in Figure 11. One
contribution to that difference is that the two leading (two
highest-$p_{\text{T}}$) lepton candidates are likely to be real, yet when they
are near each other they have a lower efficiency for satisfying the isolation
criteria, and thus are misinterpreted as fake/non-prompt leptons by the matrix
method. Analyses that are sensitive to such issues may benefit from imposing a
minimum $\Delta R$ requirement between leptons. A discrepancy is also observed
in the higher $p_{\text{T}}$ bins of Figure 11(d). Events in these bins have
three leptons with $p_{\text{T}}$ above $50\text{\,}\mathrm{GeV}$. As shown in
Figures 9 and 10, only a single $p_{\text{T}}$ bin above
$50\text{\,}\mathrm{GeV}$ is available for measuring
$\varepsilon_{\mathrm{r}}$ and $\varepsilon_{\mathrm{f}}$, due to the limited
number of events in the control regions, so variations above
$50\text{\,}\mathrm{GeV}$ may be missed. This point was not investigated
thoroughly since only a small fraction of events in the analysis were
impacted.
((a)) ((b))
((c)) ((d))
Figure 11: (a) Angular separation between the leading and second leading (in
$p_{\text{T}}$) lepton candidates, $\Delta R(\ell_{1},\ell_{2})$, (b) angular
separation between the second- and third-leading lepton candidates, $\Delta
R(\ell_{2},\ell_{3})$, (c) missing transverse momentum in the event,
$E_{\text{T}}^{\text{miss}}$, and (d) the $p_{\text{T}}$ of the third-leading
lepton in VR-$3\ell$-$1b3j$. The processes with three real leptons are
modelled with MC simulation, while the contribution from fake/non-prompt
leptons (dark red) comes from the likelihood matrix method as described above.
The hatched band shows the uncertainty from the MC statistics and the
fake/non-prompt background estimate. The rightmost bins are inclusive and
contain all events above the $x$-axis ranges. The lower panel shows the ratio
of data to the total SM prediction (sum of the real-lepton background
contributions estimated with MC samples and the fake/non-prompt lepton
contribution estimated with the matrix method). Further details are available
in Ref. [42].
((a)) ((b))
((c)) ((d))
Figure 12: Comparisons of the predicted and observed yields in
VR-$3\ell$-$1b3j$-no$Z$, with respect to the $p_{\text{T}}$ of the (a) leading
and (b) subleading lepton candidates, (c) the scalar sum of the lepton and jet
transverse momenta, $H_{\text{T}}$, and (d) the missing transverse momentum in
the event, $E_{\text{T}}^{\text{miss}}$. The processes with three real leptons
are modelled with MC simulation, while the contribution from fake/non-prompt
leptons (dark red) comes from the likelihood matrix method as described above.
The hatched band shows the uncertainty from MC statistics and the fake/non-
prompt background estimate. The rightmost (leftmost) bins are inclusive and
contain all events above (below) the $x$-axis ranges. The lower panel shows
the ratio of data to the total SM prediction (sum of the real-lepton
background contributions estimated with MC samples and the fake/non-prompt
lepton contribution estimated with the matrix method). Further details are
available in Ref. [42].
### 9.2 Model-independent search for new phenomena in multi-lepton final
states
Many interesting new models for BSM physics predict final states with three or
more leptons. The general multi-lepton search for new phenomena [44]
agnostically considers such final states. Its aim is to be sensitive to BSM
phenomena in often-overlooked corners of phase space. The background
estimation for the multi-lepton search uses MC predictions to account for
events that contain only real leptons, and the fake-factor method for events
containing at least one fake/non-prompt lepton. The dominant sources of
fake/non-prompt leptons are semileptonic heavy-flavour decays (primarily of
$b$-hadrons), light-hadron decays, and misidentification of light hadrons as
leptons. These mainly arise in $Z\text{\,+\,jets}$ and $t\bar{t}$ events.
#### 9.2.1 Fake/non-prompt lepton selection
The fake factors are measured using events with a single lepton candidate. In
order to prevent a bias in the fake factor due to trigger selection criteria
[14, 15], the selected events are required to have fired a loose single-lepton
trigger where isolation requirements are not imposed. However, due to the high
rate of events that pass such triggers, a prescale factor is applied, which
reduces the number of events available for measuring the fake factors.
Baseline lepton candidates must pass a common object selection, as detailed in
Ref. [44]. Electron candidates are required to pass either the ‘Loose’ ID WP
with calorimeter- and track-based isolation requirements, or the ‘Tight’ ID WP
with no isolation requirement. Baseline muon candidates are required to pass
the ‘Medium’ ID WP (‘HighPt’ for $p_{\text{T}}>$300\text{\,}\mathrm{GeV}$$).
Only leptons with $p_{\text{T}}>$25\text{\,}\mathrm{GeV}$$ and satisfying the
longitudinal and transverse track impact parameter requirements (see Sections
4.1 and 4.2) are considered.
More stringent selection criteria are imposed for tight lepton candidates. For
muons, where fake/non-prompt candidates are mainly muons from semileptonic
heavy-flavour decays, only the isolation criteria is modified: the tight
selection requires that muons satisfy track-based isolation criteria. For
electrons, where both light- and heavy-flavour hadrons are a non-negligible
source of fake/non-prompt candidates, both the ID and isolation criteria are
modified. Tight electrons must satisfy both the ‘Tight’ ID WP and calorimeter-
and track-based isolation requirements.
Additional selection requirements are imposed on the single-lepton sample to
ensure a high purity of fake/non-prompt leptons, which reduces the statistical
uncertainty of the computed fake factor: the $E_{\text{T}}^{\text{miss}}$ is
required to be $<$25\text{\,}\mathrm{GeV}$$ ($<$40\text{\,}\mathrm{GeV}$$) for
events with electron (muon) candidates, and the number of jets in the event is
required to be $\geq 1\leavevmode\nobreak\ (2)$ for events with electron
(muon) candidates. For muon-candidate events, there must also be at least one
jet with $p_{\text{T}}>$35\text{\,}\mathrm{GeV}$$ (the ‘tag jet’), and the
azimuthal angle between the muon candidate and this jet is required to be
$>2.7$ radians.
The fake factor is binned in both the $p_{\text{T}}$ and $|\eta|$ of the
lepton candidate. The bins are defined to have tolerable statistical
uncertainties while preventing sizeable differences between the fake-factor
values in adjacent bins. The highest muon-$p_{\text{T}}$ bin includes all muon
candidates with $p_{\text{T}}>$80\text{\,}\mathrm{GeV}$$, as there are very
few fake/non-prompt muons at higher transverse momenta.
#### 9.2.2 Fake factors
The fake factors calculated for the multi-lepton analysis are shown in Figures
13 and 14 for electrons and muons. The uncertainties in the fake factors are
categorised and evaluated as described below.
There are statistical uncertainties due to the data and MC sample sizes in
each fake-factor bin, which due to their small size are summed in quadrature
into a single uncertainty.
In the regions enriched in fake/non-prompt leptons, MC predictions are used to
subtract the real-lepton contribution from the data. The uncertainties in the
MC contributions are propagated to the fake factors. The main contributions
from real leptons in the single-lepton regions are from $Z\text{\,+\,jets}$,
$W\text{+\,jets}$ and $t\bar{t}$ events. For the $Z/W\text{+\,jets}$
processes, an uncertainty of 5% is applied to the cross-section. For the
$t\bar{t}$ process, the assumed uncertainties are ${}^{+2.4\%}_{-3.3\%}$
($\mu_{\mathrm{r}}$, $\mu_{\mathrm{f}}$) and $\pm 4.2\%$ (PDF).
Extrapolating the fake factors from the single-lepton sample to the multi-
lepton samples used in the analysis introduces an uncertainty because these
samples differ in the kinematic distributions of fake/non-prompt leptons, and
possibly also in the fake/non-prompt lepton composition. Two uncertainties are
included to address the bias caused by imposing a $E_{\text{T}}^{\text{miss}}$
upper bound in the fake/non-prompt lepton estimation sample, and by imposing a
$p_{\text{T}}$ requirement on the tag jet in the fake/non-prompt muon
estimation sample. These uncertainties are estimated by varying the
requirements on these variables upwards and downwards by
$10\text{\,}\mathrm{GeV}$.121212Only the $E_{\text{T}}^{\text{miss}}$
variation is displayed separately in the plots, as the variation of the jet
$p_{\text{T}}$ results in an uncertainty that is too small to be visible.
Plots showing the impact of these systematic effects on the fake factors are
given in Figures 13 and 14.
((a)) ((b))
((c)) ((d))
Figure 13: Measured fake factors for electrons as a function of $p_{\text{T}}$
for different $|\eta|$ ranges, and their dependence on individual variations
of parameters of the measurement, used to determine systematic uncertainties.
The ‘MC syst’ uncertainty covers variations in the model used to subtract the
real-lepton contribution in the control regions used to measure
$\varepsilon_{\text{f}}$. The combined impact of statistical and systematic
uncertainties added in quadrature is indicated by the shaded yellow area (the
grey area represents only the statistical uncertainty).
((a)) ((b))
((c))
Figure 14: Measured fake factors for muons as a function of $p_{\text{T}}$ for
different $|\eta|$ ranges, and their dependence on individual variations of
parameters of the measurement, used to determine systematic uncertainties. The
combined impact of statistical and systematic uncertainties added in
quadrature is indicated by the shaded yellow area (the grey area represents
only the statistical uncertainty).
Finally, a direct assessment of the uncertainty in the composition of the
fake/non-prompt lepton background in the multi-lepton sample is made. Since
fake/non-prompt leptons can come from both light- and heavy-flavour sources,
it is possible that the relative abundances from these sources can vary
between samples. This possibility is addressed through an additional
uncertainty which leverages the different event signatures produced by light-
and heavy-flavour hadrons, the latter consisting primarily of $b$-hadrons. To
evaluate this uncertainty, an alternative set of fake factors is computed
which, in addition to the binning in $p_{\text{T}}$ and $|\eta|$, are binned
according to the presence or absence of $b$-tagged jets in the event. This
alternative set of fake factors is shown in Figure 15. The composition
uncertainty in the total event yields is then derived from the difference
between estimates calculated with either the nominal fake factors oblivious to
the presence of $b$-tagged jets in the events, or the alternative set that
requires such a jet. This uncertainty is found to have a negligible impact on
the analysis, since most of the jets are not $b$-tagged.
((a)) ((b))
((c)) ((d))
Figure 15: Fake factors measured in bins of the lepton candidate’s
$p_{\text{T}}$ and $|\eta|$ for events with (a,c) electron candidates and
(b,d) muon candidates in events (a,b) without and (c,d) with $b$-tagged jets.
#### 9.2.3 Validation regions
Validation regions are defined using appropriate sub-selections of $ee\mu$ and
$e\mu\mu$ events. These are used to check that the computed fake factors
extrapolate correctly from the regions where they are calculated to the
regions in which they are applied. The ‘on-$Z$’ validation region requires an
SFOC lepton pair with a dilepton mass within $10\text{\,}\mathrm{GeV}$ of the
$Z$ boson mass. The ‘off-$Z$’ validation region also requires a SFOC pair of
leptons, but requires the dilepton mass to fall outside of the $Z$-mass
window. Only mixed-flavour final states are selected for these validation
regions so that which lepton to choose as the third one, assumed to be the
fake/non-prompt lepton, is unambiguous. The sources of fake/non-prompt leptons
contribute in different ratios to the on-$Z$ and off-$Z$ validation regions:
the on-$Z$ validation region is more sensitive to $Z\text{\,+\,jets}$ events
than the off-$Z$ region, while the inverse is true for $t\bar{t}$ events,
although in absolute terms, $Z\text{\,+\,jets}$ events are more numerous than
$t\bar{t}$ events in both cases. Both validation regions target, through a
$m_{\text{T}}$ requirement of
$m_{\text{T}}(\ell,E_{\text{T}}^{\text{miss}})<$40\text{\,}\mathrm{GeV}$$, a
third lepton that is likely to be fake/non-prompt. The union of the on-$Z$ and
off-$Z$ validation regions is called the ‘fakes validation region’.
The variables of primary importance for this analysis are the invariant mass
of all lepton candidates in the event ($m_{\text{inv}}$) and the
$E_{\text{T}}^{\text{miss}}$. The signal regions are separated according to
the values of these quantities, as discussed in Ref. [44]. The
$m_{\text{inv}}$ distributions in the two validation regions are shown in
Figure 16, while the $E_{\text{T}}^{\text{miss}}$ distributions are shown in
Figure 17. Lastly, the electron and muon candidate $p_{\text{T}}$
distributions are shown in the fakes validation region in Figure 18. The
comparisons in Figure 16 were also presented in Ref. [44] (albeit in
logarithmic scale) and thus include fitted normalisation factors for the
$WZ\text{\,+\,jets}$ and $ZZ\text{\,+\,jets}$ backgrounds from their
respective control regions (‘post-fit’). On the other hand, the complementary
distributions shown in Figures 17 and 18 were obtained independently of this
statistical analysis, and thus employ the unconstrained SM background
normalisations and uncertainties (‘pre-fit’ distributions). The background
estimate is consistent with the data within the statistical and systematic
uncertainties.
((a)) ((b))
Figure 16: Comparison between data and prediction for the $m_{\text{inv}}$
distribution in the (a) on-$Z$ and (b) off-$Z$ validation regions, after
fitting the normalisation factors for the $WZ\text{\,+\,jets}$ and
$ZZ\text{\,+\,jets}$ backgrounds and systematic uncertainties [44]. All
uncertainties, systematic and statistical, are included. The leftmost
(rightmost) bin is inclusive and contains all events with
$m_{\text{inv}}<$200\text{\,}\mathrm{GeV}$\hbox{
}(>$500\text{\,}\mathrm{GeV}$)$. The hatched grey area in these figures shows
the total uncertainty. Further details are available in Ref. [44].
((a)) ((b))
Figure 17: Comparison between data and prediction for the
$E_{\text{T}}^{\text{miss}}$ distribution in the (a) on-$Z$ and (b) off-$Z$
validation regions. All uncertainties, systematic and statistical, are
included. The rightmost bin is inclusive and contains all events with
$E_{\text{T}}^{\text{miss}}>$475\text{\,}\mathrm{GeV}$$. The hatched grey area
in these figures shows the total uncertainty. Further details are available in
Ref. [44].
((a)) ((b))
Figure 18: Comparison between data and prediction for the $p_{\text{T}}$
distribution in the fakes validation region (which is the union of the on-$Z$
and off-$Z$ validation regions). Shown are (a) the electron $p_{\text{T}}$ in
$e\mu\mu$ and (b) the muon $p_{\text{T}}$ in $ee\mu$ events. All
uncertainties, systematic and statistical, are included. The rightmost bin is
inclusive and contains all events with
$p_{\text{T}}>$275\text{\,}\mathrm{GeV}$$. The hatched grey area in these
figures shows the total uncertainty. Further details are available in Ref.
[44].
## 10 Conclusions
For physics analyses exploring signatures with one or more prompt leptons,
background contributions due to fake/non-prompt leptons are often difficult to
estimate in simulations. Therefore, data-driven methods are commonly used.
Three related methods have been adopted by the ATLAS Collaboration as
recommended tools: the asymptotic matrix method, the likelihood matrix method,
and the fake-factor method. All three approaches depend on defining two
categories of leptons, one of which (‘tight’) is subject to the same
identification and selection criteria as are used in the analysis. The other
category (‘loose’) adds additional lepton candidates with less stringent
selection requirements. The union of the two sets is called the ‘baseline’
sample. The criteria are typically defined such that the probability for a
real baseline lepton to satisfy the tight criteria is substantially higher
than the corresponding probability for a fake/non-prompt lepton. Then, the
relative numbers of loose and tight leptons in the analysis sample can be used
to estimate the contribution of fake/non-prompt leptons, either inclusively or
differentially in any variables of interest.
Despite their similarities, the methods each have their own strengths and
drawbacks. The asymptotic matrix method and fake-factor method provide a
fake/non-prompt lepton weight for each event, which is convenient for
analyses. However, these methods are subject to large uncertainties if the
efficiency for loose fake/non-prompt leptons to satisfy the tight criteria is
large in parts of the analysis phase space. The likelihood matrix method
returns a smaller uncertainty in such cases, and avoids any possibility of
producing a negative estimate for the event yield, but does not provide a per-
event weight, introducing difficulties e.g. for differential estimations. The
fake-factor method uses simulation rather than data to incorporate the
contribution from events where all leptons are tight. This method can
therefore be employed while the signal region for an analysis is fully
blinded, although it may induce additional simulation-related uncertainties in
the background estimate.
The systematic uncertainties for all three methods arise from similar sources,
with the largest contributions related to the extrapolation of the
efficiencies measured in the control samples to events in the analysis sample.
Differences in the fake/non-prompt lepton composition in the samples must be
accounted for and appropriate uncertainties must be assigned to this
extrapolation.
The performance of the likelihood matrix method and the fake-factor method has
been demonstrated in a differential $t\bar{t}Z$ cross section measurement and
in a model-independent search for BSM phenomena in multi-lepton final states,
respectively. In both cases, the chosen method was shown to provide a reliable
estimate of the fake/non-prompt lepton background, as measured in validation
regions.
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|
# Computational modelling of passive transport of functionalized nanoparticles
Daniela Moreno-Chaparro<EMAIL_ADDRESS>Basque Center for Applied
Mathematics, BCAM. Alameda de Mazarredo 14 , Bilbao 48400, Spain University
of the Basque Country/Euskal Herriko Unibertsitatea. Barrio Sarriena Leioa,
48940, Spain Nicolas Moreno †<EMAIL_ADDRESS>Basque Center for Applied
Mathematics, BCAM. Alameda de Mazarredo 14 , Bilbao 48400, Spain Florencio
Balboa Usabiaga Basque Center for Applied Mathematics, BCAM. Alameda de
Mazarredo 14 , Bilbao 48400, Spain Marco Ellero Basque Center for Applied
Mathematics, BCAM. Alameda de Mazarredo 14 , Bilbao 48400, Spain IKERBASQUE,
Basque Foundation for Science, Calle de Maria Dias de Haro 3, 48013,
Bilbao,Spain Zienkiewicz Center for Computational Engineering (ZCCE), Swansea
University, Bay Campus, Swansea SA1 8EN, United Kingdom
###### Abstract
Functionalized nanoparticles (NPs) are complex objects present in a variety of
systems ranging from synthetic grafted nanoparticles to viruses. The
morphology and number of the decorating groups can vary widely between
systems. Thus, the modelling of functionalized NPs typically considers
simplified spherical objects as a first-order approximation. At the nanoscale
label, complex hydrodynamic interactions are expected to emerge as the
morphological features of the particles change, and they can be further
amplified when the NPs are confined or near walls. Direct estimation of these
variations can be inferred via diffusion coefficients of the NPs. However, the
evaluation of the coefficients requires an improved representations of the NPs
morphology to reproduce important features hidden by simplified spherical
models. Here, we characterize the passive transport of free and confined
functionalized nanoparticles using the Rigid Multi-Blob (RMB) method. The main
advantage of RMB is its versatility to approximate the mobility of complex
structures at the nanoscale with significant accuracy and reduced
computational cost. In particular, we investigate the effect of functional
groups distribution, size and morphology over nanoparticle translational and
rotational diffusion. We identify that the presence of functional groups
significantly affects the rotational diffusion of the nanoparticles, moreover,
the morphology of the groups and number induce characteristic mobility
reduction compared to non-functionalized nanoparticles. Confined NPs also
evidenced important alterations in their diffusivity, with distinctive
signatures in the off-diagonal contributions of the rotational diffusion.
These results can be exploited in various applications, including biomedical,
polymer nanocomposite fabrication, drug delivery, and imaging.
††preprint: AIP/123-QED
## I functionalized nanoparticles:
Introduction
Nanoparticles (NPs) are complex structures ubiquitous on many synthetic and
biological system. NPs can be found in a variety of morphologies, ranging
from simple shapes (i.e spheres, cubes, ellipsoids) to more complex
structures. Moreover, they can exhibits a disparate number of functional
decorations with characteristic morphologies that determine their
functionality. Examples of these are organelles, viruses, and grafted
nanoparticles, all of them with a size in the order of 10 to 200nm.Choueiri
_et al._ (2016); Bao _et al._ (2021) Depending on the field, such decorations
are typically referred to as functional groups, spikes,Moreno, Chaparro, and
Usabiaga (2022) grafts, patches,Chen (2022) or hairs.Tang _et al._ (2022) For
simplicity, here, we use the functional groups (G) notation to denote any
type of decoration on the surface of the NPs core. In general, NPs have unique
transport features related to the groups morphology and number ($N_{G}$), as
illustrates in FIG 1.A. In this context, microrheological techniques emerge
as powerful tool for NPs study and characterization for various applications,
including biomedical, polymer nanocomposite fabrication, drug delivery, and
imaging,Chancellor, Seymour, and Zhao (2019) to name a few. For example, in
the field cancer tumours treatment, investigations on the diffusion of
different gold NPs and liposomesShi and Tian (2019) through the mucus and
cells have been addressed to improve drug delivery.
Figure 1: Sketch of a NP and the discretization adopted. A. NP representation
with core and functional groups , G as blue cylinders. If the number of groups
($N_{G}$) is $N_{G}=0$ the NP is only the core. B . Functionalized shapes orG
types representation, the measurement of length $l_{G}$ and width $w_{G}$ are
reduced by the radius of the core. C . Discretization of a sphere in blobs. D.
Resolution and refinement following theG types rod, tetrahedron, and the core
sphere. E . NP near a wall, the distance h is given by the wall to the center
of the NP
Computational studies on decorated particles at microscales, have already
revealed the effects of G morphology on the formation of complex structures.Li
_et al._ (2014); Karatrantos _et al._ (2017) At nanoscales, as thermal
fluctuations becomes relevant, investigations has been mostly focused on core-
only NPs (spheres), whereas the simulation of functionalized ones remains a
challenging task. The morphological features along with specific binding and
affinity interactions between the groups dramatically increase the complexity.
Different computational methods have been used to characterize the transport,
binding and interactions for non-functionalized NPs, including Brownian
dynamics for freeLi _et al._ (2014); Islam, Barua, and Barua (2017) and near
walls NPs,Lisicki _et al._ (2014) Monte Carlo,Li _et al._ (2019) coarse
-graining,Ilnytskyi (2020); Debets, Janssen, and Šarić (2020) dissipative
particle dynamics,Liu, Shah, and Tan (2012); Chen (2022) and smooth
dissipative particle dynamics.Bian _et al._ (2012); Vázquez-Quesada and
Ellero ; Vázquez-Quesada and Ellero (2022) The transport modelling of NPs
usually deals with spherical shapes, adopting the Stokes-Einstein
equations.Murray and Jackson (1992) In general, the effect of the groups
number, distribution around the core, and morphology, on the NPs transport
lack of detailed investigations. Moreover, other relevant aspects such as
their passive transport under confinement are still missing. Confined
transport may play a critical role in the design of microfluidic devicesTang
_et al._ (2022) and sensors.Willner and Vikesland (2018)
Herein, from a microrheology standpoint, we investigate how the mobility of
complex functionalized NPs is affected by the G morphology, number, and
distribution. In particular, we characterize their translational ($D_{t}$) and
rotational ($D_{r}$) diffusivity. NP’s translational and rotational
diffusivity can be derived using the Stokes-Einstein theory. NPs diffusivity
arises from the balance between thermal fluctuations and hydrodynamic
interactions with the fluid, that depend on NP morphology and fluid
viscosity.Murray and Jackson (1992) Here, we use the Rigid Multi-blob method
Usabiaga _et al._ (2016) (RMB) to model complex functionalized NPs by
discretizing them as a set of rigidly connected spherical blobs. RMB can be
applied to physical and biological systemsSprinkle _et al._ (2017); Moreno,
Chaparro, and Usabiaga (2022); Usabiaga and Delmotte (2022) at the nanoscale,
where the thermal fluctuations and hydrodynamic interactions are relevant.
This method, is suitable to model arbitrary shape objects (both free and
confined), and its principal advantage is the low computational cost of
solving a mobility problem.
We consider that the NPs are constituted by a spherical core decorated with
$N_{G}$ functional groups. We represent the NP core as a spherical shell of
multiple blobs, whereas Gs are modelled using various shapes such as rods,
spheres, and tetrahedrons. These basic set of shapes are inspired from
different real morphologies reported for synthetic and biological
nanoparticlesTang _et al._ (2022); Chen (2022); Moreno, Chaparro, and
Usabiaga (2022); Moreno, Sutisna, and Fried (2020); Choueiri _et al._ (2016),
and allows us to explore in a systematic fashion the effect of G morphology on
the nanoparticles transport. In general, the number and relative size of the
Gs can vary widely among physical systems. Thus, we also explore the effects
of number density and size (length and width) of the groups. Additionally, we
study the effects of groups distribution around the core. Since, the location
of Gs may affect the overall mobility of the nanoparticles, understanding the
effect of the groups distribution provides relevant information for
nanoparticle design and optimization, either to enhance or reduce the
diffusion. As a final compelling aspect on NPs transport, we investigate
confinement effects, by computing the effective parallel and perpendicular
diffusion near walls.
### I.1 Functionalized nanoparticles mobility
The translational diffusion coefficient of a NP is related to its
translational mobility,Einstein (1905) and similar arguments can be easily
extended to the rotational diffusion.Kim and Karrila (1991) Thus, the
diffusion coefficients, $D_{t}$ and $D_{r}$, can be expressed proportional to
the mobilities as
$\displaystyle
D_{t}=\frac{k_{B}T}{3}\mathrm{Tr}\left(\bm{M}_{t}\right),\;\;\;D_{r}=\frac{k_{B}T}{3}\mathrm{Tr}\left(\bm{M}_{r}\right),$
(1)
where $k_{B}T$ is the thermal energy and $\mathrm{Tr}$ denotes the trace
operator. The mobility components yield the linear and angular velocities of a
NP ($\bm{u}$ and $\bm{\omega}$) in response to applied forces and torques
($\bm{f}$ and $\bm{\tau}$),
$\displaystyle\left(\begin{array}[]{c}\bm{u}\\\
\bm{\omega}\end{array}\right)=\left(\begin{array}[]{cc}\bm{M}_{t}&\bm{M}_{c}\\\
\bm{M}_{c}^{T}&\bm{M}_{r}\end{array}\right)\left(\begin{array}[]{c}\bm{f}\\\
\bm{\tau}\end{array}\right).$ (8)
Theoretically, for a spherical NP of radius $R$ translational diffusion is
given by
$\displaystyle D_{t}=k_{B}T/(6\pi\eta R),$ (9)
and the rotational diffusion by
$\displaystyle D_{r}=k_{B}T/(8\pi\eta R^{3}).$ (10)
For NPs, the mobility components can be calculated using the Stokes equations
to a good approximation.Schmidt and Skinner (2004); Usabiaga _et al._ (2013)
In this limit, the fluid velocity and pressure, $\bm{v}$ and $p$, obey the
Stokes equations with viscosity $\eta$
$\displaystyle-\bm{\nabla}p+\eta\bm{\nabla}^{2}\bm{v}$ $\displaystyle=0,$ (11)
$\displaystyle\bm{\nabla}\cdot\bm{v}$ $\displaystyle=0,$ (12)
while for boundary conditions, one can assume that the fluid velocity obeys
the no-slip condition at the NPs surface and decays to zero at infinity.
We have assumed a functionalized NP behaves like a rigid body; thus, the no-
slip condition for a NP located at free point $\bm{q}$ is quite simple,
$\displaystyle\bm{v}(\bm{r})=\bm{u}+\bm{\omega}\times(\bm{r}-\bm{q})\;\;\;\mbox{for
all }\bm{r}\in\partial\Omega.$ (13)
These partial differential equations are closed by the balance of force and
torque. The integral of the fluid traction, $-\bm{\lambda}$, over the surface
of the NP balance the external forces and torques applied to the NP Pozrikidis
(1992)
$\displaystyle\int_{\partial\Omega}\bm{\lambda}\,\mathrm{d}S_{r}$
$\displaystyle=\bm{f},$ (14)
$\displaystyle\int_{\partial\Omega}(\bm{r}-\bm{q})\times\bm{\lambda}\,\mathrm{d}S_{r}$
$\displaystyle=\bm{\tau}.$ (15)
In most applications NPs are under different type of confinements. Thus, two
paradigmatic cases are i) NPs immersed in a suspension and ii) a single NP
diffusing near a large flat wall. The first case can be modeled by using a
computational domain with periodic boundary conditions. In this case the
diffusion coefficient is given byBian _et al._ (2012)
$D=D_{t}/\lambda$ (16)
where $D_{t}$ is the theoretical translational diffusion and $\lambda$ is a
drag coefficient
$\centering\lambda=(1-1.7601C^{1/3}+C-1.5593C^{2}+3.9799C^{3/8}\@add@centering$
(17) $-3.073C^{10/3}+C^{11/3})^{-1},$
where $C=4/3\pi R^{3}/L^{3}$ and $L$ is the size of the domain.
In the second case, a NP diffusing near a wall, the symmetry of the system is
broken by the presence of the boundary. Therefore, it is necessary to
distinguish between the diffusion parallel and perpendicular to the wall. If
we denote as h the distance from the centre of the NP to the wall, and s=h-R
the distance from the NP’s core to the wall, the parallel diffusion is given
by $D_{\parallel}(s)=D/\lambda_{\parallel}(s)$, and the perpendicular is
$D_{\perp}(s)=D/\lambda_{\perp}(s)$.Bian _et al._ (2012) Theoretical values
of the drag coefficients $\lambda_{\parallel}(s)$ Hasimoto (1959); Swan and
Brady (2007); Bian _et al._ (2012) and $\lambda_{\perp}(s)$Hasimoto (1959);
Swan and Brady (2007); Bian _et al._ (2012) are
$\lambda_{\perp}(s)=\frac{4}{3}\sinh\alpha\sum^{\infty}_{n=1}\frac{n(n+1)}{(2n-1)(2n+3)}$
(18) $\left[\frac{2\sinh(2n+1)\alpha+(2n+1)\sinh
2\alpha}{4\sinh^{2}(n+1/2)\alpha-(2n+1)^{2}\sinh^{2}\alpha}-1\right],$
$\lambda_{\parallel}(s)=\left[1-\frac{9}{16}\beta+\frac{1}{8}\beta^{3}-\frac{45}{256}\beta^{4}-\frac{1}{16}\beta^{5}\right]^{-1},$
(19)
where $\alpha=\cosh^{-1}(1-h/R)$ and $\beta=(1-h/R)^{-1}$ respectively.
### I.2 Rigid Multi-Blob method (RMB)
To compute the mobilities of a NP, we adopt the Rigid Multi-Blob method
(RMB).Usabiaga _et al._ (2016) We discretize the surface of the NP with $N$
markers or _blobs_ of radius $a$ with position $\bm{r}_{i}$. The blobs are
subject to constraint forces, $\bm{\lambda}_{i}$, that ensure the rigid motion
of the whole NP. Evaluating the no-slip condition at the blobs, as in
collocation methods, leads to a linear system of equations for the unknowns
$\bm{u}$, $\bm{\omega}$ and $\bm{\lambda}_{i}$,
$\displaystyle\bm{v}(\bm{r}_{i})=\sum_{j=1}^{N}\left(\bm{M}_{B}\right)_{ij}\bm{\lambda}_{j}$
$\displaystyle=\bm{u}+\bm{\omega}\times(\bm{r}_{i}-\bm{q})\;\;\;\mbox{for
}i=1,\ldots,N,$ (20) $\displaystyle\sum_{j=1}^{N}\bm{\lambda}_{j}$
$\displaystyle=\bm{f},$ (21)
$\displaystyle\sum_{j=1}^{N}(\bm{r}_{j}-\bm{q})\times\bm{\lambda}_{j}$
$\displaystyle=\bm{\tau},$ (22)
In the no-slip equation, Eq. (20), the blob mobility matrix
$\left(\bm{M}_{B}\right)_{ij}$ couples the force acting on the blob $j$ to the
flow generated at the blob $i$. We use the regularized Rotne-Prager mobility,
that has closed analytical expressionRotne and Prager (1969); Wajnryb _et
al._ (2013)
$\displaystyle\left(\bm{M}_{B}\right)_{ij}=\left(\bm{I}+\frac{a^{2}}{6}\bm{\nabla}^{2}_{\bm{r}}\right)\left(\bm{I}+\frac{a^{2}}{6}\bm{\nabla}^{2}_{\bm{r}^{\prime}}\right)\bm{G}(\bm{r},\bm{r}^{\prime})|^{\bm{r}=\bm{r}_{i}}_{\bm{r}^{\prime}=\bm{r}_{j}},$
(23)
where $\bm{G}(\bm{r},\bm{r}^{\prime})$ is the Green’s function of the Stokes
equation (i.e. the Oseen kernel). To model the NP near a wall, we use the
Rotne-Prager-Blake tensor that accounts for the hydrodynamic interactions with
the wall.Sprinkle _et al._ (2017) For reliable hydrodynamics interactions, we
recommend a minimum distance of one blob radius $r_{o}=a$, such that the blobs
do not overlap the wall.
### I.3 Morphology of functional groups and core of the nanoparticle
The morphology of a NP is given by its spherical-shape core and the functional
groups around it. This functional groups have a characteristic length,
$l_{G}$, and width, $w_{G}$. To streamline the analysis, we consider the
dimensionless size of the groups given by $l_{G}/R$ and width as $w_{G}/R$,
where $R$ is the radius of the NP’s core. For simplicity, we choose three
general shapes to construct the functional groups: rods, tetrahedrons, and
spheres. Using these general shapes, we create two composite shapes, rod-tetra
and rod-sphere, illustrated in Figure 1.B. The position of the groups in the
NP is modelled using uniform and random distributions. For the uniform case,
the Gs are distributed at equidistant positions corresponding to the vertex of
a icosahedra, whereas for random distribution, any position over the core is
allowed.
### I.4 Discretization of nanoparticles
The NPs are discretized by blobs rigidly connected. The distance between these
blobs is $r_{o}$ and defines the blob size. In FIG 1.C, we illustrate the
discretization of a solid sphere of radius $R$ into 12 connected blobs. The
general methodology to discretize the different shapes investigated is as
follows: i) For rods, the construction consists in equidistantly blobs along
an orientation angle $\alpha$ (see FIG 1.D) ii) for spheres we start with a
coarse 12-blobs model located at a distance $R$ from the center, and arbitrary
distance $r$ (see FIG 1.C). Then we conduct a recursive refinement taking the
middle point of the segments connecting two adjacent vertexes and projecting
those points radially, their new position satisfies $R^{2}=x^{2}+y^{2}+z^{2}$.
This procedure is repeated until $r\leq r_{o}$. Spheres and tetrahedrons with
different degrees of refinement are illustrated in FIG 1.D. iii) For
tetrahedron, with follow a similar iterative process, starting with coarse
surface with only four vertexes. Then the structure is refined by splitting in
half the edges between two vertexes and adding a new point in that position.
This addition must be applied to all the edges of the primary surface. This
procedure is repeated until the distance between adjacent points is smaller
than the target resolution.
### I.5 Resolution
We define the resolution ($\Phi$) of the discretization as the ratio between
the radius of NP core and the distance between blobs as $\Phi={R}/{r_{o}}$.
The optimal $\Phi$ is a selected as a trade off between accuracy and
computational cost. For simple spheres the accuracy at a given resolution we
can estimated comparing the hydrodynamic radius computed numerically with the
input radius of the object. For tetrahedron shapes, we can define different
characteristic sizes such as the width ($a$), height
($H_{\text{tetra}}={\sqrt{6}}/{3}a$), and the circumscribing sphere of radius
$R_{\text{tetra}}=l_{G}(3/8)^{1/2}$. However, to streamline the resolution
analysis for this non-spherical shapes we use its equivalent radius ($R_{e}$),
defined by the radius of a sphere with equivalent volume of the tetrahedron,
$V_{tetra}={a^{3}}/{6\sqrt{2}}$, leading to
$R_{e}=\Big{(}{3}/{4\pi}V_{\text{tetra}}\Big{)}^{{1}/{3}}.$
### I.6 Reduced translational and rotational diffusivities
For convenience, in the remaining, we discuss our findings in terms of reduced
diffusivities. For non-functionalized (only spherical core) nanoparticles, we
the theoretical translational and rotational mobilities are given by
$\bm{M}_{t}^{o}={1}/{6\pi\eta R}$ and $\bm{M}_{r}^{o}={1}/{8\pi\eta R^{3}}$,
respectively. The translational and rotational mobilities computed numerically
are simply referred as $\bm{M}_{t}|_{\text{sphere}}$ and
$\bm{M}_{r}|_{\text{sphere}}$ according the Eq. 17. Henceforth, we define the
reduced diffusivities as the ratio
$\bar{D}_{i}=\bm{M}_{i}|_{\text{sphere}}/{\bm{M}_{i}^{o}}$ where $i=t,r$. For
functionalized NP, since no theoretical values exist, we define the reduced
diffusivities in terms of the numerical mobilities of the nanoparticles and
the non-functionalized core as
$\bar{D}_{i}={\bm{M}_{i}}/{\bm{M}_{i}|_{\text{sphere}}}$. To indentify the
adequate resolution of tetrahedral shapes used for functional groups, we
conduct additional studies of freely diffusing tetrahedrons. For this shape in
particular since we do not count with an analytical expresion fo the mobility,
we express its reduced diffusivities as
$\bar{D}_{i}^{tetra}={\bm{M}_{i}|_{\text{tetra}}}/{\bm{M}_{i}^{o}}$, where
$i=t,r$ and $\bm{M}_{t}^{o}$ is the mobility of a reference sphere. The
diffusivity of confined NP is analyzed in terms of the parallel $D\parallel$
and the perpendicular $D\perp$ componets to the confining wall. To investigate
the effect of confinement, we place the NPs at different distances
$s^{\prime}$ to the wall (see FIG 1.E). We denote the diffusivities near a
wall as $D_{i}|_{s}$ and the ones of a free NP (without a wall) as
$D_{i}^{*}$, where $i=t,r$. Thus, we define the reduced translational and
rotational diffusivities for confined NPs as
$\overline{D}_{t}^{wall}={D_{t}|_{s}}/{D_{t}^{*}}$ and
$\overline{D}_{r}^{wall}={D_{r}|_{s}}/{D_{r}^{*}}$, respectively. In each
case, $\overline{D}_{t}^{wall}$ and $\overline{D}_{r}^{wall}$ are separated
into parallel and perpendicular components as explained in Eqs. 12 and 13.
## II Discussion and Results
In this section we present first the resolution tests for general shapes such
as spheres (core) and tetrahedrons (functional groups). Here, we introduce the
appropriate resolution for each shape’s accuracy and computational cost. We
also estimate such resolution effects for the whole functionalized NP
considering the core and functional groups to select the appropriate
discretization used in the simulation of NPs. Afterwards, we systematically
present the effects of type, size, distribution and number of functional
groups. Finally, we discuss the passive transport of the functionalized NP
near a walls. We compare our results on translation diffusion with existent
theoretical models for confined particles, and provide an empirical fitting
that describes the variation in rotational diffusion of the NPs.
### II.1 Resolution
#### II.1.1 Spherical shape resolution
First, we conduct resolution studies for simple spheres using six different
$\Phi$, 0.9 (12 particles), 1.8 (42 particles), 3.6 (162 particles), 7.2 (642
particles), 14.5 (2562 particles) and 29 (10242 particles). From the computed
mobilities, we define the error as
$\text{Error}_{D_{t}}={(\bm{M}_{t}-\bm{M}_{t}^{o})}/\bm{M}_{t}^{o}$ and
$\text{Error}_{D_{r}}=({\bm{M}_{r}-\bm{M}_{r}^{o}})/\bm{M}_{r}^{o}$, and in
Table 1, we summarize the results. We identify that a resolution of $14.5$
provides a reasonable approximation with errors on the order of $1\%$ in
$\bar{D}_{t}$, and $3\%$ for $\bar{D}_{r}$, for a spherical core. We must note
that aceptable results with errors of $2\%$ for $\bar{D}_{t}$, and $6\%$ for
$\bar{D}_{r}$ can be already obtained with lower resolutions ($\Phi=7.2$).
Unless otherwise stated, in the remaining we adopt $\Phi=14.5$ to investigate
the NPs diffusion.
Table 1: Resolution study for a single sphere (non-functionalized core) Resolution | | Number of
---
particles
$\bm{M}_{t}$ | $\bm{M}_{r}$ | $\frac{\bm{M}_{t}-\bm{M}_{t}^{o}}{\bm{M}_{t}^{o}}$ | $\frac{\bm{M}_{r}-\bm{M}_{r}^{o}}{\bm{M}_{r}^{o}}$
29 | 10242 | 0.0527 | 0.0391 | 0.005 | 0.016
14.5 | 2562 | 0.0524 | 0.0384 | 0.012 | 0.035
7.2 | 642 | 0.0518 | 0.0372 | 0.024 | 0.065
3.6 | 162 | 0.0503 | 0.0304 | 0.052 | 0.234
1.8 | 42 | 0.0472 | 0.0297 | 0.110 | 0.254
0.9 | 12 | 0.0420 | 0.0213 | 0.208 | 0.465
#### II.1.2 Tetrahedral shape resolution
The discretization of functional groups can define the overall resolution of
the NP as they are the smallest morphology to be resolved. However, for
typical nanoparticles with $R>l_{G}$, the number of blobs to discretize the
core will be significantly larger than the minimum required for optimal
accuracy. Therefore, we focus on finding the minimal resolution needed to
model these functional groups up to a good approximation. In table 2, we
present the computed mobilities for different resolutions
$\Phi=R_{\text{tetra}}/r_{o}$, and the convergence criteria for each case. For
these shapes, we use the relative difference of the computed mobilities with
the one for the highest resolution simulated ($\Phi=40$). For the resolution
of $\Phi=2.5$, we obtain an error of $3\%$ for $M_{t}$ and $8\%$ for $M_{r}$,
further improvement is achieved with $\Phi=4.9$ with $M_{t}$ and $M_{r}$
errors of $1\%$ and $4\%$ respectively. Higher resolutions decrease further
the error. However, the associated cost to model such a level of refinement
significantly increases. In general, for practical reasons, we identify that
values of $\Phi=2.5$ provided a reasonable approximation for tetrahedral
shapes.
Table 2: Resolutions of a discretized tetrahedron. The mobility of a sphere of radius $R_{e.t}$ for a solid tetrahedron of edge $a$ reduces the mobilities for the different resolutions. As a convergence estimator, we measure variation in the reduced mobility concerning the largest resolution simulated. Resolution | $\bm{M}_{t}/\bm{M}_{t}^{o}$ | $\bm{M}_{r}/\bm{M}_{r}^{o}$ | $R_{e.t}$ | $a$ | $\frac{\bm{M}_{t\text{max}}-\bm{M}_{t}}{\bm{M}_{t\text{max}}}$ | $\frac{\bm{M}_{r\text{max}}-\bm{M}_{r}}{\bm{M}_{r\text{max}}}$
---|---|---|---|---|---|---
Tetrahedron | – | – | 0.49 | 1.63 | 0 | 0
40 | 0.82 | 0.48 | 0.50 | 1.66 | 0 | 0
19.6 | 0.82 | 0.47 | 0.51 | 1.68 | 0.002 | 0.005
9.8 | 0.82 | 0.46 | 0.53 | 1.73 | 0.007 | 0.019
4.9 | 0.81 | 0.45 | 0.55 | 1.84 | 0.017 | 0.044
2.5 | 0.79 | 0.44 | 0.62 | 2.04 | 0.036 | 0.080
1.2 | 0.76 | 0.42 | 0.74 | 2.45 | 0.069 | 0.107
#### II.1.3 Functionalized nanoparticles resolution
Based on the previous results, we now explore four different resolutions
($\Phi=[14.5,7.2,3.6,1.8]$) for the whole functionalized NPs. In FIG 2, we
present the variation in the error for two type of nanoparticles,
corresponding to the group types with lowest (rod) and highest (tetra)
volumetric fraction. In this case, to verify the resolution we evaluate the
mobility difference between the functionalized nanoparticle $M_{NP}$ and the
single core $M_{s}$ (at the same resolution, $M_{s}-M_{NP}/M_{s}$. In general,
we identify that at a resolution of $\Phi=7.2$ the change in mobility is
already captured with a reasonable approximation. For coarser resolution
($\Phi=1.8$), the characteristic shape and aspect ratio of the group is
smeared into a single blob, thus the representation of the groups is not
properly accounted.
Figure 2: Resolution test for functionalized NPs with rods ($l_{G}/R=1$) and
tetrahedron-shaped ($w_{G}/R=0.2$) groups. The error corresponds to the
relative difference in mobilities between the functionalized NP and the single
spherical core, at the same resolution. The nanoparticles have $N_{G}=12$
uniformly distributed
### II.2 Functional groups morphology and size
Given the variety functionalized nanoparticules that can be found in the
literature, we first investigate how the diffusion of NPs with similar number
of groups can be affected by the shape and size of the groups. We compute the
reduced diffusivities $\overline{D}_{t}$ and $\overline{D}_{r}$ of
functionalized NPs with five group types uniformly placed around on the
nanoparticle’s surface, as depicted in FIG 3. For consistency, we compare NPs
with groups of equivalent size $l_{G}/R=0.5$, $N_{G}=12$ and uniformly
distributed. In general, the presence of the groups induced a reduction in the
NPs transport due to the effective larger volume. However, this reduction does
not occur in a trivial fashion based only on the aspect ratio of the groups.
Indeed, the groups shape affects the diffusion due to the added morphological
complexity. Overall, the effect of groups type on the translation and rotation
differs due to the scaling of $\overline{D}_{t}\propto R$ and
$\overline{D}_{r}\propto R^{3}$ with the radius. For example, the reduction in
$\overline{D}_{t}$ for rod and spherical groups is nearly similar, despite
their despair effective volume. In contrast, $\overline{D}_{r}$ appears as a
more distinctive parameter to discern NPs morphology.
Figure 3: Dimensionless rotational and translational diffusivity for different
type of groups.
#### II.2.1 Size of the functional groups
As discussed in the previous section, the characteristic size of the groups
influences the effective transport of the nanoparticles. Thus, we investigate
the changes in NPs mobility as the length ($l_{G}/R$) and/or width ($w_{G}/R$)
of Gs varies. In FIG 4, we present the variation of $\bar{D}_{t}$ and
$\bar{D}_{r}$ for NPs with rod and sphere-rod G, with $N_{G}=20$ randomly
distributed and different $l_{G}/R$. For this test, sphere-rod groups have a
fixed size of $w_{G}/R=0.2$ and the variation in $l_{G}$ is attained by
changing the length of the rod. As presented in FIG 3, compared to simple
rods, the sphere-rod groups exhibit the lowest mobility. However, the decay
with $l_{G}/R$ for both types is consistently preserved (see FIG 4) indicating
a strong correlation with the overall length of the group. Due to the scaling
of the rotational diffusion ($\overline{D}_{r}\propto R^{3}$), the increase in
the length of the groups induces a significant reduction ranging from 20 to 60
percent. In addition to the group length, we also inspect the effect of group
width $w_{G}/R$. In FIG 5, we compare the diffusional decay for NPs with
sphere, tetra and sphere-rod groups with three different $w_{G}/R$. Similar to
the previous case, the rate of decay shows to be preserved across the
different morphologies. In summary, we identify that regardless of the group
morphology the diffusion of the NPs scales with the group length in similar
fashion. As a consequence, for practical applications although the direct
comparison of NP diffusivity can differentiate functional groups morphology,
the estimation of the diffusional decay can provide a generic indirect
measurement of the thickness of the functional groups decorating a
nanoparticle.
Figure 4: Dimensionless rotational and translational diffusivity for different
G length $l_{G}/R$. Nanoparticles with $N_{G}$=20 randomly distributed Figure
5: Dimensionless rotational and translational diffusivity for different G
width $w_{G}/R$. Nanoparticles with $N_{G}$=20 randomly distributed
### II.3 Number and distribution of functional groups
#### II.3.1 Uniform vs random
In general, the functional groups can be uniform or randomly distributed
surface of the NPs core. Therefore, to elucidate the possible effect of Gs
placement in the nanoparticles, we now investigate this effect using rod-type
groups. We compute the mobility for NPs with $N_{G}$ ranging from $12$ to
$100$, randomly distributed. For each number of groups ten replicas of
randomly distributed groups are simulated and the average diffusion results
are compiled in FIG 6. Error bars correspond to the standard deviation. For
comparison, in FIG 6, we also include the results for NPs with uniform
distribution with $N_{G}$ equal to $12,42$, and $162$. For simplicity, the
uniform distribution of Gs is attained by localizing the groups at the vertex
of a regular tetrahedron (e.g. $N_{G}=12$ in a icosahedron). In FIG 6, the
results for uniform distribution with $N_{G}=100$ corresponds to the
interpolated value between $N_{G}=42$ and $N_{G}=162$.
Figure 6: Dimensionless rotational (in blue) and translational (in red)
diffusivities with G randomly and uniformly distributed. Error bars depict the
standard deviation of the measured mobility. For uniform distribution, we
present results for $N_{G}=[12,42,162]$, corresponding to the equidistant
vertex of regular polyhedrons. The values of 100 G are calculated with
interpolation between 42 and 162. However, it is evident that at large
$N_{G}$, randomly and uniformly distribution converges due to the packing of
the G in the core.
From FIG 6, we identify that uniform distributions lead to diffusivities of
approximately 6$\%$ (for translational) and 16$\%$ (for rotational) smaller
than the random ones. The breaking in symmetry of the randomly distributed
groups potentially enhance therefore the mobility of the NPs. If $N_{G}$
increases, both type of distributions lead to similar effective diffusivities
(difference of around 0.07$\%$ for translational and 0.5$\%$ for rotational).
At large $N_{G}$ the hydrodynamic effects of the groups overlap leading to
indistinguishable effect of location. We must highlight that the stronger
effect on rotational diffusivity at lower groups number is an interesting
feature that can be potentially used for nanoparticle characterization and
design.
#### II.3.2 Number of functional groups
In the previous section, we already showed that the increase in $N_{G}$ leads
to a reduction of the NPs diffusion, reaching a condition where random and
uniform are nearly equivalent. Now, we further investigate the effect of
groups number on the NP mobility. In FIG 7, we compile the diffusivities of
NPs with a rod, sphere-rod and tetra-rod shape groups for $N_{G}$ ranging from
$10$ to $50$ randomly distributed. The size of the groups is
$w_{G}=0.25,l_{G}=0.8$ for the tetra-rod, $w_{G}=0.2,l_{G}=0.8$ for the
sphere-rod, and $l_{G}=0.8$ for rods. We observe that although the type of G
determines the magnitude of the diffusivities, all curves follow similar trend
for the three groups. In general, the shape of the group can alter the overall
magnitude of the diffusion stemming from the enriched morphological
complexity. However, for a fixed group type, it is expected that the group
size will be a determinant of the mobility decay with $N_{G}$. In general,
such decay should scale with the characteristic size ($l_{G}$), but the direct
functional relationship cannot be trivially inferred. Therefore, we conduct
additional studies considering NPs with rod-shape groups of different lengths
$l_{G}/R$ to identify the combined effect of group size and number. In FIG 8,
we present the variation in $\overline{D}_{t}$ and $\overline{D}_{r}$ of NPs
with rod-shape groups and three different lengths $l_{G}/R=0.5,0.8,1.0$. In
FIG 8, again a characteristic diffusional decay can be identified, but in this
case as $l_{G}/R$ increases, a stronger dependence with the $N_{G}$ is
elucidated.
Figure 7: Translational and rotational reduced diffusivity of NPs with three
different types of groups morphology: rods, sphere-rod and tetra-rod. Markers
correspond to the computed diffusivities, whereas solid lines indicate the
approximated scaling obtained in equations 24 and 25. All groups have
$N_{G}=12$ uniformly distributed, the size of the groups are
$w_{G}=0.25,l_{G}=0.8$ for the tetra-rod, $w_{G}=0.2,l_{G}=0.8$ sphere-rod,
and $l_{G}=0.8$ for rods Figure 8: A. Translational and B. rotational reduced
diffusivity of NPs functionalized with rod-shape groups of different lengths
$l_{G}/R=0.5,0.8,1.0$. Markers correspond to the computed diffusivities,
whereas solid lines indicate the approximated scaling obtained in equations 24
and 25
Stemming from the diffusional decay observed when varying groups type and
length (FIG 7 and FIG 8), we can consider a power-law scaling of the diffusion
decay as $\overline{D}=1-cN_{G}^{\nu}$, where $c$ and $\nu$ depend on
geometrical features ($l_{G}$ and $w_{G}$). A numerical analysis on the
diffusional decay presented in FIG 7 reveals that the scaling (translational:
$\nu\sim-0.08l_{g}/R\pm 0.005$ and rotational: $\nu\sim-0.13l_{g}/R\pm 0.03$)
for the three groups is constant, depending only on the ratio $l_{G}/R$ that
is fixed for all the G types. In contrast, the prefactor $c$ that determines
the overall magnitude of the decay, depends on the both $l_{G}/R$ and
$w_{G}/R$. In Table III, we give a break down of the estimated $c$ and $\nu$
for each case. The prefactor $c$ has a linear Pearson correlation of
$r\approx-0.99$ with the volume of the group. Similarly, from the results
shown in FIG 8, the estimation of $c$ and $\nu$ parameters corroborate their
dependency with geometrical features ( see Table IV). From these results, we
can draw the following approximation of the diffusional decay as
$\displaystyle\overline{D}_{t}-1$ $\displaystyle\propto
v_{G}N_{G}^{-0.08l_{G}},$ (24) $\displaystyle\overline{D}_{r}-1$
$\displaystyle\propto v_{G}N_{G}^{-0.15l_{G}},$ (25)
where $v_{G}={{w_{G}}^{2}l_{G}}/{R^{3}}$. Overall, the diffusional decay of
the functionalized nanoparticles with $N_{G}$ scales with the length of the
functional group, regardless of the particular shape.
Table 3: Scaling parameters from three different group-types i.e. rod, tetra-rod and sphere-rod for translational and rotational dimensionless diffusion | $\overline{D}_{t}$ | $\overline{D}_{r}$
---|---|---
Type | $c$ | $v$ | $c$ | $v$
Rod | 2.20 | -0.08 | 2.17 | -0.16
Tetra-rod | 2.08 | -0.08 | 1.79 | -0.13
Shpere-rod | 2.04 | -0.08 | 1.67 | -0.10
Table 4: Scaling parameters from three different rod $l_{G}=0.5,0.8,1.0$ for translational and rotational dimensionless diffusion | $\overline{D}_{t}$ | $\overline{D}_{r}$
---|---|---
Rod-type | $c$ | $v$ | $c$ | $v$
$l_{G}=0.5$ | 2.13 | -0.07 | 2.24 | -0.19
$l_{G}=0.8$ | 2.15 | -0.07 | 2.07 | -0.14
$l_{G}=1.0$ | 2.13 | -0.07 | 1.82 | -0.10
### II.4 Diffusion near rigid walls
Many applications of nanoparticles involved their transport under confinement
or near walls. Under these conditions, NPs diffusion can be significantly
affected due to the restricted mobility and the asymmetry in the hydrodynamic
interactions exerted on the NP. Here, we explore the effect of a stationary
wall in the proximity of functionalized NP by estimating the parallel and
perpendicular diffusion. NPs are located at different dimensionless distance
$s^{\prime}=h/R-1$ to the wall (FIG 1.E), where $h$ is the distance from the
NP centre to the wall, and $R$ is the core radius. In FIG 9, we initially
corroborate that for spherical cores without functionalization, $N_{G}=0$, the
RMB discretization provides the correct theoretical translational diffusion
(Eqs.12 and 13). The agreement between the computed translational diffusion
and the theoretical one evidences the robustness of RMB to model confined
NPs.Sprinkle _et al._ (2017). A general expression for the theoretical
rotational diffusion near walls is not available over a larger range of
$s^{\prime}$.Swan and Brady (2007); Usabiaga _et al._ (2016) Therefore, based
on the computed rotational diffusion for plain spheres, we propose an
empirical functional form to describe the change in perpendicular and parallel
rotational diffusion with the distance of the NP to the wall. We propose
polynomial fitting function of the form
$f(s^{\prime})=a(s^{\prime-3})+b(s^{\prime-2})+c(s^{\prime-1})+1$. Using this
approximation (see FIG 9), we obtain that for parallel rotational diffusion,
$a=0.0011$, $b=-0.0125$ and $c=-0.0056$, whereas, for perpendicular, the
parameters are $a=0.0039$, $b=-0.0421$ and $c=-0.0061$. For comparison, in FIG
9 we compile the single sphere’s translational and rotational diffusion with
the corresponding theoretical approximation and the semi-empirical numerical
fitting. Overall, we observe that for simple spheres the rotational diffusion
is less affected by the wall confinement, converging to the unconfined
behaviour at shorter distance, $s^{\prime}\approx 3$.
Figure 9: A. Perpendicular direction for rotational and translational
diffusion. B. Parallel direction for rotational and translational diffusion.
Here, we compare rotational and translational according the fitting polynomial
function for rotational and the theoretical solution for translational.
For confinement studies, we focus on functionalized NPs with $N_{G}=12$ (rod-
and tetrahedron-shaped groups) uniformly distributed. Simulations for larger
$N_{G}=42$ and randomly distributed groups were also performed for comparison.
However, the deviations in those cases were smaller than $1\%$. It is
important to note that for the NPs near a wall, the RMB method does not have
precise lubrication effects. Thus, a reliable distance ($s^{\prime}$) to
capture hydrodynamic effects near the wall requires $s^{\prime}>(a+l_{G})/R$,
where $a=0.138$ is the blob radius used in our simulations and $l_{G}/R=0.5$.
We test NPs with and without functionalization at
$s^{\prime}=[0.64,1,2,3,7,10,40]$. In FIG 10, we present the perpendicular
($D\perp$) and parallel ($D\parallel$) translational diffusivity of the NP. We
must recall that the normalization of the diffusion coefficients is done with
the corresponding unconfined nanoparticle. Therefore, as the distance to the
wall increases, it is expected that the diffusion of the nanoparticles
converges to the corresponding unconfined diffusion (i.e
$\overline{D}_{t}^{wall}\sim 1$). For translational diffusion, this
convergence occurs at $s^{\prime}\sim 40$ (FIG 10), consistent with the
theoretical predictions. Our results, indicate that the translational behavior
of functionalized NPs resembles the non-functionalized ones. Moreover, the
change in $\overline{D}_{t}^{wall}$ for the two types of groups investigated,
are practically indistinguishable. Thus, is not expected that changes in
translational diffusion can serve as signature of morphological variations,
but merely indicate the presence of decorations.
Figure 10: A. Parallel and B. perpendicular translational diffusion for
functionalized NP with rods and tetra G, non-functionalized NP, and the
theoretical diffusion for spheres.
In FIG 11, we present the confinement effect on the rotational diffusion of
the functionalized NPs. Here, we also analyze the diffusion in terms of the
parallel and perpendicular component. In general, the impact of the wall in
functionalized and non-functionalized NPs conserves a similar trend, where the
translational mobility increases more rapidly than the rotational at larger
$s^{\prime}$ distances. However, in the case of functionalized nanoparticles
the deviations from the unconfined case are slightly more pronounced. These
type of variations indicate that functionalized NPs have an enhanced response
to confinement, therefore, novel methodologies for characterization,
separation, and sensing, can be potentially developed using this
characteristic response.
Figure 11: Rotational diffusion in A. parallel and B. perpendicular direction
from the wall, for different NP structures.
The source of the slight deviation at small $s^{\prime}$ evidenced for
functionalized nanoparticles in FIG 12 is the combined effect of the break in
symmetry due to the wall and the off-diagonal terms in the coupling matrix
$\bm{M}_{c}$ (Eq. 2). For free nanoparticles, these terms tend to zero for the
current RMB discretization ($1\text{e}-17$ for non-functionalized and rod-
type, and $1\text{e}-09$ for tetra-type). In RMB, the multi-blobs exhibit
small off-diagonal components for a sphere but are not zero since the discrete
sphere is not perfectly rotational invariant.Usabiaga _et al._ (2016) In
addition, the resolution ($\Phi$) of the NP model interferes with the values
of these coupling components; these is attributable to the numerical error.
Thus, to analyze the source of the measured off-diagonal terms for
functionalized NPs, we use the theoretical mobility of a sphere $M_{c}^{o}$
with radius $R=1$ as a reference case, $M_{c}^{o}=1/8\pi\eta R^{2}$. We study
the torque in the $z$ direction, computing the average of the two coupling
terms $\bm{M}_{c,xy}$ and $\bm{M}_{c,yx}$,
$\bm{M}_{c}=(\bm{M}_{c,xy}-\bm{M}_{c,yx})/2$. Hence, using the off-diagonal
terms, we estimate the ratio $M_{c}/M_{c}^{o}$, where we compute the
difference between the computed coupling matrix from functionalized tetra, rod
and the non-functionalized NP. In FIG 12. we observe that the ratio of
mobilities increases if the NP is near a wall. At a distance,
$s^{\prime}\approx 3$, for the three cases, the coupling terms decay close to
zero and similarly to a free NP. The results show that the off-diagonal
components for the functionalized NPs have higher values than the sphere at
short distances to the wall ($s^{\prime}<3$). This result evidences that the
coupled translational and rotational motion of the functionalized NPs near the
wall is enhanced. Hence, we identified that, in principle, functionalized
nanoparticle characterization could be addressed by placing the NPs near walls
and measuring the specific variations in the coupling component of the
mobility.
Figure 12: Off-diagonal terms of the coupling mobility matrix $\bm{M}_{c}$,
these terms are dimensionless with the mobility of a sphere $M_{c}^{o}$ of
radius $R=1$, $M_{c}/M_{c}^{o}$. Starting from the $s^{\prime}\approx 3$, the
mobility converges rapidly to zero, such as a free NP. From values
$s^{\prime}<3$ is evident the translational effects due to torque in $z$
direction
## III Conclusions
In this work, we have investigated the passive transport of complex
functionalized nanoparticles numerically using a variety of morphologies
common in different physical systems. We show how functional groups
distribution, shape, size, and $N_{G}$ affect the translational and rotational
diffusion of the nanoparticles. In general, we identify that the transport
properties of the functionalized NPs are significantly altered by the
morphology of the decorating groups. We observe that functional groups can
exhibit a specific reduction in mobility due to added complex morphology of
the NPsṘegarding the group distribution on the NPs surface, we identify that
random conformations facilitate the transport at a low number of groups
compared to uniform distributions due to symmetry-breaking effects. At a large
number of groups, the overlap in the hydrodynamic interaction of the groups
led to an indistinguishable effect on mobility.
The effect of the number of groups in the diffusion of the nanoparticles
exhibits a characteristic power-law decay that is governed by the length of
the groups. In contrast, the relative volume determines the overall magnitude
of the decay. In general, the characterization of the diffusional decay of
functionalized nanoparticles can provide relevant information about the degree
of functionalization and size of groups. For NPs diffunding near a wall, the
diffusion coefficient decay is similar for spherical and functionalized NP.
However, functionalized nanoparticles at a short distance from the wall are
able to have a more robust response. From a computational standpoint, we show
that the RMB method is a powerful tool to characterize and predict the
physical transport of complex functionalized NPs. As an additional outcome, we
have used RMB to provide a semi-analytical approximation of spherical
particles’ parallel and perpendicular rotational diffusivity. Our results
reveal potential avenues for nanoparticle characterization. We find that
confinement effects can be exploited to discern different particle
functionalizations, owing to the enhanced response of the rotational diffusion
of the NPs compared to bulk measurements. Additionally, we showed that
targeted modification in the morphology of the groups is a compelling design
strategy to create nanoparticles with enhanced or reduced mobility.
###### Acknowledgements.
Financial support received from the Basque Government through the BERC
2018-2021 program, by the Spanish State Research Agency through BCAM Severo
Ochoa excellence accreditation (SEV-2017-0718) and through the project
PID2020-117080RB-C55 (“Microscopic foundations of soft-matter experiments:
computational nano-hydrodynamics”) funded by AEI - MICIN and acronym “Compu-
Nano-Hydro” are gratefully acknowledged. N.M acknowledges the support from the
European Union’s Horizon 2020 under the Marie Skłodowska-Curie Individual
Fellowships grant 101021893, with acronym ViBRheo. F.B.U. acknowledges support
from “la Caixa” Foundation (ID 100010434), fellowship LCF/BQ/PI20/11760014,
and from the European Union’s Horizon 2020 research and innovation programme
under the Marie Sklodowska-Curie grant agreement No 847648.
## Data Availability Statement
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
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# Quantization-aware Interval Bound Propagation for Training Certifiably
Robust Quantized Neural Networks
Mathias Lechner1, Đorđe Žikelić2, Krishnendu Chatterjee2, Thomas A.
Henzinger2, Daniela Rus1
###### Abstract
We study the problem of training and certifying adversarially robust quantized
neural networks (QNNs). Quantization is a technique for making neural networks
more efficient by running them using low-bit integer arithmetic and is
therefore commonly adopted in industry. Recent work has shown that floating-
point neural networks that have been verified to be robust can become
vulnerable to adversarial attacks after quantization, and certification of the
quantized representation is necessary to guarantee robustness. In this work,
we present quantization-aware interval bound propagation (QA-IBP), a novel
method for training robust QNNs. Inspired by advances in robust learning of
non-quantized networks, our training algorithm computes the gradient of an
abstract representation of the actual network. Unlike existing approaches, our
method can handle the discrete semantics of QNNs. Based on QA-IBP, we also
develop a complete verification procedure for verifying the adversarial
robustness of QNNs, which is guaranteed to terminate and produce a correct
answer. Compared to existing approaches, the key advantage of our verification
procedure is that it runs entirely on GPU or other accelerator devices. We
demonstrate experimentally that our approach significantly outperforms
existing methods and establish the new state-of-the-art for training and
certifying the robustness of QNNs.
## Introduction
Quantized neural networks (QNNs) are neural networks that represent their
weights and compute their activations using low-bit integer variables. QNNs
significantly improve the latency and computational efficiency of inferencing
the network for two reasons. First, the reduced size of the weights and
activations allows for a much more efficient use of memory bandwidth and
caches. Second, integer arithmetic requires less silicon area and less energy
to execute than floating-point operations. Consequently, dedicated hardware
for running QNNs can be found in GPUs, mobile phones, and autonomous driving
computers.
Adversarial attacks are a well-known vulnerability of neural networks that
raise concerns about their use in safety-critical applications (Szegedy et al.
2013; Goodfellow, Shlens, and Szegedy 2014). These attacks are norm-bounded
input perturbations that make the network misclassify samples, despite the
original samples being classified correctly and the perturbations being barely
noticeable by humans. For example, most modern image classification networks
can be fooled when changing each pixel value of the input image by a few
percent. Consequently, researchers have tried to train networks that are
provably robust against such attacks. The two most common paradigms of
training robust networks are adversarial training (Madry et al. 2018), and
abstract interpretation-based training (Mirman, Gehr, and Vechev 2018; Wong
and Kolter 2018). Adversarial training and its variations perturb the training
samples with gradient descent-based adversarial attacks before feeding them
into the network (Madry et al. 2018; Zhang et al. 2019; Wu, Xia, and Wang
2020; Lechner et al. 2021). While this improves the robustness of the trained
network empirically, it provides no formal guarantees of the network’s
robustness due to the incompleteness of gradient descent-based attacking
methods, i.e., gradient descent might not find all attacks. Abstract
interpretation-based methods avoid this problem by overapproximating the
behavior of the network in a forward pass during training. In particular,
instead of directly training the network by computing gradients with respect
to concrete samples, these algorithms compute gradients of bounds obtained by
propagating abstract domains. While the learning process of abstract
interpretation-based training is much less stable than a standard training
procedure, it provides formal guarantees about the network’s robustness. The
interval bound propagation (IBP) method (Gowal et al. 2019) effectively showed
that the learning process with abstract interpretation can be stabilized when
gradually increasing the size of the abstract domains throughout the training
process.
Previous work has considered adversarial training and IBP for floating-point
arithmetic neural networks, however robustness of QNNs has received
comparatively much less attention. Since it was demonstrated by (Giacobbe,
Henzinger, and Lechner 2020) that neural networks may become vulnerable to
adversarial attacks after quantization even if they have been verified to be
robust prior to quantization, one must develop specialized training and
verification procedures in order to guarantee robustness of QNNs. Previous
works have proposed several robustness verification procedures for QNNs
(Giacobbe, Henzinger, and Lechner 2020; Baranowski et al. 2020; Henzinger,
Lechner, and Žikelić 2021), but none of them consider algorithms for learning
certifiably robust QNNs. Furthermore, the existing verification procedures are
based on constraint solving and cannot be run on GPU or other accelerating
devices, making it much more challenging to use them for verifying large QNNs.
In this work, we present the first abstract interpretation training method for
the discrete semantics of QNNs. We achieve this by first defining abstract
interval arithmetic semantics that soundly over-approximate the discrete QNN
semantics, giving rise to an end-to-end differentiable representation of a QNN
abstraction. We then instantiate quantization-aware training techniques within
these abstract interval arithmetic semantics in order to obtain a procedure
for training certifiably robust QNNs.
Next, we develop a robustness verification procedure which allows us to
formally verify QNNs learned via our IBP-based training procedure. We prove
that our verification procedure is complete, meaning that for any input QNN it
will either prove its robustness or find a counterexample. This contrasts the
case of abstract interpretation verification procedures for neural networks
operating over real arithmetic which are known to be incomplete (Mirman,
Baader, and Vechev 2021a). The key advantage of our training and verification
procedures for QNNs is that it can make use of GPUs or other accelerator
devices. In contrast, the existing verification methods for QNNs are based on
constraint solving so cannot be run on GPU.
Finally, we perform an experimental evaluation showing that our method
outperforms existing state-of-the-art certified $L_{\infty}$-robust QNNs. We
also elaborate on the limitations of our training method by highlighting how
the low precision of QNNs makes IBP-based training difficult.
We summarize our contribution in three points:
* •
We introduce the first learning procedure for learning robust QNNs. Our
learning procedure is based on quantization-aware training and abstract
interpretation method.
* •
We develop the first _complete_ robustness verification algorithm for QNNs
(i.e., one that always terminates with the correct answer) that runs entirely
on GPU or other neural network accelerator devices and make it publicly
available.
* •
We experimentally demonstrate that our method advances the state-of-the-art on
certifying $L_{\infty}$-robustness of QNNs.
## Related Work
#### Abstract interpretations for neural networks
Abstract interpretation is a method for overapproximating the semantics of a
computer program in order to make its formal analysis feasible (Cousot and
Cousot 1977). Abstract interpretation executes program semantics over abstract
domains instead of concrete program states. The method has been adapted to the
robustness certification of neural networks by computing bounds on the outputs
of neural networks (Wong and Kolter 2018; Gehr et al. 2018; Tjeng, Xiao, and
Tedrake 2019). For instance, polyhedra (Katz et al. 2017; Ehlers 2017; Gehr et
al. 2018; Singh et al. 2019; Tjeng, Xiao, and Tedrake 2019), intervals (Gowal
et al. 2019) , hybrid automata (Xiang, Tran, and Johnson 2018), zonotopes
(Singh et al. 2018), convex relaxations (Dvijotham et al. 2018; Zhang et al.
2020; Wang et al. 2021), and polynomials (Zhang et al. 2018b) have been used
as abstract domains in the context of neural network verification. Abstract
interpretation has been shown to be most effective for verifying neural
networks when directly incorporating them into gradient descent-based training
algorithms by optimizing the obtained output bounds as the loss function
(Mirman, Gehr, and Vechev 2018; Wong and Kolter 2018; Gowal et al. 2019; Zhang
et al. 2020).
Most of the abstract domains discussed above exploit the piecewise linear
structure of neural networks, e.g., linear relaxations such as polytopes and
zonotopes. However, linear relaxations are less suited for QNNs due to their
piecewise-constant discrete semantics.
#### Verification of quantized neural networks
The earliest work on the verification of QNNs has focused on binarized neural
networks (BNNs), i.e., 1-bit QNNs (Hubara et al. 2016). In particular,
(Narodytska et al. 2018) and (Cheng et al. 2018) have reduced the problem of
BNN verification to boolean satisfiability (SAT) instances. Using modern SAT
solvers, the authors were able to verify formal properties of BNNs. (Jia and
Rinard 2020) further improve the scalability of BNNs by specifically training
networks that can be handled by SAT-solvers more efficiently. (Amir et al.
2021) developed a satisfiability modulo theories (SMT) approach for BNN
verification.
Verification for many bit QNNs was first reported in (Giacobbe, Henzinger, and
Lechner 2020) by reducing the QNN verification problem to quantifier-free bit-
vector satisfiability modulo theory (QF_BV SMT). The SMT encoding was further
improved in (Henzinger, Lechner, and Žikelić 2021) by removing redundancies
from the SMT formulation. (Baranowski et al. 2020) introduced fixed-point
arithmetic SMT to verify QNNs. The works of (Sena et al. 2021, 2022) have
studied SMT-based verification for QNNs as well. Recently, (Mistry, Saha, and
Biswas 2022) proposed encoding of the QNN verification problem into a mixed-
integer linear programming (MILP) instance. IntRS (Lin et al. 2021) considers
the problem of certifying adversarial robustness of quantized neural networks
using randomized smoothing. IntRS is limited to $L_{2}$-norm bounded attacks
and only provides statistical instead of formal guarantees compared to our
approach.
#### Decision procedures for neural network verification
Early work on the verification of floating-point neural networks has employed
off-the-shelf tools and solvers. For instance, (Pulina and Tacchella 2012;
Katz et al. 2017; Ehlers 2017) employed SMT-solvers to verify formal
properties of neural networks. Similarly, (Tjeng, Xiao, and Tedrake 2019)
reduces the verification problem to mixed-integer linear programming
instances. Procedures better tailored to neural networks are based on branch
and bound algorithms (Bunel et al. 2018). In particular, these algorithms
combine incomplete verification routines (bound) with divide-and-conquer
(branch) methods to tackle the verification problem. The speedup advantage of
these methods comes from the fact that the bounding methods can be easily
implemented on GPU and other accelerator devices (Serre et al. 2021; Wang et
al. 2021).
#### Quantization-aware training
There are two main strategies for training QNNs: post-training quantization
and quantization-aware training (Krishnamoorthi 2018). In post-training
quantization, a standard neural network is first trained using floating-point
arithmetic, which is then translated to a quantized representation by finding
suitable fixed-point format that makes the quantized interpretation as close
to the original network as possible. Post-training quantization usually
results in a drop in the accuracy of the network with a magnitude that depends
on the specific dataset and network architecture.
To avoid a significant reduction in accuracy caused by the quantization in
some cases, quantization-aware training (QAT) models the imprecision of the
low-bit fixed-point arithmetic already during the training process, i.e., the
network can adapt to a quantized computation during training. The rounding
operations found in the semantics of QNNs are non-differentiable computations.
Consequently, QNNs cannot be directly trained with stochastic gradient
descent. Researchers have come up with several ways of circumventing the
problem of non-differentiable rounding. The most common approach is the
straight-through gradient estimator (STE) (Bengio, Léonard, and Courville
2013; Hubara et al. 2017). In the forward pass of a training step, the STE
applies rounding operations to computations involved in the QNN, i.e., the
weights, biases, and arithmetic operations. However, in the backward pass, the
rounding operations are removed such that the error can backpropagate through
the network. The approach of (Gupta et al. 2015) uses stochastic rounding that
randomly selects one of the two nearest quantized values for a given floating-
point value. Relaxed quantization (Louizos et al. 2019) generalizes stochastic
rounding by replacing the probability distribution over the nearest two values
with a distribution over all possible quantized values. DoReFA-Net (Zhou et
al. 2016) combines the straight-through gradient estimator and stochastic
rounding to train QNN with high accuracy. The authors observed that quantizing
the first and last layer results in a significant drop in accuracy, and
therefore abstain from quantizing these two layers.
Instead of having a fixed pre-defined quantization range, i.e., fixed-point
format, more recent QAT schemes allow learning the quantization range. PACT
(Choi et al. 2018) treats the maximum representable fixed-point value as a
free variable that is learned via stochastic gradient descent using a
straight-through gradient estimation. The approach of (Jacob et al. 2018)
keeps a moving average of the values ranges during training and adapts the
quantization range according to the moving average. LQ-Nets (Zhang et al.
2018a) learn an arbitrary set of quantization levels in the form of a set of
coding vectors. While this approach provides a better approximation of the
real-valued neural network than fixed-point-based quantization formats, it
also prevents the use of efficient integer arithmetic to run the network.
MobileNet (Howard et al. 2019) is a specialized network architecture family
for efficient inference on the ImageNet dataset and employs quantization as
one technique to achieve this target. HAWQ-V3 (Yao et al. 2021) dynamically
assigns the number of bits of each layer to either 4-bit, 8-bit, or 32-bit
depending on how numerically sensitive the layer is. EfficientNet-lite (Tan
and Le 2019) employs a neural architecture search to automatically find a
network architecture that achieves high accuracy on the ImageNet dataset while
being fast for inference on a CPU.
## Preliminaries
#### Quantized neural networks (QNNs)
Feedforward neural networks are functions
$f_{\theta}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ that consist of several
sequentially composed layers $f_{\theta}=l_{1}\circ\dots\circ l_{s}$, where
layers are parametrized by the vector $\theta$ of neural network parameters.
Quantization is an interpretation of a neural network $f_{\theta}$ that
evaluates the network over a fixed point arithmetic and operates over a
restricted set of bitvector inputs (Smith et al. 1997), e.g. $4$ or $8$ bits.
Formally, given an admissible input set $\mathcal{I}\subseteq\mathcal{R}^{n}$,
we define an interpretation map
$\llbracket\cdot\rrbracket_{\mathcal{I}}:(\mathbb{R}^{n}\rightarrow\mathbb{R}^{m})\rightarrow(\mathcal{I}\rightarrow\mathbb{R}^{m}),$
which maps a neural network to its interpretation operating over the input set
$\mathcal{I}$. For instance, if $\mathcal{I}=\mathbb{R}^{n}$ then $\llbracket
f_{\theta}\rrbracket_{\mathbb{R}}$ is the idealized real arithmetic
interpretation of $f_{\theta}$, whereas $\llbracket
f\rrbracket_{\text{float32}}$ denotes its floating-point 32-bit implementation
(Kahan 1996). Given $k\in\mathbb{N}$, the k-bit quantization is then an
interpretation map $\llbracket\cdot\rrbracket_{\text{int-}k}$ which uses
$k$-bit fixed-point arithmetic. We say that $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ is a $k$-bit quantized neural network
(QNN).
The semantics of the QNN $\llbracket f_{\theta}\rrbracket_{\text{int-}k}$ are
defined as follows. Let $[\mathbb{Z}]_{k}=\\{0,1\\}^{k}$ be the set of all
bit-vectors of bit-width $k$. The QNN $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ then also consists of sequentially
composed layers $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}=l_{1}\circ\dots\circ l_{s}$, where now
each layer is a function
$l_{i}:[\mathbb{Z}]_{k}^{n_{i}}\rightarrow[\mathbb{Z}]_{k}^{n_{i+1}}$ that
operates over $k$-bit bitvectors and is defined as follows:
$\displaystyle x^{\prime}_{i}$
$\displaystyle=\sum_{j=1}^{n_{i}}w_{ij}x_{j}+b_{i},$ (1) $\displaystyle
x^{\prime\prime}_{i}$
$\displaystyle=\text{round}(x^{\prime}_{i},M_{i})=\lfloor x^{\prime}_{i}\cdot
M_{i}\rfloor,\qquad\text{and}$ (2) $\displaystyle y_{i}$
$\displaystyle=\sigma_{i}(\min\\{2^{N_{i}}-1,x^{\prime\prime}_{i}\\}),$ (3)
Here, $w_{i,j}\in[\mathbb{Z}]_{k}^{n_{i}}$ and
$b_{i}\in[\mathbb{Z}]_{k}^{n_{i}}$ for each $1\leq j\leq n_{i}$ and $1\leq
i\leq n_{0}$ denote the weights and biases of $f$ which are also bitvectors of
appropriate dimension. Note that it is a task of the training procedure to
ensure that trained weights and biases are bitvectors, see below. In eq. (1),
the linear map defined by weights $w_{i,j}$ and biases $b_{i}$ is applied to
the input values $x_{j}$. Then, eq. (2) multiplies the result of eq. (1) by
$M_{i}$ and takes the floor of the obtained result. This is done in order to
scale the result and round it to the nearest valid fixed-point value, for
which one typically uses $M_{i}$ of the form $2^{-k}$ for some integer $k$.
Finally, eq. (3) applies an activation function $\sigma_{i}$ to the result of
eq. (2) where the result is first “cut-off” if it exceeds $2^{N_{i}}-1$, i.e.,
to avoid integer overflows, and then passed to the activation function. We
restrict ourselves to monotone activation functions, which will be necessary
for our IBP procedure to be correct. This is still a very general assumption
which includes a rich class of activation function, e.g. ReLU, sigmoid or tanh
activation functions. Furthermore, similarly to most quantization-aware
training procedures our method assumes that it is provided with quantized
versions of these activation functions that operate over bit-vectors.
#### Adversarial robustness for QNNs
We now formalize the notion of adversarial robustness for QNN classifiers. Let
$\llbracket
f_{\theta}\rrbracket_{\text{int-}k}:[\mathbb{Z}]_{k}^{n}\rightarrow[\mathbb{Z}]_{k}^{m}$
be a $k$-bit QNN. It naturally defines a classifier with $m$ classes by
assuming that it assigns to an input $x\in[\mathbb{Z}]_{k}^{n}$ a label of the
maximal output neuron on input value $x$, i.e.
$y=\mathsf{class}(x)=\mathsf{argmax}_{1\leq i\leq m}\llbracket
f_{\theta}\rrbracket_{\text{int-}k}(x)[i]$ with $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}(x)[i]$ being the value of the $i$-th
output neuron on input value $x$. If the maximum is attained at multiple
output neurons, we assume that $\mathsf{argmax}$ picks the smallest index
$1\leq i\leq m$ for which the maximum is attained.
Intuitively, a QNN is adversarially robust at a point $x$ if it assigns the
same class to every point in some neighbourhood of $x$. Formally, given
$\epsilon>0$, we say that $\llbracket f_{\theta}\rrbracket_{\text{int-}k}$ is
$\epsilon$-adversarially robust at point $x$ if
$\forall
x^{\prime}\in[\mathbb{Z}]_{k}^{n}.\,||x-x^{\prime}||_{\infty}<\epsilon\Rightarrow\mathsf{class}(x^{\prime})=\mathsf{class}(x),$
where $||\cdot||_{\infty}$ denotes the $L_{\infty}$-norm. Then, given a finite
dataset
$\mathcal{D}=\\{(x_{1},y_{1}),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\\}$
with $x_{i}\in[\mathbb{Z}]_{k}^{n}$ and $y_{i}\in[\mathbb{Z}]_{k}^{m}$ for
each $1\leq i\leq|\mathcal{D}|$, we say that $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ is $\epsilon$-adversarially robust with
respect to the dataset $\mathcal{D}$ if it is $\epsilon$-adversarially robust
at each datapoint in $\mathcal{D}$.
## Quantization-aware Interval Bound Propagation
In this section, we introduce an end-to-end differentiable abstract
interpretation method for training certifiably robust QNNs. We achieve this by
extending the interval bound propagation (IBP) method of (Gowal et al. 2019)
to the discrete semantic of QNNs. Our resulting quantization-aware interval
bound propagation method (QA-IBP) trains an interval arithmetic abstraction of
a QNN via stochastic gradient descent by propagating upper and lower bounds
for each layer instead of concrete values.
First, we replace each layer $l_{i}$ with two functions
$\underline{l_{i}},\,\overline{l_{i}}$:
$[\mathbb{Z}]_{k}^{n_{i}}\rightarrow[\mathbb{Z}]_{k}^{n_{0}}$ defined as
follows:
$\displaystyle\mu_{j}$
$\displaystyle=\frac{\overline{x_{j}}+\underline{x_{j}}}{2}$ $\displaystyle
r_{j}$ $\displaystyle=\frac{\overline{x_{j}}-\underline{x_{j}}}{2}$ (4)
$\displaystyle\mu_{i}$ $\displaystyle=\sum_{j=1}^{n_{i}}w_{ij}\mu_{j}+b_{i}$
$\displaystyle r_{i}$ $\displaystyle=\sum_{j=1}^{n_{i}}|w_{ij}|r_{j}$ (5)
$\displaystyle\underline{x^{\prime}_{i}}$ $\displaystyle=\mu_{i}-r_{i}$
$\displaystyle\overline{x^{\prime}_{i}}$ $\displaystyle=\mu_{i}+r_{i}$ (6)
$\displaystyle\underline{x^{\prime\prime}_{i}}$
$\displaystyle=\text{round}(\underline{x^{\prime}_{i}},k_{i})=\lfloor\underline{x^{\prime}_{i}}\cdot
M_{i}\rfloor$ (7) $\displaystyle\overline{x^{\prime\prime}_{i}}$
$\displaystyle=\text{round}(\overline{x^{\prime}_{i}},k_{i})=\lfloor\overline{x^{\prime}_{i}}\cdot
M_{i}\rfloor$ (8) $\displaystyle\underline{y_{i}}$
$\displaystyle=\max\\{0,\min\\{2^{N_{i}}-1,\underline{x^{\prime\prime}_{i}}\\}\\}$
(9) $\displaystyle\overline{y_{i}}$
$\displaystyle=\max\\{0,\min\\{2^{N_{i}}-1,\overline{x^{\prime\prime}_{i}}\\}\\}.$
(10)
As with standard QNNs, $w_{i,j}\in[\mathbb{Z}]_{k}^{n_{i}}$ and
$b_{i}\in[\mathbb{Z}]_{k}^{n_{i}}$ for each $1\leq j\leq n_{i}$ and $1\leq
i\leq n_{0}$ denote the weights and biases of $f$ which are also bit-vectors
of appropriate dimension. By the sequential composition of all layers of the
QNN we get the IBP representation of the QNN in the form of two functions
$\underline{\llbracket f_{\theta}\rrbracket_{\text{int-}k}}$ and
$\overline{\llbracket f_{\theta}\rrbracket_{\text{int-}k}}$. For a given input
sample $x$ and $\epsilon>0$ we can use the IBP representation of $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ to potentially prove the adversarial
robustness of the QNN. In particular, the input sample defines an abstract
interval domain $(\underline{x},\overline{x})$ with $\underline{x}=x-\epsilon$
and $\overline{x}=x+\epsilon$. Next, we propagate the abstract domains through
the IBP representation of the network to obtain output bounds
$\underline{y}=\underline{\llbracket
f_{\theta}\rrbracket_{\text{int-}k}}(\underline{x},\overline{x})$ and
$\overline{y}=\overline{\llbracket
f_{\theta}\rrbracket_{\text{int-}k}}(\underline{x},\overline{x})$. Finally, we
know that $\llbracket f_{\theta}\rrbracket_{\text{int-}k}$ is
$\epsilon$-adversarially robust at point $x$ if
$\underline{y_{i}}>\overline{y_{j}},\qquad\text{for }i=\mathsf{class}(x)\text{
and }\forall j\neq\mathsf{class}(x).$ (11)
#### End-to-end differentiation
We modify the IBP representation of QNNs described above to allow an end-to-
end differentiation necessary for a stochastic gradient descent-based learning
algorithm. In particular, first, we apply the straight-through gradient
estimator to propagate the error backward through the rounding operations in
Eq. 8. We do this by replacing the non-differentiable rounding operation with
the fake quantization function
$\displaystyle\text{fake\\_quant}(x_{i})$
$\displaystyle:=\text{round}(x_{i},k_{i})$ (12) $\displaystyle\frac{\partial\
\text{fake\\_quant}(x_{i})}{\partial x_{i}}$ $\displaystyle:=1.$ (13)
We also add fake quantization operations around the weights and biases in Eq.
6. The modified training graph is visualized in Figure 1.
Figure 1: Illustration of how the interval bound propagation training graph is
affected by quantization-aware training. A) Standard IBP inference graph. B)
IBP inference path with fake quantization operations inserted to model
quantized weights, biases, and computations.
For a single training sample $(x,j)$, we define the per-sample training loss
$\displaystyle L(\underline{y},\overline{y},j)=\sum_{i\neq
j}(\overline{y_{i}}-\underline{y_{j}})\mathds{1}[\underline{y_{j}}-\overline{y_{i}}\leq
0],$ (14)
where $\underline{y},\overline{y}$ are the QA-IBP output bounds QNN with
respect to the input domain $(x-\epsilon,x+\epsilon)$ and $j$ corresponds to
the label, i.e., class $j$. The loss term encourages the QA-IBP to produce
output bounds that prove adversarial robustness of the QNN $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ with respect to the input $x$ and
adversarial radius $\epsilon$.
#### Existence of robust QNNs
We conclude this section by presenting an interesting result on the existence
of robust QNNs. In particular, given $\epsilon>0$ and a finite dataset of
$1$-dimensional bit-vectors whose any two distinct datapoints are at least
$2\epsilon$ away, we prove that there exists a QNN with ReLU activations that
is $\epsilon$-robust with respect to the dataset. Furthermore, we provide an
upper bound on the number of neurons that the QNN must contain. We assume
$2\epsilon$ distance simply for the $\epsilon$-neighbourhoods of datapoints to
be disjoint so that robustness cannot impose contradicting classification
conditions.
Note that this result does not hold for real arithmetic feed-forward neural
networks with ReLU or any other continuous activation functions. Indeed, it
was observed in (Mirman, Baader, and Vechev 2021b, Corollary 5.12) that a
dataset $\\{(-2,-1),(0,1),(2,-1)\\}$ cannot be $1$-robustly classified by a
feed-forward neural network which uses continuous activation functions. The
intuition behind this impossibility result for real arithmetic networks is
that a classifier would have to be a continuous function that correctly
classifies all points in some open neighbourhoods of $x=-1$ and $x=1$.
###### Theorem 1.
Let $\epsilon>0$ and let
$\mathcal{D}=\\{(x_{1},y_{1}),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\\}$
with $x_{i}\in[\mathbb{Z}]_{k}^{1}$ and $y_{i}\in[\mathbb{Z}]_{k}^{1}$ for
each $1\leq i\leq|\mathcal{D}|$. Suppose that $||x_{i}-x_{j}||_{\infty}\geq
2\epsilon$ for each $i\neq j$. Then, there exists a QNN $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ which is $\epsilon$-robust with respect
to the dataset $\mathcal{D}$ and which consists of
$\mathcal{O}(|\mathcal{D}|\cdot\lceil\epsilon\cdot 2^{k}+1\rceil)$ neurons.
The proof of Theorem 1 is provided in the Appendix. It starts with an
observation that the set
$\mathcal{B}_{\epsilon}(x_{i})=\\{x^{\prime}\in[\mathbb{Z}]_{k}^{1}\mid||x^{\prime}-x_{i}||_{\infty}<\epsilon\\}$
is finite and consists of at most $\lceil 2\epsilon\cdot 2^{k}+1\rceil$ bit-
vectors for each $1\leq i\leq|\mathcal{D}|$. Hence, constructing a QNN
$\llbracket f_{\theta}\rrbracket_{\text{int-}k}$ that is $\epsilon$-robust
with respect to the dataset $\mathcal{D}$ is equivalent to constructing a QNN
that correctly classifies
$\mathcal{D}^{\prime}=\cup_{i=1}^{|\mathcal{D}|}\\{(x^{\prime},y_{i})\mid
x^{\prime}\in\mathcal{B}_{\epsilon}(x_{i})\\}$ which consists of
$\mathcal{O}(|\mathcal{D}|\cdot\lceil\epsilon\cdot 2^{k}+1\rceil)$ datapoints.
In the Appendix, we then design the QNN $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ that correctly classifies a dataset and
consists of at most linearly many neurons in the dataset size. Our
construction also implies the following corollary.
###### Corollary 1.
Let
$\mathcal{D}=\\{(x_{1},y_{1}),\dots,(x_{|\mathcal{D}|},y_{|\mathcal{D}|})\\}$
with $x_{i}\in[\mathbb{Z}]_{k}^{1}$ and $y_{i}\in[\mathbb{Z}]_{k}^{1}$ for
each $1\leq i\leq|\mathcal{D}|$. Then, there exists a QNN $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ which correctly classifies the dataset,
i.e. $\llbracket f_{\theta}\rrbracket_{\text{int-}k}(x_{i})=y_{i}$ for each
$1\leq i\leq|\mathcal{D}|$, and which consists of $\mathcal{O}(|\mathcal{D}|)$
neurons.
## A Complete Decision Procedure for QNN Verification
In the previous section, we presented a quantization-aware training procedure
for QNNs with robustness guarantees which was achieved by extending IBP to
quantized neural network interpretations. We now show that IBP can also be
used towards designing a complete verification procedure for already trained
feed-forward QNNs. By completeness, we mean that the procedure is guaranteed
to return either that the QNN is robust or to produce an adversarial attack.
There are two important novel aspects of the verification procedure that we
present in this section. First, to the best of our knowledge this is the first
complete robustness verification procedure for QNNs that is applicable to
networks with non-piecewise linear activation functions. Existing constraint
solving based methods that reduce verification to SMT-solving are complete but
they only support piecewise linear activation fucntions such as ReLU
(Krizhevsky and Hinton 2010). These could in theory be extended to more
general activation functions by considering more expressive satisfiability
modulo theories (Clark and Cesare 2018), however this would lead to
inefficient verification procedures and our experimental results in the
following section already demonstrate the significant gain in scalability of
our IBP-based methods as opposed to SMT-solving based methods for ReLU
networks. Second, we note that while our IBP-based verification procedure is
complete for QNNs, in general it is known that existing IBP-based verification
procedures for real arithmetic neural networks are not complete (Mirman,
Baader, and Vechev 2021a). Thus, our result leads to an interesting contrast
in IBP-based robustness verification procedures for QNNs and for real
arithmetic neural networks.
#### Verification procedure
We now describe our robustness verification procedure for QNNs. Its pseudocode
is shown in Algorithm 1. Since verifying $\epsilon$-robustness of a QNN with
respect to some finite dataset $\mathcal{D}$ and $\epsilon>0$ is equivalent to
verifying $\epsilon$-robustness of the QNN at each datapoint in $\mathcal{D}$,
Algorithm 1 only takes as inputs a QNN $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ that operates over bit-vectors of bit-
width $k$, a single datapoint $x\in[\mathbb{Z}]_{k}^{n}$ and a robustness
radius $\epsilon>0$. It then returns either ROBUST if $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ is verified to be $\epsilon$-robust at
$x\in[\mathbb{Z}]_{k}^{n}$, or VULNERABLE if an adversarial attack
$||x^{\prime}-x||_{\infty}<\epsilon$ with
$\mathsf{class}(x^{\prime})=\mathsf{class}(x)$ is found.
The algorithm proceeds by initializing a stack $D$ of abstract intervals to
contain a single element
$\\{(x-\epsilon\cdot\mathbf{1},x+\epsilon\cdot\mathbf{1})\\}$, where
$\mathbf{1}\in[\mathbb{Z}]_{k}^{n}$ is a unit bit-vector of bit-width $k$.
Intuitively, $D$ contains all abstract intervals that may contain concrete
adversarial examples but have not yet been processed by the algorithm. The
algorithm then iterates through a loop which in each loop iteration processes
the top element of the stack. Once the stack is empty and the last loop
iteration terminates, Algorithm 1 returns ROBUST.
In each loop iteration, Algorithm 1 pops an abstract interval
$(\underline{x},\overline{x})$ from $D$ and processes it as follows. First, it
uses IBP for QNNs that we introduced before to propagate
$(\underline{x},\overline{x})$ in order to compute an abstract interval
$(\underline{y},\overline{y})$ that overapproximates the set of all possible
outputs for a concrete input point in $(\underline{x},\overline{x})$. The
algorithm then considers three cases. First, if $(\underline{y},\overline{y})$
does not violate Equation (11) which characterizes violation of robustness by
a propagated abstract interval, Algorithm 1 concludes that the abstract
interval $(\underline{x},\overline{x})$ does not contain an adversarial
example and it proceeds to processing the next element of $D$. Second, if
$(\underline{y},\overline{y})$ violates Equation (11), the algorithm uses
projected gradient descent restricted to $(\underline{x},\overline{x})$ to
search for an adversarial example and returns VULNERABLE if found. Note that
the adversarial attack is generated with respect to the quantization-aware
representation of the network, thus ensuring that the input space corresponds
to valid quantized inputs. Third, if $(\underline{y},\overline{y})$ violates
Equation (11) but the adversarial attack could not be found by projected
gradient descent, the algorithm refines the abstract interval
$(\underline{x},\overline{x})$ by splitting it into two smaller subintervals.
This is done by identifying
$i^{\ast}=\mathsf{argmax}_{1\leq i\leq n}(\overline{x}[i]-\underline{x}[i])$
and splitting the abstract interval $(\underline{x},\overline{x})$ along the
$i^{\ast}$-th dimension into two abstract subintervals
$(\underline{x^{\prime}},\overline{x^{\prime}}),(\underline{x^{\prime\prime}},\overline{x^{\prime\prime}})$,
which are both added to the stack $D$.
#### Correctness, termination and completeness
The following theorem establishes that Algorithm 1 is complete, that it
terminates on every input and that it is a complete robustness verification
procedure. The proof is provided in the Appendix.
###### Theorem 2.
If Algorithm 1 returns ROBUST then $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ is $\epsilon$-robust at
$x\in[\mathbb{Z}]_{k}^{n}$. On the other hand, if Algorithm 1 returns
VULNERABLE then there exists an adversarial attack
$||x^{\prime}-x||_{\infty}<\epsilon$ with
$\mathsf{class}(x^{\prime})\neq\mathsf{class}(x)$. Therefore, Algorithm 1 is
correct. Furthermore, Algorithm 1 terminates and is guaranteed to return an
output on any input. Since Algorithm 1 is correct and it terminates, we
conclude that it is also complete.
Algorithm 1 QNN robustness verification procedure
1:Input QNN $\llbracket f_{\theta}\rrbracket_{\text{int-}k}$, datapoint
$x\in[\mathbb{Z}]_{k}^{n}$, robustness radius $\epsilon>0$
2:Output ROBUST or VULNERABLE
3:$D\leftarrow\\{(x-\epsilon\cdot\mathbf{1},x+\epsilon\cdot\mathbf{1})\\}$
4:while $D\neq\\{\\}$ do
5: $(\underline{x},\overline{x})\leftarrow$ pop item from $D$
6: $(\underline{y},\overline{y})\leftarrow\underline{\llbracket
f_{\theta}\rrbracket_{\text{int-}k}}(\underline{x},\overline{x}),\overline{\llbracket
f_{\theta}\rrbracket_{\text{int-}k}}(\underline{x},\overline{x})$ propagated
via IBP
7: if $(\underline{y},\overline{y})$ violates Equation (11) then
8: Try to generate adversarial example using projected gradient descent
restricted to $(\underline{x},\overline{x})$
9: if adversarial example found then
10: return VULNERABLE
11: else
12:
$(\underline{x^{\prime}},\overline{x^{\prime}}),(\underline{x^{\prime\prime}},\overline{x^{\prime\prime}})\leftarrow$
partition of $(\underline{x},\overline{x})$ into two abstract subintervals
13: $D\leftarrow
D\cup\\{(\underline{x^{\prime}},\overline{x^{\prime}}),(\underline{x^{\prime\prime}},\overline{x^{\prime\prime}})\\}$
14: end if
15: end if
16:end while
17:return ROBUST
## Experiments
We perform an experimental evaluation to assess the effectiveness of our
quantization-aware interval bound propagation (QA-IBP). In particular, we
first use our training procedure to train two QNNs. We then use our complete
verification procedure to verify robustness of trained QNNs and we compare our
method to the existing verification methods for QNNs of (Giacobbe, Henzinger,
and Lechner 2020) and (Henzinger, Lechner, and Žikelić 2021). Our full
experimental setup and code can be found on GitHub
111https://github.com/mlech26l/quantization˙aware˙ibp.
Method | MNIST | Fashion-MNIST
---|---|---
| $\epsilon=0$ | $\epsilon=1$ | $\epsilon=4$ | $\epsilon=0$ | $\epsilon=1$ | $\epsilon=4$
QF_BV SMT (Giacobbe, Henzinger, and Lechner 2020) | 97.1% | 92% | 0% | 85.6% | 44% | 0%
QF_BV SMT (Henzinger, Lechner, and Žikelić 2021) | 97.1% | 99%∗ | 53% | 85.6% | 76% | 36%
QA-IBP (ours) | 99.2% | 98.8% | 95.6% | 86.3% | 80.0% | 59.8%
Table 1: Certified robust accuracy of a convolutional neural network trained
with QA-IBP compared to existing methods for certifying
$L_{\infty}$-robustness of quantized neural networks reported in the
literature. ∗ Note that (Henzinger, Lechner, and Žikelić 2021) certified only
a subset of the test set due to a high per-sample runtime of their approach.
Due to this choice they reported a higher $\epsilon=1$ robust accuracy than
clean accuracy.
We train two CNNs with QA-IBP using an 8-bit quantization scheme for both
weights and activations on the MNIST (LeCun et al. 1998) and Fashion-MNIST
(Xiao, Rasul, and Vollgraf 2017) datasets. We quantize all layers; however, we
note that our approach is also compatible with mixing quantized and non-
quantized layers. Our MNIST network consists of five convolutional layers,
followed by two fully-connected layers. Our Fashion-MNIST network contains
three convolutional layers and two fully-connected layers. We apply a fixed
pre-defined fixed-point format on both weights and activations. Further
details on the network architectures can be found in the Appendix. We use the
Adam optimizer (Kingma and Ba 2015) with a learning rate of $10^{-4}$ with
decoupled weight decay of $10^{-4}$ (Loshchilov and Hutter 2019) and a batch
size of 512. We train our networks for a total of 5,000,000 gradient steps on
a single GPU. We linearly scale the value of $\epsilon$ during the QA-IBP
training from 0 to 4 and apply the elision of the last layer optimization as
reported in (Gowal et al. 2019). We pre-train our networks for 5000 steps in
their non-IBP representation with a learning rate of $5\cdot 10^{-4}$.
After training, we certify all test samples using Algorithm 1 with a timeout
of 20s per sample. We certify for $L_{\infty}$-robustness with radii 1 and 4
to match the experimental setup of (Henzinger, Lechner, and Žikelić 2021). We
report and compare the certified robust accuracy of the trained networks to
the values reported in (Henzinger, Lechner, and Žikelić 2021).
The results in Table 1 demonstrate that our QA-IBP significantly improves the
certified robust accuracy for QNNs on both datasets.
Regarding the runtime of the verification step, both (Giacobbe, Henzinger, and
Lechner 2020) and (Henzinger, Lechner, and Žikelić 2021) certified a smaller
model compared to our evaluation. In particular, (Giacobbe, Henzinger, and
Lechner 2020) report a mean runtime of their 8-bit model to be over 3 hours,
and (Henzinger, Lechner, and Žikelić 2021) report the mean runtime of their
6-bit model to be 90 and 49 seconds for MNIST/Fashion-MNIST, respectively.
Conversely, our method was evaluated with a timeout of 20 seconds and tested
on larger networks than the two existing approaches, showing the efficiency of
our approach.
### Ablation analysis
In this section, we assess the effectiveness of Algorithm 1 for certifying
robustness compared to an incomplete verification based on the QA-IBP output
bounds alone. In particular, we perform an ablation and compare Algorithm 1 to
an incomplete verification baseline. Our baseline consists of checking the
bounds obtained by QA-IBP as an incomplete verifier combined with projected
gradient descent (PGD) as an incomplete falsifier, i.e., we try to certify
robustness via QA-IBP and simultaneously try to generate an adversarial attack
using PGD. In particular, this baseline resembles a version of Algorithm 1
without any branching into subdomains. We carry our ablation analysis on the
two networks of our first experiment. We report the number of samples where
the verification approach could not determine whether the network is robust on
the sample or not. We use a timeout of 20s per instance when running our
algorithm.
| CIFAR-10
---|---
Weight decay | $\epsilon=0$ | $\epsilon=1$ | $\epsilon=4$
$1\cdot 10^{-4}$ | 30.0% | 28.4% | 22.4%
$5\cdot 10^{-5}$ | 39.6% | 34.1% | 20.6%
$1\cdot 10^{-5}$ | 83.1% | 0% | 0%
Table 2: Robust accuracy of our convolutional neural network trained with QA-
IBP on the CIFAR-10 dataset with various values for the weight decay. The
results express the robustness-accuracy tradeoff (Tsipras et al. 2018), i.e.,
the empirical observation made for non-quantized neural networks that we can
have a high robustness or a high accuracy but not both at the same time.
The results shown in Table 3 indicate that our algorithm is indeed improving
on the number of samples for which robustness could be decided. However, the
improvement is relatively small, suggesting that the QA-IBP training stems
from most of the observed gains.
Method | MNIST | Fashion-MNIST
---|---|---
| $\epsilon=1$ | $\epsilon=4$ | $\epsilon=1$ | $\epsilon=4$
IBP + Projected gradient descent | 0.35% | 3.51% | 3.96% | 19.23%
Algorithm 1 | 0.35% | 3.47% | 3.93% | 18.51%
Table 3: Percentage of samples where the robustness of the network can be
determined, i.e., certified or falsified, by the method (lower is better).
Algorithm timeout was set to 20s per instance.
### Limitations
In this section, we aim to scale our QA-IBP beyond the two gray-scale image
classification tasks studied before to the CIFAR-10 dataset (Krizhevsky and
Hinton 2010). Our setup consists of the same convolutional neural network as
used for the Fasion-MNIST. We run our setup with several values for the weight
decay rate. Similar to above, we linearly scale $\epsilon$ from 0 to 4 during
training and report the certified robust accuracy obtained by QA-IBP with an
$\epsilon$ of 1 and 4 and the clean accuracy.
The results in Table 2 express the robustness-accuracy tradeoff, i.e., the
observed antagonistic relation between clean accuracy and robustness, which
has been extensively studied in non-quantized neural networks (Zhang et al.
2019; Bubeck and Sellke 2021). Depending on the weight decay value, different
points on the trade-off were observed. In particular, our training procedure
either obtains a network that has an acceptable accuracy but no robustness or
a certifiable robust network with a significantly reduced clean accuracy.
We also trained larger models on all datasets (MNIST, Fashion-MNIST, and
CIFAR-10), but observed the same behavior of having a high accuracy but no
robustness when training, i.e., as in row with $1\cdot 10^{-5}$ in Table 2.
We found the underlying reason for this behavior to be activations of internal
neurons that clamp to the minimum and maximum value of the quantization range
in their QA-IBP representation but not in their standard representation.
Consequently, the activation gradients become zero during QA-IBP due to
falling in the constant region of the activation function. This effect is
specific to quantized neural networks due to the upper bound on the
representative range of values. Our observation hints that future research
needs to look into developing dynamic quantization ranges or weight decay
schedules that can adapt to both the standard and the QA-IBP representation of
a QNN.
## Conclusion
In this paper, we introduced quantization-aware interval bound propagation
(QA-IBP), the first method for training certifiably robust QNNs. We also
present a theoretical result on the existence and upper bounds on the needed
size of a robust QNN for a given dataset of $1$-dimensional datapoints.
Moreover, based on our interval bound propagation method, we developed the
first complete verification algorithm for QNNs that may be run on GPUs. We
experimentally showed that our training scheme and verification procedure
advance the state-of-the-art on certifying $L_{\infty}$-robustness of QNNs.
We also demonstrated the limitations of our method regarding training
stability and convergence. Particularly, we found that the boundedness of the
representable value range of QNNs compared to standard networks leads to
truncation of the abstract domains, which in turn leads to gradients becoming
zero. Our results suggest that dynamic quantization schemes that adapt their
quantization range to the abstract domains instead of the concrete activation
values of existing quantization schemes may further improve the certified
robust accuracy of quantized neural networks.
Nonetheless, our work serves as a new baseline for future research. Promising
directions on how to improve upon QA-IBP and potentially overcome its
numerical challenges is to adopt advanced quantization-aware training
techniques. For instance, dynamical quantization ranges (Choi et al. 2018;
Jacob et al. 2018), mixed-precision layers (Zhou et al. 2016; Yao et al.
2021), and automated architecture search (Tan and Le 2019) have shown
promising results for standard training QNNs and might enhance QA-IBP-based
training procedures as well. Moreover, further improvements may be feasible by
adapting recent advances in IBP-based training methods for non-quantized
neural networks (Müller et al. 2022) to our quantized IBP variant.
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## Appendix A Acknowledgments
This work was supported in part by the ERC-2020-AdG 101020093, ERC CoG 863818
(FoRM-SMArt) and the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie Grant Agreement No. 665385.
Research was sponsored by the United States Air Force Research Laboratory and
the United States Air Force Artificial Intelligence Accelerator and was
accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views
and conclusions contained in this document are those of the authors and should
not be interpreted as representing the official policies, either expressed or
implied, of the United States Air Force or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for Government
purposes notwithstanding any copyright notation herein. The research was also
funded in part by the AI2050 program at Schmidt Futures (Grant G-22-63172) and
Capgemini SE.
## Appendix B Appendix
### Proof of Theorem 1
Note that, for each $1\leq i\leq|\mathcal{D}|$, the set
$\mathcal{B}_{\epsilon}(x_{i})=\\{x^{\prime}\in[\mathbb{Z}]_{k}^{1}\mid||x^{\prime}-x_{i}||_{\infty}<\epsilon\\}$
is finite and consists of at most $\lceil 2\epsilon\cdot 2^{k}+1\rceil$ bit-
vectors. Hence, constructing a QNN $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ that is $\epsilon$-robust with respect to
the dataset $\mathcal{D}$ is equivalent to constructing a QNN that correctly
classifies the dataset
$\mathcal{D}^{\prime}=\bigcup_{i=1}^{|\mathcal{D}|}\\{(x^{\prime},y_{i})\mid
x^{\prime}\in\mathcal{B}_{\epsilon}(x_{i})\\}$
that consists of $\mathcal{O}(|\mathcal{D}|\cdot\lceil\epsilon\cdot
2^{k}+1\rceil)$ datapoints. Let $N=|\mathcal{D}^{\prime}|$. In what follows,
we fix an enumeration
$\\{(x^{\prime}_{1},y^{\prime}_{1}),\dots,(x^{\prime}_{N},y^{\prime}_{N})\\}$
of datapoints in $\mathcal{D}^{\prime}$.
We design the QNN $\llbracket f_{\theta}\rrbracket_{\text{int-}k}$ so that it
consists of the input layer with $1$ neuron, $1$ hidden layer with $5N$
neurons and the output layer with $1$ neurons. The neurons in the hidden layer
are partitioned into $N$ gadgets $G_{i}$ with $1\leq i\leq N$, where each
gadget consists of $5$ neurons. For each $G_{i}$, we connect the input layer
neuron to the $5$ neurons in the gadget and we connect the $5$ neurons in the
gadget to output layer neuron. We then use the connecting edges in order to
add the following expression to value of the output layer neuron, whenever the
value of the input layer neuron is equal to $z$:
$y^{\prime}_{i}\cdot\Big{(}\mathsf{ReLU}(z-x^{\prime}_{i}+1)-\mathsf{ReLU}(z-x^{\prime}_{i})+\mathsf{ReLU}(x^{\prime}_{i}-z)-\mathsf{ReLU}(x^{\prime}_{i}-z-1)-1\Big{)}.$
(15)
One can verify by inspection that the expression in eq. (15) evaluates to
$y^{\prime}_{i}$ if $z=x^{\prime}_{i}$ and to $0$ otherwise and that it can
indeed be encoded by $5$ neurons in the hidden layer. Hence, if we consider
the output layer neuron and take the weighted sum of all incoming edges from
each gadget, this construction ensures that for any input neuron value
$z\in[\mathbb{Z}]_{k}^{n}$ we have that the value of the output neuron of
$\llbracket f_{\theta}\rrbracket_{\text{int-}k}$ is equal to
$\llbracket
f_{\theta}\rrbracket_{\text{int-}k}(z)=\sum_{i=1}^{N}y^{\prime}_{i}\cdot\mathbb{I}(z=x^{\prime}_{i})$
with $\mathbb{I}$ an indicator function. Therefore, as datapoints in
$\mathcal{D}^{\prime}$ are distinct, one may conclude that $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}(x^{\prime}_{i})=y^{\prime}_{i}$ for each
$1\leq i\leq N$, as desired.
### Proof of Theorem 2
We first show that Algorithm 1 is correct, i.e. that if it returns ROBUST then
$\llbracket f_{\theta}\rrbracket_{\text{int-}k}$ is $\epsilon$-robust at
$x\in[\mathbb{Z}]_{k}^{n}$ and that if it returns VULNERABLE then there exists
an adversarial attack $||x^{\prime}-x||_{\infty}<\epsilon$ with
$\mathsf{class}(x^{\prime})=\mathsf{class}(x)$. Indeed, the correctness of
ROBUST outputs follows from the fact that, in order for Algorithm 1 to
terminate, it had to split the initial abstract interval
$\\{(x-\epsilon\cdot\mathbf{1},x+\epsilon\cdot\mathbf{1})\\}$ into a finite
number of abstract subintervals and to show that their propagations computed
via IBP do not contain outputs of a different class. Therefore, the
correctness of ROBUST follows by the correctness of our IBP for QNNs in
Section Quantization-aware Interval Bound Propagation. The fact that
VULNERABLE outputs are correct follows from the fact that computed adversarial
examples can be easily verified by feeding them to $\llbracket
f_{\theta}\rrbracket_{\text{int-}k}$ and by checking that the class of the
computed output differs from $\mathsf{class}(x)$.
To show that Algorithm 1 terminates, observe that in every loop iteration
Algorithm 1 either pops an item from $D$ and does not add new items, it
returns VULNERABLE or it pops an item from $D$ and replaces it by two new
abstract intervals that are obtained by splitting the popped one. Hence, all
three cases preserve the invariant that $\sum_{(\underline{x},\overline{x})\in
D}\\#((\underline{x},\overline{x}))^{2}$ strictly decreases between any two
consecutive iterations, where we use $\\#((\underline{x},\overline{x}))$ to
denote the number of bit-vectors contained in $(\underline{x},\overline{x})$.
Hence, as the number of bit-vectors of bit-width $k$ that are contained in the
initial abstract interval
$\\{(x-\epsilon\cdot\mathbf{1},x+\epsilon\cdot\mathbf{1})\\}$ is finite, we
conclude that Algorithm 1 must terminate and return an output in at most
finitely many steps.
Finally, since Algorithm 1 terminates it must return either an output ROBUST
or VULNERABLE on any input. On the other hand, since Algorithm 1 is complete,
its returned output is always correct. Hence, Algorithm 1 is guaranteed to
return either that the QNN is robust or to prove the existence an adversarial
attack, and therefore it is complete. This concludes the proof of Theorem 2.
### Experimental details
The network architectured used in our experiments can be found in Table 4, 5,
and 6. For the MNIST network, we used a Q3.5 fixed-point format for the
activations, Q5.3 for the bias terms, and Q2.6 format for the weights. For
both the Fashion-MNIST and the CIFAR-10 networks, we used a Q4.4 fixed-point
format for the activations, Q5.3 for the bias terms, and Q2.6 format for the
weights.
Layer | Parameters
---|---
Conv2D | F=64, K=5, S=2, ReLU-N
Conv2D | F=128, K=3, S=1, ReLU-N
Conv2D | F=256, K=3, S=1, ReLU-N
Conv2D | F=384, K=3, S=1, ReLU-N
Conv2D | F=512, K=3, S=2, ReLU-N
Flatten |
Fully-connected | U=128, ReLU-N
Fully-connected | U=10
Table 4: Convolutional neural network architecture used for the MNIST experiment. F represents the number of filters, K the kernel size, S the stride, and U the number of units. Layer | Parameters
---|---
Conv2D | F=64, K=5, S=2, ReLU-N
Conv2D | F=96, K=3, S=1, ReLU-N
Conv2D | F=128, K=3, S=2, ReLU-N
Flatten |
Fully-connected | U=128, ReLU-N
Fully-connected | U=10
Table 5: Convolutional neural network architecture used for the Fashion-MNIST experiment. F represents the number of filters, K the kernel size, S the stride, and U the number of units. Layer | Parameters
---|---
Conv2D | F=64, K=5, S=2, ReLU-N
Conv2D | F=96, K=3, S=1, ReLU-N
Conv2D | F=128, K=3, S=2, ReLU-N
Flatten |
Fully-connected | U=128, ReLU-N
Fully-connected | U=10
Table 6: Convolutional neural network architecture used for the CIFAR-10
experiment. F represents the number of filters, K the kernel size, S the
stride, and U the number of units.
|
# Epsilon regularity for the Navier-Stokes equations via weak-strong
uniqueness
Dallas Albritton Department of Mathematics, Princeton University, Princeton,
NJ 08544, USA<EMAIL_ADDRESS>, Tobias Barker Department of
Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
<EMAIL_ADDRESS>and Christophe Prange Cergy Paris Université,
Laboratoire de Mathématiques AGM, UMR CNRS 8088, France
<EMAIL_ADDRESS>In honor of Olga Ladyženskaja
###### Abstract.
We give a new concise proof of a certain one-scale epsilon regularity
criterion using weak-strong uniqueness for solutions of the Navier-Stokes
equations with non-zero boundary conditions. It is inspired by an analogous
approach for the stationary system due to Struwe.
## 1\. Introduction
In the regularity theory of the three-dimensional non-stationary Navier-Stokes
equations, so-called epsilon regularity theory remains the state-of-the-art.
As well as being of independent interest, epsilon regularity at one scale
(‘one-scale epsilon regularity’) has been a crucial tool for proving some of
the best results in the field. See [3], [22], [27] and [5], for example. Thus
far, there have been three main methods for proving one-scale epsilon
regularity results:
* •
direct iteration arguments (see [3]),
* •
compactness arguments (see [21], [19]),
* •
De Giorgi techniques (see [28]).
This short article is devoted to a new concise proof of a certain one-scale
epsilon regularity criterion for the three-dimensional non-stationary Navier-
Stokes equations away from boundaries. Theorem B is based on slicing
techniques and a comparison to a solution for which we have improved
integrability. To the best of our knowledge, related arguments in the Navier-
Stokes context were introduced by Struwe in [26] for the five-dimensional
stationary Navier-Stokes equations. Our paper is, in spirit, an analogue for
the non-stationary Navier-Stokes equations.
The following localized weak-strong uniqueness result is the keystone of our
paper.
###### Theorem A (first version of weak-strong uniqueness with boundary
conditions).
Let $\Omega\subset\mathbb{R}^{3}$ be a smooth bounded domain111We rely on
linear results of [10], which require $\partial\Omega$ to be of class
$C^{2,1}$. and $T>0$. There exists
$\bar{\kappa}=\bar{\kappa}_{\Omega,T}\in(0,\infty)$ and
$C(\Omega,T)\in(0,\infty)$ such that the following holds.222See (3.9) for an
estimate of $\bar{\kappa}_{\Omega,T}$. One can take
$C(\Omega,T):=K_{\Omega,T}(1+C_{0}C_{2})$. Here $C_{0}$ is defined in (3.4),
$C_{2}$ is defined in (3.7) and $K_{\Omega,T}\in(0,\infty)$ is defined in
(3.1). There exists a unique very weak solution
$U\in L^{4}(\Omega\times(0,T))$ (1.1)
to the Navier-Stokes equations on $\Omega\times(0,T)$ in the sense of
Definition 2.1 with boundary data $a$ and initial data $b$ satisfying the
integrability conditions
$a\in L^{4}(\partial\Omega\times(0,T)),\qquad b\in L^{4}(\Omega),$ (1.2)
the compatibility conditions
$\int\limits_{\partial\Omega}a(\cdot,t)\cdot
n=0,\qquad\int\limits_{\Omega}b\cdot\nabla q=0\quad\mbox{for all}\quad q\in
C^{\infty}(\mathbb{R}^{3})$ (1.3)
and the smallness condition
$\kappa:=\|a\|_{L^{4}(\partial\Omega\times(0,T))}+\|b\|_{L^{4}(\Omega)}<\bar{\kappa}.$
(1.4)
Moreover, $U\in L^{4}(0,T;L^{6}(\Omega))$, and
$\|U\|_{L^{4}(0,T;L^{6}(\Omega))}\leq C(\Omega,T)\kappa.$ (1.5)
The critical integrability in (1.5) is the main practical outcome of the
theorem.
Below, we give a second version of this localized uniqueness result, see
Theorem A’, that can be directly used to prove the epsilon regularity result
stated in Theorem B below.
###### Remark 1.1 (structure result).
We actually prove a strengthened version of the uniqueness result. Namely, we
prove that any very weak solution $U$ in the sense of Definition 2.1 with data
satisfying (1.2), (1.3) and the smallness condition (1.4) can be written as
$U=\bar{U}+V$, where $\bar{U}$ is the unique very weak solution to the Stokes
equation with boundary data $a$ and initial data $b$ and $V$ is the unique
mild solution to the perturbed Navier-Stokes equations around $\bar{U}$ in
$\Omega\times(0,T)$ with homogeneous boundary and initial data. We refer to
Section 3 below for more details.
###### Remark 1.2 (smallness assumption (1.4)).
The smallness condition (1.4) is required in Section 3 (Step 1-b) below in
order to construct a mild solution to the perturbed Navier-Stokes equations
around the solution $\bar{U}$ to the linear Stokes equations lifting the
boundary data $a$ and the initial data $b$.
###### Remark 1.3 (qualitative integrability assumption $L^{4}$).
The integrability condition (1.1) on the solution is essential to prove that
the perturbation in Section 3 (Step 2) has finite energy. The integrability
condition (1.2) on the boundary data $a$ is assumed in order to apply the
linear results of Farwig, Kozono and Sohr [10] (see Section 2 below). Notice
that according to [10], the boundary data can be taken in the larger space
$L^{4}(0,T;W^{-\frac{1}{6},6}(\partial\Omega))$. However, we do not state such
an improved result, because in the application to epsilon regularity that we
have in mind, the data $a$ satisfies (1.2).
In the second version of the localized uniqueness stated below, the main
change with respect to the previous theorem is in the conditions on the
initial data $b$. Indeed the requirement (1.3) is strong and imposes not only
that $b$ is divergence-free, but also that $b\cdot n=0$ on $\partial\Omega$.
This condition cannot be satisfied in general for data $b$ arising from
slicing in the proof of the epsilon regularity result, Theorem B below; for
details see Step 1 in Section 4. The assumptions in Theorem A’, contrary to
those of Theorem A, are immediately satisfied for the data stemming from the
slicing in the proof of Theorem B. In order to fit into the framework of
Theorem B, we take $\Omega=B_{r_{0}}$ for some
$r_{0}\in(\frac{5}{8},\frac{7}{8})$. This specific choice is only for
convenience. It is easy to extend the result to more general domains, the key
point being that one has to assume that $b$ is defined and divergence-free on
a larger domain than $\Omega$, so that it is possible to cut-off.
###### Theorem A’ (second version of weak-strong uniqueness with boundary
conditions).
Let $\Omega=B_{r_{0}}$ for some $r_{0}\in(\frac{5}{8},\frac{7}{8})$ and
$t_{0}\in(-1,-\frac{3}{4})$. There exists universal constants
$\bar{\kappa}\in(0,\infty)$ and $C\in(0,\infty)$ such that the following
holds.333See (3.14) for an estimate of $\bar{\kappa}$. One can take
$C=K(1+C_{0}C_{2}$). Here $C_{0}$ is defined in (3.4), $C_{2}$ is defined in
(3.7) and $K\in(0,\infty)$ is a universal constant (see (3.13)). There exists
a unique very weak solution
$U\in L^{4}(B_{r_{0}}\times(t_{0},0))$ (1.6)
to the Navier-Stokes equations on $B_{r_{0}}\times(t_{0},0)$ in the sense of
Definition 2.1 with boundary data $a$ and initial data $b$ satisfying the
integrability conditions
$a\in L^{4}(\partial B_{r_{0}}\times(t_{0},0)),\qquad b\in L^{4}(B_{1}),$
(1.7)
the compatibility condition
$\int\limits_{\partial B_{r_{0}}}a(\cdot,t)\cdot n=0,$ (1.8)
the incompressibility condition $\nabla\cdot b=0$ in $B_{1}$ in the sense of
distributions and the smallness condition
$\kappa:=\|a\|_{L^{4}(\partial
B_{r_{0}}\times(t_{0},0))}+\|b\|_{L^{4}(B_{1})}<\bar{\kappa}.$ (1.9)
Moreover, $U\in L^{4}(t_{0},0;L^{6}(B_{r_{0}}))$, and
$\|U\|_{L^{4}(t_{0},0;L^{6}(B_{r_{0}}))}\leq C\kappa.$ (1.10)
Again, the critical integrability is the main practical outcome of the
theorem.
We now state the epsilon regularity result. Let $(U,P)$ be a finite-energy
weak solution to the three-dimensional Navier-Stokes equations in
$Q_{1}=B_{1}(0)\times(-1,0)$ i.e.
$\displaystyle\partial_{t}U-\Delta U+U\cdot\nabla U+\nabla
P=0,\quad\nabla\cdot U=0\qquad\mbox{in}\quad Q_{1},$ in the sense of
distributions, and (1.11)
$\displaystyle\Bigg{(}\sup_{t\in(-1,0)}\int\limits_{B_{1}}|U(\cdot,t)|^{2}+\int\limits_{-1}^{0}\int\limits_{B_{1}}|\nabla
U|^{2}\Bigg{)}^{\frac{1}{2}}\leq M<\infty.$ (1.12)
###### Theorem B (epsilon regularity).
There exists $\bar{\varepsilon}\in(0,1)$ such that for any $M\in(0,\infty)$,
for any finite-energy weak solution $U$ to the Navier-Stokes equations in the
sense of (1.11)-(1.12) belonging to $C^{\infty}(B_{1}\times(-1,T))$ for all
$T\in(-1,0)$,555This assumption is there to keep technicalities to a minimum.
It makes our result applicable to rule out first-time singularities. the
following result holds.
Qualitative statement:
Assume that $U$ satisfies the smallness condition in Theorem A’, i.e.
$\|U\|_{L^{4}(Q_{1})}<\bar{\varepsilon}.$ (1.13)
Then $U\in L^{\infty}(Q_{\frac{1}{4}})$.
Quantitative statement:
Let $0<\varepsilon<\bar{\varepsilon}$. Assume that
$\|U\|_{L^{4}(Q_{1})}\leq\varepsilon.$ (1.14)
Then $U\in L^{\infty}(Q_{\frac{1}{4}})$ and in addition we have the
quantitative estimate
$\|U\|_{L^{\infty}(Q_{\frac{1}{4}})}\leq P(\varepsilon,M),$ (1.15)
where $P(\varepsilon,M)$ is a positive polynomial in $0<\varepsilon,\,M$.
This result is a mere corollary of Theorem A’.
###### Remark 1.4 (smallness condition (1.13)).
Ultimately, the reason for the space $L^{4}(Q_{1})$ in Theorem B is that the
solution of the linear Stokes equation with boundary data in
$L^{4}(\partial\Omega\times(0,T))$ and zero initial condition belongs to the
critical space $L^{4}(0,T;L^{6}(\Omega))$ (see Theorem 2.2). Notably, the
proof we present below apparently does not work in $L^{4-\varepsilon}(Q_{1})$.
###### Remark 1.5 (nonlocality and pressure).
Theorem B is in the spirit of some other epsilon regularity results that do
not involve the pressure, such as [29] and [18].
### Outline of the paper
In Section 2, we review the concept of very weak solution with $L^{p}$ initial
and boundary data. In Section 3, we prove weak-strong uniqueness, Theorems A
and A’. In Section 4, we prove the epsilon regularity criterion, Theorem B.
### Notations
For $r>0$ and $p\in(1,\infty)$, we denote by $\mathbf{A}$ the Stokes operator
realized on $L^{p}_{\sigma}(B_{r})$. Notice that
$\mathbf{A}=-\mathbb{P}\Delta_{D}$, where $\mathbb{P}$ is the Helmholtz-Leray
projection and $\Delta_{D}$ is the realization of the Laplace operator under
the Dirichlet boundary condition on $\partial B_{r}$. We also use the notation
$(e^{-t\mathbf{A}})_{t\in(0,\infty)}$ for the Stokes semigroup on $B_{r}$.
Notice that these notations do not involve explicitly the parameter $r$ in
order to lighten the notation and because the parameter $r$ will be fixed. For
further details concerning the Stokes operator and the Stokes semigroup, we
refer to [25, Chapter III.2 and IV.1] and [16]. For definitions of Sobolev
spaces on $\partial\Omega$, we refer to [25, Chapter I.3]. We define
$C_{0,\sigma}^{2}(\bar{\Omega}):=\\{f\in
C^{2}(\bar{\Omega};\mathbb{R}^{3}):\textrm{div}\,f=0,\,f|_{\partial\Omega}=0\\}.$
As is usual, the notation $C$ denotes a numerical constant possibly depending
on parameters that we do not track. This constant may change from line to
line. When a constant depends on parameters that we track, we denote this by
$C_{\alpha,\beta,\ldots}$.
## 2\. Preliminaries
Let $\Omega\subset\mathbb{R}^{3}$ be a smooth bounded domain.666Below, we
apply the results for the fixed smooth domain $\Omega=B_{r_{0}}$. Hence, we
pay no attention here to how the constants in the estimates will depend
quantitatively on the regularity of the domain. This dependence is not tracked
in [10]. See also Footnote 1. Let $-\infty<T_{1}<T_{2}<\infty$. Let $a\in
L^{1}(\partial\Omega\times(T_{1},T_{2});\mathbb{R}^{3})$, $b\in
L^{1}(\Omega;\mathbb{R}^{3})$ and $F\in
L^{1}(\Omega\times(T_{1},T_{2});\mathbb{R}^{3\times 3})$ satisfying the
compatibility conditions (1.3).
We consider the following Stokes problem:
$\displaystyle\begin{split}&\partial_{t}U-\Delta U+\nabla P=\nabla\cdot
F,\quad\nabla\cdot U=0\qquad\mbox{in}\quad\Omega\times(T_{1},T_{2}),\\\
&\bar{U}=a\qquad\mbox{on}\quad\partial\Omega\times(T_{1},T_{2}),\\\
&\bar{U}(\cdot,T_{1})=b.\end{split}$ (2.1)
This problem includes in particular the standard Navier-Stokes problem taking
$F=-U\otimes U$.
###### Definition 2.1 (very weak solution).
For $a,\,b$ and $F$ given as above, we say that $U\in
L^{1}(\Omega\times(T_{1},T_{2}))$ is a very weak solution of (2.1) if for all
$\Phi\in C^{1}_{0}([T_{1},T_{2});C_{0,\sigma}^{2}(\bar{\Omega})$ and for all
$q\in C^{\infty}(\overline{\Omega}\times[T_{1},T_{2}];\mathbb{R})$, we have
$\displaystyle\begin{split}&-\int\limits_{T_{1}}^{T_{2}}\int\limits_{\Omega}U\cdot(\partial_{t}\Phi+\Delta\Phi+\nabla
q)=-\int\limits_{T_{1}}^{T_{2}}\int\limits_{\Omega}F:\nabla\Phi\\\
&\qquad\qquad+\int\limits_{\Omega}b\cdot\Phi(\cdot,T_{1})-\int\limits_{T_{1}}^{T_{2}}\int\limits_{\partial\Omega}a\cdot(\nabla\Phi\cdot
n)-\int\limits_{T_{1}}^{T_{2}}\int\limits_{\partial\Omega}(a\cdot
n)q.\end{split}$ (2.2)
We state here an existence and uniqueness result of very weak solutions to the
Stokes system. Results in this direction were established in [9], though the
version we use below is from [10].
###### Theorem 2.2 (very weak solutions for the Stokes problem, [10, Lemma
1.2]).
Let $4\leq s,q<\infty$ such that $\frac{2}{s}+\frac{3}{q}=1$, and let
$r:=\frac{2}{3}q$. Let $a\in L^{s}(T_{1},T_{2};L^{r}(\Omega))$ and $F=b=0$.
Assume that $a$ satisfies the compatibility condition (1.3).
Then, there exists a unique very weak solution $U\in
L^{s}(T_{1},T_{2};L^{q}(\Omega))$ to the Stokes problem (2.1) in the sense of
Definition 2.1. Morever,
$\|U\|_{L^{s}(T_{1},T_{2};L^{q}(\Omega))}\leq
C_{\Omega,T_{1},T_{2},q}\|a\|_{L^{s}(T_{1},T_{2};L^{r}(\Omega))}.$ (2.3)
Notice that our boundary data $a$ is slightly less general (but general enough
for our purposes) than the data considered in [10, Lemma 1.2]. It follows that
any term in the very weak formulation of the equation (2.2) makes sense in a
classical integral sense and duality brackets are not needed. The statement of
Theorem 2.2 follows directly from [10, Lemma 1.2]. Indeed,
$W^{-\frac{1}{q},q}(\partial\Omega)$ is the dual space of
$W^{1-\frac{1}{q^{\prime}},q^{\prime}}(\partial\Omega)$ with
$\frac{1}{q}+\frac{1}{q^{\prime}}=1$, and by Sobolev embedding [4, Theorem
4.1.3], $W^{1-\frac{1}{q^{\prime}},q^{\prime}}(\partial\Omega)$ embeds into
$L^{r^{\prime}}(\partial\Omega)$ for $r=\frac{2}{3}q$ and
$\frac{1}{r}+\frac{1}{r^{\prime}}=1$. Therefore, $a\in
L^{s}(T_{1},T_{2};L^{r}(\partial\Omega))$ embeds into
$L^{s}(T_{1},T_{2};W^{-\frac{1}{q},q}(\partial\Omega))$.
###### Remark 2.3 (non-zero initial data).
We handle non-zero initial data $b$ as a separate issue. Indeed this point is
more classical than the case of rough boundary data which is treated in the
result above. For non-zero initial data, we rely on the Stokes semigroup
estimates in Lebesgue spaces, see for instance [16].
###### Remark 2.4 (on alternative linear results with rough boundary data).
It is also possible to rely on the results of Fabes, Lewis and Rivière for
boundary value problems with $L^{p}$ data obtained in [20] for the half-space
and in [8, 7] for bounded smooth domains. However, due to an additional
integrability condition on $a\cdot n$ on $\partial\Omega$ in [7, Theorem
(IV.3.3), page 643], these results require to work in $L^{4+\varepsilon}$,
$\varepsilon>0$, rather than $L^{4}$. Moreover, notice that there is a typo in
the statement of [7, Theorem (IV.3.1)]. The space for $a\cdot n$ is
$L^{\bar{q}}_{t}L^{\frac{2}{3}\bar{p}}_{x}$, not
$L^{\bar{q}}_{t}L^{\frac{3}{2}\bar{p}}_{x}$ as written in [7, Theorem IV.3.1].
Indeed, to get the estimate of the term involving the normal data, namely
$\nabla H$, one relies on Theorem IV.2.3.
###### Lemma 2.5 (Uniqueness of square integrable very weak solutions).
Let $U\in L^{2}(T_{1},T_{2};L^{2}(\Omega))$ be a very weak solution to the
Stokes problem (2.1), in the sense of Definition 2.1, with $F=a=b=0$. Then
$U\equiv 0$ on $\Omega\times(T_{1},T_{2})$.
###### Proof.
The proof relies on classical duality arguments, which we include for
completeness. Without loss of generality, let $T_{1}=-1$ and $T_{2}=0.$ Let
$f\in C^{\infty}_{0}(\Omega\times(-1,0);\mathbb{R}^{3})$ be arbitrary. Define
$\widetilde{f}(x,t)=f(x,-t)$ and let $\widetilde{\Phi}$ solve
$\partial_{t}\widetilde{\Phi}-\Delta\widetilde{\Phi}+\nabla\widetilde{q}=\widetilde{f}\quad\textrm{and}\quad\textrm{div}\,\widetilde{\Phi}=0\quad\textrm{in}\quad\Omega\times(0,\infty),\quad\widetilde{\Phi}|_{\partial{\Omega}}=0,\quad\widetilde{\Phi}(\cdot,0)=0.$
From [25, Theorem 2.7.3, IV], there is a classical solution to the above
linear problem satisfying $\widetilde{\Phi},\widetilde{q}\in
C^{\infty}(\overline{\Omega\times(\delta,2)})$ for all $\delta\in(0,2)$.
Furthermore, since $\widetilde{f}$ is supported in time away from zero we can
apply [25, Lemma 2.4.2, Chapter IV] to infer that $\widetilde{\Phi}$ is
supported in time away from zero.
We define
$\Phi(x,t)=-\widetilde{\Phi}(x,-t)\quad\textrm{and}\quad
q(x,t)=\widetilde{q}(x,-t).$
Inserting $\Phi$ and $q$ into (2.2) (where $F$, $a$ and $b$ are all zero)
gives
$\int\limits_{-1}^{0}\int\limits_{\Omega}U\cdot fdxds=0\qquad\forall f\in
C^{\infty}_{0}(\Omega\times(-1,0);\mathbb{R}^{3}).$
This implies the desired conclusion. ∎
###### Remark 2.6 (Very weak solutions and the energy equality).
Let $U\in L^{2}(T_{1},T_{2};L^{2}(\Omega))$ be a very weak solution to the
Stokes problem (2.1) in the sense of Definition 2.1, with $a=b=0$ and $F\in
L^{2}(\Omega\times(T_{1},T_{2});\mathbb{R}^{3\times 3})$. Applying [25,
Theorem 2.3.1 and Theorem 2.4.1, IV], together with Lemma 2.5, gives that
$U\in C([T_{1},T_{2}];L^{2}_{\sigma}(\Omega))\cap
L^{2}(T_{1},T_{2};W^{1,2}_{0,\sigma}(\Omega))$ with
$\|U(\cdot,t)\|_{L^{2}(\Omega)}^{2}+2\int\limits_{T_{1}}^{t}\int\limits_{\Omega}|\nabla
U|^{2}dxds=-2\int\limits_{T_{1}}^{t}\int\limits_{\Omega}F:\nabla
Udxds\quad\forall t\in[T_{1},T_{2}].$
This can also be used to show that very weak solutions to the Navier-Stokes
equations (with $a=0$) satisfy the energy equality. For results in this
direction, see [13].
## 3\. Proof of the localized weak-strong uniqueness results
### 3.1. Proof of Theorem A
Step 1: existence of a ‘strong’ solution to the Navier-Stokes system777After
completion of this paper, we became aware that in [9] the existence of a
‘strong solution’ with small non-zero boundary data is also shown. To make
this paper self-contained, we include such arguments in Step 1 below. in
$L^{4}(0,T;L^{6}(\Omega))$.
Step 1-a: linear problem.
Our goal is to lift the boundary data $a$ and the initial data $b$ satisfying
(1.2) and (1.3) by constructing a very weak solution $\bar{U}$ to the Stokes
problem (2.1) in $\Omega\times(0,T)$ with source $F=0$. By Theorem 2.2 of
Farwig, Kozono and Sohr, with $s=r=4$ and $q=6$, there exists a unique very
weak solution $U_{a}\in L^{4}(0,T;L^{6}(\Omega))$ in the sense of Definition
2.1 to the Stokes system with $F=b=0$. Hence we construct $\bar{U}$ the unique
very weak solution in the sense of Definition 2.1 to the Stokes system with
$F=0$ as follows:
$\bar{U}=U_{a}+e^{-t\mathbf{A}}b.$
Thanks to the semigroup estimates of [16], we have for $t\in(0,\infty)$,
$\|e^{-t\mathbf{A}}b\|_{L^{6}(\Omega)}\leq
C_{\Omega}t^{-\frac{1}{8}}\|b\|_{L^{4}(\Omega)}.$
Hence,
$\|\bar{U}\|_{L^{4}(0,T;L^{6}(\Omega))}\leq
K_{\Omega,T}\big{(}\|a\|_{L^{4}(\partial\Omega\times(t_{0},0))}+\|b\|_{L^{4}(\Omega)}\big{)},$
(3.1)
for $K_{\Omega,T}\in(0,\infty)$.
Step 1-b: existence of a unique strong solution $V$ to the perturbed system.
In this step, we construct a mild solution to the perturbed Navier-Stokes
system
$\displaystyle\begin{split}\partial_{t}V-\Delta V+\nabla
Q&=-\nabla\cdot(V\otimes
V)-\nabla\cdot(V\otimes\bar{U})-\nabla\cdot(\bar{U}\otimes
V)-\nabla\cdot(\bar{U}\otimes\bar{U}),\\\ \nabla\cdot
V&=0\qquad\qquad\qquad\qquad\qquad\qquad\mbox{in}\quad\Omega\times(0,T),\\\
V&=0\qquad\mbox{on}\quad\partial\Omega\times(0,T),\\\
V(\cdot,0)&=0\qquad\mbox{on}\quad\Omega.\end{split}$ (3.2)
for a small drift $\bar{U}$. We show that the Duhamel equation associated to
(3.2)
$V(\cdot,t)=-\int\limits_{0}^{t}e^{-(t-s)\mathbf{A}}\left(\nabla\cdot(V\otimes
V)+\nabla\cdot(V\otimes\bar{U})+\nabla\cdot(\bar{U}\otimes
V)+\nabla\cdot(\bar{U}\otimes\bar{U})\right)\,ds$ (3.3)
has a unique fixed point in the critical space $L^{4}(0,T;L^{6}(\Omega))$.
For $t\in(0,T)$, define
$B(D,E)(\cdot,t):=-\int\limits_{0}^{t}e^{-(t-s)\mathbf{A}}\nabla\cdot(D\otimes
E)\,ds.$
Using [17, Proposition 20], we see that
$\left\|e^{-(t-s)\mathbf{A}}\nabla\cdot(D\otimes
E)\right\|_{L^{6}(\Omega)}\leq\frac{C}{(t-s)^{\frac{3}{4}}}\|D(\cdot,t)\|_{L^{6}(\Omega)}\|E(\cdot,t)\|_{L^{6}(\Omega)},$
and hence, by Hardy-Littlewood-Sobolev’s theorem [15, Theorem 7.25],
$\left\|B(D,E)\right\|_{L^{4}(0,T;L^{6}(\Omega))}\leq
C_{0}\|D\|_{L^{4}(0,T;L^{6}(\Omega))}\|E\|_{L^{4}(0,T;L^{6}(\Omega))}.$ (3.4)
Define the linear operator
$L(D)(\cdot,t):=-\int\limits_{0}^{t}e^{-(t-s)\mathbf{A}}\nabla\cdot(D\otimes\bar{U}+\bar{U}\otimes
D)\,ds.$
Then by the same reasoning as above
$\left\|L(D)\right\|_{L^{4}(0,T;L^{6}(\Omega))}\leq
C_{1}\|D\|_{L^{4}(0,T;L^{6}(\Omega))}\|\bar{U}\|_{L^{4}(0,T;L^{6}(\Omega))},$
(3.5)
and for the source term quadratic in $\bar{U}$,
$\left\|B(\bar{U},\bar{U})\right\|_{L^{4}(0,T;L^{6}(\Omega))}\leq
C_{0}\|\bar{U}\|_{L^{4}(0,T;L^{6}(\Omega))}^{2}.$ (3.6)
Let
$C_{2}:=\min\Big{(}\frac{1}{C_{0}},\frac{1}{C_{1}}\Big{)}.$ (3.7)
Using (3.4)-(3.6), we can apply [14, Lemma 4.1]. This gives the existence of a
fixed point/strong solution $V\in L^{4}(0,T;L^{6}(\Omega))$ provided that
$\|\bar{U}\|_{L^{4}(0,T;L^{6}(\Omega))}<\frac{C_{2}}{4}.$ (3.8)
Moreover,
$\|V\|_{L^{4}(0,T;L^{6}(\Omega))}\leq
4C_{0}\|\bar{U}\|^{2}_{L^{4}(0,T;L^{6}(\Omega))}.$
In view of the linear estimate (3.1), this is achieved whenever
$\displaystyle\|\bar{U}\|_{L^{4}(0,T;L^{6}(\Omega))}\leq\ $ $\displaystyle
K_{\Omega,T}\big{(}\|a\|_{L^{4}(0,T;L^{4}(\Omega))}+\|b\|_{L^{4}(\Omega)}\big{)}$
$\displaystyle=\ $ $\displaystyle K_{\Omega,T}\kappa$ $\displaystyle<\ $
$\displaystyle K_{\Omega,T}\bar{\kappa}=\frac{C_{2}}{4},$
i.e.
$\bar{\kappa}:=\frac{C_{2}}{4K_{\Omega,T}}.$ (3.9)
Therefore, there is a strong solution such that
$\|V\|_{L^{4}(0,T;L^{6}(\Omega))}<4C_{0}K_{\Omega,T}^{2}\kappa^{2}=C_{0}C_{2}K_{\Omega,T}\kappa.$
(3.10)
The fact that $V$ is the only strong solution in $L^{4}(0,t;L^{6}(\Omega))$
follows from (3.4) and arguments in [6, Theorem (3.3)].
Step 2: weak-strong uniqueness result.
Let $W:=U-\bar{U}\in L^{4}(\Omega\times(0,T))$. Let $V\in
L^{4}(0,T;L^{6}(\Omega))$ be the strong solution constructed in Step 1-b
above. Then $V-W$ is a very weak solution to the Stokes system with $a=b=0$
and $F:=U\otimes U-(V+\bar{U})\otimes(V+\bar{U})\in L^{2}(\Omega\times(0,T))$.
Applying Remark 2.6 gives that $V-W$ has finite energy on $\Omega\times(0,T)$
with zero initial data and satisfies the energy equality
$\frac{1}{2}\int\limits_{\Omega}|V-W|^{2}(x,t)dx+\int\limits_{0}^{t}\int\limits_{\Omega}|\nabla(V-W)|^{2}dxds\\\
=\int\limits_{0}^{t}\int\limits_{\Omega}(V-W)\otimes(V+\bar{U}):\nabla(V-W)dxds\quad\forall
t\in[0,T).$
For $t\in[0,T)$, we define
$\displaystyle\mathcal{E}(t):=\sup_{s\in[0,t]}\frac{1}{2}\int\limits_{\Omega}|V-W|^{2}(x,s)dx+\int\limits_{0}^{t}\int\limits_{\Omega}|\nabla(V-W)|^{2}(x,s)dxds.$
Using the above energy equality (combined with Hölder’s inequality,
interpolation of Lebesgue spaces, the Sobolev embedding theorem and Young’s
inequality) yields that for $t\in[0,T)$ and some positive universal constant
$C$,
$\mathcal{E}(t)\leq
C\mathcal{E}(t)\int\limits_{0}^{t}\|(V+\bar{U})(\cdot,s)\|_{L^{6}(\Omega)}^{4}ds.$
Using this, together with the fact that $V+\bar{U}$ is in
$L^{4}(0,T;L^{6}(\Omega))$, allows us to use an absorbing argument to conclude
that $\mathcal{E}(t)=0$ for all $t\in(0,T)$. This concludes the proof of
Theorem A and of the structure result mentioned in Remark 1.1. Notice that the
quantitative estimate (1.5) directly follows from the linear estimate (3.1),
the estimate (3.10) for the mild solution and the definition of $\bar{\kappa}$
in (3.9).
### 3.2. Proof of Theorem A’
The proof of this result only differs from the proof of Theorem A in the
treatment of the linear evolution of the initial data $b$. Let us outline the
changes, which concern only Step 1-a of Section 3.1.
We take $\varphi\in C^{\infty}_{c}(B_{\frac{15}{16}})$ a cut-off function such
that $\varphi\equiv 1$ on $B_{\frac{7}{8}}$, $0\leq\varphi\leq 1$ and
$|\nabla\varphi|\leq 32$. Thanks to the Bogovskii operator [12] on the annulus
$B_{\frac{15}{16}}\setminus B_{\frac{7}{8}}$, there exists a divergence-free
extension $E(b)$ of $\varphi b$ defined on $\mathbb{R}^{3}$ that is compactly
supported in $B_{\frac{15}{16}}$, $E(b)=b$ on $B_{\frac{7}{8}}\subset
B_{r_{0}}$ and such that
$\|E(b)\|_{L^{4}(B_{1})}\leq C\|b\|_{L^{4}(B_{1})}.$
Let $\Gamma$ be the heat kernel on $\mathbb{R}^{3}$. We note that
$\Gamma(\cdot-t_{0})\star E(b)$ is divergence-free, which implies
$\int\limits_{\partial B_{r_{0}}}\big{(}\Gamma(\cdot-t_{0})\star
E(b)\big{)}\cdot n=0.$
Moreover, it follows from [7, Lemma IV.3.2] that
$\big{\|}\Gamma(\cdot-t_{0})\star E(b)\big{\|}_{L^{4}(\partial
B_{r_{0}}\times(t_{0},0))}\leq
C\big{(}(-t_{0})^{\frac{1}{8}}+(-t_{0})^{\frac{1}{4}}\big{)}\|E(b)\|_{L^{4}(\mathbb{R}^{3})}\leq
C\|b\|_{L^{4}(B_{1})},$ (3.11)
and
$\big{\|}\Gamma(\cdot-t_{0})\star
E(b)\big{\|}_{L^{4}(t_{0},0;L^{6}(B_{r_{0}}))}\leq
C(-t_{0})^{\frac{1}{8}}\|E(b)\|_{L^{4}(\mathbb{R}^{3})}\leq
C\|b\|_{L^{4}(B_{1})},$ (3.12)
where $C\in(0,\infty)$ denotes as usual a universal constant. We can now
construct the linear solution $\bar{U}$ as follows. Let
$\widetilde{a}=a-\Gamma(\cdot-t_{0})\star E(b)|_{\partial B_{r_{0}}}.$
By Theorem 2.2 of Farwig, Kozono and Sohr [10], with $s=r=4$ and $q=6$, there
exists a unique very weak solution $U_{\widetilde{a}}\in
L^{4}(t_{0},0;L^{6}(B_{r_{0}}))$ in the sense of Definition 2.1 to the Stokes
system with $F=b=0$ and boundary data $\widetilde{a}$. Hence we construct
$\bar{U}$ the unique very weak solution in the sense of Definition 2.1 to the
Stokes system with $F=0$ as follows:
$\bar{U}=U_{\widetilde{a}}+\Gamma(\cdot-t_{0})\star E(b).$
Hence, the estimates (2.3) and (3.12) lead to
$\|\bar{U}\|_{L^{4}(t_{0},0;L^{6}(B_{r_{0}}))}\leq
K\big{(}\|a\|_{L^{4}(\partial
B_{r_{0}}\times(t_{0},0))}+\|b\|_{L^{4}(B_{r_{0}})}\big{)},$ (3.13)
where $K\in(0,\infty)$ denotes a universal constant.
The rest of the proof, Step 1-b to Step 2 of Section 3.1 are identical,
replacing the constant $K_{\Omega,T}$ by $K$. The definition of $\bar{\kappa}$
becomes
$\bar{\kappa}:=\frac{C_{2}}{4K}.$ (3.14)
###### Remark 3.1 (on the compatibility conditions).
We emphasize that in the compatibility conditions (1.3), the condition888We
stress that this condition on $a$ comes from the linear result Theorem 2.2
taken from [10]; these conditions are also present in the work [7]. on the
boundary data $a$ is much weaker than the condition on $b$. Indeed we only ask
that $a\cdot n$ has mean zero on $\partial\Omega$, while $b\cdot n$ is
required to vanish identically on $\partial\Omega$. This fact is the essential
redeeming feature that allows the above argument to work. Notice that owing to
the fact that $\Gamma(\cdot-t_{0})\star E(b)$ is divergence-free,
$\big{(}\Gamma(\cdot-t_{0})\star E(b)\big{)}\cdot n$ has mean zero on
$\partial B_{r_{0}}$ but is not necessarily zero identically.
## 4\. Proof of the epsilon regularity result
This section is devoted to the proof of Theorem B. We directly prove the
quantitative version, i.e. the bound (1.15). We assume that $U$ is a finite-
energy weak solution to the Navier-Stokes equations, i.e. (1.11) and (1.12)
hold. In addition, we assume that $U$ belongs to
$C^{\infty}(B_{1}\times(-1,T))$ for all $T\in(-1,0)$, see Footnote 5, and
satisfies the assumption (1.14) with $0<\varepsilon\leq\bar{\varepsilon}$ and
$\bar{\varepsilon}$ defined in (4.4).
Step 1: finding good space and time scales.
We need to select a space slice and a time slice. The choices are completely
independent, so the order in which we select the slices does not matter. Let
us first select a space slice $r_{0}\in(\frac{5}{8},\frac{7}{8})$. By the
coarea formula and Fubini’s theorem, we have
$\int\limits_{-1}^{0}\int\limits_{B_{1}}|U|^{4}\,dxdt=\int\limits_{-1}^{0}\int\limits_{0}^{1}\int\limits_{\partial
B_{r}}|U|^{4}\,dS_{r}drdt=\int\limits_{0}^{1}\int\limits_{-1}^{0}\int\limits_{\partial
B_{r}}|U|^{4}\,dS_{r}dtdr.$
Therefore, the pigeonhole principle implies that there exists
$r_{0}\in(\frac{5}{8},\frac{7}{8})$ such that
$\int\limits_{-1}^{0}\int\limits_{\partial B_{r_{0}}}|U|^{4}\,dS_{r_{0}}dt\leq
4\int\limits_{-1}^{0}\int\limits_{B_{1}}|U|^{4}\,dxdt.$ (4.1)
We now select a time slice. By the pigeonhole principle, there exists
$t_{0}\in(-1,-\frac{3}{4})$ such that
$\int\limits_{B_{1}}|U|^{4}(x,t_{0})\,dx\leq
4\int\limits_{-1}^{0}\int\limits_{B_{1}}|U|^{4}\,dxdt.$ (4.2)
From now on, we call $a:=U|_{\partial B_{r_{0}}\times(-1,0)}$ and
$b:=U(\cdot,t_{0})$. By (4.1) we have $a\in L^{4}(\partial
B_{r_{0}}\times(-1,0))$ and by (4.2) we have $b\in L^{4}(B_{1})$. Moreover,
$\|a\|_{L^{4}(\partial B_{r_{0}}\times(t_{0},0))}+\|b\|_{L^{4}(B_{1})}\leq
2\sqrt{2}\|U\|_{L^{4}(Q_{1})}\leq
2\sqrt{2}\varepsilon<2\sqrt{2}\bar{\varepsilon}\leq\bar{\kappa},$
with $\bar{\kappa}$ defined by (3.14), on condition that
$\bar{\varepsilon}\leq\frac{\bar{\kappa}}{2\sqrt{2}}$.
Step 2: applying weak-strong uniqueness.
We now apply the weak-strong uniqueness result of Theorem A’ to the solution
$U$ on $B_{r_{0}}\times(t_{0},0)$ with the data $a$ and $b$ from Step 1 above.
By the quantitative estimate (1.5) it follows that
$\|U\|_{L^{4}(t_{0},0;L^{6}(B_{r_{0}}))}\leq 2\sqrt{2}\varepsilon
K(1+C_{0}C_{2}).$ (4.3)
Step 3: conclusion via Ladyženskaja-Prodi-Serrin.
We can now argue in a similar spirit to [23, 2], except we use the bound (4.3)
to set up a contraction mapping999For such a contraction mapping, see for
instance [2, Lemma 12]. related to the localized vorticity equation rather
than the velocity. In particular, there exists $\bar{\eta}$ such that for any
$0<\varepsilon<\frac{\bar{\eta}}{2\sqrt{2}K(1+C_{0}C_{2})}=:\bar{\varepsilon}$
(4.4)
we have
$\|U\|_{L^{6}(Q_{\frac{1}{2}})}\leq M(1+\varepsilon).$
With this, we can bootstrap in the same way as in [24] to obtain the estimate
(1.15). This concludes the proof of Theorem B.
###### Remark 4.1 (on the linear solution).
In this proof, notice that we are not able to assert that the solution
$\bar{U}$ to the linear Stokes problem with rough boundary data $a\in
L^{4}(B_{r_{0}}\times(t_{0},0))$ (see the structure result in Remark 1.1) is
smooth in space. However, the linear result is pivotal in order to establish
that the original solution $U$ has improved critical integrability, hence is
smooth in space. Notice that we obtain $L^{\infty}$ time integrability by
appealing to the $L^{\infty}_{t}L^{2}_{x}$ control granted by being finite-
energy. We cannot bootstrap further in time without controlling the pressure
$p$ up to time 0.
###### Remark 4.2 (on the half-space).
It remains an open problem as to whether such a proof can be done for
establishing epsilon regularity results near a boundary. Indeed, the linear
results of Farwig, Kozono and Sohr [10] and of Fabes, Lewis and Rivière [7]
ask for smoothness of the domain $\Omega$, see for instance Theorem 2.2 above.
However, we are unable to carry out the slicing procedure of Step 1 above near
a smooth boundary.
###### Remark 4.3 (comparison with Struwe [26]).
Struwe’s approach to the five-dimensional stationary Navier-Stokes equations
is technically different; it involves iterating the energy on a sequence of
balls, which we avoid. A version of our proof should work in the stationary
setting of [26] as well. There, the smallness condition in $L^{4}$ could be
replaced by smallness in $H^{1}$, since in five dimensions, the slicing
procedure yields boundary data in $H^{1}(\partial B_{r_{0}})\subset
L^{4}(\partial B_{r_{0}})$ (four-dimensional Sobolev embedding). The slicing
technique has proven useful for the six-dimensional stationary Navier-Stokes
equations [11] and advection-diffusion equation with rough drifts [1], among
others.
### Acknowledgement
DA was supported by NSF Postdoctoral Fellowship Grant No. 2002023. DA is also
grateful to ENS Paris for supporting his academic visit to Paris during which
this research was initiated. DA thanks Vladimír Šverák, who contributed an
offhanded comment in 2017 which played a role in the genesis of this paper. CP
is partially supported by the Agence Nationale de la Recherche, project BORDS,
grant ANR-16-CE40-0027-01, project SINGFLOWS, grant ANR- 18-CE40-0027-01,
project CRISIS, grant ANR-20-CE40-0020-01, by the CY Initiative of Excellence,
project CYNA (CY Nonlinear Analysis) and project CYFI (CYngular Fluids and
Interfaces).
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|
On the particle collisions during gravitational collapse of Vaidya spacetimes
Vitalii Vertogradov
Physics department, Herzen state Pedagogical University of Russia,
48 Moika Emb., Saint Petersburg 191186, Russia
SPb branch of SAO RAS, 65 Pulkovskoe Rd, Saint Petersburg 196140, Russia
<EMAIL_ADDRESS>
Summary: The center-of-mass energy can be arbitrarily high in Schwarzschild
spacetime if one considers the front collision of two particles, one of which
moves along so-called white hole geodesics and another one along a black hole
geodesic. This process can take place if one considers the gravitational
collapse model. In this paper, we consider the well-known naked singularity
formation in Vaidya spacetime and investigate the question about two particle
collision near the boundary of the collapsing cloud. The center-of-mass energy
of the front collision is considered. One particle moves away from the naked
singularity and another one falls onto a collapsing cloud. We show that the
center-of-mass energy grows unboundly if the collision takes place in the
vicinity of the conformal Killing horizon.
Key words: Gravitational collapse, Particles collision, Vaidya spacetime,
Naked singularity, Conformal symmetry
## introduction
The center-of-mass energy of two particle collision can grow unboundly in the
Kerr spacetime [1] if one of the particles is fine-tunned (so-called critical
particle [2]). This effect was firstly proposed by Ba n ados, Silk and West
and is called BSW effect. Original version of this effect declares absence of
the unbound energies in Schwarzschild and Reissner-Nordstrom spacetimes.
However, it was shown that this effect is possible in Reissner-Nordstrom-anti-
de Sitter spacetime [3]. Despite the unbound the center-of-mass energy, a
distant observer will measure small amount of the energy due to this process
in the Kerr spacetime [4] and an escaping particle will be able to carry away
arbitrarily large amount of energy in. Reissner-Nordstrom case
[bib:zaslavextract].
In spite the fact that BSW effect is absent in Schwarzschild spacetime, one
can still obtain the unbound center-of-mass energy of two colliding particles
[6]. Due to geodesic completeness, there must be geodesics which appear in our
Universe from the region inside the gravitational radius i.e. so-called white
hole geodesics. For example, geodesics for particles with negative energy in
the Kerr metric are such geodesics [7, 8]. One can imagine the following
situation: the first particle moves along the white hole geodesic away from
the gravitational radius and the second particle moves along black hole
geodesics falling onto a black hole. As the result, one can observe the front
collision in the vicinity of the event horizon and due to this process the
center-of-mass energy can grow unboundly. The problem is that the
Schwarzschild black hole is an eternal one and if one follows the geodesic
back then it must appear from the collapsing cloud. So to understand the front
collision in the vicinity of the event horizon in Schwarzschild spacetime, one
must consider the gravitational collapse problem. The nature of white hole
geodesics can be explained by the naked singularity formation due to
gravitational collapse problem [9]. The outcome of the gravitational collapse
might be not only a black hole but also a naked singularity [10, 11, 12]. The
naked singularity formation in Vaidya spacetime has been considered in [13].
The gravitational collapse of the generalized Vaidya spacetime and the naked
singularity formation has been investigated in [14, 15, 16, 17]. In the case
of the eternal naked singularity formation in Vaidya spacetime [15, 18] the
unbound center-of-mass energy is possible only in the vicinity of the
singularity.
In this paper, we consider the following model: a particle moves along the
non-spacelike, future-directed geodesic which terminates at the naked
singularity in the past. When the apparent horizon forms, the particle is in
the vicinity of the apparent horizon and outside it. At this time, the second
particle, moving along a black hole geodesic, falls onto a black hole. As the
result we have the front collision of two particles. We estimate the center-
of-mass energy of this process and find out where this process should take
place to have an unbound energy collision.
This paper is organized as follows: in sec. II we consider a well-known
gravitational collapse model of Vaidya spacetime and show the naked
singularity formation. We also show that the geodesics can originate at this
singularity. In sec. III we introduce the coordinate transformation and
consider Vaidya spacetime in conformally static coordinates. In this case we
investigate the center-of-mass energy of the two particle front collision.
Sec. IV is the conclusion.
The system of units $G=c=1$ will be used throughout the paper. We use the
signature $-\,,+\,,+\,,+$.
## The naked singularity formation in Vaidya spacetime
The Vaidya metric [19] describes a dynamical spacetime instead ofa static
spacetime as the Schwarzschild or Reissner-Nordstrom metrics do. In the real
world, astronomical bodies gain mass when theyabsorb radiation and they lose
mass when they emit radiation, which means that the space-time around them is
time-dependent. Papapetroo [20] showed that Vaidya spacetime violates the
cosmic censorship conjecture and contains the naked singularity. The line
element in Eddington-Finkelstein coordinates has the following form:
$ds^{2}=-\left(1-\frac{2M(v)}{r}\right)dv^{2}+2dvdr+r^{2}d\omega^{2}\,.$ (2.1)
Here $M(v)$ the time-depended mass of a black hole,
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is the metric on unit
sphere.
The apparent horizon equation is given by [21]:
$r_{ah}=2M(v)\,.$ (2.2)
The first shell collapses at $r=0$ at the time $v=0$ and the singularity forms
at this time. The singularity is naked if at the time of the singularity
formation $v=0$ the apparent horizon doesn’t form and there is a family of
non-spacelike, future-directed geodesics which terminate at the central
singularity in the past. Let’s prove the last statement. For this purpose, we
define the mass function as:
$M(v)=\mu v\,,\mu>0\,.$ (2.3)
Here $\mu$ is a positive constant. Here, we just show that naked singularity
is possible. The thorough investigation of this model one can find at [13].
To prove the existence of a family of non-spacelike, future-directed geodesics
which terminate at the central singularity in the past, one should consider
the null radial geodesic, which, for the metric (2.1) with the mass condition
(2.3) has the following form:
$\frac{dv}{dr}=\frac{2r}{r-2\mu v}\,.$ (2.4)
The solution $v=const.$ doesn’t suit us because of infalling matter $v=const.$
corresponds to ingoing geodesics and we are interested in outgoing ones. So,
(2.4) corresponds to outgoing geodesic if the following condition is held:
$\lim\limits_{v\rightarrow 0\,,r\rightarrow 0}\frac{dv}{dr}=X_{0}>0\,.$ (2.5)
If the value $X_{0}$ is positive and finite than the geodesic (2.4) is
outgoing one. Let’s consider the limit in (2.4):
$X_{0}=\frac{2}{1-2\mu X_{0}}\,.$ (2.6)
From this equation we obtain:
$X^{\pm}_{0}=\frac{1\pm\sqrt{1-16\mu}}{4\mu}\,.$ (2.7)
From this equation, one can see that in the case of the linear mass function
and if $\mu<\frac{1}{16}$ then the outcome of the gravitational collapse might
be the naked singularity formation. Hence, there might be particles which move
away the singularity and now we are ready to consider the front collision of
two particles.
## The front collision Effect in Vaidya spacetime
The Vaidya spacetime (2.1) is time-depended and because of it one has only one
conserved quantity - the angular momentum $L$. In the general case, the Vaidya
spacetime doesn’t possess any additional symmetry. However, for the particular
choice of the mass function, the metric (2.1) admits the conformal Killing
vector [bib:mahconformal]. In this case $M$ must have the following form [23]:
$M(v)=\mu v\,,\mu>0\,.$ (3.8)
Where $\mu$ is the positive constant. As we found out in the previous section
if $\mu<\frac{1}{16}$, the gravitational collapse might end with the naked
singularity formation. Further, in the paper, we impose the condition
$\mu<\frac{1}{16}$ because we are interested in temporal naked singularity
formation. If we take into account this condition and (3.8) then by coordinate
transformation [24]:
$\begin{split}v=r_{0}e^{\frac{t}{r_{0}}}\,,\\\
r=Re^{\frac{t}{r_{0}}}\,.\end{split}$ (3.9)
one obtains Vaidya metric in conformally static coordinates:
$ds^{2}=e^{\frac{2t}{r_{0}}}\left[-\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)dt^{2}+2dtdR+R^{2}d\Omega^{2}\right]\,.$
(3.10)
We will consider the movement in the equatorial plane $\theta=\frac{\pi}{2}$.
The metric (3.10) admits the conformal Killing vector $\frac{d}{dt}$, which is
timelike in the region:
$1-\frac{2\mu r_{0}}{R}-\frac{2R}{r_{0}}>0\,.$ (3.11)
It means that one has the conserved energy in the region (3.11) i.e.:
$E=e^{\frac{2t}{r_{0}}}\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)\frac{dt}{d\lambda}-e^{\frac{2t}{r_{0}}}\frac{dR}{d\lambda}\,.$
(3.12)
The angular momentum $L$ has the following form:
$L=e^{\frac{2t}{r_{0}}}R^{2}\frac{d\varphi}{d\lambda}\,.$ (3.13)
To find the $\frac{dR}{d\lambda}$ component of the four velocity $u^{i}$, one
should substitute (3.12) and (3.13) into the timelike condition
$g_{ik}u^{i}u^{k}=-1$. One obtains:
$e^{\frac{4t}{r_{0}}}\left(\frac{dR}{d\lambda}\right)^{2}=E^{2}-e^{\frac{2t}{r_{0}}}\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)\left(\frac{L^{2}}{r^{2}e^{\frac{2t}{r_{0}}}}+1\right)=e^{\frac{4t}{r_{0}}}P^{2}_{R}\,.$
(3.14)
Where $P_{R}=P_{R}(R,t)$ is some positive function. According to BSW effect
[1], the energy of the centre of mass $E_{c.m.}$ of two colliding particles
for the extremal Kerr black hole can grow unboundly. For this purpose, one of
the particles must be critical one [2]. In Schwarzschild spacetime, the energy
$E_{c.m.}$ is finite according to the original proposal. However, if we
consider, in Schwarzschild spacetime, the collision of two particles one of
which moves along white hole geodesic from the gravitational radius and
another one moves along a black hole geodesic and fall onto a black hole [6]
then the energy $E_{c.m.}$ of the collision can be unbound. The problem is
that the Schwarzschild metric describes the eternal black hole and the white
hole geodesic appears from the region outside the white hole gravitational
radius in past infinity. However, if one considers the physically relevant
model, then prolonging the white hole geodesic into the past, one can see that
it appears from the collapsing cloud of the matter. So to understand this
analogy of the BSW effect one, first of all, should consider the gravitational
collapse model which, in the case of Vaidya spacetime, has been done in the
previous section. We have proven that if $\mu<\frac{1}{16}$ the gravitational
collapse might end with the naked singularity formation. For our model it
means that there is a family of non-spacelike future-directed geodesics which
terminate at the central singularity in the past. Let’s consider the following
situation: the particle $1$ moves along such geodesic and when the apparent
horizon forms, this particle is in the vicinity of this horizon and outside
it. At this time the particle $2$, which falls onto a black hole, collides
with the particle $1$. Let’s calculate if the unbounded energy $E_{c.m.}$ of
the collision is possible and where this collision should take place.
For simplisity, let’s consider the collision of two particles with same mass
$m_{0}$. In this case, the energy $E_{c.m.}$ is given by:
$E_{c.m.}=m_{0}\sqrt{2}\sqrt{1-g_{ik}u^{i}_{1}u^{k}_{2}}\,.$ (3.15)
Where $u^{i}_{1}$ and $u^{i}_{2}$ are the four velocities of the particles $1$
and $2$ respectivelly. Substituting (3.12), (3.13) and (3.14) into (3.15), one
obtains:
$\frac{E_{c.m.}^{2}}{2m_{0}}=1+\frac{E_{1}\left(E_{2}+e^{\frac{2t}{r_{0}}}P_{2R}\right)}{e^{\frac{2t}{r_{0}}}\left(1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}\right)}-\frac{e^{\frac{2t}{r_{0}}}\left(E_{1}+e^{\frac{2t}{r_{0}}}P_{1R}\right)P_{2R}}{1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}}-\frac{L_{1}L_{2}}{e^{\frac{2t}{r_{0}}}R^{2}}\,.$
(3.16)
Note that only the second and third terms in the right-hand side can give us
the unbound energy $E_{c.m.}$. It is possible if:
$1-\frac{2\mu r_{0}}{R_{kh}}-\frac{2R_{kh}}{r_{0}}=0\,.$ (3.17)
Where $R=R_{kh}$ is the location of the conformal Killing horizon. Also one
should note that for outgoing particle $1$ \- $P_{1R}>0$, for ingoing particle
$2$ \- $P_{2R}<0$. So we conclude that both considered terms are positive if
we consider the region (3.11) of timelike conformal Killing vector
$\frac{d}{dt}$ and we should prove that one of them grows unboundly when
$R\rightarrow R_{kh}$. To proceed, we note that:
$\begin{split}\lim\limits_{R\rightarrow
R_{kh}}e^{\frac{2t}{r_{0}}}P_{1R}=+E\,,\\\ \lim\limits_{R\rightarrow
R_{kh}}e^{\frac{2t}{r_{0}}}P_{2R}=-E\,.\end{split}$ (3.18)
One should note that off-diagonal term in the metric (3.10) might indicate
that there are particles with negative energy. However, It was shown [25] that
there are not particles with negative energy outside the apparent horizon and
$E\geq 0$. Using this fact and by taking limit $R\rightarrow R_{kh}$ one can
see that the second term in the right-hand side of the (3.16) gives us
uncertainty $0/0$ and we won’t consider it because if it is finite then we can
neglect it. If it is infinite, then we obtain the unbound $E_{c.m.}$. However,
we focus our attention on the third term in the right-hand side of (3.16):
$\lim\limits_{R\rightarrow
R_{kh}}-\frac{e^{\frac{2t}{r_{0}}}\left(E_{1}+e^{\frac{2t}{r_{0}}}P_{1R}\right)P_{2R}}{1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}}=\frac{e^{2t}{r_{0}}2E_{1}E_{2}}{1-\frac{2\mu
r_{0}}{R}-\frac{2R}{r_{0}}}\rightarrow+\infty\,.$ (3.19)
And we can see that this term (3.19) gives us the unbound energy $E_{c.m.}$ if
the collision takes place in the vicinity of the conformal Killing horizon.
## Conclusion
In this paper, we have considered the front collision of two particles in the
Vaidya spacetime. The metric (2.1) is time-depended and to consider the
center-of-mass energy, one needs to introduce new coordinates which allow us
to write the Vaidya spacetime in conformally static form. It allows us to
consider the following model: in the case of the linear mass function, the
gravitational collapse might end up with the naked singularity. We consider
the non-spacelike geodesic which originates at this naked singularity.
Further, we assume that there are particles which move along this geodesic
away from the central singularity. Then, the other particle falls onto a
collapsing cloud and the frontal collisions of the two particles is considered
at the time of the apparent horizon formation. It means that the apparent
horizon forms and the particle, moving along a naked singularity geodesic,
finds itself outside the trapped region, in the vicinity of the apparent
horizon. We showed that the unbound center-of-mass energy is possible if the
collision takes place in the vicinity of the conformal Killing horizon. Note
that if we pick up the mass function as $M(v)=\mu v^{n}\,,n>1$ then, of
course, one has the naked singularity formation [26] but the singularity is
gravitationally weak according to the Tipler definition [27, 28] and the
spacetime doesn’t admit the conformal Killing vector anymore.
The unbound center-of-mass energy of two colliding particles near the
conformal Killing horizon is expected result. One should use this horizon to
define most physically relevant quantities. In static spacetimes, for example,
one uses the Killing horizon to define the surface gravity which is associated
with the Hawking temperature. The Killing horizon coincides with the event
horizon in the case of Schwarzschild and non-extremal Reissner-Nordstrom black
holes. However, in the dynamical case it is not easy task to define the
surface gravity [29]. One can define the surface gravity on the apparent
horizon but, according to Nielsen [30] the apparent horizon in Vaidya
spacetime is hidden inside the event horizon, although, the location of the
last one is also hard task in dynamical spacetimes.
The results, obtained in this paper, can be easily extended to generalized
Vaidya spacetime. The naked singularity formation and the mass function
conditions in this case for this metric has been proven in [14, 16, 17]. The
unbound center-of-mass is again expected if the front collision takes place in
the vicinity of the conformal Killing horizon. However, the conformal Killing
vector exists only for the following choice of the mass function:
$M(v,r)=\mu v+\nu v^{2\alpha}r^{1-2\alpha}\,,\mu>0\nu\neq
0\,,\alpha\neq\frac{1}{2}\,.$ (4.20)
Where $\alpha\in[-1\,,1]$ [31]. The generalized Vaidya spacetime admits
regular black hole solution [32], however, the question about the front
collision in this case is still open.
acknowledgments: The author says thanks to grant NUM. 22-22-00112 RSF for
financial support. The work was performed within the SAO RAS state assignment
in the part ”Conducting Fundamental Science Research”.
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# Physics Informed Neural Network for Dynamic Stress Prediction
Hamed Bolandia,b*, Gautam Sreekumarb, Xuyang Lia,b, Nizar Lajnefa, Vishnu
Boddetib
a Department of Civil and Environmental Engineering, Michigan State
University, East Lansing, MI 48824
b Department of Computer Science and Engineering, Michigan State University,
East Lansing, MI 48824
* Corresponding author, Email Address<EMAIL_ADDRESS>
###### Abstract
Structural failures are often caused by catastrophic events such as
earthquakes and winds. As a result, it is crucial to predict dynamic stress
distributions during highly disruptive events in real time. Currently
available high-fidelity methods, such as Finite Element Models (FEMs), suffer
from their inherent high complexity. Therefore, to reduce computational cost
while maintaining accuracy, a Physics Informed Neural Network (PINN), PINN-
Stress model, is proposed to predict the entire sequence of stress
distribution based on Finite Element simulations using a partial differential
equation (PDE) solver. Using automatic differentiation, we embed a PDE into a
deep neural network’s loss function to incorporate information from
measurements and PDEs. The PINN-Stress model can predict the sequence of
stress distribution in almost real-time and can generalize better than the
model without PINN.
###### keywords:
Physics Informed Neural Network, Stress Prediction, Finite Element Analysis,
Partial Differential Equation
††journal: Computer Methods in Applied Mechanics and Engineering
## 1 Introduction
A dynamic analysis is used to determine how a system will respond to general
time-dependent loads. Events such as earthquakes and explosions are typical
applications for dynamic analysis. These applications should be able to carry
out real-time analysis in the aftermath of a disaster or during extreme
disruptive events that require immediate corrections to avoid catastrophic
failures. Dynamic loading also can cause dramatic and damaging failures, which
can be avoided by evaluating the structure during the design phase. In
structural engineering, numerical analysis methods such as Finite Element
Analysis (FEA) are typically used to conduct dynamic stress analysis of
various structures and systems in which it might be hard to determine an
analytical solution. However, numerical methods such as FEA are
computationally prohibitive while being accurate. The current workflow for FEA
applications consists of: (i) modeling the geometry and its components, (ii)
specifying the material properties, boundary conditions, meshing, and loading,
(iii) dynamic analysis, which may be time-consuming based on the complexity of
the model. The complexity of this workflow and its time requirements make it
impractical for real-time applications.
Figure 1: Overview: Unlike FEM, PINN-Stress is computationally efficient,
facilitates real-time analysis and is generalizable. PINN-Stress use a
governing equation behind the equation of motion as a soft constraint in the
loss function to enforce the loss to minimize. The points with different
colors in observations correspond to the same nodes in the gusset plate.
Gusset plate image is taken from [1]
The recently introduced models [2, 3] were designed to predict static stress
distributions using deep neural network (DNN)-based methods in both intact and
damaged structural components. The primary limitations of the above data-
driven models are the incapability to produce physically consistent results
and the lack of generalizability to out-of-distribution scenarios. The concept
of physics-informed learning was introduced recently [4, 5, 6] to address the
computational cost of FEA and lack of generalizability to out-of-distribution
scenarios. There is special interest in Physics-informed Neural Networks
(PINNs), which incorporate partial differential equations (PDEs) into the
training loss function directly. However, their applications have primarily
been limited to non-engineering toy simulations. Working with engineering
problems such as those in structural engineering will require these models to
learn several factors of variation in addition to the physical equations
themselves, such as geometry. To overcome these issues, we propose a novel
model for dynamic stress prediction which is real-time and generalizable and
can therefore be used for stress prediction in seismic and explosions design.
We augment PINN with a novel neural architecture for predicting dynamic stress
distribution to achieve fast dynamic analysis and address deficiencies of
data-driven models. We model the stress distribution in gusset plates under
dynamic loading to demonstrate its utility. Gusset plates are one of the most
critical components in structural systems such as bridges and buildings. Since
gusset plates are designed for lateral loads such as earthquakes, wind, and
explosions, real-time dynamic models such as ours can help avoid catastrophic
failures. In practice, the outputted stress maps from our models can be used
by downstream applications for detecting anomalies such as cracks in the
plates. In other words, it can act as a precursor to existing vision-based
systems.
An overview of our approach is shown in Fig. 1. To summarize our
contributions, we introduce NeuroStress and PINN-Stress, two novel deep
learning models to learn dynamic stress distribution for complex geometries,
boundary conditions, and various load sequences. Loss function in PINN-Stress
uses traditional MAE loss for training. PINN-Stress uses the physics-informed
loss function described in Section 3.1. For real-life use, our models require
input from sensors placed on the plates. But since it is difficult to obtain
such data for research purposes, we generate challenging synthetic data
emulating dynamic stress prediction. Through extensive experiments on
simulated data, we show that:
1. 1.
NeuroStress and PINN-Stress can predict dynamic stress distribution with
complex geometries, boundary conditions and various load sequences faster than
traditional FEA solvers. Previous works only predict static stress
distribution;
2. 2.
NeuroStress and PINN-Stress can learn the temporal information in the data to
make accurate predictions;
3. 3.
Introducing novel spatiotemporal multiplexing to physics-informed learning and
showing its utility in dynamic stress prediction;
4. 4.
NeuroStress and PINN-Stress can predict von Mises stress distribution using
the von Mises equation. von Mises stress distribution is a primary diagnostic
tool to predict failure of a structure;
5. 5.
To the best of our knowledge, PINN-Stress is the first model that learns
governing equations behind that of motion in structures. We attribute the
generalization abilities of our architecture on unseen load sequences and
geometries to its loss function.
## 2 Related Works
Over the past few years, there has been a revolution in data-driven
applications in various engineering fields, including fluid dynamics [7, 8],
molecular dynamics simulation [9, 10] and material properties prediction [11,
12, 13, 14]. Recent studies have shown that convolutional neural networks
(CNN) and Long Term Short Memories (LSTM) can be used to build metamodels for
predicting time history responses. Modares et al. [15] studied composite
materials to identify the presence and type of structural damage using CNNs.
Nie et al. [16] developed a CNN-based method to predict the low-resolution
stress field in a 2D linear cantilever beam. Jiang et al. [17] developed a
conditional generative adversarial network for predicting low-resolution
static von Mises stress distribution in solid structures. Zhang et al. [18]
used LSTM to model nonlinear seismic responses of structures with large
plastic deformations. Do et al. [19] proposed a method for forecasting crack
propagation in risk assessment of engineering structures based on LSTM and
Multi-Layer Perceptron (MLP). Yao et al. [20] proposed a physics-guided
learning algorithm for predicting the mechanical response of materials and
structures. Das et al. [21] proposed a data-driven physics-informed method for
prognosis and applied it to predict cracking in a mortar cube specimen. Wang
et al. [22] proposed a hybrid DL model that unifies representation learning
and turbulence simulation techniques using physics-informed learning. Goswami
et al. [23] proposed a physics-informed variational formulation of DeepONet
for brittle fracture analysis. Raissi et al. [24] proposed a physics-informed
neural network that can solve supervised learning tasks while respecting any
given law of physics described by general nonlinear partial differential
equations. Haghighat et al. [25] presented physics-informed neural networks to
inversion and surrogate modeling in solid mechanics. Jin et al. [26]
investigated the ability of PINNs to directly simulate incompressible flows,
ranging from laminar to turbulent flows to turbulent channel flows. Li et al.
[27] used the Fourier transform to develop a Fourier neural operator to model
turbulent flows.
## 3 Background
To ensure that any component of an object is in equilibrium, the balance of
forces and moments acting on that component should be enforced. Stress
components acting on the face of the element can be written as equations of
equilibrium. The stress equilibrium equation can be written as a variation in
each stress term within the body since stress changes from point to point.
Considering a two-dimensional case in which stress acts in the horizontal and
vertical directions gives the following set of equations of motion:
$\frac{\partial\sigma_{xx}}{\partial x}+\frac{\partial\sigma_{xy}}{\partial
y}+b_{x}-\rho a_{x}=0$ (1)
$\frac{\partial\sigma_{yy}}{\partial y}+\frac{\partial\sigma_{xy}}{\partial
x}+b_{y}-\rho a_{y}=0$ (2)
where $\sigma_{xx}$, $\sigma_{yy}$ and $\sigma_{xy}$ denote normal stress in
horizontal and vertical directions, and shear stress respectively. $b_{x}$ and
$b_{y}$ represent body force in horizontal and vertical directions. $a_{x}$
and $a_{y}$ represent an acceleration in the horizontal and vertical
directions and $\rho$ denotes the density of the material.
### 3.1 von Mises equation
von Mises stress is a way of measuring whether a structure has begun to yield
at any point. To compare experimentally observed yield points with calculated
stresses, von Mises stress can be used mathematically as a scalar quantity. We
also predict von Mises stress since the engineering community relies heavily
on it. von Mises stress can be calculated from the predicted $\sigma_{xx}$,
$\sigma_{yy}$, and $\sigma_{xy}$ through the von Mises stress equation.
$\sigma_{vm}=\sqrt{\sigma_{xx}^{2}+\sigma_{yy}^{2}-\sigma_{xx}\sigma_{yy}+3\sigma_{yy}^{2}}$
(3)
## 4 Method
We introduce a novel architecture in this paper and augment it with a physics-
based loss function for gains in generalization.
### 4.1 Architecture
Firstly, we use a 2-layered MLP to encode the input to a larger dimensional
space. Then we introduce our spatiotemporal multiplexing (STM) module to
encode the spatial and the temporal information alternatively. We treat both
the temporal and the spatial dimensions as sequences, which may be modeled
using an appropriate deep neural architecture such as RNN, LSTM [28] or self-
attention [29]. LSTMs have demonstrated better performance than RNNs, but have
performed worse compared to self-attention. However, self-attention requires
plenty of data, which cannot be satisfied in our problem statement. Hence, as
a middle ground, we use LSTMs to model both temporal and spatial information.
Spatiotemporal multiplexing (STM): A single instance of our STM module
consists of two LSTM layers - one for temporal sequence modeling and another
for spatial sequence modeling. The input feature to an STM module is of shape
$B\times N\times T\times d$ where $B,N,T,d$ are batch size, number of spatial
nodes, number of time frames and feature dimension respectively. We reshape
this tensor into $BN\times T\times d$ and feed it as input to the first LSTM.
Here, $T$ forms the index for sequence. The output tensor from this LSTM is
reshaped to $BT\times N\times d$ before feeding it into the second LSTM for
spatial sequence modeling. We would like to point out that the idea of
multiplexing is not novel in deep learning literature [30, 31]. Our
contribution is that we are the first to introduce multiplexing in physics-
informed learning and show its utility in dynamic prediction. Our whole
architecture consists of three STM modules, totaling six LSTM layers. The
architecture is schematically shown in Fig. 2
Figure 2: Model architecture: We introduce the novel spatiotemporal
multiplexing (STM) to physics-informed learning in order to learn both spatial
and temporal information in the data. Our architecture is lightweight and
hence gives real-time performance.
### 4.2 Physics Loss Function
In order to force our model to learn the physical constraints, we minimize the
violation of the physical equations shown in Eq. 1 and 2. We also minimize the
boundary condition violation to fully enforce the underlying PDE.
Specifically, our loss function is a weighted sum of three loss terms:
$\mathcal{L}=w_{\text{data}}\mathcal{L}_{\text{data}}+w_{\text{PDE}}\mathcal{L}_{\text{PDE}}+w_{\text{bc}}\mathcal{L}_{\text{bc}}$
(4)
where $\mathcal{L}_{\text{data}}$ measures the mean absolute error (MAE)
between true and predicted labels. $\mathcal{L}_{\text{PDE}}$ measures the
violations of the physical equations by calculating the mean absolute error
between the LHS and the RHS. $\mathcal{L}_{\text{bc}}$ corresponds to boundary
condition constraints. $w_{\text{data}}$, $w_{\text{PDE}}$ and $w_{\text{bc}}$
are the weights used to balance the interplay between the three loss terms.
$\mathcal{L}_{\text{bc}}$ consists of the initial and boundary conditions at
each time step as below:
$\sigma(x,y,t=0)=0$ (5) $\sigma(x,y,(t_{0}...t_{n}))=\sigma$ (6)
Equations 5 and 6 should be satisfied for $\sigma_{xx}$, $\sigma_{yy}$ and
$\sigma_{xy}$. $x$ and $y$ are coordinates of meshes in each sample, and $t$
is the time at time steps.
### 4.3 Differentiable grid from mesh
Our physics-based loss function requires us to estimate the gradients of
stress output along $x$ and $y$ directions. But since our output is in the
form of a triangular mesh, gradient computation is not easy. Instead, we
propose to calculate gradients on a surrogate grid created using kernel
density estimation (KDE). Specifically, we calculate the stress value at a
grid vertex by adding contributions from every mesh node, weighted by a
Gaussian filter centered at this vertex and having a specific variance. By
tuning the variance of this filter, we can achieve a robust, accurate
reconstruction of the mesh along with a mask showing extrapolated regions. The
original mesh, the grid reconstructed from it, and the corresponding mask are
shown in Fig. 3. To compare the accuracy of the surrogate grid, we compare it
against the reconstruction obtained through tricontourf function in Matplotlib
package in Python. As can be observed in Fig. 3(c), the grid is accurate
within the mesh region. Now, we can estimate the gradients for the stress
outputs from these grids.
(a) (b) (c) (d)
Figure 3: Constructing grid values from mesh values (a) mesh nodes of a single
output, the color of each node represents the stress value at the
corresponding node, (b) reconstruction from Matplotlib tricontourf function,
(c) our reconstruction on a $200\times 200$ grid, (d) corresponding mask
showing interpolated regions.
## 5 Experiments and Results
### 5.1 Data Generation
Gusset plates connect beams and columns to braces in steel structures. The
behavior and analysis of these components are critical since various reports
have observed failures of gusset plates subject to lateral loads [32, 33, 34].
The boundary conditions and time-history load cases are considered to simulate
similar conditions in common gusset plate structures under external loading.
Some of the most common gusset plate configurations in practice are shown in
Fig 4.
Figure 4: Some of the most common gusset plates in practice.
We create a dataset with 71,680 unique samples by combining 14 random time-
history load cases, 1024 different geometries, and 5 most commonly found
boundary conditions in gusset plates. Boundary conditions are shown in Fig. 5,
mimicking the real gusset plates’ boundary conditions. All the translation and
rotational displacements were fixed at the boundary conditions. The range for
width and height of the plates is from 30 cm to 60 cm. Two-dimensional steel
plate structures with five edges, E1 to E5 denoting edges 1 to 5, as shown in
Fig. 6, are considered to be made of homogeneous and isotropic linear elastic
materials. Various geometries are generated by changing the position of each
node in horizontal and vertical directions, as shown in Fig. 6, which leads to
1024 unique pentagons. The material properties remain unchanged and isotropic
for all samples.
Figure 5: Different types of boundary conditions for initializing population.
Figure 6: Basic schematic topology for initializing the steel plate
geometries.
Time histories consist of 100 time-steps generated with random sine and cosine
frequencies. The frequencies range between 1 and 3 Hz, with amplitudes ranging
from 2 to 10 kN at intervals of 2 kN. All time histories in horizontal and
vertical directions are shown in Fig. 7. Each time series last for 1 second
with each time-step lasting 0.01 seconds. All the details of the input
variables used to initialize train-validation-test distribution of the
population are shown in Table 1.
Table 1: Dataset splits Split | Boundary condition | Load position | Load number | Geometry number
---|---|---|---|---
train | E2 | E4E5 | 1-8 | 1-614
train | E2E3 | E5 | 1-8 | 1-614
train | E1E2 | E4 | 1-8 | 1-614
val | E3 | E2E4 | 9-12 | 615-819
test | E1E5 | E2 | 12-14 | 820-1024
(a) (b)
Figure 7: Various load sequences in (a) horizontal and (b) vertical
directions.
#### 5.1.1 Input data
Input parameters include geometry, boundary condition, and body force in
horizontal and vertical directions, each encoded as vectors in a 3-dimensional
matrix. The size of the input matrix is $N\times M\times T$. where, $N$, $M$
and $T$ represent mesh nodes, input parameters and time, respectively. For
example, if a sample contains 200 mesh nodes, the size of the input matrix is
$200\times 5\times 100$. Fig. 8 shows how we construct the input matrix based
on the geometry, boundary conditions and body forces. This figure presents a
sample with five mesh nodes. However, all real samples in the trained model
have more than 100 mesh nodes. The first and the second columns of the input
matrix are $x$ and $y$ coordinates of the mesh nodes respectively. The third
column represents the condition of boundary constraint at each node using a
Boolean value. If there is a boundary constraint at the corresponding node,
then the value is one, otherwise is zero. The fourth and the fifth columns
represent body force sequences at each node along $x$ and $y$ directions.
Details of boundary conditions and their load positions are described in Table
1.
Figure 8: Construction of input matrix (Unit: m, N).
#### 5.1.2 Output Data
To obtain the stress distributions for each sample, we perform FEA using the
Partial Differential Equation (PDE) solver in the MATLAB toolbox.
Specifically, we use transient-planestress function of MATLAB PDE solver to
generate dynamic stress contours which will act as the ground truth for our
model. We define geometry, boundary condition, material properties, and time
histories as input, and the PDE solver returns the sequence of stress
distributions of $\sigma_{xx}$, $\sigma_{yy}$ and $\sigma_{xy}$ corresponding
to the inputs. The size of each output is mesh nodes $\times$ load sequence.
For example, if a sample contains 200 mesh nodes, the size of the output
matrix is $200\times 100$. Each of the three outputs are normalized separately
between -1 and 1 to ensure faster convergence. The input and the output
representations of the model is shown in Fig. 9.
Figure 9: Input and output representation for normal and shear stress
distribution prediction: (a) Input matrix, (b) Output ($\sigma_{xx}$), (c)
Output ($\sigma_{yy}$), (d) Output ($\sigma_{xy}$).
### 5.2 Metrics
We use Mean Absolute Error (MAE), defined in Eq. 7 as the primary training
loss and metric. To ensure that we do not overfit to a single metric, we also
use Mean Relative Percentage Error (MRPE) to evaluate the overall quality of
predicted stress distribution.
$\text{MAE}=\frac{1}{NT}\sum_{N,T}^{n,t}\left|S(n,t)-\hat{S}(n,t)\right|$ (7)
$\text{MRPE}=\frac{\text{MAE}}{\max(|S(n,t)|,|\hat{S}(n,t)|)}\times 100$ (8)
where $S(n,t)$ is the true stress value at a node $n$ at time step $t$, as
computed by FEA, and $\hat{S}(n,t)$ is the corresponding stress value
predicted by our model, $N$ is the total number of mesh nodes in each frame of
a sample, and $T$ is a total number of time steps in each sample. As mentioned
earlier, we set $T=100$ in our experiments.
### 5.3 Implementation
We implemented our model using PyTorch [35] and PyTorch Lightning. AdamW
optimizer [30] was used with an initial learning rate of $10^{-3}$. We found
that a batch size of 10 gives the best results. The computational performance
of the model was evaluated on an AMD EPYC 7313 16-core processor and one
NVIDIA A6000 48GB GPU per experiment. The time required during the training
phase for a single batch with 100 frames and a batch size of 10 for
NeuroStress and PINN-Stress were 7 and 20 milliseconds respectively. The
inference time of NeuroStress and PINN-Stress for one sample were 1 and 10
milliseconds respectively, which satisfies the real-time requirement. The most
powerful FE solvers take between 10 minutes to an hour to solve the same. We
use MATLAB PDE solver as a FE solver to compare the efficiency of our model.
We consider the minimum time for all processes of modeling geometry, meshing,
and analysis of one sample in FE solver to be about 10 minutes. MATLAB PDE
solver does not use GPU acceleration. Therefore, NeuroStress and PINN-Stress
are about $6\times 10^{5}$ and $6\times 10^{4}$ times faster than MATLAB PDE
solver.
### 5.4 Results
We implement two main models, NeuroStress and PINN-Stress. Both models are
trained on the same train dataset for 300 epochs, evaluated on the validation
dataset for fine-tuning, and we report all metrics on the test dataset. The
entire dataset contains 71,680 samples, while the train dataset contains
43,008 samples, validation and test datasets each contain 14336, forming the
60%-20%-20% split of the whole dataset. Error metrics are calculated using the
checkpoint with the least validation error. Fig. 10 shows stress distribution
prediction for $\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{xy}$ and $\sigma_{vm}$
of a randomly selected frame in a sample. PINN-Stress predictions are almost
identical to their corresponding references, and the errors in a PINN-Stress
prediction are substantially lower than those in a NeuroStress prediction.
Particularly, PINN-Stress can capture peak stress better than NeuroStress,
which is of primary importance in structural design. The importance of maximum
stress matters in the design phase since maximum stress should be less than
yield strength to avoid permanent deformation.
Figure 10: Comparison of NeuroStress and PINN-Stress predictions for
$\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{xy}$ and $\sigma_{vm}$. (Unit: MPa)
Table 2: Data split for generalization experiments
Quantity | Data split* | MRPE (%)
---|---|---
| Train | Val | Test | NeuroStress | PINN-Stress
Geometry | 1-614 | 615-819 | 820-1024 | 1.7 | 1.5
Load | 1-8 | 9-11 | 12-14 | 4.8 | 4.2
BC | E2, E2E3, E1E2 | E3 | E1E5 | 18.3 | 16
* The values in the data split column refer to indices of the corresponding generalization quantity.
## 6 Ablation Studies
### 6.1 Generalization
We investigate and compare the generalization capabilities of NeuroStress and
PINN-Stress models for varying distributions of boundary conditions, load
sequences and geometries. To that end, we collect the entire dataset and split
them into train, validation and test sets such that validation and test sets
contain unseen instances of the entity to check generalization on. For
example, for checking generalization on geometry, train set will consist of
614 geometries out of 1024, and validation and test sets will contain the
remaining (205 each). We compare the mean relative percent error (MRPE) of
each method on von Mises stress prediction. As von Mises stress identifies if
a given material is likely to yield or fracture, we use its prediction error
as the sole criterion. Figs 11 and 12 demonstrates the generalization
capability of PINN-Stress and NeuroStress to unseen load sequences and
geometries, respectively. As it can be seen, $\sigma_{xx}$, $\sigma_{yy}$,
$\sigma_{xy}$ and $\sigma_{vm}$ predictions by PINN-Stress are significantly
better than those by NeuroStress.
Figure 11: Predicting dynamic stress distribution for diverse load sequences:
Augmenting our novel architecture with a physics-based loss can induce
generalization capabilities while still remaining real-time (Unit: MPa). The
overview of our method is given in Fig. 1. Figure 12: Predicting dynamic
stress distribution for diverse geometries (Unit: MPa).
Fig 13 shows the error of each frame for a random spatial node across all time
frames for unseen load sequences and structural geometries. As it can be seen,
in both figures, the errors in PINN-Stress are less than NeuroStress,
especially in extreme peaks, which demonstrates the ability of PINN-Stress to
predict the maximum stress values. We have also compared the generalization
capability of PINN-Stress and NeuroStress over unseen load sequences and
geometries in a single spatial node across all time frames in Figs 14 and 15.
Figs 14 and 15 demonstrate the ability of our models to capture the temporal
dependencies over time frames. It can be seen that both models’ predictions
are almost identical to references in all the time frames. However, in extreme
peaks PINN-Stress outperforms NeuroStress. Table 2 shows the data split for
each experiment and the corresponding results. The lowest error in each
experiment is highlighted in bold.
In every experiment, we can observe that PINN-Stress generalizes better than
NeuroStress. However, neither method generalizes satisfyingly for various
boundary conditions. Since we only considered five different boundary
conditions in total, we ran the same experiment for different combinations of
boundary conditions. The results were similar.
(a) (b)
Figure 13: Comparison of NeuroStress and PINN-Stress errors for $\sigma_{vm}$
across 100 frames for a random spatial node in a sample. (a) unseen load
sequences and (b) unseen geometries. Figure 14: Comparison of NeuroStress and
PINN-Stress predictions for $\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{xy}$ and
$\sigma_{vm}$ across 100 frames for a sample with unseen load sequences.
Figure 15: Comparison of NeuroStress and PINN-Stress predictions for
$\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{xy}$ and $\sigma_{vm}$ across 100
frames for a sample with unseen geometries.
### 6.2 Choice of architecture
The efficiency of architecture can be attributed to several design choices we
have made. Our architecture models the temporal dependency between time frames
and the relationship between different nodes in an input via our
spatiotemporal multiplexing mechanism. As mentioned earlier, we are the first
to introduce such a design into PINNs to the best of our knowledge. Even
though self-attention has shown state-of-the-art performance in sequence
modeling, they are not suitable for tasks without large amounts of data.
Hence, we use LSTMs for sequence modeling. To demonstrate our claim, we
compare our architecture against other baseline architectures.
We compare against three architectures: Spatiotempo-Att, Tempo-LSTM, Spatio-
MLP. Spatiotempo-Att is very similar to our architecture, except the LSTM
modules in our model are replaced with self-attention modules. Tempo-LSTM is
also similar to our architecture except the LSTMs act only along the temporal
dimension. Spatio-MLP is a normal feedforward network with six layers with
LeakyReLU activation in between. It treats each time frame separately but
considers all the nodes simultaneously. We will refer to our architecture as
Spatiotempo-LSTM. To save time and resources, we train all the architectures
on 10% of training data with MAE loss. Similar to our experiments on
generalization, we report the error on von Mises stress prediction. The
results are shown in Table 3, and the best results are highlighted in bold.
Table 3: Architecture comparison Architecture
---
| Spatiotempo-Att | Tempo-LSTM | Spatio-MLP | Spatiotempo-LSTM
#Params ($K$) | 309 | 208 | 828 | 208
MRPE(%) | 19.5 | 17.5 | 25.4 | 16.6
## 7 Conclusion
We propose NeuroStress and PINN-Stress, two models for dynamic stress
prediction based on a novel architecture, with the latter augmented with
physics-informed loss function. Our models explicitly learn both spatial and
temporal information through our spatiotemporal multiplexing (STM) module.
Experiments on simulated gusset plates show that not only are our models
accurate, but adding physics-informed loss function facilitates generalization
with respect to varying load sequences and structural geometries. PINN-Stress
is also better at estimating high stress values which is of more importance to
the structural engineering community. However, collecting sufficient data
points from real gusset plates using sensors can be expensive and noisy.
Therefore, our future efforts will be directed towards achieving lower sample
complexity under noisy conditions.
## 8 Acknowledgment
This research was funded in part by National Science Foundation, United States
of America, grant number CNS 1645783. The corresponding author states that
there is no conflict of interest on behalf of all authors.
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|
11institutetext: Stuttgart Media University, Stuttgart, Germany,
11email<EMAIL_ADDRESS>
http://wiss.iuk.hdm-stuttgart.de/
# Easy and complex: new perspectives for metadata modeling using RDF-star and
Named Graphs
Florian Rupp Benjamin Schnabel Kai Eckert
###### Abstract
The Resource Description Framework is well-established as a lingua franca for
data modeling and is designed to integrate heterogeneous data at instance and
schema level using statements. While RDF is conceptually simple, data models
nevertheless get complex, when complex data needs to be represented.
Additional levels of indirection with intermediate resources instead of simple
properties lead to higher barriers for prospective users of the data. Based on
three patterns, we argue that shifting information to a meta-level can not
only be used to (1) provide provenance information, but can also help to (2)
maintain backwards compatibility for existing models, and to (3) reduce the
complexity of a data model. There are, however, multiple ways in RDF to use a
meta-level, i.e., to provide additional statements about statements. With
Named Graphs, there exists a well-established mechanism to describe groups of
statements. Since its inception, however, it has been hard to make statements
about single statements. With the introduction of RDF-star, a new way to
provide data about single statements is now available. We show that the
combination of RDF-star and Named Graphs is a viable solution to express data
on a meta-level and propose that this meta-level should be used as first class
citizen in data modeling.
###### keywords:
data modeling, RDF-star, Named Graphs, meta-level
## 1 Introduction
Data can come in many different forms and there are nearly infinite ways to
model data describing the same information. Depending on the use case,
different forms might be preferable: a simple view on data is the easiest to
work with and facilitates data reuse. On the other hand, more complex
applications need advanced data models providing the full complexity of the
data with all available details. The Resource Description Framework (RDF) [1]
provides a common basis for the publication of data in form of statements on
the Web but makes no assumptions how these statements actually look like and
how entities and their relations are used to represent the desired
information. Thus, various modeling approaches can represent the same
information. The Semantic Web aims at providing interlinked data based on the
”follow your nose” principle [4], i.e.: users and applications can get data
about a resource by dereferencing its URI.
There are two possible solutions, when data is to be provided both in simple
and complex111While it depends on the perspective, we consider data complex
that uses several layers of properties and dependent classes to describe a
resource. forms for different applications:
1. 1.
Different data representations can be provided; in this case, a mechanism is
required that allows the application to request one of these representations
(see Section 2).
2. 2.
The data is provided in a simple data model and additional information is
provided on a meta-level in the form of statements that provide further
explanations.
In this article, we discuss the latter approach. We argue that there are
benefits of using such a meta-level approach versus the provision of different
data formats: there is only one representation required, it is straight-
forward to extend existing data representations while maintaining backwards
compatibility and it fits the ”follow your nose” principle of the Semantic Web
where applications simply can ask if there is further information about a
statement.
A meta-level can be created by providing a further statement describing a
single or multiple statements. To provide statements about statements in RDF,
a mechanism is needed to identify statements. The identification can happen in
two ways: either a single statement is identified (statement-level) or a group
of statements – usually called a graph – is identified and meta-statements are
used to further describe all statements in this graph (graph-level).
For statement-level data there always has been the optional RDF reification,
for graph-level statements, Named Graphs have been introduced. Particularly
RDF reification, however, has two major shortcomings: it is inefficient for
exchanging RDF data and writing queries to access statement-level metadata is
cumbersome [7].
Now there is a new way to provide statement-level metadata within RDF itself
with the recent specification of RDF-star222https://www.w3.org/2021/12/rdf-
star.html. With increasing support in RDF databases and tools and the
specification of various serializations including TriG-star333Throughout this
paper, we use TriG-star syntax in the examples. For reference this is a list
of the implied namespaces, ex is the default or empty namespace: rdf:
http://www.w3.org/1999/02/22-rdf-syntax-ns#; owl:
http://www.w3.org/2002/07/owl#; ex: http://example.org/; dcat:
http://www.w3.org/ns/dcat#; dct: http://purl.org/dc/terms/; foaf:
http://xmlns.com/foaf/0.1/; prov: http://www.w3.org/ns/prov#; dbp:
http://dbpedia.org/property/; dbo: http://dbpedia.org/ontology/; gn:
http://www.geonames.org/ontology#; gndo:
http://d-nb.info/standards/elementset/gnd#, it provides all that is needed to
use both statement-level metadata with RDF-star and graph-level metadata with
Named Graphs.
In this article, we explore the possibilities and limitations of RDF-star and
Named Graphs with respect to the creation of data models which are simple and
easy to use, but nevertheless contain the full complexity of the data on a
meta-level so that it can be used when needed. We start with an overview on
related work focusing on providing data to applications with different
requirements. In Section 3, we identify three abstract patterns that show how
common use cases can be modeled with metadata. In Section 4, further examples
about querying and working with the meta-level are given. The paper concludes
with a discussion of the advantages and disadvantages of data modeling with a
meta-level as first-class citizen.
## 2 Related Work
Different applications have different requirements towards data. Consider a
personal blog that wants to provide a thumbnail view of a book versus a
library that needs data about a book to include in its catalog. It is common
practice in many open data portals that data can be accessed in different
formats, e.g. Dublin Core for simple data vs. MARC for all the details. In the
Semantic Web, however, this poses a problem. If a URI for a book is resolved
automatically by an application, what data should be provided? In our paper
this is not directly solved, but arguably our meta-level approach allows
applications to decide if they want to access the additional information. For
each resource there can only be one data description. The provider of the data
needs to decide, how this data is modeled and which vocabularies are used. To
address this shortcoming, the Dublin Core community developed the concept of
_Application Profiles_ [8] to describe the data model for a specific
application. Based on the notion of Application Profiles, there is a recent
RFC draft to use HTTP for the negotiation of different data representations on
content level, similar to the well-known content negotiation for different RDF
serializations [18]. Using such mechanisms, however, requires proper support
in applications and it is not yet clear if this will actually be used in
practice.
Another aspect is the provision of metadata to describe the actual data,
mostly to provide information on the _provenance_ of the data [14, 5, 6, 16,
17, 13]. There are generally two aspects to consider: (1) how can the subject,
i.e., the data, be identified to allow statements about it, and (2) how can
the provenance of data be described. As we do not focus specifically on the
description of provenance, we are mainly concerned with the former.
Besides RDF-star [12] and Named Graphs [3], other approaches have been
proposed. The oldest way to do this is _RDF reification_ , which is available
since the very first versions of RDF, cf. [15]. However, making statements
about statements using reification is hard, mainly due to its verbose syntax
and the fact that every statement needs to be reified, even when many
statements share the same meta information. This might be a reason why
reification is rarely used in real world applications444 To get evidence of
reification usage in real world application we checked a given list of SPARQL
endpoints. In favor we use the Wikidata SPARQL endpoint list at
https://www.wikidata.org/wiki/Wikidata:Lists/SPARQL˙endpoints. This
examination was done automatically with a suitable SPARQL query. The list,
however, contains 139 entries and can be, according to its small size,
considered as a sample only. Merely 78 endpoints were reachable via GET or
POST requests (on July 20th 2022). 7 out of 78 remaining endpoints were not
suitable for this study, being no actual SPARQL endpoint or the domain has
been sold. Out of 72 endpoints 5 are using reification. At just 7,7% that’s a
small percentage. This supports the thesis that reification is rarely used.
_Singleton statements_ [11] use unique identifiers which are added to the
predicate statement. Following this method, the meta-level can be expressed by
using the predicate in combination with the assigned identifier. Singleton
properties lead to compact representations of the meta-level, but obviously
affect the querying of the data. There is support for singleton properties in
some triple stores, which makes the application more feasible. Even then it is
not recommended, for instance see the documentation of
GraphDB555https://graphdb.ontotext.com/documentation/free/devhub/rdf-sparql-
star.html, Singleton Properties.).
These approaches all address statement-level metadata. There are many cases
where an identification of single statements is not needed and even not
helpful. Probably the best example is provenance information. Usually, many
statements share the same provenance when they are created together by a
single process.
For the separation of different descriptions of the same resource, as an
alternative, _Proxy entities_ [9] or resource maps in OAI-ORE [10] have been
proposed, that use proxies as distinct placeholder entities that all represent
the same actual entity but with different URIs to be able to distinguish
statements about the entity from different sources. This raises the complexity
of the data and requires applications to correctly interpret the proxy
entities.
A problem with Named Graphs is that organizing structures, for example nested
graphs or subgraphs can only be represented via additional data describing
_graph relationships_. This can be used to identify provenance information in
hierarchical graphs [5]. A proposal for the structured use of several named
graphs are _nano publications_ [2] that also explicitly support provenance.
## 3 Meta Modeling Patterns
Based on our experience with data applications, we can identify three abstract
modeling patterns that illustrate the benefits of a meta-level. For each
pattern, we formulate a problem statement, the solving approach, an example
and a conclusion. The three patterns are:
* •
Pattern 1: RDF provenance modeling in the meta-level.
* •
Pattern 2: Extending an existing data model.
* •
Pattern 3: Shifting proxy entities to the meta-level.
We will not distinguish statement-based and graph-based metadata. With RDF-
star and Named Graphs, there are now viable solutions for both levels and it
is best decided by the data modeler or the data provider which one is more
suitable. This of course requires that consuming applications understand both
approaches, i.e, TriG-star needs to be supported as serialization.
### 3.1 Pattern 1: RDF provenance modeling in the meta-level
#### Problem statement:
The provision of provenance data _in RDF_ is straight forward, for example
using PROV or Dublin Core [6]. Nevertheless provenance data is not often
provided on a meta-level _for RDF_ data. Instead, entities are introduced that
either represent the data (e.g. a dcat:CatalogRecord) or that act as a
placeholder for the actual entity for the description from one specific source
(e.g. a ore:Proxy). Both approaches are problematic when data from different
sources is to be merged due to both approaches must now fit into the same data
model.
Even unfavorable is the mixing of data about a resource and about its
description. Let’s say, we want to model the provenance of a statement that
provides a title for an entity <E>. In the following example, the information
is added directly to the instance of <E>.
<E> rdf:type Entity_E ;
dct:title "title" ;
prov:wasDerivedFrom <source> .
But what is derived here? Is it the title of the resource or the resource
itself? As all statements describe their subject, it is clearly the resource
<E> and not the title or other descriptive data about the resource.
#### Solving approach:
Provenance is a prime example where Named Graphs should be used as usually
many statements share the same provenance:
<entity> ex:ID <ID> .
:data {
<entity>
ex:data "data" ;
}
:data
prov:wasDerivedFrom <entity> ;
...
#### Example DBpedia:
The DBpedia project666https://www.dbpedia.org/. is one of the largest Linked
Open Datasets on the Web. It converts Wikipedia articles into RDF. The data is
expressed using the DBpedia ontology. The information from which Wikipedia
article data was derived is expressed using the prov:wasDerivedFrom predicate.
Additional provenance information is given by further statements such as
dbo:wikiPageID giving the ID of the Wikipedia page, dbo:wikiPageRevisionID
stating the revision ID or the number of characters of the original article in
dbo:wikiPageLength. Indeed, the provenance information is related to the
article, but is mixed with the description of the entity the article is about:
<entity>
dbp:size "63" ;
dbp:built "1889" ;
prov:wasDerivedFrom <article> ;
dbo:wikiPageID "123" ;
dct:date "2022-05-21" .
Here the statement level is not sufficient thus multiple statements share the
same provenance. A cleaner solution is a model with a meta-level. It can also
be argued that the provenance information of the data is only important for a
small subset of applications, for example when ensuring the data quality such
as the relevancy and timeliness of an article. Here is the example data using
Named Graphs:
:data {
<entity>
dbp:size "63" ;
dbp:built "1889" ;
}
<article> dbo:wikiPageID "123" .
:data
prov:wasDerivedFrom <article> ;
dct:date "2022-05-21" ;
#### Conclusion:
The meta-level is the best and most suitable way how to model provenance data
in RDF: This avoids levels of indirections, i.e. the usage of proxy resources
or specific constructs such as the notion of records. The provenance
information can be kept separate from the data and may only be queried if
needed, which makes the data model lightweight. While Named Graphs are usually
the best fit for provenance data, it has to be noted that RDF-star can also be
used – even additionally – to provide the provenance for a single statement.
This first pattern addresses the most common use case for metadata. With the
following two patterns, we would like to extend its use to new applications.
### 3.2 Pattern 2: Extending an existing data model
#### Problem statement:
Data models usually evolve over time and changes to the model are inevitable.
This poses a problem when these changes are not backwards compatible and break
existing applications. We therefore propose to use the meta-level to extend
and improve an existing data model.
For example, an existing knowledge graph could be enriched to provide
additional confidence values. Sometimes an existing data model is wrong,
unfitting or simply corrupted resulting in loss of context.
For illustration, we assume we have a simple data model where a relation of
two entities <A> and <B> can be expressed:
<A> :conformsTo <B> .
Now this model is to be enriched with additional information such as a
confidence score. To achieve this, however, various approaches are possible
without using the meta-level, for example n-ary entities. This would require a
remodeling which is a high risk of breaking existing applications, for
instance:
<A> :hasConformanceStatement <C> .
<C> a :ConformanceStatement;
:conformingTo <B>;
:confidence 0.8 .
#### Solving approach:
We propose a modeling of this new information in the meta-level without
touching the existing model to ensure backwards compatibility. The additional
information is integrated by using RDF-star.
<A> :conformsTo <B> .
<< <A> :conformsTo <B> >> :confidence 0.8 .
#### Example 1 DBpedia:
All data within DBpedia is derived from Wikipedia articles. Among other
information, many articles have a thumbnail image related to its article. In
DBpedia a thumbnail is attached to an article with the predicate
dbo:thumbnail. However, Wikipedia is a Web page where the thumbnail is an HTML
image tag. In DBpedia the src-attribute of the tag is converted to the URI
describing the thumbnail. Furthermore, an HTML image tag is described by its
caption attribute and an alternative text (alt attribute). Both are included
in DBpedia via dbp:caption and dbp:alt. However, the relation to which
thumbnail a caption or alternative text conforms is gone.777According to the
DBpedia ontology, this should actually be modeled with an n-ary entity, but at
least in the current version of DBpedia, the described problem exists. Due to
this simplification the relation is lost. This becomes really disadvantageous
if articles have multiple thumbnails where it is impossible to resolve these
relationships. While this obviously should be fixed in the data, it might be
that there are applications using the data that rely on the current data
representation. With RDF-star, however, we can provide additional information
to deliver the relationship information to applications that need it, without
changing the asserted statements in the RDF data. One way would be to add the
caption to the relationship:
<entity> dbo:thumbnail <thumbnail> ;
dbp:caption "Portrait of X" .
<< <entity> dbo:thumbnail <thumbnail> >>
dbp:caption "Portrait of X" .
This is arguably again not ideal as a caption should probably refer to a
thumbnail, not a thumbnail assignment. On the other hand, if the same
thumbnail is used in different contexts with different captions, the above
solution is good. For the sake of the argument however, let us assume that the
caption should actually be assigned to the thumbnail. So the data should look
like this:
<entity> dbo:thumbnail <thumbnail> .
<thumbnail> dbp:caption "Portrait of X" .
To actually fix the data, in this case the subject of the original statement
needs to be changed. We could provide a vocabulary for such cases that would
be universally understandable by applications supporting it, i.e. the
replaceSubjectBy-statement:
<entity> dbo:thumbnail <thumbnail> ;
dbp:caption "Portrait of X" .
<< <entity> dbp:caption "Portrait of X" >>
ex:replaceSubjectBy <thumbnail> .
This means that an application should replace the subject of the original
statement (<entity>) with the new subject <thumbnail>. As can be seen here,
RDF-star opens many interesting ways to provide a history or change requests
for statements. Consider the following example where statements are related to
each other:
<entity> dbo:thumbnail <thumbnail> .
<thumbnail> dbp:caption "Portrait of X" .
<< <thumbnail> dbp:caption "Portrait of X" >>
ex:replaced << <entity> dbp:caption "Portrait of X" >> .
In this case, the actually wrong statement <entity> dbp:caption "Portrait of
X". would only be available as part of the RDF-star triple stating that it has
been replaced. Nevertheless, it is still part of the graph and old
applications could still use it.
#### Example 2 GeoNames:
GeoNames888http://www.geonames.org/ is a graph for geographical data. It
includes alternative names, but not information on historical names. This
example shows how historical names of a city, or a place can be added to the
current data and extended including more specific information. Currently in
GeoNames there is only a distinction of gn:name and gn:alternateName. The
alternate name can contain a language tag, but it does not indicate when a
name was used in case of historical names. For example: The German city of
Chemnitz used to be called ”Karl-Marx-Stadt” between May 10th, 1953 and July
1st, 1990.
The listing below shows an excerpt of the RDF entry in
GeoNames999https://sws.geonames.org/2940132.:
<https://sws.geonames.org/2940132/>
gn:name "Chemnitz" ;
gn:alternateName "Chemnitz"@de ;
gn:alternateName "Chemnitz"@en ;
gn:alternateName "Karl-Marx-Stadt"@de .
We could add the missing information about the time span, for example by using
the Common Authority File (Gemeinsame Normdatei, GND) of the German National
Library, which provides the following data:
<https://d-nb.info/gnd/2015221-8>
gndo:preferredNameForThePlaceOrGeographicName
"Karl-Marx-Stadt";
gndo:dateOfEstablishment "10.05.1953" .
gndo:dateOfTermination "31.05.1990" ;
We can take this data from the GND and apply it to GeoNames to add the
additional data to the entities, such as date of establishment and date of
termination:
<https://sws.geonames.org/2940132/>
gn:alternateName "Karl-Marx-Stadt"@de .
<< <https://sws.geonames.org/2940132/>
gn:alternateName "Karl-Marx-Stadt"@de >>
ex:valid_from "09.05.1953"^^xsd:date ;
ex:valid_to "01.06.1990"^^xsd:date .
This way, the data is still provided in a very simple manner that nevertheless
is useful for many applications that do not need the additional information,
for example for named entity resolution. Nevertheless, the additional
information can be provided in a modular way if it is needed.
#### Conclusion:
RDF-star is a great fit when additional information about a statement needs to
be provided. In particular when a data model is already used in applications
it is an adaption to provide more complex data avoiding incompatible changes.
By doing so, a simple model retains still simple by providing an additional
context in the meta-level. The backwards compatibility is obtained as well
(see also Section 4).
### 3.3 Pattern 3: Shifting complex relations to the meta-level
#### Problem statement:
Whenever additional data about a relationship between two entities is needed,
an additional _n-ary_ entity can be created for representation. As entities
(subjects and objects in statements) are substantives, this results in graphs
with many nominalized relationships, for example:
<E> :hasSubject <SubjectAssignment1> .
<SubjectAssignment1>
:hasHeading "Data Modeling" ;
:fromVocabulary <TopicsVocabulary> .
Here, a complex entity of the class SubjectAssignment is used to represent the
subject of entity <E>. The problem is that this structure feels cumbersome for
many applications that might only be interested in the subject heading and
that do not care about the vocabulary the subject heading is coming from. Even
if uncontrolled subject headings (free tags) are used, the intermediate
subject assignment is still required as the data model is created this way.
#### Solving approach:
Similar to pattern 2, we propose to use a meta-level, but this time from the
beginning, so that the additional and perhaps even optional information can be
pushed to the meta-level:
<E> dc:subject "Data Modeling" .
<< <E> dc:subject "Data Modeling" >>
:fromVocabulary <TopicsVocabulary> .
#### Example DCAT:
In this example we demonstrate how such n-ary entities can be avoided directly
at the time of designing ontologies by shifting entities to the meta-level.
DCAT is an ontology enabling publishers to describe datasets and its
properties. The datasets can be listed in a dcat:Catalog entity. The
dcat:Dataset entity holds meta information of the actual data, which is linked
to as a dcat:Distribution entity. When listing a dataset in the catalog, the
optional entity dcat:CatalogRecord may be used to express metadata about the
listing such as the issue date.
As of the current DCAT 2 ontology
specification101010https://www.w3.org/TR/vocab-dcat-2/., the
dcat:CatalogRecord is provided for the purpose of adding additional metadata
for the description of the listing of a resource, e.g., datasets in the
catalog. The following example lists a dataset in a catalog. To add metadata
such as the issued date or the title, the CatalogRecord entity is applied
providing these information.
ex:catalog dcat:record ex:catalogRecord .
ex:dataset a dcat:Dataset .
ex:catalogRecord a dcat:CatalogRecord ;
dct:issued "05.04.2022" ;
dct:title "record title" ;
dct:description "record description" ;
foaf:primaryTopic ex:dataset .
However, the dcat:CatalogRecord entity adds metadata only to the actual
listing of the catalog and the dataset. This can be shortened by rewriting it
in the following RDF-star syntax to shift the dcat:CatalogRecord entity to the
meta-level. The attributes of this entity can be used to describe the
provenance of the relation directly:
ex:catalog dcat:dataset ex:dataset .
<< ex:catalog dcat:dataset ex:dataset >>
a dcat:CatalogRecord ;
dct:issued "05.04.2022" ;
dct:title "record title" ;
dct:description "record description" .
This information can be queried easily from the meta-level using SPARQL-star.
Here we give an example on how to query the issued date of the CatalogRecord
denoted before:
SELECT ?date WHERE {
<< ex:catalog dcat:dataset ex:dataset >> dct:issued ?date .
}
As RDF data can easily be split in several junks of statements, the metadata
could also be separated from the core data, containing only the RDF-star
statements.
#### Conclusion:
On the one hand, complexity can be shifted to the meta-level and the data
model is improved towards simpleness. In the course of this, the data may only
be queried when needed reducing the size of query results. On the other hand,
the simple data model might seem to be oversimplified to a data modeler. If
the vocabulary scheme belongs to the subject heading and is not optional, it
might seem arbitrary to put this information to the meta-level. And if the
majority of applications need the complete data, the querying and using of the
meta-level can feel more cumbersome than a more complex data model. To
conclude, there are situations where the data modeled in an entity really
should be on the same level as the rest of the actual data. In other
situations, using the meta-level might just be the ideal solution where a
simple model can be provided for simple applications and additional
information is available if needed.
## 4 Querying and constructing the meta-level with SPARQL-star
In the last section we have shown how data using the meta-level in its data
model could look like. For a broad adoption of the proposed patterns, it is
important that working with the data is effortless and data on the meta-level
can easily be found and processed when needed. For this, two aspects should be
considered:
1. 1.
How can an application find out if there is data on the meta-level?
2. 2.
How can data using the meta-level be transformed to other data models that
potentially do not use the meta-level?
#### How can an application find out if there is data on the meta-level?
For this question, it is important to adjust some expectations that an
application might have when dealing with RDF and Linked Data. A modern RDF
application that supports data on the meta-level is required
1. 1.
to support Named Graphs and RDF-star as well as at least TriG-
star111111https://w3c.github.io/rdf-star/cg-spec/2021-04-13.html“#trig-star as
standard format for any responses from a Linked Data server as well as for
data dumps,
2. 2.
to expect Named Graphs to be used for the actual data, so the internal
organisation of data, e.g. for provenance tracking, must support graph
hierarchies, such as a provenance context [5],
3. 3.
to check for the existence of a meta-level in form of Named Graphs and/or RDF-
star triples.121212https://w3c.github.io/rdf-star/cg-
spec/2021-04-13.html“#dfn-triple
The following SPARQL-star query returns all RDF-star triples:
SELECT DISTINCT ?s WHERE {
?s ?p ?o .
FILTER( isTRIPLE(?s) )
}
The function isTRIPLE is new in SPARQL-star and returns TRUE if the parameter
is an RDF-star triple. Together with a query for the existence of Named
Graphs, this enables an application to check if a meta-level is available.
#### How can data using the meta-level be transformed to other data models
that potentially do not use the meta-level?
The ability to transform data from one data model to another by means of
CONSTRUCT queries is one of the most powerful features of RDF. This is not
restricted by the use of meta-level data, as the following queries
demonstrate:
In Pattern 3 (Section 3.3) we have shown how to shift the meta information
encapsulated in an n-ary entity to the meta-level. Using SPARQL-star and
CONSTRUCT queries, it is possible to rebuild the former data model, e.g. for
the dcat:CatalogRecord. To extract the subject or object of a triple
(annotated with RDF-star), the SPARQL-star functions SUBJECT and OBJECT are
used:
CONSTRUCT {
?catalog :record ?record .
?record :primaryTopic ?dataset .
?record ?pred ?obj .
} WHERE {
?star ?pred ?obj
FILTER(isTRIPLE(?star)) .
BIND(SUBJECT(?star) as ?catalog) .
BIND(OBJECT(?star) as ?dataset) .
BIND(IRI(CONCAT(STR(?dataset), "/record")) as ?record) .
}
Here, string functions are used to coin an IRI for the new n-ary entity
?record, by appending /record to the IRI of ?dataset. This is to avoid a blank
node, but it depends on the application if and how this should done.
The other way round is also possible, i.e., to create a simple data model with
meta-level data from a complex data model:
CONSTRUCT {
?star ?pred ?obj .
} WHERE {
?catalog :record ?record .
?record :primaryTopic ?dataset .
?record ?pred ?obj .
BIND(TRIPLE(?catalog, :dataset, ?dataset) as ?star).
}
This time, the SPARQL-star function TRIPLE is used to create an RDF-star
triple from a subject, a predicate, and an object.
## 5 Discussion
With this paper, we ask the following question: can and should meta-level
concepts be used for data modeling in RDF? Besides Named Graphs and RDF-star,
several concepts have been introduced to construct a meta-level as well.
However, they all have weaknesses regarding e.g., feasibility or performance.
Named Graphs have been around for a long time now and are well-supported in
SPARQL. With RDF-star, we now see a serious contender for an RDF extension
that allows statements about statements in a concise and straight-forward way,
also with a SPARQL extension. So yes, we can use it for data modeling. But
should we?
We identified three patterns that are suitable for a more detailed discussion
of the potential benefits of meta-level modeling: (1) using the meta-level for
actual metadata, i.e. data about RDF data like provenance data; (2) using the
meta-level for the extension of existing data models; and (3) using the meta-
level from the get-go to get simpler and more modular data models.
The first pattern is arguably the most trivial one, but addresses the reasons
why Named Graphs and RDF-star have been developed in the first place. While at
least Named Graphs exist for quite some while now, we still see only rarely
data models that make use of them. Instead, mostly workarounds within vanilla
RDF are used to artificially create a meta-level when it is needed. This
should change and data about RDF data should be where it belongs: in the meta-
level.
With the second pattern, we showed that existing simple data models can be
extended by additional meta statements, with two main scenarios in mind:
First, additional data can be added to provide more complex information to
applications that need it. Second, this can be used to provide corrections and
missing information to a data model without compromising existing
applications. Adding the additional information in the meta-level does not
only preserve backwards compatibility of the data model. It also keeps the
core model simple and allows a modular distribution of additional data for the
applications that need it. This observation leads to the third pattern where
we proposed to use the meta-level to create simple and modular data models
from the beginning.
We have further shown in several examples from real world data (including
DCAT, DBpedia, and GeoNames), how the application of these patterns could look
like and how they would affect the provided data. And finally, we have shown
that working with the meta-level is straight-forward, as long as applications
expect a meta-level and support Named Graphs and RDF-star, with TriG-star as
preferred serialization. Particularly also the transformation of data into and
out of the meta-level using CONSTRUCT queries is possible.
For the question, if meta-level modeling should be used, we conclude with the
following thoughts:
#### Is the provision and management of meta-level data possible?
This depends on the technology stack that is used. Named Graphs are usually
not a problem for triple stores but might lead to problems when the meta-level
is to be preserved in other systems. For data serializations, a compatible
file format such as TRIG must be used. File-based representations of graphs
(i.e., one RDF file per graph) are also possible, but require additional
infrastructure and/or tooling to make sure all meta information is properly
preserved. RDF-star is very new, but at least a transparent serialization via
RDF reification is possible.
#### Is there a clear decision for Named Graphs and/or RDF-star?
As stated in the paper, it depends on the use case and sometimes even on the
data. It might be worthwhile to just use one of the mechanisms to reduce
complexity within a project. Alternatively, both can be used, but for
different parts of the data model, for instance Named Graphs to provide
provenance information and RDF-star to provide additional information on
entity links. And finally, this could be left to the implementation and
applications would have to expect such information being provided with either
of the mechanisms.
#### Can we still ”follow the nose”?
One of the main advantages of RDF is that it is easy to get data about a
resource and use subsequent queries to find out more or even just to check if
there are any more information. This is in our opinion the main reason why
meta-level information is not used as often as it should. Applications often
expect Turtle or even still RDF/XML as RDF serializations. They might not be
able to deal with TriG. With SPARQL, Named Graphs can be queried. However, the
application needs to know that there is useful information attached to a Named
Graph. This is similar for RDF-star. The application needs to ask (or
understand from the serialization) if there are additional statements about a
statement. On the other hand, checking for the existence of Named Graphs or
RDF-star triples is possible in SPARQL and SPARQL-star. So this is only a
matter of getting used to it.
## 6 Conclusion and Future Work
Using the meta-level actively for data modeling opens interesting new
perspectives on the task. In this paper we formulated three abstract patterns
on how and when to use the meta-level in the modeling process actively.
Therefore we gave several examples on real world applications.
Usually, the data modeler has to decide if the model should be optimized for
data consumption, i.e. as easy to digest as possible and only as complex as
necessary. However, more often than not, it is rather created with the most
complex possible data in mind. Often data that does not even exist (yet) in
reality - but certainly would be great to have at some point so the data model
should better be ready. Meta-level modeling actually provides a means to
define a simple core data model - particularly with simple, direct relations.
Additional data can be provided in a modular way, with additional RDF data
that can contain additional statements further describing resources.
For future work, we aim at using the meta-level in our own projects on
production level to gain further experience with it and to support a wider
adoption of Named Graphs and RDF-star in public data models. We are interested
in the potential of meta-level modeling for the representation of changes in
data to support data transparency and better provenance tracking. Currently we
are exploring the creation of an ontology to capture the abstract entities
appearing in meta-modeling, based on the patterns we identified so far and
potentially more patterns that will arise with further applications.
## Acknowledgements
This research was partially supported by the Volkswagen Foundation (Project:
Consequences of Artificial Intelligence on Urban Societies, Grant 98555) and
the German Research Foundation (Project: Specialized Subject Service for
Jewish Studies, Grant 286004564). We thank Magnus Pfeffer for his valuable
feedback.
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|
# Do Machine-Learning Atomic Descriptors and Order Parameters Tell the Same
Story? The Case of Liquid Water.
Edward Danquah Donkor The Abdus Salam International Center for Theoretical
Physics, Strada Costiera 11, 34151 Trieste, Italy. Alessandro Laio SISSA –
via Bonomea 265, 34136 Trieste, Italy<EMAIL_ADDRESS>Ali Hassanali The Abdus
Salam International Center for Theoretical Physics, Strada Costiera 11, 34151
Trieste, Italy<EMAIL_ADDRESS>
###### Abstract
Machine-learning (ML) has become a key workhorse in molecular simulations.
Building an ML model in this context, involves encoding the information of
chemical environments using local atomic descriptors. In this work, we focus
on the Smooth Overlap of Atomic Positions (SOAP) and their application in
studying the properties of liquid water both in the bulk and at the
hydrophobic air-water interface. By using a statistical test aimed at
assessing the relative information content of different distance measures
defined on the same data space, we investigate if these descriptors provide
the same information as some of the common order parameters that are used to
characterize local water structure such as hydrogen bonding, density or
tetrahedrality to name a few. Our analysis suggests that the ML description
and the standard order parameters of local water structure are not equivalent.
In particular, a combination of these order parameters probing local water
environments can predict SOAP similarity only approximately, and viceversa,
the environments that are similar according to SOAP are not necessarily
similar according to the standard order parameters. We also elucidate the role
of some of the metaparameters entering in the SOAP definition in encoding
chemical information.
###### keywords:
American Chemical Society, LaTeX
SISSA – via Bonomea 265, 34136 Trieste, Italy. IR,NMR,UV
## 1 Introduction
The last decade has seen a tremendous spurt in both the development and
application of machine-learning (ML) approaches to study molecular systems1,
2, 3. ML has become a mainstream component of atomistic modeling. In
particular, for the construction of interaction potentials 1, 4, 5, and for
the analysis of molecular dynamics simulations6, 7, 8, 9. Establishing an
understanding of the underlying physical and chemical principles that make ML
useful, accurate and in short, avoiding its _black box_ usage, remains an open
challenge.
A critical first step in designing a ML-based model for molecular systems, is
identifying atomic descriptors that encode information of the local
environments of a chemical moiety10, 11. Over the recent years, various
flavours of atomic descriptors have been developed including atomic-density
ones which focus on characterizing _local_ environments12, 13, topology based
descriptors relying on graph-theoretical approaches14 to extract the
connectivity patterns in data and many-body tensor network representations
which serve to characterize the global structure of materials15, to name a
few. Most of these descriptors require human intervention in the selection of
various parameters. For example, the size of the local environment or the
number of basis functions one needs to describe the local configurations.
Since ML-based descriptors are designed to capture the physical and chemical
nature of molecular systems, interpreting and understanding their meaning is
important, and can lead to a more rational and also physical choice of the
hyper-parameters entering their definition.
In this work, we address the issue of interpretability, focusing on a specific
class of local-atomic descriptors namely; the Smooth Overlap of Atomic
Positions (SOAP)13, 16. SOAP has become widely popular for identifying
structural finger prints in systems such as liquid water17, 18, 19 and
inorganic crystals20, 21 and more recently, to develop coarse-grained
intermolecular potentials22. Here we examine how SOAP fingerprints are related
to standard order parameters (Figure 1, right panel) that are typically used
to describe the local structure of liquid water23. Moreover, we study how this
relationship changes from bulk liquid water to the air-water interface; a
prototypical system used to study the environmental effects of hydrophobic
interactions24, 25, 26.
To measure the relationship between SOAP-based descriptors and standard order
parameters, we deploy a recently developed technique, dubbed as the
Information Imbalance27 (IB). IB is a statistical test that determines the
relative information content between different distance measures defined on
the same data space. If applied to two different distances, it allows
determining if these distances are equivalent, unrelated, or if one of the two
is more predictive than the other. Specifically, we explore the IB between
SOAP descriptors and a wide variety of order parameters28, 29, 30, 31, 32, 33
that have been built on physical and chemical intuition to characterize
aqueous environments.
Using the IB, we first investigate if a suitable combination of the order
parameters is able to predict the similarity of local environments measured
using SOAP features. We find that even the best combination of order
parameters is able to predict the SOAP similarity only approximately. The
quality of the prediction is better for configurations close to the surface of
water, and is significantly improved by choosing a SOAP length scale parameter
$\sigma=0.25$ Å, which is much smaller than the value typically used
($\sigma=1$ Å). This result may not be too surprising as the structural
information embedded in SOAP descriptors likely contains a richer
characterisation of the local environments in water than the standard order
parameters which are often fine-tuned to capture specific chemical
interactions.
We also investigate the reverse problem, namely if SOAP descriptors are able
to predict the similarity as measured using chemical-intuition based order
parameters. Rather surprisingly, we find that most of these standard order
parameters can be predicted rather approximately. For example, the IB between
the SOAP similarity and the similarity measured by the number of hydrogen
bonds or the Local Structural Index (LSI), appears to be close to the value
observed for distances which are unrelated to each other. Interestingly, SOAP
predicts the standard coordination number the most accurately out of all the
order parameters we examine. The IB can be improved rather marginally by
reducing the value of $\sigma$. We provide some perspectives on the possible
origins of these discrepancies.
The paper is organized as follows. We begin in Section 1 with the Methods
employed in this work, including a summary of both SOAP and order parameters
we study, as well as the Information Imbalance technique. We then move on in
Section 2 to the Results where we illustrate the relationships we unravel
between the SOAP and order parameters that are obtained using the IB method
for both bulk and interfacial water. We then end in Section 3 with some
conclusions of our work.
## 2 Methods
### 2.1 Molecular Dynamics Simulations
Molecular Dynamics (MD) simulations are carried out using the LAMMPS package.
34 We use the TIP4P/2005 35 rigid water model for most of our work. We also
repeat some of our analysis using the TIP3P36 water model which is commonly
used in bio-molecular simulations.
Our initial simulation setup consists of a bulk water system with 729 water
molecules equilibrated at ambient temperature and pressure in a box with sides
27.9$\times$ 27.9$\times$ 27.9 Å. To construct an interface, we add a vaccuum
region of 139.5 Å in the z-direction and then equilibrate within the NVT
ensemble for 10 ns at 300 K with a timestep of 2 fs. This is followed by a
production run of 20 ns. The velocity-rescaling 37 thermostat is used with a
time constant of 100 fs. In our simulations, the real space cut-off for the
Coloumb and Lennard Jones (LJ) interactions is 15 Å. Long range corrections
are treated using the Particle-Particle Particle-Mesh (PPPM)38 solver for both
the Coulomb and LJ interactions. In order to validate the use of our model for
the air-water interface, we computed the surface tension for our simulation,
obtaining a value of (67.98 $\pm$ 0.74)mN/m, consistent with previous reports.
39. For the TIP3P model, we obtain a surface tension of (47.29 $\pm$ 0.40)
mN/m, which is also consistent with previous studies. 39
### 2.2 Identifying Bulk vs Surface Water Environments
Using our MD simulations, a total of approximately 5000 local environments are
sampled from the water-surface system, on which the SOAP power spectrum is
computed using the DScribe software package40. These environments are chosen
by randomly selecting a water molecule every 4 ps. Since we were interested in
understanding how the the relationship between SOAP fingerprints and order
parameters evolves from bulk water to the hydrophobic surface of water, we
characterised the interface using the Willard-Chandler Interface (WCI)41. In
brief, a coarse-grained density field is defined as a sum of Gaussian
functions, with a specified smoothening parameter ($\xi$) centered on all the
atoms. The interface is then chosen as the set of points for which the coarse-
grained density is half of the bulk density. The $\xi$ value used for the WCI
construction was 2.4 Å consistent with previous studies. The water density as
a function of distance from the WCI is then built, yielding the distribution
shown in the Supporting Information (SI Figure 1).
For our analysis, we define various layers from the density profile as
performed in several prior works42, kessler2015structure, which allows for
identifying surface and bulk water molecules. In most of our analysis, we
focus on comparing the bulk and surface as defined by those waters in layer 4
(Bulk) and layer 1 (L1) respectively (see SI Figure 1 for a visual depiction
of these layers relative to the WCI).
### 2.3 Descriptors for the Local Structure of Water
Figure 1: SOAP descriptors and order parameters (left and right respectively)
used in this work. The left panel (top) shows a cartoon of the orientational
and radial extent of the SOAP descriptors and the bottom panel summarizes the
definition of the SOAP power spectrum - see main text for more details. The
right panel shows snapshots of local water environments and some of the
species, distance and angular based criteria that goes into defining the
various order parameters. The symbols next to each variable is what is used
throughout the manuscript to refer to each variable.
In this work, we examine the relationship between chemically inspired order
parameters and SOAP-based descriptors using the Information Imbalance method.
In the following, we begin by summarizing the theory underlying the
construction of SOAP descriptors, (see Section 1.3.1) and subsequently, the
collection of different order parameters that we examine (see Section 1.3.2).
Finally, we also discuss the principles behind the Information Imbalance
method (see Section 1.4). Figure 1 illustrates the SOAP and order parameters
along with the respective symbols that are used throughout the manuscript.
#### 2.3.1 Smooth Overlap of Atomic Positions (SOAP)
SOAP has emerged in the last few years as a powerful method to describe the
local environments of atoms and molecules, allowing for a wide range of
applications from the study of structural properties of organic molecules44,
45, 13 and very recently, the properties of liquid water19, 18, 17.
In the context of SOAP, the local density of an atomic environment $\chi$ is
written as a sum of Gaussian functions with variance $\sigma^{2}$, centered on
all species that are neighbours of the central atom:
$\rho_{\chi}(\vec{r})=\sum_{i\in\chi}\exp{\left(\frac{-|\vec{r_{i}}-\vec{r}|^{2}}{2\sigma^{2}}\right)}$
(1)
In this work, we will show that the choice of the value of $\sigma$ parameter
plays a very important role in the ability of SOAP to predict chemical
properties. Specifically, this parameter determines the resolution of chemical
details of the water hydrogen bond network.
The default value of $\sigma$ in the DScribe package is 1.0 Å which to the
best of our knowledge, is the value used in previous studies using SOAP to
study the structure of liquid-water 18, 17.
The atomic neighbour density in equation 1 can be expanded on a basis of
radial basis functions and real spherical harmonics such that:
$\rho_{\chi}(\vec{r})=\sum_{n=0}^{nmax}\sum_{l=0}^{lmax}\sum_{m=-l}^{l}c_{nlm}g_{n}(r)Y_{lm}(\theta,\phi)$
(2)
For practical purposes, one defines the environment $\chi$ by a cut-off radius
($r_{cut}$) and also limits the number of radial and angular basis functions
($n_{max}$, $l_{max}$) used. In the following work, we have examined the
sensitivity of our results to changing $r_{cut}$ from 3.7 to 5.5 Å as well as
varying $n_{max}$ and $l_{max}$ between 10-12 and 6-8 respectively (see SI
Figures 2-5). We find a marginal drop of chemical information contained in the
SOAP descriptors (as reflected in the increase in the IB between SOAP and the
order parameters) as we increase $r_{cut}$, owing to the fact that the order
parameters we have examined, consider fluctuations only within the first
solvation shell of a given water environment. We also find that there is no
significant dependence of our results to the choice of $n_{max}$ and
$l_{max}$. Unless otherwise stated, the $r_{cut}$, $n_{max}$ and $l_{max}$ are
3.7 Å, 10 and 6 respectively.
By accumulating the expansion coefficients, a rotationally invariant power
spectrum can be defined such that,
$p_{nn^{\prime}l}(\chi)=\pi\sqrt{\frac{8}{2l+1}}\sum_{m}(c_{nlm})^{\dagger}c_{n^{{}^{\prime}}lm}$
(3)
Equation 3 defines the components of the SOAP features we will use in our
analysis.
#### 2.3.2 Order parameters
Orientational Tetrahedral Order (qtet):
The $q_{tet}$ order parameter is one of the most used local structural
quantities to describe liquid water29, 46, 28, 47, 48, 49. It measures how
much a reference water environment deviates from an ordered tetrahedron whose
vertices are defined by the bond vectors between the four nearest neighbouring
oxygen atoms of a central one. It takes as input the angles between the O-O
bond vectors and gives a value of 1 for a perfectly tetrahedral environment
and closer to zero for environments which are not tetrahedral. Specifically,
the tetrahedrality of a reference water molecule is defined as:
$q_{tet}=1-\frac{3}{8}\sum_{i=1}^{3}\sum_{j=i+1}^{4}\left(\cos\phi_{ij}+\frac{1}{3}\right)^{2}$
(4)
Where $\phi_{ij}$ is the angle between the oxygen molecule of the reference
water and its nearest four neighbours (indexed with $i$ and $j$)
Translational Tetrahedral Order ($S_{k}$):
The translational tetrahedral order parameter is another measure of how much a
reference water environment deviates from a regular tetrahedron29. Whereas
$q_{tet}$ focuses on the angles between the O-O bond vectors, $S_{k}$ is
computed as the variance between the O-O distances of the four nearest water
molecules to a central water:
$S_{k}=1-\frac{1}{3}\sum_{k=1}^{4}\frac{\left(r_{k}-\bar{r}\right)^{2}}{4\bar{r}^{2}}$
(5)
$r_{k}$ is the distance between a reference water molecule and its $kth$
neighbour and $\bar{r}$ is the mean of these distances. It has been shown in
previous works that $S_{k}$ is more sensitive to the local density variations
compared to $q_{tet}$50.
Local Structural Index (LSI):
The LSI is another important variable which has been used to study the
structure of water in the bulk under different thermodynamic conditions51, 30,
52. It is obtained by ranking the O-O distances from an $ith$ central water
molecule such that $r_{1}<r_{2}<...<r_{i}<r_{i+1}<...r_{n}<3.7$ Å$<r_{n+1}$,
and then subsequently, the LSI is computed as:
$\text{LSI}=\frac{1}{n}\sum_{i=1}^{n}\left(\Delta(i)-\bar{\Delta}\right)^{2}$
(6)
where $\Delta(i)=r_{i+1}-r_{i}$ and $\bar{\Delta}$ is the arithmetic mean of
$\Delta(i)$. A high value of LSI implies a larger separation between the first
and second solvation shell and points to a more ordered water environment,
while a lower value is interpreted as a more disordered environment.
Distance to the fifth Oxygen ($d_{5}$):
The $d_{5}$ is defined as the distance between a central oxygen atom and its
fifth closest oxygen atom31, 53, 54. A large value of $d_{5}$ points to a high
separation between the first and second solvation shell of a water environment
and is interpreted as a locally ordered structure. A smaller value indicates a
smaller separation between the first and second solvation shells and with a
similar logic, a disordered local environment.
Coordination Number (C.N.):
The coordination number quantifies the average number of atoms that surround a
chosen central site within some radial cutoff. It can be computed from the
radial integral of the radial distribution function (RDF). In order to have a
smooth and continuous definition of the coordination number, we use a
switching function, commonly done in the construction of different types of
collective variables55, 56:
$C.N.=\sum_{j=1}^{N}\frac{1-\left(\frac{r_{j}}{r_{cut}}\right)^{12}}{1-\left(\frac{r_{j}}{r_{cut}}\right)^{28}}$
(7)
where $r_{j}$ is the distance between a central oxygen atom and atom $j$.
Number of Hydrogen Bonds ($N_{H.B.}$):
We adopt the definition of hydrogen bonding by Luzar and Chandler32, 33. This
definition is based on a geometrical criterion and considers two water
molecules to be hydrogen bonded when the distance between the donating and
accepting oxygen atoms (OD and OA) is within 3.5 Å and the angle formed by the
bond vector between the donating hydrogen and oxygen (HD and OD) and the bond
vector between OD and OA is less than 30∘.
Voronoi Density ($\rho_{voro}$):
A quantitative measure of the local density ($\rho_{voro}$) in water can be
extracted using a Voronoi tessellation of the water network 57, 58. The
$\rho_{voro}$ is then defined as the inverse of the Voronoi volume associated
with a single water molecule which is a sum of the atomic contributions coming
from the oxygen and two hydrogen atoms. The Voronoi tesselation is carried out
using the Voro$++$ code 59.
SOAP distance from ice (d${}_{\text{ice}}$):
From the SOAP descriptors (power spectrum) show in equation 3, it is then
possible to define distances between structures. Several groups including ours
have examined how SOAP environments in liquid water compare to different
phases of ice18, 19, 17. In this work, one of these variables we use is
comparing SOAP environments in liquid water to those in hexagonal ice (ice 1h)
which is referred to as $d_{ice}$.
$d_{ice}=\sqrt{1-\frac{\vec{p}(\chi_{\text{water}})\cdot\vec{p}(\chi_{\text{ice
1h}})}{|\vec{p}(\chi_{\text{water}})||\vec{p}(\chi_{\text{ice 1h}})|}}$ (8)
Where $\vec{p}(\chi_{\text{water}})$ and $\vec{p}(\chi_{\text{ice 1h}})$ are
the SOAP feature vectors for the liquid water and hexagonal ice environments
respectively.
### 2.4 Information Imbalance (IB)
The Information Imbalance (IB) is a recently developed method which can be
used to quantify the relative amount of information between different types of
variables which may or may not have the same measures of distance. The reader
is referred to the original work for details27. Here, we summarize the key
ideas behind the method and why it provides a powerful tool for application in
the context of the problems we address here.
Given a dataset with $N$ data points and characterised by $F$ features, we can
define distance measures A and B between the data points, such that A and B
are computed using any subset of the feature space $F$ of choice. With these
distances in hand, the IB is then defined as:
$\Delta\left(A\rightarrow B\right)=\frac{2}{N}\langle
R^{B}|R^{A}=1\rangle=\frac{2}{N^{2}}\sum_{i,j:R_{ij}^{A}=1}R_{ij}^{B}$ (9)
Where $R_{ij}^{A}$ and $R_{ij}^{B}$ are the rank matrices obtained from
distances A and B respectively, such that $R_{ij}^{A}=1$ if $j$ is the first
nearest neighbour of $i$ and $R_{ij}^{A}=2$ if $j$ is the second nearest
neighbour of $i$ and so on. With this definition, if $\Delta\left(A\rightarrow
B\right)\sim 0$ this means A can fully predict B, while
$\Delta\left(A\rightarrow B\right)\sim 1$ implies that A cannot predict B, as
the ranks estimated with B are uncorrelated to those estimated with A. It is
important to note that the IB is by definition asymmetric and thus one can
examine predictability between different distances in both directions; if for
example, $\Delta(A\rightarrow B)\sim 0.1$ and $\Delta(B\rightarrow A)\sim
0.4$, A will be able to predict B with better reliability than the reverse.
## 3 Results
### 3.1 Predicting SOAP from Order Parameters
In the ensuing analysis, we explore the relative information content between
SOAP and order parameters using the IB. We begin by determining the ability of
the order parameters to predict the full SOAP feature space and specifically,
how sensitive this predictability is to the choice of the parameter $\sigma$
used in SOAP. Furthermore, we also examine how the IB changes as one moves
from bulk water to the air-water interface.
Figure 2: Information Imbalance between combinations of order parameters and
the SOAP space. Comparison for Bulk (Red) and Surface (Black) and for
$\sigma=1.0$ Å (Dotted line ) and $\sigma=0.25$ Å (Full line). For
$\sigma=1.0$ Å one needs a minimum of about 3 order parameters to describe the
local fluctuations of the water environments contained in the full SOAP space,
both in the bulk and at the Surface. For $\sigma=0.25$, the number of order
parameters needed increases by approximately 3.
In Figure 2 we show the IB between order parameters and SOAP, which encodes
how many different order parameters are needed to predict the distances
computed with the full high-dimensional SOAP vector. In order to obtain the IB
for different combinations, we employ a greedy optimization whereby one
iteratively selects the best combination of order parameters to minimize the
IB between that combination and the SOAP space. This protocol can also be
employed to select the $d$-dimensional compact SOAP descriptor that minimizes
the Information Imbalance between SOAP and a target variable. For more
details, we refer the reader to the original manuscript on the IB method27.
Figure 2 shows that using $\sigma=1.0$ Å, for both bulk and surface, a
combination of three order parameters minimizes the IB to approximately 0.45
and 0.35 respectively. Recall, that an IB of 0.45 implies that if the data
space included, say, 1000 local environments, the first neighbour environments
according to the order parameters will be, on average the 225th neighbour
according to SOAP. This is rather far from optimal since it suggests that
local water environments predicted to be similar in the space of the order
parameters, are quite distant using SOAP.
Using $\sigma=0.25$ Å, we notice a consistent increase in the number of order
parameters needed to obtain the minimum IB; five order parameters are needed
to obtain roughly the same IB as with $\sigma=1.0$ Å. This shows that by
reducing the coarse-graining length of the Gaussian density from which the
SOAP power spectrum was built, we essentially add more chemical information
and consequently, this requires more order parameters to correctly describe.
The sensitivity of the interplay between the size of the basis set and the
choice of $\sigma$ in describing atomic environments has recently been
discussed by Pozdnyakov and co-workers60. They show that in order to obtain
the same sensitivity of the density expansion coefficients to vibrational
distortions in a single methane molecule, the $\sigma$ should be approximately
half that of the minimal interatomic distance.
Figure 3: Information Imbalance between combinations of order parameters and
the full SOAP space for $\sigma=0.25$, showing the selected variables for both
bulk and surface. It is noteworthy that the difference between the Bulk and
Surface for $\sigma=0.25$ Å is reflected in the difference between the
combinations of variables needed to minimize the IB between the order
parameters and the SOAP space. We can also see that $\rho_{voro}$ and
$d_{ice}$ are consistently important to describe the SOAP space for both the
bulk and surface whereas $q_{tet}$ for example becomes more important for the
surface due to the enhanced orientational correlations at the Surface compared
to the Bulk.
While Figure 2 shows that with $\sigma=0.25\ \text{\AA}$, the gap in the IB
between the bulk and surface becomes much less pronounced compared to
$\sigma=1.0\ \text{\AA}$, the combination of order parameters that are needed
to describe the fluctuations in the bulk and the surface are not the same. In
Figure 3, we show the order parameters that lead to the most predictive
distance in the SOAP space. $\rho_{voro}$ and the one-dimensional SOAP based
descriptor ($d_{ice}$) consistently appear to be important for both the bulk
and surface. Remarkably, the LSI parameter which is often used to study local-
water structure in the bulk61, 30, does not appear in any of the chemical
combinations that lowers the IB. Interestingly, the tetrahedral order
parameter, $q_{tet}$ plays a more important role at the surface due to the
fact that there is an enhanced orientational order. These trends are
consistently reproduced for simulations of the surface of water using the
TIP3P water model (see SI Figure 8).
### 3.2 Predicting Order Parameters from SOAP
The preceding analysis examined how well the distances in the order parameter
space can predict the full SOAP distances. Next we move on to discover the
ability of SOAP to capture the information contained in single order
parameters and at the same time, we use the IB to identify the optimal number
of SOAP components required to perform a prediction.
Figure 4 shows the IB between SOAP space and the individual order parameters
for both bulk and surface (left and right panels respectively), as a function
of the number of components. Note that for the SOAP parameters deployed here,
the full power spectrum is a vector with dimension $D=4410$. The easiest
variable to predict is the coordination number ($C.N.$) and the most difficult
to predict is the total number of hydrogen bonds ($N_{H.B.}$). Overall, the
effect of changing from $\sigma=1.0\ \text{\AA}$ to $\sigma=0.25\ \text{\AA}$
is consistent with the previous observations seen in Figure 2 and Figure 3.
The IB is reduced for smaller $\sigma$ \- the solid circles shown in both
panels correspond to the optimal IB obtained using the larger $\sigma$ value.
By increasing the local atomic resolution, the IB can decrease by up to 50%
and thus enhance the resolution of SOAP for the prediction of several order
parameters.
Figure 4: Convergence of the IB between the SOAP space and individual order
parameters with $\sigma=0.25$ Å for both bulk (left) and surface (right) . The
solid circles represent the optimized IB obtained using $\sigma=1.0$ Å . The
rate of convergence of the IB for different variables are different however,
with a minimum of about 10-20 SOAP vector components, the minimum IB is
reached. Using $\sigma=0.25$ Å significantly increases the information
contained in the SOAP space about the order parameters.
In similar spirit to the analysis presented earlier in Figure 3, the IB
between SOAP and order parameters identifies chemical coordinates such as
coordination number, $d_{ice}$, $d_{5}$ and $\rho_{voro}$, whose chemical
information is well characterized by SOAP. Furthermore, although different
order parameters plateau to the optimal IB with different rates, one only
needs a total of 10-20 components of the full SOAP power spectrum to predict
these order parameters. These trends are found in both bulk and surface liquid
water situations.
Perhaps what is more surprising is that SOAP does not seem to be able to
predict accurately the number of hydrogen bonds, LSI and $q_{tet}$. On the one
hand, this suggests that the SOAP features may not be complete in terms of
their information content of the underlying chemical system. However, many of
these order parameters are constructed based on strict geometrical cutoffs.
For example; along distances or angles or number of neighbours which is then
reflected in the IB.
To illustrate this effect more clearly, we show in Figure 5 a set of water
environments which compares and contrasts distances computed between SOAP,
$q_{tet}$ and hydrogen bonding. We begin with the top panel comparing SOAP and
$q_{tet}$. Within an $r_{cut}$ of 3.7 Å, there are two interstitial oxygen
atoms (purple arrow). Since the definition of $q_{tet}$ only looks at the
first 4 nearest oxygens, the two environments are flagged as non tetrahedral
(left) and highly tetrahedral (right). Due to the restriction of focusing on
the nearest 4 oxygens, the $q_{tet}$ parameter picks up an angle that clearly
deviates from tetrahedrality. In fact, when one restricts the SOAP computation
to only the first four neighbours, the SOAP space is completely predictive of
$q_{tet}$ (see SI Figure 10).
Figure 5: Snapshot of two sets of environments $\chi$, $\chi^{\prime}$ and
$\chi^{\prime\prime}$, $\chi^{\prime\prime\prime}$ which are nearest
neighbours in SOAP space but are distant in $q_{tet}$ (top panel) and
$N_{H.B.}$ (bottom panel) space. Since the definition of $q_{tet}$ focuses on
the first four neighbours of a central atom within the first solvation shell,
the two environments are flagged as not tetrahedral (top left) and tetrahedral
(top right), while the SOAP space predicts the two environments to be similar.
For $N_{H.B.}$, its strict angular cut-off picks up slight angle changes
($\sim$ 2∘) which labels the two environments as having different number of
hydrogen bonds. However, these are clearly two environments that are more
similar than different.
In similar spirit to the analysis presented for $q_{tet}$, the bottom panel
shows two environments that are again close in SOAP space, but far in terms of
hydrogen bonding. Specifically, the bottom left panel shows an environment
that accepts and donates two hydrogen bonds (N${}_{H.B.}=4$) while the bottom
right shows a small change in the local geometry where the angle used for the
hydrogen bonding criterion changes from 29∘ to 31∘ resulting in a defect that
now accepts two but donates only one hydrogen bond (N${}_{H.B.}=3$). In
hydrogen-bonding space, these configurations are topologically different but
in SOAP space, these are, unsurprisingly very similar.
Figure 6: Convergence of the IB between SOAP and selected order parameters, as
a function of the number of SOAP vector components. The compression is done
for all the layers of the density profile depicted in the supplementary
material. These results confirm previous reports showing that most of the
significant structural changes in the water environments occurs in the topmost
layer.
### 3.3 Predictions Within Order Parameters
Our analysis shows that there is a complex interplay of different order
parameters and that they need to be used in combination in order to predict
SOAP similarity both in the bulk and at the air-water interface. In the field
of aqueous science where the structure and dynamics of liquid water is studied
in terms of different local structures, it is quite common to synonymously
associate the different order parameters such as $d_{5}$, LSI and $q_{tet}$ .
However, these order parameters are not equivalent.
Figure 7: Information Imbalance plane 27 for the space of order parameters,
showing the IB obtained between different pairs of order parameters. In the
Bulk (left) the lowest IB obtained is $\sim$ 0.7 which is between $C.N.$ and
$d_{5}$. The rest of the variables are all not informative of each other. In
the Surface (right), the lowest IB obtained $\sim$ 0.4 is also between $C.N.$
and $d_{5}$. In general, most of the variables are uninformative of each other
both in the Bulk and on the Surface.
Figure 7 shows the IB between all pairs of order parameters for both bulk and
surface environments. For bulk environments, we observe that the lowest
imbalance obtained is approximately 0.7; between the coordination number and
$d_{5}$. Essentially all the other order parameters are not informative of
each other consistent with previous work from our group19. Moving to the
surface environments marginally improves the predictability for the
coordination number and $d_{5}$ pair. All in all, it is clear that the
different order parameters used to characterized local water structure contain
very little information about each other. For example, considering the IB
between tetrahedrality (qtet) and local density ($\rho_{voro}$) which is
closer to 1, shows that they provide different information about the local
environment. Thus, an environment that is more tetrahedral may not have a
lower local density and vice-versa.
### 3.4 Diagnostics for Water Structure
We have used the IB approach to examine the coupling between ML-based local
atomic descriptors and chemistry-inspired order parameters. We then explored
whether this could be used as a diagnostic tool to study the length-scale of
the perturbations induced by the prescence of the surface.
Earlier, we compared the IB for bulk and surface water environments, where the
latter focused on interfacial waters that reside only within $\sim$ 3.0 Å of
the WCI. Examining the relative density of water with respect to the WCI (see
SI Figure 1), one observes that the correlations albeit weaker, extend up to 1
nanometer from the surface. The extent of the thickness of this interface as
probed by both surface sensitive vibrational spectroscopy experiments and
simulations, and how it is manifested in terms of local structure, continues
to be a topic of active research. 42, 62, 63.
Since the IB appears to provide a sensitive measure of the surface water
environments, we performed the compression of the full SOAP space for selected
order parameters separately for different layers of water with respect to the
WCI (see SI Figure 1 for an illustration of how the layers are defined). The
left and right panels of Figure 6 shows the IB between SOAP and the
coordination number and $\rho_{voro}$ respectively, as a function of the
number of SOAP components. Our results show that the IB in the second and
third layer are essentially indistinguishable from the bulk. This shows that
there is essentially only a very thin layer of water ($\sim$ 3.0 Å near the
surface) whose local structural properties are significantly different,
consistent with previous studies 42.
## 4 Conclusions
In this work, we have used a recently developed statistical test (IB) to
examine the relationships between the SOAP descriptors and chemically inspired
order parameters that are used to characterize local water structure. We focus
our analysis on how these relationships evolve as one moves from bulk water to
the hydrophobic surface of water.
Examining the IB between the full SOAP space and order parameters shows that
these two classes of variables do not contain the same information. While the
chemical information encoded in the SOAP descriptors can be improved by
reducing the Gaussian smearing, the IB between the order parameters and the
SOAP descriptors obtained is far from optimal. The predictability improves
marginally for water environments sampled at the interface. Given that the
SOAP descriptors are designed to be generic and do not probe specific chemical
details, this observation may not be too surprising. The consistency between
SOAP and the order parameters could be improved by including features that
probe directly the fluctuations of the hydrogen bond network.
Order parameters like the ones we have investigated here to probe the local
structure of water, are used ubiquitously for analyzing molecular dynamics
simulations of liquid water. In particular, these are often used to assign
whether the water structure arises from a low or high density local
environment64, 51, 19. However, our analysis suggests that most of these order
parameters are somewhat independent of each other and cannot be used
synonymously. Furthermore, the SOAP descriptors seem to be probing the local
environment of water molecules in a much less specific manner. One may argue
that order parameters such as the hydrogen-bonding criterion are designed to
be very sensitive to subtle changes in local topology which cannot be captured
by SOAP. We believe that our results suggest that sensitivity analysis of
geometric criteria used in chemistry-inspired order parameters would be
appropriate.
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## 5 Supplementary Information
Figure S1: Snapshot of MD simulaton box (Left)-The water molecules are color-
coded based on their distance from the WCI. Number density profile of water
molecules as a function of the distance from the WCI (Right), the colors
correspond to the distinct structural layers that occur and have been labelled
(L1-L4) for Surface-Bulk respectively. Figure S2:
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ for $\sigma=0.25$
using ($n_{max}$,$l_{max}$) = (10,8) (Red Symbols) and ($n_{max}$,$l_{max}$) =
(10,6) (Green Symbols). For fixed $n_{max}$, the predictability does not
change with an increase in $l_{max}$ Figure S3:
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ for $\sigma=0.25$
using ($n_{max}$,$l_{max}$) = (12,6) (Red Symbols) and ($n_{max}$,$l_{max}$) =
(10,6) (Green Symbols). For fixed $l_{max}$, the predictability does not
change with an increase in $n_{max}$ Figure S4:
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ for $\sigma=0.25$
using ($n_{max}$,$l_{max}$) = (12,8) (Red Symbols) and ($n_{max}$,$l_{max}$) =
(10,6) (Green Symbols). Changing both $n_{max}$ and $l_{max}$ does not
significantly change the predictabilities Figure S5:
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ for $\sigma=0.25$
and $r_{cut}$ = 3.7 (Red), $r_{cut}$ = 4.5 (Blue) and $r_{cut}$ = 5.5 (Green).
In general, an $r_{cut}$ of 5.5 makes the SOAP consistently less informative
about the chemical variables (although very slightly). The main thing to
notice is that an $r_{cut}$ of 3.7 is able to capture as much as SOAP can, all
the local fluctuations contained in the Order Parameter. Figure S6: Evolution
of $\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ as a function
of the layer definitions showed in SI figure 1 for $\sigma=1.0$ Å (Left) and
$\sigma=0.25$ Å (Right). There is a general decrease in the IB as we go from a
larger to a smaller $\sigma$. This also confirms that much of the structural
changes that occur is at the topmost layer (L1) Figure S7: $\Delta(\text{Order
Parameters}\longrightarrow\text{SOAP}$) for the TIP3P and TIP4P/2005 water
model for the Surface layer. The optimum set of 5-variables selected are
consistent across the two models with only a difference in one selected
variable ($\text{q}_{tet}$ for the TIP4P/2005 and $\text{S}_{k}$ for TIP3P).
Figure S8: $\Delta(\text{Order Parameters}\longrightarrow\text{SOAP}$) for the
TIP3P and TIP4P/2005 water model for the Bulk layer. The results are
consistent across the two water models. Figure S9: Convergence of
$\Delta(\text{SOAP}\longrightarrow\text{Order Parameters})$ as a function of
the number of SOAP components for the TIP3P water model. The solid circles
represent the optimized IB obtained for the TIP4P/2005 model for the specified
color coded chemical variable. These results show very small differences in
the IB across the two models and hence are consistent. Figure S10: Information
Imbalance between SOAP and $\text{q}_{tet}$ for SOAP descriptors computed with
6,5 and 4 nearest neighbours. As the number of neighbours used to compute the
SOAP descriptors decreases, the ability of SOAP to predict $\text{q}_{tet}$
increases. With 4 nearest neighbours, the SOAP space is completely predictive
of $\text{q}_{tet}$
|
# FJMP: Factorized Joint Multi-Agent Motion Prediction over Learned Directed
Acyclic Interaction Graphs
Luke Rowe, Martin Ethier, Eli-Henry Dykhne, Krzysztof Czarnecki
School of Computer Science, University of Waterloo
{l6rowe, methier, ehdykhne<EMAIL_ADDRESS>
https://rluke22.github.io/FJMP Corresponding author.
###### Abstract
Predicting the future motion of road agents is a critical task in an
autonomous driving pipeline. In this work, we address the problem of
generating a set of scene-level, or joint, future trajectory predictions in
multi-agent driving scenarios. To this end, we propose FJMP, a Factorized
Joint Motion Prediction framework for multi-agent interactive driving
scenarios. FJMP models the future scene interaction dynamics as a sparse
directed interaction graph, where edges denote explicit interactions between
agents. We then prune the graph into a directed acyclic graph (DAG) and
decompose the joint prediction task into a sequence of marginal and
conditional predictions according to the partial ordering of the DAG, where
joint future trajectories are decoded using a directed acyclic graph neural
network (DAGNN). We conduct experiments on the INTERACTION and Argoverse 2
datasets and demonstrate that FJMP produces more accurate and scene-consistent
joint trajectory predictions than non-factorized approaches, especially on the
most interactive and kinematically interesting agents. FJMP ranks 1st on the
multi-agent test leaderboard of the INTERACTION dataset.
## 1 Introduction
Multi-agent motion prediction is an important task in a self-driving pipeline,
and it involves forecasting the future positions of multiple agents in complex
driving environments. Most existing works in multi-agent motion prediction
predict a set of marginal trajectories for each agent [30, 19, 46, 32, 14, 40,
21], and thus fail to explicitly account for agent interactions in the future.
This results in trajectory predictions that are not consistent with each
other. For example, the most likely marginal prediction for two interacting
agents may collide with each other, when in reality a negotiation between
agents to avoid collision is far more likely. As scene-consistent future
predictions are critical for downstream planning, recent work has shifted
toward generating a set of scene-level, or joint, future trajectory
predictions [38, 17, 5, 34, 7, 8], whereby each mode consists of a future
trajectory prediction for each agent and the predicted trajectories are
consistent with each other.
Figure 1: An illustration of the directed acyclic interaction graph, comprised
of colored nodes and red arrows. The dotted black lines denote the ground-
truth futures over a short time horizon. FJMP first produces marginal
predictions for the green (dashed) nodes, followed by a conditional prediction
for the yellow (dotted) node and a conditional prediction for the purple
(solid) node, with conditioning on the predicted future of the parent nodes in
the graph.
In this work, we focus on the problem of generating a set of joint future
trajectory predictions in multi-agent driving scenarios. Unlike marginal
prediction, the joint trajectory prediction space grows exponentially with the
number of agents in the scene, which makes this prediction setting
particularly challenging. A common approach for this setting is to
simultaneously predict the joint futures for all agents in the scene [17, 5,
34, 18, 8]; however, this approach fails to explicitly reason about future
interactions in the joint predictions. To address this limitation, recent work
has shown that decomposing the joint prediction task of two interacting agents
into a marginal prediction for the influencer agent and a conditional
prediction for the reactor agent, where the reactor’s prediction conditions on
the predicted future of the influencer, can generate more accurate and scene-
consistent joint predictions than methods that generate marginal predictions
or simultaneous joint predictions [38, 28]. However, these methods are
optimized for the joint prediction of only two interacting agents, and they do
not efficiently scale to scenes with a large number of interacting agents.
To address these limitations of existing joint motion predictors, in this work
we propose FJMP – a Factorized Joint Motion Prediction framework that
efficiently generates joint predictions for driving scenarios with an
arbitrarily large number of agents by factorizing the joint prediction task
into a sequence of marginal and conditional predictions. FJMP models the
future scene interaction dynamics as a sparse directed interaction graph,
where an edge denotes an explicit interaction between a pair of agents, and
the direction of the edge is determined by their influencer-reactor
relationship [38, 25, 27], as can be seen in Fig. 1. We propose a mechanism to
efficiently prune the interaction graph into a directed acyclic graph (DAG).
Joint future trajectory predictions are then decoded as a sequence of marginal
and conditional predictions according to the partial ordering of the DAG,
whereby marginal predictions are generated for the source node(s) in the DAG
and conditional predictions are generated for non-source nodes that condition
on the predicted future of their parents in the DAG. To enable this sequential
trajectory decoding, we adapt a lightweight directed acyclic graph neural
network (DAGNN) [39] architecture for efficiently processing predicted future
information through the DAG and decoding the marginal and conditional
trajectory predictions. Our main contributions can be summarized as follows:
* •
We propose FJMP, a novel joint motion prediction framework that generates
factorized joint trajectory predictions over sparse directed acyclic
interaction graphs. To our knowledge, FJMP is the first framework that enables
scalable factorized joint prediction on scenes with arbitrarily many
interacting agents.
* •
We validate our proposed method on both the multi-agent INTERACTION dataset
and the Argoverse 2 dataset and demonstrate that FJMP produces scene-
consistent joint predictions for scenes with up to 50 agents that outperform
non-factorized approaches, especially on the most interactive and
kinematically complex agents. FJMP achieves state-of-the-art performance
across several metrics on the challenging multi-agent prediction benchmark of
the INTERACTION dataset and ranks 1st on the official leaderboard.
## 2 Related Work
### 2.1 Motion Prediction in Driving Scenarios
Given the recent growing interest in autonomous driving, many large-scale
motion prediction driving datasets [6, 44, 13, 3, 49] have been publically
released, which has enabled rapid progress in the development of data-driven
motion prediction methods. Recurrent neural networks (RNNs) are a popular
choice for encoding agent trajectories [17, 15, 16, 10, 23, 7] and
convolutional neural networks (CNNs) are widely used in earlier works to
process the birds-eye view (BEV) rasterized encoding of the High-Definition
(HD) map [18, 5, 35, 33, 7, 15]. As rasterized HD-map encodings do not
explicitly capture the topological structure of the lanes and are constrained
by a limited receptive field, recent methods have proposed vectorized [14, 19,
32, 36], lane graph [30, 16, 9], and point cloud [46] representations for the
HD-map encoding. Inspired by the success of transformers in both natural
language processing [41, 11] and vision [12], several end-to-end transformer-
based methods have recently been proposed for motion prediction [18, 34, 32,
36, 20, 50, 21]. However, many of these transformer-based methods are
extremely costly in model size and inference speed, which makes them
impractical for use in real-world settings. FJMP adopts a LaneGCN-inspired
architecture [30] due to its strong performance on competitive benchmarks [6],
while retaining a small model size and fast inference speed.
### 2.2 Interaction Modeling for Motion Prediction
Data-driven methods typically use attention-based mechanisms [32, 34, 18, 30,
17, 50] or graph neural networks (GNNs) [5, 48, 29, 31, 14, 21, 25, 4] to
model agent interactions for motion prediction. Recent works have demonstrated
the importance of not only modeling agent interactions in the observed agent
histories but also reasoning explicitly about the agent interactions that may
occur in the future [36, 38, 1, 28, 26, 27]. MTR [36] proposes to generate
future trajectory hypotheses as an auxiliary task, where the future hypotheses
are fed into the interaction module so that it can better reason about future
interactions. Multiple works reason about future interactions through
predicted pairwise influencer-reactor relationships [38, 28, 27], where the
agent who reaches the conflict point first is defined as the influencer, and
the reactor otherwise. FJMP uses attention to model interactions in the agent
histories and constructs a sparse interaction graph based on pairwise
influencer-reactor relationships to model future interactions.
Figure 2: Illustration of the proposed FJMP framework. (a) Agent histories and
the HD-Map are first processed by a LaneGCN-inspired feature encoder. (b)
During training, the LaneGCN-encoded features are fed into an auxiliary future
proposal decoder trained with a regression loss to encourage the LaneGCN
features to be future-aware. (c) The future-aware LaneGCN-features are
processed by a GNN that predicts the pairwise influencer-reactor relationships
supervised by a focal loss. A directed interaction graph $\mathcal{G}$ is
constructed from the predicted edge probabilities and cycles are removed via
an efficient “dagification” procedure. (d). The predicted DAG and future-aware
LaneGCN features are fed into a factorized DAGNN-based trajectory decoder (red
agent removed for simplicity), which produces $K$ ($K=2$ shown above)
factorized joint futures in parallel and is supervised by a joint regression
loss.
### 2.3 Joint Motion Prediction
The majority of existing motion prediction systems generate marginal
predictions for each agent [40, 16, 37, 10, 14, 35, 33, 23, 20, 15, 2, 47, 30,
19, 46, 50, 32, 43, 21, 36, 48, 9]; however, marginal predictions lack an
association of futures across agents. Recent works have explored generating
simultaneous joint predictions [34, 5, 8, 17, 18], but these methods do not
explicitly reason about future interactions in the joint predictions. Other
works generate joint predictions for two-agent interactive scenarios by
selecting $K$ joint futures among all $K^{2}$ possible combinations of the
marginal predictions [36, 45], which quickly becomes intractable as the number
of agents in the scene increases. ScePT [7] proposes to handle the
exponentially growing joint prediction space by decomposing joint prediction
into the prediction of interactive cliques. However, the density of large
cliques imposes a severe computational burden at inference time, which
requires ScePT to upper-bound the maximum clique size to 4. To avoid the
computational burden associated with dense interaction graphs, FJMP models
future interactions as a sparse interaction graph consisting only of the
strongest interactions, which enables efficient joint decoding over
interactive scenarios with many interacting agents.
Our proposed method is most closely related to M2I [38], which first predicts
the influencer-reactor relationship between a pair of interacting agents and
then generates a marginal prediction for the influencer agent followed by a
conditional prediction for the reactor agent. However, we differ from M2I in
three critical ways. First, M2I is designed specifically to perform joint
prediction of two interacting agents, as their model design assumes one
influencer agent and one reactor agent. In contrast, FJMP naturally scales to
an arbitrary number of interacting agents, where an agent may have multiple
influencers and influence multiple reactors. Second, M2I requires a costly
inference-time procedure that does not scale to multiple agents whereby $N$
conditional predictions are generated for each marginal prediction, resulting
in $N^{2}$ joint predictions that are pruned to $K=6$ based on predicted
likelihood. On the contrary, FJMP coherently aligns the joint predictions of a
given modality through the DAG and directly produces $K=6$ factorized joint
predictions without any required pruning, which allows the system to
seamlessly scale to scenes with an arbitrarily large number of agents. Third,
M2I uses separate decoders for the marginal and conditional prediction,
whereas FJMP decodes both marginal and conditional predictions using the same
decoder, making it more parameter-efficient.
## 3 FJMP
In this section, we describe our proposed factorized joint motion prediction
framework, illustrated in Fig. 2.
### 3.1 Preliminaries
#### 3.1.1 Proposed Joint Factorization
The goal of multi-agent joint motion prediction is to predict the future
$T_{\text{fut}}$ timesteps of $N$ dynamic agents in a scene given the past
motion of the $N$ agents and the structure of the HD-Map. As there are
multiple possible futures for a given past, the joint motion prediction task
involves predicting $K>1$ modalities, whereby each modality consists of a
predicted future for each agent in the scene. We let $X$ and $Y$ denote the
past trajectories and future trajectories for all $N$ agents in the scene,
respectively, where $X_{\mathcal{S}}$ denotes the past trajectories for all
agents in the set $\mathcal{S}\subseteq[N]$, and $Y_{\mathcal{S}}$ is defined
similarly. Moreover, we let $C$ be an encoding of the HD-Map context.
We first propose to model the future scene interaction dynamics as a DAG
$\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$, where the vertices
$\mathcal{V}=[N]$ correspond to the $N$ dynamic agents in the scene, and a
directed edge $e_{mn}\in\mathcal{E}$; $m,n\in[N]$, denotes an explicit
interaction between agents $m$ and $n$ whereby $m$ is the influencer and $n$
is the reactor of the interaction. We propose to factorize the joint future
trajectory distribution $P(Y|X,C)$ over the DAG $\mathcal{G}$ as follows:
$\displaystyle
P(Y|X,C)=\prod_{n=0}^{N-1}P(Y_{\\{n\\}}|Y_{\text{pa}_{\mathcal{G}}(n)},X,C),$
(1)
where $\text{pa}_{\mathcal{G}}(n)$ denotes the set containing the parents of
node $n$ in $\mathcal{G}$. Intuitively, the proposed joint factorization can
be interpreted as an inductive bias that encourages accounting for the
predicted future of the agent(s) that influence agent $n$ when predicting the
future of agent $n$. We hypothesize that this inductive bias will ease the
complexity of learning the joint distribution when compared to methods that
produce a joint prediction for all $N$ agents simultaneously.
#### 3.1.2 Input Preprocessing
The past trajectory of a given agent is expressed as a sequence of
$T_{\text{obs}}$ states, which contains the 2D position, the velocity, and the
heading of the agent at each timestep. We denote the past state of agent $n$
at timestep $t,t\in[T_{\text{obs}}]$, by
$\mathbf{x}^{n}_{t}=[\mathbf{p}^{n}_{t},\mathbf{v}^{n}_{t},\psi^{n}_{t}]$,
where $\mathbf{p}^{n}_{t}\in\mathbb{R}^{2}$ is the position,
$\mathbf{v}^{n}_{t}\in\mathbb{R}^{2}$ is the velocity, and
$\psi^{n}_{t}\in\mathbb{R}$ is the yaw angle. We are also provided the agent
type $a^{n}$. As in LaneGCN [30], we convert the sequence of 2D positional
coordinates of each agent $n$ to a sequence of coordinate displacements:
$\hat{\mathbf{p}}_{t}^{n}=\mathbf{p}^{n}_{t}-\mathbf{p}^{n}_{t-1}$ for all
$t$. We encode the HD-Map as a lane graph with $M$ nodes, each denoting the
location of the midpoint of a lane centerline segment. Using the lane graph
construction proposed in LaneGCN, four adjacency matrices,
$\\{\mathbf{A}_{i}\\}_{i\in\\{\text{pre, suc, left, right}\\}}$,
$\mathbf{A}_{i}\in\mathbb{R}^{M\times M}$, are calculated to represent the
predecessor, successor, left, and right node connectivities in the lane graph,
respectively. Our system takes as input the $M$ lane node positional
coordinates, the lane node connectivities
$\\{\mathbf{A}_{i}\\}_{i\in\\{\text{pre, suc, left, right}\\}}$, and the
preprocessed agent history states
$[\mathbf{\hat{p}}^{n}_{t},\mathbf{v}^{n}_{t},\psi^{n}_{t}]$ for all
$n\in[N],t\in[T_{\text{obs}}]$.
### 3.2 Feature Encoder
To encode the agent history and HD-map data, we employ a LaneGCN backbone [30]
with a few key modifications. For processing the agent histories, we replace
LaneGCN’s proposed ActorNet architecture with a gated recurrent unit (GRU)
module. For processing the HD-Map, we employ the MapNet architecture, which
consists of $L$ graph convolutional operators that enrich the lane node
features by propagating them through the lane graph. We then employ the
FusionNet architecture introduced in LaneGCN [30] for fusing the map and actor
features, but we remove the actor-to-lane (A2L) and lane-to-lane (L2L)
modules, keeping only the lane-to-actor (L2A) and actor-to-actor (A2A)
modules. We observed a minimal loss in performance when removing the A2L and
L2L modules, and we benefited from the reduced parameter count. The output of
the LaneGCN feature encoder produces a set of map-aware agent features
$H=\\{\mathbf{h}_{n}\\}_{n\in[N]}$ for each agent.
#### 3.2.1 Auxiliary Proposal Decoder
While the output of the LaneGCN feature encoder provides informative map-aware
agent features, the A2A module only considers agent interactions in the
observed past trajectories. However, these features will be used downstream to
reason about agent interactions in the future, and thus we desire agent
feature representations that are future-aware – agent features that are
predictive of the future. To this end, we propose to regularize the LaneGCN
agent feature representations with an auxiliary pretext task that predicts
joint future trajectories on top of the LaneGCN-encoded agent features. We
adopt a proposal decoder $f_{\text{prop}}$, which decodes $K$ joint future
trajectories from the LaneGCN-encoded features
$\\{\hat{\mathbf{y}}_{\text{prop},k}^{n}\\}_{k\in[K]}=f_{\text{prop}}(\mathbf{h}_{n})$
and it is supervised by a joint regression loss $\mathcal{L}_{\text{prop}}$.
More details of the joint regression loss can be found in Appendix A. We
hypothesize that the proposed pretext task will regularize the LaneGCN feature
representations, so that it contains future context that will be useful for
reasoning about future interactions in the downstream modules. We note that
the proposal decoder is discarded at inference time and is only used to
regularize features during training.
### 3.3 Directed Acyclic Interaction Graph Predictor
#### 3.3.1 Interaction Graph Predictor
In order to construct the directed acyclic interaction graph, we first must
classify the future interaction label between every pair of agents in the
scene. This task can be formulated as a classification task where we classify
every edge in a fully-connected undirected interaction graph
$\mathcal{G}_{U}=\\{\mathcal{V},\mathcal{E}_{U}\\}$, where each agent
corresponds to a node in $\mathcal{V}$. Similar to [38, 27, 25, 28], given an
edge $e_{m,n}\in\mathcal{E}_{U}$, the classification task assumes three
labels: no-interaction, m-influences-n, and n-influences-m, where the ground-
truth future interaction label is heuristically determined using their ground-
truth future trajectories. Concretely, we employ a collision checker to check
for a collision between agents $m$ and $n$ at all pairs of future timesteps
$(t_{m},t_{n})$ where $|t_{m}-t_{n}|\leq\epsilon_{I}$ for some threshold
$\epsilon_{I}$. Details of the collision checker can be found in Appendix H.
We let $\mathcal{C}$ denote the set of timestep pairs where a collision is
detected. If $|\mathcal{C}|=0$, then $m$ and $n$ are not interacting and the
edge is labeled no-interaction. Otherwise, we identify the first such pair of
timesteps $(\hat{t}_{m},\hat{t}_{n})\in\mathcal{C}$ where a collision is
detected:
$\displaystyle(\hat{t}_{m},\hat{t}_{n})$
$\displaystyle=\operatorname*{arg\,min}_{(t_{m},t_{n})\in\mathcal{C}}\quad\min\\{t_{m},t_{n}\\}.$
(2)
If $|\mathcal{C}|>0$, then there exists a conflict point between the two
agents, and the influencer agent is defined as the agent who reaches the
conflict point first. Specifically, if $\hat{t}_{m}<\hat{t}_{n}$, then we
assign the edge the label m-influences-n, and otherwise we assign the edge the
label n-influences-m.
With the heuristic interaction labels, we train a classifier to predict the
interaction type on each edge of $\mathcal{G}_{U}$. We first initialize the
node features of $\mathcal{G}_{U}$ to the future-aware LaneGCN agent features
$\mathbf{h}_{n}$. We then perform a node-to-edge feature propagation step,
where for each edge $e_{m,n}$:
$\displaystyle\mathbf{h}^{e}_{m,n}=f_{\text{edge}}\Big{(}\left[\mathbf{h}_{m}||\mathbf{h}_{n}||f_{\text{dist}}(\mathbf{p}^{m}_{t_{\text{c}}}-\mathbf{p}^{n}_{t_{\text{c}}})||\mathbf{a}_{m,n}\right]\Big{)},$
(3)
where $f_{\text{edge}}$ and $f_{\text{dist}}$ are 2-layer MLPs, $||$ denotes
concatenation along the feature dimension, $t_{\text{c}}:=T_{\text{obs}}-1$ is
the present timestep, and $\mathbf{a}_{m,n}=f_{\text{type}}([a_{m},a_{n}])$ is
the output of a 2-layer MLP $f_{\text{type}}$ applied to the agent types
$a_{m},a_{n}$. We then classify the interaction label using a 2-layer MLP
$f_{\text{int}}$ with a softmax activation:
$\displaystyle\hat{r}_{m,n}=\text{softmax}(f_{\text{int}}(\mathbf{h}^{e}_{m,n})).$
(4)
The interaction classifier is trained with a focal loss
$\mathcal{L}_{\text{int}}=\mathcal{L}_{\text{focal}}^{\gamma,\alpha}(R,\hat{R})$
with hyperparameters $\gamma\text{ and }\alpha$, where $\hat{R}$ is the
predicted interaction label distributions and $R$ is the ground-truth
interaction labels. From the predicted interaction label distributions, we can
construct a directed interaction graph
$\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$ by selecting the interaction label
on each edge with the highest predicted probability. For each pair of agents,
we add a directed edge from the predicted influencer to the predicted reactor
if an interaction is predicted to exist, and no edge is added otherwise.
Model | Venue | minADE | minFDE | SMR | CrossCol | CMR
---|---|---|---|---|---|---
THOMAS [17] | ICLR 2022 | 0.416 | 0.968 | 0.179 | 0.128 | 0.252
HDGT [21] | - | 0.303 | 0.958 | 0.194 | 0.163 | 0.236
DenseTNT [19] | ICCV 2021 | 0.420 | 1.130 | 0.224 | 0.000 | 0.224
AutoBot [18] | ICLR 2022 | 0.312 | 1.015 | 0.193 | 0.043 | 0.207
HGT-Joint | - | 0.307 | 1.056 | 0.186 | 0.016 | 0.190
Traj-MAE | - | 0.307 | 0.966 | 0.183 | 0.021 | 0.188
FJMP (Ours) | - | 0.275 | 0.922 | 0.185 | 0.005 | 0.187
Table 1: Joint prediction results on the INTERACTION multi-agent test set.
Methods are sorted by the official ranking metric (CMR). For each metric, the
best method is bolded and the second-best method is underlined. Lower is
better for all metrics.
Dagification. In order to perform factorized joint prediction over the learned
directed interaction graph $\mathcal{G}$, we require $\mathcal{G}$ to be a
DAG. We propose to remove cycles from $\mathcal{G}$, or “dagify”
$\mathcal{G}$, by iterating through the cycles in $\mathcal{G}$ and removing
the edges with the lowest predicted probability. We efficiently enumerate the
cycles in $\mathcal{G}$ using Johnson’s algorithm [22], which has time
complexity $O((|\mathcal{V}|+|\mathcal{E}|)(c+1))$, where $c$ is the number of
cycles in $\mathcal{G}$. As the directed interaction graphs are typically
sparse $(|\mathcal{V}|\approx|\mathcal{E}|)$ with a small number of cycles,
for our application Johnson’s algorithm runs approximately linear in the
number of agents in the scene.
### 3.4 Factorized Joint Trajectory Decoder
Given the future-aware LaneGCN feature encodings
$H=\\{\mathbf{h}_{n}\\}_{n\in[N]}$ and the directed acyclic interaction graph
$\mathcal{G}$, we perform factorized joint prediction according to the unique
partial ordering of $\mathcal{G}$. We parameterize the factorized joint
trajectory decoder using an adapted directed acyclic graph neural network
(DAGNN) [39]. A DAGNN is a recently proposed architecture that is suited
specifically for DAG classification tasks. The originally proposed DAGNN
framework performs DAG-level classification tasks on top of the
representations of the leaf nodes in the DAG, where node features are
propagated to the leaf nodes sequentially according to the partial ordering of
the DAG. Although we desire to process the agents according to the partial
ordering of the interaction graph $\mathcal{G}$, we also aim to use the
intermediate updated node features of the DAG to generate conditional future
trajectory predictions, and thus we adapt the DAGNN design to fit this
criterion. We explain first how to produce a factorized joint prediction using
the proposed adapted DAGNN decoder, and then how the proposed decoder is
extended to produce multiple joint futures.
The factorized decoder first processes the source node(s) $\mathcal{S}$ in
parallel. For each source node $s\in\mathcal{S}$, we first decode a marginal
future trajectory prediction:
$\displaystyle\mathbf{\hat{y}}^{s}$
$\displaystyle=\text{DECODE}(\mathbf{h}_{s}),$ (5)
where DECODE is a residual block followed by a linear layer and
$\mathbf{\hat{y}}^{s}\in\mathbb{R}^{2T_{\text{fut}}}$ is the sequence of
predicted future trajectory coordinates. We then encode the predicted future
trajectories of each source node $s\in\mathcal{S}$:
$\displaystyle\mathbf{e}_{s}$
$\displaystyle=\text{ENCODE}(\mathbf{\hat{y}}^{s}),$ (6)
where ENCODE is a 3-layer MLP. For each $s\in\mathcal{S}$, the encoding of the
predicted future $\mathbf{e}_{s}$ is then fed along the outgoing edges of $s$.
Namely, after processing the source nodes $\mathcal{S}$, we update the
features of the nodes that are next in the partial ordering of $\mathcal{G}$.
For every such node $n$, we perform the following update:
$\displaystyle\mathbf{h}_{n}$
$\displaystyle\leftarrow\text{COMB}\left(\text{AGG}\left(\\{\mathbf{e}_{m}+\mathbf{a}_{mn}|m\in\text{pa}_{\mathcal{G}}(n)\\}\right),\mathbf{h}_{n}\right),$
(7)
where AGG is a neural network that aggregates the node features from $n$’s
parents and COMB is a neural network that combines this aggregated information
with $n$’s features to update the feature representation of $n$ with
conditional context about the predicted future of $n$’s parents.
$\mathbf{a}_{mn}=f^{\text{dec}}_{\text{type}}([a_{m},a_{n}])$ is the output of
a 2-layer MLP $f^{\text{dec}}_{\text{type}}$ applied to the agent types. From
here, we let $\mathbf{b}_{mn}:=\mathbf{e}_{m}+\mathbf{a}_{mn}$. Similar to
DAGNN [39] which uses additive attention, we parameterize AGG using graph
attention [42]:
$\displaystyle\mathbf{m}_{n}:=\text{AGG}\left(\\{\mathbf{b}_{mn}|m\in\text{pa}_{\mathcal{G}}(n)\\}\right)=\sum_{m\in\text{pa}_{\mathcal{G}}(n)}\alpha_{mn}\mathbf{W}_{1}\mathbf{b}_{mn},$
(8)
$\displaystyle\alpha_{mn}=\frac{\exp(\text{LeakyReLU}(\mathbf{a}^{\top}\left[\mathbf{W}_{1}\mathbf{b}_{mn}||\mathbf{W}_{2}\mathbf{h}_{n}\right]))}{\sum_{k\in\text{pa}_{\mathcal{G}}(n)}\exp(\text{LeakyReLU}(\mathbf{a}^{\top}\left[\mathbf{W}_{1}\mathbf{b}_{kn}||\mathbf{W}_{2}\mathbf{h}_{n}\right]))}.$
(9)
COMB is parameterized by a GRU recurrent module:
$\displaystyle\mathbf{h}_{n}\leftarrow\text{COMB}(\mathbf{m}_{n},\mathbf{h}_{n})=\text{GRU}(\mathbf{m}_{n},\mathbf{h}_{n}).$
(10)
As the aggregated message $\mathbf{m}_{n}$ provides conditional context for
updating the representation of $\mathbf{h}_{n}$, $\mathbf{m}_{n}$ is treated
as the input and $\mathbf{h}_{n}$ is treated as the hidden state. It is
important to note that the roles of the input and hidden state are reversed in
the original DAGNN design [39]. The updated representation $\mathbf{h}_{n}$
for node $n$ is now imbued with conditional context about the predicted future
of the parent(s) of $n$, which can now be fed into DECODE to produce
conditional future predictions for agent $n$. We sequentially continue the
process of encoding, aggregating, combining, and decoding according to the
DAG’s partial order until all nodes in the DAG have a future trajectory
prediction. The future trajectory predictions of all nodes are then
conglomerated to attain a factorized joint prediction.
Multiple Futures. To extend the DAGNN factorized decoder to produce multiple
factorized joint predictions, we simply process $K$ copies of
$H=\\{\mathbf{h}_{n}\\}_{n\in[N]}$ through the DAG in parallel. To ensure each
copy of $H$ generates a different set of futures, we concatenate a one-hot
encoding of the modality with $\mathbf{h}_{n}$ along the feature dimension,
for each $n\in[N]$, prior to it being fed into DECODE. This approach is
similar to the multiple futures approach proposed in SceneTransformer [34] and
we found it to work well in our application. We train the factorized joint
predictor to produce diverse multiple futures by training with a winner-takes-
all joint regression loss $\mathcal{L}_{\text{reg}}$. More details about the
regression loss can be found in Appendix A.
### 3.5 Training Details
We first train the interaction graph predictor separately using its own
feature encoder weights. The interaction graph predictor is trained via
gradient descent, where the loss function is defined by:
$\displaystyle\mathcal{L}_{1}=\mathcal{L}_{\text{int}}+\mathcal{L}_{\text{prop}}.$
(11)
Next, we train the factorized joint decoder using its own feature encoder
weights, where the interaction graphs $\mathcal{G}$ are generated with the
trained interaction graph predictor. The factorized joint predictor is trained
via gradient-descent, where the loss function is defined by:
$\displaystyle\mathcal{L}_{2}=\mathcal{L}_{\text{reg}}+\mathcal{L}_{\text{prop}}.$
(12)
Similar to M2I [38], we employ teacher forcing of the influencer future
trajectories during training, which helps to learn the proper influencer-
reactor dynamics.
Dataset | Actors Evaluated | Model | minFDE | minADE | SCR | SMR | iminFDE | iminADE | $\text{iminFDE}_{3}$ | $\text{iminADE}_{3}$ | $\text{iminFDE}_{5}$ | $\text{iminADE}_{5}$
---|---|---|---|---|---|---|---|---|---|---|---|---
Interaction | - | Non-Factorized | 0.643 | 0.199 | 0.004 | 0.088 | 0.688 | 0.210 | 0.784 | 0.240 | 0.854 | 0.261
| | FJMP | 0.630 | 0.194 | 0.003 | 0.084 | 0.672 | 0.206 | 0.758 | 0.232 | 0.826 | 0.252
| | $\Delta$ | 0.013 | 0.005 | 0.001 | 0.004 | 0.016 | 0.004 | 0.026 | 0.008 | 0.028 | 0.009
Argoverse 2 | Scored | Non-Factorized | 1.965 | 0.834 | - | 0.349 | 2.957 | 1.223 | 3.276 | 1.340 | 3.436 | 1.399
| | FJMP | 1.921 | 0.819 | - | 0.343 | 2.893 | 1.204 | 3.205 | 1.320 | 3.356 | 1.377
| | $\Delta$ | 0.044 | 0.015 | - | 0.006 | 0.064 | 0.019 | 0.071 | 0.020 | 0.080 | 0.022
| All | Non-Factorized | 1.995 | 0.825 | - | 0.340 | 3.302 | 1.309 | 3.759 | 1.477 | 3.952 | 1.545
| | FJMP | 1.963 | 0.812 | - | 0.337 | 3.204 | 1.273 | 3.652 | 1.439 | 3.839 | 1.504
| | $\Delta$ | 0.032 | 0.013 | - | 0.003 | 0.098 | 0.036 | 0.107 | 0.038 | 0.113 | 0.041
Table 2: Non-Factorized Baseline vs. FJMP performance on joint metrics on the
INTERACTION and Argoverse 2 validation sets. Lower is better for all metrics.
Argoverse 2 lacks agent bounding box information, so SCR is not computed.
$\Delta$ denotes the difference in performance between FJMP and the Non-
Factorized baseline.
## 4 Experiments
Datasets. We evaluate FJMP on the INTERACTION v1.2 multi-agent dataset and the
Argoverse 2 dataset, as both have multi-agent evaluation schemes for scenes
with many interacting agents and require predicting joint futures for scenes
with up to 40 and 56 agents, respectively. However, currently, only
INTERACTION has a public benchmark for multi-agent joint prediction. Argoverse
2 contains scored and focal actors, which are high-quality tracks near the ego
vehicle; and unscored actors, which are high-quality tracks more than 30 m
from the ego vehicle. We evaluate FJMP on (i) only the scored and focal
actors; and (ii) all scored, focal, and unscored actors. More details about
these datasets can be found in Appendix F.
Evaluation Metrics. We report the following joint prediction metrics: minFDE
is the final displacement error (FDE) between the ground-truth and closest
predicted future trajectory endpoint from the $K$ joint predictions; minADE is
the average displacement error (ADE) between the ground-truth and closest
predicted future trajectory from the $K$ joint predictions; SMR is the minimum
proportion of agents whose predicted trajectories “miss” the ground-truth from
the $K$ joint predictions, where a miss is defined in Appendix I; and SCR is
the proportion of modalities where two or more agents collide. The INTERACTION
test set additionally reports two joint prediction metrics: CrossCol is the
same as SCR but does not count ego collisions, and CMR is the same as SMR but
only considers modalities without non-ego collisions. For all metrics, we
evaluate $K=6$. These six joint prediction metrics do not necessarily capture
the performance on the most interactive and challenging cases in the dataset,
which is critically important for benchmarking and improving motion prediction
systems. To address this limitation, we propose two new interactive metrics:
(i) iminFDE first identifies the modality $k$ with minimum FDE over all the
agents in the scene and then computes the FDE of modality $k$ only over agents
that are interactive, which we heuristically define as agents with at least
one incident edge in the ground-truth sparse interaction graph, where
$\epsilon_{I}=2.5$ s. (ii) iminADE is defined similarly. We found that many of
the interactive cases in the datasets contain kinematically simple cases where
agents exhibit simple leader-follower behaviour. To evaluate the challenging
interactive cases, we further remove interactive agents in our evaluation that
attain less than $d$ meters in FDE with a constant velocity model. These
metrics are denoted $\textbf{iminFDE}_{\textbf{d}}$ and
$\textbf{iminADE}_{\textbf{d}}$, where we report $d=3,5$. Please see Appendix
J for more details.
Implementation Details. Our models are trained on 4 NVIDIA Tesla V100 GPUs
using the Adam optimizer [24]. The interaction graph predictor and factorized
joint decoder are trained with the same hyperparameters. For INTERACTION, we
set the batch size to 64 and train for 50 epochs with a learning rate of 1e-3,
step-decayed by a factor of 1/5 at epochs 40 and 48. For Argoverse 2, we set
the batch size to 128 and train for 36 epochs with a learning rate of 1e-3,
step-decayed by a factor of 1/10 at epoch 32. As bounding-box information is
not provided with Argoverse 2, the collision checker used to construct
interaction labels uses a predefined length/width for each agent type, as
listed in Sec. F.2. Our INTERACTION and Argoverse 2 models train in 10 and 15
hours. See Appendix G for more details.
Model | minFDE | minADE | SMR | Prop. Edges | Inf. Time (s)
---|---|---|---|---|---
Non-Factorized | 0.643 | 0.199 | 0.088 | - | 0.010
FJMP (Dense) | 0.623 | 0.193 | 0.081 | 0.180 | 0.062
FJMP | 0.626 | 0.193 | 0.083 | 0.045 | 0.038
Table 3: Comparison of sparse vs. dense interaction graphs on the INTERACTION
validation set. The FJMP model is trained and evaluated using the ground-truth
sparse interaction graphs, and FJMP (Dense) is trained and evaluated using
dense ground-truth interaction graphs attained via the M2I [38] labeling
heuristic. Prop. Edges measures the proportion of agent pairs connected in the
ground-truth training interaction graphs. Inf. Time is the inference time per
validation scene on 1 NVIDIA Tesla V100 GPU.
Methods under Comparison. We compare FJMP against the top-performing methods
on the INTERACTION multi-agent test set leaderboard [17, 21, 19, 18]. FJMP is
the only method on the leaderboard that performs factorized joint prediction.
To measure the improvement of FJMP over non-factorized approaches, we compare
FJMP against a baseline called Non-Factorized, which computes $K$ simultaneous
joint futures from the feature representations output by the proposed LaneGCN
feature encoder.
Results. The joint prediction results for the top-performing methods on the
INTERACTION multi-agent test set are shown in Tab. 1. FJMP performs the best
on minFDE, minADE, and the official ranking metric CMR, while performing
competitively on other metrics. Crucially, FJMP produces joint predictions
that are both more accurate—as demonstrated by its superior performance on
minADE and minFDE—and more scene-consistent—as demonstrated by its near-zero
collision rate—than non-factorized approaches, which highlights the benefit of
the proposed joint factorization.
Model | Prop? | TF? | minFDE | minADE | iminFDE | iminADE
---|---|---|---|---|---|---
Non-Factorized | ✗ | ✗ | 1.995 | 0.825 | 3.302 | 1.309
FJMP | ✗ | ✗ | 2.004 | 0.829 | 3.274 | 1.304
FJMP | ✗ | ✓ | 2.001 | 0.827 | 3.300 | 1.312
FJMP | ✓ | ✗ | 1.987 | 0.820 | 3.255 | 1.293
FJMP | ✓ | ✓ | 1.963 | 0.812 | 3.204 | 1.273
Table 4: Ablation study of FJMP on Argoverse 2 validation set, All setting.
Prop? denotes whether we include the proposal decoder during training. TF?
denotes whether we teacher-force the influencer trajectories during training.
Table 2 reports validation results on the INTERACTION and Argoverse 2
datasets, where we compare FJMP against the baseline method without joint
factorization. For Argoverse 2, we have two evaluation schemes: (i) we
evaluate the joint predictions of the scored and focal agents (Scored), and
(ii) we evaluate the joint predictions of the scored, unscored, and focal
agents (All) to demonstrate its scalability to scenes with a large number of
agents. The results show that the proposed joint factorized predictor
consistently provides an improvement in performance over the non-factorized
baseline. We expect that FJMP improves the most over the baseline on the
interactive cases in the dataset, as the proposed factorization directly
enables conditioning the reactor predictions on the predicted futures of their
influencers. Importantly, we note that for scenes with no predicted
interactions, the factorization becomes a product of marginal predictions and
thus FJMP reduces to the non-factorized prediction. As expected, the relative
improvement of FJMP over the baseline is larger on the interactive and
kinematically interesting cases, as demonstrated by a larger performance
improvement on the interactive minFDE/minADE metrics. This indicates that the
performance improvement from the joint factorization concentrates on the
challenging interactive cases, while still producing accurate joint
predictions for the full scene. We refer readers to Appendix M for qualitative
examples demonstrating the benefit of factorized prediction.
Table 3 uses the INTERACTION dataset to ablate the design choice of
representing the interaction graph sparsely with only the strongest pairwise
interactions as edges in the graph. We compare FJMP against a variant of FJMP
that uses a different labeling heuristic for the interaction graph, resulting
in denser interaction graphs. Namely, FJMP (Dense) uses the M2I [38]
heuristic: for each pair of agents, an interaction is defined to exist if any
pair of future trajectory coordinates in the future trajectory horizon is
within a threshold Euclidean distance of each other, where the threshold is
taken to be the sum of the lengths of the two agents. The influencer-reactor
relationship is determined by who reaches the conflict point first. We found
that the M2I heuristic often adds several unnecessary edges, especially in
congested scenes—as exemplified in Appendix K. We train and evaluate the FJMP
models in Tab. 3 using the ground-truth interaction graphs to precisely
compare the different labeling heuristics. The results show that the dense
(M2I) interaction graph improves very slightly over the sparse interaction
graph; however, we retain most of the improvement over the non-factorized
baseline with the sparse interaction graph, which indicates that modeling only
the strongest interactions is sufficient to see most of the improvement with
joint factorization. Moreover, the sparse interactions contain 75% fewer edges
than the dense interaction graph, which accelerates inference by nearly 2x.
Table 4 conducts an ablation study on Argoverse 2 where we analyze the effect
of using the auxiliary proposal decoder and teacher forcing of the
influencer’s future trajectories during training. The results indicate that
both the auxiliary proposal decoder and teacher forcing is critical for
allowing the model to reason appropriately about the influencer-reactor future
dynamics. Notably, without the proposal decoder (rows 2 and 3 in Tab. 4), FJMP
performs similarly to the non-factorized baseline, which we hypothesize is
because the LaneGCN-encoded features do not contain the necessary future
context to reason appropriately about the future interactions. Teacher forcing
also provides an additional performance benefit by removing the spurious noise
in the predicted influencer trajectories, so that the model better learns the
proper influencer-reactor dynamics during training.
## 5 Conclusion
In this paper, we propose FJMP, a factorized joint motion prediction framework
for multi-agent interactive driving scenarios. FJMP models the future scene
interaction dynamics as a sparse directed acyclic interaction graph, which
enables efficient factorized joint prediction. We demonstrate clear
performance improvements with our factorized design on the Argoverse 2 and
INTERACTION datasets and perform state-of-the-art on the challenging multi-
agent INTERACTION benchmark.
Limitations The proposed framework adopts a heuristic labeling scheme to
determine the ground-truth interaction graph. We observe a performance-
efficiency tradeoff with a denser interaction graph; however, there may exist
better heuristics for classifying future interactions that retain most of the
sparsity of the interaction graph without trading off performance. Moreover,
long chains of leader-follower behaviour in congested traffic may require
costly sequential processing with our method. Finding mechanisms to prune the
interaction graph to best trade-off performance and efficiency is a direction
we plan to explore in future work.
## 6 Acknowledgements
This work was funded by Ontario Graduate Scholarship and NSERC. We thank
Benjamin Thérien and Prarthana Bhattacharyya for their valuable insights and
discussions.
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## Appendix A Loss Functions
The loss function for training the factorized joint trajectory decoder (Sec.
3.4) is defined by:
$\displaystyle\mathcal{L}_{2}=\mathcal{L}_{\text{reg}}+\mathcal{L}_{\text{prop}},$
(13)
where
$\mathcal{L}_{\text{reg}}(\\{\hat{Y}_{k}\\}_{k\in[K]},Y):=\mathcal{L}_{\ell_{1}}(\\{\hat{Y}_{k}\\}_{k\in[K]},Y)$
is a scene-level smooth $\ell_{1}$ regression loss applied to the best
modality of $K=6$ joint modalities $\\{\hat{Y}_{k}\\}_{k\in[K]}$, where the
best modality attains the minimum loss:
$\displaystyle\mathcal{L}_{\ell_{1}}(\\{\hat{Y}_{k}\\}_{k\in[K]},Y)=\min_{k\in[K]}\frac{1}{A\cdot
T_{\text{fut}}}\sum_{a\in[A]}\sum_{t\in[T_{\text{fut}}]}\text{reg}(\hat{Y}_{t,k}^{a}-Y^{a}_{t}),$
(14)
where $Y$ denotes the ground-truth future trajectory coordinates of all $A$
agents in the scene, $\text{reg}(\mathbf{x})=\sum_{i}d(x_{i})$, $x_{i}$ is the
$i$’th element of $\mathbf{x}$, and $d(x)$ is the smooth $\ell_{1}$ loss
defined by:
$\displaystyle d(x)=\begin{cases}0.5x^{2},&\text{if $||x||_{1}\leq 1$}\\\
||x||_{1}-0.5,&\text{otherwise.}\end{cases}$ (15)
Similarly, the auxiliary decoder loss $\mathcal{L}_{\text{prop}}$ is a scene-
level smooth $\ell_{1}$ loss applied to the best of $K=15$ joint proposals
$\\{\hat{Y}^{\text{prop}}_{k}\\}_{k\in[K]}$:
$\displaystyle\mathcal{L}_{\text{prop}}(\\{\hat{Y}^{\text{prop}}_{k}\\}_{k\in[K]},Y):=\mathcal{L}_{\ell_{1}}(\\{\hat{Y}^{\text{prop}}_{k}\\}_{k\in[K]},Y).$
(16)
We use the auxiliary proposal loss $\mathcal{L}_{\text{prop}}$ for training
both the interaction graph predictor ($\mathcal{L}_{1}$ in Sec. 3.5) and the
factorized joint decoder ($\mathcal{L}_{2}$ in Sec. 3.5) as both modules
require explicit reasoning about interactions in the future trajectories, and
thus future-aware agent features are beneficial for both modules.
## Appendix B FJMP System Diagram
### B.1 Training Time
Figure 3 illustrates a high-level schematic of the FJMP architecture training
stages at training time. We note that Feature Encoder 1 and Feature Encoder 2
consist of the same architecture as described in Sec. 3.2, but use separate
weights.
(a) Interaction Graph Predictor training stage of FJMP.
(b) Factorized Joint Predictor training stage of FJMP.
Figure 3: High-level schematic of the training stages of FJMP.
### B.2 Inference Time
Figure 4 illustrates a high-level schematic of the FJMP architecture and data
flow at inference time. We note that at inference time the proposal decoders
are removed.
Figure 4: High-level schematic of the FJMP architecture at inference time.
## Appendix C Non-Factorized Baseline
We explain the non-factorized baseline described in Sec. 4 in more detail. The
non-factorized baseline uses the same feature encoder architecture as FJMP,
but the factorized joint decoder is replaced with a DECODE module consisting
of a residual block and linear layer for simultaneously decoding $K$ joint
future trajectory coordinates, where diverse futures are obtained by appending
a one-hot encoding of the modality index to the agent feature representation
before feeding it into DECODE, as is done in FJMP. The DECODE module is the
same architecture as the DECODE module used in FJMP. The non-factorized
baseline is trained with the scene-level winner-takes-all smooth $\ell_{1}$
loss $\mathcal{L}_{\text{reg}}$ that is described in Appendix A. The non-
factorized baseline is trained with the same training hyperparameters as FJMP.
## Appendix D Non-Factorized Baseline Ablation
Model | Multiple Futures Method | Hyperparameter Configuration | Feature Encoder | minFDE | minADE | SMR | SCR
---|---|---|---|---|---|---|---
LaneGCN [30] | Separate Weights | LaneGCN | LaneGCN | 0.935 | 0.300 | 0.223 | 0.233
- | One-hot Encoding | LaneGCN | LaneGCN | 0.807 | 0.264 | 0.142 | 0.010
- | One-hot Encoding | FJMP | LaneGCN | 0.713 | 0.227 | 0.113 | 0.006
Non-Factorized Baseline | One-hot Encoding | FJMP | FJMP | 0.643 | 0.199 | 0.088 | 0.004
Table 5: Ablation study of the Non-Factorized Baseline model on the
INTERACTION validation set. Multiple Futures Method denotes the method used to
attain multiple joint futures. Hyperparameter Configuration denotes the
hyperparameter settings for batch size, learning rate/step, and the number of
training epochs. Feature Encoder denotes whether we use the LaneGCN feature
encoder (LaneGCN) or the simplified LaneGCN feature encoder with fewer
components (FJMP).
In Tab. 5, we perform an ablation study on the various components of the non-
factorized baseline model on the INTERACTION dataset. First, we ablate using a
one-hot encoding for multiple futures (One-hot Encoding) compared with using
separate decoder weights for each joint future modality (Separate Weights), as
is done in LaneGCN [30]. The one-hot encoding method significantly improves
performance; this is because when using separate weights, the winner-takes-all
training process quickly converges to one future joint modality, and thus the
other decoders’ weights never receive gradients for updating their weights. As
a result, the collision rate (SCR) significantly improves when using the one-
hot encoding method. Next, we ablate using the default hyperparameter
configuration for LaneGCN compared with the FJMP hyperparameter configuration.
Namely, LaneGCN trains for 36 epochs with a batch size of 128, with the
learning rate decreasing by a factor of 10 at epoch 32. FJMP trains for 50
epochs with a batch size of 64, with the learning rate decreasing by a factor
of 5 at epochs 40 and 48. The FJMP hyperparameter configuration significantly
improves performance over the LaneGCN hyperparameter configuration. Finally,
we ablate using the modified LaneGCN feature encoder (FJMP) consisting of a
GRU for processing agent trajectories instead of LaneGCN’s proposed ActorNet
module, 2 MapNet layers instead of 4, and the A2L and L2L blocks removed.
These modifications yield further improvements in validation performance.
## Appendix E INTERACTION Ablation Study
Model | Prop? | TF? | minFDE | minADE | iminFDE | iminADE
---|---|---|---|---|---|---
Non-Factorized | ✗ | ✗ | 0.643 | 0.199 | 0.688 | 0.210
FJMP | ✗ | ✗ | 0.647 | 0.200 | 0.690 | 0.212
FJMP | ✗ | ✓ | 0.644 | 0.200 | 0.688 | 0.212
FJMP | ✓ | ✗ | 0.636 | 0.197 | 0.677 | 0.208
FJMP | ✓ | ✓ | 0.630 | 0.194 | 0.671 | 0.206
Table 6: Ablation study of FJMP on the INTERACTION validation set. Prop?
denotes whether we include the proposal decoder during training. TF? denotes
whether we teacher-force the influencer trajectories during training.
In Tab. 6, we repeat the FJMP ablation study conducted in Tab. 4 on the
INTERACTION dataset. The results are consistent with Argoverse 2, showing that
both the proposal decoder and teacher forcing are critical for performance.
## Appendix F Datasets
### F.1 INTERACTION
INTERACTION requires predicting 3 seconds into the future given 1 second of
past observations sampled at 10 Hz. INTERACTION contains 47,584 training
scenes, 11,794 validation scenes, and 2,644 test scenes. A scene consists of a
4 s sequence of observations (1 s past, 3 s future) for each agent.
INTERACTION contains pedestrians, bicyclists, and vehicles as context agents
but only requires predicting vehicles in their multi-agent challenge. As
bounding box length/width information is not provided for the
pedestrian/cyclist labels, we set the length and width to a pre-defined value
of 0.7m. We note that pedestrians and cyclists are not differentiated in the
INTERACTION dataset.
### F.2 Argoverse 2
Argoverse 2 requires predicting 6 seconds into the future given 5 seconds of
past observations sampled at 10 Hz. Argoverse 2 contains 199,908 training
scenes and 24,988 validation scenes. A scene consists of an 11 s sequence of
observations (5 s past, 6 s future) for each agent. Argoverse 2 requires
predicting 5 agent types: vehicle, pedestrian, bicyclist, motorcyclist, and
bus. As bounding box length/width information is not provided in the Argoverse
2 dataset, we use the following predefined length/width in meters for each
agent type to construct the interaction labels (length/width): vehicle
(4.0/2.0), pedestrian (0.7/0.7), bicyclist (2.0/0.7), motorcyclist (2.0/0.7),
bus (12.5/2.5).
## Appendix G Training Details
### G.1 INTERACTION
The hidden dimension of FJMP is 128 except for the GRU history encoder, which
has a hidden dimension of 256. The output of the GRU encoder is mapped to
dimension 128 with a linear layer. We set $K=6$ for the factorized decoder and
$K=15$ for the proposal decoders. For training the interaction graph
predictor, we set $\gamma=5$ and $\alpha=[1,2,4]$. We set $\epsilon_{I}=2.5$
s. During training, we center and rotate the scene on a random agent, as an
input normalization step. During validation and test time, we center and
rotate the scene on the agent closest to the centroid of the agents’ current
positions. We use 2 MapNet layers, 2 L2A layers, and 2 A2A layers, where the
L2A and A2A distance thresholds are set to 20 m and 100 m, respectively. We
use all agents in the scene for context that contains a ground-truth position
at the present timestep. As centerline information is not provided in
INTERACTION, for each lanelet we interpolate $P$ evenly-spaced centerline
points, where $P=\min\\{10,\max\\{L,R\\}\\}$ and $L,R$ are the number of
points on the lanelet’s left and right boundaries, respectively; that is, we
restrict long lanelets to have a maximum of 10 evenly-spaced centerline
points. At validation time, we consider for evaluation all vehicles that
contain a ground-truth position at both the present and final timesteps. We
train our model on the train and validation set with the same training
hyperparameters before evaluating FJMP on the INTERACTION test set.
### G.2 Argoverse 2
The details in Sec. G.1 apply to Argoverse 2 with the following exceptions.
For training the interaction graph predictor, we set $\gamma=5$ and
$\alpha=[1,4,4]$. We set $\epsilon_{I}=6$ s as interactions are comparatively
more sparse in Argoverse 2. At validation time, we center on the ego vehicle.
We increase the number of MapNet layers to 4 in Argoverse 2 to handle the
larger amount of unique roadway. The L2A threshold is set to 10 m as the
centerline points are comparatively more dense in Argoverse 2 than in
INTERACTION. We use all scored, unscored, and focal agents in the scene for
context that contains a ground-truth position at the present timestep. In the
Scored validation setting (see Tab. 2), we consider for evaluation all scored
and focal agents with a ground-truth position at both the present and final
timesteps. In the All validation setting (see Tab. 2), we consider for
evaluation all scored, unscored, and focal agents with a ground-truth position
at both the present and final timesteps.
## Appendix H Collision Checker
To construct the interaction labels as described in Sec. 3.3, a collision
checker is used to identify collisions between all pairs of timesteps in the
future trajectories. We use the collision checker provided with the
INTERACTION dataset. At each timestep, the collision checker defines each
agent by a list of circles, and two agents are defined as colliding if the
Euclidean distance between any two circles’ origins of the given two agents is
lower than the following threshold:
$\displaystyle\epsilon_{C}:=\frac{w_{i}+w_{j}}{\sqrt{3.8}},$ (17)
where $w_{i},w_{j}$ are the widths of agents $i,j$.
## Appendix I Miss Rate
Actors Evaluated | Model | $\text{SMR}_{\text{Argoverse2}}$
---|---|---
Scored | Non-Factorized | 0.264
| FJMP | 0.259
| $\Delta$ | 0.005
All | Non-Factorized | 0.259
| FJMP | 0.257
| $\Delta$ | 0.002
Table 7: Non-Factorized Baseline vs. FJMP performance on Argoverse 2 SMR
metric on the Argoverse 2 validation set. $\Delta$ denotes the difference in
performance between FJMP and the Non-Factorized baseline.
For both Argoverse 2 and INTERACTION, we use the definition of a miss used in
the INTERACTION dataset: a prediction is considered a “miss” if the
longitudinal or latitudinal distance between the prediction and ground-truth
endpoint is larger than their corresponding thresholds, where the latitudinal
threshold is $\epsilon_{\text{lat}}:=1\,\text{m}$ and the longitudinal
threshold is:
$\displaystyle\epsilon_{\text{long}}:=\begin{cases}1,&\text{if $v\leq
1.4\,\text{m/s}$}\\\ 1+\frac{v-1.4}{11-1.4},&\text{if $1.4\,\text{m/s}\leq
v\leq 11\,\text{m/s}$}\\\ 2,&\text{otherwise,}\end{cases}$ (18)
where $v$ is the ground-truth velocity at the final timestep. We note that
Argoverse 2 officially defines a miss as a prediction whose endpoint is more
than 2 m from the ground-truth endpoint; however, we report all miss rate
numbers in Tab. 2 using the miss rate definition in INTERACTION as it is a
more robust measure of miss rate that takes into account the agent’s velocity.
For completeness, we report miss rate numbers for Argoverse 2 using the
Argoverse 2 definition of a miss in Tab. 7.
## Appendix J Constant Velocity Model
(a)
(b)
Figure 5: Histogram of FDEs on interacting agents in (a) the INTERACTION
dataset, and (b) the Argoverse 2 dataset. The left y-axis corresponds to the
histogram and the right y-axis corresponds to the empirical cumulative
distribution function (CDF).
In Sec. 4, we identify the kinematically complex interactive agents in the
datasets by filtering for agents that attain at least $d$ m in FDE with a
constant velocity model. An interactive agent is defined as an agent with at
least one incident edge in the ground-truth interaction graph, where
$\epsilon_{I}=2.5$ s, as is explained in Sec. 4. In this section, we describe
the constant velocity model in more detail. The constant velocity model
computes the average velocity over the observed timesteps and unrolls a future
trajectory using the calculated constant velocity. Namely, the average
velocity is calculated as:
$\displaystyle\mathbf{v}_{\text{avg}}=\frac{1}{T_{\text{obs}}}\sum_{t\in[T_{\text{obs}}]}\mathbf{v}_{t},$
(19)
where $\mathbf{v}_{t}$ is the ground-truth velocity at timestep $t$. Using the
constant velocity model, we calculate the agent-level FDE of all interactive
agents in the INTERACTION and Argoverse 2 validation sets, respectively, where
the FDE distributions are plotted in Fig. 5. We observe that a large
proportion of the interactive agents have low FDE with a constant velocity
model, especially in the INTERACTION dataset. By filtering out these
kinematically simple agents, as is done in Sec. 4, we can assess the model’s
joint prediction performance on agents that are both interactive and
kinematically complex. In Tab. 8, we report the number of interactive agents
in the INTERACTION and Argoverse 2 validation sets that attain at least $d$ m
in FDE, for $d=0,3,5$. We note that $d=0$ corresponds to the number of
interactive agents in the respective validation sets.
Dataset | $d$ | Count
---|---|---
INTERACTION | 0 | 50967
(112994) | 3 | 21077
| 5 | 13069
Argoverse 2 | 0 | 37065
(248719) | 3 | 29421
| 5 | 26140
Table 8: Number of interactive agents in the INTERACTION and Argoverse 2
datasets that attain at least $d$ m in FDE with a constant velocity model. In
parentheses, we include the total number of evaluated agents (interactive +
non-interactive) in the respective validation sets.
## Appendix K FJMP vs. M2I Interaction Graphs
(a) Interaction graph generated with FJMP labeling heuristic.
(b) Interaction graph generated with M2I labeling heuristic.
Figure 6: Comparison of FJMP and M2I labeling heuristics on a congested scene
from the INTERACTION dataset. The ground-truth pasts are indicated in yellow
and the ground-truth futures are indicated in green. Lane boundaries are
depicted as grey lines. Each red arrow points from an influencer agent to its
corresponding reactor agent. We note that two agents at the bottom-right of
the scene are on the shoulder of the lane.
Figure 6 illustrates the ground-truth interaction graph of a congested scene
according to the FJMP and M2I heuristics, respectively. We observe that the
M2I heuristic adds several superfluous edges, which would lead to unnecessary
additional computation for the factorized decoder.
## Appendix L Interaction Graph Predictor Performance
Dataset | Edge Type | Edge Type Proportion
---|---|---
INTERACTION | no-interaction | 0.955
| m-influences-n | 0.037
| n-influences-m | 0.008
Argoverse 2 | no-interaction | 0.973
| m-influences-n | 0.015
| n-influences-m | 0.013
Table 9: Edge type proportions in the INTERACTION and Argoverse 2 training set
interaction graphs with the FJMP labeling heuristic.
Dataset | Edge Type | Edge Type Accuracy
---|---|---
INTERACTION | no-interaction | 0.992
| m-influences-n | 0.940
| n-influences-m | 0.939
Argoverse 2 | no-interaction | 0.990
| m-influences-n | 0.847
| n-influences-m | 0.859
Table 10: Accuracy of each edge type on the INTERACTION and Argoverse 2
validation sets with the FJMP interaction graph predictor.
Table 9 reports the proportion of no-interaction, m-influences-n, and
n-influences-m edges in the INTERACTION and Argoverse 2 training sets. Due to
the severe class imbalance, we employ a focal loss when training the
interaction graph predictor, as explained in Sec. 3.3.1. The edge type
accuracies of the proposed interaction graph predictor on the INTERACTION and
Argoverse 2 validation sets are reported in Tab. 10.
### L.1 Ground-truth Interaction Graph Performance
Table 11 compares the performance of FJMP with two modified versions of FJMP:
(1) we replace the predicted interaction graphs at inference time with the
ground-truth interaction graphs; and (2) we replace the predicted interaction
graphs during training and inference time with the ground-truth interaction
graphs. The results in Tab. 11 indicate that the choice of interaction graph
has a considerable effect on the performance of the factorized joint
predictor, as indicated by an additional 4 cm improvement in iminFDE with the
ground-truth interaction graph at inference time over the predicted
interaction graph. Moreover, when the model is trained and evaluated with the
ground-truth interaction graphs, we see a substantial increase in performance
over FJMP with the learned interaction graphs. This indicates that further
refinement of the interaction graph predictor may yield additional performance
improvements with our FJMP design, which we leave to future work.
Model | Train IG | Inference IG | minFDE | minADE | iminFDE | iminADE
---|---|---|---|---|---|---
FJMP | Learned | Learned | 1.963 | 0.812 | 3.204 | 1.273
FJMP | Learned | Ground-truth | 1.947 | 0.807 | 3.165 | 1.265
FJMP | Ground-truth | Ground-truth | 1.888 | 0.789 | 2.986 | 1.220
Table 11: FJMP with ground-truth vs learned interaction graphs at training and
inference time on the Argoverse 2 validation set, All setting. For each
metric, the best model is bolded. Train IG indicates the interaction graphs
that are used during training, where Learned denotes the predicted interaction
graphs from the interaction graph predictor and Ground-truth denotes the
interaction graphs obtained from the labeling heuristic. The Inference IG
column is interpreted similarly.
## Appendix M Qualitative Results
### M.1 Argoverse 2
In this section, we show qualitative results on scenes in the Argoverse 2
validation set where we show side-by-side comparisons between FJMP and the
Non-Factorized Baseline. In Fig. 7 and Fig. 8, for each row, the left panel
shows the non-factorized baseline predictions, the middle panel shows FJMP
predictions, and the right panel shows the predicted DAG. We visualize only
the best scene-level modality to avoid clutter. In Fig. 7, we show examples
where FJMP reasons properly in scenes with interactive pass-yield behaviours.
In contrast, the non-factorized baseline incorrectly predicts conservative
behaviour where the yielding vehicle avoids the passing vehicle’s trajectory.
In Fig. 8, we show qualitative examples where FJMP correctly identifies chains
of leader-follower interactions, which in turn leads to more accurate leader-
follower predictions than the non-factorized baseline. In Fig. 10, we
illustrate two failure cases of the FJMP model. In both cases, an erroneous
influencer future prediction negatively biases the downstream reactor
prediction.
### M.2 INTERACTION
Figure 9 shows qualitative results of FJMP on various scenes in the
INTERACTION dataset, with all $K=6$ scene-level modalities visualized. We
emphasize FJMP’s ability to produce accurate and scene-consistent predictions
for scenes with a large number of interacting agents.
Figure 7: Qualitative examples of left-turn interactive scenes in the
Argoverse 2 validation set. All predicted DAGs match the ground-truth DAG. In
all scenes, FJMP correctly identifies the passing vehicle as the influencer
and the left-turning vehicle as the reactor. The Non-Factorized baseline
consistently predicts overly conservative behaviour that avoids the influencer
trajectory. In contrast, FJMP consistently captures the proper left-turn
behaviour. Figure 8: Qualitative examples of leader-follower interactive
scenes in the Argoverse 2 validation set. Predicted DAGs are shown on the
right, where true positive edges are indicated in solid black and true
negative edges are shown in dotted black. In all of the above scenes, FJMP
correctly predicts chains of influencer-reactor relationships. In the first
row, the non-factorized baseline predicts conservative behaviour for the
trailing vehicle. In contrast, FJMP predicts proper leader-follower behaviour
for the trailing vehicle (leaf node in the DAG). In the second and third rows,
the right-turn mode of the trailing vehicle is missed by the non-factorized
baseline, whereas FJMP correctly identifies the right-turn mode due to
correctly identifying the leader-follower interaction. In the last row, the
non-factorized baseline predicts scene-incompliant behaviour for the trailing
vehicle whereas FJMP predicts proper leader-follower dynamics reflecting the
predicted DAG. Figure 9: Qualitative examples of FJMP on agent-dense scenes in
the INTERACTION dataset. Figure 10: Qualitative examples of failure cases of
the FJMP model. All predicted DAGs match the ground truth. In both rows, the
interaction graph is correctly predicted; however, the influencer trajectory
is erroneously predicted, which negatively biases the reactor’s prediction to
follow the influencer.
|
# SuS-X: Training-Free Name-Only Transfer of Vision-Language Models
Vishaal Udandarao
University of Cambridge
<EMAIL_ADDRESS>Ankush Gupta
DeepMind, London
<EMAIL_ADDRESS>Samuel Albanie
University of Cambridge
<EMAIL_ADDRESS>
###### Abstract
Contrastive Language-Image Pre-training (CLIP) has emerged as a simple yet
effective way to train large-scale vision-language models. CLIP demonstrates
impressive zero-shot classification and retrieval performance on diverse
downstream tasks. However, to leverage its full potential, fine-tuning still
appears to be necessary. Fine-tuning the entire CLIP model can be resource-
intensive and unstable. Moreover, recent methods that aim to circumvent this
need for fine-tuning still require access to images from the target task
distribution. In this paper, we pursue a different approach and explore the
regime of training-free “name-only transfer” in which the only knowledge we
possess about the downstream task comprises the names of downstream target
categories. We propose a novel method, SuS-X, consisting of two key building
blocks—“SuS” and “TIP-X”, that requires neither intensive fine-tuning nor
costly labelled data. SuS-X achieves state-of-the-art (SoTA) zero-shot
classification results on 19 benchmark datasets. We further show the utility
of TIP-X in the training-free few-shot setting, where we again achieve SoTA
results over strong training-free baselines. Code is available at
https://github.com/vishaal27/SuS-X.
## 1 Introduction
Vision-language pre-training has taken the machine learning community by
storm. A broad range of vision-language models (VLMs) [61, 46, 77, 1, 41]
exhibiting exceptional transfer on tasks like classification [84, 88], cross-
modal retrieval [71, 2] and segmentation [67, 30] have emerged. These models
are now the de facto standard for downstream task transfer in the field of
computer vision.
Figure 1: Training-free name-only transfer. We propose SuS-X, a framework for enhancing the zero-shot transfer abilities of VLMs like CLIP [61], BLIP [46] and TCL [76], without training. To achieve this, we propose a novel method TIP-X, which adapts these VLMs using a curated support set (SuS) that is not drawn from the target distribution. Our SuS leverages one key piece of information about the task at hand: the names of the target categories. Table 1: Taxonomy of CLIP adaptation methods for downstream classification. We underline the Zero-Shot CLIP model to signify that it is the base model that all others build on top of. ∗This method considers access to all test-set samples simultaneously, hence we still consider it zero-shot. †This method additionally uses class hierarchy maps. | Method | Does not require | Does not require | Does not require
---|---|---|---|---
| training | labelled data | target data distribution
Few-shot fine-tuning
methods | LP-CLIP [61] | ✗ | ✗ | ✗
CoOp [88] | ✗ | ✗ | ✗
PLOT [12] | ✗ | ✗ | ✗
LASP [10] | ✗ | ✗ | ✗
SoftCPT [21] | ✗ | ✗ | ✗
VT-CLIP [83] | ✗ | ✗ | ✗
VPT [19] | ✗ | ✗ | ✗
ProDA [49] | ✗ | ✗ | ✗
CoCoOp [87] | ✗ | ✗ | ✗
CLIP-Adapter [28] | ✗ | ✗ | ✗
Intermediate
methods | TIP-Adapter [84] | ✓ | ✗ | ✗
UPL [40] | ✗ | ✓ | ✗
SVL-Adapter [58] | ✗ | ✓ | ✗
TPT [52] | ✗ | ✓ | ✓
CLIP+SYN [36] | ✗ | ✓ | ✓
CaFo [82] | ✗ | ✓ | ✓
Zero-shot
methods | Zero-Shot CLIP [61] | ✓ | ✓ | ✓
CALIP [34] | ✓ | ✓ | ✓
CLIP+DN [89]∗ | ✓ | ✓ | ✓
Training-free name-only
transfer methods | CuPL [60] | ✓ | ✓ | ✓
VisDesc [53] | ✓ | ✓ | ✓
CHiLS [57]† | ✓ | ✓ | ✓
SuS-X (ours) | ✓ | ✓ | ✓
One such prominent model, CLIP [61], is trained on a web-scale corpus of 400M
image-text pairs using a contrastive loss that maximises the similarities of
paired image-text samples. CLIP pioneered the notion of zero-shot transfer in
the vision-language setting111This idea of zero-shot transfer is distinct from
the traditional zero-shot classification setup introduced by Lampert et al.
[45] in which the task is to generalise to classes not seen during training.:
classification on unseen datasets. For a given classification task, CLIP
converts the class labels into classwise textual prompts. An example of such a
prompt is “A photo of a $<$CLASS$>$.”, where $<$CLASS$>$ is replaced by the
ground-truth text label for each class. It then computes similarities between
the query image and text prompts of all classes. The class whose prompt yields
the maximal similarity with the query image is then chosen as the predicted
label.
The zero-shot performance of CLIP is however limited by its pre-training
distribution [27, 64, 24, 55]. If the downstream dataset distribution diverges
too strongly from the distribution of images seen during pretraining, CLIP’s
zero-shot performance drastically drops [24]. To mitigate this, several lines
of work propose to adapt CLIP on diverse downstream tasks—Tab. 1 provides a
brief summary of these methods. Most of them employ fine-tuning on either
labelled or unlabelled subsets of data from the target task. However, fine-
tuning such an over-parameterised model can be unstable and lead to
overfitting [17, 28]. Furthermore, having access to the true distribution of
the target task can be prohibitive in data-scarce environments [13, 4, 42] and
online learning settings [16, 69].
To alleviate these issues, in this paper, we aim to adapt CLIP and other VLMs
for downstream classification in a name-only (requires only category
names222We use category and class interchangeably in this paper., but no
samples from the target task) and training-free fashion. We propose SuS-X (see
Fig. 1), consisting of two novel building blocks: (i) SuS (Support Sets), our
dynamic support set curation strategy that forgoes the need for samples from
the target task, and (ii) TIP-X, our main framework for performing zero-shot
classification while being training-free. For a given downstream task, we
first curate a support set by leveraging the task category labels, either in a
parametric manner _i.e_., generating images from large-scale text-to-image
models (_e.g_., Stable Diffusion [63]) or non-parametric manner _i.e_.,
retrieving real-world images from a large vision-language data bank (_e.g_.,
LAION-5B [65]). We then use the curated support set as a proxy few-shot
dataset to inform our downstream predictions using TIP-X, in a similar vein to
recent few-shot adaptation methods [28, 84].
Our extensive experiments show that SuS-X outperforms zero-shot methods on 19
benchmark datasets across three VLMs, namely, CLIP, BLIP and TCL by 4.60%,
5.97% and 11.37% absolute average accuracy respectively. We further extend the
TIP-X framework to the few-shot regime, outperforming previous SoTA methods in
the training-free domain. Our main contributions are three-fold: (1) We
propose SuS-X, a SoTA method in the training-free name-only transfer setting
for downstream adaptation of VLMs, (2) We present SuS, an effective strategy
for curating support sets using parametric or non-parametric methods to
mitigate the lack of data samples available from the target task distribution,
and (3) We propose TIP-X, a novel training-free method for adapting VLMs to
downstream classification in both the name-only transfer and few-shot regimes.
## 2 Related Work
Vision-Language (VL) Foundation Models. In the past few years, there has been
a Cambrian explosion in large-scale VL foundation models [6]. In a seminal
work, Radford et al. [61] introduced CLIP, a large VLM trained on a massive
corpus (400M image-text pairs acquired from the web) that exhibits strong
downstream visual task performance. The introduction of CLIP inspired further
development of VLMs [46, 1, 41, 20, 85, 79, 76, 11, 74, 29, 31, 47, 50, 78],
each pre-trained on web-scale datasets to learn joint image-text
representations. These representations can then be applied to tackle
downstream tasks like semantic segmentation [67, 30], object detection [33,
23], image captioning [54, 3] and generative modelling [63, 62], In this work,
we adapt such VLMs in a training-free setting to diverse downstream tasks.
Adaptation of VL models. The paradigm shift introduced by CLIP is its ability
to do image classification in a zero-shot transfer setting [61]. In this
setup, none of the target dataset classes are known a-priori and the task is
to adapt implicitly at inference time to a given dataset. Since CLIP’s
training objective drives it to assign appropriate similarities to image-text
pairs, it acquires the ability to perform zero-shot classification directly.
Inspired by CLIP’s zero-shot success, further work has sought to improve upon
its performance. In Tab. 1, we characterise some of these methods along three
major axes: (i) if the method requires training, (ii) if the method requires
labelled samples from the target task, and (iii) if the method requires
samples from the target task distribution333Note that (iii) subsumes (ii).
(ii) refers to access to labelled data samples from the target dataset whereas
(iii) refers to a more general setting where the samples from the target
dataset can be unlabelled. We distinguish between the two for clarity..
In this work, we focus on the training-free name-only transfer regime—our goal
is to adapt VLMs to target tasks without explicit training or access to
samples from the target distribution. Instead, we assume access only to
category names of target tasks. This formulation was recently considered for
semantic segmentation, where it was called name-only transfer [66]—we likewise
adopt this terminology. To the best of our knowledge, only two other
concurrent approaches, CuPL [60] and VisDesc [53], operate in this regime.
They use pre-trained language models to enhance textual prompts for zero-shot
classification. By contrast, SuS-X pursues a support set curation strategy to
adapt VLMs using knowledge of category names. These approaches are
complementary, and we find that they can be productively combined. Two other
related works operating purely in the zero-shot setting are: (1) CALIP [34],
which uses parameter-free attention on image-text features, and (2) CLIP+DN
[89], which uses distribution normalisation. We compare with these four
baselines in Sec. 4.
## 3 SuS-X: Training-Free Name-Only Transfer
Figure 2: SuS-X for training-free name-only transfer. SuS-X consists of two
core building blocks. (1) SuS (top right), a dynamic support set that we
construct to infuse visual information into the VLM based only on knowledge of
target category names. We construct support sets either in a parametric
(generating images using Stable Diffusion) or non-parametric (retrieving
images from LAION-5B) manner. (2) TIP-X (bottom right), our novel training-
free method that leverages image-text distances to compute similarities
between the support set and the test images. These similarities act as
attention weights for the support set labels, and can directly be combined
with the original logits from the VLM for classification.
We describe the two main building blocks of SuS-X—(1) Support Set (SuS)
construction, and (2) training-free inference using our novel TIP-X method.
Fig. 2 depicts our overall training-free name-only transfer framework.
### 3.1 SuS Construction
We follow recent adaptation methods [84, 28] that use a small collection of
labelled images to provide visual information to CLIP. However, differently
from these methods, rather than accessing labelled images from the target
distribution, we propose two methods (described next) to construct such a
support set (SuS) without such access.
(I) Stable Diffusion Generation. Our first method leverages the powerful text-
to-image generation model, _Stable Diffusion_ [63]. We employ specific
prompting strategies for generating salient and informative support images.
Concretely, given a set of downstream textual class labels,
$\mathcal{T}=\\{t_{1},t_{2},\dots,t_{C}\\}$, where $C$ denotes the number of
categories, we prompt Stable Diffusion to generate $N$ images per class. In
this way, we construct our support set of size $NC$, with each image having
its associated class label.
By default, we prompt Stable Diffusion using the original CLIP prompts,
_i.e_., “A photo of a $<$CLASS$>$.”, where $<$CLASS$>$ is the class text
label. To further diversify the generation process, we follow CuPL [60] to
first generate customised textual prompts for each class by prompting GPT-3
[8] to output descriptions of the particular class. We then feed this
customised set of prompts output by GPT-3 into Stable Diffusion for generating
images. For example, to generate images from the “dog” class, we prompt GPT-3
to describe “dogs”, and then prompt Stable Diffusion with the resulting
descriptions. In section 4.4, we compare the performance of the default
(called Photo) and this augmented prompting procedure (called CuPL). Unless
otherwise specified, all our experiments with Stable Diffusion support sets
use the CuPL strategy.
(II) LAION-5B Retrieval. Our second method leverages the large-scale vision-
language dataset, LAION-5B [65]. It contains 5.85 billion image-text pairs,
pre-filtered by CLIP. Using LAION-5B, we retrieve task-specific images using
class text prompts for constructing the support set. Concretely, given textual
class labels, $\mathcal{T}=\\{t_{1},t_{2},\dots,t_{C}\\}$, we rank all images
in LAION-5B by their CLIP image-text similarity to each text class label
$t_{i}$, where $i\in[1,C]$. We then use the top $N$ image matches as our
support set for class $i$, resulting in an $NC$-sized support set of images
with their associated class labels. Note that curating supporting knowledge by
search is a classical technique in computer vision [26] that was recently
revisited in the task of semantic segmentation [67]. Here we adapt this idea
to the name-only transfer classification setting. For efficient retrieval, we
leverage the approximate nearest neighbour indices released by the
authors444https://huggingface.co/datasets/laion/laion5B-index. Similar to the
Stable Diffusion generation approach, we experiment with both Photo and CuPL
prompting strategies for curating our LAION-5B support set (see Sec. 4.4). By
default, we use Photo prompting for all our experiments with LAION-5B support
sets.
Remark. Note that SuS can be seen as a visual analogue to CuPL [60], where,
for each class, we augment VLMs with rich, relevant images, instead of the
customised textual descriptions generated in CuPL.
### 3.2 TIP-X Inference
Given our support set from the previous section, our task is to now leverage
it in a training-free inference scheme to inform CLIP’s zero-shot predictions.
We first briefly review the zero-shot CLIP classification pipeline, discuss
the recently proposed TIP-Adapter [84] for training-free adaptation, and
highlight a critical shortcoming in its method due to uncalibrated intra-modal
embedding distances, which we address in our method—TIP-X.
Zero-shot CLIP. For classification into $C$ classes, CLIP converts class
labels into text prompts and encodes them with its text encoder. Collectively,
the encoded prompt vectors can be interpreted as a classifier weight matrix
$W\in\mathbb{R}^{C{\times}d}$, where $d$ is embedding dimension. For a test
set ${T}{=}{\\{y_{1},y_{2},...,y_{t}\\}}$ comprising $t$ test images, CLIP’s
image encoder is applied to produce test image features:
$\begin{gathered}f_{i}=\texttt{CLIPImageEncoder}(y_{i}),i\in[1,t],f_{i}\in\mathbb{R}^{d}\\\
f=\texttt{Concat}([f_{1},f_{2},\dots,f_{t}]),f\in\mathbb{R}^{t\times
d}\end{gathered}$ (1)
Using $W$ and $f$, CLIP performs classification by computing zero-shot logits
(ZSL) via a dot product:
$\texttt{ZSL}=fW^{T}$ (2)
TIP-Adapter. Given a $CK$-sized $K$-shot labelled dataset
$D=\\{x_{1},x_{2},\dots,x_{CK}\\}$555Note that a $K$-shot labelled dataset for
$C$ classes has a size $CK$. from the target domain, TIP-Adapter [84] encodes
$D$ using CLIP’s image encoder:
$\begin{gathered}F_{i}=\texttt{CLIPImageEncoder}(x_{i}),i\in[1,CK],F_{i}\in\mathbb{R}^{d}\\\
F=\texttt{Concat}([F_{1},F_{2},\dots,F_{CK}]),F\in\mathbb{R}^{CK\times
d}\end{gathered}$ (3)
It then converts each of the few-shot class labels to one-hot vectors
$L\in\mathbb{R}^{CK{\times}C}$. Next, it computes an affinity matrix to
capture the similarities between $F$ and $f$:
$A=\exp(-\beta(1-fF^{T}))$ (4)
where $\beta$ is a hyperparameter that modulates “sharpness”. Finally, these
affinities are used as attention weights over $L$ to produce logits that are
blended with ZSL using a hyperparameter, $\alpha$:
$\texttt{TL}=\alpha AL+fW^{T}$ (5)
Motivating TIP-X. TIP-Adapter gains from the affinity computation between the
test and few-shot image samples (see Eq. 4). This similarity is computed in
CLIP’s image space. However, prior research [80, 48, 70] has demonstrated the
existence of a modality gap between CLIP’s image and text spaces. This leads
us to question if doing image-image similarity comparisons in CLIP’s image
space is optimal.
(a) Intra-modal and inter-modal CLIP cosine similarities. We observe quite
distinct intra-modal and inter-modal cosine similarity distributions.
(b) Intra-modal degrees of freedom. Different intra-modal similarities can
satisfy same inter-modal constraints, leaving room for poor calibration.
Figure 3: Our two-fold analysis motivating TIP-X
Fig. 3(a) shows the pairwise image-image, text-text and image-text cosine
similarities of the ImageNet validation set CLIP embeddings. Clearly, the
intra-modal and inter-modal similarities are distributed differently—the
inter-modal similarities have small variance and mean, whereas the intra-modal
similarities have larger means and variances. This mismatch happens because
contrastive training of CLIP maximises the inter-modal cosine similarities of
paired samples without regard to intra-modal similarities. This implies that
the intra-image CLIP embedding similarities employed by TIP-Adapter may not
reflect the true intra-image similarities. Fig. 3(b) illustrates this idea
with a simple example. Consider two image embeddings that are required to be a
distance $r$ away from a particular text embedding. The two image embeddings
can satisfy this condition by being very close to each other or very far apart
from each other. Fig. 3(b) shows that this constraint can be satisfied by any
two arbitrary points on a hypersphere of radius $r$. While we expect loose
constraints to be imposed via transitivity, we nevertheless expect a lower
quality of calibration in intra-modal (_e.g_., image-image) comparisons.
TIP-X to the rescue. To get around the problem of uncalibrated intra-modal
embedding distances in TIP-Adapter, we propose to use inter-modal distances as
a bridge. More specifically, rather than computing similarities between the
test features ($f{\in}\mathbb{R}^{t{\times}d}$) and few-shot features
($F{\in}\mathbb{R}^{CK\times d}$) in the image embedding space ($fF^{T}$), we
use the image-text space. We first construct signatures by computing
similarities of $f$ and $F$ with the text classifier weights $W$:
$\begin{gathered}S=\texttt{softmax}(FW^{T}),S\in\mathbb{R}^{CK\times C}\\\
s=\texttt{softmax}(fW^{T}),s\in\mathbb{R}^{t\times C}\end{gathered}$ (6)
These signatures comprise probability distributions encoding inter-modal
affinities between the few-shot features and class text vectors, and likewise
for the test features. We then construct our affinity matrix
$M\in\mathbb{R}^{t\times CK}$ by measuring the KL-divergence between the
signatures as follows:
$\begin{gathered}M_{i,j}=\texttt{KL}(s_{i}||S_{j}),i\in[1,t],j\in[1,CK]\end{gathered}$
(7)
where $s_{i}$ represents the $i^{th}$ test signature for the $t$ test samples,
and $S_{j}$ represents the $j^{th}$ few-shot signature. Since we are working
with discrete probability distributions, we compute the KL-divergence as
$\texttt{KL}(P||Q)=\sum_{i}P_{i}\log\frac{P_{i}}{Q_{i}}$.
The construction of the affinity matrix $M$ can be seen as analogous to the
affinity computation in TIP-Adapter (Eq. 4). However, our affinity matrix
construction removes direct reliance on the uncalibrated image-image
similarities.
Finally, before using our affinity matrix $M$ as attention weights for $L$
(one-hot encoded class labels), we rescale (denoted by $\psi$) the values of
$M$ to have the same range (min, max values) as the TIP-Adapter affinities
($A$). Further, since our affinity matrix $M$ consists of KL-divergence
values, the most similar samples will get small weights since their KL-
divergence will be low (close to 0). To mitigate this, we simply negate the
values in $M$. We then blend our predicted logits with TL using a scalar
$\gamma$:
$\texttt{TXL}=fW^{T}+\alpha AL+\gamma\psi(-M)L$ (8)
The entire TIP-X method is shown in Fig. 2 (bottom right).
### 3.3 SuS-X: Combining SuS and TIP-X
Since our constructed support sets act as pseudo few-shot datasets, we
directly replace the few-shot features $F$ in the TIP-X framework with the
features of our support set. We call our method SuS-X-LC if we combine TIP-X
with the LAION-5B curated support set, and SuS-X-SD when combined with the
Stable Diffusion generated support set. These methods enable training-free
name-only adaptation of zero-shot VLMs.
## 4 Experiments
First, we evaluate SuS-X against strong baselines in the training-free zero-
shot/name-only transfer regimes, across three VLMs. Next, we illustrate the
adaptation of TIP-X into the few-shot training-free regime. Finally, we ablate
and analyse our method to provide additional insights.
### 4.1 Training-free name-only transfer evaluation
Datasets. For a comprehensive evaluation, we test on 19 datasets spanning a
wide range of object, scene and fine-grained categories: ImageNet [18],
StanfordCars [43], UCF101 [68], Caltech101 [25], Caltech256 [32], Flowers102
[56], OxfordPets [59], Food101 [7], SUN397 [75], DTD [14], EuroSAT [37],
FGVCAircraft [51], Country211 [61], CIFAR-10 [44], CIFAR-100 [44], Birdsnap
[5], CUB [72], ImageNet-Sketch [73] and ImageNet-R [38]. Previous few-shot
adaptation methods [81, 28, 86] benchmark on a subset of 11 of these 19
datasets. We report results on the 19-dataset suite in the main paper and
compare results using only the 11-dataset subset in the supp. mat.
Experimental Settings. We compare against six baselines. For zero-shot CLIP,
we use prompt ensembling with 7 different prompt templates following [61,
84]666The 7 prompt templates are: “itap of a $<$class$>$.”, “a origami
$<$class$>$.”, “a bad photo of the $<$class$>$.”, “a photo of the large
$<$class$>$.”, “a $<$class$>$ in a video game.”, “art of the $<$class$>$.”,
and “a photo of the small $<$class$>$.”.. We run
CuPL777https://github.com/sarahpratt/CuPL,
VisDesc888https://github.com/sachit-menon/classify_by_description_release
(name-only transfer) and
CLIP+DN999https://github.com/fengyuli2002/distribution-normalization (zero-
shot transfer) using their official code. We also experiment with augmenting
the CuPL prompts with the original prompt ensemble, and call it CuPL+e. For
CALIP (zero-shot transfer), in the absence of public code at the time of
writing, we aim to reproduce their results using our own implementation. For
our proposed methods, we report results using both SuS-X-LC and SuS-X-SD. For
both methods, we use a fixed number of support samples per dataset (see supp.
mat. for details). For CALIP and SuS-X, we conduct a hyperparameter search on
the dataset validation sets. In Sec. 4.4 we perform a hyperparameter
sensitivity test for a fair evaluation. By default, we use the ResNet-50 [35]
backbone as CLIP’s image encoder for all models.
Table 2: Training-free adaptation of CLIP on 19 datasets with RN50 visual backbone. The best and second best results for each dataset are bolded and underlined, respectively. Individual results for all 19 datasets are available in the supp. mat. ∗Average reported across 19 datasets. †Our re-implementation. | Method | Average∗ | ImageNet [18] | ImageNet-R [38] | ImageNet-Sketch [73] | EuroSAT [37] | DTD [14] | Birdsnap [5]
---|---|---|---|---|---|---|---|---
Zero-shot | Zero-shot CLIP [61] | 52.27 | 60.31 | 59.34 | 35.42 | 26.83 | 41.01 | 30.56
CALIP [34] | – | 60.57 | – | – | 38.90 | 42.39 | –
CALIP [34]† | 52.37 | 60.31 | 59.33 | 36.10 | 26.96 | 41.02 | 30.68
CLIP+DN [89] | 53.02 | 60.16 | 60.37 | 35.95 | 28.31 | 41.21 | 31.23
Name-only | CuPL [60] | 55.50 | 61.45 | 61.02 | 35.13 | 38.38 | 48.64 | 35.65
CuPL+e | 55.76 | 61.64 | 61.17 | 35.85 | 37.06 | 47.46 | 35.80
VisDesc [53] | 53.76 | 59.68 | 57.16 | 33.78 | 37.60 | 41.96 | 35.65
SuS-X-SD (ours) | 56.73 | 61.84 | 61.76 | 36.30 | 45.57 | 50.59 | 37.14
SuS-X-LC (ours) | 56.87 | 61.89 | 62.10 | 37.83 | 44.23 | 49.23 | 38.50
(a)
(b)
(c)
Figure 4: (a) Comparison of SuS-X with Zero-shot CLIP. (b) Results of
training-free few-shot classification. (c) Performance comparison of SuS-X
across visual backbones.
Main Results. In Tab. 2, we compare both variants of SuS-X with the baselines.
We report an average across 19 datasets. We also include results on ImageNet,
EuroSAT, DTD, Birdsnap, ImageNet-R and ImageNet-Sketch (results on all 19
datasets in the supp. mat.). SuS-X methods outperform zero-shot CLIP by 4.6%
on average across all 19 datasets. We observe striking gains of 18%, 8% and 7%
on EuroSAT, DTD and Birdsnap respectively. We also outperform the SoTA
training-free adaptation methods—CuPL+ensemble and VisDesc by 1.1% and 3.1% on
average respectively. To further probe where we attain the most gains, we plot
the absolute improvement of our models over zero-shot CLIP in Fig. 4(a). We
observe large gains on fine-grained (Birdsnap, CUB, UCF101) and specialised
(EuroSAT, DTD) datasets, demonstrating the utility of SuS-X in injecting rich
visual knowledge into zero-shot CLIP (additional fine-grained classification
analysis in supp. mat.). We further compare SuS-X to few-shot methods that use
labelled samples from the true distribution in the supp. mat.—despite being at
a disadvantage due to using no target distribution samples, SuS-X is still
competitive with these methods.
### 4.2 Transfer to different VLMs
We evaluate transfer to VLMs other than CLIP, namely TCL [76] and BLIP [46].
We only retain image and text encoders of these models for computing features,
while preserving all other experimental settings from Sec. 4.1. Tab. 3 shows
our SuS-X methods strongly outperform all baseline methods across both VLMs—we
improve on zero-shot models by 11.37% and 5.97% on average across 19 datasets.
This demonstrates that our method is not specific to CLIP, but can improve
performance across different VLMs.
Table 3: SuS-X generalises to different VLMs. ∗Average reported across 19 datasets. VLM | Method | Average∗ | ImageNet | EuroSAT | DTD | Birdsnap
---|---|---|---|---|---|---
TCL | Zero-shot | 31.38 | 35.55 | 20.80 | 28.55 | 4.51
CuPL | 34.79 | 41.60 | 26.30 | 42.84 | 6.83
CuPL+e | 32.79 | 41.36 | 25.88 | 41.96 | 6.60
VisDesc | 33.94 | 40.40 | 21.27 | 34.28 | 5.69
SuS-X-SD | 41.49 | 52.29 | 28.75 | 48.17 | 13.60
SuS-X-LC | 42.75 | 52.77 | 36.90 | 46.63 | 17.93
BLIP | Zero-shot | 48.73 | 50.59 | 44.10 | 44.68 | 10.21
CuPL | 51.11 | 52.96 | 39.37 | 52.95 | 12.24
CuPL+e | 51.36 | 53.07 | 41.48 | 53.30 | 12.18
VisDesc | 49.91 | 50.94 | 42.25 | 47.45 | 11.69
SuS-X-SD | 53.20 | 55.93 | 45.36 | 56.15 | 16.95
SuS-X-LC | 54.64 | 56.75 | 51.62 | 55.91 | 23.78
### 4.3 Adapting to the few-shot regime
A key component of our SuS-X method is TIP-X. In the previous section, we
showcased SoTA results in the training-free name-only transfer regime. Due to
its formulation, TIP-X can directly be extended to the few-shot regime, where
our support sets are labelled samples from the target dataset rather than
curated/generated samples. To evaluate TIP-X on such real-world support sets,
we conduct training-free few-shot classification using TIP-X. We compare
against the SoTA method in this regime—TIP-Adapter [84]. We report results on
the 11-dataset subset used by TIP-Adapter on five different shot settings of
the $K$-shot classification task: 1, 2, 4, 8 and 16.
We present average accuracy results on all shots in Fig. 4(b)—TIP-X
outperforms both Zero-shot CLIP and TIP-Adapter (absolute gain of 0.91% across
shots). Notably, on OxfordPets, we achieve 2.1% average gain. This further
demonstrates the generalisability of the TIP-X method in transferring to the
few-shot training-free setting.
### 4.4 Analysis
We conduct several ablations and provide additional visualisations to offer
further insight into the SuS-X method.
Component Analysis. SuS-X consists of two major building blocks—SuS
construction and TIP-X. We compare the performance difference (with average
accuracy across 19 datasets) of using SuS with TIP-Adapter instead of TIP-X in
Tab. 4. We use both default ensemble prompts and CuPL prompts for CLIP’s text
classifier to break down the performance gains further. We note that both SuS
and TIP-X are crucial for achieving the best results.
(a)
(b)
(c)
(d)
Figure 5: Support samples from the generated SuS-SD, retrieved SuS-LC and true
training distribution for ImageNet. By randomising the image order in each
subfigure, we pose a challenge question—can you match the three images for
each subfigure to their source i.e. SuS-SD, SuS-LC or ImageNet train set? The
answers are provided at the bottom of the page1111footnotemark: 11.
Transfer to different visual backbones. We evaluate the scalability of our
model across different CLIP visual backbones— Fig. 4(c) shows that both SuS-X
variants consistently improve upon zero-shot CLIP across ResNet and
VisionTransformer backbones of varying depths and sizes.
SuS size. We study the effect of varying support set size for SuS-LC and SuS-
SD—we generate three different support sets with random seeds for support
sizes of 1, 5, 10, 25, 50, 75 and 100 samples. From Fig. 6, we observe two
broad trends—some tasks benefit (ImageNet-R, DTD) from having more support set
samples while others do not (Country211, Flowers102). We suggest that this is
connected to the domain gap between the true data distribution and support set
samples—if the domain gap is large, it is inimical to provide a large support
set, whereas if the domains are similar, providing more support samples always
helps.
(a)
(b)
Figure 6: Effect of support size.
SuS visualisation. We visualise samples from both support set construction
methods on ImageNet in Fig. 5. It is hard to distinguish between the true
ImageNet samples and the SuS samples—we can therefore construct support sets
to mimic the true data distribution, with access to only the category names. A
caveat is that the support set does not always capture the domain
characteristics of the true distribution, leading to a domain gap (lighting
conditions, diverse scene backgrounds, confounding objects etc). To fully
close the gap to using true few-shot datasets as support sets [28, 84],
further research into exact unsupervised domain matching of support sets and
few-shot datasets is required.
Prompting strategies for SuS construction. Tab. 5 depicts the performance of
Photo and CuPL prompting—best results are achieved with the LC-Photo and SD-
CuPL strategies. We further compare the diversity of images produced by the
two strategies on ImageNet121212We compute diversity as 1 minus the mean of
the average pairwise image cosine-similarities within a class. A larger value
implies low cosine similarities across images within a class, implying more
diverse images. Alternatively, a smaller value implies less diverse
images.—from Tab. 5, it is evident that CuPL prompting leads to more diverse
support sets as compared to Photo prompting.
Table 4: Component Analysis of SuS-X. Text Prompts | Method | SuS | TIP-X | Average Accuracy
---|---|---|---|---
Default | Zero-shot CLIP | ✗ | ✗ | 52.27
SuS-TIP-SD | ✓ | ✗ | 53.49 (+1.22%)
SuS-X-SD | ✓ | ✓ | 53.69 (+1.42%)
SuS-TIP-LC | ✓ | ✗ | 53.83 (+1.56%)
SuS-X-LC | ✓ | ✓ | 54.20 (+1.93%)
CuPL+e | CuPL+e | ✗ | ✗ | 55.76 (+3.49%)
SuS-TIP-SD | ✓ | ✗ | 56.63 (+4.36%)
SuS-X-SD | ✓ | ✓ | 56.73 (+4.46%)
SuS-TIP-LC | ✓ | ✗ | 56.72 (+4.45%)
SuS-X-LC | ✓ | ✓ | 56.87 (+4.60%)
Table 5: Prompting strategies for SuS construction. SuS
method | Average Acc. | ImageNet Acc. | Diversity
---|---|---|---
Photo | CuPL | Photo | CuPL | Photo | CuPL
LC | 56.87 | 56.20 | 61.89 | 61.79 | 0.28 | 0.32
SD | 56.32 | 56.73 | 61.79 | 61.84 | 0.17 | 0.20
Hyperparameter Sensitivity. We perform a sensitivity test for our $\gamma$
hyperparameter (refer Eq. 8) on ImageNet-R, OxfordPets, and DTD. We fix
$\alpha$ and $\beta$ to be 1, and run a sweep over $\gamma\in[0,1]$. From Tab.
6, we observe that moderate values of $\gamma$ are typically preferred, and
the variance of the accuracy values is small. However, note that for DTD, the
optimal $\gamma$ is slightly larger (0.75)—this is due to its specialised
nature which requires more guidance from the specialised support set to inform
pre-trained CLIP. Previous few-shot adaptation works [28, 84] observed similar
results. For more hyperparameter ablations, see the supp. mat.
Table 6: Hyperparameter sensitivity for $\boldsymbol{\gamma}$ Dataset | $\boldsymbol{\gamma}$ value
---|---
$\boldsymbol{0}$ | $\boldsymbol{0.1}$ | $\boldsymbol{0.2}$ | $\boldsymbol{0.3}$ | $\boldsymbol{0.5}$ | $\boldsymbol{0.75}$ | $\boldsymbol{1}$
ImageNet-R | 60.87 | 60.98 | 61.03 | 61.05 | 61.00 | 60.89 | 60.65
OxfordPets | 76.76 | 77.17 | 77.58 | 77.44 | 77.17 | 77.17 | 76.90
DTD | 47.16 | 47.16 | 47.51 | 47.69 | 47.87 | 47.96 | 47.60
1212footnotetext: (a)LC,SD,Train,(b)SD,Train,LC,(c)Train,LC,SD,(d)SD,Train,LC
### 4.5 Limitations and broader impact
While demonstrating promising results, we note several limitations of our
approach. (1) To perform name-only transfer, we rely on CLIP to have seen
related concepts during pre-training. For concepts that are so rare that they
do not appear during pre-training, transfer will not be feasible. (2) We
employ LAION-5B [65] as a source of knowledge. While reasonable for a proof of
concept, this data is relatively uncurated and may contain harmful content. As
such, our approach is not suitable for real-world deployment without careful
mitigation strategies to address this concern. Similar arguments apply to
Stable Diffusion [63].
## 5 Conclusion
In this paper, we studied the training-free name-only transfer paradigm for
classification tasks. We systematically curated support sets with no access to
samples from the target distribution and showed that they help improve CLIP’s
zero-shot predictions by providing rich, task-specific knowledge. We further
motivated the TIP-X framework through the observation that CLIP’s intra-modal
embedding spaces are not optimal for computing similarities. With these two
building blocks, we demonstrated superior performance to prior state-of-the-
art.
Acknowledgements. This work was supported by the Isaac Newton Trust and an
EPSRC access-to-HPC grant. SA would like to acknowledge the support of Z.
Novak and N. Novak in enabling his contribution. VU would like to thank
Gyungin Shin, Surabhi S. Nath, Jonathan Roberts, Vlad Bogolin, Kaiqu Liang and
Anchit Jain for helpful discussions and feedback.
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## Appendix A Dataset Details
We enumerate the validation and testing split sizes of all datasets in Tab. 7.
We make two small modifications to the standard datasets as described in CoOp
[86]: (1) We discard the “BACKGROUND Google” and “Faces easy classes” from the
Caltech101 dataset, and (2) For the UCF101 dataset, we consider the middle
frame of each video as our image sample.
Table 7: Dataset details for the 19 datasets used in this study. Dataset | Classes | Val | Test
---|---|---|---
UCF-101 | 101 | 1898 | 3783
CIFAR-10 | 10 | 10000 | 10000
CIFAR-100 | 100 | 10000 | 10000
Caltech101 | 100 | 1649 | 2465
Caltech256 | 257 | 6027 | 9076
ImageNet | 1000 | 50000 | 50000
SUN397 | 397 | 3970 | 19850
FGVCAircraft | 100 | 3333 | 3333
Birdsnap | 500 | 7774 | 11747
StanfordCars | 196 | 1635 | 8041
CUB | 200 | 1194 | 5794
Flowers102 | 102 | 1633 | 2463
Food101 | 101 | 20200 | 30300
OxfordPets | 37 | 736 | 3669
DTD | 47 | 1128 | 1692
EuroSAT | 10 | 5400 | 8100
ImageNet-Sketch | 1000 | 50889 | 50889
ImageNet-R | 200 | 30000 | 30000
Country211 | 211 | 10550 | 21100
## Appendix B Details about Support Set Curation Strategies
We include further technical details about our two support set curation
strategies—Stable Diffusion Generation and LAION-5B Retrieval.
Stable Diffusion Generation. For all our experiments with the Stable Diffusion
model, we use the stable-diffusion-v1-4 checkpoint with a 9.5 guidance scale
[39], 85 diffusion steps and $512{\times}512$ output resolution. We then
downscale these images to CLIP’s input resolution of $224{\times}224$.
LAION-5B Retrieval. For all our experiments, we rank all images in the
LAION-5B corpus based on their image-text similarity with the given class
textual prompt. We use the LAION-5B pre-constructed index that leverages the
CLIP-ViT-L/14 model. Finally, since the images might be of varying
resolutions, we pre-process them to CLIP’s input resolution of
$224{\times}224$.
## Appendix C Few-shot Learning with TIP-X
In Sec. 4.3, we adapt TIP-X to the few-shot training-free adaptation regime,
and compare with the SoTA model TIP-Adapter. We now show the extended results
on all 11 datasets in Fig. 7. On average, we outperform TIP-Adapter by
$0.91\%$ across all shots.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
Figure 7: Results for the training-free few-shot regime across 11 datasets.
## Appendix D Details about Support Set Sizes
For our main results in Sec. 4.1, we use a fixed number of support set samples
per dataset. In Tab. 8, we enumerate the number of support set samples used
per dataset. As shown in Sec. 4.4, the support set size can impact performance
significantly—the nature of these impacts are dataset-specific.
Table 8: Support Set Sizes Dataset | Support Set Size
---|---
UCF-101 | 5858
CIFAR-10 | 50
CIFAR-100 | 4700
Caltech101 | 101
Caltech256 | 3084
ImageNet | 36000
SUN397 | 397
FGVCAircraft | 7900
Birdsnap | 39000
StanfordCars | 980
CUB | 400
Flowers102 | 3162
Food101 | 3434
OxfordPets | 2627
DTD | 188
EuroSAT | 150
ImageNet-Sketch | 42000
ImageNet-R | 10200
Country211 | 844
## Appendix E Details about Baselines
For our main zero-shot/name-only training-free CLIP-based experiments, we use
six main baselines—Zero-shot CLIP [61], CALIP [34], CLIP+DN [89], VisDesc
[53], CuPL [60] and CuPL+e.
Zero-shot CLIP. For Zero-shot CLIP, we directly use the model weights released
by OpenAI and the official repository for reproducing results on different
datasets131313https://github.com/openai/CLIP. For benchmarking all our
results, we use the 7-prompt ensemble set used by TIP-Adapter [84] for all
datasets. The 7 prompt templates in the ensemble are: “itap of a
$<$class$>$.”, “a origami $<$class$>$.”, “a bad photo of the $<$class$>$.”, “a
photo of the large $<$class$>$.”, “a $<$class$>$ in a video game.”, “art of
the $<$class$>$.”, and “a photo of the small $<$class$>$.”.
CALIP details. Due to the unavailability of publicly released code at the time
of writing this paper, we re-implement the CALIP baseline, following the
description in [34]. We provide access to our re-implementation as part of our
released codebase.
CLIP+DN details. For CLIP+DN, we use the official
code141414https://github.com/fengyuli2002/distribution-normalization released
by the authors on all datasets. As specified in the paper, we (i) use 100
random unlabeled validation samples for the mean estimation for DN, and (ii)
report the average accuracy across 5 different random seeds.
VisDesc details. For VisDesc, we use the official
code151515https://github.com/sachit-menon/classify_by_description_release
released by the authors on all datasets. We use their default prompt settings
for generating the GPT-3 descriptors.
CuPL details. For CuPL, we use the official
code161616https://github.com/sarahpratt/CuPL released by the authors on all
datasets. The list of pre-prompts used as inputs to GPT-3 for different
datasets are listed in Tab. 9 and Tab. 10.
CuPL+e details. For CuPL+e, we simply concatenate the 7-prompt ensemble
embeddings of each class with the custom GPT-3 generated CuPL embeddings of
that particular class. We then average all the embeddings within a class to
generate the textual embedding for that class. Then, we proceed as standard to
construct the classifier weight matrix by stacking all class text embeddings.
### E.1 Transfer to other VLMs
We can transfer all the aforementioned baselines to different VLMs by simply
swapping out CLIP’s frozen image and text encoders with those of TCL [76] and
BLIP [46]. For the TCL171717https://github.com/uta-smile/TCL experiments, we
use the standard ViT-B/16 base model that is fine-tuned for retrieval on MS-
COCO, released by the authors here. For the
BLIP181818https://github.com/salesforce/BLIP experiments, we use the standard
ViT-B/16 base model fine-tuned for retrieval on MS-COCO, released by the
authors here.
Table 9: CuPL hand-written prompts (1/2) Dataset | GPT-3 prompts
---|---
UCF101 | “What does a person doing {} look like”
“Describe the process of {}”
“How does a person {}”
CIFAR10 | “Describe what a {} looks like”
“How can you identify {}?”
“What does {} look like?”
“Describe an image from the internet of a {}”
“A caption of an image of {}: ”
CIFAR100 | “Describe what a {} looks like”
“How can you identify {}?”
“What does {} look like?”
“Describe an image from the internet of a {}”
“A caption of an image of {}: ”
Caltech101 | “Describe what a {} looks like”
“What does a {} look like”
“Describe a photo of a {}”
Caltech256 | “Describe what a {} looks like”
“What does a {} look like”
“Describe a photo of a {}”
ImageNet | “Describe what a {} looks like”
“How can you identify {}?”
“What does {} look like?”
“Describe an image from the internet of a {}”
“A caption of an image of {}: ”
SUN397 | “Describe what a {} looks like”
“How can you identify a {}?”
“Describe a photo of a {}”
FGVCAircraft | “Describe a {} aircraft”
Birdsnap | “Describe what a {}, a species of bird, looks like”
“What does a {} look like”
“Visually describe a {}, a type of bird”
“A caption of an image of a {}, a type of bird”
“Describe the appearance of a {}”
“What are the prominent features to identify a {} bird”
StanfordCars | “How can you identify a {}”
“Description of a {}, a type of car”
“A caption of a photo of a {}:”
“What are the primary characteristics of a {}?”
“Description of the exterior of a {}”
“What are the identifying characteristics of a {}, a type of car?”
“Describe an image from the internet of a {}”
“What does a {} look like?”
“Describe what a {}, a type of car, looks like”
Table 10: CuPL hand-written prompts (2/2) Dataset | GPT-3 prompts
---|---
CUB | “Describe what a {}, a species of bird, looks like”
“What does a {} look like”
“Visually describe a {}, a type of bird”
“A caption of an image of a {}, a type of bird”
“Describe the appearance of a {}”
“What are the prominent features to identify a {} bird”
Flowers102 | “What does a {} flower look like”
“Describe the appearance of a {}”
“A caption of an image of {}”
“Visually describe a {}, a type of flower”
Food101 | “Describe what a {} looks like”
“Visually describe a {}”
“How can you tell that the food in this photo is a {}?”
OxfordPets | “Describe what a {} pet looks like”
“Visually describe a {}, a type of pet”
DTD | “What does a {} material look like?”
“What does a {} surface look like?”
“What does a {} texture look like?”
“What does a {} object look like?”
“What does a {} thing look like?”
“What does a {} pattern look like?”
EuroSAT | “Describe an aerial satellite view of {}”
“How does a satellite photo of a {} look like”
“Visually describe a centered satellite view of a {}”
ImageNet-Sketch | “Describe how a black and white sketch of a {} looks like”
“A black and white sketch of a {}”
“Describe a black and white sketch from the internet of a {}”
ImageNet-R | “An art drawing of a {}”
“Artwork showing a {}”
“A cartoon a {}”
“An origami of a {}”
“A deviant art photo depicting a {}”
“An embroidery of a ”
“A graffiti art showing a {}”
“A painting of a {}”
“A sculpture of a {}”
“A black and white sketch of {}”
“A toy of a {}”
“A videogame of a {}”
Country211 | “Visually describe what {} looks like”
“What does the landscape of {} look like”
“Describe a photo taken in {}”
“How does a typical photo taken in {} look like”
## Appendix F More SuS Visualisations
In Fig. 8, we provide further support set samples across different datasets
curated using both SuS-LC and SuS-SD methods.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Figure 8: Support samples from the generated SuS-SD, retrieved SuS-LC and true
training distribution for different datasets. For each subfigure, the ordering
of figures is—SuS-LC, SuS-SD, Train. We label each figure with its source
dataset and class name.
## Appendix G Hyperparameter Settings
We provide the hyperparameter settings for obtaining our main results from
Sec. 4.1 in Tab. 11. For our hyperparameters, we conduct a search over
$[0.1,50]$ for $\alpha$, $[1,50]$ for $\beta$ and $[0.1,30]$ for $\gamma$. In
the main paper, we have a hyperparameter sensitivity test which ensures that
the variance in accuracy values is not too large as we vary our
hyperparameters.
Table 11: Hyperparameter settings for the 19 datasets. Dataset | $\boldsymbol{\alpha}$ | $\boldsymbol{\beta}$ | $\boldsymbol{\gamma}$
---|---|---|---
UCF-101 | 0.10 | 8.59 | 0.10
CIFAR-10 | 5.09 | 5.41 | 0.10
CIFAR-100 | 0.10 | 1.49 | 0.10
Caltech101 | 0.10 | 1.27 | 0.10
Caltech256 | 0.10 | 12.76 | 0.10
ImageNet | 10.08 | 39.46 | 0.10
SUN397 | 2.60 | 8.35 | 0.10
FGVCAircraft | 2.60 | 24.52 | 0.69
Birdsnap | 48.53 | 22.55 | 0.69
StanfordCars | 0.10 | 1.58 | 0.10
CUB | 0.10 | 8.84 | 0.10
Flowers102 | 0.10 | 2.72 | 0.10
Food101 | 17.56 | 49.02 | 0.10
OxfordPets | 10.08 | 41.91 | 1.29
DTD | 5.09 | 23.79 | 0.70
EuroSAT | 2.60 | 1.00 | 0.10
ImageNet-Sketch | 30.04 | 38.48 | 0.69
ImageNet-R | 2.60 | 30.65 | 0.70
Country211 | 12.57 | 22.31 | 0.10
Results without tuning. We also report the results on all 19 datasets without
tuning our hyperparameters in Tab. 12. For this, we fix the hyperparameters to
be ${\alpha}{=}{0.1}$, ${\beta}{=}{1.0}$, ${\gamma}{=}{0.1}$. Even without
hyperparameter tuning, we see large gains over Zero-shot CLIP.
Table 12: Zero-shot/name-only results with fixed hyperparameters (no hyperparameter tuning) | UCF101 | CIFAR-10 | CIFAR-100 | Caltech101 | Caltech256 | ImageNet | SUN397 | FGVCAircraft | Birdsnap | StanfordCars | CUB | Flowers102 | Food101 | OxfordPets | DTD | EuroSAT | ImageNet-Sketch | ImageNet-R | Country211 | Average (11 subset) | Average (19 datasets)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
ZS-CLIP | 55.56 | 73.10 | 40.58 | 85.92 | 78.98 | 60.31 | 59.11 | 16.71 | 30.56 | 56.33 | 41.31 | 62.89 | 74.11 | 81.82 | 41.01 | 26.83 | 35.42 | 59.34 | 13.42 | 56.41 | 52.27
SuS-X-SD-P | 61.41 | 74.68 | 43.45 | 89.57 | 80.46 | 61.64 | 62.96 | 18.84 | 36.20 | 57.19 | 48.90 | 66.18 | 77.45 | 85.17 | 48.76 | 37.11 | 36.05 | 61.69 | 14.26 | 60.57 | 55.89
SuS-X-SD-C | 61.51 | 74.65 | 43.53 | 89.53 | 80.50 | 61.65 | 62.95 | 19.11 | 36.36 | 57.18 | 48.84 | 66.26 | 77.53 | 85.17 | 48.35 | 37.27 | 35.88 | 61.69 | 14.25 | 60.59 | 55.91
SuS-X-LC-P | 61.49 | 74.62 | 44.30 | 89.57 | 80.56 | 61.80 | 63.02 | 20.04 | 36.75 | 57.19 | 48.81 | 66.87 | 77.36 | 85.31 | 47.87 | 37.49 | 36.25 | 61.62 | 14.20 | 60.73 | 56.01
SuS-X-LC-C | 60.51 | 74.61 | 44.07 | 89.49 | 80.59 | 61.53 | 62.94 | 19.23 | 36.25 | 57.05 | 49.02 | 66.83 | 77.35 | 82.27 | 47.04 | 36.78 | 35.76 | 60.91 | 14.21 | 60.09 | 55.60
Analysis of hyperparameters. From Tab. 11, we note that for some datasets, the
weight for the inter-modal distance term $\gamma$ is dominated by the weight
for the intra-modal distance term $\alpha$. We analyse this in depth, and show
that despite this disparity, using inter-modal distances still brings gains.
Tab. 13 reports results on these datasets (for which ${\alpha}{>>}{\gamma}$)
using their optimal hyperparameters (${\alpha}{>}{\gamma}$), fixed
hyperparameters (${\alpha}{=}{\gamma}{=}{0.1}$), and removed inter-modal
contributions (${\gamma}{=}{0}$). In most cases, it is beneficial to use small
inter-modal distance contributions over neglecting them (see green rows).
Hence, we conclude that both these terms are important for bringing the large
performance gains of our model.
Table 13: Analysis of $\alpha$ and $\gamma$ values. Dataset | Optimal | Fixed&Equal | Inter-modal only
---|---|---|---
${\alpha}{>}{\gamma}$ | ${\alpha}{=}{\gamma}{=}{0.1}$ | ${\alpha}{=}{\text{optimal}},{\gamma}{=}{0}$
ImageNet | 61.89 | 61.80 | 61.30
ImageNet-Sketch | 37.83 | 36.25 | 36.10
ImageNet-R | 62.10 | 61.62 | 61.30
OxfordPets | 86.59 | 85.31 | 85.00
Birdsnap | 38.50 | 36.75 | 37.70
Food101 | 77.62 | 77.53 | 77.55
## Appendix H Discussion on SuS vs CuPL/VisDesc
As discussed in the main paper, CuPL and VisDesc are two name-only transfer
methods that leverage a large pre-trained language model (GPT-3) to enhance
the textual prompts used for zero-shot classification. On the other hand, our
SuS construction strategies endow the zero-shot model with rich visual
information to discriminate between different categories.
We note that text alone cannot model the rich information in the world [15,
9]. Consider a task of classifying between two bird species—“Florida Scrub
Jay” and “Blue Jay”. The difference is all in the subtle visual details—blue
jays have a crest and distinct black markings on their necks. This level of
rich visual information is hard to extract from textual descriptions of class
names. Hence, the main advantage of SuS is in imparting this expressive visual
information for discriminating between fine-grained categories. We verify this
empirically in Fig. 9 depicting large gains in fine-grained datasets like
Birdsnap, Flowers102, OxfordPets etc (Full results in Tab. 17 below.).
Figure 9: Improvement for fine-grained tasks using SuS.
## Appendix I Compute Cost Comparison
We compare the computational requirements of our SuS-X and the baselines in
Tab. 14—for each method, we measure the time and memory requirements for one
ImageNet class _i.e_. on 50 test images. For CuPL, VisDesc and SuS-X, we
measure the construction time required for curating the enhanced textual
prompts and support sets. Note that in practical applications, it is typical
to cache the curated support sets/prompts for each class, thereby amortising
costs across queries. We note that our SuS-X models offer the most competetive
performance-efficiency tradeoff when comparing the compute requirements and
accuracy values.
Table 14: Compute requirements of different methods. These numbers are with our tests on a single Nvidia A100-80GB GPU with one ImageNet class _i.e_. 50 test images. Method | Construction | Inference | GPU | ImageNet
---|---|---|---|---
Time | Time | Memory | Accuracy
Zero-shot | – | 10.22ms | 2.2GB | 60.32
CALIP | – | 121.26ms | 24GB | 60.57
CLIP+DN | – | 10.22ms | 2.2GB | 60.16
VisDesc | $\sim$3s | 10.22ms | 2.2GB | 59.68
CuPL+e | $\sim$3s | 10.22ms | 2.2GB | 61.64
SuS-X-SD | $\sim$60s | 10.50ms | 3.2GB | 61.84
SuS-X-LC | $\sim$2s | 10.50ms | 3.2GB | 61.89
## Appendix J Diversity of CuPL and Photo prompting strategies
In this section, we describe in detail the computation of the diversity metric
used in Sec. 4.4.
We assume access to a support set $S$ of size $NC$, where there are $C$
classes and $N$ support samples per class. We denote the support subset of a
given class $i$ as $S_{i}=\\{s_{i,1},s_{i,2},\dots,s_{i,N}\\}$, where
$s_{i,j}$ denotes the $j^{th}$ support sample for class $i$. Corresponding to
these support subsets, we denote the features of $S_{i}$ as $F_{i}$ (using
CLIP’s image encoder):
$\begin{gathered}F_{i,j}=\texttt{CLIPImageEncoder}(s_{i,j}),F_{i,j}\in\mathbb{R}^{d},i\in[1,C],j\in[1,N]\\\
F_{i}=\texttt{Concat}([F_{i,1},F_{i,2},\dots,F_{i,N}]),F_{i}\in\mathbb{R}^{N\times
d}\end{gathered}$
We now compute the mean pairwise cosine similarity between all support samples
within a class i.e. for class $i$, we compute:
$\texttt{PCS}_{i}=\frac{\sum_{j=1}^{N}\sum_{k=1}^{N}F_{i,j}F_{i,k}^{T}}{N^{2}}$
The intuition is that if all the support samples within a class are similar to
each other, then the support set is less diverse. Hence, a higher value of
$\texttt{PCS}_{i}$ implies a lower diversity. We then compute the mean PCS
over all classes as:
$\texttt{MPCS}=\frac{\sum_{i=1}^{C}\texttt{PCS}_{i}}{C}$
Finally, we define diversity to be:
$\texttt{Diversity}=1-\texttt{MPCS}$
## Appendix K Further Analyses
We conduct some further ablation studies to analyse our novel TIP-X method
with more rigour. Due to lack of space in the main paper, we include these
ablations here, however these are vital analyses which delineate important
properties of our method.
### K.1 Contribution of intra-model and inter-modal distances
In Sec. 3.2, we describe our TIP-X method that utilises image-text distances
as a bridge for computing image-image intra-modal similarities. We refer to
the main equation for computing TIP-X logits again, highlighting the
importance of each term:
$\texttt{TXL}=\underbrace{fW^{T}}_{\text{1. zero-shot
component}}+\underbrace{\alpha AL}_{\text{2. intra-modal distance
component}}+\underbrace{\gamma\psi(-M)L}_{\text{3. inter-modal distance
component}}$
Zero-shot CLIP utilises only the zero-shot term (1) above. TIP-Adapter
utilises the zero-shot and intra-modal distance terms (1+2). Our method uses
all three terms (1+2+3). We further ablate this design choice to break down
the gains brought forth from each individual term. In Tab. 15, we show the
performance gains from each of these terms with our best performing SuS-X-LC
model across 19 datasets. We observe large gains from inter-modal and intra-
modal distances independently over just using the zero-shot term. Further,
both these distances provide complementary information to each other, and
hence can be productively combined leading to the best results.
Table 15: Contribution of intra-modal and inter-modal distances. Dist. terms used | 1 | 1+3 | 1+2 | 1+2+3
---|---|---|---|---
(Zero-shot) | (Inter-modal) | (Intra-modal) | (Both)
Average Acc. | 52.27 | 56.30 | 56.56 | 56.87
Gain | 0 | +4.03 | +4.29 | +4.60
### K.2 Comparing name-only SuS-X to few-shot methods
In Sec. 4.1 of the main paper, we showcased SoTA results with our SuS-X model
in the name-only setting. Recollect that in this setting, we use no images
from the true target distribution. Here, we evaluate how well our SuS-X model
fares against methods that use image samples from the true target
distribution. We compare our best performing SuS-X-LC method (uses no images
from target distribution) with 16-shot TIP-Adapter and 16-shot TIP-X (both
using 16 labelled images per class). From Tab. 16, we see that SuS-X-LC is
competitive (in green) against these few-shot adaptation methods, despite
using no target task images. There are however cases where SuS-X-LC severely
underperforms the few-shot methods—this is due to the domain gap between the
SuS images and the true labelled images (refer Sec. 4.4).
Table 16: SuS-X against few-shot labelled methods. Dataset | Zero-shot | SuS-X-LC | TIP-Adapter | TIP-X
---|---|---|---|---
(name-only, ours) | (few-shot) | (few-shot, ours)
ImageNet | 60.31 | 61.89 | 62.01 | 62.16
Food101 | 74.11 | 77.62 | 75.82 | 75.96
OxfordPets | 81.82 | 86.59 | 84.50 | 87.52
Caltech101 | 85.92 | 89.65 | 90.43 | 90.39
Flowers102 | 62.89 | 67.97 | 89.36 | 90.54
FGVCAircraft | 16.71 | 21.09 | 29.64 | 29.61
### K.3 Intuitions for best performing configurations
From Tab. 5 of the main paper, we note that our best name-only results are
achieved with the LC-Photo and SD-CuPL SuS construction strategies. A natural
question arises: “Why do the two SuS construction methods require different
prompting strategies for achieving their best results?”. We attempt to answer
this question via careful inspection of the support sets curated from these
strategies. For this case study, we inspect the support sets for the CIFAR-10
dataset.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 10: Uncovering the intuitions for different prompting configurations.
We showcase some support samples using different prompting configurations for
two CIFAR-10 classes—airplane and bird. The key takeaways upon inspecting
these samples are enumerated below.
From Fig. 10, we can draw two key takeaways regarding the best prompting
strategies for the two SuS curation methods:
1. 1.
LAION-5B retrieval. The support sets constructed with CuPL prompts are largely
divergent from the “true” distribution of natural semantic images of the
target concepts/classes. This can be noted from the right panels of the first
two rows in Fig. 10—this disparity in the retrieved support set images leads
to a large domain gap to the target distribution, hence resulting in poorer
performance than the Photo prompting strategy. Further, since the LAION-5B
support sets consist of natural images _i.e_. images available on the web, the
LAION-5B Photo support set images are closer to the true target distribution
of images.
2. 2.
Stable Diffusion Generation. The support sets generated using Stable Diffusion
represent a synthetic data distribution _i.e_. there is an innate distribution
shift from the target distribution images owing to the target datasets
(mostly) consisting of natural images. Hence, the Stable Diffusion support
sets are inherently at a disadvantage compared to the LAION-5B support sets.
However, within the constructed Stable Diffusion support sets, the CuPL
prompting strategy mildly wins over the Photo strategy since it helps generate
a more diverse set of images (consisting of more expansive lighting
conditions, background scenes etc.)—this diversity helps reduce the domain gap
to the target dataset to a small extent. This phenomenon of added diversity in
synthetic datasets aiding downstream performance has also been noted in
previous works [36].
## Appendix L Extended Results on all Datasets
In Tab. 17, we report the accuracies obtained on each of the 19 individual
datasets for all our baselines, and our SuS-X model variants with CLIP as te
VLM. We also report the average accuracy obtained on the 11 dataset subset
used in previous CLIP adaptation works [84, 28, 86]. In Tab. 18, we report all
the results with the TCL model as the VLM, and in Tab. 19, we report the
results with the BLIP model as the VLM.
Table 17: Training-free zero-shot/name-only results across model configurations and datasets. We report average results using both the 11 dataset subset used by previous works on few-shot adaptation [84, 28, 86] and the entire 19 dataset suite. For the CALIP baseline, we report numbers from the original paper (denoted with a subscript o) as well as our re-implementation (denoted with a subscript (r)). We refer to the Zero-shot CLIP model as ZS-CLIP and CuPL+ensemble baseline as CuPL+e. We use the CuPL+ensemble prompts for CLIP’s text classifier in this experiment. For both variants of our models, we append P or C to the name to distinguish between Photo and CuPL prompt strategies. For instance, SuS-X-LC-P refers to the SuS-X model with LC curation using the Photo strategy. All models use the ResNet-50 visual backbone. The best results for each dataset are bolded and the second best are underlined. This table contains the full set of values used for generating Fig. 4(a) and populating Tab. 2 in the paper. | UCF101 | CIFAR-10 | CIFAR-100 | Caltech101 | Caltech256 | ImageNet | SUN397 | FGVCAircraft | Birdsnap | StanfordCars | CUB | Flowers102 | Food101 | OxfordPets | DTD | EuroSAT | ImageNet-Sketch | ImageNet-R | Country211 | Average (11 subset) | Average (19 datasets)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
ZS-CLIP | 55.56 | 73.10 | 40.58 | 85.92 | 78.98 | 60.31 | 59.11 | 16.71 | 30.56 | 56.33 | 41.31 | 62.89 | 74.11 | 81.82 | 41.01 | 26.83 | 35.42 | 59.34 | 13.42 | 56.41 | 52.27
CALIPo | 61.72 | – | – | 87.71 | – | 60.57 | 58.59 | 17.76 | – | 56.27 | – | 66.38 | 77.42 | 86.21 | 42.39 | 38.90 | – | – | – | 59.45 | –
CALIPr | 55.61 | 73.15 | 40.62 | 86.20 | 79.08 | 60.31 | 59.10 | 16.71 | 30.68 | 56.32 | 41.40 | 63.01 | 74.13 | 81.84 | 41.01 | 26.96 | 36.10 | 59.32 | 13.45 | 56.47 | 52.37
CLIP+DN | 55.60 | 74.49 | 43.73 | 87.25 | 79.24 | 60.16 | 59.11 | 17.43 | 31.23 | 56.55 | 43.03 | 63.32 | 74.64 | 81.92 | 41.21 | 28.31 | 35.95 | 60.37 | 13.76 | 56.86 | 53.02
CuPL | 58.97 | 74.13 | 42.90 | 89.29 | 80.29 | 61.45 | 62.55 | 19.59 | 35.65 | 57.28 | 48.84 | 65.44 | 76.94 | 84.84 | 48.64 | 38.38 | 35.13 | 61.02 | 13.27 | 60.30 | 55.50
CuPL+e | 61.45 | 74.67 | 43.35 | 89.41 | 80.57 | 61.64 | 62.99 | 19.26 | 35.80 | 57.23 | 48.77 | 65.93 | 77.52 | 85.09 | 47.45 | 37.06 | 35.85 | 61.17 | 14.27 | 60.45 | 55.76
VisDesc | 58.47 | 73.22 | 39.69 | 88.11 | 79.94 | 59.68 | 59.84 | 16.26 | 35.65 | 54.76 | 48.31 | 65.37 | 76.80 | 82.39 | 41.96 | 37.60 | 33.78 | 57.16 | 12.42 | 58.30 | 53.76
SuS-X-SD-P | 61.72 | 74.71 | 44.14 | 89.65 | 80.62 | 61.79 | 62.96 | 19.17 | 36.59 | 57.37 | 49.12 | 67.97 | 77.59 | 86.24 | 49.35 | 38.11 | 36.58 | 62.10 | 14.26 | 61.08 | 56.32
SuS-X-SD-C | 61.54 | 74.69 | 44.63 | 89.53 | 80.64 | 61.84 | 62.95 | 19.47 | 37.14 | 57.27 | 49.12 | 67.72 | 77.58 | 85.34 | 50.59 | 45.57 | 36.30 | 61.76 | 14.27 | 61.76 | 56.73
SuS-X-LC-P | 61.49 | 74.95 | 44.48 | 89.57 | 80.62 | 61.89 | 63.01 | 21.09 | 38.50 | 57.17 | 48.86 | 67.07 | 77.62 | 86.59 | 49.23 | 44.23 | 37.83 | 62.10 | 14.24 | 61.72 | 56.87
SuS-X-LC-C | 61.43 | 74.76 | 44.12 | 89.61 | 80.63 | 61.79 | 62.94 | 20.34 | 37.07 | 57.06 | 48.86 | 67.60 | 77.58 | 85.22 | 49.47 | 37.16 | 36.45 | 61.39 | 14.26 | 60.93 | 56.20
Table 18: Training-free zero-shot/name-only results across model configurations using the TCL [76] architecture. For our SuS-X models, we only use the two best configurations from the previous CLIP experiment i.e. SuS-X-SD with CuPL strategy and SuS-X-LC with Photo strategy. This table contains the full set of values used for populating Tab. 3 in the paper. | UCF101 | CIFAR-10 | CIFAR-100 | Caltech101 | Caltech256 | ImageNet | SUN397 | FGVCAircraft | Birdsnap | StanfordCars | CUB | Flowers102 | Food101 | OxfordPets | DTD | EuroSAT | ImageNet-Sketch | ImageNet-R | Country211 | Average (11 subset) | Average (19 datasets)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
ZS-TCL | 35.29 | 82.33 | 50.86 | 77.65 | 61.90 | 35.55 | 42.12 | 2.25 | 4.51 | 1.53 | 7.63 | 28.30 | 24.71 | 20.63 | 28.55 | 20.80 | 24.24 | 46.05 | 1.42 | 28.84 | 31.38
CuPL | 41.23 | 81.75 | 52.63 | 81.66 | 65.91 | 41.60 | 49.35 | 3.48 | 6.83 | 2.11 | 10.20 | 26.10 | 23.62 | 22.15 | 42.84 | 26.30 | 25.67 | 53.61 | 4.07 | 32.77 | 34.79
CuPL+e | 41.63 | 82.07 | 52.66 | 81.29 | 66.46 | 41.36 | 49.98 | 3.51 | 6.60 | 2.11 | 9.80 | 26.91 | 24.84 | 21.17 | 41.96 | 25.88 | 26.36 | 53.36 | 3.68 | 34.82 | 32.79
VisDesc | 42.53 | 82.30 | 51.89 | 77.00 | 66.51 | 40.40 | 51.18 | 3.21 | 5.69 | 2.91 | 8.96 | 25.13 | 27.16 | 24.58 | 34.28 | 21.27 | 27.05 | 49.26 | 3.57 | 31.77 | 33.94
SuS-X-SD-C | 47.66 | 82.92 | 55.19 | 81.38 | 66.52 | 52.29 | 49.98 | 9.21 | 13.60 | 2.31 | 9.72 | 30.98 | 48.87 | 65.96 | 48.17 | 28.75 | 32.22 | 58.95 | 3.66 | 42.32 | 41.49
SuS-X-LC-P | 50.28 | 83.14 | 57.47 | 81.38 | 66.80 | 52.77 | 49.97 | 10.98 | 17.93 | 2.57 | 9.77 | 30.04 | 48.06 | 69.96 | 46.63 | 36.90 | 36.28 | 57.58 | 3.72 | 43.59 | 42.75
| *We use the official TCL-base checkpoint from here for these results.
Table 19: Training-free zero-shot/name-only results across model configurations using the BLIP [46] architecture. For our SuS-X models, we only use the two best configurations from the previous CLIP experiment i.e. SuS-X-SD with CuPL strategy and SuS-X-LC with Photo strategy. This table contains the full set of values used for populating Tab. 3 in the paper. | UCF101 | CIFAR-10 | CIFAR-100 | Caltech101 | Caltech256 | ImageNet | SUN397 | FGVCAircraft | Birdsnap | StanfordCars | CUB | Flowers102 | Food101 | OxfordPets | DTD | EuroSAT | ImageNet-Sketch | ImageNet-R | Country211 | Average (11 subset) | Average (19 datasets)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
ZS-BLIP | 50.49 | 86.68 | 61.72 | 92.13 | 82.17 | 50.59 | 54.22 | 5.40 | 10.21 | 54.71 | 14.95 | 40.15 | 54.21 | 59.04 | 44.68 | 44.10 | 43.69 | 70.93 | 5.84 | 49.97 | 48.73
CuPL | 56.09 | 86.06 | 61.99 | 92.41 | 83.45 | 52.96 | 59.16 | 5.85 | 12.24 | 54.64 | 18.53 | 43.97 | 56.14 | 72.00 | 52.95 | 39.37 | 44.83 | 72.27 | 6.26 | 53.23 | 51.11
CuPL+e | 55.61 | 86.33 | 62.16 | 92.29 | 83.59 | 53.07 | 59.38 | 6.27 | 12.18 | 54.89 | 18.63 | 43.72 | 57.10 | 71.73 | 53.30 | 41.48 | 45.34 | 72.40 | 6.42 | 53.53 | 51.36
VisDesc | 53.42 | 86.78 | 60.47 | 92.04 | 81.53 | 50.94 | 55.85 | 6.30 | 11.69 | 54.64 | 16.65 | 42.71 | 58.50 | 69.22 | 47.45 | 42.25 | 43.30 | 68.62 | 6.01 | 52.12 | 49.91
SuS-X-SD-C | 57.28 | 87.56 | 63.60 | 92.33 | 83.66 | 55.93 | 59.46 | 10.14 | 16.95 | 54.89 | 18.95 | 44.38 | 62.75 | 74.68 | 56.15 | 45.36 | 46.51 | 73.85 | 6.45 | 55.76 | 53.20
SuS-X-LC-P | 59.90 | 88.28 | 64.43 | 92.29 | 83.61 | 56.75 | 59.39 | 11.82 | 23.78 | 54.94 | 19.24 | 43.97 | 64.14 | 79.72 | 55.91 | 51.62 | 48.53 | 73.42 | 6.44 | 57.31 | 54.64
| *We use the official BLIP-base checkpoint from here for these results.
## Appendix M Results with different Visual Backbones
All our main results use the ResNet-50 [35] visual backbone for CLIP’s image
encoder. In Tab. 20, we compare the accuracies obtained on all 19 datasets
using 2 different visual backbone model classes—ResNets [35] (ResNet-50,
ResNet-101) and Vision Transformers [22] (ViT-B/32, ViT-B/16). We observe that
the accuracy values monotonically improve as we increase the model capacity.
Table 20: Training-free name-only results across visual backbones. For this experiment, we use the default versions of our SuS-X models: SuS-X-LC with Photo strategy and SuS-X-SD with CuPL strategy. This experiment uses the CuPL prompts for CLIP’s text classifier. This table contains the raw data for generating Fig. 4(c) of the paper. | | UCF101 | CIFAR-10 | CIFAR-100 | Caltech101 | Caltech256 | ImageNet | SUN397 | FGVCAircraft | Birdsnap | StanfordCars | CUB | Flowers102 | Food101 | OxfordPets | DTD | EuroSAT | ImageNet-Sketch | ImageNet-R | Country211 | Average (11 subset) | Average (19 datasets)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
RN50 | SuS-X-LC | 59.98 | 74.79 | 44.22 | 89.29 | 80.29 | 61.66 | 62.70 | 21.87 | 38.56 | 56.92 | 48.90 | 66.91 | 77.21 | 86.35 | 50.06 | 43.99 | 37.25 | 61.97 | 13.21 | 61.54 | 56.64
SuS-X-SD | 59.48 | 74.21 | 44.33 | 89.25 | 80.27 | 61.65 | 62.58 | 19.92 | 37.00 | 57.14 | 49.10 | 67.32 | 77.02 | 85.09 | 51.00 | 47.69 | 37.25 | 61.73 | 13.30 | 61.65 | 56.59
RN101 | SuS-X-LC | 60.03 | 77.51 | 46.72 | 92.09 | 81.96 | 62.11 | 61.50 | 22.92 | 39.87 | 61.20 | 45.82 | 59.28 | 78.52 | 88.44 | 51.18 | 39.23 | 40.05 | 69.07 | 11.45 | 61.50 | 57.31
SuS-X-SD | 57.84 | 76.97 | 46.01 | 92.09 | 81.96 | 62.18 | 61.61 | 21.66 | 35.60 | 61.05 | 45.93 | 60.90 | 78.41 | 86.56 | 51.95 | 39.23 | 40.47 | 68.94 | 11.41 | 61.23 | 56.88
ViT-B/32 | SuS-X-LC | 63.49 | 89.32 | 65.25 | 93.18 | 84.73 | 64.73 | 65.49 | 23.01 | 40.77 | 61.19 | 53.03 | 68.01 | 80.31 | 87.95 | 52.25 | 53.91 | 43.10 | 70.55 | 14.91 | 64.87 | 61.85
SuS-X-SD | 63.20 | 88.39 | 64.84 | 93.18 | 84.73 | 64.71 | 65.47 | 21.66 | 38.97 | 61.12 | 53.52 | 68.17 | 80.24 | 86.81 | 51.89 | 53.91 | 43.27 | 70.42 | 14.91 | 64.58 | 61.55
ViT-B/16 | SuS-X-LC | 66.72 | 90.94 | 68.66 | 93.91 | 87.41 | 70.00 | 67.85 | 30.51 | 47.71 | 65.90 | 56.96 | 73.08 | 86.08 | 91.58 | 55.32 | 58.06 | 49.34 | 78.20 | 19.19 | 69.00 | 66.18
SuS-X-SD | 66.59 | 89.88 | 68.47 | 93.96 | 87.45 | 69.88 | 67.73 | 28.68 | 45.53 | 66.13 | 57.11 | 73.81 | 86.08 | 90.57 | 54.55 | 57.49 | 49.51 | 78.22 | 19.28 | 68.68 | 65.84
## Appendix N Results with different Text-to-Image Generation Models
We also experiment with different text-to-image generation models for support
set generation to showcase the generalisability and robustness of our method’s
results. Tab. 21 depicts SuS-X-SD results by generating support sets using
different text-to-image generation models. The results presented in the main
paper all use the Stable-Diffusion-v1.4 model, but we also note similar gains
across three other generative models.
Table 21: SuS-X-SD Results with additional T2I models. T2I Model | ImageNet | EuroSAT | DTD | OxfordPets | Average
---|---|---|---|---|---
ZS-CLIP (baseline) | 60.31 | 26.83 | 41.01 | 81.82 | 52.49
StableDiffusion-1.4 (from main paper) | 61.84 | 45.57 | 50.59 | 85.34 | 60.84 (+8.35%)
Kandinsky2.1 | 61.83 | 44.96 | 49.17 | 85.47 | 60.36 (+7.87%)
OpenJourney-4 | 61.81 | 45.00 | 50.71 | 85.17 | 60.67 (+8.18%)
Protogen-2.2 | 61.82 | 48.67 | 50.35 | 85.26 | 61.52 (+9.03%)
## Appendix O Fine-tuning SuS-X
Despite our work’s main focus being the training-free adaptation regime, we
explore some preliminary results with fine-tuning SuS-X on a few datasets. We
compare both the training-free and the fine-tuned variants of SuS-X with other
CLIP adaptation methods that use full or partial (parameter-efficient fine-
tuning) in Tab. 22. We note for some datasets, full/partial fine-tuning
methods perform better than training-free SuS-X. However, due to the domain
gap between StableDiffusion/LAION-5B curated data and real test data, the
gains are not large (confirming prior work [36, sariyildiz2023fake]). Further,
we note that full fine-tuning and SuS-X are complementary, allowing a large
boost in performance for SuS-X-F. On the other hand, we emphasise that the
goal of our work is to keep the approach flexible and scalable—one can apply
SuS-X to an arbitrary number of rare categories without training. This
training-free approach can particularly benefit when the categories of
interest vary frequently, rendering repetitive fine-tuning inefficient.
Moreover, fine-tuning forces the model to fit a very specific task
distribution, enforcing forgetting of the model’s pre-trained performance on a
wide array of tasks. Since SuS-X only requires target task class names and
does not fine-tune the model, we can cache the task-specific support sets
a-priori and switch them dynamically based on the task at hand, without
causing catastrophic forgetting of CLIP’s pre-trained knowledge.
Table 22: Fine-tuning methods vs SuS-X. Method | ZS-CLIP | FT-CLIP | CoOp [88] | CLIP-Adapter [28] | SuS-X | SuS-X-F
---|---|---|---|---|---|---
(No adaptation) | (Full fine-tuning) | (PromptTuning) | (Adapters) | (Ours) | (Ours)
ImageNet | 60.31 | 60.35 | 60.96 | 61.61 | 61.89 | 63.22
EuroSAT | 26.83 | 55.37 | 52.12 | 57.00 | 44.23 | 59.22
DTD | 41.01 | 50.35 | 45.66 | 49.29 | 49.23 | 52.30
OxfordPets | 81.82 | 84.51 | 85.99 | 85.06 | 86.59 | 87.77
|
[table]capposition=top capbtabboxtable[][]
# Latent Graph Inference
using Product Manifolds
Antiquus S. Hippocampus, Natalia Cerebro & Amelie P. Amygdale
Department of Computer Science
Cranberry-Lemon University
Pittsburgh, PA 15213, USA
<EMAIL_ADDRESS>
&Ji Q. Ren & Yevgeny LeNet
Department of Computational Neuroscience
University of the Witwatersrand
Joburg, South Africa
<EMAIL_ADDRESS>
Coauthor
Affiliation
Address
email Use footnote for providing further information about author (webpage,
alternative address)— _not_ for acknowledging funding agencies. Funding
acknowledgements go at the end of the paper.
###### Abstract
Graph Neural Networks usually rely on the assumption that the graph topology
is available to the network as well as optimal for the downstream task. Latent
graph inference allows models to dynamically learn the intrinsic graph
structure of problems where the connectivity patterns of data may not be
directly accessible. In this work, we generalize the discrete Differentiable
Graph Module (dDGM) for latent graph learning. The original dDGM architecture
used the Euclidean plane to encode latent features based on which the latent
graphs were generated. By incorporating Riemannian geometry into the model and
generating more complex embedding spaces, we can improve the performance of
the latent graph inference system. In particular, we propose a computationally
tractable approach to produce product manifolds of constant curvature model
spaces that can encode latent features of varying structure. The latent
representations mapped onto the inferred product manifold are used to compute
richer similarity measures that are leveraged by the latent graph learning
model to obtain optimized latent graphs. Moreover, the curvature of the
product manifold is learned during training alongside the rest of the network
parameters and based on the downstream task, rather than it being a static
embedding space. Our novel approach is tested on a wide range of datasets, and
outperforms the original dDGM model.
## 1 Introduction
Graph Neural Networks (GNNs) have achieved state-of-the-art performance in a
number of applications, from travel-time prediction (Derrow-Pinion et al.
(2021)) to antibiotic discovery (Stokes et al. (2020)). They leverage the
connectivity structure of graph data, which improves their performance in many
applications as compared to traditional neural networks (Bronstein et al.
(2017)). Most current GNN architectures assume that the topology of the graph
is given and fixed during training. Hence, they update the input node
features, and sometimes edge features, but preserve the input graph topology.
A substantial amount of research has focused on improving diffusion using
different types of GNN layers. However, discovering an optimal graph topology
that can help diffusion has only recently gained attention (Topping et al.
(2021); Cosmo et al. (2020); Kazi et al. (2022)).
In many real-world applications, data can have some underlying but unknown
graph structure, which we call a latent graph. That is, we may only be able to
access a pointcloud of data. Nevertheless, this does not necessarily mean the
data is not intrinsically related, and that its connectivity cannot be
leveraged to make more accurate predictions. The vast majority of Geometric
Deep Learning research so far has relied on human annotators or simplistic
pre-processing algorithms to generate the graph structure to be passed to
GNNs. Furthermore, in practice, even in settings where the correct graph is
provided, it may often be suboptimal for the task at hand, and the GNN may
benefit from rewiring (Topping et al. (2021)). In this work, we drop the
assumption that the graph adjacency matrix is given and study how to learn the
latent graph in a fully-differentiable manner, using product manifolds,
alongside the GNN diffusion layers. More elaborately, we incorporate
Riemannian geometry to the discrete Differentiable Graph Module (dDGM)
proposed by Kazi et al. (2022). We show that it is possible and beneficial to
encode latent features into more complex embedding spaces beyond the Euclidean
plane used in the original work. In particular, we leverage the convenient
mathematical properties of product manifolds to learn the curvature of the
embedding space in a fully-differentiable manner.
##### Contributions
1) We explain how to use model spaces of constant curvature for the embedding
space. To do so, we outline a principled procedure to map Euclidean GNN output
features to constant curvature model space manifolds with non-zero curvature:
we use the hypersphere for spherical space and the hyperboloid for hyperbolic
space. We also outline how to calculate distances between points in these
spaces, which are then used by the dDGM sparse graph generating procedure to
infer the edges of the latent graph. Unlike the original dDGM model which
explored using the Poincaré ball with fixed curvature for modeling hyperbolic
space, in this work we use hyperboloids of arbitrary negative curvature. 2) We
show how to construct more complex embedding spaces that can encode latent
data of varying structure using product manifolds of model spaces. The
curvature of each model space composing the product manifold is learned in a
fully-differentiable manner alongside the rest of the model parameters, and
based on the downstream task performance. 3) We test our approach on 15
datasets which includes standard homophilic graph datasets, heterophilic
graphs, large-scale graphs, molecular datasets, and datasets for other real-
world applications such as brain imaging and aerospace engineering. 4) It has
been shown that traditional GNN models, such as Graph Convolutional Networks
(GCNs) (Kipf & Welling (2017)) and Graph Attention Networks (GATs) (Veličković
et al. (2018)) struggle to achieve good performance in heterophilic datasets
(Zhu et al. (2020)), since in fact homophily is used as an inductive bias by
these models. Amongst other models, Sheaf Neural Networks (SNNs) (Hansen &
Gebhart (2020); Bodnar et al. (2022); Barbero et al. (2022b; a)) have been
proposed to tackle this issue. We show that latent graph inference enables
traditional GNN models to give good performance on heterophilic datasets
without having to resort to sophisticated diffusion layers or model
architectures such as SNNs. 5) To make this work accessible to the wider
machine learning community, we have created a new PyTorch Geometric layer.
## 2 Background
In this section we discuss relevant background for this work. We first provide
a literature review regarding recent advances in latent graph inference using
GNNs as well as related work on manifold learning and graph embedding. Next,
we give an overview of the original Differentiable Graph Module (DGM)
formulation, but we recommend referring to Kazi et al. (2022) for further
details.
### 2.1 Related Work
Latent graph and topology inference is a standing problem in Geometric Deep
Learning. In contrast to algorithms that work on sets and that apply a shared
pointwise function such as PointNet (Qi et al. (2017)), in latent graph
inference we want to learn to optimally share information between nodes in the
pointcloud. Some contributions in the literature have focused on applying pre-
processing steps to enhance diffusion based on an initial input graph (Topping
et al. (2021); Gasteiger et al. (2019); Alon & Yahav (2021); Wu et al.
(2019)). Note, however, that this area of research focuses on improving an
already existing graph which may be suboptimal for the downstream task. This
paper is more directly related to work that addresses how to learn the graph
topology dynamically, instead of assuming a fixed graph at the start of
training. When the underlying connectivity structure is unknown, architectures
such as transformers (Vaswani et al. (2017)) and attentional multi-agent
predictive models (Hoshen (2017)), simply assume the graph to be fully-
connected, but this can become hard to scale to large graphs. Generating
sparse graphs can result in more computationally tractable solutions (Fetaya
et al. (2018)) and avoid over-smoothing (Chen et al. (2020a)). For this a
series of models have been proposed, starting from Dynamic Graph Convolutional
Neural Networks (DGCNNs) (Wang et al. (2019)), to other solutions that
decouple graph inference and information diffusion, such as the Differentiable
Graph Modules (DGMs) in Cosmo et al. (2020) and Kazi et al. (2022). Note that
latent graph inference may also be referred to as graph structure learning in
the literature. A survey of similar methods can be found in Zhu et al. (2021),
and some additional classical methods include LDS-GNN (Franceschi et al.,
2019), IDGL (Chen et al., 2020b), and Pro-GNN (Jin et al., 2020).
In this work, we extend the dDGM module proposed by Kazi et al. (2022) for
learning latent graphs using product manifolds. Product spaces have primarily
been studied in the manifold learning and graph embedding literature (Cayton
(2005); Fefferman et al. (2013); Bengio et al. (2012)). Recent work has
started exploring encoding the geometry of data into rich ambient manifolds.
In particular, hyperbolic geometry has proven successful in a number of tasks
(Liu et al. (2019); Chamberlain et al. (2017); Sala et al. (2018)). Different
manifold classes have been employed to enhance modeling flexibility, such as
products of constant curvature spaces (Gu et al. (2019)), matrix manifolds
(Cruceru et al. (2021)), and heterogeneous manifolds (Di Giovanni et al.
(2022)). We will leverage these ideas and use product manifolds to generate
the embedding space for constructing our latent graphs.
### 2.2 An Overview of the Discrete Differentiable Graph Module
Kazi et al. (2022) proposed a general technique for learning an optimized
latent graph, based on the output features of each layer, onto which to apply
the downstream GNN diffusion layers. Here, we specifically focus on the dDGM
module (not the cDGM), which is much more computationally efficient and
recommended by the authors. The main idea is to use some measure of similarity
between the latent node features to generate latent graphs which are optimal
for each layer $l$. We can summarize the architecture as
$\mathbf{\hat{X}}^{(l+1)}=f_{\mathbf{\Theta}}^{(l)}(concat(\mathbf{X}^{(l)},\mathbf{\hat{X}}^{(l)}),\mathbf{A}^{(l)})\rightarrow\mathbf{A}^{(l+1)}\sim\mathbf{P}^{(l)}(\mathbf{\hat{X}}^{(l+1)}))\rightarrow\mathbf{X}^{(l+1)}=g_{\bm{\phi}}(\mathbf{X}^{(l)},\mathbf{A}^{(l+1)}).$
The node features in layer $l$, $\mathbf{X}^{(l)}$, are transformed into
$\mathbf{\hat{X}}^{(l+1)}$ through a function $f_{\mathbf{\Theta}^{(l)}}$,
which has learnable parameters, and compared using a similarity measure
$\varphi(T)$, which is parameterized by a scalar learnable parameter $T$. On
the other hand, $g_{\bm{\phi}}$ is a diffusion function which in practice
corresponds to multiple GNN layers stacked together. $g_{\bm{\phi}}$ diffuses
information based on the inferred latent graph connectivity structure
summarized in $\mathbf{A}^{(l+1)}$, which is an unweighted sparse matrix. The
dDGM module generates a sparse $k$-degree graph using the Gumbel Top-k trick
(Kool et al. (2019)), a stochastic relaxation of the kNN rule, to sample edges
from the probability matrix
$\mathbf{P}^{(l)}(\mathbf{X}^{(l)};\mathbf{\Theta}^{(l)},T)$, where each entry
corresponds to
$p_{ij}^{(l)}(\mathbf{\Theta}^{(l)})=\exp(\varphi(f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{i}^{(l)}),f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{j}^{(l)});T))=\exp(\varphi(\mathbf{\hat{x}}_{i}^{(l+1)},\mathbf{\hat{x}}_{j}^{(l+1)};T)).$
(1)
The main similarity measure used in Kazi et al. (2022) was to compute the
distance based on the features of two nodes in the graph embedding space. They
assumed that the latent features laid in an Euclidean plane of constant
curvature $K_{\mathbb{E}}=0$, so that
$p_{ij}^{(l)}=\exp(-T\mathfrak{d}_{\mathbb{E}}(f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{i}^{(l)}),f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{j}^{(l)})))=\exp(-T\mathfrak{d}_{\mathbb{E}}(\mathbf{\hat{x}}_{i}^{(l+1)},\mathbf{\hat{x}}_{j}^{(l+1)})),$
where $\mathfrak{d}_{\mathbb{E}}$ denotes distance in Euclidean space. Then,
based on
$\textrm{argsort}(\log(\mathbf{p}_{i}^{(l)})-\log(-\log(\mathbf{q})))$, where
$\mathbf{q}\in\mathbb{R}^{N}$ is uniform i.i.d in the interval $[0,1]$, we can
sample the edges
$\mathcal{E}^{(l)}(\mathbf{X}^{(l)};\mathbf{\Theta}^{(l)},T,k)=\\{(i,j_{i,1}),(i,j_{i,2}),...,(i,j_{i,k}):i=1,...,N\\},$
where $k$ is the number of sampled connections using the Gumbel Top-k trick.
This sampling approach follows the categorical distribution
$\frac{p_{ij}^{(l)}}{\Sigma_{r}p_{ir}^{(l)}}$ and
$\mathcal{E}(\mathbf{X}^{(l)};\mathbf{\Theta}^{(l)},T,k)$ is represented by
the unweighted adjacency matrix
$\mathbf{A}^{(l)}(\mathbf{X}^{(l)};\mathbf{\Theta}^{(l)},T,k)$. Note that
including noise in the edge sampling approach will result in the generation of
some random edges in the latent graphs which can be understood as a form of
regularization. In this work, we generalize Equation 1 to measure similarities
based on distances but dropping the assumption used in Kazi et al. (2022), in
which they limit themselves to fixed-curvature spaces, specifically to
Eucliden space where $K_{\mathbb{E}}=0$. We will use product manifolds of
model spaces of constant curvature to improve the similarity measure $\varphi$
and construct better latent graphs.
## 3 Method: Product Manifolds for Latent Graph Inference
In this section, we first introduce model spaces, which are a special type of
Riemannian manifolds, and explain how to map Euclidean GNN output features to
model spaces with non-zero curvature. In case the reader is unfamiliar with
the topic, additional details regarding Riemannian manifolds can be found in
Appendix A. Then, we mathematically define product manifolds and how to
calculate distances between points in the manifold. Next, we introduce scaling
metrics which help us learn the curvature of each model space composing the
product manifold. A discussion on product manifold curvature learning can be
found in Appendix B. The intuition behind the method is that we can consider
the embedding space represented by the product manifold as a combination of
more simple spaces (model spaces of constant curvature), and compute distances
between the latent representations mapped onto the product manifold by
considering distances in each model space individually and later aggregating
them in a principled manner. This allows to generate diverse embedding spaces
which at the same time are computationally tractable.
### 3.1 Constant Curvature Model Spaces
Curvature is effectively a measure of geodesic dispersion. When there is no
curvature geodesics stay parallel, with negative curvature they diverge, and
with positive curvature they converge. Euclidean space,
$\mathbb{E}^{d_{\mathbb{E}}}_{K_{\mathbb{E}}}=\mathbb{R}^{d_{\mathbb{E}}}$, is
a flat space with curvature $K_{\mathbb{E}}=0$. Note that here we use
$d_{\mathbb{E}}$ to denote dimensionality. On the other hand, hyperbolic and
spherical space, have negative and positive curvature, respectively. We define
hyperboloids as
$\mathbb{H}^{d_{\mathbb{H}}}_{K_{\mathbb{H}}}=\\{\mathbf{x}_{p}\in\mathbb{R}^{d_{\mathbb{H}}+1}:\langle\mathbf{x}_{p},\mathbf{x}_{p}\rangle_{\mathcal{L}}=1/K_{\mathbb{H}}\\},$
where $K_{\mathbb{H}}<0$ and $\langle\cdot,\cdot\rangle_{\mathcal{L}}$ is the
Lorentz inner product
$\langle\mathbf{x},\mathbf{y}\rangle_{\mathcal{L}}=-x_{1}y_{1}+\sum_{j=2}^{d_{\mathbb{H}}+1}x_{j}y_{j},\,\,\,\forall\,\mathbf{x},\mathbf{y}\in\mathbb{R}^{d_{\mathbb{H}}+1},$
and hyperspheres as
$\mathbb{S}^{d_{\mathbb{S}}}_{K_{\mathbb{S}}}=\\{\mathbf{x}_{p}\in\mathbb{R}^{d_{\mathbb{S}}+1}:\langle\mathbf{x}_{p},\mathbf{x}_{p}\rangle_{2}=1/K_{\mathbb{S}}\\},$
where $K_{\mathbb{S}}>0$ and $\langle\cdot,\cdot\rangle_{2}$ is the standard
Euclidean inner product
$\langle\mathbf{x},\mathbf{y}\rangle_{2}=\sum_{j=1}^{d_{\mathbb{S}}+1}x_{j}y_{j},\,\,\,\forall\,\mathbf{x},\mathbf{y}\in\mathbb{R}^{d_{\mathbb{S}}+1}.$
Table LABEL:tab:_summary_spaces provides a summary of relevant operators in
Euclidean, hyperbolic, and spherical spaces with arbitrary curvatures. The
closed forms for the distances between points in hyperbolic and spherical
space use the arccosh and the arccos, their domains are
$\\{x\in\mathbb{R}:x\geq 1\\}$ and $\\{x\in\mathbb{R}:-1\leq x\leq 1\\}$,
respectively. We apply clipping to avoid giving inputs close to the domain
limits and prevent from instabilities during training.
Table 1: Relevant operators (exponential maps and distances between two
points) in Euclidean, hyperbolic, and spherical spaces with arbitrary constant
curvatures.
Space Model | $exp_{\mathbf{x}_{p}}(\mathbf{x})$ | $\mathfrak{d}(\mathbf{x},\mathbf{y})$
---|---|---
$\mathbb{E}$, Euclidean | $\mathbf{x}_{p}+\mathbf{x}$ | $||\mathbf{x}-\mathbf{y}||_{2}$
$\mathbb{H}$, hyperboloid | $\cosh{\left(\sqrt{-K_{\mathbb{H}}}||\mathbf{x}||\right)}\mathbf{x}_{p}+\sinh{\left(\sqrt{-K_{\mathbb{H}}}||\mathbf{x}||\right)}\frac{\mathbf{x}}{\sqrt{-K_{\mathbb{H}}}||\mathbf{x}||}$ | $\frac{1}{\sqrt{-K_{\mathbb{H}}}}\,\textrm{arccosh}\,{\left(K_{\mathbb{H}}\langle\mathbf{x},\mathbf{y}\rangle_{\mathcal{L}}\right)}$
$\mathbb{S}$, hypersphere | $\cos{\left(\sqrt{K_{\mathbb{S}}}||\mathbf{x}||\right)}\mathbf{x}_{p}+\sin{\left(\sqrt{K_{\mathbb{S}}}||\mathbf{x}||\right)}\frac{\mathbf{x}}{\sqrt{K_{\mathbb{S}}}||\mathbf{x}||}$ | $\frac{1}{\sqrt{K_{\mathbb{S}}}}\arccos{\left(K_{\mathbb{S}}\langle\mathbf{x},\mathbf{y}\rangle_{2}\right)}$
The latent output features produced by the neural network layers are in
Euclidean space and must be mapped to the relevant model spaces before
applying the distance metrics. We use the appropriate exponential map (refer
to Table LABEL:tab:_summary_spaces). To map Euclidean data to the hyperboloid,
we use the hyperboloid north pole, that is, the origin
$\mathbf{o}^{\mathbb{H}}_{K_{\mathbb{H}}}:=(\frac{1}{\sqrt{-K_{\mathbb{H}}}},0,...,0)=(\frac{1}{\sqrt{-K_{\mathbb{H}}}},\mathbf{0})$
as a reference point to perform tangent space operations. Using the trick
described in Chami et al. (2019)111Note that in this paper they define
curvature differently., if $\mathbf{x}$ is an Euclidean feature we can
consider $concat(0,\mathbf{x})$ to be a point in the manifold tangent space at
$\mathbf{o}^{\mathbb{H}}_{K_{\mathbb{H}}}$. Therefore, we can obtain the
mapped features
$\mathbf{\overline{x}}=exp_{\mathbf{o}^{\mathbb{H}}_{K_{\mathbb{H}}}}^{\mathbb{H}}(concat(0,\mathbf{x}))$
via
$\mathbf{\overline{x}}=\left(\frac{1}{\sqrt{-K_{\mathbb{H}}}}\cosh{\left(\sqrt{-K_{\mathbb{H}}}||\mathbf{x}||\right)},\sinh{\left(\sqrt{-K_{\mathbb{H}}}||\mathbf{x}||\right)}\frac{\mathbf{x}}{\sqrt{-K_{\mathbb{H}}}||\mathbf{x}||}\right).$
Similarly for the hypersphere using as reference point
$\mathbf{o}^{\mathbb{S}}_{K_{\mathbb{S}}}:=(\frac{1}{\sqrt{K_{\mathbb{S}}}},0,...,0)$,
$\mathbf{\overline{x}}=\left(\frac{1}{\sqrt{K_{\mathbb{S}}}}\cos{\left(\sqrt{K_{\mathbb{S}}}||\mathbf{x}||\right)},\sin{\left(\sqrt{K_{\mathbb{S}}}||\mathbf{x}||\right)}\frac{\mathbf{x}}{\sqrt{K_{\mathbb{S}}}||\mathbf{x}||}\right).$
### 3.2 Product Manifolds
We define a product manifold as the Cartesian product
$\mathcal{P}=\bigtimes_{i=1}^{n_{\mathcal{P}}}\mathcal{M}_{K_{i}}^{d_{i}},$
where $K_{i}$ and $d_{i}$ are the curvature and dimensionality of the manifold
$\mathcal{M}_{K_{i}}^{d_{i}}$, respectively. We write points
$\mathbf{x}_{p}\in\mathcal{P}$ using their coordinates
$\mathbf{x}_{p}=concat\left(\mathbf{x}_{p}^{(1)},\mathbf{x}_{p}^{(2)},...,\mathbf{x}_{p}^{(n_{\mathcal{P}})}\right):\mathbf{x}_{p}^{(i)}\in\mathcal{M}_{K_{i}}^{d_{i}}.$
Also, the metric of the product manifold decomposes into the sum of the
constituent metrics $g_{\mathcal{P}}=\sum_{i=1}^{n_{\mathcal{P}}}g_{i},$
hence, $(\mathcal{P},g_{\mathcal{P}})$ is also a Riemannian manifold if
$(\mathcal{M}_{K_{i}}^{d_{i}},g_{i}),\,\forall i$ are all Riemannian manifolds
in the first place. Note that the signature of the product space, that is, its
parametrization, has several degrees of freedom: the number of components
used, as well as the type of model spaces, their dimensionality, and
curvature. If we restrict $\mathcal{P}$ to be composed of the Euclidean plane
$\mathbb{E}_{K_{\mathbb{E}}}^{d_{\mathbb{E}}}$, hyperboloids
$\mathbb{H}_{K_{j}^{\mathbb{H}}}^{d_{j}^{\mathbb{H}}}$, and hyperspheres
$\mathbb{S}_{K_{k}^{\mathbb{S}}}^{d_{k}^{\mathbb{S}}}$ of constant curvature,
we can write an arbitrary product manifold of model spaces as
$\mathcal{P}=\mathbb{E}_{K_{\mathbb{E}}}^{d_{\mathbb{E}}}\times\left(\bigtimes_{j=1}^{n_{\mathbb{H}}}\mathbb{H}_{K_{j}^{\mathbb{H}}}^{d_{j}^{\mathbb{H}}}\right)\times\left(\bigtimes_{k=1}^{n_{\mathbb{S}}}\mathbb{S}_{K_{k}^{\mathbb{S}}}^{d_{k}^{\mathbb{S}}}\right)=\mathbb{E}\times\left(\bigtimes_{j=1}^{n_{\mathbb{H}}}\mathbb{H}_{j}\right)\times\left(\bigtimes_{k=1}^{n_{\mathbb{S}}}\mathbb{S}_{k}\right),$
(2)
where $K_{\mathbb{E}}=0$, $K_{j}^{\mathbb{H}}<0$, and $K_{k}^{\mathbb{S}}>0$.
The rightmost part of Equation 2 is included to simplify the notation.
$\mathcal{P}$ would have a total of $1+n_{\mathbb{H}}+n_{\mathbb{S}}$
component spaces, and total dimension
$d_{\mathbb{E}}+\Sigma_{j=1}^{n_{\mathbb{H}}}d_{j}^{\mathbb{H}}+\Sigma_{k=1}^{n_{\mathbb{S}}}d_{k}^{\mathbb{S}}$.
As shown in Gallier & Quaintance (2020), in the case of a product manifold as
defined in Equation 2, the geodesics, exponential, and logarithmic maps on
$\mathcal{P}$ are the concatenation of the corresponding notions of the
individual model spaces.
### 3.3 Distances and Scaling Metrics for Product Manifolds
To compute distances between points in the product manifold we can add up the
square distances for the coordinates in each of the individual manifolds
$\mathfrak{d}_{\mathcal{P}}(\mathbf{\overline{x}}_{p_{1}},\mathbf{\overline{x}}_{p_{2}})^{2}=\mathfrak{d}_{\mathbb{E}}\left(\mathbf{x}_{p_{1}}^{(1)},\mathbf{x}_{p_{2}}^{(1)}\right)^{2}+\sum_{j=1}^{n_{\mathbb{H}}}\mathfrak{d}_{\mathbb{H}_{j}}\left(\mathbf{\overline{x}}_{p_{1}}^{(1+j)},\mathbf{\overline{x}}_{p_{2}}^{(1+j)}\right)^{2}+\sum_{k=1}^{n_{\mathbb{S}}}\mathfrak{d}_{\mathbb{S}_{k}}\left(\mathbf{\overline{x}}_{p_{1}}^{(1+n_{\mathbb{H}}+k)},\mathbf{\overline{x}}_{p_{2}}^{(1+n_{\mathbb{H}}+k)}\right)^{2},$
where the overline denotes that the adequate exponential map to project
Euclidean feature entries to the relevant model space has been applied before
computing the distance. In practice, this would be equivalent to mapping the
feature outputs to the product manifold and operating on $\mathcal{P}$
directly. As suggested in Tabaghi et al. (2021), instead of directly updating
the curvature of the hyperboloid and hypersphere model spaces used to
construct the product manifold, we can set $K_{j}^{\mathbb{H}}=-1,\,\forall
j$, and $K_{k}^{\mathbb{S}}=1,\,\forall k,$ and use a scaled distance metric
instead. To do so, we introduce learnable coefficients
$\alpha_{j}^{\mathbb{H}}$ and $\alpha_{k}^{\mathbb{S}}$,
$\leavevmode\resizebox{365.68546pt}{}{$\mathfrak{d}_{\mathcal{P}}(\mathbf{\overline{x}}_{p_{1}},\mathbf{\overline{x}}_{p_{2}})^{2}=\mathfrak{d}_{\mathbb{E}}\left(\mathbf{x}_{p_{1}}^{(1)},\mathbf{x}_{p_{2}}^{(1)}\right)^{2}+\sum_{j=1}^{n_{\mathbb{H}}}\left(\alpha_{j}^{\mathbb{H}}\mathfrak{d}_{\mathbb{H}_{j}}\left(\mathbf{\overline{x}}_{p_{1}}^{(1+j)},\mathbf{\overline{x}}_{p_{2}}^{(1+j)}\right)\right)^{2}+\sum_{k=1}^{n_{\mathbb{S}}}\left(\alpha_{k}^{\mathbb{S}}\mathfrak{d}_{\mathbb{S}_{k}}\left(\mathbf{\overline{x}}_{p_{1}}^{(1+n_{\mathbb{H}}+k)},\mathbf{\overline{x}}_{p_{2}}^{(1+n_{\mathbb{H}}+k)}\right)\right)^{2}$},$
(3)
which is equivalent to learning the curvature of the non-Euclidean model
spaces, but computationally more tractable and efficient (for further details
on how this coefficients are updated refer to Appendix C.2). This newly
defined distance metric can then be applied to calculate the probability of
there existing an edge connecting latent features
$p_{ij}^{(l)}(\mathbf{\Theta}^{(l)})=\exp\left(-T\mathfrak{d}_{\mathcal{P}}\left(\overline{f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{i}^{(l)})},\overline{f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{j}^{(l)}})\right)\right).$
Hence,
$\mathcal{E}^{(l)}(\mathbf{X}^{(l)};\mathbf{\Theta}^{(l)},T,k,\mathfrak{d}_{\mathcal{P}})=\\{(i,j_{i,1}),(i,j_{i,2}),...,(i,j_{i,k}):i=1,...,N\\}.$
As discussed in Kazi et al. (2022), the logarithms of the edge probabilities
are used to update the dDGM. This is done by incorporating an additional term
to the network loss function which will be dependent on
$\log
p_{ij}^{(l)}(\mathbf{\Theta}^{(l)})=-T\mathfrak{d}_{\mathcal{P}}\left(\overline{f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{i}^{(l)})},\overline{f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{j}^{(l)}})\right)=-T\mathfrak{d}_{\mathcal{P}}\left(\overline{\mathbf{x}}_{p_{i}}^{(l)},\overline{\mathbf{x}}_{p_{j}}^{(l)}\right)$
(4)
where the additional graph loss is given by
$L_{GL}=\sum_{i=1}^{N}\left(\delta\left(y_{i},\hat{y}_{i}\right)\sum_{l=1}^{l=L}\sum_{j:(i,j)\in\mathcal{E}^{(l)}}\log
p_{ij}^{(l)}\right),$ and
$\delta\left(y_{i},\hat{y}_{i}\right)=\mathds{E}(ac_{i})-ac_{i}$ is a reward
function based on the expected accuracy of the model. The loss function to
update the dDGM model, $L_{GL}$, is identical to the original loss proposed by
Kazi et al. (2022), for a brief review one may refer to Appendix C.1. Note
that after passing the input $\mathbf{x}_{i}^{(l)}$ through the dDGM
parameterized function $f_{\mathbf{\Theta}}^{(l)}$, the output
$f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{i}^{(l)})=\mathbf{\hat{x}}_{i}^{(l+1)}=\mathbf{x}_{p_{i}}^{(l)}$
has dimension
$d_{\mathbb{E}}+\Sigma_{j=1}^{n_{\mathbb{H}}}d_{j}^{\mathbb{H}}+\Sigma_{k=1}^{n_{\mathbb{S}}}d_{k}^{\mathbb{S}}$
and must be subdivided into $1+n_{\mathbb{H}}+n_{\mathbb{S}}$ subarrays for
each of the component spaces. Each subarray must be appropriately mapped to
its model space. Hence, the overline in
$\overline{f_{\mathbf{\Theta}}^{(l)}(\mathbf{x}_{i}^{(l)})}$ in Equation 4.
Finally, Figure 1 summarizes the method described in this section. Note that
$\mathbf{x}_{p}$ in Figure 1, would correspond to the concatenation of the
origins of each model space composing the product manifold.
Figure 1: Diagram depicting mapping procedure from GNN Euclidean output to
latent manifold $\mathcal{P}$. The appropriate exponential map is used to map
the points from the tangent plane to the manifold. We construct the latent
graph based on the distances on the learned manifold.
## 4 Experiments and Results
The main objective of the experimental validation is to show that latent graph
inference can benefit from using products of models spaces. To do so, we
compare the performance of the dDGM module when using single model spaces
against Cartesian products of model spaces. The model spaces are denoted as:
Euclidean (dDGM-E/dDGM∗-E, which is equivalent to the original architecture
used by Kazi et al. (2022)), hyperbolic (dDGM-H/dDGM∗-H), and spherical space
(dDGM-S/dDGM∗-S). The asterisk sign in the model name denotes that the dDGM∗
module is tasked with generating the latent graph without having access to the
original adjacency matrix. dDGM models take as input $\mathbf{X^{(0)}}$ and
$\mathbf{A^{(0)}}$, whereas dDGM∗ models only have access to
$\mathbf{X^{(0)}}$. To refer to product manifolds, we simply append all the
model spaces that compose the manifold to the name of the module, namely, the
dDGM-SS module embedding space is a torus. In practice, we will use the same
dimensionality but different curvature for each of the Cartesian components of
the product manifolds. Lastly, if a GNN model uses the dDGM module, we name
the diffusion layers and the latent graph inference module after. For example,
GCN-dDGM-E refers to a GCN that instead of using the original dataset graph
for diffusion, incorporates latent graph inference to the network and uses the
Euclidean plane as embedding space.
Note that we only use a single latent graph inference module per neural
network, that is, networks diffuse information based on only one latent graph.
This is in line with previous work (Kazi et al. (2022)). Additionally, in
Appendix E.1, we investigate the effect of leveraging multiple latent graphs
in the same network and conclude that in general it is better to use a single
latent graph due to computational efficiency and diminishing returns. The
study regarding computational efficiency can be found in Appendix C.3. In
particular, we compare the runtime speedup obtained using symbolic matrices as
compared to standard dense PyTorch matrices. We observe that as more product
manifolds and dDGM modules are included, the runtime speedup obtained using
symbolic matrices becomes increasingly large. Moreover, without using symbolic
matrices standard GPUs (we use NVIDIA P100 and Tesla T4) run out of memory for
datasets with $\mathcal{O}(10^{4})$ nodes such as PubMed, Physics, and CS.
Hence, we recommend using symbolic matrices to help with scalability. Model
architecture descriptions for all experiments can be found in Appendix G.
### 4.1 Homophilic and Heterophilic Benchmark Graph Datasets
We first focus on standard graph datasets widely discussed in the Geometric
Deep Learning literature such as Cora, CiteSeer (Yang et al. (2016); Lu &
Getoor (2003); Sen et al. (2008)), PubMed, Physics and CS (Shchur et al.
(2018)), which have high homophily levels ranging between 0.74 and 0.93. We
also present the results for several heterophilic datasets, which have
homophily levels between 0.11 and 0.23. In particular, we work with Texas,
Wisconsin, Squirrel, and Chameleon (Rozemberczki et al., 2021). Results of
particular interest for these datasets are recorded in Table 2. Additional
experiments are available in Appendix E.2, E.3, and E.4, in which we perform
an in depth exploration of different hyperparameters, compare dDGMs and dDGM∗s
for all datasets, and try many product manifold combinations. Referring back
to Table 2, we can see that product manifolds consistently outperform latent
graph inference systems which only leverage a single model space for modeling
the embedding space. Also note, that unlike in the work by Kazi et al. (2022),
we do find single hyperbolic model spaces to often outperform inference
systems that use the Euclidean plane as embedding space. This shows that
indeed mapping the Euclidean output features of the GNN layers to hyperbolic
space using the exponential map before computing distances is of paramount
importance (Kazi et al. (2022) ignored the exponential maps required to map
features to the Poincaré ball).
Table 2: Results for heterophilic and homophilic datasets combining GCN
diffusion layers with the latent graph inference system. We display results
using model spaces as well as product manifolds to construct the latent
graphs. The First, Second and Third best models for each dataset are
highlighted in each table. $k$ denotes the number of connections per node when
implementing the Gumbel Top-k sampling algorithm. Additional $k$ values are
tested in Appendix E.2. Note that the models which use the Euclidean plane
(former dDGM) as embedding space, denoted with an E in the table, are
equivalent to those presented in Kazi et al. (2022).
| HETEROPHILIC DATASETS | | HOMOPHILIC DATASETS |
---|---|---|---|---
| Texas | Wisconsin | Squirrel | Chameleon | | Cora | CiteSeer | PubMed | Physics | CS
Homophily level | 0.11 | 0.21 | 0.22 | 0.23 | Homophily level | 0.81 | 0.74 | 0.80 | 0.93 | 0.80
Nodes | 183 | 251 | 5,201 | 2,277 | Nodes | 2,708 | 3,327 | 18,717 | 34,493 | 18,333
Features | 1,703 | 1,703 | 2,089 | 2,325 | Features | 1,433 | 3,703 | 500 | 8,415 | 6,805
Edges | 295 | 466 | 198,498 | 31,421 | Edges | 5,278 | 4,676 | 44,327 | 247,962 | 81,894
Classes | 5 | 5 | 5 | 5 | Classes | 7 | 6 | 3 | 5 | 15
Average Degree | 3.22 | 3.71 | 76.33 | 27.60 | Average Degree | 3.9 | 2.77 | 4.5 | 14.38 | 8.93
Former dDGM
Model | Accuracy $(\%)$ $\pm$ Standard Deviation | Model | Accuracy $(\%)$ $\pm$ Standard Deviation |
$k$ | 2 | 10 | 3 | 5 | $k$ | 7 | 7 | 7 | 5 | 7
GCN-dDGM∗-E | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}80.00{\scriptstyle\pm 8.31}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}88.00{\scriptstyle\pm 5.65}}}$ | $34.35{\scriptstyle\pm 2.34}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}48.90{\scriptstyle\pm 3.61}}}$ | GCN-dDGM-E | $82.11{\scriptstyle\pm 4.24}$ | $72.35{\scriptstyle\pm 1.92}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}87.69{\scriptstyle\pm 0.67}}}$ | $95.96{\scriptstyle\pm 0.40}$ | $87.17{\scriptstyle\pm 3.82}$
dDGM with Riemannian Geometry and Single Model Spaces (Ours)
Model | Accuracy $(\%)$ $\pm$ Standard Deviation | Model | Accuracy $(\%)$ $\pm$ Standard Deviation |
$k$ | 2 | 10 | 3 | 5 | $k$ | 7 | 7 | 7 | 5 | 7
GCN-dDGM∗-H | $79.44{\scriptstyle\pm 7.88}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}89.03{\scriptstyle\pm 1.89}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}35.00{\scriptstyle\pm 2.35}}}$ | $48.28{\scriptstyle\pm 4.11}$ | GCN-dDGM-H | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}84.68{\scriptstyle\pm 3.31}}}$ | $70.43{\scriptstyle\pm 4.95}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}87.74{\scriptstyle\pm 0.72}}}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}96.06{\scriptstyle\pm 0.42}}}$ | $88.78{\scriptstyle\pm 2.24}$
GCN-dDGM∗-S | $73.88{\scriptstyle\pm 9.95}$ | $85.33{\scriptstyle\pm 4.98}$ | $33.12{\scriptstyle\pm 2.22}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}48.63{\scriptstyle\pm 3.12}}}$ | GCN-dDGM-S | $80.44{\scriptstyle\pm 5.26}$ | $72.89{\scriptstyle\pm 2.00}$ | $87.13{\scriptstyle\pm 0.66}$ | $95.91{\scriptstyle\pm 0.41}$ | $84.16{\scriptstyle\pm 2.78}$
dDGM with Riemannian Geometry and Product Manifolds (Ours)
Model | Accuracy $(\%)$ $\pm$ Standard Deviation | Model | Accuracy $(\%)$ $\pm$ Standard Deviation |
$k$ | 2 | 10 | 3 | 5 | $k$ | 5 | 3 | 10 | 10 | 3
GCN-dDGM∗-HH | $78.89{\scriptstyle\pm 8.53}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}88.00{\scriptstyle\pm 3.26}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}34.38{\scriptstyle\pm 1.07}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}48.33{\scriptstyle\pm 4.14}}}$ | GCN-dDGM-HH | $76.09{\scriptstyle\pm 7.11}$ | $71.27{\scriptstyle\pm 2.09}$ | $87.50{\scriptstyle\pm 0.91}$ | $94.73{\scriptstyle\pm 2.83}$ | $82.91{\scriptstyle\pm 3.00}$
GCN-dDGM∗-SS | $73.89{\scriptstyle\pm 8.62}$ | $74.66{\scriptstyle\pm 18.85}$ | $34.06{\scriptstyle\pm 2.20}$ | $48.28{\scriptstyle\pm 3.07}$ | GCN-dDGM-SS | $65.96{\scriptstyle\pm 9.46}$ | $59.16{\scriptstyle\pm 5.96}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}87.82{\scriptstyle\pm 0.59}}}$ | $90.72{\scriptstyle\pm 5.26}$ | $59.31{\scriptstyle\pm 7.18}$
GCN-dDGM∗-EH | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}81.67{\scriptstyle\pm 7.05}}}$ | $86.67{\scriptstyle\pm 3.77}$ | $34.37{\scriptstyle\pm 1.72}$ | $47.58{\scriptstyle\pm 3.85}$ | GCN-dDGM-EH | $82.32{\scriptstyle\pm 4.71}$ | $72.89{\scriptstyle\pm 1.64}$ | $87.41{\scriptstyle\pm 0.80}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}96.03{\scriptstyle\pm 0.37}}}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}91.37{\scriptstyle\pm 1.28}}}$
GCN-dDGM∗-ES | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}81.11{\scriptstyle\pm 10.30}}}$ | $76.00{\scriptstyle\pm 11.31}$ | $33.38{\scriptstyle\pm 1.86}$ | $47.49{\scriptstyle\pm 3.60}$ | GCN-dDGM-ES | $81.44{\scriptstyle\pm 5.80}$ | $71.87{\scriptstyle\pm 3.20}$ | $87.50{\scriptstyle\pm 0.65}$ | $95.41{\scriptstyle\pm 1.73}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}90.87{\scriptstyle\pm 0.82}}}$
GCN-dDGM∗-HS | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}81.11{\scriptstyle\pm 9.69}}}$ | $86.67{\scriptstyle\pm 1.89}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}34.65{\scriptstyle\pm 2.45}}}$ | $47.84{\scriptstyle\pm 2.67}$ | GCN-dDGM-HS | $82.59{\scriptstyle\pm 4.50}$ | $72.77{\scriptstyle\pm 2.76}$ | $85.89{\scriptstyle\pm 4.29}$ | $95.84{\scriptstyle\pm 0.29}$ | $89.43{\scriptstyle\pm 2.37}$
GCN-dDGM∗-EHH | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}81.11{\scriptstyle\pm 5.09}}}$ | $77.60{\scriptstyle\pm 8.62}$ | $33.19{\scriptstyle\pm 1.92}$ | $44.27{\scriptstyle\pm 2.96}$ | GCN-dDGM-EHH | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}86.63{\scriptstyle\pm 3.25}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}75.42{\scriptstyle\pm 2.39}}}$ | $39.93{\scriptstyle\pm 1.35}$ | $95.63{\scriptstyle\pm 1.36}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}92.86{\scriptstyle\pm 0.96}}}$
GCN-dDGM∗-EHS | $79.44{\scriptstyle\pm 6.11}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}89.33{\scriptstyle\pm 1.89}}}$ | $34.17{\scriptstyle\pm 2.23}$ | $47.58{\scriptstyle\pm 4.54}$ | GCN-dDGM-EHS | $83.58{\scriptstyle\pm 4.39}$ | $69.98{\scriptstyle\pm 2.70}$ | $87.05{\scriptstyle\pm 1.38}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}96.21{\scriptstyle\pm 0.44}}}$ | $89.93{\scriptstyle\pm 3.86}$
Vanilla Architectures
Model | Accuracy $(\%)$ $\pm$ Standard Deviation | Model | Accuracy $(\%)$ $\pm$ Standard Deviation |
MLP∗ | $77.78{\scriptstyle\pm 10.24}$ | $85.33{\scriptstyle\pm 4.99}$ | $30.44{\scriptstyle\pm 2.55}$ | $40.35{\scriptstyle\pm 3.37}$ | MLP∗ | $58.92{\scriptstyle\pm 3.28}$ | $59.48{\scriptstyle\pm 2.14}$ | $85.75{\scriptstyle\pm 1.02}$ | $94.91{\scriptstyle\pm 0.30}$ | $87.80{\scriptstyle\pm 1.54}$
GCN | $41.66{\scriptstyle\pm 11.72}$ | $47.20{\scriptstyle\pm 9.76}$ | $24.19{\scriptstyle\pm 2.56}$ | $32.56{\scriptstyle\pm 3.53}$ | GCN | $83.11{\scriptstyle\pm 2.29}$ | $69.97{\scriptstyle\pm 2.06}$ | $85.75{\scriptstyle\pm 1.01}$ | $95.51{\scriptstyle\pm 0.34}$ | $87.28{\scriptstyle\pm 1.54}$
We have shown that the latent graph inference system enables GCN diffusion
layers to achieve good performance for heterophilic datasets. We hypothesize
that for this to be possible, it should be able to generate homophilic latent
graphs, on which GCNs can easily diffuse. In Table 3 we display the homophily
levels of the learned latent graphs, which corroborates our intuition. As we
can see from the results all models are able to generate latent graphs with
higher homophily than those of the original dataset graphs. The latent graph
inference system seems to find it easier to increase the homophily levels of
smaller datasets, which is reasonable since there is less information to
reorganize. There is a clear correlation between model performance in terms of
accuracy (Table 2) and the homophily level that the dDGM∗ modules are able to
achieve for the latent graphs (Table 3).
Table 3: Homophily level of the learned latent graphs. Latent graph inference
modules which use different manifolds to generate their respective latent
graphs achieve different homophily levels. Also, depending on weight
initialization, the inference system can converge to slightly different latent
graphs.
| Texas | Wisconsin | Squirrel | Chameleon
---|---|---|---|---
Original Graph Homophily | 0.11 | 0.21 | 0.22 | 0.23
$k$ | 2 | 10 | 3 | 5
Model | Latent Graph Homophily $h$ $\pm$ Standard Deviation
GCN-dDGM∗-E | $0.89{\scriptstyle\pm 0.02}$ | $0.69{\scriptstyle\pm 0.01}$ | $0.33{\scriptstyle\pm 0.00}$ | $0.37{\scriptstyle\pm 0.02}$
GCN-dDGM∗-H | $0.89{\scriptstyle\pm 0.01}$ | $0.66{\scriptstyle\pm 0.01}$ | $0.33{\scriptstyle\pm 0.00}$ | $0.46{\scriptstyle\pm 0.05}$
GCN-dDGM∗-S | $0.86{\scriptstyle\pm 0.03}$ | $0.64{\scriptstyle\pm 0.01}$ | $0.32{\scriptstyle\pm 0.00}$ | $0.45{\scriptstyle\pm 0.04}$
GCN-dDGM∗-HH | $0.91{\scriptstyle\pm 0.01}$ | $0.66{\scriptstyle\pm 0.02}$ | $0.32{\scriptstyle\pm 0.00}$ | $0.37{\scriptstyle\pm 0.01}$
GCN-dDGM∗-SS | $0.85{\scriptstyle\pm 0.02}$ | $0.51{\scriptstyle\pm 0.01}$ | $0.27{\scriptstyle\pm 0.00}$ | $0.31{\scriptstyle\pm 0.01}$
GCN-dDGM∗-EH | $0.90{\scriptstyle\pm 0.01}$ | $0.65{\scriptstyle\pm 0.01}$ | $0.32{\scriptstyle\pm 0.00}$ | $0.43{\scriptstyle\pm 0.03}$
GCN-dDGM∗-ES | $0.91{\scriptstyle\pm 0.00}$ | $0.63{\scriptstyle\pm 0.02}$ | $0.27{\scriptstyle\pm 0.02}$ | $0.32{\scriptstyle\pm 0.04}$
GCN-dDGM∗-HS | $0.91{\scriptstyle\pm 0.02}$ | $0.67{\scriptstyle\pm 0.02}$ | $0.33{\scriptstyle\pm 0.00}$ | $0.36{\scriptstyle\pm 0.08}$
GCN-dDGM∗-EHH | $0.90{\scriptstyle\pm 0.05}$ | $0.59{\scriptstyle\pm 0.04}$ | $0.43{\scriptstyle\pm 0.03}$ | $0.55{\scriptstyle\pm 0.03}$
GCN-dDGM∗-EHS | $0.91{\scriptstyle\pm 0.03}$ | $0.66{\scriptstyle\pm 0.03}$ | $0.32{\scriptstyle\pm 0.01}$ | $0.45{\scriptstyle\pm 0.01}$
For example, in the case of Texas we achieve the highest homophily levels for
the latent graphs of between $0.85\pm 0.02$ and $0.91\pm 0.03$, and also some
of the highest accuracies ranging from $73.88\pm 9.95\%$ to $81.67\pm 7.05\%$.
For Wisconsin the homophily level is lower than that for Texas, but this can
be attributed to the fact that $k=10$, inevitably creating more connections
with nodes from other classes. Also, in Wisconsin there are two classes with
substantially less nodes than the rest, meaning that a high accuracy can be
achieved even if those are misclassified. On the other hand, for Squirrel,
although the latent graph inference system still manages to increase homophily
from $0.22$ in the original graph to between $0.27\pm 0.00$ and $0.43\pm 0.03$
in the latent graph, the increase is not big as compared to the other datasets
and we can see how this also has an effect on performance. In Table 2 the
maximum accuracy for Squirrel is of $35.00\pm 2.35\%$. Note that this is still
substantially better than using a MLP or a standard GCN, which obtain
accuracies of $30.44\pm 2.55\%$ and $24.19\pm 2.56\%$, respectively. The same
discussion applies to Chameleon. Figure 2 displays how the graph connectivity
is modified during the training process. This shows that the inference system
is able to dynamically learn an optimal connectivity structure for the latent
graph based on the downstream task, and modify it accordingly during training.
Additional latent graph plots for the different datasets can be found in
Appendix F.
(a) Epoch 1, $h=0.39$.
(b) Epoch 5, $h=0.46$.
(c) Epoch 10, $h=0.50$.
(d) Epoch 100, $h=0.74$.
(e) Epoch 500, $h=0.81$.
(f) Epoch 1000, $h=0.93$.
Figure 2: Latent graph homophily level, $h$, evolution as a function of
training epochs for Texas. The latent graphs are produced during the training
process for the GCN-dDGM∗-EH model with $k=2$.
### 4.2 Real-World Applications
Next, we test the latent graph inference system on real-world applications: on
the TadPole dataset (Marinescu et al. (2020)), which was also discussed by
Kazi et al. (2022), and the Aerothemodynamics dataset, which we have created
for this work. The TadPole dataset contains information about brain images for
different patients and the task is to classify each patient into three
classes: Normal Control, Alzheimer’s Disease and Mild Cognitive Impairment. On
the other hand, the Aerothermodynamics dataset was inspired by recent research
discussing potential applications of machine learning to aerospace engineering
(Maheshwari et al. (2018); Sáez de Ocáriz Borde et al. (2021); de Ocáriz Borde
et al. (2021)), and challenges networks to classify different regions of a
shock wave around a rocket (refer to Appendix D for more information on the
dataset). Note that since neither of these datasets have a graph structure
(they are pointclouds), only the dDGM∗ can be utilized in this case. Results
are given in Table 4. We use GAT diffusion layers and compare the performance
using single model spaces and product manifolds. Almost all models using the
latent graph inference system outperform the performance of the MLP. It is
important to note that since the datasets do not provide an input graph it
would not be possible to use GAT models without using dDGM∗ modules. Again, we
find that using product manifolds to model the latent space of potentially
complex real-world datasets proves beneficial and boosts model accuracy.
In principle, the Aerothermodynamics datasets classifies shock regions based
on the flow absolute velocity into 4 regions as recorded in Table 4. However,
we tested increasing the number of classes by further subdividing the flow
into a total of 7 regions, as shown in Figure 4. We found that interestingly,
the latent graph inferred by the model does not only cluster nodes with
similar labels together, but it actually organizes the latent graph in order
based on the absolute velocities (which are not explicitly given to the model:
the velocities are used to create the labels but values are not provided to
the model as input). This suggests that the graph generation system is
organizing the latent graph based on some inferred high level understanding of
the physics of the problem. Additional latent graphs for these datasets are
provided in Appendix F.2.
Figure 3: Results for the TadPole and the Aerothermodynamics datasets using
GAT diffusion layers and different latent graph inference modules.
| TadPole | Aerothermodynamics
---|---|---
Nodes | 564 | 1,456
Features | 30 | 1
Classes | 3 | 4
Model | Accuracy $(\%)$ $\pm$ Standard Deviation
GAT-dDGM∗-E | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}90.36{\scriptstyle\pm 3.21}}}$ | $89.20{\scriptstyle\pm 1.81}$
GAT-dDGM∗-H | $88.75{\scriptstyle\pm 3.91}$ | $86.36{\scriptstyle\pm 2.99}$
GAT-dDGM∗-S | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}90.89{\scriptstyle\pm 4.55}}}$ | $86.21{\scriptstyle\pm 5.37}$
GAT-dDGM∗-HH | $89.68{\scriptstyle\pm 5.70}$ | $77.47{\scriptstyle\pm 13.81}$
GAT-dDGM∗-SS | $89.82{\scriptstyle\pm 4.79}$ | $71.26{\scriptstyle\pm 14.40}$
GAT-dDGM∗-EH | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}90.36{\scriptstyle\pm 4.16}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}89.90{\scriptstyle\pm 0.55}}}$
GAT-dDGM∗-ES | $89.46{\scriptstyle\pm 5.56}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}90.34{\scriptstyle\pm 3.90}}}$
GAT-dDGM∗-HS | $86.43{\scriptstyle\pm 5.82}$ | $88.73{\scriptstyle\pm 4.72}$
GAT-dDGM∗-EHH | $87.68{\scriptstyle\pm 9.95}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}91.72{\scriptstyle\pm 0.98}}}$
GAT-dDGM∗-EHS | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}92.68{\scriptstyle\pm 3.52}}}$ | $88.74{\scriptstyle\pm 3.39}$
MLP∗ | $87.68{\scriptstyle\pm 3.52}$ | $81.03{\scriptstyle\pm 8.56}$
Figure 4: Latent graph obtained by the GAT-dDGM∗-EHH model with $k=7$
including more subclasses for different absolute velocity simulation regions
for the Aerothermodynamics dataset.
### 4.3 Scaling to Large Graphs
All datasets considered so far are relatively small. In this section we work
with datasets from the Open Graph Benchmark (OGB) which contain large-scale
graphs and require models to perform realistic out-of-distribution
generalization. In particular, we use the OGB-Arxiv and the OGB-Products
datasets. As discussed in Appendix C.3.3, for training on these datasets we
use graph subsampling techniques, and Graph Attention Network version 2
(GATv2) diffusion layers (Brody et al. (2021)); since we do not expect
overfitting we can use more expressive layers. OGB-Arxiv and OGB-Products have
a total of $40$ and $47$ classes, respectively. Previous datasets only
considered multi-class classification for between 3 to 15 classes. This, added
to the fact that the datasets have orders of magnitude more nodes and edges,
makes the problems in this section considerably more challenging. For OGB-
Arxiv using a MLP and a GATv2 model without leveraging latent graph inference
we obtain accuracies of $63.49\pm 0.15\%$ and $61.93\pm 1.62\%$, respectively.
The best model with latent graph inference, GATv2-dDGM∗-EHS, achieves an
accurary of $65.06\pm 0.09\%$. For OGB-Products, the MLP and a GATv2 results
are $66.05\pm 0.20\%$ and $62.02\pm 2.60\%$, and for the best model,
GATv2-dDGM-E, we record an accuracy of $66.59\pm 0.30\%$. From the results
(more in Appendix E.5), we conclude that latent graph inference is still
beneficial for this larger datasets but there is substantial room for
improvement. Graph subsampling interferes with embedding space learning.
## 5 Discussion and Conclusion
In this work we have incorporated Riemannian geometry to the dDGM latent graph
inference module by Kazi et al. (2022). First, we have shown how to work with
manifolds of constant arbitrary curvature, both positive and negative. Next,
we have leveraged product manifolds of model spaces and their convenient
mathematical properties to enable the dDGM module to generate a more complex
homogeneous manifold with varying curvature which can better encode the latent
data, while learning the curvature of each model space composing the product
manifold during training.
We have evaluated our method on many and diverse datasets, and we have shown
that using product manifolds to model the embedding space for the latent graph
gives enhanced downstream performance as compared to using single model spaces
of constant curvature. The inference system has been tested on both homophilic
and heterophilic benchmarks. In particular, we have found that using optimized
latent graphs, diffusion layers like GCNs are able to successfully operate on
datasets with low homophily levels. Additionally, we have tested and proven
the applicability of our method to large-scale graphs. Lastly, we have shown
the benefits of applying this procedure in real-world problems such as brain
imaging based data and aerospace engineering problems. All experiments
discussed in the main text are concerned with transductive learning; however,
the method is also applicable to inductive learning, see Appendix E.6.
The product manifold embedding space approach has provided a computationally
tractable way of generating more complex homogenous manifolds for the latent
features’ embedding space. Furthermore, the curvature of the product
components is learned rather than it being a fixed hyperparameter, which
allows for greater flexibility. However, the number of model spaces to
generate the product manifold must be specified before training. It would be
interesting to devise an approach for the network to independently add more
model spaces to the product manifold when needed. Also, we are restricting our
approach to product manifolds based on model spaces of constant curvature due
to their suitable mathematical properties. Such product manifolds do not cover
all possible arbitrary manifolds in which the latent data could be encoded and
hence, there could still be, mathematically speaking, more optimal manifolds
to represent the data. It is worth exploring whether approaches to generate
even more diverse and computationally tractable manifolds would be possible.
##### Future Work
Lastly, there are a few limitations intrinsic to the dDGM module, irrespective
of the product manifold embedding approach introduced in this work. Firstly,
although utilizing symbolic matrices can help computational efficiency
(Appendix C.3), the method still has quadratic complexity. Kazi et al. (2022)
proposed computing probabilities in a neighborhood of the node and using tree-
based algorithms to reduce it to $\mathcal{O}(n\log n)$. Moreover, the Gumbel
Top-k sampling approach restricts the average node degree of the latent graph
and requires manually adjusting the $k$ value through testing. A possible
solution could be to use a distance based sparse threshold approach in which
an unweighted edge is created between two nodes if they are within a threshold
distance of each other in latent space. This is similar to the Gumbel Top-k
trick, but instead of choosing a fixed number of closest neighbors, we connect
all nodes within a distance. This could help better capture the heterogeneity
of the graph. However, we actually tested this approach and found it quite
unstable. Note that although we do not have the $k$ parameter anymore, we must
still choose a threshold distance. Another avenue to help with scalability,
improve computational complexity, and facilitate working with large-scale
graphs would be to use a hierarchical perspective. Inspired by brain
interneurons (Freund & Buzsáki (1996)), we could introduce fictitious
connector inducing nodes in different regions of the graph, use those nodes to
summarize different regions of large graphs, and apply the kNN algorithm or
the Gumbel Top-k trick to the fictitious connector inducing nodes. This way
the computational complexity would still be quadratic, but proportional to the
number of interconnectors. Similar techniques have been applied to Gaussian
Processes (Galy-Fajou & Opper (2021); Wu et al. (2021)) and Set Transformers
(Lee et al. (2019)).
## Acknowledgements
Haitz Sáez de Ocáriz Borde expresses gratitude to the Rafael del Pino
Foundation for their generous financial assistance throughout his academic
journey at the University of Cambridge. Additionally, Haitz Sáez de Ocáriz
Borde extends appreciation to the G-Research Grant for enabling his
participation in the International Conference on Learning Representations
(ICLR 2023) held in Kigali, Rwanda.
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## Appendix A Riemannian Manifolds
In Riemannian geometry, we define a Riemannian manifold or Riemannian space
$(\mathcal{M},g)$ as a real and differentiable manifold $\mathcal{M}$ in which
each tangent space has an associated inner product $g$, that is, a Riemannian
metric, which must vary smoothly when considering points in the manifold. The
Riemannian manifold $\mathcal{M}\subseteq\mathbb{R}^{N}$ (lives in the ambient
space $\mathbb{R}^{N}$) is a collection of real vectors, and is locally
similar to a linear space. The Riemannian metric generalizes inner products
for Riemannian manifolds. It also allows to define geometric notions on a
Riemannian manifold such as lengths of curves, curvature, and angles to name a
few. If the point $\mathbf{x}_{p}\in\mathcal{M}$, then we can denote the
tangent space at $\mathbf{x}_{p}$ as $T_{\mathbf{x}_{p}}\mathcal{M}$, which
has the same dimensionality as $\mathcal{M}$. $T_{\mathbf{x}_{p}}\mathcal{M}$
is the collection of all tangent vectors at ${\mathbf{x}_{p}}$. Moreover,
$g_{\mathbf{x}_{p}}:T_{\mathbf{x}_{p}}\mathcal{M}\times
T_{\mathbf{x}_{p}}\mathcal{M}\rightarrow\mathbb{R}$ is given by a positive-
definite inner product in the tangent space and depends smoothly on
${\mathbf{x}_{p}}$. A geodesic represents the shortest smooth path between two
points in a Riemannian manifold and generalizes the notion of a straight line
in Euclidean space. The length of a smooth continuously differentiable curve
$\gamma\leavevmode\nobreak\ :\leavevmode\nobreak\ t\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ \gamma(t)\leavevmode\nobreak\
\in\leavevmode\nobreak\ \mathcal{M},t\leavevmode\nobreak\
\in\leavevmode\nobreak\ [0,1]$ is given by
$L(\gamma)=\int_{0}^{1}||\gamma^{\prime}(t)||\,dt.$ (5)
Note that $L(\gamma)$ is unchanged by a monotone reparametrization. The
geodesic distance between two points
$\mathbf{x}_{p_{i}},\mathbf{x}_{p_{j}}\in\mathcal{M}$ is defined as the
infimum (greatest lower bound) of the length taken over all piecewise
continuously differentiable curves, such that
$\gamma_{\mathbf{x}_{p_{i}},\mathbf{x}_{p_{j}}}=\underset{\gamma}{\textrm{argmin}}\,L(\gamma):\gamma(0)=\mathbf{x}_{p_{i}},\gamma(1)=\mathbf{x}_{p_{j}}.$
(6)
The norm of a tangent vector $\mathbf{v}\in T_{\mathbf{x}_{p}}\mathcal{M}$ is
given by
$||\mathbf{v}||=\sqrt{g_{\mathbf{x}_{p}}(\mathbf{v},\mathbf{v})}.$ (7)
Moving from a point $\mathbf{x}_{p}\in\mathcal{M}$ with initial constant
velocity $\mathbf{v}\in T_{\mathbf{x}_{p}}\mathcal{M}$ is formalized by the
exponential map
$exp_{\mathbf{x}_{p}}:T_{\mathbf{x}_{p}}\mathcal{M}\rightarrow\mathcal{M},$
(8)
which gives the position of the geodesic at $t=1$ so that
$exp_{\mathbf{x}_{p}}(\mathbf{v})=\gamma(1).$ (9)
There is a unique unit speed geodesic $\gamma$ which satisfies
$\gamma(0)=\mathbf{x}_{p}$ and $\gamma^{\prime}(0)=\mathbf{v}$. On the other
hand, and less relevant to the work at hand, the logarithmic map is the
inverse of the exponential map
$log_{\mathbf{x}_{p}}=exp_{\mathbf{x}_{p}}^{-1}:\mathcal{M}\rightarrow
T_{\mathbf{x}_{p}}\mathcal{M}.$ (10)
In geodesically complete Riemannian manifolds, both the exponential and
logarithm maps are well-defined (Needham (1997)).
In general, it can be impossible to find a solution for the geodesic between
two points for an arbitrary manifold. As discussed in the main text, we want
to be able to move beyond constant curvature spaces and compute the similarity
measure $\varphi$ for latent features which may reside in a more general and
learnable manifold. This will enable us to more accurately model data which
may comprise varying structure, beyond that which can be represented in
Euclidean space, and also beyond hierarchical or cyclical data which can be
linked to constant curvature hyperbolic and spherical spaces, respectively.
That is, by varying structure we refer to data which may present different
underlying patterns in different regions of space. Generating more complex
manifolds we intend to minimize the distortion incurred in the data and to
represent the points in a more suitable manifold for the downstream task.
However, we must still be able to map the Euclidean output features of the
network to our learnable manifold and to compute distances between points. To
generate a more complex and learnable manifold while still having a closed
form solution for the exponential map and geodesics, we will introduce a
product manifold embedding space composed of multiple copies of simple model
spaces with constant curvature. Although product manifolds based on constant
curvature models still fall under the homogeneous manifold category (Kowalski
et al. (1989)), they allow to model more complex embedding spaces for the dDGM
module than those that can be represented using only constant curvature
homogeneous spaces.
For example, the standard torus, which can be obtained by multiplying two
spheres, is a homogeneous manifold. Nevertheless, some points on the manifold
have positive Gaussian curvature (outer part of the torus, elliptic points),
some have negative (inner part, hyperbolic points), and others have zero
(parabolic points). This is because for the torus the Euler characteristic
(Beltramo et al. (2021)) is zero so it must always have regions of negative
Gaussian curvature. This is to counterbalance the regions of positive
curvature guaranteed in Hilbert’s theorem. The main point is that using
product manifolds of constant curvature spaces we can generate manifolds with
regions with different Gaussian curvature that can help us better represent
data structures which may not be purely Euclidean, hyperbolic, or spherical.
For all points on the manifold, we can define a normal vector that is at right
angles to the surface. By generating intersections between normal planes (that
contain the normal vector) and the surface we can compute normal sections. In
general, different normal sections will have different curvatures. We refer to
$\kappa_{1}$ and $\kappa_{2}$ as the principal curvatures. They correspond to
the maximum and minimum values that the curvature can take at a given point
(Kühnel (2005)). The Gaussian curvature $K$ is the product of the two
$K=\kappa_{1}\kappa_{2}.$ (11)
A Riemannian manifold is said to have constant Gaussian curvature $K$ if
$sec(P)=K$ for all two-dimensional linear subspaces $P\subset
T_{\mathbf{x}_{p}}\mathcal{M}$ and for all $\mathbf{x}_{p}\in\mathcal{M}$.
Manifolds can be classified into three classes depending on their curvature:
flat space, positively curved space, and negatively curved space.
## Appendix B Further Discussion on Curvature Learning
Next, we aim to provide a more detailed explanation on the topic of curvature
learning for product manifolds and the implemented procedure using learnable
distance-metric-scaling coefficients. Let us discuss product manifolds which
are solely generated based on Cartesian products of the same model spaces,
such as
$\mathcal{P}_{\mathbb{H}}=\bigtimes_{j=1}^{n_{\mathbb{H}}}\mathbb{H}_{K_{j}^{\mathbb{H}}}^{d_{j}^{\mathbb{H}}},$
(12)
which uses Cartesian products of hyperboloids. Likewise, taking Cartesian
products of hyperspheres we would obtain
$\mathcal{P}_{\mathbb{S}}=\bigtimes_{k=1}^{n_{\mathbb{S}}}\mathbb{S}_{K_{k}^{\mathbb{S}}}^{d_{k}^{\mathbb{S}}}.$
(13)
For these cases, although
$\mathfrak{d}_{\mathbb{H}_{K_{j}^{\mathbb{H}}}^{d_{j}^{\mathbb{H}}}}\left(\mathbf{\overline{x}}_{p_{1}}^{(j)},\mathbf{\overline{x}}_{p_{2}}^{(j)}\right)$
and
$\mathfrak{d}_{\mathbb{S}_{K_{k}^{\mathbb{S}}}^{d_{k}^{\mathbb{S}}}}\left(\mathbf{\overline{x}}_{p_{1}}^{(k)},\mathbf{\overline{x}}_{p_{2}}^{(k)}\right)$
are in practice computed all using hyperboloids and hyperspheres with the same
fixed curvature $K_{j}^{\mathbb{H}}=-1,\,\forall j$, and
$K_{k}^{\mathbb{S}}=1,\,\forall k$, the scaling coefficients control the
curvature of the spaces individually,
$\mathfrak{d}_{\mathcal{P_{\mathbb{H}}}}(\mathbf{\overline{x}}_{p_{1}},\mathbf{\overline{x}}_{p_{2}})=\sqrt{\sum_{j=1}^{n_{\mathbb{H}}}\left(\alpha_{j}^{\mathbb{H}}\mathfrak{d}_{\mathbb{H}_{K_{j}^{\mathbb{H}}}^{d_{j}^{\mathbb{H}}}}\left(\mathbf{\overline{x}}_{p_{1}}^{(j)},\mathbf{\overline{x}}_{p_{2}}^{(j)}\right)\right)^{2}},$
(14)
and,
$\mathfrak{d}_{\mathcal{P_{\mathbb{S}}}}(\mathbf{\overline{x}}_{p_{1}},\mathbf{\overline{x}}_{p_{2}})=\sqrt{\sum_{k=1}^{n_{\mathbb{S}}}\left(\alpha_{k}^{\mathbb{S}}\mathfrak{d}_{\mathbb{S}_{K_{k}^{\mathbb{S}}}^{d_{k}^{\mathbb{S}}}}\left(\mathbf{\overline{x}}_{p_{1}}^{(k)},\mathbf{\overline{x}}_{p_{2}}^{(k)}\right)\right)^{2}}.$
(15)
That is, $\alpha_{j}^{\mathbb{H}}$ and $\alpha_{k}^{\mathbb{S}}$ scale the
distances generated based on unit hyperboloids and hyperspheres, respectively.
Using the scaled metrics, we are effectively still computing Cartesian
products of hyperboloids and hyperspheres with different curvatures, so that
$\mathcal{P}_{\mathbb{H}}=\bigtimes_{j=1}^{n_{\mathbb{H}}}\mathbb{H}_{K_{j}^{\mathbb{H}}}^{d_{j}^{\mathbb{H}}}\neq\bigtimes_{j=1}^{n_{\mathbb{H}}}\mathbb{H}_{K_{\mathbb{H}}}^{d_{j}^{\mathbb{H}}},$
(16)
$\mathcal{P}_{\mathbb{S}}=\bigtimes_{k=1}^{n_{\mathbb{S}}}\mathbb{S}_{K_{k}^{\mathbb{S}}}^{d_{k}^{\mathbb{S}}}\neq\bigtimes_{k=1}^{n_{\mathbb{S}}}\mathbb{S}_{K_{\mathbb{S}}}^{d_{k}^{\mathbb{S}}}.$
(17)
This guarantees we are retrieving equivalent values to those which would be
generated by model spaces with different curvatures. This is done to avoid
backpropagating through operators such as exponential maps and closed form
solutions for distances in the different model spaces.
Considering again an arbitrary product manifold of all model spaces as in
Equation 2, we can control the curvature through the following derivatives
$\frac{\partial\mathfrak{d}_{\mathcal{P}}}{\partial\alpha_{j}^{\mathbb{H}}},$
(18)
for the hyperboloid terms, and
$\frac{\partial\mathfrak{d}_{\mathcal{P}}}{\partial\alpha_{k}^{\mathbb{S}}},$
(19)
for the hyperspheres. In the case of Euclidean space the curvature is always
$K_{\mathbb{E}}=0$, so there is no need to learn it. Also, using a Cartesian
product of Euclidean spaces would be equivalent to using a single Euclidean
space of greater dimensionality
$\mathcal{P}_{\mathbb{E}}=\bigtimes_{i=1}^{n_{\mathbb{E}}}\mathbb{E}_{K_{i}^{\mathbb{E}}}^{d_{i}^{\mathbb{E}}}=\mathbb{E}_{K_{\mathbb{E}}}^{d_{\mathbb{E}}},$
(20)
where $K_{i}^{\mathbb{E}}=K_{\mathbb{E}}=0,\,\forall i$, and
$d_{\mathbb{E}}=\sum_{i}^{n_{\mathbb{E}}}d_{i}^{\mathbb{E}}$. Hence, in
Equation 2 we used a single Euclidean space. The behavior described in
Equation 20 can be better appreciated by comparing distance
$\mathfrak{d}_{\mathcal{P_{\mathbb{E}}}}$ obtained using
$\mathcal{P}_{\mathbb{E}}$ and
$\mathfrak{d}_{\mathbb{E}_{K_{\mathbb{E}}}^{d_{\mathbb{E}}}}$ from
$\mathbb{E}_{K_{\mathbb{E}}}^{d_{\mathbb{E}}}$:
$\mathfrak{d}_{\mathcal{P_{\mathbb{E}}}}^{2}=\sum_{i=1}^{n_{\mathbb{E}}}\left(\mathfrak{d}_{\mathbb{E}_{K_{i}^{\mathbb{E}}}^{d_{i}^{\mathbb{E}}}}\right)^{2}=\sum_{i=1}^{n_{\mathbb{E}}}\left(\mathfrak{d}_{\mathbb{E}_{K_{\mathbb{E}}}^{d_{i}^{\mathbb{E}}}}\right)^{2},$
(21)
$\mathfrak{d}_{\mathcal{P_{\mathbb{E}}}}^{2}=\mathfrak{d}_{\mathbb{E}_{K_{\mathbb{E}}}^{d_{1}^{\mathbb{E}}}}^{2}+\mathfrak{d}_{\mathbb{E}_{K_{\mathbb{E}}}^{d_{2}^{\mathbb{E}}}}^{2}+...+\mathfrak{d}_{\mathbb{E}_{K_{\mathbb{E}}}^{d_{n_{\mathbb{E}}}^{\mathbb{E}}}}^{2}.$
(22)
Considering that the equation for independent distances in each Euclidean
space is
$\mathfrak{d}_{\mathbb{E}_{K_{\mathbb{E}}}^{d_{i}^{\mathbb{E}}}}=\sqrt{\sum_{a}\left(\mathbf{x}_{p_{1}}^{(a)}-\mathbf{x}_{p_{2}}^{(a)}\right)^{2}},$
(23)
we obtain,
$\mathfrak{d}_{\mathcal{P_{\mathbb{E}}}}=\sqrt{\sum_{a}\left(\mathbf{x}_{p_{1}}^{(a)}-\mathbf{x}_{p_{2}}^{(a)}\right)^{2}+\sum_{b}\left(\mathbf{x}_{p_{1}}^{(b)}-\mathbf{x}_{p_{2}}^{(b)}\right)^{2}+...+\sum_{z}\left(\mathbf{x}_{p_{1}}^{(z)}-\mathbf{x}_{p_{2}}^{(z)}\right)^{2}},$
(24)
which is analogous to taking the second power of the difference of two points
in another Euclidean space with more dimensions, so that
$\mathfrak{d}_{\mathcal{P_{\mathbb{E}}}}=\mathfrak{d}_{\mathbb{E}_{K_{\mathbb{E}}}^{d_{\mathbb{E}}}}.$
(25)
This behavior is only applicable to Euclidean space (Gu et al. (2019)), when
multiplying other model spaces
$\mathcal{P}_{\mathbb{H}}=\bigtimes_{j=1}^{n_{\mathbb{H}}}\mathbb{H}_{K_{j}^{\mathbb{H}}}^{d_{j}^{\mathbb{H}}}\neq\mathbb{H}_{K_{\mathbb{H}}}^{d_{\mathbb{H}}},$
(26)
$\mathcal{P}_{\mathbb{S}}=\bigtimes_{k=1}^{n_{\mathbb{S}}}\mathbb{S}_{K_{k}^{\mathbb{S}}}^{d_{k}^{\mathbb{S}}}\neq\mathbb{S}_{K_{\mathbb{S}}}^{d_{\mathbb{S}}},$
(27)
even when the curvature is the same for all hyperboloids and hyperspheres,
$K_{j}^{\mathbb{H}}=K_{\mathbb{H}},\,\forall j$ and
$K_{k}^{\mathbb{S}}=K_{\mathbb{S}},\,\forall k$. For example, multiplying two
hyperspheres will result in a hypertorus, not a higher-dimensional
hypersphere.
As a final remark, note that any loss function which is dependent on the
distance function induced by the Riemannian metric of a Riemannian manifold is
locally smooth on $\mathcal{M}$ (where $\mathcal{M}$ could be a product
manifold $\mathcal{P}$) and can be optimized by first order methods.
## Appendix C Training Procedure
In this appendix we discuss key concepts to train the dDGM module. We detail
how to backpropagate through the discrete sampling method and introduce an
additional loss term to make this possible. We also we provide additional
implementation details for updating learnable distance-metric-scaling
coefficients and dealing with distance functions during training. Lastly, we
describe the two approaches used in this work to make training computationally
tractable for larger graphs: symbolic handling of distance metrics and graph
subsampling.
### C.1 Backpropagation through the dDGM
The baseline node feature learning part of the architecture is optimized based
on the downstream task loss: for classification we use the cross-entropy loss
and for regression the mean squared error loss. Nevertheless, we must also
update the graph learning dDGM parameters. To do so, we follow the approach
proposed by Kazi et al. (2022) and apply a compound loss that rewards edges
involved in a correct classification and penalizes edges which result in
misclassification. We define the reward function,
$\delta\left(y_{i},\hat{y}_{i}\right)=\mathds{E}(ac_{i})-ac_{i}$ (28)
as the difference between the average accuracy of the $i$th sample and the
current prediction accuracy, where $y_{i}$ and $\hat{y}_{i}$ are the predicted
and true labels, and $ac_{i}=1$ if $y_{i}=\hat{y}_{i}$ or $0$ otherwise. Based
on $\delta\left(y_{i},\hat{y}_{i}\right)$ we obtain the loss employed to
update the graph learning module
$L_{GL}=\sum_{i=1}^{N}\left(\delta\left(y_{i},\hat{y}_{i}\right)\sum_{l=1}^{l=L}\sum_{j:(i,j)\in\mathcal{E}^{(l)}}\log
p_{ij}^{(l)}\right).$ (29)
$L_{GL}$’s gradient approximates the gradient of the expectation
$\mathds{E}_{(\mathcal{G}^{(1)},...,\mathcal{G}^{(L)})\sim(\mathbf{P}^{(1)},..,\mathbf{P}^{(L)})}\sum_{i=1}^{N}\delta\left(y_{i},\hat{y}_{i}\right),$
(30)
with respect to the parameters of the graphs in all the layer $\theta_{GL}$.
So that,
$\frac{dL_{GL}}{d\theta_{GL}}\approx\frac{d}{d\theta_{GL}}\mathds{E}_{(\mathcal{G}^{(1)},...,\mathcal{G}^{(L)})\sim(\mathbf{P}^{(1)},..,\mathbf{P}^{(L)})}\sum_{i=1}^{N}\delta\left(y_{i},\hat{y}_{i}\right).$
(31)
The expectation $\mathds{E}(ac_{i})^{(t)}$ is calculated based on
$\mathds{E}(ac_{i})^{(t)}=\beta\mathds{E}(ac_{i})^{(t-1)}+(1-\beta)ac_{i},$
(32)
with $\beta=0.9$ and $\mathds{E}(ac_{i})^{(t=0)}=0.5$. For regression we use
the R2 score instead of the accuracy.
### C.2 Updating Learnable Distance-metric-scaling Coefficients
The scaling metrics are learnable, so that we can indirectly adjust the
curvature of each model space without having to backpropagate through the
exponential map functions and the formulas for the distances. If we simply
take the derivative of the graph loss and update the distance-metric-scaling
coefficients during training $\alpha^{(t)}=\alpha^{(t-1)}-lr\frac{\partial
L_{GL}}{\partial\alpha^{(t-1)}},$ ($lr$ being the learning rate) this can
result in negative values for the coefficients that multiply the distances in
different model spaces, which would be mathematically incorrect since
distances are by definition positive or zero. To solve this issue we learn
$\tilde{\alpha}$, instead of $\alpha$. The two are related by the following
equation, $\alpha=S\left(\tilde{\alpha}\right),$ where $S$ is the sigmoid
function. So that,
$\alpha^{(t)}=S(\tilde{\alpha}^{(t)})=S\left(\tilde{\alpha}^{(t-1)}-lr\frac{\partial
L_{GL}}{\partial\tilde{\alpha}^{(t-1)}}\right),$ which means that
$0<\alpha^{(t)}<1$. Using ReLU instead of $S$ could allow $\alpha^{(t)}$ to
take arbitrarily large positive values, but we would have gradient problems if
$\tilde{\alpha}$ becomes negative. Given that using the sigmoid function
bounds the maximum value that the scaling metrics can take, we must multiply
the Euclidean plane distance by its own scaling coefficient. Since the Gumbel
Top-k trick selects the closest points to generate unweighted edges, rather
than storing the actual geodesics, if we scale the Euclidean plane
contribution down when necessary, this would equate to having other model
spaces with much bigger curvature than that which is actually possible due to
the scaling coefficient being now bounded by zero and one. For example, if we
were to have a product manifold based on the Cartesian product of the
Euclidean plane and a hypersphere, if the model learns a scaling metric close
to zero for the Euclidean plane geodesics, this would be equivalent to having
a hypersphere with a really large curvature since edges would only be
generated based on the distances calculated on the hypersphere. Finally, note
that in practice we use the Adam optimizer instead of simple gradient descent
for training.
### C.3 Computational Efficiency
In this section we discuss some of the implementation techniques used to make
computation more efficient. We cover two main topics: symbolic handling of
distance metrics and graph subsampling. One of the main computational
limitations of the approach described in this work is that we must compute
distances between all points to generate the latent graph. Although the
discrete graph sampling method used by dDGM is more computationally efficient
than its continuous counterpart, cDGM — because it generates sparse graphs
that make convolutional operators lighter — we quickly run into memory
problems for graph datasets with $\mathcal{O}(10^{4})$ nodes. Starting from a
pointcloud, we must compute the distances between all points to determine
whether a connection should be established. This is problematic, since the
computational complexity scales quadratically as a function of the number of
nodes in the graph, which can rapidly become intractable as we increase the
size of our graph.
Most of the experiments were performed using NVIDIA Tesla T4 Tensor Core GPUs
with 16 GB of GDDR6 memory, NVIDIA P100 GPUs with 16 GB of CoWoS HBM2 memory,
or NVIDIA Tesla K80 GPUs with 24 GB of GDDR5 memory. All these GPUs have
limited memories that are easily exceeded during backpropagation for datasets
other than Cora and CiteSeer (Yang et al. (2016); Lu & Getoor (2003); Sen et
al. (2008)), which have 2,708 nodes and 3,327 nodes, respectively. For
example, using the standard PyTorch Geometric implementation we are not able
to backpropagate for the PubMed dataset which has 18,717 nodes.
#### C.3.1 Symbolic Handling of Distance Metrics
To avoid memory overflows we resort to Kernel Operations (KeOps) (Charlier et
al. (2021)), which makes it possible to compute reductions of large arrays
whose entries are given by a mathematical formula. We can classify matrices in
three categories: dense, sparse, and symbolic.
Dense matrices are dense numerical arrays $M_{ij}=M[i,j]$ which put a heavy
load on computer memory. When they increase in size, they can struggle to fit
into RAM or GPU memory. This is what happens in our case, when calculating
distances between points. Sparse matrices are typically used to try to address
this problem. Sparse matrices use lists of indices $(i_{n},j_{n})$ and
associated values $M_{n}$. Hence, in sparse matrices we only store the values
for non-zero entries. The main limitation of this approach is that the
computing speedup obtained by using sparse matrices is highly dependent on
sparsity. To obtain significant improvements in performance using sparse
encoding, the original matrix should effectively be more than $99\%$ empty
(Charlier et al. (2021)). This significantly constrains the applicability of
sparse encoding.
Alternatively, KeOps uses symbolic matrices, to represent matrices which can
be summarized using an underlying common mathematical structure. In this setup
the matrix entries $M_{ij}$ are represented as a function of vectors
$\mathbf{x}_{i}$ and $\mathbf{x}_{j}$, so that
$M_{ij}=F(\mathbf{x}_{i},\mathbf{x}_{j}).$ (33)
Even if these objects are not necessarily sparse, they can be represented
using only small data arrays $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$, which can
result into large improvements in computational efficiency, avoiding memory
overflow. In our case, the matrices we are working with are fully populated
and using sparse tensors would not enhance performance. On the other hand, the
symbolic approach implemented using KeOps enables us to represent the original
dense matrix containing all the distances between all the points in a much
more compact way, and to work with larger graphs than Cora and Citeseer (Yang
et al. (2016); Lu & Getoor (2003); Sen et al. (2008)), namely, PubMed, CS, and
Physics (Shchur et al. (2018)). Finally, note that in our case the function
used by the symbolic matrix is the distance metric appropiate for whichever
manifold we are using to construct the latent graph.
$M_{ij}=\mathfrak{d}(\mathbf{x}_{i},\mathbf{x}_{j}).$ (34)
#### C.3.2 Quantitative Study of Runtime Speedup
We quantify the runtime speedup obtained by using symbolic handling of the
distance metrics when generating latent graphs for Cora and CiteSeer. Note,
however, that improved execution time is not the only benefit of symbolic
handling. As discussed in Section C.3.1, symbolic matrices also help avoid
memory overflow for larger graphs. Indeed, this is the reason we choose Cora
and CiteSeer to conduct these experiments: using other datasets we experience
GPU memory overflow when using dense matrices, and hence we cannot compare
dense to symbolic matrix performance. It should also be highlighted that using
both dense and symbolic matrices gives the same results in terms of accuracy
since they are equivalent mathematically, the difference lies in the
computational efficiency of each method.
We display the results in Table 4 and Table 5, in which we record execution
times for a 100 epochs using a NVIDIA P100 GPU. We evaluate the effect of
increasing the number of dDGM layers on the runtime. We also compare the dDGM
module using Euclidean (GCN-dDGM-E) and a product manifold of Euclidean,
hyperbolic, and spherical space (GCN-dDGM-EHS) to generate the latent graphs.
Table 4: Results for runtime speedup quantification using symbolic as compared
to dense matrices. These results are for the GCN-dDGM-E model using $k=3$ and
training for 100 epochs using NVIDIA P100 GPU.
Model | No. dDGMs | Matrix | Dataset | Runtime (s) $\pm$ Standard Deviation
---|---|---|---|---
GCN-dDGM-E | 1 | Dense | Cora | $2.98{\scriptstyle\pm 0.01}$
1 | Symbolic | Cora | $2.64{\scriptstyle\pm 0.01}$
1 | Dense | CiteSeer | $3.50{\scriptstyle\pm 0.02}$
1 | Symbolic | CiteSeer | $2.73{\scriptstyle\pm 0.02}$
2 | Dense | Cora | $4.87{\scriptstyle\pm 0.19}$
2 | Symbolic | Cora | $4.03{\scriptstyle\pm 0.03}$
2 | Dense | CiteSeer | $6.16{\scriptstyle\pm 0.58}$
2 | Symbolic | CiteSeer | $4.13{\scriptstyle\pm 0.02}$
3 | Dense | Cora | $6.53{\scriptstyle\pm 0.15}$
3 | Symbolic | Cora | $5.38{\scriptstyle\pm 0.00}$
3 | Dense | CiteSeer | $8.42{\scriptstyle\pm 0.69}$
3 | Symbolic | CiteSeer | $5.51{\scriptstyle\pm 0.04}$
Table 5: Results for runtime speedup quantification using symbolic as compared
to dense matrices. These results are for the GCN-dDGM-EHS model using $k=3$
and training for 100 epochs using NVIDIA P100 GPU.
Model | No. dDGMs | Matrix | Dataset | Runtime (s) $\pm$ Standard Deviation
---|---|---|---|---
GCN-dDGM-EHS | 1 | Dense | Cora | $6.17{\scriptstyle\pm 0.41}$
1 | Symbolic | Cora | $4.27{\scriptstyle\pm 0.06}$
1 | Dense | CiteSeer | $7.92{\scriptstyle\pm 0.33}$
1 | Symbolic | CiteSeer | $4.38{\scriptstyle\pm 0.02}$
2 | Dense | Cora | $10.67{\scriptstyle\pm 0.32}$
2 | Symbolic | Cora | $7.18{\scriptstyle\pm 0.03}$
2 | Dense | CiteSeer | $14.15{\scriptstyle\pm 0.07}$
2 | Symbolic | CiteSeer | $7.48{\scriptstyle\pm 0.16}$
3 | Dense | Cora | $15.32{\scriptstyle\pm 0.13}$
3 | Symbolic | Cora | $10.08{\scriptstyle\pm 0.06}$
3 | Dense | CiteSeer | $20.85{\scriptstyle\pm 0.42}$
3 | Symbolic | CiteSeer | $10.31{\scriptstyle\pm 0.04}$
We can see that as we add more dDGMs the difference between using dense and
symbolic matrices becomes more substantial. Likewise, the benefit from using
symbolic matrices becomes more apparent when using product manifolds, this is
because more distances must be computed. The product manifolds are calculated
based on the Cartesian product of three model spaces and hence, to obtain the
overall distance between each of the nodes, we must compute geodesics in all
constant curvature manifolds independently. Another observation that we can
make is that the computation time for CiteSeer increases substantially as
compared to Cora using dense matrices. CiteSeer only has 995 more nodes than
Cora, yet using 3 dDGM-EHS layers the execution time for 100 epochs increases
by 5.53 seconds for dense matrices. This clearly shows that using dense
matrices can quickly become hard to scale for larger graphs, which can have
orders of magnitude more nodes than CiteSeer. In line with the literature,
using symbolic matrices is more computationally tractable for larger graphs
Kazi et al. (2022).
Also, we run some additional experiments to quantify the increase in
computation time as a function of $k$, that is, the number of edges per latent
graph node when applying the Gumbel Top-k trick. As we can see in Table 6 and
Table 7, $k$ does not seem to have a statistically significant impact on the
execution time. As before, we find that using symbolic matrices is
consistently more efficient.
Table 6: Results for runtime speedup quantification using symbolic as compared
to dense matrices for different $k=1-30$. These results are for the GCN-dDGM-E
model and training for 100 epochs using a Tesla T4 GPU.
GCN-dDGM-E
---
$k$ | Matrix | Dataset | Runtime (s) $\pm$ Standard Deviation
1 | Dense | Cora | $4.22{\scriptstyle\pm 0.14}$
1 | Symbolic | Cora | $2.84{\scriptstyle\pm 0.02}$
1 | Dense | CiteSeer | $5.26{\scriptstyle\pm 0.06}$
1 | Symbolic | CiteSeer | $2.95{\scriptstyle\pm 0.02}$
2 | Dense | Cora | $4.23{\scriptstyle\pm 0.06}$
2 | Symbolic | Cora | $2.99{\scriptstyle\pm 0.18}$
2 | Dense | CiteSeer | $5.29{\scriptstyle\pm 0.03}$
2 | Symbolic | CiteSeer | $2.98{\scriptstyle\pm 0.01}$
3 | Dense | Cora | $4.24{\scriptstyle\pm 0.06}$
3 | Symbolic | Cora | $2.94{\scriptstyle\pm 0.02}$
3 | Dense | CiteSeer | $5.54{\scriptstyle\pm 0.24}$
3 | Symbolic | CiteSeer | $3.04{\scriptstyle\pm 0.03}$
5 | Dense | Cora | $4.25{\scriptstyle\pm 0.11}$
5 | Symbolic | Cora | $2.93{\scriptstyle\pm 0.01}$
5 | Dense | CiteSeer | $5.41{\scriptstyle\pm 0.17}$
5 | Symbolic | CiteSeer | $3.02{\scriptstyle\pm 0.11}$
7 | Dense | Cora | $4.27{\scriptstyle\pm 0.02}$
7 | Symbolic | Cora | $2.94{\scriptstyle\pm 0.03}$
7 | Dense | CiteSeer | $5.44{\scriptstyle\pm 0.16}$
7 | Symbolic | CiteSeer | $3.11{\scriptstyle\pm 0.18}$
10 | Dense | Cora | $4.30{\scriptstyle\pm 0.14}$
10 | Symbolic | Cora | $2.94{\scriptstyle\pm 0.04}$
10 | Dense | CiteSeer | $5.48{\scriptstyle\pm 0.21}$
10 | Symbolic | CiteSeer | $3.01{\scriptstyle\pm 0.03}$
20 | Dense | Cora | $4.67{\scriptstyle\pm 0.52}$
20 | Symbolic | Cora | $3.19{\scriptstyle\pm 0.05}$
20 | Dense | CiteSeer | $5.49{\scriptstyle\pm 0.03}$
20 | Symbolic | CiteSeer | $3.14{\scriptstyle\pm 0.05}$
30 | Dense | Cora | $4.23{\scriptstyle\pm 0.01}$
30 | Symbolic | Cora | $3.21{\scriptstyle\pm 0.20}$
30 | Dense | CiteSeer | $5.25{\scriptstyle\pm 0.03}$
30 | Symbolic | CiteSeer | $3.35{\scriptstyle\pm 0.03}$
Table 7: Results for runtime speedup quantification using symbolic as compared
to dense matrices for different $k=1-30$. These results are for the GCN-dDGM-
EHS model and training for 100 epochs using a Tesla T4 GPU.
GCN-dDGM-EHS
---
$k$ | Matrix | Dataset | Runtime (s) $\pm$ Standard Deviation
1 | Dense | Cora | $5.46{\scriptstyle\pm 0.02}$
1 | Symbolic | Cora | $4.50{\scriptstyle\pm 0.02}$
1 | Dense | CiteSeer | $7.27{\scriptstyle\pm 0.80}$
1 | Symbolic | CiteSeer | $4.97{\scriptstyle\pm 0.56}$
2 | Dense | Cora | $5.37{\scriptstyle\pm 0.03}$
2 | Symbolic | Cora | $4.56{\scriptstyle\pm 0.02}$
2 | Dense | CiteSeer | $6.69{\scriptstyle\pm 0.02}$
2 | Symbolic | CiteSeer | $4.71{\scriptstyle\pm 0.03}$
3 | Dense | Cora | $5.36{\scriptstyle\pm 0.02}$
3 | Symbolic | Cora | $4.57{\scriptstyle\pm 0.16}$
3 | Dense | CiteSeer | $6.77{\scriptstyle\pm 0.06}$
3 | Symbolic | CiteSeer | $5.06{\scriptstyle\pm 0.73}$
5 | Dense | Cora | $5.57{\scriptstyle\pm 0.22}$
5 | Symbolic | Cora | $4.58{\scriptstyle\pm 0.07}$
5 | Dense | CiteSeer | $6.74{\scriptstyle\pm 0.02}$
5 | Symbolic | CiteSeer | $4.70{\scriptstyle\pm 0.03}$
7 | Dense | Cora | $5.62{\scriptstyle\pm 0.29}$
7 | Symbolic | Cora | $4.47{\scriptstyle\pm 0.01}$
7 | Dense | CiteSeer | $6.75{\scriptstyle\pm 0.05}$
7 | Symbolic | CiteSeer | $4.99{\scriptstyle\pm 0.58}$
10 | Dense | Cora | $5.92{\scriptstyle\pm 0.73}$
10 | Symbolic | Cora | $4.70{\scriptstyle\pm 0.04}$
10 | Dense | CiteSeer | $7.06{\scriptstyle\pm 0.43}$
10 | Symbolic | CiteSeer | $4.83{\scriptstyle\pm 0.03}$
20 | Dense | Cora | $5.44{\scriptstyle\pm 0.02}$
20 | Symbolic | Cora | $4.77{\scriptstyle\pm 0.13}$
20 | Dense | CiteSeer | $6.76{\scriptstyle\pm 0.03}$
20 | Symbolic | CiteSeer | $4.86{\scriptstyle\pm 0.18}$
30 | Dense | Cora | $5.63{\scriptstyle\pm 0.30}$
30 | Symbolic | Cora | $4.74{\scriptstyle\pm 0.03}$
30 | Dense | CiteSeer | $6.93{\scriptstyle\pm 0.22}$
30 | Symbolic | CiteSeer | $5.33{\scriptstyle\pm 0.66}$
#### C.3.3 Training on Large Graphs
Although symbolic handling of distance metrics is certainly necessary, for
larger graphs such as the graphs for node property prediction of the Open
Graph Benchmark (OGB) (Hu et al. (2020)) which have
$\mathcal{O}(10^{5})-\mathcal{O}(10^{8})$ nodes and
$\mathcal{O}(10^{6})-\mathcal{O}(10^{9})$ edges, we must combine KeOps with
graph subsampling techniques to make backpropagating computationally
tractable.
We apply a neighbor sampler to track message passing dependencies for the
subsampled nodes. This allows computation to be more lightweight. Based on the
message passing equation
$\mathbf{x}_{i}^{(l+1)}=\phi\Big{(}\mathbf{x}_{i}^{(l)},\bigoplus_{j\in\mathcal{N}(v_{i})}\psi(\mathbf{x}_{i}^{(l)},\mathbf{x}_{j}^{(l)})\Big{)},$
(35)
to calculate $\mathbf{x}_{i}^{(l+1)}$ we must aggregate, and hence, have
stored the node features of its neighbors when subsampling the graph. Note
that unlike in the original node prediction setup in which we give as input
all the nodes and predict properties for all nodes in the complete graph, here
we have a different number of input and output nodes, which gives rise to a
bipartite structure for multi-layer minibatch message passing. Such a
bipartite graph, which samples only the necessary input and output nodes from
the original graph, is called a message flow graph (Ladkin & Leue (2005)). For
every node that we compute in a given batch we will need to track its message
flow graph alongside all relevant dependencies.
## Appendix D The Aerothermodynamics Dataset
We generate a multi-class classification dataset based on Computational Fluid
Dynamics (CFD) simulations conducted for the rocket designs developed for the
Karman Space Programme. Specifically, the dataset is generated based on the
shock wave velocity distribution around the nose of a rocket at 10 degrees of
angle of attack. The simulation meshgrid has varying degrees of resolution,
and the shock wave is not symmetric due to the angle of attack. Although CFD
software uses a meshgrid to discretize space and run the aerothermodynamics
simulations, it can be challenging to extract the connectivity of the original
graph, since most often the software is designed to only output a pointcloud.
Moreover, given that across a shock wave, the static pressure, temperature,
and gas density increases almost instantaneously and there is an abrupt
decrease in the flow area, the original graph can present high heterophily.
Hence, latent graph inference can be beneficial.
For the dataset, we only consider the shock region at the leading edge of the
rocket. To obtain the class labels, we separate the shock flow absolute
velocity into 4 regions: $\leq\leavevmode\nobreak\ 300$ m/s, $300-450$ m/s,
$450-650$ m/s and $>650$ m/s. The original simulation has 207,745 datapoints,
but we only focus on the shock around the nose of the rocket and apply graph
coarsening, see Figure 5.
Figure 5: Pointcloud plot of shock intensity regions. The different regions
are represented using different colors which correspond to each target class.
This is the dataset after applying graph coarsening.
The network input is a pointcloud with pressure values. Note that the
coordinates with respect to the rocket are not given to the network. The model
is tasked with classifying the shock into four intensity regions which are
based on the absolute velocity distribution. Table 8 summarizes the dataset.
Table 8: Summary of properties for Aerothermodynamics dataset
| | Aerothermodynamics
---|---|---
Homophily level | | N/A
Nodes | | 1,456
Features | | 1
Edges | | N/A
Classes | | 4
Average Degree | | N/A
Learning | | Transductive
Task | | Classification
Network Type | | CFD simulation
## Appendix E Additional Experiments and Results
In this appendix we include additional experiments for the latent graph
inference system. In Section E.1 we explore using more than one latent graph
inference system per GNN model. In Section E.2 we include additional results
for the homophilic graph datasets in which we vary the value of $k$ for the
graph generation algorithm. In Section E.3 we experiment with using a greater
number of model spaces to generate product manifolds for the embedding space.
Section E.4 includes results for heterophilic datasets, Section E.5 for OGB,
and Section E.6 for inductive learning. The First, Second and Third best
models for each dataset are highlighted in each table.
### E.1 Number of dDGM Modules
In these experiments, we investigate whether there is any benefit in stacking
multiple dDGM modules. Using multiple dDGMs effectively means that within the
model, the network layers would be learning based on different latent graphs.
In principle, according to the results in Table 9 and 10, there is no clear
improvement in performance for Cora and CiteSeer. In fact, the accuracy of the
models can decrease. It is only for PubMed that there is sometimes some
improvement in the accuracy. These results align with previous studies by Kazi
et al. (2022). Finally, we also run a few experiments for Physics and CS.
Again, we find no substantial improvement using an additional dDGM module, see
Table 11. Given our findings, we use a single dDGM module, since it is more
computationally efficient.
Table 9: Results for Cora, CiteSeer, and PubMed using more than one dDGM∗
(taking a pointcloud as input) latent graph inference module.
| | | Cora | CiteSeer | PubMed
---|---|---|---|---|---
Model | $k$ | Layers | Accuracy $(\%)$ $\pm$ Standard Deviation
GCN-dDGM∗-E | 3 | dDGM∗/GCN/GCN/GCN | $62.48{\scriptstyle\pm 3.24}$ | $62.47{\scriptstyle\pm 3.20}$ | $83.89{\scriptstyle\pm 0.70}$
GCN-dDGM∗-E | 3 | dDGM∗/GCN/dDGM∗/GCN/GCN | $54.22{\scriptstyle\pm 3.79}$ | $59.76{\scriptstyle\pm 2.18}$ | $85.74{\scriptstyle\pm 2.06}$
GCN-dDGM∗-EH | 3 | dDGM∗/GCN/GCN/GCN | $61.07{\scriptstyle\pm 10.18}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}65.13{\scriptstyle\pm 2.19}}}$ | $86.67{\scriptstyle\pm 0.69}$
GCN-dDGM∗-EH | 3 | dDGM∗/GCN/dDGM∗/GCN/GCN | $51.96{\scriptstyle\pm 4.53}$ | $55.36{\scriptstyle\pm 5.35}$ | $86.19{\scriptstyle\pm 1.10}$
GCN-dDGM∗-EHS | 3 | dDGM∗/GCN/GCN/GCN | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}66.87{\scriptstyle\pm 3.27}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}63.80{\scriptstyle\pm 2.94}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}87.03{\scriptstyle\pm 0.70}}}$
GCN-dDGM∗-EHS | 3 | dDGM∗/GCN/dDGM∗/GCN/GCN | $50.66{\scriptstyle\pm 4.26}$ | $51.66{\scriptstyle\pm 3.24}$ | $86.32{\scriptstyle\pm 0.70}$
GCN-dDGM∗-E | 5 | dDGM∗/GCN/GCN/GCN | $60.63{\scriptstyle\pm 3.01}$ | $63.28{\scriptstyle\pm 3.35}$ | $86.82{\scriptstyle\pm 0.69}$
GCN-dDGM∗-E | 5 | dDGM∗/GCN/dDGM∗/GCN/GCN | $52.15{\scriptstyle\pm 4.21}$ | $54.46{\scriptstyle\pm 3.19}$ | $79.04{\scriptstyle\pm 4.45}$
GCN-dDGM∗-EH | 5 | dDGM∗/GCN/GCN/GCN | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}63.67{\scriptstyle\pm 6.48}}}$ | $61.59{\scriptstyle\pm 12.28}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}86.90{\scriptstyle\pm 1.16}}}$
GCN-dDGM∗-EH | 5 | dDGM∗/GCN/dDGM∗/GCN/GCN | $45.30{\scriptstyle\pm 3.06}$ | $51.69{\scriptstyle\pm 4.33}$ | $82.02{\scriptstyle\pm 5.42}$
GCN-dDGM∗-EHS | 5 | dDGM∗/GCN/GCN/GCN | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}64.82{\scriptstyle\pm 3.68}}}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}64.62{\scriptstyle\pm 3.35}}}$ | $86.85{\scriptstyle\pm 0.97}$
GCN-dDGM∗-EHS | 5 | dDGM∗/GCN/dDGM∗/GCN/GCN | $45.07{\scriptstyle\pm 4.21}$ | $45.42{\scriptstyle\pm 3.00}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}86.92{\scriptstyle\pm 0.83}}}$
MLP∗ | N/A | Linear/Linear/Linear | $58.92{\scriptstyle\pm 3.28}$ | $59.48{\scriptstyle\pm 2.14}$ | $85.75{\scriptstyle\pm 1.02}$
Table 10: Results for Cora, CiteSeer, and PubMed using more than one dDGM
(leveraging the original graph connectivity structure) latent graph inference
module.
| | | Cora | CiteSeer | PubMed
---|---|---|---|---|---
Model | $k$ | Layers | Accuracy $(\%)$ $\pm$ Standard Deviation
GCN-dDGM-E | 3 | dDGM/GCN/GCN/GCN | $80.44{\scriptstyle\pm 4.60}$ | $70.18{\scriptstyle\pm 1.46}$ | $87.45{\scriptstyle\pm 0.72}$
GCN-dDGM-E | 3 | dDGM/GCN/dDGM/GCN/GCN | $70.44{\scriptstyle\pm 5.81}$ | $69.85{\scriptstyle\pm 4.22}$ | $86.67{\scriptstyle\pm 0.48}$
GCN-dDGM-EH | 3 | dDGM/GCN/GCN/GCN | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}83.65{\scriptstyle\pm 5.25}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}72.89{\scriptstyle\pm 1.64}}}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}87.62{\scriptstyle\pm 0.64}}}$
GCN-dDGM-EH | 3 | dDGM/GCN/dDGM/GCN/GCN | $77.30{\scriptstyle\pm 7.30}$ | $72.32{\scriptstyle\pm 2.09}$ | $87.02{\scriptstyle\pm 0.79}$
GCN-dDGM-EHS | 3 | dDGM/GCN/GCN/GCN | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}84.70{\scriptstyle\pm 2.96}}}$ | $69.98{\scriptstyle\pm 2.70}$ | $87.35{\scriptstyle\pm 0.90}$
GCN-dDGM-EHS | 3 | dDGM/GCN/dDGM/GCN/GCN | $79.07{\scriptstyle\pm 4.35}$ | $71.99{\scriptstyle\pm 1.94}$ | $86.93{\scriptstyle\pm 0.71}$
GCN-dDGM-E | 5 | dDGM/GCN/GCN/GCN | $82.74{\scriptstyle\pm 4.42}$ | $71.78{\scriptstyle\pm 1.50}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}87.60{\scriptstyle\pm 0.69}}}$
GCN-dDGM-E | 5 | dDGM/GCN/dDGM/GCN/GCN | $72.70{\scriptstyle\pm 6.03}$ | $68.25{\scriptstyle\pm 4.57}$ | $86.59{\scriptstyle\pm 0.87}$
GCN-dDGM-EH | 5 | dDGM/GCN/GCN/GCN | $82.32{\scriptstyle\pm 4.71}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}73.56{\scriptstyle\pm 2.75}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}87.67{\scriptstyle\pm 0.76}}}$
GCN-dDGM-EH | 5 | dDGM/GCN/dDGM/GCN/GCN | $76.04{\scriptstyle\pm 4.23}$ | $72.11{\scriptstyle\pm 2.33}$ | $86.49{\scriptstyle\pm 0.73}$
GCN-dDGM-EHS | 5 | dDGM/GCN/GCN/GCN | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}83.58{\scriptstyle\pm 4.39}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}74.00{\scriptstyle\pm 1.68}}}$ | $87.12{\scriptstyle\pm 0.72}$
GCN-dDGM-EHS | 5 | dDGM/GCN/dDGM/GCN/GCN | $76.67{\scriptstyle\pm 5.55}$ | $70.42{\scriptstyle\pm 1.66}$ | $82.57{\scriptstyle\pm 0.81}$
GCN | N/A | GCN/GCN/GCN | $83.11{\scriptstyle\pm 2.29}$ | $69.97{\scriptstyle\pm 2.06}$ | $85.75{\scriptstyle\pm 1.01}$
Table 11: Results for Physics and CS using two dDGM latent graph inference
modules with a product manifold combining Euclidean, hyperbolic, and spherical
model spaces.
| | | Physics | CS
---|---|---|---|---
Model | $k$ | Layers | Accuracy $(\%)$ $\pm$ Standard Deviation
GCN-dDGM-EHS | 5 | dDGM/GCN/GCN/GCN | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}96.17{\scriptstyle\pm 0.30}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}92.06{\scriptstyle\pm 0.83}}}$
GCN-dDGM-EHS | 5 | dDGM/GCN/dDGM/GCN/GCN | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}96.18{\scriptstyle\pm 0.31}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}87.00{\scriptstyle\pm 4.00}}}$
GCN | N/A | GCN/GCN/GCN | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}95.51{\scriptstyle\pm 0.34}}}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}87.28{\scriptstyle\pm 1.54}}}$
### E.2 Homophilic Graph Datasets Extended Results
In this section we include extended results for the Cora, CiteSeer, PubMed,
Physics, and CS datasets. Table 12 presents results using single model spaces
for the embedding space, and Table 13 and Table 14 using product manifolds. As
discussed in the main text, the dDGM∗ does not use the original dataset graph
as inductive bias, since it is only provided with a pointcloud. On the other
hand, the dDGM does use the original graph. Different $k$ values are applied.
Table 12: Results for classical homophilic datasets combining GCN diffusion
layers with the dDGM∗ and dDGM latent graph inference system and using single
model spaces to construct the latent graphs.
| | Cora | CiteSeer | PubMed | Physics | CS
---|---|---|---|---|---|---
Homophily level | | 0.81 | 0.74 | 0.80 | 0.93 | 0.80
Nodes | | 2,708 | 3,327 | 18,717 | 34,493 | 18,333
Features | | 1,433 | 3,703 | 500 | 8,415 | 6,805
Edges | | 5,278 | 4,676 | 44,327 | 247,962 | 81,894
Classes | | 7 | 6 | 3 | 5 | 15
Average Degree | | 3.9 | 2.77 | 4.5 | 14.38 | 8.93
Model | $k$ | Accuracy $(\%)$ $\pm$ Standard Deviation
GCN-dDGM∗-E | 3 | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}62.48{\scriptstyle\pm 3.24}}}$ | $62.47{\scriptstyle\pm 3.20}$ | $83.89{\scriptstyle\pm 0.70}$ | $94.03{\scriptstyle\pm 0.45}$ | $76.05{\scriptstyle\pm 6.89}$
GCN-dDGM∗-H | 3 | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}63.82{\scriptstyle\pm 3.69}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}63.76{\scriptstyle\pm 3.40}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}87.15{\scriptstyle\pm 0.69}}}$ | $93.24{\scriptstyle\pm 0.45}$ | $76.26{\scriptstyle\pm 6.39}$
GCN-dDGM∗-S | 3 | $32.78{\scriptstyle\pm 5.90}$ | $21.96{\scriptstyle\pm 6.32}$ | $41.60{\scriptstyle\pm 5.43}$ | $58.37{\scriptstyle\pm 1.44}$ | $54.52{\scriptstyle\pm 17.30}$
GCN-dDGM∗-E | 5 | $60.63{\scriptstyle\pm 3.01}$ | $63.28{\scriptstyle\pm 3.35}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}86.82{\scriptstyle\pm 0.69}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}95.13{\scriptstyle\pm 0.50}}}$ | $80.46{\scriptstyle\pm 2.34}$
GCN-dDGM∗-H | 5 | $61.48{\scriptstyle\pm 3.68}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}64.76{\scriptstyle\pm 2.88}}}$ | $85.00{\scriptstyle\pm 0.69}$ | $94.83{\scriptstyle\pm 0.51}$ | $82.69{\scriptstyle\pm 1.44}$
GCN-dDGM∗-S | 5 | $42.04{\scriptstyle\pm 5.01}$ | $22.20{\scriptstyle\pm 6.23}$ | $40.38{\scriptstyle\pm 5.44}$ | $54.70{\scriptstyle\pm 2.01}$ | $70.67{\scriptstyle\pm 13.69}$
GCN-dDGM∗-E | 7 | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}62.30{\scriptstyle\pm 5.27}}}$ | $61.36{\scriptstyle\pm 7.46}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}86.86{\scriptstyle\pm 0.68}}}$ | $93.71{\scriptstyle\pm 3.23}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}86.03{\scriptstyle\pm 5.27}}}$
GCN-dDGM∗-H | 7 | $61.44{\scriptstyle\pm 10.83}$ | $62.05{\scriptstyle\pm 5.74}$ | $85.19{\scriptstyle\pm 5.07}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}95.05{\scriptstyle\pm 0.30}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}84.20{\scriptstyle\pm 4.70}}}$
GCN-dDGM∗-S | 7 | $41.71{\scriptstyle\pm 14.22}$ | $20.75{\scriptstyle\pm 2.84}$ | $40.49{\scriptstyle\pm 1.57}$ | $55.20{\scriptstyle\pm 13.26}$ | $73.46{\scriptstyle\pm 8.17}$
GCN-dDGM∗-E | 10 | $61.37{\scriptstyle\pm 3.85}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}64.27{\scriptstyle\pm 3.97}}}$ | $85.48{\scriptstyle\pm 4.40}$ | $91.82{\scriptstyle\pm 3.62}$ | $81.42{\scriptstyle\pm 4.60}$
GCN-dDGM∗-H | 10 | $61.63{\scriptstyle\pm 4.56}$ | $63.62{\scriptstyle\pm 5.22}$ | $84.92{\scriptstyle\pm 4.85}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}95.02{\scriptstyle\pm 0.46}}}$ | $80.36{\scriptstyle\pm 5.42}$
GCN-dDGM∗-S | 10 | $34.04{\scriptstyle\pm 9.65}$ | $19.96{\scriptstyle\pm 4.14}$ | $39.84{\scriptstyle\pm 1.07}$ | $50.52{\scriptstyle\pm 2.77}$ | $69.55{\scriptstyle\pm 5.94}$
MLP∗ | N/A | $58.92{\scriptstyle\pm 3.28}$ | $59.48{\scriptstyle\pm 2.14}$ | $85.75{\scriptstyle\pm 1.02}$ | $94.91{\scriptstyle\pm 0.30}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}87.80{\scriptstyle\pm 1.54}}}$
| | Cora | CiteSeer | PubMed | Physics | CS
GCN-dDGM-E | 3 | $80.44{\scriptstyle\pm 4.60}$ | $70.18{\scriptstyle\pm 1.46}$ | $87.45{\scriptstyle\pm 0.72}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}96.03{\scriptstyle\pm 0.41}}}$ | $85.98{\scriptstyle\pm 2.59}$
GCN-dDGM-H | 3 | $80.31{\scriptstyle\pm 4.30}$ | $69.78{\scriptstyle\pm 1.56}$ | $85.79{\scriptstyle\pm 0.73}$ | $95.29{\scriptstyle\pm 0.43}$ | $85.95{\scriptstyle\pm 2.58}$
GCN-dDGM-S | 3 | $73.26{\scriptstyle\pm 4.33}$ | $67.98{\scriptstyle\pm 2.21}$ | $81.39{\scriptstyle\pm 1.00}$ | $91.60{\scriptstyle\pm 0.55}$ | $76.76{\scriptstyle\pm 4.65}$
GCN-dDGM-E | 5 | $82.74{\scriptstyle\pm 4.42}$ | $71.78{\scriptstyle\pm 1.50}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}87.60{\scriptstyle\pm 0.69}}}$ | $95.96{\scriptstyle\pm 0.40}$ | $87.88{\scriptstyle\pm 2.55}$
GCN-dDGM-H | 5 | $80.80{\scriptstyle\pm 4.43}$ | $67.53{\scriptstyle\pm 2.12}$ | $87.58{\scriptstyle\pm 0.71}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}96.06{\scriptstyle\pm 0.42}}}$ | $87.65{\scriptstyle\pm 2.56}$
GCN-dDGM-S | 5 | $80.81{\scriptstyle\pm 2.30}$ | $71.81{\scriptstyle\pm 1.20}$ | $77.65{\scriptstyle\pm 1.01}$ | $95.91{\scriptstyle\pm 0.41}$ | $80.73{\scriptstyle\pm 4.62}$
GCN-dDGM-E | 7 | $82.11{\scriptstyle\pm 4.24}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}72.35{\scriptstyle\pm 1.92}}}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}87.69{\scriptstyle\pm 0.67}}}$ | $95.50{\scriptstyle\pm 1.25}$ | $87.17{\scriptstyle\pm 3.82}$
GCN-dDGM-H | 7 | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}84.68{\scriptstyle\pm 3.31}}}$ | $70.43{\scriptstyle\pm 4.95}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}87.74{\scriptstyle\pm 0.72}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}96.06{\scriptstyle\pm 0.46}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}88.78{\scriptstyle\pm 2.24}}}$
GCN-dDGM-S | 7 | $80.44{\scriptstyle\pm 5.26}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}72.89{\scriptstyle\pm 2.00}}}$ | $87.13{\scriptstyle\pm 0.66}$ | $95.76{\scriptstyle\pm 0.43}$ | $84.16{\scriptstyle\pm 2.78}$
GCN-dDGM-E | 10 | $81.67{\scriptstyle\pm 5.74}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}72.44{\scriptstyle\pm 2.81}}}$ | $85.45{\scriptstyle\pm 4.32}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}95.99{\scriptstyle\pm 0.49}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}88.19{\scriptstyle\pm 3.51}}}$
GCN-dDGM-H | 10 | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}83.15{\scriptstyle\pm 3.20}}}$ | $71.54{\scriptstyle\pm 1.46}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}87.69{\scriptstyle\pm 0.76}}}$ | $95.84{\scriptstyle\pm 0.73}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}88.49{\scriptstyle\pm 2.21}}}$
GCN-dDGM-S | 10 | $82.70{\scriptstyle\pm 3.22}$ | $72.29{\scriptstyle\pm 2.87}$ | $87.38{\scriptstyle\pm 0.69}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}96.03{\scriptstyle\pm 0.44}}}$ | $85.20{\scriptstyle\pm 4.07}$
GCN | N/A | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}83.11{\scriptstyle\pm 2.29}}}$ | $69.97{\scriptstyle\pm 2.06}$ | $85.75{\scriptstyle\pm 1.01}$ | $95.51{\scriptstyle\pm 0.34}$ | $87.28{\scriptstyle\pm 1.54}$
Table 13: Results for classical homophilic datasets combining GCN diffusion
layers with the dDGM∗ module and using product manifolds to construct the
latent graphs.
| | Cora | CiteSeer | PubMed | Physics | CS
---|---|---|---|---|---|---
Homophily level | | 0.81 | 0.74 | 0.80 | 0.93 | 0.80
Nodes | | 2,708 | 3,327 | 18,717 | 34,493 | 18,333
Features | | 1,433 | 3,703 | 500 | 8,415 | 6,805
Edges | | 5,278 | 4,676 | 44,327 | 247,962 | 81,894
Classes | | 7 | 6 | 3 | 5 | 15
Average Degree | | 3.9 | 2.77 | 4.5 | 14.38 | 8.93
Model | $k$ | Accuracy $(\%)$ $\pm$ Standard Deviation
GCN-dDGM∗-HH | 3 | $57.00{\scriptstyle\pm 9.78}$ | $62.50{\scriptstyle\pm 4.25}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}87.43{\scriptstyle\pm 0.40}}}$ | $92.20{\scriptstyle\pm 3.47}$ | $74.88{\scriptstyle\pm 4.65}$
GCN-dDGM∗-SS | 3 | $38.26{\scriptstyle\pm 10.94}$ | $24.20{\scriptstyle\pm 10.12}$ | $41.59{\scriptstyle\pm 0.71}$ | $54.43{\scriptstyle\pm 10.00}$ | $55.29{\scriptstyle\pm 7.33}$
GCN-dDGM∗-EH | 3 | $61.07{\scriptstyle\pm 10.18}$ | $65.13{\scriptstyle\pm 2.19}$ | $86.67{\scriptstyle\pm 0.69}$ | $94.85{\scriptstyle\pm 0.55}$ | $85.91{\scriptstyle\pm 2.88}$
GCN-dDGM∗-ES | 3 | $62.07{\scriptstyle\pm 4.08}$ | $64.31{\scriptstyle\pm 3.15}$ | $86.85{\scriptstyle\pm 1.02}$ | $95.01{\scriptstyle\pm 0.55}$ | $78.67{\scriptstyle\pm 4.44}$
GCN-dDGM∗-HS | 3 | $59.55{\scriptstyle\pm 10.90}$ | $63.95{\scriptstyle\pm 2.35}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}87.04{\scriptstyle\pm 0.79}}}$ | $95.02{\scriptstyle\pm 0.45}$ | $76.55{\scriptstyle\pm 9.89}$
GCN-dDGM∗-EHH | 3 | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}70.85{\scriptstyle\pm 4.30}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}68.86{\scriptstyle\pm 2.97}}}$ | $39.93{\scriptstyle\pm 1.35}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}95.21{\scriptstyle\pm 0.34}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}92.22{\scriptstyle\pm 1.09}}}$
GCN-dDGM∗-EHS | 3 | $66.87{\scriptstyle\pm 3.27}$ | $63.80{\scriptstyle\pm 2.94}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}87.03{\scriptstyle\pm 0.70}}}$ | $95.03{\scriptstyle\pm 0.39}$ | $88.64{\scriptstyle\pm 1.90}$
GCN-dDGM∗-HH | 5 | $59.03{\scriptstyle\pm 4.86}$ | $63.47{\scriptstyle\pm 1.70}$ | $86.30{\scriptstyle\pm 0.99}$ | $94.31{\scriptstyle\pm 0.47}$ | $76.48{\scriptstyle\pm 4.30}$
GCN-dDGM∗-SS | 5 | $41.33{\scriptstyle\pm 10.54}$ | $21.48{\scriptstyle\pm 3.81}$ | $40.70{\scriptstyle\pm 1.31}$ | $53.43{\scriptstyle\pm 9.15}$ | $58.47{\scriptstyle\pm 5.79}$
GCN-dDGM∗-EH | 5 | $63.67{\scriptstyle\pm 6.48}$ | $61.59{\scriptstyle\pm 12.28}$ | $86.90{\scriptstyle\pm 1.16}$ | $93.94{\scriptstyle\pm 3.37}$ | $86.00{\scriptstyle\pm 3.04}$
GCN-dDGM∗-ES | 5 | $63.44{\scriptstyle\pm 4.18}$ | $63.10{\scriptstyle\pm 2.71}$ | $86.83{\scriptstyle\pm 0.82}$ | $93.92{\scriptstyle\pm 2.38}$ | $75.86{\scriptstyle\pm 11.18}$
GCN-dDGM∗-HS | 5 | $64.44{\scriptstyle\pm 4.00}$ | $62.80{\scriptstyle\pm 3.90}$ | $85.17{\scriptstyle\pm 4.72}$ | $94.81{\scriptstyle\pm 0.23}$ | $77.93{\scriptstyle\pm 4.47}$
GCN-dDGM∗-EHH | 5 | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}69.74{\scriptstyle\pm 5.09}}}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}66.81{\scriptstyle\pm 2.04}}}$ | $39.93{\scriptstyle\pm 1.35}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}95.25{\scriptstyle\pm 0.36}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}90.46{\scriptstyle\pm 1.03}}}$
GCN-dDGM∗-EHS | 5 | $64.82{\scriptstyle\pm 3.68}$ | $64.62{\scriptstyle\pm 3.35}$ | $86.85{\scriptstyle\pm 0.97}$ | $94.30{\scriptstyle\pm 2.09}$ | $88.48{\scriptstyle\pm 1.77}$
GCN-dDGM∗-HH | 7 | $54.58{\scriptstyle\pm 10.40}$ | $62.02{\scriptstyle\pm 5.73}$ | $86.91{\scriptstyle\pm 0.89}$ | $94.08{\scriptstyle\pm 3.14}$ | $79.88{\scriptstyle\pm 5.77}$
GCN-dDGM∗-SS | 7 | $40.22{\scriptstyle\pm 13.00}$ | $20.54{\scriptstyle\pm 3.42}$ | $40.49{\scriptstyle\pm 1.39}$ | $58.67{\scriptstyle\pm 16.44}$ | $50.94{\scriptstyle\pm 14.36}$
GCN-dDGM∗-EH | 7 | $59.92{\scriptstyle\pm 8.92}$ | $64.79{\scriptstyle\pm 2.32}$ | $86.97{\scriptstyle\pm 0.45}$ | $94.18{\scriptstyle\pm 2.46}$ | $80.82{\scriptstyle\pm 6.38}$
GCN-dDGM∗-ES | 7 | $61.44{\scriptstyle\pm 3.23}$ | $62.41{\scriptstyle\pm 2.88}$ | $86.48{\scriptstyle\pm 0.58}$ | $92.13{\scriptstyle\pm 5.81}$ | $74.84{\scriptstyle\pm 7.24}$
GCN-dDGM∗-HS | 7 | $59.11{\scriptstyle\pm 8.15}$ | $64.86{\scriptstyle\pm 3.03}$ | $85.21{\scriptstyle\pm 4.74}$ | $94.72{\scriptstyle\pm 0.92}$ | $83.23{\scriptstyle\pm 3.95}$
GCN-dDGM∗-EHH | 7 | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}70.37{\scriptstyle\pm 4.72}}}$ | $62.50{\scriptstyle\pm 11.69}$ | $39.93{\scriptstyle\pm 1.35}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}95.16{\scriptstyle\pm 0.37}}}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}91.58{\scriptstyle\pm 0.91}}}$
GCN-dDGM∗-EHS | 7 | $63.30{\scriptstyle\pm 3.93}$ | $64.54{\scriptstyle\pm 1.89}$ | $86.99{\scriptstyle\pm 0.79}$ | $92.85{\scriptstyle\pm 2.49}$ | $86.96{\scriptstyle\pm 2.54}$
GCN-dDGM∗-HH | 10 | $57.59{\scriptstyle\pm 8.05}$ | $59.43{\scriptstyle\pm 5.46}$ | $86.43{\scriptstyle\pm 0.70}$ | $92.50{\scriptstyle\pm 3.35}$ | $73.13{\scriptstyle\pm 8.55}$
GCN-dDGM∗-SS | 10 | $36.30{\scriptstyle\pm 8.23}$ | $21.20{\scriptstyle\pm 2.88}$ | $39.83{\scriptstyle\pm 1.17}$ | $50.52{\scriptstyle\pm 2.77}$ | $53.85{\scriptstyle\pm 16.78}$
GCN-dDGM∗-EH | 10 | $59.63{\scriptstyle\pm 11.04}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}65.49{\scriptstyle\pm 2.86}}}$ | $84.54{\scriptstyle\pm 5.77}$ | $93.55{\scriptstyle\pm 4.53}$ | $72.79{\scriptstyle\pm 5.79}$
GCN-dDGM∗-ES | 10 | $63.04{\scriptstyle\pm 4.21}$ | $62.65{\scriptstyle\pm 6.29}$ | $86.71{\scriptstyle\pm 0.89}$ | $93.42{\scriptstyle\pm 3.22}$ | $74.85{\scriptstyle\pm 7.59}$
GCN-dDGM∗-HS | 10 | $64.24{\scriptstyle\pm 5.38}$ | $62.73{\scriptstyle\pm 2.59}$ | $85.24{\scriptstyle\pm 4.58}$ | $95.03{\scriptstyle\pm 0.64}$ | $75.03{\scriptstyle\pm 6.18}$
GCN-dDGM∗-EHH | 10 | $69.63{\scriptstyle\pm 4.00}$ | $64.70{\scriptstyle\pm 4.61}$ | $39.93{\scriptstyle\pm 1.35}$ | $95.02{\scriptstyle\pm 0.39}$ | $83.81{\scriptstyle\pm 11.41}$
GCN-dDGM∗-EHS | 10 | $63.61{\scriptstyle\pm 3.71}$ | $64.52{\scriptstyle\pm 2.77}$ | $86.60{\scriptstyle\pm 0.66}$ | $90.03{\scriptstyle\pm 4.86}$ | $84.52{\scriptstyle\pm 7.39}$
MLP∗ | N/A | $58.92{\scriptstyle\pm 3.28}$ | $59.48{\scriptstyle\pm 2.14}$ | $85.75{\scriptstyle\pm 1.02}$ | $94.91{\scriptstyle\pm 0.30}$ | $87.80{\scriptstyle\pm 1.54}$
Table 14: Results for classical homophilic datasets combining GCN diffusion
layers with the dDGM module and using product manifold to construct the latent
graphs.
| | Cora | CiteSeer | PubMed | Physics | CS
---|---|---|---|---|---|---
Homophily level | | 0.81 | 0.74 | 0.80 | 0.93 | 0.80
Nodes | | 2,708 | 3,327 | 18,717 | 34,493 | 18,333
Features | | 1,433 | 3,703 | 500 | 8,415 | 6,805
Edges | | 5,278 | 4,676 | 44,327 | 247,962 | 81,894
Classes | | 7 | 6 | 3 | 5 | 15
Average Degree | | 3.9 | 2.77 | 4.5 | 14.38 | 8.93
Model | $k$ | Accuracy $(\%)$ $\pm$ Standard Deviation
GCN-dDGM-HH | 3 | $77.82{\scriptstyle\pm 5.46}$ | $71.27{\scriptstyle\pm 2.09}$ | $87.35{\scriptstyle\pm 0.65}$ | $94.04{\scriptstyle\pm 3.54}$ | $82.91{\scriptstyle\pm 3.00}$
GCN-dDGM-SS | 3 | $67.00{\scriptstyle\pm 10.23}$ | $59.16{\scriptstyle\pm 5.96}$ | $86.83{\scriptstyle\pm 0.60}$ | $90.30{\scriptstyle\pm 5.90}$ | $59.31{\scriptstyle\pm 7.18}$
GCN-dDGM-EH | 3 | $83.65{\scriptstyle\pm 5.25}$ | $72.89{\scriptstyle\pm 1.64}$ | $87.62{\scriptstyle\pm 0.64}$ | $96.07{\scriptstyle\pm 0.27}$ | $91.37{\scriptstyle\pm 1.28}$
GCN-dDGM-ES | 3 | $81.19{\scriptstyle\pm 6.63}$ | $71.87{\scriptstyle\pm 3.20}$ | $86.47{\scriptstyle\pm 2.30}$ | $95.32{\scriptstyle\pm 0.42}$ | $90.87{\scriptstyle\pm 0.82}$
GCN-dDGM-HS | 3 | $80.70{\scriptstyle\pm 2.96}$ | $72.77{\scriptstyle\pm 2.76}$ | $87.29{\scriptstyle\pm 0.85}$ | $96.10{\scriptstyle\pm 0.35}$ | $89.43{\scriptstyle\pm 2.37}$
GCN-dDGM-EHH | 3 | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}86.59{\scriptstyle\pm 3.33}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}75.42{\scriptstyle\pm 2.39}}}$ | $49.17{\scriptstyle\pm 19.39}$ | $96.06{\scriptstyle\pm 0.34}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}92.86{\scriptstyle\pm 0.96}}}$
GCN-dDGM-EHS | 3 | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}85.84{\scriptstyle\pm 2.96}}}$ | $69.98{\scriptstyle\pm 2.70}$ | $87.35{\scriptstyle\pm 0.90}$ | $95.96{\scriptstyle\pm 0.42}$ | $89.93{\scriptstyle\pm 3.86}$
GCN-dDGM-HH | 5 | $76.09{\scriptstyle\pm 7.11}$ | $72.65{\scriptstyle\pm 2.23}$ | $87.36{\scriptstyle\pm 0.78}$ | $95.93{\scriptstyle\pm 0.37}$ | $80.73{\scriptstyle\pm 4.62}$
GCN-dDGM-SS | 5 | $65.96{\scriptstyle\pm 9.46}$ | $65.37{\scriptstyle\pm 5.90}$ | $87.21{\scriptstyle\pm 0.75}$ | $90.40{\scriptstyle\pm 6.90}$ | $70.60{\scriptstyle\pm 9.31}$
GCN-dDGM-EH | 5 | $82.32{\scriptstyle\pm 4.71}$ | $73.56{\scriptstyle\pm 2.75}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}87.67{\scriptstyle\pm 0.76}}}$ | $94.48{\scriptstyle\pm 4.33}$ | $91.11{\scriptstyle\pm 1.57}$
GCN-dDGM-ES | 5 | $81.44{\scriptstyle\pm 5.80}$ | $71.57{\scriptstyle\pm 2.08}$ | $87.41{\scriptstyle\pm 1.14}$ | $96.11{\scriptstyle\pm 0.40}$ | $89.31{\scriptstyle\pm 2.31}$
GCN-dDGM-HS | 5 | $82.59{\scriptstyle\pm 4.50}$ | $72.51{\scriptstyle\pm 2.52}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}87.79{\scriptstyle\pm 1.08}}}$ | $95.13{\scriptstyle\pm 2.41}$ | $90.98{\scriptstyle\pm 1.00}$
GCN-dDGM-EHH | 5 | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}86.63{\scriptstyle\pm 3.25}}}$ | $73.95{\scriptstyle\pm 2.97}$ | $44.53{\scriptstyle\pm 13.84}$ | $96.16{\scriptstyle\pm 0.37}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}92.30{\scriptstyle\pm 1.05}}}$
GCN-dDGM-EHS | 5 | $83.58{\scriptstyle\pm 4.39}$ | $74.00{\scriptstyle\pm 1.68}$ | $87.12{\scriptstyle\pm 0.72}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}96.17{\scriptstyle\pm 0.30}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}92.06{\scriptstyle\pm 0.83}}}$
GCN-dDGM-HH | 7 | $79.70{\scriptstyle\pm 5.55}$ | $71.30{\scriptstyle\pm 2.94}$ | $85.08{\scriptstyle\pm 5.15}$ | $95.99{\scriptstyle\pm 0.44}$ | $86.27{\scriptstyle\pm 2.66}$
GCN-dDGM-SS | 7 | $63.62{\scriptstyle\pm 9.19}$ | $61.39{\scriptstyle\pm 5.61}$ | $87.41{\scriptstyle\pm 0.55}$ | $91.68{\scriptstyle\pm 5.50}$ | $64.05{\scriptstyle\pm 13.45}$
GCN-dDGM-EH | 7 | $82.89{\scriptstyle\pm 4.31}$ | $71.84{\scriptstyle\pm 2.50}$ | $87.40{\scriptstyle\pm 0.59}$ | $96.06{\scriptstyle\pm 0.52}$ | $91.22{\scriptstyle\pm 0.93}$
GCN-dDGM-ES | 7 | $80.85{\scriptstyle\pm 4.75}$ | $71.08{\scriptstyle\pm 4.14}$ | $85.54{\scriptstyle\pm 5.04}$ | $95.24{\scriptstyle\pm 2.88}$ | $89.21{\scriptstyle\pm 2.56}$
GCN-dDGM-HS | 7 | $81.78{\scriptstyle\pm 5.25}$ | $72.41{\scriptstyle\pm 1.86}$ | $87.27{\scriptstyle\pm 0.62}$ | $96.03{\scriptstyle\pm 0.37}$ | $89.90{\scriptstyle\pm 1.06}$
GCN-dDGM-EHH | 7 | $85.65{\scriptstyle\pm 3.54}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}74.67{\scriptstyle\pm 2.01}}}$ | $48.04{\scriptstyle\pm 16.30}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}96.18{\scriptstyle\pm 0.39}}}$ | $90.74{\scriptstyle\pm 2.88}$
GCN-dDGM-EHS | 7 | $83.70{\scriptstyle\pm 3.88}$ | $73.02{\scriptstyle\pm 2.57}$ | $87.61{\scriptstyle\pm 0.67}$ | $96.09{\scriptstyle\pm 0.38}$ | $90.64{\scriptstyle\pm 1.26}$
GCN-dDGM-HH | 10 | $82.18{\scriptstyle\pm 2.90}$ | $72.47{\scriptstyle\pm 2.81}$ | $87.50{\scriptstyle\pm 0.91}$ | $94.73{\scriptstyle\pm 2.83}$ | $86.48{\scriptstyle\pm 0.88}$
GCN-dDGM-SS | 10 | $66.15{\scriptstyle\pm 1.61}$ | $61.51{\scriptstyle\pm 8.14}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}87.82{\scriptstyle\pm 0.59}}}$ | $90.72{\scriptstyle\pm 5.26}$ | $69.62{\scriptstyle\pm 10.14}$
GCN-dDGM-EH | 10 | $81.44{\scriptstyle\pm 3.94}$ | $73.10{\scriptstyle\pm 2.26}$ | $87.41{\scriptstyle\pm 0.80}$ | $96.03{\scriptstyle\pm 0.37}$ | $90.98{\scriptstyle\pm 1.15}$
GCN-dDGM-ES | 10 | $80.07{\scriptstyle\pm 5.40}$ | $72.86{\scriptstyle\pm 2.51}$ | $87.50{\scriptstyle\pm 0.65}$ | $95.41{\scriptstyle\pm 1.73}$ | $90.25{\scriptstyle\pm 1.79}$
GCN-dDGM-HS | 10 | $83.78{\scriptstyle\pm 3.32}$ | $72.52{\scriptstyle\pm 2.82}$ | $85.89{\scriptstyle\pm 4.29}$ | $95.84{\scriptstyle\pm 0.29}$ | $88.64{\scriptstyle\pm 2.97}$
GCN-dDGM-EHH | 10 | $84.53{\scriptstyle\pm 3.84}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}74.33{\scriptstyle\pm 2.30}}}$ | $39.93{\scriptstyle\pm 1.35}$ | $95.63{\scriptstyle\pm 1.36}$ | $91.54{\scriptstyle\pm 2.09}$
GCN-dDGM-EHS | 10 | $83.50{\scriptstyle\pm 4.64}$ | $71.82{\scriptstyle\pm 1.68}$ | $87.05{\scriptstyle\pm 1.38}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}96.21{\scriptstyle\pm 0.44}}}$ | $91.42{\scriptstyle\pm 1.08}$
GCN | N/A | $83.11{\scriptstyle\pm 2.29}$ | $69.97{\scriptstyle\pm 2.06}$ | $85.75{\scriptstyle\pm 1.01}$ | $95.51{\scriptstyle\pm 0.34}$ | $87.28{\scriptstyle\pm 1.54}$
### E.3 More Complex Product Manifolds Results
The main objective of this section is trying to test the limits of our
approach. In Table 15 we display the results multiplying up to five model
spaces for the CS dataset. From the results, we can see that adding more
product manifolds can result in improved performance. The GCN-dDGM-EHHSS
network with $k=5$ obtains an accuracy of $93.10\pm 0.74\%$, as compared to
the best single model space based model GCN-dDGM-E with $k=5$ which achieves a
result of only $87.88\pm 2.55\%$.
Table 15: Results using more complex product manifolds for the CS dataset. We
multiply up to five model spaces to generate the product manifolds.
| | CS
---|---|---
Model | $k$ | Accuracy $(\%)$ $\pm$ Standard Deviation
GCN-dDGM-E | 3 | $85.98{\scriptstyle\pm 2.59}$
GCN-dDGM-H | 3 | $85.95{\scriptstyle\pm 2.58}$
GCN-dDGM-S | 3 | $76.76{\scriptstyle\pm 4.65}$
GCN-dDGM-EH | 3 | $91.37{\scriptstyle\pm 1.28}$
GCN-dDGM-ES | 3 | $90.87{\scriptstyle\pm 0.82}$
GCN-dDGM-HS | 3 | $89.43{\scriptstyle\pm 2.37}$
GCN-dDGM-EHH | 3 | $92.86{\scriptstyle\pm 0.96}$
GCN-dDGM-EHS | 3 | $89.93{\scriptstyle\pm 3.86}$
GCN-dDGM-EHHH | 3 | $92.86{\scriptstyle\pm 1.04}$
GCN-dDGM-EHHS | 3 | $92.70{\scriptstyle\pm 0.66}$
GCN-dDGM-EHHSH | 3 | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}92.96{\scriptstyle\pm 0.46}}}$
GCN-dDGM-EHHSS | 3 | $91.51{\scriptstyle\pm 2.51}$
GCN-dDGM-E | 5 | $87.88{\scriptstyle\pm 2.55}$
GCN-dDGM-H | 5 | $87.65{\scriptstyle\pm 2.56}$
GCN-dDGM-S | 5 | $80.73{\scriptstyle\pm 4.62}$
GCN-dDGM-EH | 5 | $91.11{\scriptstyle\pm 1.57}$
GCN-dDGM-ES | 5 | $89.31{\scriptstyle\pm 2.31}$
GCN-dDGM-HS | 5 | $90.98{\scriptstyle\pm 1.00}$
GCN-dDGM-EHH | 5 | $92.30{\scriptstyle\pm 1.05}$
GCN-dDGM-EHS | 5 | $92.06{\scriptstyle\pm 0.83}$
GCN-dDGM-EHHH | 5 | $92.63{\scriptstyle\pm 0.63}$
GCN-dDGM-EHHS | 5 | $92.82{\scriptstyle\pm 1.34}$
GCN-dDGM-EHHSH | 5 | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}92.91{\scriptstyle\pm 0.66}}}$
GCN-dDGM-EHHSS | 5 | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}93.10{\scriptstyle\pm 0.74}}}$
GCN | N/A | $87.28{\scriptstyle\pm 1.54}$
### E.4 Heterophilic Graph Datasets Extended Results
In Table 16 we display the results using both the dDGM and the dDGM∗ module
for heterophilic datasets. The dDGM module uses the original dataset graph as
inductive bias for the generation of the latent graphs. As expected, this
leads to worse results than those using the dDGM∗ for heterophilic graphs,
since the original graph is not good for diffusion using GCNs. It is better to
start directly from a pointcloud and completely ignore the original adjacency
matrix $\mathbf{A}^{(0)}$ since it does not provide the model with a good
inductive bias.
Table 16: Results for heterophilic datasets combining GCN diffusion layers
with the dDGM∗ latent graph inference system. We display results using model
spaces as well as product manifolds to construct the latent graphs.
| Texas | Wisconsin | Squirrel | Chameleon
---|---|---|---|---
Homophily level | 0.11 | 0.21 | 0.22 | 0.23
Nodes | 183 | 251 | 5,201 | 2,277
Features | 1,703 | 1,703 | 2,089 | 2,325
Edges | 295 | 466 | 198,498 | 31,421
Classes | 5 | 5 | 5 | 5
Average Degree | 3.22 | 3.71 | 76.33 | 27.60
$k$ | 2 | 10 | 3 | 5
Model | Accuracy $(\%)$ $\pm$ Standard Deviation
GCN-dDGM∗-E | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}80.00{\scriptstyle\pm 8.31}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}88.00{\scriptstyle\pm 5.65}}}$ | $34.35{\scriptstyle\pm 2.34}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}48.90{\scriptstyle\pm 3.61}}}$
GCN-dDGM∗-H | $79.44{\scriptstyle\pm 7.88}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}89.03{\scriptstyle\pm 1.89}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}35.00{\scriptstyle\pm 2.35}}}$ | $48.28{\scriptstyle\pm 4.11}$
GCN-dDGM∗-S | $73.88{\scriptstyle\pm 9.95}$ | $85.33{\scriptstyle\pm 4.98}$ | $33.12{\scriptstyle\pm 2.22}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}48.63{\scriptstyle\pm 3.12}}}$
GCN-dDGM∗-HH | $78.89{\scriptstyle\pm 8.53}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}88.00{\scriptstyle\pm 3.26}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}34.38{\scriptstyle\pm 1.07}}}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}48.33{\scriptstyle\pm 4.14}}}$
GCN-dDGM∗-SS | $73.89{\scriptstyle\pm 8.62}$ | $74.66{\scriptstyle\pm 18.85}$ | $34.06{\scriptstyle\pm 2.20}$ | $48.28{\scriptstyle\pm 3.07}$
GCN-dDGM∗-EH | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}81.67{\scriptstyle\pm 7.05}}}$ | $86.67{\scriptstyle\pm 3.77}$ | $34.37{\scriptstyle\pm 1.72}$ | $47.58{\scriptstyle\pm 3.85}$
GCN-dDGM∗-ES | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}81.11{\scriptstyle\pm 10.30}}}$ | $76.00{\scriptstyle\pm 11.31}$ | $33.38{\scriptstyle\pm 1.86}$ | $47.49{\scriptstyle\pm 3.60}$
GCN-dDGM∗-HS | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}81.11{\scriptstyle\pm 9.69}}}$ | $86.67{\scriptstyle\pm 1.89}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}34.65{\scriptstyle\pm 2.45}}}$ | $47.84{\scriptstyle\pm 2.67}$
GCN-dDGM∗-EHH | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}81.11{\scriptstyle\pm 5.09}}}$ | $77.60{\scriptstyle\pm 8.62}$ | $33.19{\scriptstyle\pm 1.92}$ | $44.27{\scriptstyle\pm 2.96}$
GCN-dDGM∗-EHS | $79.44{\scriptstyle\pm 6.11}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}89.33{\scriptstyle\pm 1.89}}}$ | $34.17{\scriptstyle\pm 2.23}$ | $47.58{\scriptstyle\pm 4.54}$
GCN-dDGM-E | $60.56{\scriptstyle\pm 8.03}$ | $70.67{\scriptstyle\pm 10.49}$ | $29.87{\scriptstyle\pm 2.46}$ | $44.19{\scriptstyle\pm 3.85}$
GCN-dDGM-H | $58.89{\scriptstyle\pm 9.36}$ | $72.00{\scriptstyle\pm 5.56}$ | $29.56{\scriptstyle\pm 2.49}$ | $44.01{\scriptstyle\pm 4.08}$
GCN-dDGM-S | $59.44{\scriptstyle\pm 8.62}$ | $62.66{\scriptstyle\pm 16.11}$ | $30.58{\scriptstyle\pm 2.34}$ | $45.46{\scriptstyle\pm 2.35}$
GCN-dDGM-HH | $57.22{\scriptstyle\pm 5.58}$ | $60.00{\scriptstyle\pm 19.87}$ | $30.29{\scriptstyle\pm 1.37}$ | $44.19{\scriptstyle\pm 3.78}$
GCN-dDGM-SS | $59.44{\scriptstyle\pm 6.11}$ | $49.33{\scriptstyle\pm 12.36}$ | $30.15{\scriptstyle\pm 2.40}$ | $45.29{\scriptstyle\pm 1.87}$
GCN-dDGM-EH | $62.78{\scriptstyle\pm 9.31}$ | $65.33{\scriptstyle\pm 4.99}$ | $30.00{\scriptstyle\pm 2.58}$ | $43.09{\scriptstyle\pm 3.42}$
GCN-dDGM-ES | $60.56{\scriptstyle\pm 8.03}$ | $69.33{\scriptstyle\pm 6.80}$ | $30.44{\scriptstyle\pm 2.38}$ | $45.68{\scriptstyle\pm 2.66}$
GCN-dDGM-HS | $57.78{\scriptstyle\pm 10.00}$ | $72.00{\scriptstyle\pm 3.26}$ | $30.06{\scriptstyle\pm 2.66}$ | $43.30{\scriptstyle\pm 4.67}$
GCN-dDGM-EHH | $57.77{\scriptstyle\pm 10.88}$ | $44.80{\scriptstyle\pm 10.55}$ | $28.55{\scriptstyle\pm 4.28}$ | $41.01{\scriptstyle\pm 7.68}$
GCN-dDGM-EHS | $58.89{\scriptstyle\pm 7.53}$ | $76.00{\scriptstyle\pm 3.26}$ | $30.27{\scriptstyle\pm 2.95}$ | $41.15{\scriptstyle\pm 9.84}$
MLP∗ | $77.78{\scriptstyle\pm 10.24}$ | $85.33{\scriptstyle\pm 4.99}$ | $30.44{\scriptstyle\pm 2.55}$ | $40.35{\scriptstyle\pm 3.37}$
GCN | $41.66{\scriptstyle\pm 11.72}$ | $47.20{\scriptstyle\pm 9.76}$ | $24.19{\scriptstyle\pm 2.56}$ | $32.56{\scriptstyle\pm 3.53}$
### E.5 Results for Large Graphs from the Open Graph Benchmark
Table 17 and Table 18 display results for the OGB-Arxiv and OGB-Products
datasets. Although the OGB-Products dataset is considerably larger than the
OGB-Arxiv both in terms of the number of nodes and edges, our models achieve
slightly better performance. This may be due to the fact that the OGB-Products
dataset is an undirected graph, whereas the OGB-Arxiv dataset is directed. The
dDGM module generates undirected edges between nodes by construction (if we
neglect the noise in the Gumbel Top-k trick), which may be affecting
performance in the case of the OGB-Arxiv dataset.
Table 17: Results for OGB-Arxiv dataset using GATv2 diffusion layers and
different latent graph inference modules.
| OGB-Arxiv
---|---
Nodes | 169,343
Features | 128
Edges | 1,166,243
Classes | 40
$k$ | 20
Model | Accuracy $(\%)$ $\pm$ Standard Deviation
GATv2-dDGM∗-E | $64.34{\scriptstyle\pm 0.40}$
GATv2-dDGM∗-H | $61.30{\scriptstyle\pm 0.50}$
GATv2-dDGM∗-S | $64.41{\scriptstyle\pm 0.64}$
GATv2-dDGM∗-HH | $64.24{\scriptstyle\pm 0.17}$
GATv2-dDGM∗-SS | $64.05{\scriptstyle\pm 0.29}$
GATv2-dDGM∗-EH | $64.08{\scriptstyle\pm 1.01}$
GATv2-dDGM∗-ES | $64.13{\scriptstyle\pm 0.61}$
GATv2-dDGM∗-HS | $64.45{\scriptstyle\pm 0.12}$
GATv2-dDGM∗-EHS | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}65.06{\scriptstyle\pm 0.09}}}$
GATv2-dDGM-E | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}64.65{\scriptstyle\pm 0.01}}}$
GATv2-dDGM-H | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}65.05{\scriptstyle\pm 0.10}}}$
GATv2-dDGM-S | $64.60{\scriptstyle\pm 0.16}$
GATv2-dDGM-HH | $64.00{\scriptstyle\pm 0.36}$
GATv2-dDGM-SS | $64.35{\scriptstyle\pm 0.36}$
GATv2-dDGM-EH | $64.00{\scriptstyle\pm 0.76}$
GATv2-dDGM-ES | $64.37{\scriptstyle\pm 0.04}$
GATv2-dDGM-HS | $61.00{\scriptstyle\pm 1.12}$
GATv2-dDGM-EHS | $64.25{\scriptstyle\pm 0.50}$
MLP∗ | $63.49{\scriptstyle\pm 0.15}$
GATv2 | $61.93{\scriptstyle\pm 1.62}$
Table 18: Results for OGB-Products dataset using GATv2 diffusion layers and
different latent graph inference modules.
| | OGB-Products
---|---|---
Nodes | | 2,449,029
Features | | 100
Edges | | 61,859,140
Classes | | 47
Model | $k$ | Accuracy $(\%)$ $\pm$ Standard Deviation
GATv2-dDGM-E | 3 | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}66.59{\scriptstyle\pm 0.30}}}$
GATv2-dDGM-H | 3 | $62.22{\scriptstyle\pm 0.25}$
GATv2-dDGM-EH | 3 | $65.51{\scriptstyle\pm 0.30}$
GATv2-dDGM-E | 5 | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}66.25{\scriptstyle\pm 0.71}}}$
GATv2-dDGM-H | 5 | $63.95{\scriptstyle\pm 0.42}$
GATv2-dDGM-EH | 5 | $65.62{\scriptstyle\pm 0.20}$
MLP∗ | N/A | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}66.05{\scriptstyle\pm 0.20}}}$
GATv2 | N/A | $62.02{\scriptstyle\pm 2.60}$
### E.6 Inductive Learning: Results for the QM9 and Alchemy datasets
The datasets discussed in the main task were solely concerned with tranductive
learning. For completeness, we show that the latent graph inference system
based on product manifolds is also applicable to inductive learning. Molecules
are naturally represented as graphs, with atoms as nodes and bonds as edges.
Prediction of molecular properties is a popular application of GNNs in
chemistry Wieder et al. (2020); Stark et al. (2021); Li et al. (2021); Godwin
et al. (2021); Zhang et al. (2021). Specifically, we work with the QM9
(Ramakrishnan et al. (2014); Ruddigkeit et al. (2012)) and Alchemy (Morris et
al. (2020)) datasets which are well known in the Geometric Deep Learning
literature. Table 19 displays the results. These tasks are substantially
different to the ones previously discussed because they involve inductive
learning and regression, whereas before all tasks focused on transductive
learning and multi-class classification.
Table 19: Results for the QM9 and Alchemy datasets using the dDGM module.
| QM9 | Alchemy
---|---|---
No. graphs | 133,885 | 119,487
Targets | 12 | 12
$k$ | 5 | 5
Model | R2 score ($\times 100$) $\pm$ Standard Deviation
GCN-dDGM∗-E | $96.23{\scriptstyle\pm 1.55}$ | $96.11{\scriptstyle\pm 1.43}$
GCN-dDGM∗-H | $97.50{\scriptstyle\pm 1.04}$ | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}96.12{\scriptstyle\pm 1.59}}}$
GCN-dDGM∗-S | $96.52{\scriptstyle\pm 2.30}$ | $94.52{\scriptstyle\pm 2.45}$
GCN-dDGM∗-HH | $96.53{\scriptstyle\pm 2.03}$ | $95.06{\scriptstyle\pm 1.59}$
GCN-dDGM∗-SS | $95.51{\scriptstyle\pm 1.23}$ | $92.01{\scriptstyle\pm 2.04}$
GCN-dDGM∗-EH | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}97.79{\scriptstyle\pm 1.24}}}$ | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}96.52{\scriptstyle\pm 1.89}}}$
GCN-dDGM∗-ES | $94.03{\scriptstyle\pm 2.13}$ | $94.10{\scriptstyle\pm 2.54}$
GCN-dDGM∗-HS | $96.59{\scriptstyle\pm 1.53}$ | $96.00{\scriptstyle\pm 1.01}$
GCN-dDGM∗-EHS | $96.78{\scriptstyle\pm 1.56}$ | $96.03{\scriptstyle\pm 2.59}$
GCN-dDGM-E | $\mathbf{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}97.79{\scriptstyle\pm 1.34}}}$ | $96.10{\scriptstyle\pm 2.01}$
GCN-dDGM-H | $96.69{\scriptstyle\pm 1.34}$ | $96.11{\scriptstyle\pm 1.04}$
GCN-dDGM-S | $95.53{\scriptstyle\pm 2.30}$ | $91.98{\scriptstyle\pm 2.40}$
GCN-dDGM-HH | $96.78{\scriptstyle\pm 1.05}$ | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}96.45{\scriptstyle\pm 2.54}}}$
GCN-dDGM-SS | $96.45{\scriptstyle\pm 2.32}$ | $93.67{\scriptstyle\pm 3.01}$
GCN-dDGM-EH | $\mathbf{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}98.00{\scriptstyle\pm 1.34}}}$ | $96.04{\scriptstyle\pm 1.45}$
GCN-dDGM-ES | $96.52{\scriptstyle\pm 1.54}$ | $95.02{\scriptstyle\pm 1.23}$
GCN-dDGM-HS | $96.51{\scriptstyle\pm 1.23}$ | $95.39{\scriptstyle\pm 1.01}$
GCN-dDGM-EHS | $\mathbf{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}97.99{\scriptstyle\pm 1.02}}}$ | $96.02{\scriptstyle\pm 2.10}$
MLP∗ | $82.20{\scriptstyle\pm 3.80}$ | $76.56{\scriptstyle\pm 1.50}$
GCN | $94.35{\scriptstyle\pm 1.30}$ | $95.15{\scriptstyle\pm 1.43}$
## Appendix F Latent Graph Learning Plots
In this appendix we include additional plots for the learned latent graphs for
the heterophilic datasets discussed in the main text as well as for the
TadPole and the Aerothermodynamics dataset.
### F.1 Learned Latent Graphs for Heterophilic Datasets
In Figure 6, 7, 8, and 9, we display the original graphs provided by the
heterophilic datasets, and compare them to the latent graphs generated by the
dDGM modules. From the plots we can clearly see the high homophily levels of
the Texas latent graph in Figure 6, for which we obtain four distinct
clusters. In the case of the bigger datasets in Figure 8, and 9, the algorithm
is still able to create clusters but there is mixing between classes.
(a) Original graph, $h=0.11$.
(b) Learned latent graph, $h=0.93$.
Figure 6: Texas, original vs learned latent graph. The learned latent graph
displayed was learned using the GCN-dDGM∗-EH model with $k=2$ in the Gumbel
Top-k trick.
(a) Original graph, $h=0.21$.
(b) Learned latent graph, $h=0.64$.
Figure 7: Wisconsin, original vs learned latent graph. The learned latent
graph displayed was learned using the GCN-dDGM∗-EHS model with $k=10$ in the
Gumbel Top-k trick.
(a) Original graph, $h=0.22$.
(b) Learned latent graph, $h=0.32$.
Figure 8: Squirrel, original vs learned latent graph. The learned latent graph
displayed was learned using the GCN-dDGM∗-S model with $k=3$ in the Gumbel
Top-k trick. Note that both graphs (a) and (b) have the same number of nodes,
but in (a) nodes are displayed more closely packed due to the connectivity
structure of the graph.
(a) Original graph, $h=0.23$.
(b) Learned latent graph, $h=0.42$.
Figure 9: Chameleon, original vs learned latent graph. The learned latent
graph displayed was learned using the GCN-dDGM∗-E model with $k=3$ in the
Gumbel Top-k trick. Note that both graphs (a) and (b) have the same number of
nodes, but in (a) nodes are displayed more closely packed due to the
connectivity structure of the graph.
Note that the generated latent graphs are dependent on both the downstream
task and the diffusion layers we are using, that is, the GCNs. Since we have
created a fully-differentiable system the models are optimizing the latent
graph generation together with the rest of the model parameters. Hence, if we
were to change the downstream task or the diffusion layers we would expect
different latent graphs.
In Figure 2 from Section 4.1 we showed the latent graph learning evolution
plots as a function of training epochs for the Texas dataset. We provide
additional plots for other datasets. This shows that the latent graph
inference system is applicable across a wide range of datasets and that is
learns to organize the connectivity of the latent graph during training.
(a) Epoch 1, $h=0.34$.
(b) Epoch 5, $h=0.43$.
(c) Epoch 10, $h=0.46$.
(d) Epoch 100, $h=0.52$.
(e) Epoch 500, $h=0.62$.
(f) Epoch 1000, $h=0.64$.
Figure 10: Latent graph homophily level, $h$, evolution as a function of
training epochs for Wisconsin. The latent graphs are produced during the
training process for the GCN-dDGM∗-EHS model with $k=10$.
(a) Epoch 1, $h=0.20$.
(b) Epoch 100, $h=0.26$.
(c) Epoch 500, $h=0.31$.
(d) Epoch 1000, $h=0.32$.
Figure 11: Latent graph homophily level, $h$, evolution as a function of
training epochs for Squirrel using the GCN-dDGM∗-S model with $k=3$. Recall
that each node in the graph is represented with a point and each class is
assigned a different color. In (a) there is no structure, after 1,000 epochs
in (d) the algorithm has been able to organize the graph structure to separate
some of the classes, but there is still a substantial amount of mixing.
(a) Epoch 1, $h=0.19$.
(b) Epoch 100, $h=0.36$.
(c) Epoch 500, $h=0.40$.
(d) Epoch 1000, $h=0.42$.
Figure 12: Latent graph homophily level, $h$, evolution as a function of
training epochs for Chameleon using the GCN-dDGM∗-ES model with $k=5$.
Lastly, we display the learned latent graphs using original heterophilic graph
as inductive bias for both the Texas and Wisconsin datasets. In Figure 13 and
Figure 14, we compare the final inferred latent graphs for the datasets when
the original heterophilic dataset graph is used as inductive bias, against
starting from a pointcloud. Models that use the dDGM module, which makes use
of the original graph, are not able to achieve high homophily levels as
compared to those using the dDGM∗ module and ignoring the original graph. This
explains the difference in performance in Table 16 from Appendix E.4.
(a) $h=0.93$.
(b) $h=0.60$.
Figure 13: Comparison of final inferred latent graph (after 1,000 epochs of
training) for the Texas dataset by (a) the GCN-dDGM∗-E and (b) the GCN-dDGM-E
model, both with $k=2$.
(a) $h=0.65$.
(b) $h=0.36$.
Figure 14: Comparison of final inferred latent graph (after 1,000 epochs of
training) for the Wisconsin dataset by (a) the GCN-dDGM∗-EHS and (b) the GCN-
dDGM-EHS model, both with $k=10$.
### F.2 Learned Latent Graphs for Real-World datasets
Next we displayed the learned latent graph for the TadPole and the
Aerothermodynamics datasets, see Figure 15, Figure 16, Figure 17, and Figure
18.
(a) Epoch 1, $h=0.44$.
(b) Epoch 5, $h=0.42$.
(c) Epoch 10, $h=0.46$.
(d) Epoch 100, $h=0.72$.
(e) Epoch 400, $h=0.88$.
(f) Epoch 800, $h=0.92$.
Figure 15: Latent graph homophily level, $h$, evolution as a function of
training epochs for TadPole. The latent graphs shown here were obtain using
the GCN-dDGM∗-H model with $k=3$.
(a) Epoch 1, $h=0.43$.
(b) Epoch 10, $h=0.46$.
(c) Epoch 100, $h=0.65$.
(d) Epoch 800, $h=0.90$.
Figure 16: Latent graph homophily level, $h$, evolution as a function of
training epochs for TadPole obtained using the GCN-dDGM∗-EHS model with $k=7$.
Nodes with different colors correspond to different target classes. Starting
from a unstructured graph in (a), the algorithm is able to generate a highly
homophilic graph after 800 epochs in (d).
(a) Epoch 1, $h=0.35$.
(b) Epoch 10, $h=0.87$.
(c) Epoch 500, $h=0.86$.
(d) Epoch 1000, $h=0.88$.
Figure 17: Latent graph homophily level, $h$, evolution as a function of
training epochs for Aerothermodynamics. The latent graphs shown here were
obtain using the GAT-dDGM∗-EHH model with $k=7$.
(a) Epoch 1, $h=0.85$.
(b) Epoch 10, $h=0.79$.
(c) Epoch 500, $h=0.84$.
(d) Epoch 1000, $h=0.87$.
Figure 18: Latent graph homophily level, $h$, evolution as a function of
training epochs for Aerothermodynamics. The latent graphs shown here were
obtain using the GAT-dDGM∗-H model with $k=7$.
## Appendix G Model Architectures
For reproducibility, in this appendix we include a summary of all the models
used to obtain the results discussed in this paper. This includes the GNN
network archictectures as well as the internal structure of the dDGM modules.
### G.1 Network Architectures
In this section we provide the neural network architectures used for the
experimental validation as well as other training specifications.
#### G.1.1 Networks for Classical Graph Datasets
All neural network models used for homophilic graph datasets, MLP, GCN, and
GCN-dDGMs, follow the architecture depicted in Table 20. We apply a learning
rate of $lr=10^{-2}$ and a weight decay of $wd=10^{-4}$. Models are trained
for about 1,500 epochs.
Table 20: Summary of model architectures for experiments using classical graph
datasets. Most experiments only use the first dDGM module.
| | Model
---|---|---
| | MLP | GCN | GCN-dDGM
No. Layer parameters | Activation | Layer type
| | N/A | N/A | dDGM
(No. features, 32) | ELU | Linear | Graph Conv | Graph Conv
| | N/A | N/A | dDGM
(32, 16) | ELU | Linear | Graph Conv | Graph Conv
| | N/A | N/A | dDGM
(16, 8) | ELU | Linear | Graph Conv | Graph Conv
(8, 8) | ELU | Linear | Linear | Linear
(8, 8) | ELU | Linear | Linear | Linear
(8, No. classes) | - | Linear | Linear | Linear
#### G.1.2 Networks for Heterophilic Graph Datasets
The networks used for heterophilic datasets follow the architecture summarized
in Table 21. We apply $lr=10^{-2}$ and $wd=10^{-3}$, and we train the models
for about 1,000 epochs.
Table 21: Summary of model architectures for experiments using heterophilic
graph datasets. The dDGM module uses different $k$ values.
| | | Model
---|---|---|---
| | | MLP | GCN | GCN-dDGM
No. Layer parameters | BatchNorm | Activation | Layer type
| | | N/A | N/A | dDGM
(No. features, 16) | No | ELU | Linear | Graph Conv | Graph Conv
(16, 8) | No | ELU | Linear | Graph Conv | Graph Conv
(8, 8) | Yes | ELU | Linear | Linear | Linear
(8, No. classes) | No | - | Linear | Linear | Linear
#### G.1.3 Networks for OGB-Arxiv
Table 22 shows the networks used for the OGB-Arxiv dataset. We use
$lr=10^{-3}$, $wd=0$, and train for 100 epochs. For graph subsampling, we
sample up to 1,000 neighbours per node and use a batch size of 1,000.
Table 22: Summary of model architectures for experiments for the OGB-Arxiv
dataset using GATv2 diffusion layers. As specified in the table, $k=20$ is
used for all experiments.
| | | Model
---|---|---|---
| | | MLP | GATv2 | GATv2-dDGM
No. Layer parameters | BatchNorm | Activation | Layer type
| | | N/A | N/A | dDGM ($k=20$)
($\textrm{No. features}=128$, 40) | No | SiLU | Linear | Graph Attention v2 | Graph Attention v2
(40, 40) | No | SiLU | Linear | Graph Attention v2 | Graph Attention v2
(40, 40) | No | SiLU | Linear | Graph Attention v2 | Graph Attention v2
(40, 40) | Yes | SiLU | Linear | Linear | Linear
(40, 100) | Yes | SiLU | Linear | Linear | Linear
(100, $\textrm{No. classes}=40$) | No | - | Linear | Linear | Linear
#### G.1.4 Networks for OGB-Products
Table 23 shows the networks used for the OGB-Products dataset. We use
$lr=10^{-2}$, $wd=0$, and train for 30 epochs. For graph subsampling, we
sample up to 200 neighbors per node and use a batch size of 1,000.
Table 23: Summary of model architectures for experiments for the OGB-Products
dataset using GATv2 diffusion layers, with $k=3$ for the dDGM module.
| | | Model
---|---|---|---
| | | MLP | GATv2 | GATv2-dDGM
No. Layer parameters | BatchNorm | Activation | Layer type
| | | N/A | N/A | dDGM ($k=3$)
($\textrm{No. features}=100$, 100) | No | SiLU | Linear | Graph Attention v2 | Graph Attention v2
(100, 100) | No | SiLU | Linear | Graph Attention v2 | Graph Attention v2
(100, 47) | No | SiLU | Linear | Graph Attention v2 | Graph Attention v2
(47, $\textrm{No. classes}=47$) | No | - | Linear | Linear | Linear
#### G.1.5 Networks for Inductive Learning
Table 24 summarizes the networks used for the QM9 and Alchemy datasets. We use
$lr=5\times 10^{-2}$, $wd=10^{-3}$, and train for 30 to 60 epochs.
Table 24: Summary of model architectures for experiments for the QM9 and
Alchemy datasets. These models incorporate a global mean pool layer since
predictions are at the graph level, rather than at the node level as in all
other datasets.
| | | Model
---|---|---|---
| | | MLP | GCN | GCN-dDGM
No. Layer parameters | BatchNorm | Activation | Layer type
| | | N/A | N/A | dDGM
(No. features, 20) | Yes (GraphNorm) | SiLU | Linear | Graph Conv | Graph Conv
(20, 20) | Yes (GraphNorm) | SiLU | Linear | Graph Conv | Graph Conv
| | | Global Mean Pool
(20, 20) | No | SiLU | Linear | Linear | Linear
(20, 20) | No | SiLU | Linear | Linear | Linear
(20, No. prediction targets) | No | - | Linear | Linear | Linear
#### G.1.6 Networks for Brain Imaging
Table 25 summarizes the networks used for the TadPole dataset. We use
$lr=10^{-3}$ and $wd=2\times 10^{-4}$, and train for about 800 epochs.
Table 25: Summary of model architectures for TadPole.
| | | Model
---|---|---|---
| | | MLP | GCN-dDGM | GAT-dDGM
No. Layer parameters | BatchNorm | Activation | Layer type
| | | N/A | dDGM | dDGM
(No. features, 32) | No | ELU | Linear | Graph Conv | Graph Attention
(32, 16) | No | ELU | Linear | Graph Conv | Graph Attention
(16, 8) | No | ELU | Linear | Graph Conv | Graph Attention
(8, 8) | Yes | ELU | Linear | Linear | Linear
(8, 8) | Yes | ELU | Linear | Linear | Linear
(8, No. classes) | No | - | Linear | Linear | Linear
#### G.1.7 Networks for Aerospace Engineering
Table 26 summarizes the networks used for the Aerothermodynamics dataset. We
use $lr=10^{-2}$ and $wd=0$, and train for about 1,000 epochs.
Table 26: Summary of model architectures for Aerothermodynamics.
| | | Model
---|---|---|---
| | | MLP | GCN-dDGM | GAT-dDGM
No. Layer parameters | BatchNorm | Activation | Layer type
| | N/A | N/A | dDGM | dDGM
(No. features, 32) | No | ELU | Linear | Graph Conv | Graph Attention
(32, 16) | No | ELU | Linear | Graph Conv | Graph Attention
(16, 8) | No | ELU | Linear | Graph Conv | Graph Attention
(8, 8) | Yes | ELU | Linear | Linear | Linear
(8, 8) | Yes | ELU | Linear | Linear | Linear
(8, No. classes) | No | - | Linear | Linear | Linear
### G.2 dDGM Architectures
The dDGM and dDGM∗ internal parameterized mapping functions are slightly
modified for different datasets. All architectures are recorded in Table 27,
28, 29, 30, 31, 32, and 33.
Table 27: dDGM∗ and dDGM architectures for classical homophilic datasets.
| | dDGM∗ | dDGM
---|---|---|---
No. Layer parameters | Activation | Layer type
(No. features, 32) | ELU | Linear | Linear
(32, 16 per model space) | ELU | Linear | Graph Conv
(16 per model space, 4 per model space) | Sigmoid | Linear | Graph Conv
Table 28: dDGM∗ and dDGM architectures for heterophilic datasets.
| | | dDGM∗ | dDGM
---|---|---|---|---
No. Layer parameters | BatchNorm | Activation | Layer type
(No. features, 32) | Yes | ELU | Linear | Linear
(32, 4 per model space) | Yes (dDGM∗)/ No (dDGM) | ELU | Linear | Graph Conv
(4 per model space, 4 per model space) | Yes (dDGM∗)/ No (dDGM) | Sigmoid | Linear | Graph Conv
Table 29: dDGM∗ and dDGM architectures for OGB-Arxiv.
| | | dDGM∗ | dDGM
---|---|---|---|---
No. Layer parameters | BatchNorm | Activation | Layer type
(No. features, 40) | Yes | SiLU | Linear | Linear
(40, 32 per model space) | Yes (dDGM∗)/ No (dDGM) | SiLU | Linear | Graph Conv
(32 per model space, 16 per model space) | Yes (dDGM∗)/ No (dDGM) | SiLU | Linear | Graph Conv
Table 30: dDGM∗ and dDGM architectures for OGB-Products.
| | | dDGM∗ | dDGM
---|---|---|---|---
No. Layer parameters | BatchNorm | Activation | Layer type
(No. features, 32 per model space) | Yes (dDGM∗)/ No (dDGM) | SiLU | Linear | Graph Conv
(32 per model space, 16 per model space) | Yes (dDGM∗)/ No (dDGM) | SiLU | Linear | Graph Conv
Table 31: dDGM∗ and dDGM architectures for QM9 and Alchemy.
| | | dDGM∗ | dDGM
---|---|---|---|---
No. Layer parameters | BatchNorm | Activation | Layer type
(No. features, 3 per model space) | Yes | SiLU | Linear | Graph Conv
(3 per model space, 3 per model space) | Yes | SiLU | Linear | Graph Conv
Table 32: dDGM∗ and dDGM architectures for TadPole.
| | | dDGM∗ | dDGM
---|---|---|---|---
No. Layer parameters | BatchNorm | Activation | Layer type
(No. features, 16 per model space) | Yes (dDGM∗)/ No (dDGM) | ELU | Linear | Graph Conv
(16 per model space, 4 per model space) | Yes (dDGM∗)/ No (dDGM) | Sigmoid (dDGM∗)/ ELU (dDGM) | Linear | Graph Conv
Table 33: dDGM∗ and dDGM architectures for Aerothermodynamics.
| | | dDGM∗ | dDGM
---|---|---|---|---
No. Layer parameters | BatchNorm | Activation | Layer type
(No. features, 32) | No | ELU | Linear | Linear
(32, 16 per model space) | Yes (dDGM∗)/ No (dDGM) | ELU | Linear | Graph Conv
(16 per model space, 4 per model space) | Yes (dDGM∗)/ No (dDGM) | Sigmoid (dDGM∗)/ ELU (dDGM) | Linear | Graph Conv
|
# Generic Networks of Votings
Ewa Zawiślak-Sprysak
USUMA GmbH Institut für Marktforschung und Sozialforschung
E-mail<EMAIL_ADDRESS>
Paweł Zawiślak
Department of Mathematics and Mathematical Economics
SGH Warsaw School of Economics
E-mail<EMAIL_ADDRESS>
###### Abstract
In this paper, we analyse results of the $15^{\textrm{th}}$ International
Henryk Wieniawski Violin Competition by comparing the properties of its
results network to the properties of _generic networks of votings_.
Suppose that a competition Comp is given. In this competition, $k$ contestants
are rated by $n$ jurors. Suppose that the Borda count is used as a voting
method, i.e. every juror gives $k$ points for the best contestant, $k-1$
points for the second best contestant, and so on. In particular, the worts
contestant gets $1$ point.
For such a competition, we create a weighted network $N(\textrm{{Comp}})$ in
the following way. The node set of $N(\textrm{{Comp}})$ corresponds to jurors
and the link set of $N(\textrm{{Comp}})$ consists of all links
$\\{J_{s},J_{t}:s\neq t\\}$. For link $l_{st}$ connecting nodes $J_{s}$ and
$J_{t}$, we assign the weight $w(l_{st})=w_{st}$, where
$w_{st}=\mathcal{LF}_{2}(\alpha_{s}\alpha_{t}^{-1})\textrm{.}$
Here $\mathcal{LF}_{2}$ is a Lehmer norm on the permutation group $S_{k}$,
whereas $\alpha_{s}$ and $\alpha_{t}$ denotes the votes of jurors $J_{s}$ and
$J_{t}$, respectively. In particular, for $i=1,2,\ldots,k$, $\alpha_{s}(i)$ is
the number of points given to the $i$-th contestant by juror $J_{s}$. The
similar holds for juror $J_{t}$. Note that $\alpha_{s}$ and $\alpha_{t}$ can
be considered as elements of $S_{k}$.
Suppose now that the probability measure $\mathbb{P}$ is given on space $V$ of
all possible votings of a single juror, i.e. on space $S_{k}$. Suppose that
every juror votes independently according to $\mathbb{P}$. We repeat such a
voting process $100$ times and for every $j=1,2,\ldots,100$, we create a
network $N_{j}$ in the way described above.
In this paper, we compare some statistical properties of networks $N_{j}$, for
probability measures $\mathbb{P}$ being the convex combinations of two Dirac
probability measures and a uniform probability measure, to the properties of
network of jurors’ votings in the $2016$ Wieniawski Competition.
†† _JEL Classification System_ : C02, C15, D71, D85.†† _Key words and phrases_
: generic network, Lehmer norm, Borda count.
## 1 Introduction
There where many controversies concerning the results of the
$15^{\textrm{th}}$ International Henryk Wieniawski Violin Competition (2016).
Both Gazeta Wyborcza, one of the most popular Polish newspaper, see [D], and
Ruch Muzyczny, the most significant Polish music journal, see [JC], raised the
possibility that the jurors of the most recent Wieniawski Competition formed
cliques.
The results of the 2016 Wieniawski Competition were analysed in [SZ], where
they were compared to the results of $16^{\textrm{th}}$ (2010) and
$17^{\textrm{th}}$ (2015) International Chopin Piano Competitions. The metods
of network theory, see for example [J], [Ne], [NBW] and [W], were applied to
compare the voting results of the three aforementioned music competitions. For
these three competitions weighted networks, see [H], $W^{2016}$, $C^{2010}$
and $C^{2015}$ were created and some numerical properties of these network
were compared.
Weighted networks are used in biology, see [H], in stock markets analysis, see
[CGQS] and [GDKO], as well as in the studies on the structural and functional
organisation of the human brain, see [PF]. Usually, the weight of link
$l_{st}$ connecting nodes $s$ and $t$ are given by some kind of a correlation
coefficient related to some rankings or processes asocciated to nodes. In
[SZ], the weights $w_{st}$ of links $l_{st}$ were given by $w_{st}=\tau_{st}$,
where $\tau_{st}$ were _Kendall’s $\tau$ coefficients_, see [K1], [K2] and
[A], of the voting results of jurors $J_{s}$ and $J_{t}$. For stock market
networks, this correlation was measured by the _Pearson correlation
coefficient_ , see [BS].
In the case of rankings, many measures of disarray has been studied in the
literature. In the case of the _Borda count_ method of voting, see [HM], [Nu]
and [O] for a description of this method, votes can be regarded as elements of
permutations group $S_{k}$ ($k$ is a number of contestants). The best known
measures of dissarray are _Kendall’s $\tau$ correlation coefficient_ and
_Spearman’s $\rho$ correlation coefficient_, see [S], as well as metric
measures such as _Kendall distance_ , _Spearman distance_ , _Hamming distance_
and _Footrule distance_ , see [DG] and [QY]. The weighted versions of the
Kendall distance and the Footrule distance were considered in [KV] and [PP].
The classical measure of dissarray mentioned above has such a property that
changing the first two positions in the Borda ranking has the same impact on
the measure as changing the last two positions. On the other hand, the
weighted generalisations of the Kendall distance and the Footrule distance
proposed in [KV] fail to be metrics (for some choices of weights).
In [Z], the _Lehmer factorial norm_ is considered. This is a symmetric,
right–invariant norm on the permutation group $S_{n}$ satisfying the triangle
inequality and thus determining the metric on $S_{n}$. Additionally, this norm
allows for distinguishing changes in the first positions and in the last
positions of rankings. We use this norm to create networks related to the
results of the $15^{\textrm{th}}$ International Henryk Wieniawski Violin
Competition. We compare the properties of these networks to the average
properties of networks related to randomly choosen results of a competition
with the same numbers of contestants and jurors.
This article is organised in the following way. In Section 2, we present basic
definitions concernig permutation groups and we set the notations used in this
paper. We also recall the definition and basic properties of the Lehmer
factorial norm. In Section 3, we analyse the results of the $15^{\textrm{th}}$
International Henryk Wieniawski Violin Competition using the network approach.
In Section 4, we describe the procedure of generating random networks of
votings. These _generic networks_ are used later in Section 5 for determinig
the model of generating random networks with properties best fitting the
properties of the network related to the results of the $15^{\textrm{th}}$
International Henryk Wieniawski Violin Competition. Section 6 contains
conclusions and open questions related to the subject of these studies. All
tables and figures are included in Section 7.
## 2 Basic definitions and notations, the Lehmer norm
In this section, we presrent some basic definitions used in this paper and we
set some notations. We also refer to the definition and basic properties of
the Lehmer factorial norm.
For a natural number $n>0$ by $[n]$, we denote the set $\\{1,2,\ldots,n\\}$
and by $S_{n}$ – the group of all permutations of $[n]$. Permutation
$\sigma\in S_{n}$ is denoted by
$\sigma=(\sigma(1),\sigma(2),\ldots,\sigma(n))\textrm{.}$
In particular $\varepsilon_{n}=(1,2,\ldots,n)$ denotes the identity
permutation.
By $\sigma^{-1}$, we denote the inverse permutation to $\sigma$, and by
$\sigma\tau$ – the composition of $\sigma$ and $\tau$, defined by
$(\sigma\tau)(i)=\sigma(\tau(i))$ for $i=1,2,\ldots,n$. By $\bar{\sigma}$, we
denote the reverse permutation to $\sigma$ given by
$\bar{\sigma}(i)=\sigma(n+1-i)$ for $i=1,2,\ldots,n$.
For $s=1,2,\ldots,n-1$ let
$\alpha_{n}^{s}=(1,2,\ldots,s-1,s+1,s,s+2,\ldots,n)\textrm{,}$
so $\alpha_{n}^{s}$ is adjacent transposition, $(s,s+1)$ in the cycle
notation.
For permutation $\sigma\in S_{n}$, its _Lehmer code_
$\operatorname{lc}(\sigma)$, see [L1], [L2] and [G], is defined by
$\operatorname{lc}(\sigma)=[c_{1}(\sigma),c_{2}(\sigma),\ldots,c_{n}(\sigma)]$
where numbers $c_{i}(\sigma)$ are given by
$c_{i}(\sigma)=\left|\\{j\in[n]:j>i\textrm{ and
}\sigma(j)<\sigma(i)\\}\right|$
for $i=1,2,\ldots,n$.
###### Definition 2.1 (Definition 3.4 in [Z]).
Let $\sigma\in S_{n}$ be a permutation with the Lehmer code
$\operatorname{lc}(\sigma)=[c_{1}(\sigma),c_{2}(\sigma),\ldots,c_{n}(\sigma)]\textrm{.}$
_Lehmer factorial norm_ $\mathcal{LF}_{2}:S_{n}\to\mathbb{N}$ is given by
$\mathcal{LF}_{2}(\sigma)=\sum_{i=1}^{n}\left[2^{n-i}-2^{n-i-
c_{i}(\sigma)}\right]\textrm{.}$
In the next theorem, we refer to some basic properties of the Lehmer norm.
###### Theorem 2.2 (Theorem 3.6 in [Z]).
Norm $\mathcal{LF}_{2}$ satisfies the following:
1. (i)
$\mathcal{LF}_{2}(\varepsilon_{n})=0$ is minimal and $\varepsilon_{n}$ is the
only permutation with this property.
2. (ii)
$\mathcal{LF}_{2}(\bar{\varepsilon}_{n})=2^{n}-(n+1)$ is maximal and
$\bar{\varepsilon}_{n}$ is the only permutation with this property.
3. (iii)
$\mathcal{LF}_{2}(\alpha_{n}^{s})=2^{n-1-s}$ for $s=1,2,\ldots,n-1$, and
therefore
$\mathcal{LF}_{2}(\alpha_{n}^{1})>\mathcal{LF}_{2}(\alpha_{n}^{2})>\ldots>\mathcal{LF}_{2}(\alpha_{n}^{n-1})\textrm{.}$
4. (iv)
$\mathcal{LF}_{2}(\sigma)=\mathcal{LF}_{2}(\sigma^{-1})$ for all $\sigma\in
S_{n}$.
5. (v)
$\mathcal{LF}_{2}(\sigma\tau)\leq\mathcal{LF}_{2}(\sigma)+\mathcal{LF}_{2}(\tau)$
for all $\sigma,\tau\in S_{n}$.
Note that properties (i), (iv) and (v) imply that $\mathcal{LF}_{2}$
determines the metric on $S_{n}$. Indeed, the function
$d_{\textrm{L}}:S_{n}\times S_{n}\to\mathbb{N}$ given by
$d_{\textrm{L}}(\sigma,\tau)=\mathcal{LF}_{2}(\sigma\tau^{-1})$
is a metric. We call it the _Lehmer distance_.
Note also that $d_{\textrm{L}}$, considered as a distance on rankings,
distinguishes changes in high places in the competitions from changes in low
places.
## 3 Results of the $15^{\textrm{th}}$ Wieniawski Competition. A network
approach
In this section, we analyse the results of the $15^{\textrm{th}}$
International Henryk Wieniawski Violin Competition using the network approach.
In this paper we, consider simply undirected networks. A _(simply undirected)
network_ is a pair $N=(N(N),L(N))$ consisting of set $N(N)$ of _nodes_ ,
usually finite, and set $L(N)$ of _links_ , where every link $l\in L(N)$ is a
subset of $N(N)$ consisting of two different elements. Networks are often
called _graphs_ in the literature, nodes and links - _vertices_ and _edges_ ,
_sites_ and _bonds_ , or _actors_ and _ties_ , respectively. Let $N$ be a
network, and suppose that there is a map $w:L(N)\to\mathbb{R}$. Triple
$(N(N),L(N),w)$ is called a _weighted network_.
A good introduction to the concept of networks can be found in [Ne] and [NBW],
whereas [W] contains the same ideas described in the languange of graphs.
Methods of weighted networks can be found in [H].
There were $11$ jurors and $7$ contestants in the final stage of the
$15^{\textrm{th}}$ Wieniawski Competition. The jurors rated the contestants in
the final according to the Reverse Borda count: $1$ point for the best and $7$
points for the worst. The winner was the contenstant with the lower sum of
points. The results are presented in Table 1.
We define the weighted network $N(\textrm{{W}})$ in the following way. The
nodes set of $N(\textrm{{W}})$ corresponds to the jurors set
$\\{J_{i}:i\in[11]\\}$, whereas the links set consists of all links
$\\{J_{s},J_{t}:i\neq j\in[11]\\}$. For link $l_{st}$ connecting $J_{s}$ with
$J_{t}$, we assign weight $w(l_{st})=w_{st}$, where
$w_{st}=\mathcal{LF}_{2}(\alpha_{s}\alpha_{t}^{-1})\textrm{.}$
Here $\alpha_{s}$ and $\alpha_{t}$ denote the votes of jurors $J_{s}$ and
$J_{t}$, respectively. In particular, for $i=1,2,\ldots,7$, $\alpha_{s}(i)$ is
the number of points given to the $i$-th contestant by juror $J_{s}$. The
similar holds for juror $J_{t}$. Note that $\alpha_{s}$ and $\alpha_{t}$ can
be considered as elements of $S_{7}$. Network $N(\textrm{{W}})$ is presented
in Figure 1.
For $p=1,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.1,0.0$, we create networks
$N(\textrm{{W}})_{p}$ removing from $N(\textrm{{W}})$ links $l_{st}$ with
weights satisfying the condition
$w(l_{st})>p\cdot\max{\\{\mathcal{LF}_{2}(\sigma):\sigma\in S_{7}\\}}=p\cdot
120\textrm{.}$
Note that for $p$ decreasing, $N(\textrm{{W}})_{p}$ contains links connecting
jurors voting more and more consitently. They will be used later in Section 5
. Networks $N(\textrm{{W}})_{0.4}$, $N(\textrm{{W}})_{0.3}$,
$N(\textrm{{W}})_{0.2}$ and $N(\textrm{{W}})_{0.1}$ are presented in Figure 2.
For better understanding of the consistency of jurors’ voting, for
$s=0,0.05,0.10,0.15,0.20,0.25,0.30,0.35,0.40,0.45,0.50$, we create networks
$N(\textrm{{W}})^{s}$ removing from $N(\textrm{{W}})$ links $l_{st}$, with
weights satisfying the condition
$|w(l_{st})-0.5\cdot\max{\\{\mathcal{LF}_{2}(\sigma):\sigma\in
S_{7}\\}}|<s\cdot\max{\\{\mathcal{LF}_{2}(\sigma):\sigma\in S_{7}\\}}\textrm{,
i.e.}$ $|w(l_{st})-60|<s\cdot 120\textrm{.}$
Note that when $s$ is increasing, then $N(\textrm{{W}})^{s}$ contains only the
links connecting jurors that vote more and more consitently and more and more
inconsistently, since we remove the links connecting jurors voting
independently. Networks $N(\textrm{{W}})^{0.2}$, $N(\textrm{{W}})^{0.25}$,
$N(\textrm{{W}})^{0.3}$, $N(\textrm{{W}})^{0.35}$, $N(\textrm{{W}})^{0.4}$ and
$N(\textrm{{W}})^{0.45}$ are presented in Figure 3.
## 4 Generic networks of votings
In this section, we describe the procedure of generating random networks of
votings. These _generic networks_ will be used later in Section 5 for
determining the model of generating random networks with properties best
fitting the properties of network $N(\textrm{{W}})$. We determine this best
fitting model to check the hypothesis that jurors of $15^{\textrm{th}}$
International Henryk Wieniawski Violin Competition voted controversially.
Namely, their votings were neither consistent nor random.
Consider the space of all possible votings of a single juror in W \- the final
stage of the $15^{\textrm{th}}$ Wieniawski Competition. This space can be seen
as $S_{7}$. Let $\mathbb{P}$ denote the probability measure on $S_{7}$. We
consider the measures of the form
$\mathbb{P}=\mathbb{P}(d,\alpha,\beta)=\alpha\mathbb{P}_{\sigma_{1}}+\beta\mathbb{P}_{\sigma_{2}}+(1-\alpha-\beta)\mathbb{P}_{\textrm{uniform}}$
where:
* •
$\alpha,\beta\in[0,1]$ satisfy the condition $\alpha+\beta\leq 1$,
* •
$d\in[0,1]$,
* •
$\mathbb{P}_{\textrm{uniform}}$ is the uniform probability measure on space
$S_{7}$,
* •
$\mathbb{P}_{\sigma_{1}}$ and $\mathbb{P}_{\sigma_{2}}$ are the Dirac
probability measures on $S_{7}$ centred at permutations $\sigma_{1}$ and
$\sigma_{2}$ respectively, where $\sigma_{1}$ and $\sigma_{2}$ are randomly
chosen in such a way that they satisfy the condition
$\mathcal{LF}_{2}(\sigma_{1}\sigma_{2}^{-1})\simeq
d\cdot\max{\\{\mathcal{LF}_{2}(\sigma):\sigma\in S_{7}\\}}\textrm{.}$
$\mathbb{P}$ defined in this way is a convex combination of
$\mathbb{P}_{\sigma_{1}}$, $\mathbb{P}_{\sigma_{2}}$ and
$\mathbb{P}_{\textrm{uniform}}$. The condition bonding $\sigma_{1}$ and
$\sigma_{2}$ is a metric analogue of the condition for the correlation
coefficient of $\sigma_{1}$ and $\sigma_{2}$. In this paper, we consider
$\alpha,\beta=0,0.05,0.1,\ldots,0.95,1$ and $d=0,0.1,\ldots,0.9,1$. Set
$N_{\textrm{max}}=\max{\\{\mathcal{LF}_{2}(\sigma):\sigma\in S_{7}\\}}$. The
notation $\simeq$ means that $\sigma_{1}$ and $\sigma_{2}$ are randomly chosen
from all possible pairs of permutations satisfying condition
$\mathcal{LF}_{2}(\sigma_{1}\sigma_{2}^{-1})\in\left[d\cdot
N_{\textrm{max}}-0.05\cdot N_{\textrm{max}},d\cdot N_{\textrm{max}}+0.05\cdot
N_{\textrm{max}}\right]$.
The procedure for generating the random network of votings is as follows:
1. (i)
choose a repetition number $j=1,2,\ldots,100$,
2. (ii)
choose $d=0,0.1,\ldots,0.9,1$,
3. (iii)
randomly choose such $\sigma_{1}$ and $\sigma_{2}$ that
$\mathcal{LF}_{2}(\sigma_{1}\sigma_{2}^{-1})\simeq d\cdot N_{\textrm{max}}$,
4. (iv)
choose such $\alpha,\beta=0,0.05,0.1,\ldots,0.95,1$ that $\alpha+\beta\leq 1$,
5. (v)
for every $s=1,2,\ldots,11$ randomly, according to
$\mathbb{P}=\alpha\mathbb{P}_{\sigma_{1}}+\beta\mathbb{P}_{\sigma_{2}}+(1-\alpha-\beta)\mathbb{P}_{\textrm{uniform}}$,
choose $\alpha_{n}^{s}\in S_{7}$ – this is the vote of juror $J_{s}$,
6. (vi)
create weighted network $N(d,\alpha,\beta,j)$ according to the procedure
described in Section 3,
7. (vii)
for $p=1,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.1,0.0$, create network
$N(d,\alpha,\beta,j)_{p}$.
## 5 Fitting the parameters
In this section, we determine such parameters $d$, $\alpha$ and $\beta$ that
the statistical properties of a family of networks
$\\{N(d,\alpha,\beta,j):j=1,2,\ldots,100\\}$ fit best the properties of the
network $N(\textrm{{W}})$.
According to the constructions of networks $N(\textrm{{W}})_{p}$, their
connected components, as $p$ decreases, correspond to groups of jurors voting
in a more and more consistent way. Table 2 contains the number of connected
components of networks $N(\textrm{{W}})_{p}$.
Let $\textrm{C}(N)$ denote the number of connected components of network $N$.
We determine the numbers $d_{\textrm{e}}^{\textrm{min}}$,
$d_{\textrm{m}}^{\textrm{min}}$, $d_{\textrm{e}}^{\textrm{max}}$ and
$d_{\textrm{m}}^{\textrm{max}}$ given by
$d_{\textrm{e}}^{\textrm{min}}=\min_{d,\alpha,\beta}\left\\{\sqrt{\sum_{p}\left(\textrm{C}\left(N(\textrm{{W}})_{p}\right)-\frac{\sum_{j=1}^{100}\textrm{C}(N(d,\alpha,\beta,j)_{p})}{100}\right)^{2}}\right\\}\textrm{,}$
$d_{\textrm{m}}^{\textrm{min}}=\min_{d,\alpha,\beta}\left\\{\sum_{p}\left|\textrm{C}\left(N(\textrm{{W}})_{p}\right)-\frac{\sum_{j=1}^{100}\textrm{C}(N(d,\alpha,\beta,j)_{p})}{100}\right|\right\\}\textrm{,}$
$d_{\textrm{e}}^{\textrm{max}}=\max_{d,\alpha,\beta}\left\\{\sqrt{\sum_{p}\left(\textrm{C}\left(N(\textrm{{W}})_{p}\right)-\frac{\sum_{j=1}^{100}\textrm{C}(N(d,\alpha,\beta,j)_{p})}{100}\right)^{2}}\right\\}\textrm{
and }$
$d_{\textrm{m}}^{\textrm{max}}=\max_{d,\alpha,\beta}\left\\{\sum_{p}\left|\textrm{C}\left(N(\textrm{{W}})_{p}\right)-\frac{\sum_{j=1}^{100}\textrm{C}(N(d,\alpha,\beta,j)_{p})}{100}\right|\right\\}$
respectively.
Note that $d_{\textrm{e}}^{\textrm{min}}$ and $d_{\textrm{m}}^{\textrm{min}}$
minimalise the Euclidean distance and the Manhattan distance between the
number of connected components of $N(\textrm{{W}})_{p}$ and the average number
of connected components of $N(d,\alpha,\beta,j)_{p}$, respectively.
Similarily, $d_{\textrm{e}}^{\textrm{max}}$ and
$d_{\textrm{m}}^{\textrm{max}}$ maximalise these distances. Table 3 contains
parameters $d$, $\alpha$ and $\beta$ of models for which these distances are
the smallest. The sum
$\sqrt{\sum_{p}\left(\textrm{C}\left(N(\textrm{{W}})_{p}\right)-\frac{\sum_{j=1}^{100}\textrm{C}(N(d,\alpha,\beta,j)_{p})}{100}\right)^{2}}$
varies between $d_{\textrm{e}}^{\textrm{min}}=14.29699$ and
$d_{\textrm{e}}^{\textrm{max}}=24.75912$, whereas the sum
$\sum_{p}\left|\textrm{C}\left(N(\textrm{{W}})_{p}\right)-\frac{\sum_{j=1}^{100}\textrm{C}(N(d,\alpha,\beta,j)_{p})}{100}\right|$
varies between $d_{\textrm{m}}^{\textrm{min}}=34.28$ and
$d_{\textrm{m}}^{\textrm{max}}=75.02$. In both cases best fitted models are
those with parameter $d=1$. Note that $d=1$ means that $\sigma_{1}$ and
$\sigma_{2}$, chosen during the procedure described in Section 4, satisfy
$\mathcal{LF}_{2}(\sigma_{1}\sigma_{2}^{-1})\simeq N_{max}$, therefore being
almost revers permutations. This fact confirms the hypothesis that jurors of
the $15^{\textrm{th}}$ International Henryk Wieniawski Violin Competition were
far away from being consistent.
## 6 Conclusions and recommendations for further research
This section contains conclusions and open questions related to the subject of
these studies.
The results of voting include a lot of information about preferences of voters
and their structure. The application of network theory can highlight
properties of networks constructed on the basis of jurors’ votings. The
obtained networks may be used to describe homogeneity or heterogeneity of
jurors’ votings.
During this research some questions arose.
1. 1.
How do statistical coefficients of networks $N(d,\alpha,\beta,j)_{p}$ depend
on $d$, $\alpha$ and $\beta$?
2. 2.
How does behaviour of generic networks depend on the number of jurors and the
number of contestants?
3. 3.
What are the asymptotic (with number of jurors and/or contestants tending to
$\infty$) properties of generic networks?
4. 4.
How do best fitting parameters $d$, $\alpha$ and $\beta$ change when
increasing the number of repeats?
5. 5.
How will properties of generic networks change when we randomly choose
permutation $\sigma_{1}$ and $\sigma_{2}$ with
$\mathcal{LF}_{2}(\sigma_{1}\sigma_{2}^{-1})$ precisely set from all possible
values of $\mathcal{LF}_{2}$?
These questions are a good starting point for further research.
### Acknowledgements
While working on this paper, the second author was partially supported by the
SGH fund KAE/S21 and by the NCN fund UMO-2018/31/B/HS4/01005.
All calculations and figures were prepared with R 4.0.3, see[R].
The second author is grateful to his whife for her strong and loving support
as well as for her inspiring _”try to think nonstandard”_.
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## 7 Tables and Figures
Table 1: Final results of the $15^{\textrm{th}}$ International Henryk Wieniawski Violin Competition | J1 | J2 | J3 | J4 | J5 | J6 | J7 | J8 | J9 | J10 | J11
---|---|---|---|---|---|---|---|---|---|---|---
A | $7$ | $3$ | $2$ | $7$ | $7$ | $4$ | $3$ | $7$ | $7$ | $7$ | $7$
B | $4$ | $7$ | $7$ | $2$ | $2$ | $7$ | $7$ | $2$ | $5$ | $6$ | $5$
C | $5$ | $5$ | $5$ | $3$ | $6$ | $6$ | $5$ | $5$ | $6$ | $1$ | $6$
D | $3$ | $6$ | $4$ | $5$ | $1$ | $5$ | $4$ | $4$ | $3$ | $5$ | $1$
E | $1$ | $4$ | $6$ | $1$ | $3$ | $3$ | $6$ | $3$ | $4$ | $3$ | $4$
F | $6$ | $2$ | $1$ | $6$ | $4$ | $2$ | $1$ | $6$ | $1$ | $2$ | $2$
G | $2$ | $1$ | $3$ | $4$ | $5$ | $1$ | $2$ | $1$ | $2$ | $4$ | $3$
Table 2: Number of connected componets of networks $N(\textrm{{W}})_{p}$ treshold parameter $p$ | component number
---|---
1 | 1
0.9 | 1
0.8 | 1
0.7 | 1
0.6 | 2
0.5 | 2
0.4 | 3
0.3 | 7
0.2 | 8
0.1 | 10
0 | 11
Table 3: Best fitting parameters Euclidean Distance | maximal value - $24.75912$
---|---
$d=1$, $\alpha=0.4$, $\beta=0.6$ | $14.29699$
$d=1$, $\alpha=0.45$, $\beta=0.55$ | $14.29773$
$d=1$, $\alpha=0.5$, $\beta=0.5$ | $14.30108$
$d=1$, $\alpha=0.55$, $\beta=0.45$ | $14.30696$
$d=1$, $\alpha=0.6$, $\beta=0.4$ | $14.30696$
Manhattan Distance | maximal value - $75.02$
$d=1$, $\alpha=0.1$, $\beta=0.9$ | $34.28$
$d=1$, $\alpha=0.15$, $\beta=0.85$ | $34.32$
$d=1$, $\alpha=0.85$, $\beta=0.15$ | $34.32$
$d=1$, $\alpha=0.9$, $\beta=0.1$ | $34.4$
$d=0.9$, $\alpha=0.1$, $\beta=0.9$ | $34.78$
Figure 1: Network $N(\textrm{{W}})$, green – small weights, red – large
weights Figure 2: Voting networks based on the final results of the
$15^{\textrm{th}}$ International Henryk Wieniawski Violin Competition
(a) Network $N(\textrm{{W}})_{0.4}$
(b) Network $N(\textrm{{W}})_{0.3}$
(c) Network $N(\textrm{{W}})_{0.2}$
(d) Network $N(\textrm{{W}})_{0.1}$
Figure 3: Voting networks based on the final results of the $15^{\textrm{th}}$
International Henryk Wieniawski Violin Competition
(a) Network $N(\textrm{{W}})^{0.2}$
(b) Network $N(\textrm{{W}})^{0.25}$
(c) Network $N(\textrm{{W}})^{0.3}$
(d) Network $N(\textrm{{W}})^{0.35}$
(e) Network $N(\textrm{{W}})^{0.4}$
(f) Network $N(\textrm{{W}})^{0.4}$
|
# SLAN: Self-Locator Aided Network for Cross-Modal Understanding
Jiang-Tian Zhai1 Qi Zhang211footnotemark: 1 Tong Wu2 Xing-Yu Chen2 Jiang-Jiang
Liu122footnotemark: 2
Bo Ren2 Ming-Ming Cheng1
CS Indicates equal contributions.Work done when interning at Tencent Youtu
Lab. Nankai University1 Tencent Youtu Lab2
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Learning fine-grained interplay between vision and language allows to a more
accurate understanding for Vision-Language tasks. However, it remains
challenging to extract key image regions according to the texts for semantic
alignments. Most existing works are either limited by text-agnostic and
redundant regions obtained with the frozen detectors, or failing to scale
further due to its heavy reliance on scarce grounding (gold) data to pre-train
detectors. To solve these problems, we propose Self-Locator Aided Network
(SLAN) for cross-modal understanding tasks without any extra gold data. SLAN
consists of a region filter and a region adaptor to localize regions of
interest conditioned on different texts. By aggregating cross-modal
information, the region filter selects key regions and the region adaptor
updates their coordinates with text guidance. With detailed region-word
alignments, SLAN can be easily generalized to many downstream tasks. It
achieves fairly competitive results on five cross-modal understanding tasks
(_e.g.,_ 85.7% and 69.2% on COCO image-to-text and text-to-image retrieval,
surpassing previous SOTA methods). SLAN also demonstrates strong zero-shot and
fine-tuned transferability to two localization tasks.
## 1 Introduction
Recent years have witnessed growing interest in exploring relationships
between vision and language modalities. A wide range of applications have been
boosted by its rapid development, such as multi-modal search engines [8, 10,
4] and recommender systems [29, 30, 7]. It motivates researchers to find
semantic correspondence between two modalities and bridging their visual-
semantic discrepancy. Some earlier works [27, 13, 21, 12] focused on learning
joint embeddings for these two modalities, while more recent ones [14, 22, 41]
have turned to considering latent vision-language alignments at the level of
regions and words.
|
---|---
(a) Image-text Retrieval | (b) Image Caption
|
(c) Object Detection | (d) Phrase Grounding
Figure 1: Visualization on four different tasks. We visualize the activation
map for text-to-image retrieval task in (a). As for the caption task in (b),
we visualize regions selected by our model. Besides cross-modal understanding
task, SLAN can transfer to localization tasks, shown in (c) and (d), and we
list the confidence score for each region.
In order to achieve fine-grained cross-modal alignments, some works [18, 17,
23] use object detectors to extract key regions in images. Treated as black
boxes, the detectors only support for fixed vocabulary object detection.
Meanwhile, the extracted regions cannot adapt to different text information
due to the freezing parameters of the detectors. To alleviates the problem,
VinVL [41] applies a pre-trained object detector with more than $2000$ classes
and attributes to enrich local visual representations. However, the extended
label set still limits the perceptive capability of object detectors for
cross-modal understanding compared to free-form text from real-world scenes.
Recently, more works have attempted to apply learnable region locators for
cross-modal tasks, which extract regions of interest conditioned on different
texts. Unlike previous methods using frozen object detectors, MDETR [14]
builds an end-to-end framework on datasets with region-to-word annotations.
GLIP [22] directly proposes grounded language-image pre-training for learning
object-level, language-aware, and semantic-rich visual representation. These
methods demonstrate their effectiveness in cross-modal reasoning by
introducing trainable locators. However, in order to supervise the training of
locators, these methods require a certain amount of region-to-word grounding
annotations (gold data), which are based on burdensome and expensive
annotation efforts. It limits their applications on existing larger scale of
cross-modal datasets which have abundant but coarse-grained image and text
pairs.
To address the problems above, we propose Self-Locator Aided Network (SLAN)
for cross-modal understanding. The designed self-locator is capable of
accurately locating regions of interest based on different texts.
Specifically, the self-locator consists of a region filter to select important
regions and a region adaptor to update coordinates of regions with text
guidance. By incorporating the self-locator into our framework, SLAN performs
context-aware region extraction and cross-modal feature fusion. Moreover, SLAN
is trained solely on datasets with paired images and texts, making it scalable
to larger pre-training settings for further performance improvements. With
fine-grained region-word alignments, SLAN has a more detailed understanding of
interactions in vision and language modalities.
To sum up, our contributions have four aspects:
* •
We propose a framework termed SLAN to capture fine-grained interplay between
vision and language modalities. A self-locator is introduced to perform text-
guided region adaptation, enabling dynamic region-word alignments for cross-
modal understanding tasks.
* •
We demonstrate that SLAN can be easily applied to large-scale pre-training on
cross-modal datasets, because it is free from training with gold data.
* •
We naturally generalize SCAN to some localization tasks, such as object
detection and phrase grounding, due to its ability to locate key regions in
images.
* •
Experiments on five cross-modal understanding and two localization tasks
demonstrate the effectiveness of our method. For example, SLAN achieves state-
of-the-art performance on COCO image-text retrieval.
## 2 Related Work
### 2.1 Vision-language Task
Figure 2: Overall framework of our proposed SLAN, consisting of two unimodal
encoders for embedding images and text and a self-locator for fine-grained
region-word alignments. The self-locator performs saliency prediction with a
region filter and progressive regression with a region adaptor. The learned
vision and language features with cross-modal awareness are used for
downstream tasks.
Many efforts have been made to explore relationships between visual and
textual modalities, then apply the knowledge to various downstream multi-modal
tasks. Some previous methods propose loss functions and network structure to
learn semantic vision-language alignments. DeViSE [11] first introduces a
linear layer to unify image and text embeddings. TBNN[31] learns cross-modal
features and applies contrastive losses. The instance loss is introduced in
[42] to train the dual encoder. Other methods introduce some prior tools or
knowledge to assist the image-text matching analysis. For example, SGG [35]
considers the internal relations between visual regions and compares them with
the text structure. ViSTA [9] applies OCR [3] to extract text from images for
richer cross-modal labels.
In recent years, a prevailing direction applies cross-modal pre-training on
some larger datasets. CLIP [27] pre-trains with 400M image-text pairs
collected from the web, building global relations between images and texts.
BLIP [20] benefits from large-scale web data by filtering out noisy ones, and
performs vision-language understanding and generation tasks. Beit-3 [32]
adopts mask-then-predict self-supervised training on large-scale monomodal
data (_i.e.,_ images, and texts) and multi-modal data (_i.e.,_ image-text
pairs) to learn internal cross-modal dependencies.
Aside from these attempts, exploring local relations between words in text and
objects in the image works efficiently in cross-modal pre-training. It helps
localize more accurate objects according to corresponding words, and provides
cues for downstream tasks.
### 2.2 Localization for Vision-language Task
There are two kinds of methods whose differences are whether the detection
module are frozen or trained to adapt to cross-modal tasks. The first kind
introduces a frozen object detector to extract detailed visual
representations. SCAN uses Faster R-CNN pre-trained on Visual Genomes to
choose key regions for matching with word embeddings. Instead of using a
traditional object detector trained with relatively few classes and data, some
later works (e.g., VinVL [41], Oscar [23]) increase the number of detection
labels and introduce some attribute information to complement previous visual
concepts.
The other kind relies on fine-grained annotations of the cross-modal dataset
to perform pre-training. MDETR [14] introduces a modulated detector with
multi-modal datasets, which have precise alignments between phrases in text
and objects in the images. GLIP [22] applies grounded pre-training to learn
object-level, language-aware, and semantic-rich visual representations. These
methods link object detection with phrase grounding (region-word alignments),
enabling models to learn from richer semantic knowledge from both modalities.
However, these methods require cross-modal data with fine-grained annotations,
limiting their application on larger-scale pre-training settings.
## 3 SLAN
We will introduce our proposed SLAN in this section. As shown in Figure 2, our
framework consists of three components, two unimodal encoders and a self-
locator. We first briefly introduce two unimodel encoders. Then we introduce
the detailed structures of SLAN which adaptively select informative regions
with text guidance. Finally, we list our pre-training objectives.
### 3.1 Unimodal Encoding
We introduce the vision and text encoder to learn visual and textual
representations with ${D}$ dimensions. For text feature extraction, we use
BERT [15] as our text encoder, encoding words into a shared semantic space. A
text [CLS] token is added to word embeddings to summarize the whole sentence.
For image feature extraction, we encode images and obtain the vision feature
map $V$ with high-level semantics.
### 3.2 Self-locator for Cross-modal Understanding
Since fine-grained region-word alignments are important for cross-modal
relation exploration, our self-locator is built on DETR [5] to output original
regions. Region embeddings are extracted from the feature map $V$. A vision
[CLS] token is then obtained from global average pooling of region embeddings.
Different from most traditional object detection tasks that use the pre-
defined label set, cross-modal tasks usually have a wider vocabulary and free-
form textual expressions. Therefore, our self-locator is adapted to introduce
a region filter for region saliency prediction and a region adaptor for
progressive region regression for cross-modal tasks. By replacing fixed
vocabulary prediction with region saliency prediction, our self-locator
assigns each region a saliency score to estimate the probability that the
region is useful for the alignment process. For traditional detection
settings, the regression targets are annotated region coordinates. Since there
is no grounding (gold) annotations in our setting, we propose progressive
region regression supervised by a weighted summation of updated regions.
#### 3.2.1 Vision Decoder: Pyramid Feature Extraction
Our proposed self-locator is designed for regression in a coarse-to-fine
manner, requiring visual features of multi-scale and pyramid hierarchy.
Considering these characteristics, we adopt a vision decoder after the global
visual feature to extract multi-scale feature maps. Let
$F_{i}~{}(i\in{1,2,...,L})$ denotes the $i$-th level of decoder features
separately, where $L$ is the number of layers of the self-locators. Then
$F_{i}$ is fed to the $i$-th level of self-locator regression.
#### 3.2.2 Region Filter: Region Saliency Prediction
When describing images, people usually focus on limited salient regions in the
images. However, original detection models (e.g., DETR) output a relatively
large number of region proposals (e.g., 100) for images, which are mostly
redundant for text descriptions. If we directly select all detected regions,
it will lead to unnecessary computational cost and might cause the model to
learn some meaningless region-to-word pairs. The strategy to control the
maximum number of selected regions. The strategy is as follows: (a) Normalize
all saliency scores of these regions. After this process, the scores are
represented as $S=\\{S_{1},...,S_{k}\\},S_{i}\in\mathbb{R}$, with the maximum
value of 1. (b) Sort these regions in descending order according to their
saliency scores. (c) Select regions with saliency scores above the threshold
$h$ ($S_{i}>h$). (d) If the number of selected regions is greater than hyper-
parameter $T$, the self-locator picks the first $T$ regions. Finally, we
weight region embeddings by the scores.
Figure 3: Illustration of one level of the region adaptor to update each
region’s coordinate with text guidance. We use the feature map from vision
decoder to extract region embeddings and explore latent region-word
alignments. Algorithm 1 Self-localization
1:Image $x$, region embeddings $E_{0}^{G}$, text embeddings $E_{0}^{T}$,
pyramid feature map $F_{i}$, neighbour size $(NH_{i},NW_{i})$, total region
regression layers $L$.
2:Updated regions $G_{out}$, region supervision on detection head
$\overline{G}$, visual token $v_{cls}$, textual token $t_{cls}$.
3:$G_{0},S,E_{0}^{G}$ ← Detection($x$)
4:$G_{0},E_{0}^{G}$ ← Region Saliency Prediction ($G_{0},S,E_{0}^{G}$)
5:for $i\in$ $\\{1,2,...,L\\}$ do
6: $E^{G}_{i},E^{T}_{i}$ ← Cross attention($E^{G}_{i-1},E^{T}_{i-1}$)
7: $R_{i}$ ← $(NH_{i},NW_{i})$
8: $E^{Ne}_{i}$ ← Neighbour Embedding($R_{i},G_{i-1}$)
9: $Sim_{i}$ ← Similarity($E^{Ne}_{i}$,$E^{T}_{i}$)
10: $\Delta$$x_{i}$,$\Delta$$y_{i}$ ← $Offset$($Sim_{i}$)
11: $p_{w_{i}},p_{h_{i}}$ are learnable parameters.
12: $G_{i}$ ←
Update($G_{i-1}$,$\Delta$$x_{i}$,$\Delta$$y_{i}$,$p_{w_{i}},p_{h_{i}}$)
13: $E^{G}_{i}$ ← Embedding($G_{i},f_{i}$)
14:end for
15:$v_{cls},t_{cls}$ ← ExtractCLS($E^{G}_{out},E^{T}_{out}$)
16:$G_{out}$ ← $G_{L}$, $\overline{G}$ ← $(\sum_{i=1}^{L}G_{i})/L$
#### 3.2.3 Region Adaptor: Progressive Region Regression
This module aims at adjusting the coordinates of proposed regions to align
with words with the same semantics. The difficulty comes from no annotated
text-referenced regions as ground truths. We transform this problem into the
$L$ cascaded coarse-to-fine progressive regression. We set $L$ to 3 in
default. As shown in Figure 3, the $i$-th level of the region regression
receives three inputs: word embeddings $E_{i-1}^{T}\in\mathbb{R}^{N^{T}\times
D}$, region embeddings $E_{i-1}^{G}\in\mathbb{R}^{N^{G}\times D}$ with their
coordinates $G_{i}\in\mathbb{R}^{N^{G}\times 4}$, and a global decoder feature
map $F_{i}\in\mathbb{R}^{H_{i}\times W_{i}\times D}$, where $N^{T}$ and
$N^{G}$ denotes the number of words and selected regions respectively.
We describe the procedure for progressive region regression in Algorithm 1.
The cross-modality multi-head attention layers fuse region and word embeddings
and model their interactions as follows:
$\begin{aligned} Attn&=\frac{E_{i-1}^{G}E_{i-1}^{T\top}}{\sqrt{D}}\\\
E^{G}_{i}&=Softmax(Attn)E^{T}_{i-1}\\\
E^{T}_{i}&=Softmax(Attn^{\top})E^{G}_{i-1}\\\ \end{aligned},$ (1)
where $D$ denotes the dimension of embeddings. With cross-modal semantics, the
updated vision-aware word embeddings $E^{T}_{i}$ are able to guide region
coordinate updates by searching for highly correlated regions around the
original one. Specifically, the neighborhood of the region $g=(x,y,w,h)$ is
defined as a region of size $(NH_{i},NW_{i})$ centered on it, where $NH_{i}$
and $NW_{i}$ are pre-defined parameters for the $i$-th level in region
regression. And the neighborhood is split to $K\times K$ regions to compute
region-word similarities, as shown in Figure 3, where each region embedding is
extracted with RoIAlign and average pooling from $F_{i}$. With different
response scores to words, neighbor regions aggregate context information to
the central one. The coordinate update for the central region is in the form
of weighted summation of coordinates of neighbor center points as follows:
$\displaystyle w^{\prime}$ $\displaystyle=p_{w}w,~{}~{}h^{\prime}=p_{h}h,$ (2)
$\displaystyle x^{\prime}$ $\displaystyle=x+\Delta x,~{}~{}y^{\prime}=y+\Delta
y,$ $\displaystyle\Delta x$
$\displaystyle=\sum_{j=0}^{K^{2}-1}s_{j}H_{j}(\lfloor\frac{j}{K}\rfloor-\lfloor\frac{K}{2}\rfloor),$
$\displaystyle\Delta y$
$\displaystyle=\sum_{j=0}^{K^{2}-1}s_{j}W_{j}((j~{}mod~{}K)-\lfloor\frac{K}{2}\rfloor),$
where $\lfloor\cdot\rfloor$ is the round down operation, $p_{w}$ and $p_{h}$
are learnable parameters, and $s_{j}$ is the maximum similarity between the
embedding of the $j$-th neighbor region and all word embeddings.
For each original region $g$, let $g_{i}$ denotes the updated version after
the $i$-th layer of region regression. We take the average of them as the
ground truth and apply the $L_{1}$ and GIoU regression loss:
$\begin{aligned} \overline{g}&=\frac{\sum_{i=1}^{L}g_{i}}{L}\\\
\mathcal{L}_{reg}(g)&=\mathcal{L}_{L1}(g,\overline{g})+\mathcal{L}_{GIoU}(g,\overline{g})\\\
\end{aligned}.$ (3)
### 3.3 Pre-training Objectives with SLAN
Our SLAN pretrains on image-text pairs and learns fine-grained region-word
alignments, supervised by three common losses.
Image-Text Matching Loss (ITM) predicts whether a given image-text pair is
positive or not, which can be viewed as a binary classification problem. The
visual and textual [CLS] tokens $(v_{cls},t_{cls})$ are concatenated and sent
to a linear layer $f_{c}$. The ITM loss is formalized as follows:
$\displaystyle\mathcal{L}_{itm}(I,T)$
$\displaystyle=H(f_{c}(cat(v_{cls},t_{cls})),y_{v,t}),$ (4)
where $y_{v,t}$ denotes the matching relation (1 for matched and 0 for
unmatched), and $H$ is the cross-entropy loss for classification. We directly
select positive pairs from the dataset and build hard negative samples with
batch sampling, following ALBEF [21].
Image-Text Contrastive Loss (ITC) ensures that visual and textual embeddings
share the same semantic space and the positive (matched) image-text pairs are
pulling closer than negative (unmatched) ones. We use two queues $I_{q},T_{q}$
to save the latest visited image and text samples. For each image-text pair
$(I,T)$, the softmax-normalized cross-modal similarity is computed as as:
$\begin{aligned}
p_{i2t}(I,T,T_{q})&=\frac{exp(sim(I,T)/\tau)}{\sum_{T^{\prime}\in
T_{q}}exp(sim(I,T^{\prime})/\tau)}\\\
p_{t2i}(T,I,I_{q})&=\frac{exp(sim(T,I)/\tau)}{\sum_{I^{\prime}\in
I_{q}}exp(sim(T,I^{\prime})/\tau)}\\\ \end{aligned},$ (5)
where $\tau$ is a temperature parameter and $sim(\cdot)$ measures cross-modal
similarity, which is implemented by the dot product between image and text
embeddings. Following ALBEF [21], we compute ITC loss as:
$\displaystyle\mathcal{L}_{itc}(I,T)$
$\displaystyle=-log(p_{i2t}(I,T,T_{q}))-log(p_{t2i}(T,I,I_{q})).$ (6)
Language Modeling Loss (LM) encourages the model to predict masked words with
context information. We randomly mask $15\%$ text tokens and apply the masked
language modeling loss as follows:
$\displaystyle\mathcal{L}_{lm}(I,T)$
$\displaystyle=H(p_{mask}(I,T),y_{mask}),$ (7)
where $y_{mask}$ denotes the masked word to predict and $p_{mask}(I,T)$ is its
predicted probability.
The full pre-training objective is the combination of the downstream loss and
our constraint on progressive region regression, computed as follows:
$\displaystyle\mathcal{L}_{all}$
$\displaystyle=\mathcal{L}_{ds}+\mathcal{L}_{reg}$ (8)
where $\mathcal{L}_{all}$ is the downstream loss and $\mathcal{L}_{reg}$
denotes the summation of the regression loss in Equation 3 for all regions.
$\displaystyle\mathcal{L}_{ds}(I,T)$
$\displaystyle=\mathcal{L}_{itm}(I,T)+\mathcal{L}_{itc}(I,T)+\mathcal{L}_{lm}(I,T).$
(9)
## 4 Experiments
We first pre-train our method on a combined dataset of 14M image-text pairs
from five datasets: COCO [24], Visual Genome [16] (excluding COCO images),
Conceptual Captions [6], Conceptual [6], and SBU Captions [25]. The data
statistics are described in supplementary. We evaluate the proposed SLAN by
comparing it to other state-of-the-art cross model methods on several
downstream tasks. The ablation studies are further conducted to study how each
component of our method influences the performance.
Method | Pre-train | Zero-shot | Fine-tune
---|---|---|---
# Images | Image $\rightarrow$ Text | Text $\rightarrow$ Image | Image $\rightarrow$ Text | Text $\rightarrow$ Image
| | R@1 | R@5 | R@10 | R@1 | R@5 | R@10 | R@1 | R@5 | R@10 | R@1 | R@5 | R@10
ALIGN [21] | 1.8B | 88.6 | 98.7 | 99.7 | 75.7 | 93.8 | 96.8 | 95.3 | 99.8 | 100.0 | 84.9 | 97.4 | 98.6
FILIP [38] | 300M | 89.8 | 99.2 | 99.8 | 75.0 | 93.4 | 96.3 | 96.6 | 100.0 | 100.0 | 87.1 | 97.7 | 99.1
BLIP [20] | 14M | 94.8 | 99.7 | 100.0 | 84.9 | 96.7 | 98.3 | 96.6 | 99.8 | 100.0 | 87.2 | 97.5 | 98.8
BEIT-3 [32] | 21M | 94.9 | 99.9 | 100.0 | 81.5 | 95.6 | 97.8 | 98.0 | 100.0 | 100.0 | 90.3 | 98.7 | 99.5
Ours | 14M | 96.0 | 100.0 | 100.0 | 86.1 | 97.0 | 98.5 | 98.1 | 100.0 | 100.0 | 90.2 | 99.0 | 99.6
Table 1: Comparison with state-of-the-art image-text retrieval methods on Flickr30k. We use Recall@k scores as the evaluation metric under zero-shot and fine-tuning settings. Method | Pre-training | Retrieval(COCO) | Caption(COCO) | VQA(VQAv2) | NLVR(NLVR2)
---|---|---|---|---|---
Data | I2T R@1 | T2I R@1 | B@4 | M | C | S | test-dev | test-std | dev | test-P
Oscar [23] | 6.5M | 73.5 | 57.5 | 37.4 | 30.7 | 127.8 | 23.5 | 73.6 | 73.8 | 79.1 | 80.3
VinVL [41] | 8.85M | 75.4 | 58.8 | 38.5 | 30.4 | 130.8 | 23.4 | 76.5 | 76.6 | 82.6 | 83.9
SimVLM [33] | 1.8B | - | - | 40.6 | 33.7 | 143.3 | 25.4 | 80.0 | 80.3 | 84.5 | 85.1
GLIPv2-H [40] | 16M | - | - | - | - | 131.0 | - | 74.6 | 74.8 | - | -
CoCa [39] | 4.8B | - | - | 40.9 | 33.9 | 143.6 | 24.7 | 82.3 | 82.3 | 86.1 | 87.0
BLIP [20] | 14M | 82.4 | 65.1 | 40.4 | - | 136.7 | - | 78.2 | 78.3 | 82.1 | 82.2
Ours | 14M | 85.7 | 69.2 | 43.7 | 34.1 | 144.3 | 25.6 | 83.4 | 83.5 | 90.4 | 91.3
Table 2: Comparison on more downstream tasks. For COCO retrieval, I2T and T2I
represent image to text and text to image retrieval task, respectively. For
COCO image captioning, we report BLEU@4 (B@4), METEOR (M), CIDEr (C), and
SPICE (S) on the Karpathy test split. For VQA, we evaluate the vqa-score on
VQAv2 test-dev and test-standard (test-std) splits. For NLVR, we report
accuracy on NLVR2 development set (dev) and public test set (test-P).
### 4.1 Implementation Details
We choose $BERT_{base}$ [15] as our text encoder, which is initialized from
HuggingFace [34]. For the vision encoder, we explore four options: one is the
CNN model ResNet50, and three kinds of ViT: ViT-Base, ViT-Large and ViT-Huge,
which are all random initialized. As for the neighbour size for each region
adaptor, we use a ratio $r_{i}$ to denote them: ($NH_{i}$, $NW_{i}$) =
($r_{i}H_{i}$, $r_{i}W_{i}$), where $r_{1},r_{2},r_{3}=1,0.5,0.25$,
respectively. We pre-train SLAN for 20 epochs. For different options of the
vision encoder, the batch size is set to 1280, 960, 640, 640 for ResNet50,
Vit-B, Vit-L and Vit-H, respectively. The AdamW optimize is adopted with the
initial learning rate 3e-4, and the learning rate is linearly decayed to 0. We
resize the input images to 224$\times$224.
### 4.2 Comparison on The Downstream Tasks
We compare our approach with state-of-the-art methods on five challenging
cross-modal understanding tasks, including image-text retrieval, image
captioning, visual question answering, natural language visual reasoning,
zero-shot video-text retrieval. Besides, we transfer our method to two
localization tasks: object detection and phrase grounding. The default vision
encoder is Vit-Huge, if not specified.
#### 4.2.1 Image-Text Retrieval
Given an image, the task expects to retrieve the corresponding text from the
text gallery through the input image, and vice versa. We evaluate our method
on Flickr30k [26] under zero-shot and fine-tune settings with Karpathy split
and the performance is evaluated in terms of Recall@k. The comparative results
are shown in Table 1. Specifically, on the same pre-training setting, SLAN
also outperform BLIP [20] by 3.3% in average recall@1 on COCO. This
performance gain explains the efficiency of learning fine-grained alignments
from coarse annotations.
#### 4.2.2 Image Captioning
Given an input image, this task generates a sentence description to describe
the image in detail. We use COCO Karpathy split to fine-tune and evaluate our
method. Our SLAN outperforms most previous methods under this efficient
setting, as shown in Table 2.
Method | R@1 $\uparrow$ | R@5 $\uparrow$ | R@10 $\uparrow$ | MdR $\downarrow$
---|---|---|---|---
ClipBERT [19] | 22.0 | 46.8 | 59.9 | 6
VideoCLIP [36] | 30.9 | 55.4 | 66.8 | -
FiT† [2] | 43.3 | 65.6 | 74.7 | 2
BLIP† [20] | 43.3 | 65.6 | 74.7 | 2
Ours† | 46.8 | 70.5 | 83.6 | 1.5
Table 3: Comparisons with state-of-the-art methods for text-video retrieval on the 1k test split of the MSRVTT [37] dataset. †denotes the zero-shot settings, while others are fine-tuned ones. Method | Backbone | Params(M) | Pretrain Data(M) | Object Detection(COCO) | Phrase Grounding(Flickr30k)
---|---|---|---|---|---
Image-Text | Region-Word | Zero-shot | Fine-tune | R@1 | R@5 | R@10
DETR [5] | ResNet50 | 42 | 0 | 0 | - | 42.0 | - | - | -
MDETR [14] | ResNet101 | 185 | 0 | 0.2 | - | - | 84.3 | 93.9 | 95.8
GLIP [22] | Swin-Large | 430 | 24 | 3 | 49.8 | 60.8 | 87.1 | 96.9 | 98.1
GLIPv2 [40] | Swin-Huge | 870 | 16 | 3 | - | 60.2 | 87.7 | 97.3 | 98.5
Ours | ResNet50 | 322 | 14 | 0 | 46.9 | 59.2 | 86.8 | 96.6 | 97.4
Ours | Vit-Base | 383 | 14 | 0 | 47 | 59.6 | 87.4 | 96.9 | 98.2
Ours | Vit-Large | 601 | 14 | 0 | 48.5 | 60.5 | 89.1 | 98.0 | 98.9
Ours | Vit-Huge | 929 | 14 | 0 | 50.1 | 63.5 | 90.6 | 98.6 | 99.3
Table 4: Comparison on two localization tasks: Object Detection on COCO and Phrase Grounding on Flickr30k. The pre-training data covers: image-text pairs and word-specific region annotations. we evaluate zero-shot and fine-tune settings on object detection. We use recall@k scores to evaluate phrase grounding task. Trainable | Adaptor | | COCO | | Flickr30k
---|---|---|---|---|---
Detector | Number | | TR@1 | IR@1 | | TR@1 | IR@1
✘ | 0 | | 68.5 | 53.5 | | 85.0 | 74.1
✔ | 0 | | 69.1 | 53.8 | | 86.7 | 76.2
✔ | 1 | | 70.0 | 57.2 | | 88.3 | 77.4
✔ | 2 | | 70.8 | 57.5 | | 88.7 | 78.1
✔ | 3 | | 72.1 | 58.3 | | 90.3 | 78.9
Table 5: Effect of training detection module and region adaptor. ✘ in the first coloum denotes applying a frozen detection module and no self-locator. TR@1 and IR@1 denote recall@1 of image to text and text to image retrieval. To evaluate the effect of the self-locator against a frozen detection module, we load the pre-trained weight from COCO Detection to compare with our method for line 1. The remaining experiments train from scratch. We use Vit-Base as the vision encoder. Top K | Threshold | | COCO | | Flickr30k
---|---|---|---|---|---
| TR@1 | IR@1 | | TR@1 | IR@1
- | - | | 69.4 | 54.1 | | 85.9 | 74.7
10 | - | | 70.6 | 56.8 | | 87.5 | 77.3
10 | 0.3 | | 71.2 | 57.6 | | 89.1 | 78.2
10 | 0.5 | | 72.1 | 58.3 | | 90.3 | 78.9
Table 6: Different settings of the region filter.
#### 4.2.3 Visual Question Answering
Visual Question Answering (VQA) [1] requires the model to predict an answer
from an image and a question. We follow [20] and treat VQA as an open-ended
question-generation task. We fuse the image embedding with the question
embedding and send them to the question decoder to get the result. As shown in
Table 2, SLAN outperform other method by at least 1.1% on VQAv2 test-dev and
test-std with less or equal pre-training data.
#### 4.2.4 Natural Language Visual Reasoning
Natural Language Visual Reasoning (NLVR2) [28] measures whether a sentence
describes a pair of images. We extract the image and text embeddings from the
image-text input, which are fused by a cross-attention layer. We use a binary
classification module to predict their relations. As shown in Table 2, our
results surpass others by a large margin, showing the importance of learning
fine-grained cross-modal alignments.
#### 4.2.5 Zero-shot Video-Text Retrieval
Besides the image-text tasks mentioned above, our method can generalize to the
video-text retrieval task. We randomly select $m$ frames of the video input
and concatenate them to get an image-text sequence, then we feed them directly
into our image-text retrieval model. The performance in Table 3 is comparable
to others, demonstrating the cross-modal knowledge learned in our method is
semantic-rich.
#### 4.2.6 Localization Tasks
We conduct two localization tasks: object detection on COCO, and phrase
grounding on Flickr30k. For the text input on the object detection task, we
use a prompt composed of concatenated labels of COCO (e.g. detect: person,
bicycle, car, … , toothbrush). We adopt the output from the last layer of the
Table 4 has exciting results about SLAN on localization tasks. For example, in
the task of object detection with Vit-Base as the backbone, SLAN achieves
comparable results to GLIP using a larger backbone and 3M gold data, i.e.,
Swin-Large, only slightly worse on object detection task, but still slightly
better on grounding task. When applying a larger backbone Vit-Huge, our method
significantly outperforms all comparison methods.
### 4.3 Ablation Study
#### 4.3.1 Effect of Self-locator
Importance of learnable detection module. As shown in the first row of Table
5, we replace our self-locator with a frozen detector pre-trained on COCO
Detection, and the second row is the result of our learnable detector. We do
not load the detector with pre-trained weights, but only fine-tune on the
downstream task datasets. Our method improves on average about 0.5% and 2% on
COCO and Flickr30k’s image-to-text and text-to-image retrieval tasks,
respectively.
Number of region adaptors for region regression. The region adaptor performs
progressive regression on the regions output by the detector to provide more
accurate region localization for cross-modal understanding tasks. As shown in
Table 5, as the number of region adaptors increases from 0 to 3, the retrieval
performance can be significantly improved by an average of more than 3%.
Region filter for saliency prediction. Table 6 illustrates how the region
filter affects the performance on COCO and Flickr30k retrieval tasks.
Learnable detector is trained from scratch and the number of region adaptors
is set to 3. The first and second rows show that when the regions are sorted
by saliency score and selected by a certain number, we can achieve an
performance gain of 2% on each dataset. When using the saliency score
threshold, our region filter is able to remove redundant regions that
negatively affect cross-modal adaptation with higher performance.
#### 4.3.2 Computational Cost
Table 7 illustrates the computational cost of our method and other state-of-
the-art methods. Our method has the smallest amount of parameters and FLOPS
because in this experiments our vision backbone is a relatively lightweight
ResNet50. However, our retrieval performance significantly outperforms other
methods. As far as our concerned, our method is efficient and high-
performance.
Method | Backbone | Params(M) | FLOPS(G) | COCO
---|---|---|---|---
TR@1 | IR@1
BLIP | Vit-Base | 370 | 558 | 81.9 | 64.3
BLIP | Vit-Large | 810 | 1594 | 82.4 | 65.1
Coca | Vit-Huge | 2100 | 4103 | 83.0 | 65.5
Beit-3 | Vit-Huge | 1900 | - | 84.8 | 67.2
Ours | ResNet50 | 322 | 324 | 85.1 | 68.9
Table 7: Comparison of model parameter and FLOPS on cross-modal retrieval
task. The FLOPS is calculated with the image of 384x384 resolution. The
Backbone denotes the vision encoder.
### 4.4 Visualization Analysis
#### 4.4.1 Text-guided Region Adaptation
As shown in Figure 4, our region adaptor produces text-specific results with
relatively high confidence. When we change the detailed description of the
sentence, e.g., “a man in a red coat” to “a man in black pants”, the
interesting phenomenon is that the attention regions of our self-locator are
also shifted accordingly with relatively high confidence.
#### 4.4.2 Region adaptation during training phases
SLAN learns to localize accurate regions given input sentences. As illustrated
in Figure 5, as training progresses, the regions of interest outputed by our
model gradually become accurate. This demonstrates that our model can be
trained without grounding annotation and can gradually learn fine-grained
image-text alignment information.
|
---|---
Figure 4: Illustration of text-specific region adaptation. We colorize three words per sentence and use the corresponding colors to mark the regions with the highest matching scores. This denotes the interpretable region adaptation of our method, which brings fine-grained cross-modal feature fusion for downstream tasks. | |
---|---|---
0 | 1/2 | 1
Figure 5: Visualization of region proposal during pre-training. The numbers
under images denote the relative training duration, where $1$ represents the
whole pre-training procedure. The referenced text is ”A man in a green shirt
is doing the trick on a skateboard.” Figure 6: Coarse-to-fine process of the
region adaptation. We also list the matching score between the regions and
counterpart words.
#### 4.4.3 Coarse-to-fine Region Adaptation
To verify the calibration effect of region adaptation, we visualize an image
with its text in Figure 6. Model locates more accurate regions of interest
with higher similarity scores after three levels of region adaptor. This shows
that our self-locator hierarchically refines the related regions of provided
words.
## 5 Conclusions and Future Work
In this paper, we present the Self-Locator Aided Network (SLAN), which
incorporates the self-locator to adapt proposed regions for cross-modal
alignments without extra grounding (region-to-word) annotations. Experimental
results show that our method achieves a strong performance on many cross-modal
understanding tasks. Besides, the self-locator can also be transferred to some
localization tasks, and demonstrates its potential on them. We hope SLAN
provides a new direction for learning fine-grained cross-modal alignments
without relying on expensive grounding annotations.
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## Appendix A Appendices
### A.1 Details of Pre-training Dataset
We list the statistic of each dataset used for pre-training. The overall pre-
training dataset has 14M image-text pairs, without any fine-grained region-to-
word annotations.
| #Image | #Text
---|---|---
COCO | 113K | 567K
Visual Genome | 100K | 769K
SBU captions | 860K | 860K
Conceptual Captions | 3M | 3M
Conceptual | 10M | 10M
Table 8: The number of images and texts in our pre-training datasets.
### A.2 Load Detection Weights for Pre-training
Pre-trained | | COCO | | Flickr30k
---|---|---|---|---
Detector | | TR@1 | IR@1 | | TR@1 | IR@1
✘ | | 85.1 | 68.9 | | 97.4 | 88.2
✔ | | 85.6 | 67.7 | | 97.6 | 88.4
Table 9: Comparison on loading the pre-trained detection module. We set the
vision encoder as ResNet50.
We pre-train our SLAN from scratch in default. Besides, we also attempt to
load the pre-trained weights on COCO Detection [24] for the detection module
(DETR [5]), which provides more prior knowledges to learn the overall
architecture. As shown in Table 9, loading pre-trained detection module brings
0.5% and 0.8% on recall@1 of COCO image-text retrieval, while training from
scratch also performs relatively well. This also demonstrates that our SLAN
can learn rich cross-modal alignments without any pre-defined information.
Feature | | COCO | | Flickr30k
---|---|---|---|---
Source | | TR@1 | IR@1 | | TR@1 | IR@1
Encoder | | 82.4 | 66.7 | | 96.2 | 87.5
Decoder | | 85.1 | 68.9 | | 97.4 | 88.2
Table 10: Different source of multi-scale vision feature on SLAN.
### A.3 Multi-scale Vision Feature
We ablate with two kinds of multi-scale vision feature used in the region
adaptor. The default setting in the paper is to introduce a vision decoder to
provide rich visual semantic knowledge. To prove its necessity, we use the
vision encoder for generating multi-scale vision feature and list results in
Table 10. The experiment with decoder vision feature outperforms the other one
with more than 2% on recall@1 of COCO image-text retrieval.
### A.4 Visualization of Region Filter
|
---|---
Before Region Filter | After Region Filter
Figure 7: Illustration of the image before and after our region filter. The
caption is “A guy wearing shorts and a white t-shirt is skateboarding down the
road, while someone sits and watches him from the curb”.
As shown in Figure 7, our region filter selects salient regions for further
adaptation with words. It effectively focuses on informative regions.
BLIP | | | | |
---|---|---|---|---|---
Ours | | | | |
| Text: A lonely man heads off on a mountain trail wearing a
| bright red jacket.
Figure 8: Text-to-image retrieval on Flickr30k. We list top-5 retrieved images given query text for two methods, and highlight the ground truth with red box. |
---|---
MDETR | Ours
Figure 9: We compare our method with MDETR on phrase grounding in Flickr30k.
The query text is: A group of people are playing soccer on the field.
### A.5 Visualization of image-text retrieval and Phrase Grounding
We list the retrieval results of our SLAN and BLIP [20] for comparison. SLAN
can capture more fine-grained relation between the query text and images,
(_e.g.,_ red jacket in retrieved images). This leads to more accurate
retrieval performance and our method successfully marks the ground truth as
the top-1 retrieval result.
To show the detail-aware localization ability of SLAN, we also compare with
MDETR [14] in Figure 9. Both methods recognize the people in the image, while
our SLAN also locates the soccer with relatively high confidence.
|
# The Vanishing Decision Boundary Complexity and the Strong First Component
Hengshuai Yao
Sony AI
University of Alberta
<EMAIL_ADDRESS>
###### Abstract
We show that unlike machine learning classifiers, there are no complex
boundary structures in the decision boundaries for well-trained deep models.
However, we found that the complicated structures do appear in training but
they vanish shortly after shaping. This is a pessimistic news if one seeks to
capture different levels of complexity in the decision boundary for
understanding generalization, which works well in machine learning.
Nonetheless, we found that the decision boundaries of predecessor models on
the training data are reflective of the final model’s generalization. We show
how to use the predecessor decision boundaries for studying the generalization
of deep models. We have three major findings. One is on the strength of the
first principle component of deep models, another about the singularity of
optimizers, and the other on the effects of the skip connections in ResNets.
Code is at https://github.com/hengshu1/decision_boundary_github.
## 1 Introduction
Decision boundary is very useful for understanding the generalization of
machine learning classifiers. A decision boundary that can explain the
generalization of deep neural networks is an attractive and long-standing
challenge. Many efforts are related to this problem [25, 30, 21, 48, 2, 9, 53,
39, 40, 10, 62, 54, 51, 7]. We are going to review and discuss in details
later in Section 5. In a nutshell, most works rely on the adversarial samples
to characterize the decision boundary structures of deep neural networks.
We aim to understand the decision boundary on the training samples. Not only
is our method much simpler in methodology and computation, but also
understanding the test performance of classifiers from the training samples is
a fundamental problem. We found that, surprisingly, the decision boundaries of
well trained deep models are approximately linear. It is well known that for
the decision boundaries of machine learning classifiers, their complexity
grows as fitting more closely to the training data. However, such complex
structures do not exist for well trained deep models. The first plot of Figure
1 shows the case for VGG19 [61]. 111Following [62], we use the CIFAR-10 data
set [34] to visualize the decision boundary. Li et. al. [38] also used this
data set to understand generalization by visualization (via loss contour
plots). Near the end of training, the decision boundary is indeed shaping but
the complex boundary structures vanish shortly after they are formed.
Interestingly, the predecessor models (especially those achieving over 99.0%
training accuracy) demonstrate an evolutionary process in the decision
boundary, as shown by the rest of the plots. 222A video of the full evolution
of the decision boundary is provided in the supplementary materials. The
boundary complexity vanishing behavior has not been observed before in both
deep learning and machine learning to the best of our knowledge.
The major contributions of this paper are as follows.
Figure 1: The decision boundary undergoes an evolutionary process. The
boundaries are plotted for the final model and a number of predecessor models,
using the first two principle components found by running PCA over the
embedding features of all the training cats and dogs. The heat map is
plotted by first building a mapping from PCA(2) to the embedding space. See
the flowchart in equation 1. The Yellow/Dark-purple corresponds to the
probability of CAT/DOG close to one. Model: VGG19. Optimizer: SGD-anneal-lr.
* •
Our results show that the predecessor decision boundaries on the training set
are indicative of the final model’s generalization performance. Visually, the
models that are more self-centered, compact and have few overlapping samples
near the boundaries generalize better.
* •
We found that deep networks decide only with the first principle component in
the end. The first principle component (out of hundreds of PCA components)
grows stronger and stronger during training. Eventually, the first component
itself supports a wholly 100% training accuracy. This indicates that the first
principle component is associated with overfitting.
* •
There is a singularity phenomenon for optimizers. We observed that just a
little reduction in the rank of an auto-correlation matrix leads to a
significant generalization improvement when comparing deep learning
optimizers. For training the same model, the optimizers with a low rank of
this matrix generalize better than the optimizers with full-rank ones.
* •
The explained variance of the first component is much larger for VGG19 than
for ResNet18 and DLA. This shows the skip connections (used in both ResNets
and DLA) have an effect of balancing the dominance of the first component to
reduce the variances in label prediction. Furthermore, VGG19’s auto-
correlation matrix is singular, while for ResNets and DLA, the matrix is full
rank. This shows the skip connections effectively use all the feature
dimensions and extract more linearly independent features.
There are other interesting observations in this paper. In particular, there
is a trade-off between cluster size and class splitting for the learning rate
of the SGD optimizer. Adam has similar variances in the first two principle
directions to SGD with small learning rates. This explains why Adam has an
inferior generalization to SGD with the annealed learning rate.
## 2 Decision Boundary Evolution
### 2.1 The Decision Boundary Complexity Disappears
Let us start with how Figure 1 is plotted. The VGG19 model [61] is trained
with 200 epochs of SGD and a batch size of 128. The learning rate starts with
0.1 and decays according to the Cosine rule [44]. This leads to models that
are very well trained. The training accuracy is 100% and the test accuracy is
always above 93% across many runs in our experiments. For VGG19, the CAT class
has the lowest test accuracy, about 87%, while the highest of the other
classes is about 96%. For ResNet18, CAT is 91.5% while the best of the other
classes is about 97%. DOG is the second most mis-classified class on the test
set. The mis-classification rates between CAT and DOG are the highest among
all the class pairs, e.g., see [12]. It thus makes sense to focus on the
boundary between CAT and DOG.
Given a final and well-trained model, Principle Component Analysis (PCA) [70,
1] is performed on the joint feature matrix of cats and dogs. The feature
matrix is generated using the embedding features from the last layer of VGG19.
We visualize the training cats and dogs according to their two major
components in the first plot of Figure 1. To understand how the decision
changes in this space, we need to know the likelihood of any point in this
space being CAT or DOG.
Let PCA(2) $\subset\mathcal{R}^{2}$ be the space of the first two principle
components. Denote a vector in PCA(2) by $x$. We generate a feature vector
$\psi(x)\in\mathcal{R}^{d}$ such that it is the mean of the nearest neighbors
of $x$ in PCA(2):
$\psi(x)=\frac{1}{n}\sum_{i=1}^{n}\phi(x_{i}(x)),\quad
x_{i}(x)\in\mbox{PCA(2)},$
where $\\{x_{i}(x)\\}$ are the nearest neighboring training samples of $x$ in
the PCA(2) space. Note that $\phi(\cdot)$ is an inverse mapping of the
dimension reduction by PCA. For the training samples $\\{x_{i}\\}$, their
embedding feature vectors can be queried from the networks. We take their
original feature vectors for the mapping of $\phi(\cdot)$, instead of the
results from the inverse transform by PCA which have large approximation
errors instead. In this way, we build a mapping $F$ that generalizes $\phi$.
In particular, $F:$ PCA(2) $\to\mathcal{R}^{d}$, which is able to map any $x$
in PCA(2) back to the embedding space.
The dimension $d$ is just that of the embedding space. For example, $d$ is 512
for VGG19. This gives a way of querying the neural networks model for any
point in PCA(2). This process can be summarized by the flow
$x\stackrel{{\scriptstyle F}}{{\to}}\psi(x)\to classifier(\psi(x)),$ (1)
where the last step is simply performed by a linear operation,
$\psi(x)W^{T}+b$. Here $W,b$ are the parameters of the last layer. The softmax
operation over CAT and DOG is also performed in the $classifier$ function. The
probability of CAT is thus computed for each point in PCA(2). These
probabilities generate a heat map, which is plotted in the same space with the
training samples. This finishes the process of generating the first plot of
Figure 1. The rest of the plots in the figure are produced in the same way for
some immature models during training. We call them the predecessor models in
the view that they are earlier states of the final model.
There are a few observations of this figure. For the final model (100%
training accuracy), there is no complex structure in the decision boundary.
Instead, the boundary is approximately linear. Going back in training, we can
see that there are some blurry regions in the decision boundaries of the
predecessor models. These regions correspond to where the decision is
ambiguous. Especially note that they are located near where the cats and dogs
overlap with each other. Even though the overall training accuracy is descent
(e.g., 99.48%), there are still a significant number of dogs in the CAT area,
and vice versa. The objects of the same classes become more and more centered
and their spread size has a decreasing trend. This is especially true when
training is near the end (e.g., over 99% training accuracy) in our
observation.
The first observation is quite surprising given what we know for machine
learning classifiers. Usually classifiers that overfit have complex boundary
structures. 333See
https://en.wikipedia.org/wiki/Overfitting#/media/File:Overfitting.svg for an
illustration of how overfitting contributes to the complexity of the decision
boundary in machine learning. How come the well-trained deep models do not
have such complexity in their decision boundaries? It did not make sense to us
in the beginning. We further plotted the cats and dogs in the decision space,
which is shown by Figure 9 (see Appendix 7.1). This confirms the decision
boundary undergoes an evolution (in both the PCA space and the decision
space), and finally the boundary has no complicated structures indeed.
### 2.2 Predecessor Boundaries are Indicative
We have shown so far that the final model has no complex decision boundary
structures but during training they do appear when training is near the end.
Why is it important? We found that the decision boundaries of the predecessor
models are indicative of the final model’s generalization performance.
Figure 2 shows the decision boundaries of CAT versus all the other classes in
the PCA(2) space. For each class pair, a joint feature matrix is formed by
querying the network’s embedding feature output using the training samples of
the two classes and PCA is performed afterwards. The model is the 99.87%
predecessor model, which is the same one as used in Figure 1 and Figure 9.
Previously [12] showed that CAT and DOG have the poorest generalization on the
test set. Here this figure shows the CAT-DOG decision boundary is the most
crowded. In particular, this plot shows that there are many cats near the
boundary (the more cats in an area the more solid is the blue color because
the color is plotted using alpha equal to 0.6 transparency). Especially note
that many cats cross into the area of DOG. The CAT-DOG plot in Figure 1 (the
last plot in the first row; dogs are in the foreground instead there) shows
there are a significant number of dogs that cross the boundary with CAT too.
All the other classes have some sort of boundary with CAT. In particular, CAT
has a significant number of samples near the boundary with BIRD (fluffy vs.
feather), DEER (four-leg), HORSE (four-leg), and FROG (pointy ears vs. eyes
atop). This explains their interference on the test set [12] hereby using the
training data.
Figure 2: Decision boundaries between CAT and all the other classes. This is
plotted using the 99.87%-model (in Figure 1) and all the training objects.
Model: VGG19. Optimizer: SGD-anneal-lr. See the text in Section 2.2 for
details.
The above method can be extended to compare the decision boundaries of the
classes in a 3D space. This is shown in Appendix 7.2.
### 2.3 Rethinking the Decision Boundary for DL
One myth in deep learning is why deep neural networks overfit with such a
level of over-parameterization but they still generalize so much better than
machine learning classifiers. This seems to contradict with the common belief
in machine learning that over-parameterized models are usually more complex
and they easily catch the noises in data, leading to a low bias but a high
variance and thus a poor generalization. This dilemma seems odd indeed. See
[62] for a similar feel.
Belkin et. al. [3] has a nice study on the bias-variance trade-off especially
for deep learning. They uncovered a double descent phenomenon for deep
learning. The classical bias-variance trade-off for machine learning is
described by the well-known “U” shape of the test risk in terms of the
capacity of the model. Belkin et. al.’s discovery is a double-“U” shape.
Following the first “U” shape, as the capacity of the model keeps growing into
over-parameterization, the test risk deceases again due to the combined effect
of a decreasing variance and a low bias. Somepalli et. al. [62] confirmed this
finding with a study of the width parameter of network layers. They also
studied in particular the transition between under- and over-parameterized
models. Their finding is that the “instabilities” in the decision boundary are
the main reason of double descent. Note, though, the decision boundary they
used is in the space of individual samples, which is different from ours. See
Section 5 for detailed discussions.
Our results indicate that the standard understanding of decision boundary in
machine learning explains deep neural networks with some differences. 444This
is implied by the double descent phenomenon too. Recall our results show that
the final decision boundary is approximately linear and there is no complex
boundary structures. Linear decision boundary is the simplest and it
generalizes well with good features. Perceptrons [46, 49], Support Vector
Machine [11, 65], Kernel methods [6, 58, 29] and Logistic Regression [47, 22]
all greatly advanced Artificial Intelligence and some of them are still widely
used in practice. Multi-layered Perceptrons [56], LSTM [17] and convolution
neural networks [35, 61, 24, 18] all use linear operation as an elementary
composition at layers. Before the recent success of deep reinforcement
learning, reinforcement learning algorithms with linear function approximation
have supported the field for decades [68, 67, 5, 36, 64, 66, 4], and it has
shown excellent generalization with sparse and CMAC encoded features [63].
Silver’s Ph.D thesis is built on Computer Go programs from linear value
function approximation [59]. Some program evaluates the Go game board from a
linear combination of millions of binary features [60]. There is nothing wrong
with linear methods across fields and in fact they generalize very well. Their
limitation is the requirement of good features.
For deep neural networks, the linearity of the final decision boundary is
surprising given the huge number of parameters. We think this is the most
fascinating part of deep neural networks: the excessive over-parameterization
does not lead to highly complex decision boundary structures. This finding
suggests that deep neural networks are structured in a way that enables
learning very good features who finally land in a linear space.
For the final model, the large margins between classes like CAT and DOG
indicate that all the classes are linearly separable in both the embedding
space and the decision space thanks to the deeply learned features. Thus the
decision boundary of the final well-trained model has a very simple linear
structure. This can be observed from the final model in Figure 1 and in Figure
9 for the CAT-DOG plot, and the 99.87% model for the CAT-PLANE, CAT-CAR and
CAT-SHIP in Figure 2. All the class pairs 555Other class pairs are not shown
because their boundaries look more clean. can be easily separated by a
straight line in the PCA(2) space for the corresponding model.
DL decision boundary != ML decision boundary: Overfitting in deep learning has
a very different effect on the decision boundary from machine learning.
Overfitting in machine learning leads to complex decision boundaries. As the
complexity grows to some level, the generalization of machine learning
classifiers deteriorates. However, our results show that the decision boundary
of the final well-trained model by deep learning is linear in the embedding
space. In our opinion this is one reason that deep neural networks generalize
much better than machine learning classifiers in many application areas. In
fact, “overfitting” in deep learning has a beneficial effect for
generalization as discussed in Section 3.2.
It is common to examine the final models for machine learning classifiers once
training finishes. However, the decision boundary of the final well-trained
deep models cannot explain class interference [12], e.g., why CAT and DOG
generalize much more poorly than the other classes even though they are also
linearly separable in the PCA(2) space. Our main result is that the
predecessor boundaries are indicative of the final model’s generalization. The
slowness in carving a clear separation between classes in training is strongly
correlated with the poor generalization between them at test time. We will dig
deeper into this slowness remark when studying the auto-correlation matrix in
Section 3.2.
## 3 Insights into Optimizer Generalization
### 3.1 From the Predecessor Boundary
Creating algorithms and models that generalize better is the heartbeat of deep
learning. We show that comparing the predecessor decision boundaries on the
training data gives us insights into their generalization performance. This
section compares the effects of optimizers on models, and the next section is
focused on the architectures.
We start with comparing models trained with different optimizers. SGD-anneal-
lr. The learning rate starts with an initial value of 0.1, and then decays
according to a Cosine rule. This is the optimizer used for the results
presented in Section 2. The momentum rate is 0.9, and the weight decay is
$0.0005$. The same momentum rate and weight decay are also used in the
following optimizers. For SGD-big-lr and SGD-small-lr, the learning rate is
0.01 and 0.0001, respectively. Adam [33]: with default parameters in PyTorch.
The model generating the results presented in this section is VGG19.
Figure 3: This shows the effect of the learning rate for SGD, in shaping the
decision boundary between CAT and DOG. The Adam optimizer is also compared.
Model: VGG19. See the text in Section 3.1 for details.
Figure 3 shows that the effect of the learning rate on the decision boundary.
The overall test accuracy and the recall rates of CAT and DOG are shown in the
plots as well. All optimizers are trained 200 epochs.
First, let’s take a look at the plots using the training data in the first
column. Small learning rates are very good at splitting classes with clean
boundaries. However, the resulting cluster per class is much larger in size
than using the big learning rate. Note the different ranges in the $x,y$ axes.
Though, for the big learning rate, the splitting is poorer. The annealed
learning rate leads to much smaller clusters with a good splitting and an even
much better generalization. Note the large CAT areas behind the foreground DOG
in the second-row plots. It is well known that small learning rates generalize
poorly in deep learning. The reason was previously explained by the sharpness
of the minima [28, 32, 13, 31, 23, 12]. This is the first time uncovering the
effect of the learning rate on the cluster size.
Why is a big cluster bad? A big cluster due to small learning rates means the
variances of the training samples are high (along the first two principle
components). Intuitively, a big class cluster means that there are lots of
gaps among the training samples and on the edges. Such gaps are areas where
there are no training samples and thus the uncertainty is high there. At
testing time, the samples are likely to slip into these gaps, which incur high
variances in the label predictions. To validate this, we generate the same
plot using the test data, shown at the second column of the figure (except the
last row, which will be detailed later). For SGD-small-lr, there are many test
samples on and across the decision boundary, which is very different from the
plot on the training set. The color plot is generated using the transparency
parameter alpha equal to 0.2 for DOG (foreground) and 0.6 for CAT
(background). So intense red color means there are lots of dogs for the areas.
The big-lr optimized model, instead, has much fewer dogs (and cats) near the
test boundary. The SGD-anneal-lr optimizer further gathers most test cats and
dogs in two corners, leading to the highest recall rates for the test cats and
dogs as well as the overall test accuracy.
The Adam optimizer has an interesting boundary structure. In particular, the
spread range of the training cats (in blue) is small in the $x$-axis (the
first component direction), similar to the annealed learning rate. Note the
different ranges in the $x$-axis. However, the spread of both cats and dogs in
the $y$-axis and the spread of dogs in the $x$-axis are both much wider, in
fact, similar to SGD-small-lr. In addition, the boundaries of all the SGD
optimizers are roughly aligned with the axes. However, for Adam, the training
samples are instead rotated. There are also lots of overlapping samples near
the boundary for the test data. These observations help understand why the
final model by Adam is inferior to SGD-anneal-lr in generalization.
Finally, the last row shows two predecessor models of Adam. This shows that
the predecessor models on the training data are reflective of the decision
boundary on the test data for Adam optimizer too. First, as shown by the last
second row, the final model has very clear boundary structures on the training
data, which is quite different from on the test data. Second, the testing plot
is more similar to the training plot of the 99.5%-model than to that of the
final model. This means predecessor models give us more clues about
generalization than the final model, which is consistent with the case of the
SGD optimizer. This is another verification that the predecessor models are
indicative of generalization.
### 3.2 The Singularity of Optimizers
In Section 3.1, we plotted the predecessor boundaries for understanding the
generalization capabilities of SGD and Adam optimizers. This section presents
deeper insights into the generalization of the optimizers in terms of spectral
properties. This draws on the “slowness remark” at the end of Section 2.3.
Quantifying the spectral properties of the last layer gives us insights into
how fast the training has converged. It appears that, as we will show later,
the rate of convergence in training is strongly correlated with the
generalization performance.
Figure 4: Optimizer profiling in terms of generalization and the spectrum of
singular values of the auto-correlation matrix. The colored text number is the
test accuracy of a VGG19 model trained by a corresponding optimizer. The
snapshot in the middle shows the first two singular values. See the text in
Section 3.2 for the description of the “singularity” phenomenon and other
observations.
Let a feature matrix be $\Phi\in\mathcal{R}^{n,d}$, where $n$ and $d$ are the
numbers of samples and embedding features, respectively. The squared singular
values of $\Phi$ are just the eigenvalues of matrix $A=\Phi^{T}\Phi$. This
matrix $A$ is fairly important. In fact, $A$ is often called the auto-
correlation matrix, which is often used for studying the convergence rate of
SGD with linear function approximation in neural networks (known as the
Widrow-Hoff or the Delta rule) [69] and signal processing (known as the least-
mean-squares filter) [43].
Optimizing the networks up to convergence reveals the generalization
capabilities of the optimizers because at convergence the data is also
overfit. Thus we cover all the optimizers (in torch.optim) that can be trained
to converge by learning rate annealing. Optimizers that can not be well
trained may be at a random state even training finishes. For example, SGD-big-
lr is in an oscillation state at the end of training. Taking the auto-
correlation matrix of such models thus compares different states of the
optimizers, which is not good and thus it was avoided by us at first.
We train the same model (VGG19) using different optimizers. Each optimizer is
applied with learning rate annealing in 200 epochs with a mini-batch size of
128. Figure 4 profiles ten optimizers in terms of the final test accuracy. For
each optimizer, we also plot the spectrum of the singular values of the auto-
correlation matrix for the CAT and DOG samples in the training set. The
majority of the optimizers are on the right side. RMSprop [27] and NAdam [14]
(combining Nesterov accelerated gradient [52] with Adam) are on the left side.
If we follow the green arrows from both sides, there is a trend of
generalization improving whilst the rank of this matrix gets close to some
intermediate number. (We will discuss the RAdam optimizer later).
There is an interesting singularity phenomenon. Reducing the rank just a
little (by Adadelta [72] and Adam [33]) leads to a significant generalization
improvement (91.9% and 92.6%) over the full-rank cases (ASGD and Adagrad, both
90.6% test accuracy). AdamW [45] and the SGD-anneal-lr further reduce the
rank, which leads to more generalization improvement. The RMSProp optimizer
reduces the rank to less than 10% of the feature dimension and yet the
generalization is descent. A large first singular value ($\sigma_{1}$) is a
strong indicator of poor generalization, e.g., see ASGD [55], Adagrad [15], as
shown by the small blue window that sits in the middle of the plot.
Figure 5: Similar to Figure 4, except all the non-SGD optimizers are their
default settings without learning rate annealing. Every optimizer generalizes
worse than their counterpart in Figure 4. Note the optimizers in Figure 4 are
aided with learning rate annealing and they have higher training accuracies
(all above 99.9%). Thus this shows that all the (non-SGD) optimizers
generalize better when the data is overfit. The observations for Figure 4
follow similarly for the green arrow trend and the singularity of optimizers.
In general, this figure shows when comparing different optimizers on the same
model architecture, the rank of the auto-correlation matrix of the most
interfering class pair is indicative of the generalization performance, and so
is the $\sigma_{1}$. In fact, for SGD-anneal-lr, AdamW, NAdam and RMSprop, the
tailing singular values are all very close to zero. This suggests these
optimizers automatically filters out a significant number of noisy components
in the embedding space. This is interesting because in machine learning, it is
well known that dimension reduction captures major features in the data [8,
57, 16, 22, 42], which leads to models that are more robust with better
generalization.
We noted that the RAdam optimizer [41] is an exception. It has a full-rank
auto-correlation matrix and yet the generalization is very good (93%). We
think this is due to that the first and second singular values are of similar
magnitudes, and the second principle component helps with the generalization.
Out of the ten compared optimizers, RAdam is the only one that has large and
close first two singular values. Adamax [33] is also more balanced between the
first and second singular values than ASGD and Adagrad and generalizes better
than them.
We did run another experiment of training all the non-SGD optimizers without
learning rate annealing. None of them is able to train to 100% accuracy. Their
training accuracies are typically between 99.3% to 99.7%, except Adagrad
reaches 99.91%. This means all of them are not overfit comparing to SGD-lr-
anneal (100% training accuracy). Their test accuracy is shown in Figure 5.
Surprisingly, the overfitting SGD-lr-anneal model beats all these less
overfitting optimizers in generalization.
Note that Figure 4 has learning rate annealing applied to all the non-SGD
optimizers. They typically reach a training accuracy higher than 99.99%, which
is very much overfitting. Comparing the test accuracy numbers across the two
figures, we can see that for each of the non-SGD optimizers, the overfitting
implementation (Figure 4) has a better generalization than the less
overfitting one (Figure 5). This verifies that overfitting in deep learning is
distinctive — many optimizers generalize better when they are trained very
close to 100% accuracy. “Overfitting” in deep learning has a different effect
from machine learning, and perhaps we need a better name for it in deep
learning. See Section 2.3 for more discussions about this.
The relation of the test accuracy versus the rank of the auto-correlation
matrix (as indicated by the two arrow directions) in Figure 5 is similar to
that in Figure 4, except that there is randomness for the ordering of AdamW
and RAdam due to the models are not convergent. Without learning rate
annealing, Adam reduces the rank significantly. It generalizes better than
ASGD and Adagrad (both are full rank). However, it is still inferior to SGD-
lr-anneal, which has a larger rank. This suggests that there may be an
intermediate, optimal rank number, and it’s not like “the lower the rank, the
better generalization”.
Figure 6: Comparing the 99.5% predecessor decision boundaries of VGG19,
ResNet18 and DLA on the training data. For the VGG19 model, the samples are
less compact, while for the ResNet18 and DLA models, the samples are
distributed in a much more oval shape. The boundary complexity of ResNet18 is
lower than DLA.
## 4 Understanding ResNet
The Residual network (ResNet) is an extremely popular architecture for making
deep layered network models. At the time of writing, the ResNet paper [24]
received 136837 citations. The skip connection proposed in ResNet is widely
adopted in designing deep models for many applications nowadays. However, why
ResNets generalize well is still poorly understood. This section provides a
study into this question starting from the predecessor decision boundaries. In
particular, we compare VGG19, ResNet18 and DLA [71], optimized with SGD-
anneal-lr. DLA is included because it extends ResNets into a tree structured
hierarchy of layers with skip connections both within and across trees. We aim
to understand the effects of the skip connections on the decision boundary and
generalization.
Figure 6 shows the decision boundaries of VGG19, ResNet18 and DLA, plotted
using their predecessor model at the 99.5% training accuracy. The overall
accuracy and the CAT and DOG recall rates on the test set are shown in the
figure as well. The test accuracy of the final models is also summarized in
the table. In terms of the generalization performance, ResNet18 is better than
DLA, and DLA is better than VGG19. Both ResNet18 and DLA models have better
self-centerness in that the samples are more clustered, while for VGG19 the
samples form a less oval shape. The ResNet18 model has cleaner boundaries than
DLA.
Figure 7: Evolution of the largest explained variances $\sigma_{1}^{2}$,
$\sigma_{2}^{2}$ and $\sigma_{3}^{2}$ for the training cats and dogs.
Optimizer: SGD-anneal-lr.
We were wondering why ResNets and DLA have nicer boundaries. How does the skip
connection help with the decision boundary in particular? Figure 7 shows the
evolution of the explained variance during training for VGG19, ResNet18 and
DLA. The plot confirms that the first component becomes very much dominant in
the end because the feature variances are mainly explained by $\sigma_{1}$ .
In particular, $\sigma_{1}$ grows larger and larger for each model in
training. Eventually the first component itself supports 100% training
accuracy for all the three models.
VGG19 has a much bigger $\sigma_{1}$ than ResNet18 and DLA. The growth curve
of $\sigma_{1}$ is also much more steep for VGG19. This shows that skip
connections have an effect of reducing the dominance of the first component.
Between ResNet18 and DLA, the explained variances are fairly close.
Figure 8: Spectral profiling of ResNet18 with different optimizers. One
difference from VGG19 (in Figure 4) is that here all the auto-correlation
matrices are full rank. Bringing the auto-correlation matrix close to
singularity in general still leads to better generalization. For example, SGD-
anneal-lr continues to perform the best due to that it brings the matrix near-
singular.
Figure 8 shows the spectral profile of ten optimizers for ResNet18. Note that
for VGG19, the majority of the optimizers have a singular auto-correlation
matrix, as shown in Figure 4. However, with ResNet18, this matrix is full rank
for most of the optimizers. This shows that the skip connections in ResNets
have an effect of extracting more linearly independent features. This
indicates that the feature dimension quota (512) is not effectively used by
VGG19; and many features are linearly dependent on each other. For example,
for SGD-anneal-lr, ResNet18’s smallest singular value is several orders larger
than VGG19. We checked the rank of the auto-correlation matrix. For VGG, it is
439 for all class pairs. For ResNet18, it is 512 (full rank) for all pairs.
Note this is reverse to the comparison of optimizers on the same model. In
that case, reducing the rank of the auto-correlation matrix tends to improve
the generalization of optimizers. However, for different architectures, it
appears models that extract more linearly independent features generalize
better.
About the ordering of the optimizers, in this full-rank case, though certain
optimizers (ADAMW and NADAM) in the rectangle are not exactly ordered in the
traversal direction of the upper green arrow in Figure 4, their test
accuracies are close. If we view the optimizers in the rectangle as a group,
the green arrow traversal ordering of the generalization still holds the same
as VGG19. RMSProp still outperforms ASGD and Adagrad with a much lower rank.
Thus in general, for optimizing the same model, the optimizers that have a
(near-)singular or low-rank auto-correlation matrix tend to generalize well.
The largest singular values may play a role too. For example, the zoom view in
Figure 8 shows that ASGD and Adagrad have the top-2 $\sigma_{1}$s and they
have the poorest generalization. The ordering of $\sigma_{1}$ is not the same
as the generalization performance though. Adagrad (with a larger $\sigma_{1}$)
actually generalizes better than ASGD. Note that Adamax is more balanced
between the first and second largest singular values than ASGD and Adagrad for
a better generalization. This was also observed for RAdam in the VGG19
experiment. In a summary, the generalization performance of optimizers is
strongly correlated with the spectral properties of this correlation matrix,
and a due evaluative measure is yet to be discovered.
## 5 Related Work
The popular approach in characterizing the decision boundary of deep neural
networks is by adversarial attacks, which applies some perturbation to input
images such that the label predictions are changed. This technique is known as
Generative Adversarial Networks (GAN) [19, 20, 50]. The connection between GAN
and decision boundary is that in order to generate adversarial samples, they
need to cross the decision boundary.
This fact has been used in a few works on the decision boundary of deep
classifiers, e.g., see [25, 30, 48, 39]. For example, Karimi et. al. [30]
generated adversarial samples, which are ambiguous to classifiers, i.e., with
equal probabilities of label predictions given an object. Mickisch et. al.
[48] used DeepFool [50] to generate adversarial samples and study the decision
boundary.
Guan and Loew [21] proposed a metric to evaluate the complexity of decision
boundary using adversarial samples that are generated near the boundary. The
key of this metric is to form a feature matrix of the adversarial samples, and
then compute the Shannon Entropy of its eigenvalues.
Lei et. al. [37] presented a theoretical analysis on the decision boundary
(DB) variability by ($\epsilon,\eta$)-data DB variability using a subset of
training data ($\eta\%$) for a reconstruction error of $\epsilon$. They also
studied the DB variability with respect to algorithms, such as training time,
sample sizes, and label noise ratios.
Somepalli et. al. [62] proposed a special decision boundary, which is defined
by plotting the predicted labels of synthetic samples in a 2D plane spanned by
three (real) basis images.
The decision boundary is useful beyond understanding the generalization of
deep neural networks. See Appendix 7.3 for other use cases.
## 6 Conclusion
The decision boundary structure is transient for deep models. Our major
findings are: (1) The tail of the boundary evolution is indicative of the
generalization performance of deep models. (2) The first component of deep
models tend to grow dominant in training and in the end, it suffices to fit
the data very well by itself. (3) Optimizers have a singularity phenomenon of
rank reduction in the auto-correlation matrix and improved generalization. (4)
The skip connections have the effects of balancing the dominance of the first
principle component and effectively using all feature dimensions.
## 7 Appendix
### 7.1 Decision Boundary in the Decision Space
The decision boundary evolution in the decision space is shown in Figure 9.
With the final model, the samples are not only separated well but the model is
very affirmative — almost all the dogs and cats are predicted with a
probability close to one. Thus studying the final model entails no decision
boundary structure, and at a moment it seemed impossible to us to use the
training data for studying the decision boundary. Indeed, Ortiz-Jimenez et.
al. [53] showed that “the decision boundaries of a neural networks can only
exist when the classifier is trained with some features that hold them
together”. It is common to define the decision boundary as the set of samples
for which the label prediction is equiprobable [48, 30]. The problem of
applying this definition is that there are no training samples that are
ambiguous.
Figure 9: The decision boundary vanishing phenomenon illustrated in the
decision space. The $x$-$y$ plane shows the predicted probabilities of CAT and
DOG for the training cats and dogs (5000 each class). The final classifier
clearly separates the two classes without ambiguity and there is no complex
decision boundary structure. However, the complexity does appear for the
predecessor models. Model: VGG19. Optimizer: SGD-anneal-lr.
### 7.2 Decision Boundary in 3D Space
Visualization in 3D is convenient because it allows to zoom and rotate when
inspecting the boundaries between classes. We form a feature matrix from the
training objects of a class triple. Then PCA is applied to give three major
components, which are the space to plot the samples. Figure 10 shows a few
triples. The complexity of the boundaries in terms of the overlapping of
training samples near the boundaries are consistent with the generalization
performance on the test set.
Figure 10: Decision boundaries in the PCA(3) space from the feature matrix of
the class triples in the training set. In particular, plot (1) shows planes
are clearly separated from both cats and dogs. (2) CAT-DOG boundary is more
crowded than CAT-FROG boundary, and CAT-FROG boundary is more complex than
DOG-FROG boundary. (3) Cars are clearly separated from cats and dogs. (4) CAR-
TRUCK is more complex than CAR-PLANE. (5) Cats are clearly separated from cars
and trucks. (6) SHIP-TRUCK boundary is more complex than SHIP-CAR boundary,
probably because SHIP is closer in size to TRUCK than CAR. These triples were
chosen to show because CAR, TRUCK, PLANE, and SHIP are all metallic bodies and
they interfere. These observations are consistent with the cross-class
generalization between classes on the test set [12].
### 7.3 Use Cases of Decision Boundary
Besides understanding the generalization of deep neural networks, there are
also other use cases for the decision boundary. For example, Heo et. al. [26]
used boundary samples from adversarial attacks to train a student network for
the purpose of knowledge distillation.
Alfarra et al. [2] found that the decision boundary of a simplest networks
(affine-relu-affine) is a polytope, and showed how to use this geometric
representation of the decision boundary for network pruning (i.e., reducing
number of parameter by sparse matrices) and adversarial attacks.
Choi et. al. [10] generated synthetic boundary supporting samples for the
purpose of model quantization without access to the original training data.
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|
# PaCMO: Partner Dependent Human Motion Generation in Dyadic Human Activity
using Neural Operators
Md Ashiqur Rahman1, Jasorsi Ghosh1, Hrishikesh Viswanath1, Kamyar
Azizzadenesheli2, Aniket Bera1
1Department of Computer Science, Purdue University, West Lafayette, Indiana.
2Nvidia Corporation, Santa Clara, CA.
{rahman79, ghosh117<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
We address the problem of generating 3D human motions in dyadic activities. In
contrast to the concurrent works, which mainly focus on generating the motion
of a single actor from the textual description, we generate the motion of one
of the actors from the motion of the other participating actor in the action.
This is a particularly challenging, under-explored problem, that requires
learning intricate relationships between the motion of two actors
participating in an action and also identifying the action from the motion of
one actor. To address these, we propose partner conditioned motion operator
(PaCMO), a neural operator-based generative model which learns the
distribution of human motion conditioned by the partner’s motion in function
spaces through adversarial training. Our model can handle long unlabeled
action sequences at arbitrary time resolution. We also introduce the
"Functional Fréchet Inception Distance" $\text{F}^{2}\text{ID}$ metric for
capturing similarity between real and generated data for function spaces. We
test PaCMO on NTU RGB+D and DuetDance datasets and our model produces
realistic results evidenced by $\text{F}^{2}\text{ID}$ score and the conducted
user study.
## 1 Introduction
Generating 3D human motion is widely studied and is a core component of
animation, games, human-robot interaction, and AR/VR applications. Despite
many efforts from the community, the problem of modeling 3D human motion is
still poorly understood. Most of the works are focused on generating the
motion of a single person [29, 45, 50, 2]. These works often generate human
motion without any interaction with the surrounding objects and rarely model
multi-agent interactions. Modeling the motion of multiple people in an
interactive setting is particularly difficult.
Recent works [29, 45, 43, 30] primarily deal with generating motion of a
single human with or without conditioned by action labels and sentences
describing the action. The advancement of joint language-image models (such as
CLIP) enables the production of motion sequences for unseen actions [45, 14].
But these models can only generate motion sequences of a fixed length at a
specific time resolution. To synthesize lengthy (long duration) human motions
many prior works assume an autoregressive or Markovian assumption [1, 9, 54,
19, 2], which lead to a static state after a certain time. Many recent works
[29, 43, 30, 45, 3] have utilized attention mechanisms [47] to capture non-
local dependencies in long motion sequences. Though these models can generate
long sequences but often in practice the produced sequences are of fixed
length at a particular resolution. They can only generate human motion at a
fixed time resolution which depends on the training data. But time continuous
3D human motion generation is crucial for animation and video games. At the
same time, the problem of multi-person motion generation in interactive
situations is under-explored. Though there have been attempts to address this
problem in recent works [43, 24], these methods require explicit action labels
or sentences describing the action which may be impractical for AR/VR or HRI
applications where we do not have explicit action labels. Rather, in many
cases, we need to model the motion of digital humans interacting with real
humans in arbitrary dyadic activities. In these situations, we need to
generate the motion of a virtual human only from the motion of participating
humans such that virtual humans can correctly collaborate with the intended
activity of the human partner.
To address these problems, inspired by the recent development of generative
modeling in functional spaces [33], we propose PaCMO, a conditional generative
adversarial neural operator model to generate 3D human motion for dyadic
activity from the movements of the other interacting human, i.e. the partner.
To the best of our knowledge, PaCMO is the first to propose a time-continuous
generative architecture to solve this problem. We model human motion sequence
as a stochastic function of time. Given the motion of one of the actors in a
dyadic activity, our model learns a push-forward measure from a Gaussian
random field [6] to the motion sequence of the other actor. Gaussian random
field or Gaussian processes (GP) are continuous functions such that any finite
collection of points on the function constitutes a multi-variate Gaussian
distribution. Unlike traditional deep neural networks, neural operators are
resolution invariant. As a result, our model can generate continuous time
human motion and is not restricted to the resolution of the training data. And
in evaluation time, PaCMO can generate motion at a resolution demanded by the
application (e.g., matching require frame-rate for video game application).
PaCMO can efficiently model global dependencies in long sequences and is not
limited by the receptive field of discrete convolution [50]. Also, PaCMO
outputs the entire motion sequence at once, making it suitable for real-time
applications and also avoiding the issue with autoregressive and Markovian
models.
Evaluating generative models is nontrivial and subject to human perception.
Fréchet Inception Distance (FID) [13] is a well used metric to quantitatively
evaluate generative models. To obtain a closed-form solution of Fréchet
distance, features from both generated and real data are modeled using
multivariate Gaussian distributions. But finite dimensional multivariate
Gaussian distributions cannot model features from infinite dimensional spaces.
Also, such FID is not resolution invariant, as a fixed finite dimensional
Gaussian distribution cannot model feature vectors of different sizes. As a
result, FID is ill-suited to evaluate generative models in infinite
dimensional spaces. To address this, we remove the finite-dimensional
assumption and propose Functional FID or $\text{F}^{2}\text{ID}$, which is
designed for general vector spaces and is resolution invariant.
In this work, we consider human motion as a sequence of skeletal poses but our
model does not depend on any particular representation of human motion and can
easily be extended to other representations, such as SMPL [23]. We have
evaluated our model on two widely used datasets: NTU-RGB+D-120 [22] and
DuetDance[18] by both quantitative metrics and perceptual human evaluations.
Our contributions can be summarized as follows:
* •
We propose PaCMO, a novel deep neural operator for generating partner-
dependent 3D human motion in interactive dyadic human activities without any
action label or sentence describing the action.
* •
We introduce a novel conditional generative neural operator which is a
resolution-invariant conditional generator for infinite dimensional spaces.
* •
We propose ($\text{F}^{2}\text{ID}$), an evaluation metric for generative
models in function spaces.
* •
Our model PaCMO can generate promising results outperforming baselines, taking
the first step in partner-dependent human motion generation in an interactive
setting.
In the rest of the paper, in section 2 we discuss related works for human
motion generation and introduce neural operators to the readers. In section 3
we introduce the novel PaCMO architecture and ($\text{F}^{2}\text{ID}$) score.
We discuss the details of our experiment with results in section 4.
## 2 Related Works
### 2.1 Human Motion Generation
Human motion prediction and generation are well-studied problems in the fields
of kinesiology, robotics, computer vision, and computer graphics. Especially
the field of human motion generation has experienced rapid development
following success in generative modeling such as GAN [7], VAE [16],
normalizing flow [15, 27], and diffusion [5] based models. The task of human
motion generation can be broadly divided into two categories : (a) conditional
human motion generation, where the generation is guided by external variables
such as action labels, action descriptions, or speech. (b) unconditional human
motion generation, where we aim to model the entire space of human motion.
Earlier works in unconstrained motion generation primarily depend on
autoregressive models [10, 51, 54]. CSGN [50] learns a mapping from latent
vector sampled from the Gaussian process to human motion sequence by a series
of convolution operations in an adversarial way. But due to the finite
receptive field, it has the inherent limitation to capture very long-term
dependencies and can only generate human motion at a fixed predetermined time
resolution. NeMF [11] directly extends the recent developments in neural
implicit represent action [31, 42] to generative modeling but such models are
extremely limited as the whole generated process which is infinite-dimensional
(continuous human motion) depends only on a finite-dimensional latent vector.
Generating human motion from action label or description largely dominates the
field on conditional motion generation [9, 2, 1]. Both ACTOR [30] and TEMOS
[29] use a transformer architecture to generate human motion, whereas the
former uses a pre-trained language model DistilBERT [36] to encode the
sentence description. GOAL [44] generates human motion with the specific goal
of grasping objects. TEACH [3] is also a text-conditioned human motion
generator but it provides temporal control over the generated motion.
MotionCLIP [45, 14] leverages CLIP [32] to generate unseen actions and body
shapes.
The problem of interactive multi-person motion generation is particularly
challenging and requires joint modeling of multiple human motions. ActFormer
[43] provides a framework to generate action-conditioned single and multi-
person motion sequences using a transformer architecture. MUGL [24] uses a
duration-aware autoregressive model to generate a multi-person motion sequence
from an action label.
Figure 1: The PaCMO generator architecture. It takes the motion of one actor
in a dyadic activity as the condition and generates the motion of the other
partner. The blue circles represent the concatenation operation in function
spaces. Figure 2: The PaCMO discriminator architecture. It takes the motion
of two people involved in an interactive dyadic activity as input and maps it
to a real number $r\in\mathbb{R}$.
### 2.2 Neural Operators
Neural Operators [20, 21, 17] are a new paradigm in machine learning that
learns a mapping between function spaces. Traditional neural networks can only
map between finite-dimensional euclidean space i.e., can only process finite-
dimensional vectors or sets. Neural operator overcomes this limitation and is
discretization invariant. Among the different parameterizations of the neural
operator, Fourier Neural Operators (FNO) [20] have proven to be extremely
effective in achieving state-of-the-art results in learning to solve different
partial differential equations (PDE). An FNO $\mathcal{G}$ can be expressed as
$\mathcal{G}:\\{a:\mathcal{D}_{A}\rightarrow\mathbb{R}^{d_{a}}\\}\rightarrow\\{u:\mathcal{D}_{u}\rightarrow\mathbb{R}^{d_{u}}\\}$
where $a$ is the input function and $u$ is the output function. A neural
Operator architecture consists of three principal components.
1. 1.
Point-wise operator $P$ which transforms the input function $a$ to higher
dimension spaces by mapping it to a function with higher co-domain dimension.
2. 2.
A series of $n$ stacked non-linear neural integral operator layer
$\\{L_{i}\\}_{i=1}^{i=n}$, where $n$ is a hyper-parameter of the model. Each
layer $L_{i}$ is composed of a global kernel operator and a point-wise
operator. The layer $L_{i}$ maps input function $v_{i}$ to the function
$v_{i+1}$, which can be described as
$v_{i+1}=L_{i}(v_{i})=\sigma(\int\kappa_{i}(x,y)v_{i}(x)d\mu(y)+W_{i}v_{i})$
where $\mu$ is the measure over the domain of function $v_{i}$. The kernel
integration is usually computed in the Fourier domain as
$\mathcal{F}^{-1}(R_{i}\cdot\mathcal{F}(v_{i}))$, where $\mathcal{F}$ is the
Fourier transform and $R_{i}$ is a complex-valued matrix representing kernel
$\kappa_{i}$. And the $\sigma$ is a non-linearity of choice. In practice GELU
[12] is widely used [20, 34].
3. 3.
The series of integral operators are followed by a projection operator $Q$. It
projects the output function of $L_{n}$ to match the desired output function
$u$.
Due to the ability to capture complex nonlinear dynamics, FNOs are being
employed to solve complex real-world problems such as CO2 injection [49],
weather modeling [28], and seismology [52]. Recently proposed GANO [33]
introduces a generative adversarial approach to learn probability on function
spaces generalizing GAN which is the first principled approach for generative
models in function spaces.
The problem of generating human motion for interactive activities conditioned
by the motion of other interacting humans is not well-explored. Often in
human-robot interaction, AR/VR, and VFX design, we need to generate the motion
of digital humans depending on the motion of the real actors. To the best of
our knowledge, PaCMO is the first work to address such an important problem
and we do that by introducing a conditional generative neural operator
architecture.
## 3 Methodology
In this section, we provide the mathematical formulation of our problem. Then
we formalize a general construction for the conditional generative adversarial
neural operator. Next, we propose the ($\text{F}^{2}\text{ID}$) score, an
adoption of Fréchet inception distance (FID) score for infinite dimensional
vector (function) spaces.
### 3.1 Problem Formulation
We consider human motion as a sequence of skeletal poses. In skeletal
representation with $J$ joints, the human pose or gesture can be represented
as a vector
$g=[x^{1},y^{1},z^{1},x^{2},y^{2},z^{2},\dots
x^{J},y^{J},z^{J}]\in\mathbb{R}^{(3\times J)}$
where $(x^{i},y^{i},z^{i})$ represents the location of $i$th joint in 3D
euclidean space. Let $\mathcal{M}$ be the motion space of humans such that for
$m\in\mathcal{M},m:t\rightarrow\\{g\in\mathbb{R}^{(3\times J)}\\}$, where
$t\in\mathbb{R}$ is the time. Then the probability space
$(\mathcal{M}\times\mathcal{M},\sigma(\mathcal{M})\times\sigma(\mathcal{M}),\mathcal{P}_{\mathcal{M}\times\mathcal{M}})$
represents the probability space of motions for all dyadic human activities.
In other words, for any
$(m_{1},m_{2})\sim\mathbb{P}_{\mathcal{M}\times\mathcal{M}}$, motion sequences
$m_{1}$ and $m_{2}$ form a valid dyadic human activity when
$\mathbb{P}_{\mathcal{M}\times\mathcal{M}}\in\mathcal{P}_{\mathcal{M}\times\mathcal{M}}$.
Also we assume that $(m_{1},m_{2})$ and $(m_{2},m_{1})$ represents the same
dyadic activity i.e., $m_{1}$ and $m_{2}$ are exchangeable.
Here we want to learn the conditional probability distribution
$\mathbb{P}_{\mathcal{M}|\mathcal{M}}$ such that for any $m_{2}\in\mathcal{M}$
and $m_{1}\sim\mathbb{P}_{\mathcal{M}|m_{2}}$, $m_{1},m_{2}$ constitute a
valid dyadic human activity. Note that we do not assume any particular time
discretization or length of the motion sequence.
### 3.2 Conditional Generative Adversarial Neural Operator (cGANO)
Let $\mathcal{U},\mathcal{A},$ and $\mathcal{C}$ be Polish function spaces
such that for any
$\forall
u\in\mathcal{U},u:\mathcal{D}_{\mathcal{U}}\rightarrow\mathbb{R}^{d_{\mathcal{U}}},~{}~{}\forall
a\in\mathcal{A},a:\mathcal{D}_{\mathcal{A}}\rightarrow\mathbb{R}^{d_{\mathcal{A}}},$
$\text{and~{}}\forall
c\in\mathcal{C},c:\mathcal{D}_{\mathcal{C}}\rightarrow\mathbb{R}^{d_{\mathcal{C}}}$
We define the space of operators $\bm{G}$ (Generators) such that for any
$\mathcal{G}\in\bm{G},\mathcal{G}:\mathcal{A}\times\mathcal{C}\rightarrow\mathcal{U}$
and we also define the space of functional $\bm{L}$ (Discriminators) that for
any $d\in\bm{L},d:\mathcal{U}\times\mathcal{C}\rightarrow\mathbb{R}$.
Let the probability space $(\mathcal{A},\sigma(A),\mathcal{P}_{\mathcal{A}})$
is induced by Gaussian Random Field (GRF) and
$(\mathcal{U}\times\mathcal{C},\sigma(\mathcal{U})\times\sigma(\mathcal{C}),\mathcal{P}_{\mathcal{U}\times\mathcal{C}})$
denotes the probability space on the product space
$\mathcal{U}\times\mathcal{C}$. We want to model the conditional probability
$\mathbb{P}_{\mathcal{U}|\mathcal{C}}$ with the push-forward measure of
$\mathbb{P}_{\mathcal{A}}$ under the map $\mathcal{G}$ given $\mathcal{C}$
i.e., $\bm{G}_{\mathcal{C}}\sharp\mathbb{P}_{\mathcal{A}}$. Hence we define
the Wasserstein distance between the two distributions as follows
$W(\mathbb{P}_{\mathcal{U}\times\mathcal{C}},\mathbb{P}_{\mathcal{C}}\cdot\bm{G}_{\mathcal{C}}\sharp\mathbb{P}_{\mathcal{A}})=\sup_{\begin{subarray}{c}d:d\in\bm{L}\\\
Lip(d)\leq
1\end{subarray}}\mathbb{E}_{\mathbb{P}_{\mathcal{U}\times\mathcal{C}}}[d]-\mathbb{E}_{\mathcal{G}\sharp\mathbb{P}_{\mathcal{A}},\mathbb{P}_{\mathcal{C}}}[d]$
(1)
where $\mathbb{P}_{\mathcal{C}}$ comes from the joint probability measure
$\mathbb{P}_{\mathcal{U}\times\mathcal{C}}$. We aim to find
$\mathcal{G}\in\bm{G}$ that minimizes the Wasserstein distance between the
true and generated probability measure. Assuming
$(\mathcal{U}\times\mathcal{C})^{*}$ denotes the dual space of
$\mathcal{U}\times\mathcal{C}$, we express the Lipshitz constraint as [8, 33]
$Lip(d)\leq 1\iff\|\partial d\|_{(\mathcal{U}\times\mathcal{C})^{*}}\leq 1$
Following the construction of GANO, we can rewrite the objective without the
constraint as
$\inf_{\mathcal{G}\in\bm{G}}\sup_{d\in\bm{L}}\mathbb{E}_{\mathbb{P}_{\mathcal{U}\times\mathcal{C}}}[d]-\mathbb{E}_{\mathbb{P}_{\mathcal{C}},\mathcal{G}\sharp\mathbb{P}_{\mathcal{A}}}[d]+\lambda\mathbb{E}_{\mathbb{P}^{\prime}_{\mathcal{U}\times\mathcal{C}}}(\|\partial
d\|_{(\mathcal{U}\times\mathcal{C})^{*}}-1)^{2}$ (2)
where
$\mathbb{P}^{\prime}_{\mathcal{U}\times\mathcal{C}}:=\lambda\mathbb{P}_{\mathcal{C}}\cdot\mathcal{G}\sharp\mathbb{P}_{\mathcal{A}}+(1-\lambda)\mathbb{P}_{\mathcal{U}\times\mathcal{C}}$
given $\lambda\sim U(0,1)$.
Now we construct the operator on the product space of
$\mathcal{A}\times\mathcal{C}$. Let’s decompose $\bm{G}$ as
$\bm{G}:=\bm{G}^{\prime}\circ\bm{G}_{con}$
where $\mathcal{G}_{con}$ defines concatenation operator such that for any
$\mathcal{G}_{con}\in\bm{G}_{con},\mathcal{G}_{con}:\mathcal{A}\times\mathcal{C}\rightarrow\mathcal{V}$.
Here the function space $\mathcal{V}$ is defined such that for any
$v\in\mathcal{V},v:(D_{\mathcal{A}}\times\mathcal{D}_{\mathcal{C}})\rightarrow\mathbb{R}^{d_{\mathcal{A}}+d_{\mathcal{C}}}$
and
$v(x\oplus y)=a(x)\oplus
c(y)~{}~{}\forall(x,y)\in(D_{\mathcal{A}}\times\mathcal{D}_{\mathcal{C}}).$
Where $\oplus$ denotes the concatenation of functions and $\bm{G}^{\prime}$ is
the space of operators mapping $\mathcal{V}$ to the target function space
$\mathcal{U}$.
In many practical scenarios, we want to learn the conditional probability
measure of the function space where the conditional variables are finite-
dimensional (e.g., action labels). In other words, we want to model
$\mathbb{P}_{\mathcal{U}|s}$ where $s\in\mathcal{D}_{s}$, where
$\mathcal{D}_{s}$ is a finite-dimensional vector space.
In such cases, we learn a mapping from $\mathcal{D}_{s}$ to $\mathcal{C}$. We
define the mapping space $\mathcal{E}$ such that for any
$E\in\mathcal{E},E:\mathcal{D}_{\mathcal{C}}\times\mathcal{D}_{s}\rightarrow\mathbb{R}^{d_{\mathcal{C}}}$.
For any $s\in D_{s}$ the map $E$ maps it to $c_{s}\in\mathcal{C}$ such that
$c_{s}(y)=E(y\oplus s)~{}~{}~{}~{}\forall y\in\mathcal{D}_{\mathcal{C}}$
In this work, we use U-NO [34], a U-shaped Fourier neural operator
architecture for the generator (see Fig. 1). For the discriminator, we also
use the U-NO architecture followed by a functional layer (see Fig. 2). A
functional layer maps functions to real numbers which can be described as
following
$r=\int_{D_{v}}\kappa_{f}(x)v_{n}(x)dx$
where $v:\mathcal{D}_{v}\rightarrow\mathbb{R}$ is the input function, $r$ is a
real number, and $\kappa_{f}:\mathcal{D}_{v}\rightarrow\mathbb{R}$ is a
learnable function in the functional layer implemented by a fully connected
neural network.
For conditional human motion generation, the function space $\mathcal{C}$ and
$\mathcal{U}$ is the 3D human motion space i.e.,
$\mathcal{C}=\mathcal{U}=\mathcal{M}$, i.e., for any operator
$\mathcal{G}\in\bm{G}$
$\mathcal{G}:\mathcal{A}\times\mathcal{M}\rightarrow\mathcal{M}$
where for any Gaussian process $a\in\mathcal{A}$, $m_{2}\in\mathcal{M}$, and
$m_{1}=\mathcal{G}(a,m_{2})$, $(m_{1},m_{2})$ constitutes a valid dyadic human
motion for two person.
### 3.3 Construction of ($\text{F}^{2}\text{ID}$)
Quantitative evaluation of high-dimensional generative models is an open
problem. Among different proposed metrics, Fréchet Inception Distance or FID
[13] is well used. Though it was first introduced to evaluate generative
models for images, it has been adapted for other generative tasks (e.g.,
motion, voice). FID measures the Wasserstein-2 distance between the
distribution of inception features of the generated and real images. For other
modalities, features from the real and generated data are extracted using
suitable models. To get a closed-form solution of Wasserstein-2 distance, the
extracted features from real and generated data are modeled by two
multivariate Gaussian distributions. Let $(\mu,\Sigma)$,
$(\mu^{\prime},\Sigma^{\prime})$ be the mean and covariance of real and
generated data respectively. Then the FID score is calculated as
$\displaystyle FID$
$\displaystyle=W_{2}^{2}({\mathcal{N}}(\mu,\Sigma),{\mathcal{N}}(\mu^{\prime},\Sigma^{\prime}))^{2}$
$\displaystyle=\lVert\mu-\mu^{\prime}\rVert_{2}^{2}+\operatorname{tr}\left(\Sigma+\Sigma^{\prime}-2\left(\Sigma^{\frac{1}{2}}\cdot\Sigma^{\prime}\cdot\Sigma^{\frac{1}{2}}\right)^{\frac{1}{2}}\right).$
But multi-variate Gaussian distributions are not suitable to model data (or
features) from infinite-dimensional function spaces or processes (both
stochastic and deterministic). Therefore, the FID score is not a suitable
metric for evaluating generative models in function spaces.
To address this problem, we introduce Functional FID ($\text{F}^{2}\text{ID}$)
scores. Here, we model the extracted feature by the Gaussian process. A
Gaussian process, $GP(m,k)$, is defined by a mean function $m$ and a co-
variance function $k$. Wasserstein-2 distance between Gaussian processes is
well defined and has a closed-form solution under the assumption that they are
indexed over a compact domain $X\subset\mathbb{R}^{n}$ with a metric $d(.,.)$,
$m\in L^{2}(x)$, and $k\in L^{2}(X\times X)$ [25]. If the features extracted
from real and generated data are modeled by two Gaussian processes
$GP(m_{1},k_{1})$ and $GP(m_{2},k_{2})$ with associated covariance operator
$K_{1}$ and $K_{2}$ respectively. Then the $\text{F}^{2}\text{ID}$ score is
defined as
$\text{F}^{2}\text{ID}=d_{2}(m_{1},m_{2})+Tr(K_{1}+K_{2}-2(K_{1}^{\frac{1}{2}}\cdot
K_{2}\cdot K_{1}^{\frac{1}{2}}))^{\frac{1}{2}}$ (3)
where for any operator $T$, $Tr(T)$ is the trace of the operator $T$ and we
take $d(.,.)$ as the metric induced by $L^{2}$ norm. In this work, we define
$GP$ over time $t$. Given the features from the generated and real motions, we
model them by the Gaussian processes $GP_{1}(m_{1},k_{1})$ and
$GP_{2}(m_{2},k_{2})$ respectively using moment matching. We evaluate mean and
covariance functions over a finite discretization of the domain. Finally, we
apply the Eqn.3 to calculate $\text{F}^{2}\text{ID}$ (details in
supplementary). Finite resolution computation of Wasserstein-2 distance is
proven to converge to true Wasserstein-2 distance with the increase of
discretization resolution [25, Theorem 8].
Figure 3: Samples of conditional generation of human motion by PaCMO. The top
row shows the input condition motion (Red and Green) on the left and the
complete motion on the right by combining the generated motion (in Magenta and
Blue) and input motion (Red and Green). The following rows only show the
complete motion combining the condition motions and generated motions from
PaCMO. Different task labels are indicated on the left. Though PaCMO does not
take action labels or descriptions as input, we can observe that PaCMO
generated motions interact with the input motions depending on the action.
## 4 Experiments
NTU-RGB+D
---
Methods | $\text{F}^{2}\text{ID}$ $\downarrow$ | Diversity Score $\rightarrow$ | MMD-A $\downarrow$ | MMD-S$\downarrow$ | APE (root joint)$\downarrow$ | AVE(root joint)$\downarrow$
Real Motion | 0.001 | 1.90 | 0 | 0 | 0 | 0
PaCMO | 0.002 | 1.91 | 5.8e-3 | 3.8e-3 | 0.03 | 0.23
GRU baseline | 0.351 | 1.92 | 7.4e-3 | 4.4e-3 | 0.05 | 0.35
Table 1: Performance NTU-RGB+D 120 dataset. $(\downarrow)$ denotes the lower
the score the better and $(\rightarrow)$ denotes the closer to the real
dataset the better. Each metric is calculated five times and the mean is
reported. DuetDance
---
Methods | $\text{F}^{2}\text{ID}$ $\downarrow$ | Diversity Score$\rightarrow$ | MMD-A $\downarrow$ | MMD-S$\downarrow$ | APE (root joint)$\downarrow$ | AVE(root joint)$\downarrow$
Real Motion | 1.6e-4 | 0.261 | 0 | 0 | 0 | 0
PaCMO | 4.0e-4 | 0.264 | 6.7e-3 | 6.67e-3 | 0.01 | 0.01
GRU baseline | 494.9e-4 | 0.190 | 23.36e-3 | 23.35e-3 | 0.04 | 0.08
Table 2: Performance DuetDance dataset. $(\downarrow)$ denotes the lower the
score the better and $(\rightarrow)$ denotes the closer to the real dataset
the better. Each metric is calculated five times and the mean is reported.
### 4.1 Datasets
For the purpose of our work, we need an interactive dyadic human activity
dataset with skeletal pose annotation. Unfortunately, such publicly available
datasets are scarce. We use two existing datasets: NTU-RGB+D 120[22] and the
DuetDance [18] datasets.
* •
NTU-RGB+D 120 is the extended version of original NTU-RGB+D [39] dataset. It
contains 120 action types (such as handshake, pushing) of 106 subjects. The
provided skeleton data contains 3D locations of 25 major body joints. For our
purpose, we only considered actions with two persons resulting in 26K
sequences (motion pairs). Though we do not clip any motion to restrict the
length, we are not modeling the movement of the finger, as it requires finer
precision noise-free data [44] and we exclude two joints from each hand
representing the tip of the finger and thumb.
* •
DuetDance The dataset is created from complex dance poses such as Cha-cha,
Jive, Rumba, Salsa, and Samba from the curated video tutorial on Youtube. Due
to the nature of the actions, the motion encompasses long-term correlations
between physical motions. From the curated videos the skeletal positions are
calculated using LCRNet++ [35]. The skeletal data contain 3D location of 15
major body joints. The dataset consists of one very long sequence ($\sim$ 48K
time steps) for each of the dance types. So to create more training instances,
we divide one long sequence into smaller sequences non-overlapping sequences.
### 4.2 Hyperparameters
We use U-shaped neural operator architecture [34] in both the generator and
discriminator network similar to GANO [33]. We use seven stacked integral
operator layers with GELU as non-linearity (see Fig. 1 and 2). We divide each
dataset into training and evaluation sets randomly at a ratio of $8:2$. For
each dataset, we train our model for 200 epochs with a single NVIDIA A100 GPU
on the training set. We used an ADAM optimizer with an initial learning rate
of $10^{-4}$ which is halved after every 50 epochs. We train our model on the
skeletal data represented as joint locations in 3D space. While training we do
not limit the length of the motion sequence by clipping. To maintain the
symmetry of the human body, We also add additional constraints on the
generator by penalizing asymmetric limb generations i.e., restricting the
generator to produce left and right legs of the same length with soft
constraints. Empirically we have found that these soft constraints increase
the convergence rate and the realism of the generated motions.
### 4.3 Experiment Results
#### 4.3.1 Evaluation Metrics
* •
$\text{F}^{2}\text{ID}$: We used our proposed $\text{F}^{2}\text{ID}$ score as
one of the the evaluation metric. To extract features from the motion
sequences, we have trained an auto-encoder neural operator to extract features
from human motion. The auto-encoder operator can be defined as
$\mathcal{G}:(\mathcal{M}\times\mathcal{M})\rightarrow\mathcal{M}\times\mathcal{M}$.
The auto-encoder also is composed of seven non-linear integral operator layers
and the output of the bottleneck layer is taken as a feature. We extract
features from randomly sampled (with replacement) 1000 generated and real
dyadic human motions and calculate the $\text{F}^{2}\text{ID}$. The process is
repeated five times and the mean $\text{F}^{2}\text{ID}$ score is reported.
* •
Diversity Score: Diversity score represents the variance in the generated
data. We randomly sample (with replacement) 1000 motion from the generator and
extract the features using the auto-encoder. We divide the features into two
sets $S$ and $S^{\prime}$ each containing 500 motions. Then the diversity
score is calculated as
$\text{Diversity Score
}=\frac{1}{5e2}\sum_{i=1}^{5e2}L_{2}(S_{i},S^{\prime}_{i})^{2}$
where $L_{2}(.,.)$ is the $L_{2}$ norm. The average diversity score of five
repeated calculations is reported.
* •
Maximum Mean Discrepancy (MMD) Maximum Mean Discrepancy measures the distance
between two distributions and is used to measure the discrepancy between
generated and real data [4, 53, 46]. Following the construction in [24], we
also used the kernel trick to calculate the MMD. We also present two MMD-based
metric MMD-A which measures the dissimilarity per time step basis and MMD-S
which measure the dissimilarity based on the whole sequence. We use the
official implementation of the work MUGL [24].
* •
Average Position Error (APR) Average position error for joint $j$ between
generated motion $\\{m_{i}^{\prime}\\}_{i=0}^{n}$ and the corresponding ground
truth real motion $\\{m_{i}\\}_{i=0}^{n}$is defined as
$APE(j)=\frac{1}{N}\sum_{i=0}^{n}\frac{1}{T_{i}}\sum_{t=1}^{T_{i}}||m^{\prime
j}_{i}(t)-m^{j}_{i}(t)||_{2}$
where $m^{j}_{i}(t)$ is the position of $jth$ joint of motion $m_{i}$ at time
$t$ and $T_{i}$ is length of motion $m_{i}$
* •
Average Variance Error (AVE) Following the above notation, AVE can be defined
as
$AVE(j)=\frac{1}{N}\sum_{i=0}^{n}||\sigma(m^{\prime
j}_{i})-\sigma(m^{j}_{i})||_{2}$
where $\sigma(m^{j}_{i})$ is the variance of joint $j$ over time. We calculate
both APR and AVE without any normalization of the data.
It is pointed out in recent studies [24] that feature-based metrics (e.g.,
accuracy) are not reliable to evaluate action sequences. As a result, we kept
both feature-based (FID, Diversity score) metrics and metrics calculated
directly from the generated and real data (MMD, AVE, APE) as our evaluation
metrics. We report the mean value for every metric over five repeated
calculations on the randomly sampled data from the evaluation set.
#### 4.3.2 Baseline
The problem of partner-conditioned human motion generation in function spaces
is new and we do not have any method to compare with. Previously, LSTM-based
models are used to generate body motion from audio [41]. Here we also design a
GRU base model to predict our desired human motion from the motion of their
partner in the action. The model consists of five stacked GRU layers with a
$200$ dimensional hidden state. This is followed by three layered fully
connected neural networks with GELU activation. This type of sequence-to-
sequence (seq-to-seq) architecture has been used successfully in many complex
real-world problems such as language translation, weather prediction [37, 38,
26, 40]. So, this model serves as an appropriate deterministic baseline for
this problem.
| Real data | | PaCMO Generated
---
data
| Perceived as
---
Realistic
0.81 | 0.75
| Perceived as
---
Unrealistic
0.19 | 0.25
Table 3: Result of the survey of real and PaCMO generated motion.
#### 4.3.3 Results
We present the performance of PaCMO on the DuetDance dataset shown in Table 2
and on the NTU-RGD+D 120 dataset shown in Table 1. $\text{F}^{2}\text{ID}$ is
most important in evaluating the quality of the generated motions. In both the
datasets, PaCMO achieves lower $\text{F}^{2}\text{ID}$ vastly outperforming
the baseline and the diversity score of PaCMO is closer to the real dataset.
Unlike motion generation from the action label where the input condition is
just a fixed label, here the input condition is the motion of the partner for
a dyadic activity and contains much variability even within a single action
category. As a result, both GRU base seq-to-seq models and PaCMO achieved a
diversity score close to the real dataset. For feature-free evaluation metrics
(MMD, APE, AVE), PaCMO achieves excellent results outperforming the baseline.
This shows that the motion generated by PaCMO is very realistic and close to
the real dataset.
#### 4.3.4 Perceptual Evaluation
To bolster the finding in the previous section, we also conduct a visual
perceptual evaluation using of 50 web-based participants. In this study, we
provide the participants with both real and PaCMO-generated motions and ask
them to label them as real or fake (computer generated). Each participant is
asked to label 14 different randomly selected motions from both real and
generated data. The result of the study is shown in Table 3. We can observe
that our generated motion is labeled as real at a very high rate ($76\%$),
which makes it suitable for real applications such as AR/VR and animation. For
qualitative evaluation, we present a few motions generated by PaCMO along the
condition motion (see Fig. 3) and we observe that the generated motions are
realistic and in sync with the given motion of the partner as a condition.
## 5 Conclusion, Limitations, and Future
This work addresses an important yet under-explored problem of partner-
dependent human motion generation in dyadic human activities. And as a
solution, we propose PaCMO, a novel generative neural operator architecture.
Unlike existing conditional generative approaches, PaCMO is resolution
invariant, able to model global long-term dependencies, and also does not
depend on any particular representation of human poses. We also propose
$\text{F}^{2}\text{ID}$ as a suitable evaluation metric for generative models
for infinite dimensional spaces. Our work is the first step in generating
continuous conditional human motion generation. In the future, we plan to
extend the work from dyadic activities to multi-person group activities. Also,
we aim to model complected human motion such as grasping and emotions.
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## Appendix A Supplementary Material
### A.1 Calculation of $\text{F}^{2}\text{ID}$
In this section we will discuss the approximation of Wasserstein-2 distance
for Gaussian processes and $\text{F}^{2}\text{ID}$.
Data: Features of real data $F_{t}$, features of generated data $F_{g}$, the
grid size $r$
Result: $\text{F}^{2}\text{ID}$ score
1 model $F_{t}$ by $GP(m_{1},k_{1})$;
2 model $F_{g}$ by $GP(m_{2},k_{2})$;
3 g[] $\leftarrow$ $r\text{ evenly spaced number on the interval }[0,1]$;
4 Initialize matrix $M_{t}$ and $M_{g}$ of size $r\times r$;
5 for _i in $\\{1,2,\dots r\\}$_ do
6 for _j in $\\{1,2,\dots r\\}$_ do
7 $M_{t}[i][j]$ $\leftarrow$ $\frac{1}{r}k_{1}(g[i],g[j])$ ;
8 $M_{g}[i][j]$ $\leftarrow$ $\frac{1}{r}k_{2}(g[i],g[j])$ ;
9
10
11L $\leftarrow$
$M_{t}+M_{g}-2(M_{t}^{\frac{1}{2}}M_{g}M_{t}^{\frac{1}{2}})^{\frac{1}{2}}$;
12 $\text{F}^{2}\text{ID}$=
$\frac{1}{r}\sum_{i=1}^{r}(m_{1}(g[i])-m_{2}(g[i]))^{2}+\sum_{i=1}^{r}L[g[i]][g[i]]$;
Algorithm 1 Calculation of $\text{F}^{2}\text{ID}$
Let $GP(m,k)$ be a Gaussian process where the mean function $m\in L^{2}(X)$,
and the covariance function $k\in L^{2}(X\times X)$. For the simplicity of
calculation, without any loss of generality we will assume that $X$ is a
compact subset of $\mathbb{R}^{d}$. We also assume that $\lambda(X)=1$ with
Lebesgue measure $\lambda$ on $R^{d}$. We assume that the function in
$L^{2}(X\times X)$ are discretized uniformly at the finite set of point
$x_{i}\in X_{i}$.
For numerical feasibility, we choose only a finite set of abnormal basis
$\\{e_{1},e_{2},..e_{r}\\}$ for the function spaces $L^{2}(X)$. The domain $X$
is discretized uniformly with a grid size of $r$ such that
$\lambda(X_{i})=\frac{1}{r}~{}~{}\forall i\in\\{1,2,\dots r\\}$ and
$\bigcup_{i=0}^{r}X_{i}=X$.
The basis function $e_{i}$ is defined to be equal to $\frac{1}{c}\sqrt{r}$ at
the point $x_{i}$ and $0$ outsize the domain $X_{i}$. Such choice constants
maintains the basis functions at unit norm. The $c$ is the universal constant
depends only the particular choice of $e_{i}$ [55], we assume it to be
$\approx 1$ for our calculation.
The covariance operator $K$ associated with covariance function $k$ is defined
as
$[K\phi](x)=\int_{X}k(x,s)\phi(s)ds$ (4)
Applying the operator $K$ on the the basis function $e_{i}$, we get
$\displaystyle[Ke_{i}](x)$ $\displaystyle=\int_{X}k(x,s)e_{i}(s)ds$
$\displaystyle=\int_{X_{i}}k(x,s)ds$
$\displaystyle\approx\frac{1}{\sqrt{r}}k(x,x_{i})~{}~{}~{}[\text{where
}x_{i}\in X_{i}]$
Here we assume that for a fixed $x$ the covariance function $k(x,s)$ is
constant $\forall s\in X_{i}$.
The trace of the operator $K$ can be defined as
$Tr(K)=\sum_{i=1}^{r}\langle Ke_{i},e_{i}\rangle$
Now,
$\displaystyle\begin{split}\langle
Ke_{i},e_{i}\rangle&=\int_{X}[Ke_{i}](x)e_{i}(x)dx\\\
&=\int_{X}\frac{1}{\sqrt{r}}k(x,x_{i})e_{i}(x)dx\\\
&\approx\int_{X_{i}}k(x,x_{i})dx\\\ &=\frac{1}{r}k(x_{i},x_{i})\end{split}$
(5)
where we assumes that for a fixed $x_{i}$, $k(x,x_{i})$ is constant $\forall
x\in X_{i}$. Now the trace can be written as
$Tr(K)\approx\frac{1}{r}\sum_{i=1}^{r}k(x_{i},x_{i})$
where $x_{i}\in X_{i}$. Assuming a finite set of standard basis we can
construct covariance matrix $M_{K}$ associated with the covariance operator
$K$ defined as follows
$M_{K}[i,j]=\langle Ke_{i},e_{j}\rangle$
where $i,j\in\\{1,2,\dots r\\}$. Following the same procedure in Eqn. 5, we
can show that $M_{K}[i,j]\approx\frac{1}{r}k(x_{i},x_{j})$ where $x_{i}\in
X_{i}$ and $x_{i}\in X_{j}$.
We present a pseudo-code for calculating $\text{F}^{2}\text{ID}$ in Algorithm
1. We assume that the feature functions are defined on the interval $[0,1]$.
The $F_{t}=\\{f_{t}^{1},f_{t}^{2},...,f_{t}^{n}\\}$ and
$F_{g}=\\{f_{g}^{1},f_{g}^{2},...,f_{g}^{n}\\}$ are the set of $n$ feature
functions of the real and generated motion respectively where
$f:[0,1]\rightarrow\mathbb{R}$. The feature functions are extracted from the
bottleneck layer of the auto-encoder neural operator. The auto-encoder neural
operator follows the architecture of PaCMO generator without the skip
connections. Each of the feature functions $f^{i}$ for $i\in\\{1,2,...,n\\}$
for both real and generated is represented using a finite set of basis as
following
$f^{i}(x)=\sum_{j=1}^{r^{i}}F^{i}[j]e^{i}_{j}$
where $r^{i}$ is resolution (discretization grid size) of the feature function
$f^{i}$ which depends on the input resolution feed to the auto-encoder neural
operator. The $e^{i}_{j}$ s are the corresponding basis function which depend
on $r^{i}$ and $F^{i}$ s are the coefficients. We model the the features from
real data by Gaussian process $GP(m_{1},k_{1})$ where
$m_{1}(x)=\mathbb{E}[f_{t}]\approx\frac{1}{n}\sum_{i}^{n}f_{t}^{i}(x)$
and
$\displaystyle k_{1}(x,x^{\prime})$
$\displaystyle=\mathbb{E}[\big{(}f_{t}(x)-m_{1}(x)\big{)}\big{(}f_{t}(x^{\prime})-m_{1}(x^{\prime})\big{)}]$
$\displaystyle\approx\frac{1}{n}\sum_{i}^{n}(f^{i}_{t}(x)-m_{1}(x))(f^{i}_{t}(x^{\prime})-m_{1}(x^{\prime}))$
Here the expectations are taken over the distributions of feature functions
for real data and approximated by the the set $F_{t}$. In the same way we
model the feature functions of the generated data by $GP(m_{2},k_{2})$. To
approximate the 2-Wasserstein metric between these Gaussian processes we fix a
resolution $r$. And the matrix operations are computed using Scipy [48].
### A.2 More Qualitative Results
Figure 4: Each rows represents a complete dyadic motion with the condition
motion (in Red and Green) and motion generated by PaCMO (in Magenta and Blue).
|
# Identification of Rare Cortical Folding Patterns using Unsupervised Deep
Learning
Louise Guillon<EMAIL_ADDRESS>Joël Chavas Audrey Bénézit
Marie-Laure Moutard Denis Rivière Jean-François Mangin Université Paris-
Saclay, CEA, CNRS, NeuroSpin, Baobab, Gif-sur-Yvette, France Service de
Neurologie et Réanimation Pédiatrique. Hôpital Raymond Poincaré. APHP.
Garches, France Service de Neuropédiatrie, Hôpital Trousseau, Hôpitaux
Universitaires de l’Est Parisien, Sorbonne Université, Paris, France
###### Abstract
Like fingerprints, cortical folding patterns are unique to each brain even
though they follow a general species-specific organization. Some folding
patterns have been linked with neurodevelopmental disorders. However, due to
the high inter-individual variability, the identification of rare folding
patterns that could become biomarkers remains a very complex task. This paper
proposes a novel unsupervised deep learning approach to identify rare folding
patterns and assess the degree of deviations that can be detected. To this
end, we preprocess the brain MR images to focus the learning on the folding
morphology and train a $\beta-VAE$ to model the inter-individual variability
of the folding. We compare the detection power of the latent space and of the
reconstruction errors, using synthetic benchmarks and one actual rare
configuration related to the central sulcus. Finally, we assess the
generalization of our method on a developmental anomaly located in another
region. Our results suggest that this method enables encoding relevant folding
characteristics that can be enlightened and better interpreted based on the
generative power of the $\beta-VAE$. The latent space and the reconstruction
errors bring complementary information and enable the identification of rare
patterns of different nature. This method generalizes well to a different
region on another dataset. Code is available at https://github.com/neurospin-
projects/2022_lguillon_rare_folding_detection.
###### keywords:
folding patterns , cortical folding , cortical sulci , anomaly detection ,
unsupervised learning , $\beta-VAE$
## 1 Introduction
During gestation, the human cortex folds and gets its convoluted shape
composed of gyri—the ridges of white matter—that are delimited by furrows— the
sulci. Historically, their shape and characteristics have been described by
neuroanatomists based on specimens. In the human population, stability of the
folding patterns is observed with an overall similarity of location, shape and
arrangements [45]. This stability is important enough to enable to define a
road map and a nomenclature of sulci and to develop methods that automate
sulci recognition [51, 9]. Despite this homogeneity, each brain displays a
unique cortical folding, acting as a fingerprint [67]. Fig. 1A and B. show
examples of the variability in the central sulcus region, which is one of the
most stable. The folding variability is so complex that it has long been
overlooked.
However, thanks to advances in the neuroimaging field, studies have tried to
characterize sulci with elementary shapes that amount to building blocks of
alternative patterns. For instance, the central sulcus is typically composed
of one or several knobs [69]. Similarly, the mid-fusiform sulcus presents an
omega pattern [68]. In contrast, some very rare patterns have also been
described, such as the interruption of the central sulcus that can be found in
only about 1% of the population [38]. Four examples of interrupted central
sulci in the right hemisphere are presented in Fig. 1C.
Figure 1: Central sulcus region variability. A. Localization of the studied
region of interest (ROI) on a 3D view of one right hemisphere. The colored
ribbons represent sulci, defined as a negative cast of the furrows. The
central sulcus is red. B. Examples of non-interrupted central sulci. C.
Examples of interrupted central sulci.
Folding patterns have also proved to be very interesting as they are related
to function. For example, the central sulcus divides the cortex into the motor
and the sensory areas, and specific parts of the central sulcus have been
correlated with the cortical areas of the tongue, foot, and hand among others
[38, 23]. Specifically, the central sulcus main knob has been linked to the
hand motricity and is called the ”hand knob” [69]. In another region,
cingulate sulcus patterns have been associated with the inhibitory control
[10].
Specific patterns were also correlated to neurodevelopmental disorders. The
Power Button Sign (PBS), a rare configuration of the precentral sulcus, may be
associated with a certain type of epilepsy [42]. Patterns in the superior
temporal sulcus (STS), central, intraparietal and frontal regions could be
related to autism [34, 2, 29]. Hence, deciphering sulcal complexity and having
a better understanding of the underlying shape variability is of great
interest: folding patterns could become biomarkers of neurodevelopmental
disorders.
Folding patterns can be analyzed with two approaches. On one hand,
morphometric features can be extracted such as the depth, the surface
curvature, or the opening of each sulcus. On the other hand, one can look
directly at the shapes of the sulci. Working on shapes rather than on
morphometric values is particularly interesting as they constitute ”trait
features” opposite to ”state features” [13]. Unlike state features that can
evolve during the lifespan, trait features remain fixed afterbirth. For
example, the sulcal opening is a state feature because it increases with aging
[33, 31]. In return, the pattern of the cingulate sulcus area is a trait
feature because it is stable throughout life after infancy. This difference
between trait and state features has also been demonstrated in the study of
the effects of handedness on the central sulcus shape. For example, forced
dextral subjects show similarities to sinistral subjects in shape, but changes
in elongation mimicking dextrals, occur when they are constrained to use the
right hand for writing [61]. Different strategies can be adopted for exploring
the shapes: a finite number of shapes can be considered, using clustering for
instance [43, 18], or shapes can be represented in a continuous way, such as
manifold-based analyses [61], [66]. For both strategies, a first step is
required to represent the folding patterns. Folding shapes’ complexity can be
reduced based on the similarity between different sulci [62, 61, 66] or
between sulcal graphs [30, 43]. These similarity measures are then either
directly analyzed, or projected to a lower dimensional space.
Advances in machine learning are now opening up new possibilities for studying
folding patterns, identifying typical or rare patterns, and hopefully,
emerging sulcal biomarkers. Recently, a neural network classifier was used to
map geometric shapes to the broken-H shape pattern in the orbitofrontal region
[52]. However, this method requires having pre-identified the geometric shape
to be mapped, which makes it difficult for unknown patterns to emerge. To
tackle this issue, unsupervised deep learning techniques seem to be promising.
Thus, two unsupervised deep learning models, a $\beta-VAE$ and SimCLR, were
compared in the task of identifying typical patterns in the cingulate region
[26]. Other works have focused on the task of identifying abnormal folding
patterns thanks to unsupervised deep learning in the region of the superior
temporal sulcus branches [25]. Anomaly detection has been a subject of great
interest in the domain of biomedical imaging: many studies, including for
brain MR images, have tried to identify abnormal samples [15, 20]. A common
framework is to use auto-encoders as they implement a latent space with fewer
dimensions than the input which makes it hard to encode uncommon features.
Then, the identification of anomalies is usually performed based on the
reconstruction error rather than in the latent space.
In this work, we investigate whether an unsupervised deep learning model can
learn normal folding variability to identify deviating regional patterns; and
if so, what granularity of deviations can be detected? Here, we define
granularity as the characteristics and properties of the anomalies, such as
their size or nature. The analysis of the granularity that can be identified
aims to characterize the abnormal features that can be detected and at what
level of detail. We also seek to describe which space is the most relevant to
identify deviating patterns: is it based on the reconstruction error, in the
input space, that is to say in our case, the folding space, or is it the
latent space?
Folding mechanisms may lead to both global and regional anomalies and these
two scales have led to correlations with function disorders [21]. Here, we
focus on regional patterns rather than on a global representation.
Specifically, we concentrate on the central sulcus which is a good candidate
for our work. Indeed, it is one of the first folds to appear and it is stable
enough to be a first step in modeling inter-individual variability. More
importantly, usually long and continuous, the central sulcus can be
interrupted in very rare cases, making interrupted central sulci relevant
patterns to assess our method. Finally, this region is of clinical interest as
it is linked to hand motricity and asymmetries have been described [61, 5].
To perform our study, we worked on the HCP database [65]. From the MR images,
we focused on the folding morphology of the central sulcus area with a
specific preprocessing that does not require labeling the sulci of the studied
subjects. Indeed, in the future, we wish to be able to apply our methodology
to new databases whether the sulci are labeled or not. Even if automatic
recognition tools exist and are efficient, some errors may remain and thus
lead to the selection of another sulcus and to a contaminated learning set,
especially regarding unusual patterns. We then trained a $\beta-VAE$ to learn
the inter-individual variability. Due to the small number of interrupted
central sulci and to be able to characterize the detected granularity, we
designed synthetic benchmarks of rare patterns to assess our methodology more
reliably. We then investigated the detection power of our methodology both on
the latent space and on the folding space, using either our synthetic outliers
or actual interrupted central sulci, an actual rare pattern. Finally, we
assessed the generalization of our approach on another dataset presenting
abnormal folding patterns in the cingulate region.
We stress out that rare patterns are not necessarily abnormal and associated
with some disorders. However, whether they have a link with pathologies or
not, rare patterns are interesting objects to study as they can constitute
traces of neurodevelopmental processes. Therefore, in this article, the only
abnormal pattern that we study is the one in the cingulate region. We consider
our synthetic benchmarks and interrupted central sulci as rare patterns.
## 2 Material and methods
### 2.1 Database
We used T1 weighted MR images of the Human Connectome Project (HCP) dataset
[65]. Data were acquired on a single Siemens Skyra Connectom scanner at an
isotropic resolution of 0.7mm. Subjects are healthy controls from 22 to 36
years old. In the context of our study we considered only the right
hemispheres of the right-handed subjects leading to a total of 1001 subjects.
The long-term goal of this work is to identify rare folding patterns that have
not been characterized yet. However, we first need to assess our method. To do
so, we decided to work on a rare pattern already described, the interrupted
central sulci (CS). A previous study identified in this database seven sulci
in the right hemisphere and two in the left [38]. The identification was based
on the depth profiles of the sulcal pit maps, which are defined as the locally
deepest point of the cortical surface [36], and it was then visually
confirmed. We chose to work on the right hemisphere rather than on the left in
order to have the highest number of rare patterns.
### 2.2 Folding representation
In this work, we consider the folds as the skeleton of a negative cast of the
brain, that is to say, voxels located in the cerebrospinal fluid, (Fig.2A.5)
[37], which can be represented as ribbons located between the gyri (green
ribbons in Fig.2A.7). Folding or sulcal patterns are defined as the
combination and arrangements of shapes of one or several folds.
Figure 2: Overview of the BrainVISA/Morphologist pipeline’s main steps and of
the folds representation. A. Main steps of BrainVISA/Morphologist pipeline.
1\. Raw T1-w MRI, 2. Bias-corrected image, 3. Segmentation of the brain, 4.
Segmentation of the hemispheres and of the grey and white matter, 5. Skeleton
representation of the folding graph, representing a negative cast of the 4. 6.
Mesh representation of the white matter of the right hemisphere, 7. Folding
graph that represents the folds (in green) as the negative cast of the white
matter of the right hemisphere (white mesh). B. Folds representation. 1\.
Example of a central sulcus, which is composed of several elementary entities
called simple surfaces (SS). (Orientation: A: Anterior, P: Posterior, S:
Superior, I: Inferior). 2. Corresponding schematic representation of the
sulcus represented in 1, which is formed by four simple surfaces. Depth
variation caused by the buried gyrus and the presence of two branches lead to
the division into four different simple surfaces. 3. Corresponding folding
graph.
##### BrainVisa/Morphologist pipeline
Structural MR images hold numerous pieces of information beyond the morphology
of cortical folding. In order to focus on the folding characteristics, we
developed a preprocessing pipeline. The raw MR images are first processed by
the BrainVISA/Morphologist software (https://brainvisa.info/). This pipeline
is composed of several steps that include skull stripping, bias correction,
segmentation of the brain and of the hemispheres, skeletonization of the grey
matter and the cerebrospinal fluid union (Fig.2A) [51]. This step leads to so-
called skeletons, 3D images representing only the folding which is then
segmented into simple surfaces (SS) depending on various parameters such as
the sulcal depth or topological properties (Fig.2B) [37]. For example, in
Fig.2B, small branches (SS2 and SS4) are represented as different simple
surfaces from the main ones, SS1 and SS3. The depth variation resulting from
the buried gyrus leads to two distinct simple surfaces (SS1 and SS3).
Therefore, in this case, the central sulcus is composed of four simple
surfaces. All in all, the obtained outputs are 3D images that correspond to a
negative cast of the brain (first step of Fig.3).
Figure 3: Pipeline. A mask of the central sulcus area is defined based on a
distinct manually labeled dataset. HCP is processed with Morphologist to
obtain folding graphs, which are used to obtain 3D images of skeletons (1).
The Chamfer distance is applied to the skeletons to obtain geodesic distance
maps (2). Distance maps are then downsampled and cropped according to the mask
(3) and are fed as input to a $\beta-VAE$.
##### From skeletons to distance maps
Skeleton-based images have proved to be relevant and were the object of
previous studies for fold recognition [9, 8], as well as for folding patterns
representation [25, 26]. However, in this work, we applied an additional
preprocessing step to convert our skeleton images into distance maps. Indeed,
we believe that skeletons have several shortcomings that could limit the
performance of our approach. First, skeletons are binary images representing
the folding patterns; hence, they are very sparse images. Therefore, only very
few voxels hold explicit sulcal information in skeletons. As shown in Fig.4,
on average, sulci represent 3500 voxels in our region of interest (ROI), which
stand for less than 5% of the voxels. We argue that it would be interesting to
have a representation where the information devoted to folding patterns is
more distributed. In addition, skeleton images are not smooth and the local
regions of interest in our application correspond to high frequencies making
the skeleton details harder to represent and reconstruct. It is also complex
to reconstruct folding patterns in skeletons as they are not continuous
images, so there is no notion of the proximity of a voxel to a sulcus. This
makes the reconstruction error and the gradient-based learning less efficient.
Finally, distance maps are built based on the whole hemisphere; therefore, if
we work on a ROI, they give, especially near the border of the ROI,
information about objects outside the ROI, which is not the case for crops
based on skeletons. Therefore, to tackle these shortcomings, we convert the
resulting skeletons into distance maps based on the Chamfer distance, which
approximates the Euclidean distance. Sulci are considered objects, and the
further away a sulcus is, the larger the value of a voxel is. An example of a
distance map is presented in Fig.3 step 2.
##### Focusing on a single region: crop definition
As we are interested in capturing the local folding patterns variability, such
as the hand knob, rather than the global hemisphere-wide arrangements of
sulci, we chose to focus our study on a sub-region of the right hemisphere,
the central sulcus area. To define the ROI, we learned a mask of the central
sulcus over a manually labeled dataset comprising 62 healthy controls [9].
Subjects are first affinely registered to the ICBMc2009 space and resampled to
an isotropic resolution of 1 mm. Then, for each subject, a mask is incremented
for all the voxels of the central sulcus represented as a set of simple
surfaces. The resulting mask is slightly dilated by 5 mm to include potential
central sulcus locations not represented in our database. We crop the distance
maps of the HCP subjects according to the mask bounding box using the same
affine normalization procedure (step 3 of Fig.3). The mask is applied on the
fly during the training of our network.
##### Distance maps and folds visualization
Data visualization, and shape characterization in particular, can be performed
directly based on the distance maps. However, this type of input enables only
2D slice views. To better visualize the folds of our crops, we binarize our
distance maps with an empirically defined threshold of 0.4 and convert them to
meshes.
### 2.3 Learning a Representation of the Folding Variability
#### 2.3.1 beta-VAE
In order to identify rare patterns we first seek to model the inter-individual
variability. In the outlier detection field based on unsupervised methods,
auto-encoder (AE) models are widely used as they implement a latent space,
also known as the bottleneck, that has far fewer dimensions than the input
space. Usually, training is performed only on control subjects. The assumption
is that the model learns a representation of the normal variability and that
at inference when facing outliers, it will not be able to encode and
reconstruct them as well as control data. To overcome some shortcomings of
simple convolutional AE and to regularize the latent space, variational auto-
encoder (VAE) was introduced [32]. Its strength also lies in its generative
power, enabling not only to reconstruct but also to generate new data. Other
AE-based models have been used for anomaly detection in the biomedical field
such as Generative Adversarial Networks (GAN) [55, 54], which were then
transposed to brain images [58]. A comparison of AE models showed that VAE was
one of the most efficient in brain MR images [3]. In the context of
representation learning of folding patterns, VAE was proved to be well adapted
[25, 26].
In the VAE framework, a sample of input space $\mathcal{X}$ is mapped to a
distribution in a latent space $\mathcal{Z}$ of $L$ dimensions, by an encoder
$\theta$. A vector z is then drawn from this distribution and reconstructed by
a decoder $\phi$. The objective function seeks to minimize both the
reconstruction error and the Kullback-Liebler (KL) divergence
($\mathcal{D}_{KL}$). The model is thus trained to maximize:
$\mathcal{L}(\theta,\phi;\textbf{x},\textbf{z},\beta)=\mathbb{E}_{q_{\phi}(\textbf{z}|\textbf{x})}[\textup{log}p_{\theta}(\textbf{x}|\textbf{z})]-\beta\mathcal{D}_{KL}(q_{\phi}(\textbf{z}|\textbf{x})||p(\textbf{z}))$
(1)
where $p(\textbf{z})$ refers to the prior distribution (in this work, a
reduced centered Gaussian distribution) which is approximated with
$q_{\phi}(\textbf{z}|\textbf{x})$, the posterior distribution. $\beta-VAE$ is
an extension of the VAE where the KL divergence is weighted by $\beta$ [27].
#### 2.3.2 Training procedure
##### Preprocessing
The input data of the model are the previously defined just cropped, then
masked distance maps. For augmentation purposes, random rotations between
[-10°, 10°], centered on the mask center, are drawn from a uniform
distribution at each epoch and applied to the whole brain, before applying the
mask that strictly remains at the same position. Such rotations are also
sought to limit the edge effects. More precisely, the central sulcus is
surrounded by two main folds, the precentral and the postcentral sulci. Parts
of these sulci are included in the ROI. Therefore, rotating the distance map
under the mask enables to capture a wider context and to try to limit their
influence. We observe that skeleton voxels equal 0 in the initial distance
maps $X$ and the values increase with the distance to a sulcus, possibly
ranging up to 10 mm. Potential reconstruction errors near the sulci would be
minor compared to the voxels located far from them at the edge of the mask,
whereas we wish the model to concentrate more on the sulci. Thus, to limit the
impact of distance variability far from the sulci, we perform a normalization
according to equation 2, resulting in values between [0, 1], with the highest
values on the folds and a saturation at about 4-5 mm which corresponds to half
of the typical width of gyri. An example is shown in Annex 1 of the
supplementary. Finally, we apply a small padding, resulting in samples of
dimensions 80 x 80 x 96.
$X_{norm}=1-[2\frac{1}{1+e^{-X}}-1]$ (2)
##### Training
Dataset was split into train, validation, and test sets of respectively 640,
161, and 200 subjects. Training is only performed on control data, all
identified interrupted central sulci (CS) were added to the test set. The
interrupted central sulci were identified based on the detection of the two
main sulcal pits of the central sulcus, between which a depth profile was
computed to determine the depth of the ”pli de passage frontal-moyen” (PPFM),
usually a buried gyrus in the sulcus. Subjects with a shallow PPFM were then
manually inspected to determine whether the surrounding central sulci were
interrupted [38]. However, we point out that there may remain some undetected
interrupted central sulci in the training set as all subjects were not
individually inspected. To model the normal inter-individual variability, we
used a classic convolutional $\beta-VAE$ of depth 3. In order to choose the
best values for $\beta$ and latent space dimension L, we performed a
gridsearch ($\beta$=2-10, L=4-150). To assess each parameter configuration, we
used two criteria. Our first criterion is the reconstruction quality. Indeed,
we seek to leverage the reconstruction and generative power of the $\beta-
VAE$, hence the reconstruction quality must be sufficient. Our second
criterion is the detection power on a proxy for the interrupted central sulci.
The pre-central and post-central sulci demonstrate some similarities with the
central sulcus in terms of orientation, size, and shape. However, they tend to
be more interrupted and to present a higher number of ramifications.
Therefore, we used the HCP dataset crops of these two other regions as fake
outliers. We selected only pre- and post-central sulci which presented some
ambiguities with the central sulcus based on the procedure described in Annex
2 of the supplementary. Finally, our ambiguous set was composed of 28
precentral sulci and 18 postcentral sulci. For each hyperparameter
combination, we trained a $\beta-VAE$ on the train set, then a linear SVM was
trained to classify between the latent codes of the validation samples and of
the pre- or post-central sulci. We kept the hyperparameters that led to the
best classification results and good reconstructions (based on reconstruction
error and visual inspection).
### 2.4 Generating synthetic rare patterns
One of the challenges of our work is the lack of consensual rare patterns to
evaluate our methodology. In addition, it would be interesting to be able to
quantify the degree of deviation that our model is able to detect. Therefore,
several sets of synthetic rare patterns were generated to be used as
benchmarks. Both benchmarks were generated from the test set subjects.
#### 2.4.1 Deletion benchmark
Our first benchmark consists of subjects for whom we have erased one simple
surface (SS). Erasing small simple surfaces could be a good proxy to simulate
rare patterns because some fold branches may be missing in some people, or a
sulcus may be shorter or absent. Large simple surfaces are less likely to be
missing but allow us to assess the degree of deviation that can be detected.
Deleting simple surfaces directly on the distance maps would not be
interesting as the voxels next to the simple surface indicate the SS position.
To tackle this issue, the suppression was done during the generation of the
raw skeletons. The distance map is then computed based on the pruned
skeletons. To analyze the granularity of anomaly that can be detected by our
method, we generated several benchmarks which vary according to the size of
the deleted simple surface (SS). As such, we created four sets where SS size
was between 200-500 voxels, 500-700 voxels, 700-1000 voxels, and simple
surfaces of more than 1000 voxels. In the following, we name each set with the
minimum number of voxels: for instance, 200 corresponds to the benchmark where
simple surfaces of size between 200 and 500 were erased. To be deleted, simple
surfaces must have a number of voxels included inside the mask corresponding
to the range of the different sets. If several simple surfaces meet the
criteria, one is randomly chosen to be erased. Otherwise, a subject may not
have a simple surface satisfying the requirements. In such cases, the subject
is not included in the benchmarks. Finally, from the 200 test subjects,
benchmark 200 contains 180 subjects; benchmark 500, 68; benchmark 700, 108 and
benchmark 1000, 151 subjects. To have a better representation of the amount of
deleted sulci, Fig.4 shows the simple surface sizes distribution in the
central sulcus region. The figure shows that our crops contain a large
majority of very small simple surfaces (less than 500 voxels) and far fewer
large simple surfaces. The smaller simple surfaces are mostly part of the
precentral and postcentral sulci, representing more than 85% of the surfaces
between 200 and 500 voxels. On the contrary, larger simple surfaces correspond
to the central sulcus. Therefore, beyond deleting simple surfaces of varying
sizes, the nature of the sulci and thus the location, are also different,
especially between the set 200 and the others. The right part of Fig.4 shows
the number of voxels corresponding to skeletons in our crops. It demonstrates
the progressive intensity of anomalies when deleting simple surfaces from 200
voxels to more than 1000 voxels. Indeed, when simple surfaces of more than
1000 voxels are deleted, it corresponds to a third or a quarter of the
skeleton crop. Distance maps are then generated according to 2.2. An example
is presented in Fig.5.
Figure 4: Skeleton’s description of the test set. Left: Stacked histogram
representing the distribution of simple surfaces sizes for the test subjects
for the three main sulci of our crop, the central sulcus (S.C._right), the
precentral sulcus (S.Pe.C._right) and the postcentral sulcus (S.Po.C._right).
(Note: The labeling used is automatic and therefore not entirely reliable, but
these labels are sufficient to draw conclusions regarding the SS size
distribution.) Right: Distribution of the number of skeletons’ fold voxels for
the test subjects when the mask is applied to the crops. Figure 5: Deletion
benchmarks. Visualization of original sulcal pattern and its altered versions
from the four deletion benchmarks showing patterns with increasing simple
surface size deleted. Upper row: Mesh visualization. Middle and bottom rows:
distance maps on axial view, visualization at depths 15 and 37.
#### 2.4.2 Asymmetry benchmark
Our second benchmark corresponds to the equivalent crop but in the left
hemisphere. Left hemisphere distance maps are generated according to the same
methodology as the right. Like our control crops of the right hemisphere, we
computed a left central sulcus mask on the labeled dataset. To enforce the
exact same crop size, we adapted the mask to match the adequate dimensions by
adding or deleting a few voxels. Once the crops were obtained, they were
flipped. During training, the right central sulcus mask was applied on the
fly. We emphasize that we did not use the interhemispheric plane-symmetric
coordinates but a mask specifically designed for the left central sulcus. This
is especially important since there is a slight asymmetry in the position of
the central sulcus between the two hemispheres [16]. An example is presented
in Fig.6.
Figure 6: Asymmetry benchmark. Visualization of the original sulcal pattern
and its flipped contralateral version for two subjects.
### 2.5 Identifying Outliers
Once the model has learned a representation of the inter-individual
variability, outliers identification can be performed at two levels.
Traditionally, anomaly detection with AE is done based on the reconstruction
error and an error map can be obtained comparing the input and the output [55,
54, 48]. But one can also wonder about the distribution of outliers in the
latent space. Are the outliers distributed differently? To answer this
question, we investigated the detection power in the outliers’ distribution in
the latent space based on the reconstruction errors performed in the input
space—which we call folding space in our case, as we study folding patterns.
For both approaches, control test images and outlier images (deletion
benchmarks, asymmetry benchmark and interrupted sulci) are encoded and
reconstructed by our trained model.
#### 2.5.1 A specification on data
As mentioned in 2.3.2, our control test set comprises 200 subjects. However,
when studying our different outliers sets, data subsets were different since
some subjects did not have any SS meeting the benchmark’s criteria.
* 1.
Deletion benchmarks: to avoid any bias, we used only control subjects with a
simple surface meeting the benchmark’s criteria for each benchmark. Therefore,
for benchmark 200, we used 90 controls that have a simple surface between 200
and 500 voxels but that has not been erased, and 90 benchmark subjects, for
whom simple surfaces were actually erased. Resulting in
$n_{control}^{200}=n_{deletion}^{200}=90$,
$n_{control}^{500}=n_{deletion}^{500}=34$,
$n_{control}^{700}=n_{deletion}^{700}=54$ and $n_{control}^{1000}=75$ and
$n_{deletion}^{1000}=76$.
* 2.
Asymmetry benchmark: all subjects have their asymmetric counterpart. Hence,
100 subjects were randomly picked among the subjects from the test set for
whom we took their asymmetric version. Resulting in
$n_{control}=n_{asymmetry}=100$.
* 3.
Interrupted central sulci: the whole test set is used as control data, leading
to $n_{control}=200$ and $n_{interrupted}=7$.
#### 2.5.2 On the Latent Space
##### A hint from the visualization
For both of our benchmarks and the interrupted central sulci, we first sought
to have a visualization of data distribution in the latent space. Therefore we
projected encoded data into a smaller space of two dimensions with UMAP
algorithm [40, 39]. This projection enables us to get a first hint as to how
outliers are represented.
##### Assessing the detection power on the benchmarks
However, the UMAP algorithm drastically reduces dimensions, leading to some
information loss. We tried to assess whether relevant information regarding
folding patterns was encoded in the latent space. Therefore, we trained linear
support-vector machines (SVM) [47] on the latent codes with stratified cross-
validation to classify between control data and benchmark. Performance is
assessed based on the ROC curve.
##### Quantifying the marginality of interrupted central sulci
As interrupted central sulci are very few, we cannot use classification as we
did for the benchmarks. Classic machine learning out-of-distribution
algorithms are more suited. Therefore, to quantify whether the interrupted
sulci are likely to be detected from their location in this reduced space, we
applied two classic algorithms, One-Class SVM (OCSVM) [56, 47], and isolation
forest [35, 47] based on the data coordinates in the UMAP space. However,
interrupted central sulci may not be the rarest pattern, and other folding
configurations may be very scarce. Therefore, we also looked at control
subjects repeatedly predicted as outliers by these algorithms.
##### Travelling through the latent space
Finally, to better understand the encoded properties and the learned
representations, we leverage the generative power of the $\beta-VAE$. We
computed average representations from different sets of data points, taking
the mean for each dimension of the latent space. We then reconstructed these
vectors. To further analyze the latent space, we traveled through it, going
from one point, either the average pattern or a subject, to another point in
the latent space, linearly interpolating vectors and reconstructing them.
#### 2.5.3 On the Folding Space
Outlier identification in the folding space relies on the model’s error. The
reconstruction errors’ distributions were compared visually and assessed with
the Kolmogorov-Smirnov test for the benchmarks and with the Mann-Whitney
U-test for interrupted central sulci. For both cases, the null hypothesis was
that the two distributions were identical.
The other strength of analyzing this space rather than the latent space is
that the model’s errors can help understand and locate the rare patterns’
characteristics. To localize the errors, we commonly look at the residuals,
which are the difference between the input and the reconstruction of the
model. This corresponds to what the model has missed or added. To
differentiate these two types of errors, we looked at them independently,
computing the difference between the input and the output, i.e., the model’s
omissions, and between the output and the input, i.e., the model’s additions.
It is particularly interesting in the case of interrupted sulci, as we could
expect that the model makes them continuous.
### 2.6 Generalization to another region
To assess the reproducibility in another region and to ensure that our
framework is not limited to the central sulcus, we transposed our methodology
to the isolated corpus callosum dysgenesis (CCD) which leads to a cortex
anomaly located in the cingulate region. This disorder is a congenital
malformation that results in a complete or partial absence of the corpus
callosum. The corpus callosum is composed of fibers that connect the two
hemispheres.
#### 2.6.1 Database
The dataset includes 7 children between 9 and 13 years old presenting an
isolated CCD and 7 matched control children [12]. Among the patients, 3
present a complete agenesis, 3 a partial agenesis, and one a hypoplasia,
corresponding to ”a homogeneous reduction of the callosal size” [63]. In this
case, the corpus callosum is completely formed, but abnormally small [6]. For
all children, the CCD was not associated with other malformations or
developmental disorders. As presented before, we used T1-w MR images obtained
from a Siemens Tim Trio 3T scanner with an isotropic resolution of 1mm.
#### 2.6.2 Transposition of the method
The described anatomical anomalies associated with CCD include ”sulci
radiating on hemisphere medial surface, complete or partial absence of the
callosomarginal sulcus and of the cingulate gyrus” [12]. Therefore, we
transposed our method to the cingulate sulcus region. Using the same
methodology as presented before, we computed a mask of the cingulate sulcus
(gathering the calloso-marginal anterior and posterior fissure in the
BrainVISA nomenclature), resulting in crops of dimensions 30 x 128 x 125 and
30 x 130 x 108, which were padded up to 32 x 128 x 128 and 32 x 136 x 112
respectively for the right and left hemispheres. We used the same data split
as before. We used the hyperparameters obtained with the gridsearch on the
central sulcus region for training. Choosing these parameters may lead to sub-
optimal performances but enables us to have a first validation of our
methodology. Analyses of the latent and the folding spaces are performed
following the method described above for the central sulcus. Since the corpus
callosum connects the two hemispheres, CCD can be studied equally in both
hemispheres. Therefore, we conducted our experiments in the right and in the
left hemisphere.
## 3 Results
### 3.1 Training Results
Each training lasted for approximately 1 hour on an Nvidia Quadro RTX5000 GPU.
We obtained with our gridsearch $\beta$ = 2 and L = 75.
### 3.2 Assessment on Synthetic Known Anomalies
#### 3.2.1 On the Latent Space
UMAP latent space visualizations for the four deletion benchmarks are
presented in Fig. 7. For the benchmark 200, benchmark data are rather
homogeneously distributed among control data, suggesting that simple surfaces
of sizes between 200 and 500 voxels are too subtle to be encoded differently.
Indeed, as shown in Fig. 5, small, simple surfaces can correspond to tiny
branches that display a high variability in the population. Therefore these
synthetic anomalies may be included in the normal variability. The
distribution of benchmark 500 seems to be not completely similar to the
control’s, but the restricted number of subjects makes it hard to conclude.
However, the trend becomes more pronounced for benchmarks 700 and 1000 where
fake anomalies are gradually gathered and their distributions are different
from the controls. These results are confirmed by the ROC curves (Fig.7). Even
when using all the latent dimensions, classification results are very poor for
benchmark 200 (AUC = 0.51), supporting that the deleted branches may be too
melted into the inter-individual variability. Classification performances are
also very low for benchmark 500 (AUC = 0.70). They start to be slightly better
for benchmark 700 (AUC = 0.81) but are very good only for benchmark 1000 (AUC
= 0.96).
Figure 7: Deletion benchmarks results. For each row, controls are represented
in green and benchmark data in pink. Left column: UMAP projection of benchmark
and control data. Middle column: ROC curves of classification of control and
benchmark data. Right column: reconstruction error distributions and p-value
of the Kolmogorov-Smirnov test with the null hypothesis that the two samples
come from the same distribution.
For the asymmetry benchmark, UMAP visualization demonstrates a good separation
between the right and the left hemisphere (Fig.8A), which is verified by the
classification of the whole latent space (AUC=0.82). These results suggest
that specific shape features are encoded among other properties in the latent
space.
To better understand the asymmetry characteristics encoded by the model, we
leveraged the generative power of our $\beta-VAE$. Fig. 8B. and C. show the
average patterns for the right (green) and the left hemisphere (blue) as
encoded by our model. The hand knob of the right central sulcus seems to be
slightly higher and shallower than in the left hemisphere. Moreover, the
double-knob configuration appears more prominent in the left hemisphere. To
further highlight the main differences between the two hemispheres, we
selected the most important dimensions for the classifier, here dimensions 9
and 36. In Fig. 9A., control and benchmark data are represented according to
these two dimensions. Even if the separation is not well marked, we can
observe a trend represented by the arrow. We tried to understand the features
encoded by the 9th dimension. We took the average for all 75 dimensions of the
latent space, and we traveled from the minimum to the maximum of the 9th
dimension and reconstructed the resulting vector. Fig. 9B. 1, 2, and 3
represent the reconstructions. These interpolations confirm the trend observed
previously. We observe a double-knob configuration in the left hemisphere. The
view from underneath and the side view enable visualizing the pli de passage
frontal moyen (PPFM). A pli de passage is a gyrus that connects two gyri and
which is buried in the depth of some furrows [38]. Fig.2B.1. and 2. propose a
visualization of a ”pli de passage” located in the central sulcus, the PPFM.
According to the different views from Fig.9B. 1, 2 and 3, it seems that the
PPFM is smaller in the right hemisphere and located higher in the central
sulcus.
Figure 8: Asymmetry benchmark results. Controls are represented in green and
benchmark data in blue. A. UMAP projection of benchmark and control data, ROC
curves of classification of control and benchmark data, and reconstruction
error distributions. B. Averages for the control subjects, i.e. right
hemispheres (in green), and for the highlighted asymmetry subjects, i.e. left
hemispheres (in blue). These averages are also placed on the UMAP dimensions.
C. 1. and 2. Respectively side and bottom views of the averages of B. The
single star indicates a single-knob configuration, and the two stars indicate
the second knob of a double-knob configuration. C. 3. Superposition of the two
averages respectively in upper and bottom view. Figure 9: Travelling through
the $9^{th}$ dimension of the latent space. A. Visualization of controls and
asymmetry benchmark according to the most important features of the
classifier. B. Interpolations along the $9^{th}$ dimension. 1, 2, and 3,
respectively correspond to the upper, bottom and side view of these
interpolations. C. Superposition of extreme interpolations.
#### 3.2.2 On the Folding Space
We then investigated whether the folding space, i.e, reconstruction errors,
was relevant for identifying outliers. For deletion benchmarks, we observe a
similar trend as in the latent space. For deletion 200, we cannot see a
difference of distributions (p-value = 0.38). However, from deletion 500 we
can see a stall with the deletion benchmarks having significantly higher
reconstruction errors (p-values of 0.044, 5.3e-14 and 1.2e-24 for benchmarks
500, 700 and 1000 respectively) (Fig. 7). On the contrary, for the asymmetry
benchmark, there is no significant difference, nor a trend, in the
reconstruction error distributions (Fig. 8).
### 3.3 Application on the Case of Interrupted Central Sulcus
#### 3.3.1 On the Latent Space
The UMAP projection from the latent space is shown in Fig. 10A. On this
distribution, we can observe that most interrupted central sulci are at the
margin of the point cloud except for one. Thus, it appears that the
representation learned by our model enables to project rare patterns at the
margin of the population. Interestingly, when we look at the pattern of each
one of the interrupted sulci, it seems that a specific pattern, the ”T-shape”
pattern [38] is specifically located on one side of the representation. Fig.
10B. shows the assessment of the marginality of the interrupted sulci based on
an OCSVM and isolation forest. Error margins correspond to various UMAP
projections, suggesting that the ability to detect interrupted CS in the UMAP
space is very dependent on the UMAP projection. Interrupted CS detection is
within the confidence interval, but the curves are close to the superior bound
suggesting a tendency. However, interrupted CS positions in the UMAP space are
not enough to detect them: detecting 5 interrupted CS out of 7 would lead to
more than 40% of false positives. Nevertheless, some other patterns considered
as controls and detected as outliers might also be rare.
Figure 10: Interrupted central sulci on UMAP space. A. Interrupted central
sulci shape distribution in the UMAP space. The 3D folding patterns of the
subjects are positioned according to their location in the UMAP space. For
instance, the pattern located in the lower left corner corresponds to subject
510225 in the UMAP representation. Subjects with interrupted sulci on the
upper left of the UMAP visualization seem to correspond to an interruption
with a T-shape pattern. B. Outlier detection performances using OCSVM and
isolation forest on the interrupted CS. C. Controls and interrupted CS
reconstruction error distributions.
Fig. 11 presents the controls’ patterns most often predicted as outliers by
the OCSVM. First, we note that the outliers are logically located at the
border of the distribution. Moreover, we observe distinct patterns in
different regions of the UMAP space. We visually highlighted the subjects of
the four regions. Analyzing the corresponding crops’ meshes, we observe
similarities within the groups. Group B seems to demonstrate a very wide open
knob. In addition, the knobs are well defined by the upper and the bottom part
of the sulcus. On the contrary, the sulci of group C appear to have larger
knobs than usual but they show more continuity with the upper and the bottom
parts. The pattern of group D seems to correspond to a rather flat central
sulcus with a close, long and continuous postcentral sulcus. The shape
characteristics of A are less obvious but the sulci give the impression of
having several small knobs, two or even three in the two bottom cases and a
small part of the precentral inferior opposite to an upper part of the
postcentral sulci. Fig. 12 provides a better understanding of these features.
For each pattern, we go from the centroid to one of the subjects in each group
by interpolating and generating samples. Fig. 12A. presents the interpolations
from the centroid to the several-knobs pattern. We gradually see the upper
part of the hand knob curving and becoming more pronounced until forming a
first knob at the top of the sulcus. Another knob in the bottom part appears
similarly. Likewise, patterns B, C and D vary progressively until they match
the centroid’s shape.
Figure 11: Control subjects identified as outliers. A, B, C and D correspond
to groups of visually similar patterns. The UMAP projection is the same as the
one in Fig.10. Control subjects identified as outliers are in pink and
subjects with interrupted central sulci are still represented in red. Figure
12: Travelling through the latent space from the centroid to the margin of the
UMAP space. The centroid is the centroid of HCP controls. Then, for each row,
interpolations between the centroid and one of the patterns of each group are
computed and then reconstructed.
#### 3.3.2 On the Folding Space
When analyzing the detection power on interrupted CS in the folding space, we
first note that the reconstruction errors’ distributions seem to be different
between HCP controls and interrupted CS (p-value = 0.0011). This result
suggests that our model has more difficulties to reconstruct the input and
that reconstruction error could constitute a relevant metric to detect rare or
abnormal patterns. However, having only seven subjects strongly limits our
conclusions and this should be replicated with more data.
Observing the reconstructions and the residual maps of Fig. 13 gives clues
into the way our model has encoded the interrupted CS. First, we can note that
the reconstruction quality is quite good visually. The model’s omissions
appear to be quite noisy (blue small fold pieces). The arrow points out an
omission beyond the noise which corresponds to a perpendicular branch pointing
toward the frontal cortex. Such a pattern might be an uncharacteristic
configuration. It is interesting to note that in six out of seven cases, the
model transformed interrupted sulci into continuous patterns. This is
highlighted by the ”output-input” visualizations. Unlike the omissions, the
model additions are rather localized. Moreover, the asterisks show where the
model has filled the interrupted sulci. Such visualization could be useful to
identify rare patterns like interruptions or perpendicular branches.
Figure 13: Reconstructions and residuals for all seven interrupted sulci. A.
For all rows, distance maps are converted to meshes for an easier
visualization. First row: input data. Second row: reconstruction of the model.
Third row: Reconstruction of the model with the difference between the input
and the output, i.e. the model’s omissions (in blue). The purple arrow
highlights an omission corresponding to a perpendicular branch pointing toward
the frontal cortex. Last row: Reconstruction of the model with the difference
between the output and the input, i.e. the model’s additions (in purple). B.
Rotated view of the reconstructions represented with asterisks in the last row
of A.
### 3.4 Application to corpus callosum dysgenesis
#### 3.4.1 On the Latent Space
We first compare distributions of CCD children (n=7) with control children
(n=7) acquired in the same conditions and with HCP adult subjects (n=200).
UMAP projections, presented in Fig.14A., give different results depending on
the hemisphere. For the right hemisphere, it seems that most children controls
are included in the distribution of adult controls (hcp_test in green). Five
out of the seven subjects having a CCD are located at the margin of the
controls, suggesting that their latent representation differs from the average
cingulate sulcus pattern. However, two subjects, one with a complete and one
with a partial agenesis, are in the middle of the controls. In the left
hemisphere, only three control children are clearly in the control adult
distribution. The other four are closer to the CCD subjects but they seem to
be still distinct. Indeed, CCD subjects are gathered very close to each other.
This could be due to the fact that there may be an age effect between
children’s and adults’ brains or a site effect (different scanners,
resolution), which we tried to reduce by using skeleton-based images but which
may still remain. Nevertheless, we can still observe a difference in
distribution between control children and CCD subjects.
Figure 14: Results on corpus callosum dysgenesis (CCD) subjects. First row:
right hemisphere. Bottom row: left hemisphere. For both rows: A. UMAP
projections of CCD subjects, control children and HCP test. B. Reconstruction
error distributions for the CCD subjects, control children and HCP test. C.
Reconstruction error variations for the CCD subjects, control children and HCP
test. Significant differences between populations according to the Mann-
Whitney test are indicated with an asterisk.
#### 3.4.2 On the Folding Space
Regarding reconstruction error distributions (Fig.14B.), we observe for both
hemispheres that control children seem to have the same distribution as adult
controls, which is confirmed by Fig.14C. (p-value=0.034 and 0.017 respectively
for right and left hemisphere). On the contrary, CCD subjects present higher
reconstruction errors that are significantly different from both HCP controls
(p-value=3.6e-06 for the two hemispheres) and children controls
(p-value=0.0011 for the two hemispheres). Therefore, it seems that there is a
complete individual separability of the CCD patients which is very promising
and should be replicated with more data.
The reconstructions presented in Fig.15 highlight the singularities of CCD.
The model’s additions mostly make the cingulate more continuous than
initially. The model’s omissions are mainly small branches perpendicular to
the cingulate sulcus that are radially oriented.
Figure 15: Right cingulate sulcus reconstructions and residuals for the CCD
subjects and one control. A. CCD subjects. B. One control subject from the
same cohort. For both A. and B.: each column corresponds to a subject. For all
rows, distance maps are converted to meshes for an easier visualization. First
row: input data. Second row: reconstruction of the model. Third row:
Reconstructions of the model with the difference between the input and the
output, i.e. the model’s omissions. Last row: Reconstructions of the model
with the difference between the output and the input, i.e. the model’s
additions. The arrows highlight interesting features added or missed by the
model.
## 4 Discussion
This work proposed a methodology to study rare folding patterns which was
applied to the central sulcus region and to a described rare pattern,
interrupted central sulci. Specifically, we represented folding patterns with
distance maps and leveraged the generative power of the $\beta-VAE$ to have a
better understanding of the learned representations. In addition, we proposed
a way to study the granularity of deviations that can be identified and we
brought to light several rare patterns in the region. We also compared the
identification power of both the latent space and the folding space. Finally,
we assessed the generalization of our methodology on a developmental anomaly
in another region.
### 4.1 Latent Space and Folding Space, Two Complementary Information
In many anomaly detection works applied to medical images, the detection is
performed based on the reconstruction error rather than in the latent space
[54, 3, 4, 64]. However, both of these spaces have their interest and could
bring complementary information. In our work, we studied four types of rare
patterns, two synthetic types, deletion and asymmetry benchmarks, and two
actual rare patterns. These four categories differ from control data by their
own characteristics and thus help to study the granularity detected, that is
to say, the typology of rare features that can be identified. For instance,
the asymmetry benchmark includes more double-knob configurations. Depending on
the size of the deleted simple surface, deletion benchmarks represent
different features: benchmarks 200 and 500 represent mainly a missing branch
with increasing size, which may represent the normal variability of branches.
Benchmark 700 could look like an interrupted sulcus in some cases or in
others, like benchmark 1000, an unlikely configuration. Interrupted central
sulci present a clear interruption and a rare arrangement of the shapes
forming the central sulcus. Last, CCD subjects demonstrate a missing sulcus or
missing sulcal parts and branches with different orientations.
These different kinds of deviations from the norm provide clues to the
characteristics of rare patterns that can be identified respectively in the
latent space or in the folding space of our model. As a matter of fact, the
identification performances in the latent and in the folding space vary
depending on the kind of patterns. For deletion benchmarks, the folding space,
based on the reconstruction error, seems to enable the identification of
unusual patterns from smaller modifications: different distributions are
observed from 500 deleted voxels. Whereas in the latent space, the detection
requires at least 1000 deleted voxels. Likewise, for the interrupted central
sulci, despite the small number of samples, their detection seems to be easier
on the basis of reconstruction error than in the latent space. Similar results
were obtained on CCD subjects even if the latent representation was
encouraging. In return, the error distribution of the asymmetry benchmark is
not different from that of the controls, but the benchmark is well detected in
the latent space. Therefore, the latent space could be more sensitive to shape
arrangements than the folding space. The lack of difference in the error
distributions may be due to the fact that the voxel-to-voxel differences
between the right and left central sulci are local and subtle and could be
embedded in the normal variability. In addition, the reconstruction error is
for the entire image. Therefore, in the case of small and very local
deviations from the norm, the reconstruction error alone is likely to be
insufficient. A way to limit such effects could be to use a more local error,
applied to sub-regions or patches for instance.
The difference in the outlier detection performance may also lie in the way
our model encodes the outliers. Based on our results, we can consider several
cases. First, a rare configuration is represented by several samples present
in the training set. This would be the case with the asymmetry benchmark.
Indeed, there are more double-knob configurations in the left hemisphere but
single and double-knob patterns coexist on both sides. In such a case, the
distribution support of the left and right hemispheres are the same, but the
densities differ, which could lead to a projection of the outlier at the
margin of the latent space but to a good reconstruction. Second, the rare
configuration is almost never represented in the training set and the model
has not detected and thus encoded its local specificity. Then, the subject
would be encoded with a ”default” representation and projected in the middle
of the other subjects. This would be consistent with the results of [25],
where major anomalies (different parts of the brain from the one considered in
the train set) were projected in the middle of the point cloud and
reconstructed as the average reconstruction. It could be the case of the
benchmark deletion 500 and of the interrupted central sulcus that is projected
in the point cloud. Last, the outlier configuration is almost never
represented in the training set but the model has detected the rare
characteristic. The subject is then projected at the margin of the point cloud
and the decoder has not learned this part of the latent space leading to a
poor reconstruction (interrupted central sulci, CCD subjects).
Nevertheless, in all cases, a strength of the folding space is the possibility
to localize the reconstruction errors and, in some cases, the unusual
features. If not too noisy, reconstruction errors can be very informative. For
example, in the case of interrupted central sulci, looking at the model’s
addition permits clearly localizing what is atypical in a subject (Fig.13).
Similarly, in the case of the CCD subjects, the reconstruction errors
highlight the presence of radial small branches that are typical of this brain
disorder [12]. But some noise remains, and it might be interesting to add an
additional constraint to represent only errors that correspond to a minimum
number of contiguous voxels. This could lead to a good explanation of the
abnormality which is of major importance in the field and especially when
applied to medical images. Other explanation methods exist, directly on the
network such as Grad-CAM [57] or on an OC-SVM applied on the learned features
[59] for instance; but the use of the reconstruction error is immediate and
easy to implement. Hence, the latent space and the folding space, based on
reconstruction error, can provide complementary information and both can be
used to identify rare patterns.
### 4.2 Data size limitations and unknown number of rare patterns
The method should be further qualified because of the low number of our
examples of rare patterns. While the study of a known rare pattern is
interesting and important, having only seven samples severely limits our
conclusions. Similarly, the poor results of benchmarks 500 and 700 in the
latent space could be due to their small size, and having larger benchmark
datasets could lead to increased performances.
Also, we assessed our method in the CS area on the benchmarks and on an
existing rare pattern, but because few rare patterns have been described in
this region, there may be other rare configurations in what we consider the
control population. For instance, three morphologic variants in the central
sulcus region have been introduced, representing 2.9%, 7.0% and 1.8% of the
studied population, opposed to 78.2% of ”omega” shape, i.e. the central sulcus
knob and 10.1% of ”epsilon” shape which corresponds to the double-knob
configuration [14]. This multiplication of rare patterns in the populations
would make the identification of interrupted central sulci more difficult.
### 4.3 Relevance of synthetic benchmarks
Moreover, we can wonder about the relevance of our synthetic benchmarks.
Although synthetic rare patterns are of high interest as they enable to
quantify the performances on different degrees of deviations from the norm,
few works have been interested in them to our knowledge [25, 41]. But the use
of fake deviations raises the question: do they constitute adequate rare
patterns? Few studies introduced rare folding patterns based on the
arrangement of their shapes such as the PBS [42], an interrupted central
sulcus [38] or a flat central sulcus [60]. Here, we emphasize their advantage
in the study of our understanding of the brain: they are evidence of
neurodevelopmental processes and then stable throughout life. But other
abnormal sulcal features have been studied and found to be important and
correlated with neurodevelopmental disorders, such as the depth, which
demonstrated anomalies in autism spectrum disorder [44, 17] or Williams
syndrome [19] for instance. Despite being another subject of study, a
benchmark corresponding to central sulcus depth variations could be
interesting to assess whether our framework can be extended to detect such
anomalies.
Regarding the current benchmarks we use, we said that small erased SS could
remain undetected as this deletion could be embedded in the normal
variability. However, there may be several categories of deletion deviations.
Some may be minor, as a small SS representing a tiny branch. On the opposite,
some small SS, for instance one corresponding to depth change, representing
the presence of a pli de passage, and thus leading to an interrupted central
sulcus would be expected to be a major feature of the topology. Hence, our
criterion, only based on the size of the SS may be insufficient and it could
be interesting to add another one, such as topological criteria.
In any case, having an unusual feature (e.g., a missing simple surface or
unusual depth) that can be incrementally increased, or comparing several types
of features, helps characterize the detection power of a model and the
features likely to be detected.
### 4.4 Learning Relevant Representations
When dealing with sulcal patterns and their high complexity, it may be easier
to use representations of the folding which attempt to gather several subjects
with similar patterns. Local averages of sulci, also called moving averages,
enable to concentrate on the main features of the different patterns and are
thus very useful to analyze folding patterns [61, 66, 22, 26]. From a graph-
based representation of the sulci, the identification of patterns can be done
after computing similarity and applying a clustering [43]. Our approach
proposes another method to learn sulcal representations. From our cropped
distance maps, the $\beta-VAE$ learns a mapping to a latent representation
which can then be reconstructed. Therefore, rather than explicitly computing
pairwise similarity between the subjects, gathering them, and then analyzing
the patterns, we hope that our $\beta-VAE$ directly learns shapes that can be
combined and arranged in patterns. The representations learned by our model
seem to be relevant and consistent with some morphological characteristics of
the central sulcus area.
First, the reconstruction of the average representation of the right central
sulcus is composed of an upper knob whereas the left average tends more
towards a double-knob configuration (respectively green and blue sulci in
Fig.8B). This is one of the main known asymmetries in terms of patterns and it
appears early in the development. It has been detected in infants of 30 weeks
postmenstrual age [66] and in adults [61].
We also observed differences in terms of curvature of the hand-knob, with a
hand-knob more pronounced in the left hemisphere than in the right (Fig.8B.).
Considering that we study a right-handed population, this could be related to
handedness. With the lateralization of the hand motricity, we expect the motor
area of the right hand in the left hemisphere and particularly the precentral
gyrus to be more developed for right-handed subjects, pushing backward the
upper part of the central sulcus which would result in a knob more pronounced.
This interpretation is consistent with a study on one-handed subjects that
showed that subjects born without a hand had a flatter central sulcus
contralateral to the missing hand [60].
Another interesting property that was successfully encoded is the PPFM. This
pli de passage was first described in 1888 [11] and has been a source of
growing interest due to its link with the motor hand area [7] and in the
context of understanding the formation of the knob regarding evolutionary
questions [28]. Our model was able to encode the PPFM in the latent space as
well as its asymmetry characteristics. Indeed, we observed that the PPFM is
smaller in the right hemisphere which corresponds to central sulcus depth
variations described in [1]. This is also consistent as the PPFM has been
correlated to the hand. Therefore, right-handed subjects tend to have a more
developed hand area in the left hemisphere and thus a larger PPFM.
Hence, our latent space has learned relevant normal characteristics that are
consistent with the region’s morphology. It has also enabled to propose four
other groups of likely rare patterns (Fig.11). The pattern representing a
rather flat central sulcus is indeed a non-typical configuration. Less than 2%
of the studied subjects were reported to have such a configuration in [14].
Moreover, flat central sulci appeared as the most important feature when
comparing controls to congenital one-handed subjects who tended to demonstrate
flatter central sulci [60], confirming that flat central sulci are less
frequent patterns. The groups representing large knobs and wide open knobs
(Fig.11B. and C.) are also an atypical configuration that is present at one
extremity of the axis representing the most extreme variations in Human and is
closer to configurations we observe in Chimpanzees [22].
### 4.5 Generative power of $\beta-VAE$ and comparison with other strategies
Since our proposed framework is able to encode relevant features regarding
folding patterns, the generative power of the $\beta-VAE$ can be exploited.
Indeed, reconstructions and interpolations are tools to understand the folding
variability. We have just mentioned that the learned patterns were relevant
and consistent with those obtained by other methods, but our method has the
advantage of being able to reconstruct and interpolate. For instance,
interpolations along the main axis of asymmetry variations highlight the
evolution from a right to a left hemisphere. It can also be useful to
understand the folding process and in particular the formation of interrupted
central sulci. As a matter of fact, on Fig.12C., an interruption of the
central sulcus happens when interpolating from the central subject to one
control outlier. When observing the PPFM, we can see that the PPFM increases
until reaching the surface of the brain and thus interrupting the central
sulcus. Jointly, the inferior sulcal part connects to the precentral sulcus.
Such observations may provide additional clues in our understanding of the
folding processes.
But other deep learning models could be interesting to study folding patterns.
For instance, $\beta-VAE$ reconstructions are known to be blurry contrary to
GAN’s. Currently, this shortcoming is limited as we seek to have a simpler
representation of folding patterns, still, for more subtle details, another
model may be better suited. In addition, in the anomaly detection field,
models that add constraints on controls distribution are quite appealing. For
instance, deep One-Class Classification and its derivatives have been proposed
to push control data into the smallest hypersphere in the latent space [53].
This could help increase the detection performance in the latent space.
Nevertheless, no matter the architecture or the framework, an important limit
to understanding what our model has really encoded is the high number of
latent dimensions.
One can also wonder about the representation of the folding patterns. In the
introduction, we mentioned two main strategies: clustering and manifold.
Usually, these two approaches are applied to a continuous space. Nevertheless,
if we consider sulcal shapes as symbolic entities that can be combined and
arranged, we could represent folding patterns based on a discrete space rather
than a continuous one. As such, VQ-VAE [46] seems to be an interesting
representation to compare with our present results.
Finally, this framework of outlier detection based on training on control
subjects alone may be sensitive to outliers present in the training set.
Having a contaminated dataset could severely limit the detection performances,
at least in the folding space which is based on the reconstruction error. It
has been reported in a brain tumor detection problem that having 3% of
outliers in the training set (about 1000 samples) leads to a decrease of 5% of
the AUROC and to a 13% decrease if the contamination reaches 12% of the
training set [4]. Therefore, one serious shortcoming of our paradigm is that
we do not know the outliers we are looking for. Applying our framework to a
control population alone in order to bring out rare patterns may limit the
different patterns that can be identified. A way to tackle this issue and to
increase the patterns detected would be to exploit the presence of outliers in
the training set as proposed in [50]. In their technique, the authors
introduce an iterative joint training where they assign labels (anomalous or
control) to the examples, and then optimize the network’s parameters to better
identify the anomalies. Such a method could also enable to project the
outliers more at the margin of the latent space. The impact of the presence of
outliers during training on the latent space has not yet been investigated to
our knowledge. If, as we suggested before, outliers present in the training
phase are encoded at the margin of the distribution, i.e. in a different area
of the latent space, it could be interesting to deepen our analysis, based on
clustering for instance.
### 4.6 Generalization of the approach: towards an analysis of the whole
brain?
This work has shown that our approach had successfully encoded some relevant
features of the folding patterns in the central sulcus region but it is
attractive to think about the behavior and results we could obtain in other
parts of the brain. Here, we assessed the generalizability of the framework on
another dataset and another region. Our results suggest that our method can
well transpose in other brain regions. Specifically, even if we use the
hyperparameters ($\beta$ and L) optimized for another area, the learned
representations still enable us to distinguish between control and outlier
subjects. This is all the more interesting that the two studied regions are
rather different. The central sulcus is one of the first folds to form and is
rather stable, contrary to the cingulate region that is more variable [62].
Therefore, it seems that no matter the folding variability of the zone, our
framework can be applied. This encouraging result raises a question regarding
the procedure to adopt to extend our analysis to the whole brain. A way could
be to define a set of regions, consistent with the cytoarchitecture and
function and to train our $\beta-VAE$ on each region. In particular, some
areas seem to be interesting from a clinical point of view [49, 70, 24, 10,
29]. Our future works may thus focus on proposing an adequate methodology to
tackle the whole brain.
On the other hand, when we applied this framework to CCD subjects, we also
operated a domain shift. Indeed, the dataset to explore included exclusively
children while the $\beta-VAE$ was trained on young adults. Despite folding
patterns being reported as trait features [13], such an age variation may have
an impact. In addition, beyond dealing with children, the site and the scanner
are different. Such differences have been reported to affect the
generalizability and the performances on various targeted tasks. In terms of
distributions in the latent space, despite the fact that the distribution of
the controls does not seem to completely overlap the distribution of the HCP
controls, the patients still seem to present a different distribution than
both controls’ populations. Moreover, the domain shift does not seem to have
an effect on the folding space where controls reconstruction errors are not
significantly different from HCP contrary to CCD subjects that have
significantly higher reconstruction errors. However, having only seven
subjects makes it difficult to conclude on the importance of these age and
site effects for our task. We will explore these questions in further studies.
To conclude, this study proposed a framework to identify rare and abnormal
folding patterns based on the modeling of the inter-individual variability.
With a new representation of folding patterns, we proposed a model that was
able to encode relevant folding characteristics. The use of synthetic rare
patterns enlightened the identification power of our model on both the latent
space and the folding space. Finally, we successfully generalized our approach
to another clinical brain anomaly in the cingulate region. Our results open up
several avenues of work such as the definition of new synthetic benchmarks
that match the characteristics of other known anomalies, the use of other deep
learning models that exploit the presence of outliers in the training set, or
the use of our framework to better understand the folding process.
## 5 Funding
This work was supported by the European Union’s Horizon 2020 Research and
Innovation Programme under Grant Agreement No. 945539 (HBP SGA3), the
ANR-19-CE45-0022-01 IFOPASUBA, the ANR-14-CE30-0014-02 APEX the
ANR-20-CHIA-0027-01 FOLDDICO.
## 6 Acknowledgments
The authors thank G. Dehaene for her involvment in the scanning procedure of
the children cohort and C. Langlet for his help in the discussions about this
study. Data were provided in part by the Human Connectome Project funded by
the NIH.
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|
# From concentration to quantitative regularity: a short survey of recent
developments for the Navier-Stokes equations
Tobias Barker Department of Mathematical Sciences, University of Bath, Bath
BA2 7AY, UK<EMAIL_ADDRESS>and Christophe Prange Cergy Paris
Université, Laboratoire de Mathématiques AGM, UMR CNRS 8088, France
<EMAIL_ADDRESS>
###### Abstract.
In this short survey paper, we focus on some new developments in the study of
the regularity or potential singularity formation for solutions of the 3D
Navier-Stokes equations. Some of the motivating questions are: Are certain
norms accumulating/concentrating on small scales near potential blow-up times?
At what speed do certain scale-invariant norms blow-up? Can one prove explicit
quantitative regularity estimates? Can one break the criticality barrier, even
slightly? We emphasize that these questions are closely linked together. Many
recent advances for the Navier-Stokes equations are directly inspired by
results and methods from the field of nonlinear dispersive equations.
_In honor of Carlos Kenig’s 70th birthday_
Keywords Navier-Stokes equations, norm concentration, quantitative estimates,
regularity criteria, supercritical norms, slight criticality breaking,
Kolmogorov scales.
Mathematics Subject Classification (2010) 35A99, 35B44, 35B65, 35Q30, 76D05
###### Contents
1. 1 Introduction
2. 2 Weak concentration
3. 3 Local-in-space smoothing
1. 3.1 Lin type compactness methods
2. 3.2 Caffarelli-Kohn-Nirenberg type methods
3. 3.3 Scaled local energy methods
4. 4 Strong concentration
5. 5 Three facets of quantitative regularity
1. 5.1 Quantitative blow-up rates
2. 5.2 Quantitative regularity estimates
3. 5.3 Quantitative estimates of dissipative/Kolmogorov scales
6. 6 A general strategy for quantitative estimates of dissipative scales
1. 6.1 The toy model $L^{5}_{t,x}$
2. 6.2 The case of $L^{\infty}_{t}L^{3}_{x}$
7. 7 Quantitative regularity in the Type I case
1. 7.1 Three facets of the quantitative regularity in the Type I case
2. 7.2 Quantitative estimates of dissipative scales in the Type I case
3. 7.3 A comparison with Tao’s strategy
8. 8 Some further developments
9. 9 Mild criticality breaking and a conjecture of Tao
10. 10 Summary of selected results
## 1\. Introduction
This short survey paper is concerned with recent developments in the study of
the regularity or potential singularity formation for solutions of the 3D
Navier-Stokes equations
$\partial_{t}U-\Delta U+U\cdot\nabla U+\nabla P=0,\qquad\nabla\cdot U=0.$
We mostly focus on the whole-space case $\mathbb{R}^{3}$ or localize away from
physical boundaries. We will mainly concentrate on two topics, concentration
of solutions on small scales on the one hand and quantitative regularity on
the other hand, and show how these subjects are related. The study of these
questions is recent for the Navier-Stokes equations, all the main results in
this paper were published in the last five years.
In 2003 Escauriaza, Seregin and Šverák [29] were able to the blow-up of the
critical borderline norm $L^{3}$ for potentially singular solutions of 3D
Navier-Stokes. This paper was followed by a tremendous amount of works proving
analogous results for many kind of evolution equations with a scaling
symmetry111Divergence of critical norms near maximal time of existence for
PDEs with a scaling symmetry does not follow from local well-posedness theory,
see [58]. [58, 41, 42, 44, 28, 62, 61, 15]… These results are all qualitative,
with the exception of the work by Merle and Raphaël [58] for nonlinear
Schrödinger, which gives a quantitative blow-up rate for a critical norm.
A breakthrough for the Navier-Stokes equations was achieved by Tao in 2019
[84]. He was able to explicitly quantify the rate of blow-up of the critical
$L^{3}$ norm near a potential first-time singularity
$\limsup_{t\rightarrow
T^{*}}\frac{\|U(\cdot,t)\|_{L^{3}(\mathbb{R}^{3})}}{\big{(}\log\log\log(T^{*}-t)\big{)}^{c}}=\infty,$
(1.1)
for a universal constant $c\in(0,\infty)$. Previously, only abstract
quantitative results were known, which were based on abstract quantification
of the seminal (qualitative) result of Escauriaza, Seregin and Šverák [29] and
the use of persistence of singularities.
Some techniques described in this paper are directly inspired by methods
introduced for nonlinear dispersive equations. In this vein, let us mention
the ‘stacking of scales scheme’ used to prove quantitative regularity
estimates in Section 6 and Section 7 inspired by [58], and the ‘mild
criticality breaking’ result in Section 9 inspired by [13].
### Explicit quantitative estimates: for what purpose?
Blow-up rates and quantitative regularity estimates are two sides of the same
coin. Let us outline four motivations for the study of quantitative regularity
and blow-up rates:
1. (1)
In the field of PDEs, there seems to be very few quantitative rates for
critical norms near the maximal time of existence.
2. (2)
Quantitative regularity estimates with explicit bounds under a priori
boundedness of critical norms enable to break (to some limited extent) the
criticality barrier. For instance in Section 9, we use the result of Tao [84]
to derive a new regularity criteria in terms of a slightly supercritical
Orlicz norm.
3. (3)
Blow-up rates for critical norms such as (1.1) may enable to rule out certain
blow-up scenarios for which the numerically computed growth of the $L^{3}$
norm is too slow. However, the extremely slow triple logarithmic rate in (1.1)
means testing it may be beyond computing capacities, as is emphasized in Hou’s
recent paper [34].
4. (4)
As is well understood for dispersive equations, finding blow-up rates and
concentration estimates is a first step toward understanding potential blow-up
profiles.
5. (5)
In turbulence theory, cascade processes are the dominant feature of the
inertial range, where nonlinear inertial effects dominate (or are in balance
with) viscous dissipative effects. Below certain scales though, so-called
‘Kolmogorov scales’, for very high wavenumbers or small spatial scales,
dissipative effects dominate. Estimating those dissipative scales
quantitatively is one of the main objectives of the quantitative regularity
theory, see Section 5.3 and Objective C.
### Outline of the paper
This survey paper is partly based on several talks given in the past two
years. Further comments and topics are found in the habilitation thesis [71].
The first three sections are devoted to ‘weak concentration’ (Section 2),
‘strong concentration’ (Section 4) and a fundamental tool for the quantitative
regularity, namely local-in-space smoothing (Section 3). The rest of the paper
is concerned with quantitative regularity and blow-up rates for critical
norms. Section 5 describes three facets of quantitative regularity: blow-up
rates, quantitative regularity estimates and quantitative estimates for
dissipative scales. Moreover, two cases are studied, the case of $L^{5}_{t,x}$
which is a toy model, and the case of $L^{\infty}_{t}L^{3}_{x}$ studied by
Tao. Section 6 focuses on the strategy to estimate the dissipative scales in a
quantitative manner. Section 7 concentrates on the Type I case. Section 8
reviews some recent developments in the wake of [84]. Section 9 shows that the
scaling barrier can be slightly broken thanks to good quantitative estimates
in the critical case. This section is based on the paper [8]. There a
(partial) answer to a question asked in [84] about the blow-up of certain
Orlicz norms is given. Finally in Section 10, we summarize some results in two
tables.
Notice that $C$ and $c$ are universal positive constants that may change from
line to line.
## 2\. Weak concentration
In this paper we distinguish between:
‘weak concentration’:
of norms near a potential blow-up time $T^{*}$; these results assert the
existence of points $x(t)$ (or a sequence of points $x_{n}$) and scales
$\lambda(t)$ (or a sequence of scales $\lambda_{n}$) such that certain norms
accumulate on $B_{x(t)}(\lambda(t))$ as $t\rightarrow T^{*}$;
‘strong concentration’:
of norms near a potential space-time blow-up point $(x^{*},T^{*})$;222A ‘blow-
up/singular point’ $(x^{*},T^{*})$ is a point for which the solution is
unbounded in any parabolic cylinder
$Q_{(x^{*},T^{*})}(r)=B_{x^{*}}(r)\times(T^{*}-r^{2},T^{*})$ centered at that
point. Conversely, a ‘regular point’ is a point which is not a singular point.
It is known that determining whether or not the singular points occur for
Leray-Hopf solutions is equivalent to the Millenium problem. these results
assert the existence of scales $\lambda(t)$ (or a sequence of scales
$\lambda_{n}$) such that certain norms accumulate on shrinking balls
$B_{x^{*}}(\lambda(t))$ as $t\rightarrow T^{*}$.
#### A short review of concentration for certain nonlinear PDEs
The first results on concentration near potential singularities date back to
more than 30 years ago. They mainly fall into the class of ‘weak
concentration’. The study of ‘mass concentration’ i.e. concentration of the
$L^{2}$ norm for the nonlinear Schrödinger equations has triggered a lot of
developments in this direction, in the wake of the seminal results by
Weinstein [86], Merle and Tsutsumi [59], Nawa [64, Theorem B and Theorem C]
and [65, Theorem B], Merle [56, 57]. These results were followed by many
others: Bourgain[11], Nawa and Tsutsumi [66], Hmidi Keraani [32, Corollary
1.8] for nonlinear Schrödinger, Kenig, Ponce and Vega [43, Corollary 1.4] for
the KdV equation, Merle and Zaag [60, Theorem 1, (ii)] for the semilinear wave
equation… We also refer to the books by Cazenave [17, Section 6.5], Tao [82]
and Sulem and Sulem [80, Section 5.2.4 and Section 14.3.2].
There are also a number of results that fall into the category of ‘strong
concentration’ results, especially for the nonlinear Schrödinger equation, for
radially symmetric solutions: Merle and Tsutsumi [59], Tsutsumi [85, Theorem
1.1], Holmer Roudenko [33, Theorem 1.2]…
The topic of concentration is also strongly tied to proving the blow-up of
certain critical norms. The recent paper of Mizoguchi and Souplet on the
semilinear heat equation states a strong concentration property for a Type I
singularity [62, Lemma 3.1] that is key to proving the blow-up of a critical
norm in that case; see also Miura and Takahashi [61] without the Type I
assumption.
As for fluids, we mention two results dealing with the possible energy
concentration in solutions of Euler and Navier-Stokes equations. These results
are a bit orthogonal to the concentration results on which we focus in this
paper, but are also interesting directions. Chae and Wolf [19] proved that for
3D Euler under a Type I condition there is no concentration of the energy into
isolated points at possible blow-up times. For 3D Navier-Stokes, Arnold and
Craig [3, Theorem 4.5] gave a lower bound on the energy concentrating set.
#### Weak concentration for the Navier-Stokes equations
Li, Ozawa and Wang [51, Theorem 1.2] prove what we believe is the first
concentration result for potential blow-up solutions of the 3D Navier-Stokes
equations. It is proved that for any Leray-Hopf solution $U$ that first blows-
up at time $T^{*}\in(0,\infty)$, one has for any $q\in[1,\infty]$, the
following concentration of the $L^{q}$ norm:
$\|U(\cdot,t)\|_{L^{q}(|\cdot-
x_{n}|\lesssim\omega(t_{n})^{-1})}\gtrsim\omega(t_{n})^{1-\frac{3}{q}}$ (2.1)
where $t_{n}\rightarrow T^{*}$, $x_{n}\in\mathbb{R}^{3}$ and
$\omega(t_{n}):=\|U(\cdot,t_{n})\|_{L^{\infty}(\mathbb{R}^{3})}\stackrel{{\scriptstyle
t\rightarrow T^{*}}}{{\longrightarrow}}\infty$. The proof is based on: (i)
selecting a sequence of times $t_{n}$ on which one has large growth of the
$L^{\infty}$ norm of the solution, (ii) showing that the low frequencies
$\lesssim\omega(t_{n})$ contribute to a large part of the growth of the
$L^{\infty}$ norm of $U$.
Let us make two comments on this result. First, it is a concentration result
that holds for any Leray-Hopf solution that blows-up. The price to pay for
this generality is the fact that the concentration is weak, i.e. not localized
in space. Second, the concentration (2.1) holds for any norm in the Lebesgue
scale, no matter whether the norm is subcritical ($q\in(3,\infty]$), critical
($q=3$) or supercritical ($q\in[1,3)$). Notice that in this last case, the
lower bound in (2.1) goes to zero as $t\rightarrow T^{*}$. In the case of
$q\in(3,\infty]$, one can bound the right hand side of (2.1) from below by
$(T^{*}-t_{n})^{-\frac{1}{2}(1-\frac{3}{q})}$ thanks to Leray’s [50] lower
bound, see (5.1) below, and get the concentration on a ball of size
$\sqrt{T^{*}-t_{n}}$.
Existence results of mild solutions with $L^{q}_{uloc}$ initial data also
enable to prove weak concentration for potential blow-up solutions to the
Navier-Stokes equations. This was remarked in [53, Corollary 1.1], where the
existence of mild solutions in $L^{q}_{uloc}$ is combined with a simple
scaling argument to yield that for every $t\in(0,T^{*})$, there exists
$x(t)\in\mathbb{R}^{3}$, such that for any $q\in[3,\infty]$,333Let us stress
that although the paper [53] is actually concerned with the existence of mild
solutions for data in $L^{q}_{uloc}$ in the half-space, we state (2.2) for the
whole-space. This concentration result is implied by the existence result of
mild solutions in the whole-space from [55] and the argument of [53, Corollary
1.1].
$\|U(\cdot,t)\|_{L^{q}(|\cdot-x(t)|\leq\sqrt{T^{*}-t})}\gtrsim(T^{*}-t)^{-\frac{1}{2}(1-\frac{3}{q})}.$
(2.2)
This strategy is robust enough to apply to the half-space $\mathbb{R}^{3}_{+}$
as in [53].
In [39, Theorem 1.6 (i)], Kang, Miura and Tsai prove a statement that can be
read as weak concentration result for the supercritical $L^{2}$ norm near
potential singularities of the Navier-Stokes equations: there exists
$\gamma_{univ}\in(0,\infty)$, $S\in(0,\infty)$ and a function
$x=x(t)\in\mathbb{R}^{3}$ such that for all $t\in(0,T^{*})$,
$\frac{1}{\sqrt{T^{*}-t}}\int\limits_{B_{x(t)}\big{(}\sqrt{\frac{T^{*}-t}{S}}\big{)}}|U(x,t)|^{2}\,dx>\gamma_{univ}.$
(2.3)
This result is in the vein of the one of Bradshaw and Tsai [12, Theorem 8.2],
see also Grujić and Xu [31, Theorem 4.1].
The bottom line is that ‘weak concentration’ results are in general a
consequence of global regularity results or local well-posedness results and
hence use a limited amount of specific structure of the equations. In order to
have ‘strong concentration’ results, one needs more localized regularity
results at the price of additional a priori assumptions such as Type I. This
is the topic of the next section.
## 3\. Local-in-space smoothing
The idea behind local-in-space smoothing is very natural. Assume that one is
given a rough initial data, for instance finite-energy, that happens to be
more regular, in the sense critical or subcritical, on some fixed ball, say
$B_{0}(1)$ to fix the ideas. For Navier-Stokes the general question becomes:
> Is the local smoothing due to the heat part of the equation strong enough to
> compensate for the nonlocal effects of the pressure that tend to propagate
> irregularities of the solutions from large-scale spatial scales to
> $B_{0}(1)$?
The answer is yes in many situations, when the local ‘regular’ data is taken
in critical or subcritical Lebesgue, Lorentz or Besov spaces. More
surprisingly, such results remain even true for the Navier-Stokes equations in
the half-space with no-slip boundary condition, where nonlocal effects are
known to be strong, see for instance [36, 37, 76]. When the answer is yes, we
call such results ‘local-in-space short-time smoothing’. The solution $U$ then
satisfies bounds of the type
$\sup_{t\in(0,S(M))}t^{\frac{1}{2}}\|U(\cdot,t)\|_{L^{\infty}(B_{0}(\frac{1}{2})))}\lesssim
1$ (3.1)
under the condition that $\|U_{0}\|_{L^{3}(B_{0}(1))}\lesssim 1$ and for a
short time $S=S(M)\ll 1$. This line of research was pioneered by Jia and
Šverák in the seminal paper [35]. We find three main classes of methods that
we sketch below. For more details and in particular statements of theorems, we
refer to [35, Theorem 3.1, Section 2 and 3], [6, Theorem 1, Section 2 and 4],
[71, Chapter 5], [39, Theorem 1.1 and Section 3] and [46, Theorem 1.6 and
Section 5].
Let us stress that local-in-space smoothing is a versatile tool that proved to
be efficient in many situations, such as the existence proof of self-similar
solutions [35], strong concentration (see below Section 4) and quantitative
regularity (see below Section 5 and Section 7). For the last point in
particular, it is important to have quantitative versions of local-in-space
smoothing, see for instance [9, Theorem 5.1].
Local-in-space smoothing also helps us gain understanding of a very natural
question: for which initial data does the associated solution exhibit improved
regularity properties?
### 3.1. Lin type compactness methods
We work with $U$ a global-in-time local energy solution to the Navier-Stokes
equations on $\mathbb{R}^{3}\times(0,\infty)$ with initial data $U_{0}\in
L^{2}_{uloc}(\mathbb{R}^{3})$ with some mild decay at space infinity in order
to rule out parasitic solutions and get a formula for the pressure.444The
framework here is that of global solutions. However, it is possible to
localize the results and state them for suitable solutions, see for instance
[38, 46, 2]. Assume that $U_{0}|_{B_{0}(1)}\in L^{q}(\mathbb{R}^{3})$, with
$q\in[3,\infty]$.555If the data is locally in the critical space $L^{3}$, one
requires in addition smallness of the data.
The compactness method proceeds in two steps. One first decomposes the
solution $U$ to the Navier-Stokes equations into a mild solution $a$
originating from the critical or subcritical data $U_{0}|_{B_{0}(1)}$666One
needs to properly extend this data as a compactly supported divergence-free
function. and a perturbation $V$ solving a perturbed Navier-Stokes system,
with critical or subcritical drift terms and initial data locally zero in
$B_{0}(1)$. The perturbation has a small energy locally in
$B_{0}(\frac{1}{2})$ near the initial time and can be extended by zero
backward in time. Hence the regularity of $V$ falls into the realm of epsilon-
regularity results, which is the second step of this method.
The method for establishing the epsilon-regularity for the perturbed equation
is inspired from the compactness method [52] that Lin used to prove epsilon-
regularity for the Navier-Stokes equations. Since the equation for the
perturbation $V$ is a Navier-Stokes equation with drift terms, one needs to
discriminate between subcritical and critical drifts:
* •
For subcritical drifts ($q\in(3,\infty]$), one has improved regularity for the
limit equation in the compactness argument, hence one can directly prove local
space-time Hölder regularity near initial time of the perturbation $V$. This
was done in [35].
* •
For critical drifts ($q=3$), there is no improved Hölder regularity at the
limit in general. One therefore aims at first proving a subcritical Morrey
bound for the perturbation that just misses boundedness; see [38]. Then one
can combine this subcritical information for $V$ with subcritical information
for the mild solution $a$ away from initial time to apply standard epsilon-
regularity results for the Navier-Stokes equations to give boundedness of $V$
up to the initial time; see for instance [38, Section 5.3]. It is also
possible using information from the initial data to bootstrap the regularity
of the perturbation to be Hölder continuous near the initial time; see for
instance [6, Section 3].
In the half-space with no-slip boundary condition, we were able to use the
compactness method to prove local-in-space smoothing for subcritical and
critical data in the Lebesgue scale; see [2]. This work relies on the new
estimates for the harmonic pressure obtained in [54]. The compactness scheme
is convenient, because in the critical case, the smallness of the drift can be
incorporated in the scheme, which avoids proving regularity for the limiting
Stokes equation with drift terms, as is done for the whole-space in [35, Lemma
2.2] in the subcritical case or [38, Section 4] in the critical case.
The bottom line is that the compactness method is flexible to handle both the
global and the local settings and the regularity away or near boundaries.
### 3.2. Caffarelli-Kohn-Nirenberg type methods
The general scheme of the method is exactly as the one previous of the
previous method: decomposition $U=a+V$ and smallness of the local energy of
the perturbation $V$ (first step) and epsilon regularity for the perturbed
Navier-Stokes equation with drift terms (second step). The difference is in
the way the epsilon regularity is proved. In the paper [6, Section 2], we
prove an epsilon regularity result for the perturbed Navier-Stokes equations
with critical drift terms by using an iteration scheme _à la_ Caffarelli, Kohn
and Nirenberg [14]. The criticality of the drift $a$ associated to $L^{3}$
data causes difficulties, as was the case for the compactness method, and
therefore only enables to propagate a subcritical Morrey bound for the
perturbation $V$.
The main advantage of this method lies in the fact that given the current
state of the art, it is the only method that manages to handle data locally
small in the borderline endpoint Besov space $B^{-1+\frac{3}{q}}_{q,\infty}$
for $q\in(3,\infty)$; see [6, Appendix C]. The results can also be localized,
see [6, Theorem 3].
### 3.3. Scaled local energy methods
This approach was started by Kang, Miura and Tsai [39]. Roughly speaking, it
relies on working with the scale-invariant energy and pressure
$E_{r}(t):=\frac{1}{r}\int\limits_{B_{0}(r)}|U(\cdot,t)|^{2}+\frac{1}{r}\int\limits_{0}^{t}\int\limits_{B_{0}(r)}|\nabla
U|^{2}+\frac{1}{r^{2}}\int\limits_{0}^{t}\int\limits_{B_{0}(r)}|p|^{\frac{3}{2}},$
and to propagate the smallness at initial time, i.e. of
$\sup_{r\in(R,1)}E_{r}(0)$ where $R$ is a given non negative scale, forward-
in-time via local energy estimates and a nonlinear Gronwall-type inequality
for
$\mathscr{E}_{R,\hat{R}}(t)=\sup_{r\in[R,\hat{R}]}\sup_{s\in(0,t)}E_{r}(s),$
for a well-chosen parameter $\hat{R}$.
There are two main advantages of this method. The first advantage is a
technical one. The first step of the previous two methods, which consists in
splitting the solution into $U=a+V$ where $a$ is the mild solution and $V$ the
perturbation is not needed here any longer. One directly works with the
solution. The second advantage is that it enables to directly prove local-in-
space smoothing for small scaled local kinetic energy [39, Theorem 1.1].
All the local-in-space smoothing results in the local setting mentioned so far
are for suitable solutions. Hence the pressure is a priori assumed to be in
the Lebesgue space $L^{\frac{3}{2}}$. Building upon the method of Kang, Miura
and Tsai [39], Kwon [46, Theorem 1.6] was able to extend the local-in-space
smoothing result for data locally in $L^{3}$ to the class of dissipative
solutions, for which the pressure is barely a distribution. Moreover in [46,
Theorem 1.6] the time interval for which the local-in-space smoothing occurs
is independent of the pressure.
## 4\. Strong concentration
As mentioned above, see the preamble of Section 2, strong concentration is the
accumulation of norms near potential blow-up points, on balls centered at a
singularity, whereas ‘weak concentration’ shows accumulation near blow-up
times and is not well localized in space.
Given the current state of the art, strong concentration can be proved in two
cases:
* •
either under some symmetry assumption on the solution, such as radial symmetry
for solutions of the nonlinear Schrödinger equation, see for instance [59, 85,
33],
* •
or under a Type I assumption, see for instance [62, Lemma 3.1] for the
semilinear heat equation.
In principle, strong concentration results for the Navier-Stokes equations are
simply deduced from local-in-space smoothing results by a contradiction
argument, on condition that one has a good workable notion of Type I.777The
case of Type I a priori control contains many still unresolved blow-up
Ansätze, such as Backward Discretely Self-Similar solutions (see [18]). We
assume that the solution satisfies the following generalized Type I bound: for
fixed $M,\ T^{*}\in(0,\infty)$ and a fixed radius $r_{0}\in(0,\infty]$,
$\displaystyle\begin{split}&\sup_{\bar{x}\in\mathbb{R}^{3}}\sup_{r\in(0,r_{0})}\sup_{T^{*}-r^{2}<t<T^{*}}r^{-\frac{1}{2}}\Bigg{(}\int\limits_{B_{\bar{x}}(r)}|U(x,t)|^{2}dx\Bigg{)}^{\frac{1}{2}}\leq
M.\end{split}$ (4.1)
In order to make sense of the condition (LABEL:ec6.gentypeI) in the case when
$r_{0}>\sqrt{T^{*}}$, we may extend $U$ by zero in negative times.888Notice
that it is not immediatly clear that the generalized notion of Type I
(LABEL:ec6.gentypeI) is implied by more classical notions of Type I, such as
the ODE blow-up Type I $\sqrt{T^{*}-t}|U(x,t)|\leq M^{\prime}$ (4.2) It turns
out to be true, see [78] and the review article [77, pages 844-849]. This
makes (LABEL:ec6.gentypeI) a good notion. In the half-space with no-slip
boundary conditions such an implication remains true, but is much harder to
prove due to the strong nonlocality of the pressure. It is proved in [7]. That
condition ensures that blow-up profiles belong to the class of local energy
solutions in which local-in-space smoothing results are proved, see Section 3.
In [6], strong concentration of the critical $L^{3}$, $L^{3,\infty}$ and Besov
norms $B^{-1+\frac{3}{q}}_{q,\infty}$ ($q\in(3,\infty)$) are proven. Figure 1
explains the idea to deduce strong concentration of the $L^{3}$ norm from
local-in-space smoothing for critical $L^{3}$ data. For a solution $U$
satisfying the Type I bound (LABEL:ec6.gentypeI), first blowing-up at $T^{*}$
and such that the space-time point $(0,T^{*})$ is a singularity, we have
$\|U(\cdot,t)\|_{L^{3}\big{(}B_{0}\big{(}\sqrt{\frac{T^{*}-t}{S(M)/2}}\big{)}\big{)}}>\gamma_{univ},$
(4.3)
for all $t\in(t_{*}(T^{*},M,r_{0}),T^{*})$ and $S(M)$ a time appearing in the
local-in-space smoothing estimate (3.1).
Figure 1. Quantitative local-in-space short-time smoothing and concentration
near potential Type I singularities
In [39, Theorem 1.6 (ii)] Kang, Miura and Tsai prove the strong concentration
of the scaled energy on concentrating balls. Their result holds under the
generalized Type I condition (LABEL:ec6.gentypeI). There exists
$\gamma_{univ}\in(0,\infty)$ and $S(M)\in(0,\infty)$ such that for all
$t\in(t_{*}(T^{*},M,r_{0}),T^{*})$,
$\frac{1}{\sqrt{T^{*}-t}}\int\limits_{B_{0}\big{(}\sqrt{\frac{T^{*}-t}{S(M)}}\big{)}}|U(x,t)|^{2}\,dx\gtrsim
1.$ (4.4)
Notice that under the stronger ODE blow-up Type I condition (4.2), this result
simply follows from (4.3) and interpolation.
Figure 6 on page 6 summarizes some results about weak and strong concentration
for the Navier-Stokes equations, as well as local-in-space smoothing.
The bottom line is that strong concentration of certain scale-invariant
quantities is at the heart of the quantitative regularity, in particular for
proving quantitative estimates of dissipative scales; see Section 5.3, Section
6 and Section 7.2 below. From now on the paper is concerned with such
questions relating to quantitative regularity.
## 5\. Three facets of quantitative regularity
In our view, there are three main objectives: (i) quantitative blow-up rates,
see Objective A below, (ii) quantitative regularity estimates, see Objective B
below, (iii) quantitative estimates of dissipative scales, see Objective C
below. The last objective is in some sense more fundamental, since the two
others will in general follow from it. We formulate the general objectives in
a somewhat loose way. In the rest of the paper, we will explain how the
examples fit into that abstract framework. In particular in this section we
illustrate the objectives in two cases: the case of a priori control in
$L^{5}(\mathbb{R}^{3}\times(-1,0))$ which is a critical non-borderline
space,999In this case the qualitative regularity is the classical
Ladyženskaja-Prodi-Serrin criteria, see for instance [79, 81]. and the case of
a priori control in $L^{\infty}(-1,0;L^{3}(\mathbb{R}^{3}))$101010In this case
the qualitative regularity is the result of Escauriaza, Seregin and Šverák
[29]. which is a critical borderline space.
### 5.1. Quantitative blow-up rates
It is known since the seminal work of Leray [50] that subcritical norms blow-
up with an algebraic rate near a potential blow-up time $T^{*}$, i.e.
$\|U(\cdot,t)\|_{L^{q}(\mathbb{R}^{3})}\gtrsim(T^{*}-t)^{-\frac{3}{2}(\frac{1}{3}-\frac{1}{q})}\quad\mbox{for}\
q\in(3,\infty)\ \mbox{and}\ t\in[0,T^{*}).$ (5.1)
###### Objective A.
Show that for certain critical spaces $L^{p}_{t}\mathcal{A}_{x}$ and
$\mathcal{B}$,111111Here $\mathcal{A},\,\mathcal{B}$ are certain Banach spaces
contained in $\mathcal{S}^{\prime}(\mathbb{R}^{3})$ and $p\in[1,\infty]$.
Notice that $\mathcal{B}$ is scaling-invariant, as well as
$L^{p}_{t}\mathcal{A}_{x}$. there exists an explicit positive function
$\mathscr{F}$ or $\mathscr{F}_{p,\mathcal{A}}$ on $[0,\infty)$ such that
$\mathscr{F}(s),\ \mathscr{F}_{p,\mathcal{A}}(s)\stackrel{{\scriptstyle
s\rightarrow 0}}{{\longrightarrow}}\infty$ and for a smooth solution with
enough decay121212Under this assumption the solution is smooth on $(0,T)$ for
any $T<T^{*}$, hence is a classical solution; see [9, Section 1.4] for a
definition. $U$ to the Navier-Stokes equations on
$\mathbb{R}^{3}\times(0,T^{*})$ blowing-up at time $T^{*}$,
$\frac{\|U\|_{L^{p}(0,t;\mathcal{A})}}{\mathscr{F}(T^{*}-t)}\gtrsim
1\quad\mbox{when}\ t\ \mbox{is sufficiently close to}\ T^{*}.$ (5.2)
or
$\frac{\|U(\cdot,t)\|_{\mathcal{B}}}{\mathscr{F}_{p,\mathcal{A}}(T^{*}-t)}\gtrsim
1\quad\mbox{when}\ t\ \mbox{is sufficiently close to}\ T^{*}.$ (5.3)
#### The toy model $L^{5}_{t,x}$
Using the same reasoning as (5.5), the quantitative estimate (5.6) combined
with Leray’s blow-up rate (5.1) imply that
$\|U\|_{L^{5}(0,t;L^{5}(\mathbb{R}^{3}))}\gtrsim\sqrt{T^{*}}(\log(T^{*}-t))^{\frac{1}{5}},\quad\mbox{for
all}\quad t\in(\tfrac{T^{*}}{2},T^{*}).$
Not surprisingly, this estimate is compatible with (and actually a consequence
of) the Leray blow-up rate (5.1) in the case $q=5$.
#### The case of $L^{\infty}_{t}L^{3}_{x}$
The critical borderline case $q=3$ was open until 2019 and a remarkable paper
of Tao [84]. There Tao proves the rate (1.1). Notice that this rate follows
from the lower bound (5.2) proved in [84] with $p:=\infty$,
$\mathcal{A}:=L^{3}(\mathbb{R}^{3})$ and $\mathscr{F}(s):=(\log\log\log
s)^{c}$, for a universal constant $c\in(0,\infty)$.
### 5.2. Quantitative regularity estimates
###### Objective B.
Show that for certain critical spaces $L^{p}(-1,0;\mathcal{A})$ and
$\mathcal{B}$,131313Here $\mathcal{A},\,\mathcal{B}$ are certain Banach spaces
contained in $\mathcal{S}^{\prime}(\mathbb{R}^{3})$ and $p\in[1,\infty]$.
Notice that $\mathcal{B}$ is scaling-invariant, as well as
$L^{p}(-1,0;\mathcal{A})$. there exists an explicit positive function
$\mathscr{G}$ on $[0,\infty)$,141414It is hard to give a general statement
covering all results in this line of research. Notice that Theorem B in [9] is
a quantitative regularity result without a priori control of a global scale-
invariant quantity. Such a result takes the following form
$\|U\|_{L^{\infty}(\mathbb{R}^{3}\times(-t_{*},0))}\leq\mathscr{G}\big{(}\sup_{t_{j}\nearrow
0}\|U(\cdot,t_{j})\|_{L^{3}(\mathbb{R}^{3})}\big{)},$ where
$t_{*}=t_{*}(\sup_{t_{j}\nearrow
0}\|U(\cdot,t_{j})\|_{L^{3}(\mathbb{R}^{3})})$. That result quantifies
Seregin’s 2012 liminf qualitative criteria [73], and is hence a result that
goes beyond the critical case. For more details about [9, Theorem B], we also
refer to [71, Theorem 6.3 and Section 6.3.4]. Furthermore, in the same vein,
we get regularity at time $t=0$ for large $L^{3}$ data at $t=-1$ if the
profile at time $t=0$ is quantitatively small; see [1, Theorem 4.1 (i)] for a
qualitative statement and [9, Proposition 4.4] for a quantitative statement.
such that for a critically bounded smooth solution $U$ (with enough decay) of
the Navier-Stokes equations on $\mathbb{R}^{3}\times(-1,0)$,
$\|U\|_{L^{\infty}(\mathbb{R}^{3}\times(-\frac{1}{2},0))}\leq\mathscr{G}(\|U\|_{L^{p}(-1,0;\mathcal{A})},\|U(\cdot,0)\|_{\mathcal{B}}).$
(5.4)
We previously remarked that certain qualitative regularity results in terms of
critical norms can be quantified abstractly, see [9, Introduction]. This is
the case of the Escauriaza, Seregin and Šverák [29] result that can be
abstractly quantified via the use of the persistence of singularities [72].
The focus of this survey paper is to derive an explicit formula for
$\mathscr{G}$.
Let us also note that a general quantitative result such as (5.4) can be
combined with the Leray blow-up rate (5.1) to yield quantitative blow-up rates
of the form (5.2). Indeed, by scaling151515We recall that the Navier-Stokes
equations are invariant under the scaling $U_{\lambda}=\lambda
U(\lambda\cdot,\lambda^{2}\cdot)$, for $\lambda>0$. Here we take
$\lambda=\sqrt{t}$.
$\displaystyle\begin{split}\frac{1}{\sqrt{t}}\mathscr{G}(\|U\|_{L^{\infty}(0,t;L^{3}(\mathbb{R}^{3}))})&\
=\frac{1}{\sqrt{t}}\mathscr{G}(\|U_{\sqrt{t}}\|_{L^{\infty}(0,1;L^{3}(\mathbb{R}^{3}))})\\\
&\
\geq\frac{1}{\sqrt{t}}\|U_{\sqrt{t}}\|_{L^{\infty}(\mathbb{R}^{3}\times(\frac{1}{2},1))}\\\
&\ =\|U\|_{L^{\infty}(\mathbb{R}^{3}\times(\frac{t}{2},t))}\\\ &\ \geq
C(T^{*}-t)^{-\frac{1}{2}}.\end{split}$ (5.5)
#### The toy model $L^{5}_{t,x}$
Using the strategy described in Section 6 below,161616A strategy based on
energy estimates on the level of the vorticity equation is described in [9,
Section 1.2.2]. It also yields a single exponential bound in
$L^{5}(\mathbb{R}^{3}\times(-1,0))$, but has an additional dependence in terms
of the initial data for the vorticity $\omega(\cdot,-1)$. Notice that $L^{5}$
energy estimates can be applied directly, see [63]. one can prove the
following quantitative estimate:
$\|U\|_{L^{\infty}(\mathbb{R}^{3}\times(-\frac{1}{2},0))}\lesssim\exp\big{(}\|U\|_{L^{5}(\mathbb{R}^{3}\times(-1,0))}^{5}\big{)},$
(5.6)
where $C\in(0,\infty)$ is a universal constant. Here
$p=5,\quad\mathcal{A}=L^{5}(\mathbb{R}^{3})\quad\mbox{and}\quad\mathscr{G}(A,B)\simeq\exp(A^{5})$
in the notation of the general form estimate (5.4).
#### The case of $L^{\infty}_{t}L^{3}_{x}$
Tao shows that for classical solutions to the Navier-Stokes equations on
belonging to the critical space $L^{\infty}(-1,0;L^{3}(\mathbb{R}^{3}))$,
$\|U\|_{L^{\infty}(\mathbb{R}^{3}\times(-\frac{1}{2},0))}\lesssim\exp\exp\exp\big{(}\|U\|_{L^{\infty}(-1,0;L^{3}(\mathbb{R}^{3}))}^{c}\big{)},$
(5.7)
where $c\in(0,\infty)$ is a universal constant. Here
$p:=\infty,\quad\mathcal{A}:=L^{3}(\mathbb{R}^{3})\quad\mbox{and}\quad\mathscr{G}(A,B)\simeq\exp(\exp(\exp(A^{c})))$
in the notation of the general form estimate (5.4).
### 5.3. Quantitative estimates of dissipative/Kolmogorov scales
###### Objective C.
Show that for certain critical spaces $L^{p}(-1,0;\mathcal{A})$ and
$\mathcal{B}$,171717As in footnote 13, $\mathcal{A},\,\mathcal{B}$ are certain
Banach spaces contained in $\mathcal{S}^{\prime}(\mathbb{R}^{3})$ and
$p\in[1,\infty]$. there exists an explicit positive function $\mathscr{H}$ on
$[0,\infty)$ such that for a critically bounded smooth solution with enough
decay $U$ to the Navier-Stokes equations on $\mathbb{R}^{3}\times(-1,0)$,
dissipative effects take over the nonlinearity181818This is on a formal level.
On a practical level, this is where well-posedness theory or epsilon
regularity takes over and gives regularity. for physical scales
$\lambda\leq\mathscr{H}(\|U\|_{L^{\infty}(-1,0;\mathcal{A})},\|U(\cdot,0)\|_{\mathcal{B}})$
or Fourier scales
$N\geq\mathscr{H}(\|U\|_{L^{\infty}(-1,0;\mathcal{A})},\|U(\cdot,0)\|_{\mathcal{B}}).$
The threshold between the scales where diffusive effects dominate vs. where
nonlinear effects dominate is measured by the smallness vs. concentration of
certain scale-critical quantities $\mathscr{S}$. Once this threshold is
estimated in a quantitative way, regularity criteria (epsilon-regularity,
local smoothing…) imply the quantitative bounds stated in Objective A. The
estimate of dissipative scales is hence the heart of the matter.
Figure 2 summarizes the general strategy in the three main cases described in
this survey paper.
the global scale-critical standing assumptions
---
$U\in L^{5}_{t,x}$ | $U\in L^{\infty}_{t}L^{3,\infty}_{x}$ and $U(\cdot,0)\in L^{3}$ | $U\in L^{\infty}_{t}L^{3}_{x}$
see Section 5.3 | see Section 7.2 | see Section 5.3
prevent the following scale-critical quantities
${\displaystyle(-t)^{\frac{1}{5}}\|U(\cdot,t)\|_{L^{5}}}$ | ${\displaystyle\sqrt{-t}\int\limits_{B_{O\big{(}\|U\|_{L^{\infty}_{t}L^{3,\infty}_{x}}\sqrt{-t}\big{)}}}|\omega(\cdot,t)|^{2}}$ | ${\displaystyle N^{-1}|P_{N}U(t,x)|}$
defined by (5.9) | defined by (7.5) | defined by (5.11)
to concentrate to close to final time | to concentrate for large frequencies
i.e. for times $t_{*}<t<0$ | $N\geq N_{*}$
no concentration, i.e. smallness, implies regularity
Figure 2. Quantitative regularity via concentration: a summary
#### The toy model $L^{5}_{t,x}$
For solutions of the Navier-Stokes equations critically bounded in
$L^{5}(\mathbb{R}^{3}\times(-1,0))$, we work with the following scale-
invariant Weissler-Kato type norms
$(-t)^{\frac{1}{5}}\|U(\cdot,t)\|_{L^{5}(\mathbb{R}^{3})},\quad\mbox{for}\quad
t\in(-1,0).$ (5.8)
If the quantity defined by (5.8) is small for all sufficiently small times
$0>t>t_{*}(A)$, then Caffarelli, Kohn and Nirenberg type epsilon-regularity
results imply the quantitative bound (5.6).191919Notice that well-posedness
theory could also be used in replacement of epsilon-regularity.
Objective C in the $L^{5}_{t,x}$ case:
If the following statement202020The $\varepsilon$ in (5.9) comes from epsilon-
regularity criteria.
$\displaystyle\mathscr{S}(t):=(-t)^{\frac{1}{5}}\|U(\cdot,t)\|_{L^{5}(\mathbb{R}^{3})}<\varepsilon$
(5.9)
fails for a certain $t\in(-1,0)$, find a quantitative upper bound
$t_{*}\in(-1,0)$ for $t$.
In can be shown, applying the general strategy outlined in Section 6, that
$-t>\frac{1}{2}\exp\big{(}-\tfrac{32\|U\|_{L^{5}(\mathbb{R}^{3}\times(-1,0))}^{5}}{\varepsilon^{5}}\big{)}=:-t_{*}(A).$
(5.10)
Hence, $\mathscr{H}(A,B):=-t_{*}(A)=\exp(-\frac{32A^{5}}{\varepsilon^{5}})$
with the notation of Objective C.
#### The case of $L^{\infty}_{t}L^{3}_{x}$
Tao works with globally defined quantities due to a Fourier based approach.
His analysis is based on the following scale-invariant quantities
$\mathscr{S}(N;x,t):=N^{-1}|P_{N}U(x,t)|,\quad\mbox{for}\quad(x,t)\in\mathbb{R}^{3}\times(-1,0),$
(5.11)
where $P_{N}$ is the Littlewood-Paley projection on the frequency
$N\in(0,\infty)$. Indeed, if the quantity $\mathscr{S}(N;x,t)$ defined by
(5.11) is small in terms of
$A:=\|U\|_{L^{\infty}_{t}L^{3}_{x}(\mathbb{R}^{3}\times(-1,0))}$, uniformly in
$(x,t)\in\mathbb{R}^{3}\times(-\frac{1}{2},0)$ and for high frequencies $N\geq
N_{*}(A)$, then
$\|U\|_{L^{\infty}_{x,t}(\mathbb{R}^{3}\times(-\frac{1}{8},0))}$ can be
estimated explicitly in terms of $A$ and $N_{*}$. Related observations were
made previously by Cheskidov and Shvydkoy [25, 26] and Cheskidov and Dai [24],
but without the bounds explicitly stated. There the frequency $N_{*}$ is
called the Kolmogorov scale and denoted $\Lambda$. If
$N^{-1}\|P_{N}U\|_{L^{\infty}_{x,t}}\ll 1$ is small, then
$N\|P_{N}U\|^{2}_{L^{\infty}_{t,x}}\ll N^{2}\|P_{N}U\|_{L^{\infty}_{t,x}}$ so
that the diffusion dominates the nonlinearity, heuristically at least, since
some of the frequencies in the paraproduct are neglected, see [83].
From this perspective, Tao’s aim is the following.
Tao’s Objective C:
Assume
$A:=\|U\|_{L^{\infty}_{t}L^{3}_{x}(\mathbb{R}^{3}\times(-1,0))}<\infty.$
If the following statement
$N^{-1}\|P_{N}U\|_{L^{\infty}_{x,t}(\mathbb{R}^{3}\times(-\frac{1}{2},0))}<\varepsilon(A)\,\,\,\textrm{for}\,\,\textrm{all}\,\,\,N\geq
N_{*},$ (5.12)
fails for $\varepsilon(A)=A^{-c}$ and a certain frequency $N$, find a
quantitative upper bound $N_{*}(A)$ for $N$.
In Tao’s paper [84, Theorem 5.1], it is shown that
$\mathscr{H}(A,B):=N_{*}(A)\simeq\exp\exp\exp(A^{c})$ (5.13)
with the notations of Objective C.
## 6\. A general strategy for quantitative estimates of dissipative scales
In this part we describe a general strategy to fulfill Objective C, i.e. to
estimate quantitatively the dissipative scales. The scheme is based on the
study of the concentration of certain scale-invariant quantities
$\mathscr{S}$, such as the Weissler-Kato norm (5.9) in the case of
$L^{5}_{t,x}$ or the frequency bubbles (5.11) in the case of
$L^{\infty}_{t}L^{3}_{x}$. One can summarize the idea as follows:
1. (Step-1)
Propagation of concentration
If a certain scale-invariant quantity $\mathscr{S}$ concentrates at a given
scale, then it will concentrate at many disjoint scales. One relies on various
tools such as bilinear estimates or local-in-space smoothing when applying
quantitative Carleman inequalities. For this, one needs to work in space-time
regions where one has good quantitative regularity properties: epochs and
annuli of quantitative regularity.
2. (Step-2)
Summation of scales and coercivity of the standing assumption
The standing critical assumption
$\|U\|_{L^{\infty}(-1,0;\mathcal{A})}<\infty\quad\mbox{or}\quad\|U(\cdot,0)\|_{\mathcal{B}}<\infty$
implies an upper bound on the number of disjoint scales where the scale-
invariant quantity $\mathscr{S}$ concentrates, which directly yields a
quantitative estimate of the estimate of dissipative scales $\mathcal{H}$ in
Objective C.
### 6.1. The toy model $L^{5}_{t,x}$
Let us now sketch the proof in the toy model case. Assume that
$A:=\|U\|_{L^{5}(\mathbb{R}^{3}\times(-1,0))}<\infty$
and that (5.9) fails for a certain $t$. In order to estimate the dissipative
scales quantitatively, see Objective C above, we argue in the following two-
step way:
1. (Step-1)
Backward propagation of Weissler-Kato norm concentration
Using bilinear estimates for the Oseen tensor [16] we get
$\|U(\cdot,t)\|_{L^{5}(\mathbb{R}^{3})}>\frac{\varepsilon}{(-t)^{\frac{1}{5}}}\
\Rightarrow\
\|U(\cdot,t^{\prime})\|_{L^{5}(\mathbb{R}^{3})}>\frac{\varepsilon/2}{(-t^{\prime})^{\frac{1}{5}}},\quad\mbox{for
all}\quad t^{\prime}\in(-1,2t).$
2. (Step-2)
Summation of scales and coercivity of the standing assumption
Integrating the concentration for $t^{\prime}\in(-1,2t)$,
$\|U\|_{L^{5}(\mathbb{R}^{3}\times(-1,0))}^{5}\geq\int\limits_{-1}^{2t}\|U(\cdot,t^{\prime})\|_{L^{5}(\mathbb{R}^{3})}^{5}\,dt^{\prime}\geq-\frac{\varepsilon^{5}}{32}\log(-2t).$
Hence, we get the estimate (5.10), which is the upper bound for the times
where the Weissler-Kato norm concentrates.
Figure 3. Quantitative regularity via concentration of scale-critical
quantities: the toy model $L^{5}_{t,x}$
### 6.2. The case of $L^{\infty}_{t}L^{3}_{x}$
We now sketch the strategy of Tao to prove his main Objective C described on
page 5.12. A slightly different summary of the strategy is given in [9,
Section 1.1] and in [84, Section 1]. Assume that
$A:=\|U\|_{L^{\infty}(-1,0;L^{3}_{x}(\mathbb{R}^{3}))}<\infty$ (6.1)
and that (5.12) fails for $\varepsilon(A)=A^{-c}$ and a certain frequency $N$.
Hence there exists $(x_{0},t_{0})\in\mathbb{R}^{3}\times(-\frac{1}{2},0)$ such
that
$N^{-1}|P_{N}u(x_{0},t_{0})|>A^{-c}.$ (6.2)
We call this the ‘initial concentration’.
1. (Step-1)
Propagation of concentration
The idea is to transfer the initial concentration (6.2) in space and time in
order to get a lower bound on the $L^{3}(\mathbb{R}^{3})$ norm at time zero.
That propagation of concentration, or ‘frequency bubbling’, relies on:
1. (i)
_Backward frequency bubbling_
For all $n\in\mathbb{N}$, there exists a frequency $N_{n}\in(0,\infty)$,
$(x_{n},t_{n})\in\mathbb{R}^{3}\times(-1,t_{n-1})$ such that
$N_{n}^{-1}|P_{N_{n}}u(x_{n},t_{n})|>A^{-c}$ (6.3)
with
$x_{n}=x_{0}+O((-t_{n})^{\frac{1}{2}}),\quad
N_{n}\simeq|-t_{n}|^{-\frac{1}{2}}.$ (6.4)
2. (ii)
_Transfer of concentration in Fourier space to physical space_
In order to use quantitative Carleman inequalities, see the next point, one
needs to transfer the information on the concentration in Fourier space to
physical space quantities, namely a scale-invariant enstrophy. To do this it
seems important to have a priori control a global scale-invariant norm, such
as (6.1).
3. (iii)
_Large-scale and forward-in-time propagation of concentration_
Using quantitative versions of the Carleman inequalities in [29], see [84,
Proposition 4.2 and Proposition 4.3], Tao shows that the lower bounds on the
scale-invariant enstrophy can be transferred to a lower bound on the $L^{3}$
norm of $U$ at the final moment of time $0$. The applicability of the
quantitative Carleman inequalities to the vorticity equation requires the
‘epochs of regularity’ in the previous step and the existence of ‘good spatial
annuli’ where the solution enjoys good quantitative estimates. Specifically,
Tao shows that for certain admissible time scales $S$,212121These admissible
time scales are related to where one has backward concentration in frequency
space, see step ‘Backward frequency bubbling’ above and [84, Proof of Theorem
5.1]. one has the concentration of the $L^{3}$ norm on the annulus
$\\{S^{\frac{1}{2}}\leq|\cdot-x_{0}|\leq\exp(A^{c})S^{\frac{1}{2}}\\}$, i.e.
$\int\limits_{S^{\frac{1}{2}}\leq|x-x_{0}|\leq\exp(A^{c})S^{\frac{1}{2}}}|U(x,0)|^{3}dx\geq\exp(-\exp(A^{c})).$
(6.5)
2. (Step-2)
Summation of scales and coercivity of the standing assumption
Summing (6.5) over admissible time scales $S$ such that the annuli
$\\{S^{\frac{1}{2}}\leq|\cdot-x_{0}|\leq\exp(A^{c})S^{\frac{1}{2}}\\}$ are
disjoint, one eventually obtains
$A^{3}\geq\int\limits_{\mathbb{R}^{3}}|U(x,0)|^{3}dx\gtrsim\log(N)\exp(-\exp(A^{c})).$
This concludes the proof of the triple exponential upper bound (5.13) for
$N$.222222Notice that due to the stacking of scales, we always get at least
one exponential in the quantitative estimates.
Let us note that ideas in a similar spirit were used by Merle and Raphaël in
[58] to prove a quantitative rate of blow-up of a critical norm for radial
solutions of a supercritical nonlinear Schrödinger equation; see [58, Theorem
2]. In particular a lower bound on annuli analogous to (6.5) is obtained and
the final $\log$ blow-up rate is obtained by a similar stacking of scales
argument; see [58, Section 4.2].
## 7\. Quantitative regularity in the Type I case
We focus here on the quantitative regularity results in the Type I case proved
in [9]. Let us emphasize that the regularity in the Type I case is proved only
for axisymmetric flows232323In the half-space, this appears to still be an
open problem even without swirl. [22, 21, 45]. We now show how the results in
[9] fulfill the three objectives stated in Section 5.
### 7.1. Three facets of the quantitative regularity in the Type I case
#### Answer to Objective A
###### Result A ([9, Theorem A], localized quantitative rate of blow-up).
Assume that $U$ is a smooth solution with sufficient decay to the Navier-
Stokes equations and that $T^{*}$ is a first blow-up time.
Assume in addition that $(0,T^{*})$ is a Type I singular point i.e.
$A:=\|U\|_{L^{\infty}(0,T^{*};L^{3,\infty}(\mathbb{R}^{3}))}<\infty.$
Then the above assumptions imply that there exists $S(A)\simeq A^{-30}$ such
that for any $t\in(\frac{T^{*}}{2},T^{*})$ and
$R\in\Big{(}\sqrt{\tfrac{T^{*}-t}{S(A)}},e^{A^{1022}}\sqrt{T^{*}}\Big{)}$
(7.1)
we have
$\int\limits_{|x|<R}|U(x,t)|^{3}dx\geq\frac{\log\big{(}\frac{R^{2}}{A^{802}(T^{*}-t)}\big{)}}{\exp(\exp(A^{1025}))}.$
(7.2)
Estimate (7.2) is written in a scale-invariant form. Notice that this estimate
implies an estimate of the form (5.2). Indeed taking
$R=(T^{*}-t)^{\frac{1}{2}-\delta}$ for $\delta>0$ and small, we have (5.2)
with $\mathcal{B}:=L^{3}(B_{0}((T^{*}-t)^{\frac{1}{2}-\delta})))$,
$p:=\infty$, $\mathcal{A}:=L^{3,\infty}(\mathbb{R}^{3})$,
$\mathscr{F}_{p,\mathcal{A}}\simeq-\log\big{(}A^{802}(T^{*}-t)^{2\delta}\big{)}$
and $|T^{*}-t|\lesssim_{A}1$.
Notice that contrary to (1.1), it is stated in (7.2) that the norm blows-up
not only along a subsequence but pointwise for $t\rightarrow T^{*}$. This is
due to the fact that the quantitative regularity under the a priori Type I
control, see Result B, just involves the $L^{3}$ norm at the final time and
not the whole $L^{\infty}_{t}L^{3}_{x}$ norm of the solution as the estimate
(5.7) of [84].
#### Answer to Objective B
###### Result B ([9, Proposition 2.1, estimate (52)], localized quantitative
regularity).
Assume that $U$ is a smooth solution with sufficient decay to the Navier-
Stokes equations.
Assume in addition that $U$ satisfies the Type I bound
$A:=\|U\|_{L^{\infty}(-1,0;L^{3,\infty}(\mathbb{R}^{3}))}<\infty.$
Then, letting
$\displaystyle\begin{split}t_{*}=\ &t_{*}(A,U(\cdot,0))\\\ :=\
&-A^{-c}\exp\bigg{(}-\exp\exp(A^{1024})\int\limits_{B_{0}(\exp(A^{1023}))}|U(\cdot,0)|^{3}\bigg{)},\end{split}$
the following quantitative boundedness holds
$\|U\|_{L^{\infty}(B_{0}(A^{c}\sqrt{-t_{*}})\times(t_{*}/2,0))}\lesssim\frac{A^{-c}}{\sqrt{-t_{*}}},$
(7.3)
with $c\in(0,\infty)$ a universal constant.
Notice that (7.3) involves the a priori Type I control in
$L^{\infty}_{t}L^{3,\infty}_{x}$ and the control of the $L^{3}$ norm at the
final time. Given the current state of the art, such assumptions are needed to
prove the regularity; a mere Type I assumption is at this point not enough to
beat the scaling except in the axisymmetric case and the self-similar
case.242424This is due to an additional scalar structure in those cases. For
more insights on the qualitative regularity in borderline endpoint critical
spaces such as the Lorentz space $L^{3,\infty}$ or the Besov space
$B^{-1+\frac{3}{p}}_{p,\infty}$, see [1].252525These spaces are sometimes also
referred to as ‘ultracritical spaces’. Unlike $L^{3}$, a function in these
spaces can have a simultaneous presence in terms of its norm at an arbitrary
amount of disjoint scales/frequencies, see [4, Footnote 7]
Estimate (7.3) also implies a localized quantitative bound in the spirit of
(5.4) with $\mathcal{B}:=L^{3}(B_{0}(\exp(A^{1023})))$, $p:=\infty$,
$\mathcal{A}:=L^{3,\infty}(\mathbb{R}^{3})$,
$\mathscr{G}(A,B)\simeq A^{c}\exp\big{(}\exp(\exp(A))B),\ \mbox{where}\
A:=\|U\|_{L^{\infty}(-1,0;L^{3,\infty}(\mathbb{R}^{3}))}\\\ \mbox{and}\
B:=\int\limits_{B_{0}(\exp(A^{1023}))}|U(\cdot,0)|^{3}.$
#### Answer to Objective C
###### Result C ([9, Proposition 2.1, estimate (50)], quantitative estimates
of dissipative/Kolmogorov scales).
Assume that $U$ is a smooth solution with sufficient decay to the Navier-
Stokes equations.
Assume in addition that $U$ satisfies the Type I bound
$A:=\|U\|_{L^{\infty}(-1,0;L^{3,\infty}(\mathbb{R}^{3}))}<\infty.$
Then the threshold determining where the dissipative effects dominate the
nonlinear effects is estimated as follows:
$\displaystyle\begin{split}t_{*}=\ &t_{*}(A,U(\cdot,0))\\\ :=\
&-A^{-c}\exp\bigg{(}-\exp\exp(A^{1024})\int\limits_{B_{0}(\exp(A^{1023}))}|U(\cdot,0)|^{3}\bigg{)}.\end{split}$
(7.4)
It is not surprising that the definition of $t_{*}$ already appears in Result
B, since Result B is (almost) an immediate consequence of Result C and a
regularity criteria (namely a local-in-space short-time regularity result).
More precisely, $-t_{*}$ defined by (7.4) is the time after which the
following scale-invariant enstrophy262626Here, as is usual,
$\omega=\nabla\times U$.
$\mathscr{S}(A;t):=(-t)^{\frac{1}{2}}\int\limits_{B_{0}\big{(}A^{c}(-t)^{\frac{1}{2}}\big{)}}|\omega(x,t)|^{2}dx$
(7.5)
is smaller than a given small number $\varepsilon(A)=A^{-c}$.
Estimate (7.4) also implies a quantitative bound like (C) with
$\mathcal{B}:=L^{3}(B_{0}(\exp(A^{1023})))$, $p:=\infty$,
$\mathcal{A}:=L^{3,\infty}(\mathbb{R}^{3})$,
$\mathscr{H}(A,B)\simeq A^{-c}\exp\big{(}-\exp\exp(A^{1024})B\big{)},\
\mbox{where}\ A:=\|U\|_{L^{\infty}(-1,0;L^{3,\infty}(\mathbb{R}^{3}))}\\\
\mbox{and}\ B:=\int\limits_{B_{0}(\exp(A^{1023}))}|U(\cdot,0)|^{3}.$
### 7.2. Quantitative estimates of dissipative scales in the Type I case
The general scheme to get the estimate (7.4) is a physical space parallel to
Tao’s Fourier based strategy described in Section 6.2. Assume that
$\mathscr{S}(A;t)$ defined by (7.5) concentrates, i.e. is not small, for a
certain time $t$. Then, the goal is to find an upper bound $t_{*}$ for $t$.
More precisely, assume that
$\mathscr{S}(A;t):=(-t)^{\frac{1}{2}}\int\limits_{B_{0}\big{(}A^{c}(-t)^{\frac{1}{2}}\big{)}}|\omega(x,t)|^{2}dx>A^{-c},$
(7.6)
for a $t\in(-1,0)$ is not too close to $-1$. We call this the ‘initial
concentration’.
Figure 4. Quantitative regularity in the Type I case via concentration of a
scale-invariant enstrophy: (Step-1) backward, large-scale and forward
propagation of enstrophy concentration Figure 5. Quantitative regularity in
the Type I case via concentration of a scale-invariant enstrophy: (Step-2)
summation of scales
1. (Step-1)
Propagation of concentration
For this step we refer to Figure 5. The idea is to transfer the initial
concentration (7.6) in space and time in order to get a lower bound on the
localized $L^{3}$ norm at time zero. That propagation of concentration relies
on:
1. (i)
_Backward propagation of concentration_
For all $t^{\prime}\in(-1,t)$ such that $-t^{\prime}$ is not too close to
$-t$, we have
$(-t^{\prime})^{\frac{1}{2}}\int\limits_{B_{0}\big{(}A^{c}(-t^{\prime})^{\frac{1}{2}}\big{)}}|\omega(x,t^{\prime})|^{2}dx>A^{-c}.$
(7.7)
2. (ii)
_Large-scale and forward-in-time propagation of concentration_
It is shown that for certain admissible time scales $S$,272727These admissible
time scales are related to where one has backward concentration, see [9,
equation (106)-(107)]. one has the concentration of the $L^{3}$ norm on the
annulus $\\{S^{\frac{1}{2}}\leq|\cdot|\leq\exp(A^{c})S^{\frac{1}{2}}\\}$, i.e.
$\int\limits_{S^{\frac{1}{2}}\leq|\cdot|\leq\exp(A^{c})S^{\frac{1}{2}}}|U(x,0)|^{3}dx\geq\exp(-\exp(A^{c})).$
(7.8)
The role of the Type I bound is to show that the solution $U$ obeys good
quantitative estimates in certain space-time regions, epochs of quantitative
regularity and annuli of quantitative regularity, which is needed to apply the
Carleman inequalities to the vorticity equation, see [29], see [84,
Proposition 4.2 and Proposition 4.3].
2. (Step-2)
Summation of scales and coercivity of the standing assumption
We refer to Figure 5 for this step. Summing (7.8) over all permissible
disjoint annuli finally gives us the desired lower bound for $-t^{\prime}$. We
note that the localized $L^{3}$ norm of $U$ at time $0$ plays a distinct role
to that of the Type I condition described in the previous step. Its sole
purpose is to bound the number of permissible disjoint annuli that can be
summed. This concludes the proof of the single exponential upper bound in
${\displaystyle\int\limits_{B_{0}(\exp(A^{1023}))}|U(\cdot,0)|^{3}}$ for $t$,
see (7.4).
Notice that quantitative local-in-space smoothing, see Section 3 above, is a
fundamental tool to achieve the backward propagation of enstrophy
concentration in (Step-1) above as well as to go from Result C to Result B.
### 7.3. A comparison with Tao’s strategy
We outline here the two main differences between: on the one hand Tao’s
strategy [84] for the quantitative regularity in the case of a priori control
of the solution in the borderline critical space $L^{\infty}_{t}L^{3}_{x}$ and
on the other hand the strategy of [9] for the quantitative regularity in the
case of a priori control of the solution in the borderline endpoint critical
space $L^{\infty}_{t}L^{3,\infty}_{x}$ with additional control of the $L^{3}$
norm of $U(\cdot,0)$. There are two main aspects, see below. The first aspect
is the decisive difference that also explains the second point. Let us state
these two differences for the sake of clarity.
#### Fourier space vs. physical space
We already underlined this aspect above. Tao works with the frequency bubbles
of concentration (5.11), which are scale-invariant quantities defined in
Fourier space. Therefore these quantities involve the solution $U(\cdot,t)$ on
the whole-space $\mathbb{R}^{3}$. On the contrary, the analysis of [9] relies
on the localized scale-invariant enstrophies (7.5). Those quantities are
defined in physical space and involve the solution $U$ only on the ball
$B_{0}(A^{c}(-t)^{\frac{1}{2}})$. Therefore, one can obtain localized results,
such as the blow-up rate (7.2) of Result A.
#### Global scale-critical vs. criticality along a sequence of times/at a
given time
This aspect is a consequence of the point made above about Fourier vs.
physical space approach. In Tao’s work, there is a step, see ‘Transfer of
concentration in Fourier space to physical space’ in Section 6.2, that
consists in transferring the concentration of frequency bubbles to
concentration of a scale-invariant enstrophy, see [84, equation (5.6)]. That
localized $L^{2}$ norm of the vorticity is needed to work with quantitative
Carleman inequalities in order to propagate the concentration at large scales
and forward in time. The process of transfer is based on the fact that the
solution is bounded in $L^{\infty}(-1,0;L^{3}(\mathbb{R}^{3}))$. In the case
of the strategy described in Section 7.2 above in the Type I case, this step
is not needed.
This remark has several implications:
1. (1)
The scheme developed in the Type I case, see Section 7.2, enables to handle
the case when a global scale-critical standing assumption is lacking, which is
the case in [9, Theorem B] that quantifies Seregin’s 2012 criteria [73].
2. (2)
In a related vein, we get regularity at time $t=0$ for large $L^{3}$ data at
$t=-1$ if the profile at time $t=0$ is quantitatively small; see [1, Theorem
4.1 (i)] for a qualitative statement and [9, Proposition 4.4] for a
quantitative statement.
## 8\. Some further developments
There are many interesting research developments in the wake of the two papers
[84, 9]. Let us review some of them.
Very recently, the paper [5] was able to give a localized version of Tao’s
blow-up rate (1.1). For a smooth suitable weak solution $U$ to the Navier-
Stokes equations on $B(0,4)\times(0,T^{*})$, that possesses a singular point
$(x_{0},T^{*})\in B(0,4)\times\\{T^{*}\\}$, then for all $\delta>0$
sufficiently small
$\limsup_{t\uparrow
T^{*}}\frac{\|U(\cdot,t)\|_{L^{3}(B(x_{0},\delta))}}{\Big{(}\log\log\log\Big{(}\frac{1}{(T_{*}-t)^{\frac{1}{4}}}\Big{)}\Big{)}^{\frac{1}{1129}}}=\infty.$
In that sense, this bound is a true quantification of the qualitative
Escauriaza, Seregin and Šverák [29] criteria, which is also localized but
qualitative. The main difficulty with the localization is related to showing
that the local solution possesses quantitative annuli and epochs of
regularity, which is required in (Step-1) (Step-1)(iii).
The quantitative regularity for solutions $U\in L^{\infty}_{t}L^{d}_{x}$ to
the Navier-Stokes equations in higher dimensions $d\geq 4$ was handled by
Palasek in [69]. This work gives an effective quantification of the
qualitative result by Dong and Du [27]. For the blow-up rate, one pays the
price of an additional logarithm compared to the result in dimension three
(1.1).
Palasek [68] was also able to improve upon the triple logarithmic rate (1.1)
obtained by Tao in [84]: in the case of axisymmetric solutions for instance,
the triple logarithm is replaced by a double logarithm. Without any symmetry
assumption on the solution, a similar improvement can be obtained by replacing
the $L^{3}$ norm by the scale-invariant norm
$\|r^{1-\frac{3}{q}}U\|_{L^{\infty}_{t}L^{q}_{x}}$ for $q\in(3,\infty)$ and
$r:=|x_{h}|$.
Finally, let us mention a related research line that aims at quantifying the
regularity of axisymmetric solutions satisfying a critical or slightly
supercritical Type I a priori bound. In the wake of the result of Pan [70], De
Giorgi methods were intensively used to improve upon the regularity beyond the
Type I case (see [22, 21, 45, 48]) by slightly breaking the a priori scale-
invariant assumption. This research was carried by Seregin [74, 75] and Chen,
Tsai and Zhang [23]. In this last paper, a double-logarithmic quantitative
blow-up rate for the $\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{3})$ norm of
$U$ is obtained; see [23, Theorem 1.4]. Using Harnack inequalities instead of
the Carleman inequalities used by Tao [84], Ożański and Palasek [67] recover
the blow-up rate of the $L^{3,\infty}(\mathbb{R}^{3})$ norm of $U$; see [67,
Corollary 1.2]. In addition, they obtain a quantitative bound of the form
(5.4) in terms of the $L^{\infty}_{t}L^{3,\infty}_{x}$ norm of the solution
only; see [67, Theorem 1.1]. For other developments, see [47].
## 9\. Mild criticality breaking and a conjecture of Tao
Recently, in [84, Remark 1.6], Tao conjectured that if a solution first loses
smoothness at time $T^{*}>0$, then the Orlicz norm
$\|U(\cdot,t)\|_{L^{3}(\log\log\log L)^{-c}(\mathbb{R}^{3})}$ must blow-up as
$t$ tends to $T^{*}$. Result 9.1 provides a positive answer to Tao’s
conjecture, at the cost of one additional logarithm.
###### Result D (blow-up of slightly supercritical Orlicz norms; [8, Theorem
2]).
There exists a universal constant $\theta\in(0,1)$ such that the following
holds.
Let $U$ be a Leray-Hopf solution to the Navier-Stokes equations on
$\mathbb{R}^{3}\times(0,\infty)$ with initial data $U_{0}\in
L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$. Assume that $U$ first blows-
up at $T^{*}\in(0,\infty)$. Then
$\lim\sup_{t\uparrow
T^{*}}\int\limits_{\mathbb{R}^{3}}\frac{|U(x,t)|^{3}}{\Big{(}\log\log\log\big{(}(\log(e^{e^{3e^{e}}}+|U(x,t)|))^{\frac{1}{3}}\big{)}\Big{)}^{\theta}}dx=\infty.$
(9.1)
As far as we know, Result 9.1 is the first result of this type for the Navier-
Stokes equations concerned with slight criticality breaking in borderline
spaces. Previously, it was shown for non-borderline spaces by Chan and Vasseur
[20] that if $U$ is a Leray-Hopf solution satisfying
$\int\limits_{0}^{\infty}\int\limits_{\mathbb{R}^{3}}\frac{|U|^{5}}{\log(1+|U|)}dxdt<\infty$
then $U$ is smooth on $\mathbb{R}^{3}\times(0,\infty)$. Subsequent
improvements were obtained in [49] and [10]; see also [63]. Let us mention
that the techniques used in these papers cannot be used to treat the
borderline case considered in Result 9.1.
#### A new method for transferring subcriticality of the data forward in time
The method for proving Result 9.1 relies on the following lemma and on a
careful tuning of the parameters (estimating the $L^{3-\mu}$ norm for a well-
chosen parameter $\mu$). Lemma E is directly inspired by the recent result of
Bulut [13] for a nonlinear supercritical defocusing Schrödinger equation.
###### Result E (‘mild criticality breaking’; [8, Theorem 1]).
For all $M,\,A\in[1,\infty)$ sufficiently large, there exists
$\delta(M,A)\in(0,\frac{1}{2}]$ such that the following holds. Let $U$ be a
suitable weak Leray-Hopf solution to the Navier-Stokes equations on
$\mathbb{R}^{3}\times(0,\infty)$ with initial data $U_{0}\in
L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$.
Assume that
$\|U_{0}\|_{L^{2}},\ \|U_{0}\|_{L^{4}}\leq M,$
and that
$\|U\|_{L^{\infty}(0,\infty;L^{3-\delta(M,A)}(\mathbb{R}^{3}))}\leq A.$ (9.2)
Then, the above assumptions imply that $U$ is smooth on
$\mathbb{R}^{3}\times(0,\infty)$. Moreover, there is an explicit formula for
$\delta(M,A)$, see [8, equation (26)], and $\delta(M,A)\rightarrow 0$ when
$M\rightarrow\infty$ or $A\rightarrow\infty$.
We call Result E a ‘mild breaking of the criticality’, or a ‘mild
supercritical regularity criteria’ as opposed to strong criticality breaking
results obtained for instance in the axisymmetric case [70, 74, 75, 23], see
Section 8. Indeed, the supercritical space $L^{\infty}_{t}L^{3-\delta(M,A)}$
in which we break the scaling depends on the size $A$ of the solution in this
supercritical space via $\delta(M,A)$. In other words this can be considered
as a non-effective regularity criteria, hence the term ‘mild’. Moreover, given
a solution $U$, assume that you knew all the $L^{\infty}_{t}L^{3-\delta}_{x}$
norms for $\delta\rightarrow 0$. Then the question whether Lemma E applies to
$U$ or not becomes a question about how fast
$\|U\|_{L^{\infty}(0,\infty;L^{3-\delta}(\mathbb{R}^{3}))}$
grows when $\delta\rightarrow 0$. Of course one would have regularity if the
solution was a priori bounded in the critical space $L^{\infty}_{t}L^{3}_{x}$.
The result shows that with $L^{4}$ initial data one can relax the exponent $3$
to a slightly supercritical $3-\delta(M,A)$. Let us also remark that the
initial condition $U_{0}\in L^{4}(\mathbb{R}^{3})$ can be replaced by any
subcritical initial condition $U_{0}\in L^{3+}(\mathbb{R}^{3})$.
The main idea of the proof of Lemma E is to transfer subcritical information
from the initial time forward in time. In a nutshell, subcritical energy
estimates are combined with quantitative regularity estimates as obtained by
Tao [84]. Hence the growth of the subcritical norm along the evolution can be
estimated.282828Result E can be abstractly quantified using persistence of
singularities, see [8, Introduction]. In that perspective the main objective
is the following.
###### Objective D.
Prove that there exists $\delta(M,A)\in(0,\frac{1}{2}]$ and
$K(M,A)\in[1,\infty)$ such that for all $U_{0}\in L^{2}(\mathbb{R}^{3})\cap
L^{4}(\mathbb{R}^{3})$ and any suitable weak Leray-Hopf solution associated to
the initial data $U_{0}$, if
$\|U_{0}\|_{L^{2}},\ \|U_{0}\|_{L^{4}}\leq M,$
and
$\|U\|_{L^{\infty}(0,\infty;L^{3-\delta(M,A)}(\mathbb{R}^{3}))}\leq A,$
then
$\|U\|_{L^{\infty}(0,T;L^{4}(\mathbb{R}^{3}))}\leq K(M,A).$ (9.3)
for any $T>0$.
This then obviously implies the result stated in Result E. The crucial point
is that $K(M,A)$ is uniform in time.
Let us emphasize that the only a priori globally controlled quantity is a
supercritical $L^{\infty}_{t}L^{3-}$ norm. We are not aware of any regularity
mechanism enabling to break the critically barrier based on the sole knowledge
of such a supercritical bound. Therefore, the idea, following Bulut [13], is
to transfer the subcritical information coming from the initial data $U_{0}\in
L^{4}(\mathbb{R}^{3})$ to arbitrarily large times by using three ingredients:
1. (1)
the control of the critical $L^{\infty}_{t}L^{3}_{x}$ norm via interpolation
between the supercritical norm $L^{\infty}_{t}L^{3-\delta(M,A)}_{x}$ and the
subcritical $L^{\infty}_{t}L^{4}_{x}$ norm
$\displaystyle\begin{split}\|U\|_{L^{\infty}(0,T;L^{3}(\mathbb{R}^{3}))}\leq\
&\|U\|_{L^{\infty}(0,T;L^{3-\delta}(\mathbb{R}^{3}))}^{\frac{3-\delta}{3+3\delta}}\|U\|_{L^{\infty}(0,T;L^{4}(\mathbb{R}^{3}))}^{\frac{4\delta}{3+3\delta}}\\\
\leq\
&A^{\frac{3-\delta}{3+3\delta}}K^{\frac{4\delta}{3+3\delta}}\,;\end{split}$
2. (2)
the quantitative control of the critical non borderline $L^{5}_{t,x}$ norm
(see [8, Proposition 3]) in terms of the critical norm
$\|U\|_{L^{\infty}(0,\infty;L^{3}(\mathbb{R}^{3}))}$, and the supercritical
$L^{2}$ and subcritical $L^{4}$ norms of the initial data $U_{0}$
$\|U\|_{L^{5}(0,T;L^{5}(\mathbb{R}^{3}))}\leq
C(M)\exp\exp\exp\big{(}C_{univ}\big{(}A^{\frac{3-\delta}{3+3\delta}}K^{\frac{4\delta}{3+3\delta}}\big{)}^{c}\big{)};$
this hinges on the quantitative bounds on solutions belonging to the critical
space $L^{\infty}_{t}L^{3}_{x}$, which were established by Tao in [84], see
(5.7); this step enables the slicing of the interval $(0,T)$ into a
$T$-independent number $m$ of disjoint epochs $I_{j}=(t_{j},t_{j+1})$,
$\displaystyle\varepsilon^{5}m=\sum_{j=1}^{m}\|U\|_{L^{5}(I_{j};L^{5}(\mathbb{R}^{3}))}^{5}\leq\
$ $\displaystyle\|U\|_{L^{5}(0,T;L^{5}(\mathbb{R}^{3}))}^{5}$
$\displaystyle\leq\ $ $\displaystyle
C(M)\exp\exp\exp\Big{(}\big{(}A^{\frac{3-\delta}{3+3\delta}}K^{\frac{4\delta}{3+3\delta}}\big{)}^{c}\Big{)}\,;$
3. (3)
an $L^{4}$ energy estimate [8, Proposition 4] under the $L^{5}_{t,x}$ control
of $U$, which enables the transfer the subcritical information from time
$t_{j}$ to $t_{j+1}$
$\displaystyle\mathscr{E}_{4,t_{j+1}}\leq\ $
$\displaystyle\|U(\cdot,t_{j})\|_{L^{4}(\mathbb{R}^{3})}^{4}+C\|U\|_{L^{5}(\mathbb{R}^{3}\times
I_{j})}\mathcal{E}_{4,t_{j+1}}$ $\displaystyle\leq\ $
$\displaystyle\|U(\cdot,t_{j+1})\|_{L^{4}(\mathbb{R}^{3})}^{4}+C\varepsilon\mathscr{E}_{4,t_{j+1}},$
where $\mathscr{E}_{4,t_{j+1}}$ is the $L^{4}$ energy, see [8, equation (13)],
and eventually to $T$
$\displaystyle\|U\|_{L^{\infty}(0,T;L^{4}(\mathbb{R}^{3}))}^{4}=\max_{1\leq
j\leq m+1}\\{\|U\|_{L^{\infty}(I_{j};L^{4}(\mathbb{R}^{3}))}^{4}\\}\leq
64M^{4}2^{m}.$
One then designs the number $K(M,A)$ and $\delta(M,A)$ to bound the right hand
side above.
## 10\. Summary of selected results
Figure 6 on page 6 summarizes some results about weak and strong concentration
for the Navier-Stokes equations, as well as local-in-space smoothing. Figure 7
on page 7 summarizes qualitative and quantitative regularity results for the
Navier-Stokes equations.
| weak concentration | local-in-space smoothing | strong concentration
---|---|---|---
local energy solutions/Type I | | Jia, Šverák (2014) [35] _subcritical_ Barker, Prange (2020) [6] _critical, whole-space_ Kang, Miura, Tsai (2020) [38] _critical, whole-space_ Kang, Miura, Tsai (2020) [39] _scaled energy, whole-space_ Albritton, Barker, Prange (2021) [2] _critical, half-space_ | Barker, Prange (2020) [6] _whole-space_ Albritton, Barker, Prange (2021) [2] _half-space_ Barker, Prange (2021) [9] _blow-up of_ $L^{3}\big{(}B_{0}(O((T^{*}-t)^{\frac{1}{2}-}))\big{)}$ Kang, Miura, Tsai (2021) [39] _scale-invariant energy_
beyond Type I | Li, Ozawa, Wang (2018) [51] Maekawa, Miura, Prange (2020) [53] Kang, Miura, Tsai (2021) [39] _scale-invariant energy_ | |
Figure 6. Weak and strong concentration: a selection of results
| critical | borderline critical | borderline endpoint critical | slightly supercritical | supercritical
---|---|---|---|---|---
| | $L^{\infty}_{t}L^{3}_{x}$ | Type I | ‘log’ breaking | $\sup_{k}\|U(\cdot,t_{k})\|_{L^{3}}<\infty$
qualitative | Ladyženskaja-Prodi-Serrin (60’s) | Escauriaza, Seregin, Šverák (2003) [29], Kenig, Koch (2011) [40], Gallagher, Koch, Planchon (2013) [30]… | | $\int\limits_{0}^{T}\int\limits_{\mathbb{R}^{3}}\frac{|U|^{5}}{\log(1+|U|)}\,dxds<\infty$ Chan, Vasseur (2007) [20], Bjorland, Vasseur (2011) [10], Lei, Zhou (2013) [49]… | Seregin (2012) [73], Albritton and Barker [1, Theorem 4.1 (i)]
| | | | Barker, Prange (2021) [8] _blow-up of an Orlicz norm_ |
explicit quantitative | | Tao (2019) [84] | Barker, Prange (2021) [9] _localized blow-up of $L^{3}$_ | | Barker, Prange (2021) [9] _quantification of Seregin’s 2012 result and of Albritton and Barker’s 2019 result_
increasing level of criticality
Figure 7. Qualitative vs. abstract and explicit quantitative regularity: a
selection of results
### Acknowledgement
Both authors thank the Institute of Advanced Studies of Cergy Paris University
for their hospitality. CP is partially supported by the Agence Nationale de la
Recherche, project BORDS, grant ANR-16-CE40-0027-01, project SINGFLOWS, grant
ANR- 18-CE40-0027-01, project CRISIS, grant ANR-20-CE40-0020-01, by the CY
Initiative of Excellence, project CYNA (CY Nonlinear Analysis) and project
CYFI (CYngular Fluids and Interfaces).
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14
# Online Unrelated-Machine Load Balancing and Generalized Flow with Recourse
Ravishankar Krishnaswamy<EMAIL_ADDRESS>Microsoft Research India. Shi
Li<EMAIL_ADDRESS>Department of Computer Science and Engineering,
Universify of Buffalo. Varun Suriyanarayana
<EMAIL_ADDRESS>Cornell University.
###### Abstract
We consider the online unrelated-machine load balancing problem with recourse,
where the algorithm is allowed to re-assign prior jobs. We give a
$(2+\epsilon)$-competitive algorithm for the problem with $O_{\epsilon}(\log
n)$ amortized recourse per job. This is the first $O(1)$-competitive algorithm
for the problem with reasonable recourse, and the competitive ratio nearly
matches the long-standing best-known offline approximation guarantee. We also
show an $O(\log\log n/\log\log\log n)$-competitive algorithm for the problem
with $O(1)$ amortized recourse. The best-known bounds from prior work are
$O(\log\log n)$-competitive algorithms with $O(1)$ amortized recourse due to
[GKS14], for the special case of the restricted assignment model.
Along the way, we design an algorithm for the online generalized network flow
problem (also known as network flow problem with gains) with recourse. In the
problem, any edge $uv$ in the network has a gain parameter $\gamma_{uv}>0$ and
$\theta$-units of flow sent across $uv$ from $u$’s side becomes
$\gamma_{uv}\theta$ units of flow on the $v$’th side. In the online problem,
there is one sink, and sources come one by one. Upon arrival of a source, we
need to send 1 unit flow from the source. A recourse occurs if we change the
flow value of an edge. We give an online algorithm for the problem with
recourse at most $O(1/\epsilon)$ times the optimum cost for the instance with
capacities scaled by $\frac{1}{1+\epsilon}$. The $(1+\epsilon)$-factor
improves upon the corresponding $(2+\epsilon)$-factor of [GKS14], which only
works for the ordinary network flow problem. As an immediate corollary, we
also give an improved algorithm for the online $b$-matching problem with
reassignment costs.
## 1 Introduction
Load balancing is one of the fundamental problems in online algorithms, due to
its real-world motivations, as well as its clean formulation leading to the
development of several techniques in online algorithms. In this paper we study
the power of _recourse/re-assignments_ for the online load balancing problem
_on unrelated machines_. In the $\mathsf{OLBwR}$ (Online unrelated machine
Load Balancing with Recourse) problem, we are given a set $M$ of $m$ machines,
and $n$ jobs $[n]$ arrive online. Job $t\in[n]$, which arrives at time $t$, if
assigned to machine $i$, would induce a load of $p_{it}$ on the machine. The
goal is to assign jobs to machines to minimize the maximum load of any
machine, which is the sum of loads of jobs assigned to it. The algorithm can
also re-assign prior jobs, and we separately track the _recourse_ to be the
total number of re-assignments over the course of arrivals.
The above is a very natural problem to study, since jobs/demands typically
arrive in an online manner, and moreover, real-world systems (see, e.g.
[Ala09]) often _migrate jobs_ between servers to achieve better balance.
However, since migrating jobs is a disruptive operation, we seek to minimize
the total number of movements while also ensuring nearly balanced assignments
at all times. We formalize the twin objectives as follows.
###### Definition 1.1 ($\alpha$-competitive, $\beta$-amortized recourse
Algorithms).
Let $\mathbf{Opt}^{t}$ denote the maximum load of any machine in an optimal
assignment for the first $t$ jobs. Then, we say that an algorithm is
$\alpha$-competitive with $\beta$-amortized recourse if the maximum load over
all machines of the algorithm’s assignment is most $\alpha\,\mathbf{Opt}^{t}$,
and the total number of re-assignments done over the course of the first $t$
job arrivals is at most $\beta t$.
This problem has received significant attention since the 1990s. The classical
online load balancing problem has the same model as $\mathsf{OLBwR}$, with the
restriction that the online algorithm’s assignments are irrevocable, i.e., no
recourse is allowed. A series of works introduced several elegant ideas
culminating with tight $\Theta(\log m)$-competitive algorithms [AAF+97], with
matching lower bound for any randomized online algorithm. The power of
recourse has also been studied for the load balancing problem, albeit from the
perspective of handling job departures [Wes00], where it is evident that no
online algorithm can have non-trivial competitive ratios without recourse. To
the best of our knowledge, [GKS14] were the first to study the power of
recourse to getting improved guarantees for job arrivals (the same setting as
ours), and showed that $O(1)$-amortized recourse can yield $O(\log\log
mn)$-competitive algorithms for online load balancing for the _restricted
assignment problem_ (where each job can only be assigned to a subset $N(j)$ of
machines, but it has the same processing time of $p_{j}$ on any of these
machines). This bound represents an exponential improvement when compared to
the $\Omega(\log m)$ lower bounds on the competitive ratio for the same model
when no recourse is allowed. _The central focus of this paper is in trying to
understand if we can get similar improvements for the $\mathsf{OLBwR}$ on
unrelated machines._ We obtain the following results:
###### Theorem 1.2.
For any constant $\epsilon>0$, there is an efficient deterministic
$(2+\epsilon)$-competitive algorithm for $\mathsf{OLBwR}$ on unrelated
machines with $O\left(\frac{\log
n\log(1/\epsilon)}{\epsilon^{5}}\right)$-amortized recourse.
Note that we are able to get a competitive ratio bound nearly matching the
best known offline approximation factor [ST93, LST90] for this classical
problem, with a small amount of recourse.
###### Theorem 1.3.
There is an efficient randomized $O(\log\log n/\log\log\log n)$-competitive
algorithm for $\mathsf{OLBwR}$ on unrelated machines with $O(1)$-amortized
recourse.
Our algorithms work by maintaining a $(1+\epsilon)$-competitive fractional
assignment online, and then rounding it in an online manner while ensuring
that both steps do not modify the solution too much. Prior to our work, no
algorithm that maintained constant-competitive fractional solutions with
bounded recourse was known. Indeed, a crucial observation central to our
result is that the fractional assignments for unrelated machines can be seen
as a special case of the _generalized flow_ (a.k.a min-cost flow with gains)
problem. Here, we are given a directed graph $G=(V,E)$ with cost $c_{e}$ and
capacities $\mu_{e}$ on the edge set $E$. However, unlike classical flows
where the same amount of flow exits the edge as entering, in the generalized
problem, the amount of flow exiting an edge is a scalar multiple of the amount
entering. This is captured by _gain factors_ $\gamma_{e}$ on the edges, which
represent the _extent to which one unit of flow originating at one end-vertex
of the edge gets scaled when it reaches the other end-vertex_. We are given a
set $S$ of sources, each of which wants to send one unit of flow, and a sink
$\tau$ which can absorb the flows. The goal is to maintain minimum cost flows
which can send unit from the sources while ensuring flow conservation at all
vertices $v\in V\setminus S\setminus\\{\tau\\}$. Since we seek to maintain
online fractional solutions for $\mathsf{OLBwR}$, we introduce and study the
$\mathsf{OGNF}$ (Online Generalized Network Flow) problem. Here, the sources
arrive one by one, and the algorithm needs to pay the incremental cost of
sending one unit flow from the new source, on top of the existing flow. The
goal is to minimize the total cost in comparison with the offline optimum.
Informally, we obtain the following result, and state the formal version in
Theorem 2.1.
###### Theorem 1.4.
There is an efficient deterministic
$O\left(\frac{1}{\epsilon}\right)$-competitive algorithm for $\mathsf{OGNF}$
when the algorithm can violate edge capacities by a factor of $(1+\epsilon)$.
In fact, the existing $\mathsf{OLBwR}$ results for restricted assignment
follow a similar template. However, the reduction in [GKS14] only applies to
the restricted assignment setting, and results in an online version of the
classical flow problem (without gains). Theorem 2.1 represents a two-fold
advancement over this. Firstly, [GKS14] presents $O(1)$-competitive algorithms
with $(2+\epsilon)$-factor capacity violations, while our results improve this
to $(1+\epsilon)$ violation, and secondly, our algorithms can handle arbitrary
gain factors, while the GKS-algorithms only applies to unit gains.
We now discuss our final algorithmic contribution. Indeed, since our results
improve the capacity violation even for the regular flow problem, this
immediately translates to an improvement for the so-called Online $b$-Matching
with Reassignment Costs ($\mathsf{ObMwRC}$) problem, where we have a bipartite
graph $G=(L\uplus R,E)$. Right vertices $v\in R$ each have capacities
$b_{v}\in\mathbb{Z}_{\geq 0}$, and the goal is to assign the left vertices
while respecting the capacities. It is guaranteed that $G$ has a valid
$b$-matching: a matching where every $u\in L$ is matched once and every $v\in
R$ is matched at most $b_{v}$ times. In the online problem, the left vertices
arrive one by one. When a left-vertex $u\in L$ arrives, it specifies its
neighbors $N(u)\subseteq R$ in $G$, and a _reassignment cost_ $c_{u}\geq 0$.
Each time we re(assign) a $u\in L$ to a (new) vertex $v\in R$, we pay a cost
of $c_{u}$. We need to always maintain an assignment which violates the
capacities by a small amount while incurring a small cost.
###### Theorem 1.5.
There is an efficient deterministic algorithm for $\mathsf{ObMwRC}$ which,
with $O\left(\frac{1}{\epsilon}\right)$ amortized recourse, maintains a
matching where each $v\in L$ is matched once and each $v\in R$ is matched at
most ${\left\lceil(1+\epsilon)b_{v}\right\rceil}$ times.
This improves over the prior work [GKS14] where the capacity violation was a
factor of $(2+\epsilon)$.
Finally, one may ask what we can do in the fully-dynamic model for the online
load balancing problem with recourse. That is, jobs may arrive and depart. In
Appendix C, we show that in this case, even the offline algorithm which knows
the whole sequence of the job arrivals and departures ahead of time need to
incur an amortized recourse of $n^{\Omega(1/\alpha)}$, in order to achieve an
$\alpha$-competitive ratio. Thus, to circumvent the negative result, one needs
to consider a different measurement for recourse for the fully dynamic model.
### 1.1 Our Techniques, at a High Level
Load Balancing. [GKS14] solved the restricted assignment problem using the
following method: for every job $j$ with load $p_{j}$, they create a source
$j$ of demand $p_{j}$; there is also a vertex $v_{i}$ for every machine $i$.
The source $j$ has a directed edge to $v_{i}$ if and only if $j$ can be
scheduled on $i$. Finally, there are directed edges of capacity $T^{*}$, the
optimal makespan bound, from each $v_{i}$ to a common sink $\tau$. It is then
easy to see that feasible flows in this network correspond to fractional
solutions for the restricted assignment problem. [GKS14] therefore use their
online flow algorithm to them maintain $O(1)$-competitive fractional solutions
with $O(1)$-recourse. They then _round_ the fractional solution using a
combination of randomized rounding and bucketing jobs based on their size to
obtain the $O(\log\log mn)$-competitive ratio. However, note that both steps
(computing the fractional solution as well as bucketing) crucially use the
fact that a job has equal processing time $p_{j}$ on all its valid machines,
and it is not clear how to extend either to the unrelated machines setting.
To circumvent these issues, we instead view the fractional solution to the
unrelated machines problem as a _generalized flow_ instance, which is what
motivated our study of the $\mathsf{OGNF}$ problem in the first place. We can
then immediately use Theorem 1.4 to maintain $(1+\epsilon)$-competitive
fractional solutions with $O\left(\frac{1}{\epsilon}\right)$-recourse. As for
the rounding step, we first present a cleaner algorithm with
$O(1)$-approximation ratio, that delivers the key ideas. We view each machine
as $O(\log n)$ sub-machines, with sub-machine $v_{ik}$ only catering to jobs
$j$ of size $p_{i,j}\approx T^{*}/2^{k}$. Then, suppose that in some
fractional solution $\\{x_{i,j}\\}$, a total of $f_{ik}$ units of jobs is
assigned to sub-machine $v_{ik}$. Then, if we are able to assign roughly
${\left\lceil f_{ik}\right\rceil}$ jobs in our rounded solution, we can
guarantee that our load will not exceed that of the fractional solution by too
much! With this intuition, we can solve the rounding challenge as follows: we
maintain an instance of online $b$-matching where jobs correspond to left
vertices, and sub-machines $v_{ik}$ correspond to right vertices. Each right
vertex also has a capacity _equal to the total fractional allocation in the
online LP solution, rounded up to the next integer_. Then, when the fractional
solution is updated, we update the capacities of right vertices, and simply
run a low-recourse algorithm for the online $b$-matching instance to recover
our overall assignment.
The online rounding algorithm with $(2+O(\epsilon))$-approximation ratio is
more involved. We use the grouping idea of [ST93] for the
$(2+\epsilon)$-approximation for the generalized assignment problem. We
maintain a 2-level partition of fractional jobs assigned to every machine $i$,
according to non-increasing order of job sizes: They are first partitioned
into _buckets_ , each of which contain roughly $\Theta(1/\epsilon)$ factional
jobs, and then every bucket is partitioned into _segments_ , each of which
contains $1-\Theta(\epsilon)$ fractional jobs. The bipartite matching instance
can be constructed from the segments: Each segment corresponds to a vertex on
the right side. A good fractional matching is guaranteed to exist by existence
of the fractional assignments. As the segments are created by job sizes, a
matching gives a $(2+O(\epsilon))$-approximate assignment, as in [ST93].
Our $L(n)=O\left(\frac{\log\log n}{\log\log\log n}\right)$-competitive
algorithm with $O(1)$-amortized recourse in Theorem 1.3 uses many ideas from
the $L(m)$-competitive online rounding algorithm of [LX21]. The offline
version of our algorithm works as follows. A job $j$ is big on machine $i$ if
$p_{ij}>T^{*}/\log n$ and small otherwise. Then a job is big if $1/2$
fractional of it is big on its assigned machines in the fractional solution,
and small otherwise. For small jobs, independent rounding already gives an
$O(1)$-approximation ratio. For big jobs $j$, with first round $x_{ij}$ values
so that each $x_{ij}$ becomes either $0$ or at least $1/\log n$. This makes
the support of $x$ for big jobs sparse. Then we attempt to assign a big job to
a machine randomly using the new $x_{ij}$ values. The assignment fails if the
target machine is too overloaded. Finally, we use the deterministic
2-approximation rounding algorithm to round the failed jobs. To analyze the
recourse, we use two crucial lemmas: A job $j$ fails with
$1/\operatorname{poly}\log(n)$ probability, and with high probability, a
connected component induced by failed jobs has size
$\operatorname{poly}\log(n)$. Therefore, even if we reassign everything in
such a connected component, the recourse is small.
Generalized Flow. Indeed [GKS14] studied the special case of this problem with
unit gains, and showed that the natural algorithm of satisfying the
requirement of a new source by sending the unit flow along the shortest path
to the sink in the residual graph, actually works. Here, the residual graph
might have some forward arcs and some backward arcs, but the GKS algorithm,
assigns a cost of $c_{e}$ for forward arcs as well as backward arcs in the
residual graph, since that is the cost of the augmentation incurs in their
online model. This is in contrast to the classical residual graph for offline
flow problems where backward arcs have opposite sign of costs. Now, they
define a quantity ${\sf height}(s)$, which is the _cost of the shortest path
to the sink in the residual graph_ , and show that it is a good estimate of
the augmentation cost of $s$ when it arrives. Moreover, they show that ${\sf
height}(s)$ is non-decreasing over arrival of subsequent sources. Finally,
they show how we can construct a good dual solution to the offline LP
formulation of the min-cost flow problem using the ${\sf height}(s)$ values,
using which they relate the online and offline costs.
How do we extend these ideas to the generalized flow problem? Intuitively,
when a new source arrives, a natural strategy might be to augment along the
shortest path in the residual graph. While this is a reasonable idea, observe
that the concept of augmenting paths is very different when there are gains.
Indeed, we could potentially augment along a path in the residual graph from
the source to a cycle which does not even contain the sink, provided the
product of gains on the cycle is $<1$ (these are called _flow-absorbing
cycles_ in literature). Keeping this in mind, we generalize the notion of
height above to be the _minimum cost way to send one unit flow out of the
source_ , as opposed to the cost of the shortest path from source to sink. We
then show that with these modifications, we can seamlessly apply the duality-
based proof technique of [GKS14]. Finally, to improve the capacity violation
factor to $(1+\epsilon)$, we actually set the cost of backward arcs to $0$ in
the residual graph as opposed to a positive number. Up to a factor of 2 in the
recourse, the two definitions are equivalent. This is also the reason that in
the definition of our problem, we only incur costs for increasing flow values.
However, this minor modification in defining the heights helps us perform a
tighter analysis to get the improved the capacity violation bound.
### 1.2 Related Work
It is well known that without any form of recourse, the best possible
competitive ratio for online load balancing is $O(\log n)$ [ANR95]. When
arrivals and departures are allowed, [ABK94] give a lower bound of
$O(\sqrt{n})$. Philip and Westbrook [PW93] considered the same case and showed
$O(\log n)$-competitive algorithms with $O(1)$-recourse for the special case
of unit-size restricted assignment (aka online matching). Westbrook [Wes00]
subsequently designed $O(1)$-competitive algorithms with $O(\log n)$-recourse
for the same setting. The case of unrelated machines in the fully-dynamic case
was considered in [AAPW01] where the authors designed $O(\log n)$-competitive
algorithms with $O(\log n)$-recourse. In the special case of identical
machines, [IC03] gave a lower bound $\sqrt{3}$ and an upper bound of $1.923$
is known from [Alb99]. [SSS09] presented a class of $(1+\epsilon)$-competitive
algorithms with reassignments allowed, where the mitigation factor grows as
$\epsilon$ gets smaller. When there are only arrivals and no departures, but
recourse is allowed, [GKS14] showed that one can achieve a $O(\log\log nm)$
competitive ratio with $O(1)$-amortized recourse in the restricted assignment
setting. [BEM22] extended this result to general re-assignment costs. Both
papers also gave constant approximations with similar amount of reassignments
in the special case of OGNF with all gains $\gamma_{e}=1$.
There is also significant work on designing offline algorithms for generalized
flow problems (see e.g., Chapter 6 of [Wil19]), and these algorithms have also
been used to design fast approximation algorithms for makespan minimization
[GMW07]. We believe our work is the first to address the online version of
generalized flow.
#### Organization:
In Section 2 we present the $O(\frac{1}{\epsilon})$ competitive online
algorithm for generalized flow with a $(1+\epsilon)$-capacity violation,
proving Theorem 1.4. Using this algorithm, in Section 3 we design an
$O(1)$-competitive algorithm with $O(\log n)$ amortized recourse; the
$(2+\epsilon)$-competitive rounding algorithm that proves Theorem 1.2 is
deferred to Appendix B. In Section 4, we present the $O(\frac{\log\log
n}{\log\log\log n})$-competitive algorithm for the same problem with $O(1)$
amortized recourse, proving Theorem 1.3. In Appendix C we give the lower bound
on the recourse required in the fully dynamic model. Missing proofs can be
found in Appendix A.
#### Notations
For a real number $a$, we use $(a)_{+}$ to denote $\max\\{a,0\\}$. For a graph
$H$ and a vertex $v$ in $H$, we shall use $N_{H}(v)$ to denote the set of
neighbors of $v$ in $H$. For a directed graph $H$ and a vertex $v$ in $H$, we
use $\delta^{+}_{H}(v)$ and $\delta^{-}_{H}(v)$ to denote the sets of outgoing
and incoming edges of $v$ in $H$ respectively. When $H$ is $G$ in the context,
we omit the subscript. For online load balancing, we shall identify jobs with
time steps: jobs are denoted as $[n]$, where job $t\in[n]$ arrives at time
$t$.
## 2 The Online Generalized Network Flow ($\mathsf{OGNF}$) Problem
In the generalized network flow problem, we are given a digraph $G=(V,E)$ with
sources $S\subseteq V$ and a sink $\tau\in V\setminus S$, where the sources
$S$ do not have incoming edges and the sink $\tau$ does not have outgoing
edges in $G$. We are given vectors $\mu,c,\gamma\in\mathbb{R}_{>0}^{E}$, where
for every $e\in E$, $\mu_{e}$, $\gamma_{e}$ and $c_{e}$ denote the capacity,
gain and cost of the edge $e$ respectively. Every source $s\in S$ has
$a_{s}>0$ units of supply. As in the ordinary network flow problem, our goal
is for each $s\in S$ to send $a_{s}$ units flow in the network satisfying the
flow conservation and edge capacity constraints, so as to minimize the cost.
The generalization comes from the gain vector $\gamma$: when a vertex $u$
sends $\theta$ unit flow along an edge $e=uv$, $v$ will receive
$\gamma_{e}\theta$ units flow from the edge. Therefore, the problem can be
formulated as the following LP, where we assume $a_{v}=0$ for every $v\in
V\setminus S\setminus\\{\tau\\}$.
$\min\sum_{e\in E}c_{e}x_{e}\qquad\text{s.t.}$ (1)
$\displaystyle\sum_{e\in\delta^{+}(v)}x_{e}-\sum_{e\in\delta^{-}(v)}\gamma_{e}x_{e}$
$\displaystyle=a_{v}$ $\displaystyle\forall v\in V\setminus\\{\tau\\}$
$\displaystyle 0\leq x_{e}$ $\displaystyle\leq\mu_{e}$ $\displaystyle\forall
e\in E$
In the LP, $x_{e}$ for an edge $e=uv\in E$ indicates the amount of flow sent
by $u$ through $e$. Both the capacity $\mu_{e}$ and the per-unit cost $c_{e}$
are defined w.r.t the flow on the sender’s side of $e$. We call any
${\mathbf{x}}\in\mathbb{R}_{\geq 0}^{E}$ satisfying the constraints in the LP
a valid flow for the instance. Notice that unlike the ordinary network flow
problem, the balance parameter (i.e., the $a$-value) of the sink $\tau$ can
not be decided by those of other vertices. Therefore we leave $a_{\tau}$
undefined and do not impose the flow conservation constraint on $\tau$.
Online Generalized Network Flow Problem ($\mathsf{OGNF}$): We are initially
given an instance of the problem $(G=(V,E),S,\tau,\mu,c,\gamma)$ with
$S=\varnothing$. In each time step $t=1,2,\cdots,T$, a new source $s_{t}\notin
V$ arrives. We assume $a_{s_{t}}=1$, as this suffices for our purpose. Along
with $s_{t}$, we are given the outgoing edges of $s_{t}$, as well as their
$\mu,c$ and $\gamma$ values. After the arrival of $s_{t}$, we add it to $S$
and $V$, and its outgoing edges to $E$.
For any time $t$, we need to maintain a valid flow
${\mathbf{x}}^{(t)}\in\mathbb{R}_{\geq 0}^{E}$ for the instance at time $t$.
The cost incurred at this time step is defined as $\sum_{e\in
E}c_{e}\big{(}x^{(t)}_{e}-x^{(t-1)}_{e}\big{)}_{+}$, where we assume undefined
variables have value $0$. Our goal is to design an online algorithm with a
small cost.
We elaborate more on the definition of our cost; for better clarity, we focus
on the ordinary network flow problem (that is, we have $\gamma_{e}=1$ for all
$e\in E$). [GKS14] defined the cost incurred at step $t$ to be $\sum_{e\in
E}c_{e}\cdot|x^{(t)}_{e}-x^{(t-1)}_{e}|$ in their model to accurately capture
the notion of recourse for flow problems. In this cost model, decreasing the
flow value along an edge would incurs a positive cost. This is in contrast to
classical offline algorithms for flow where decreasing the flow value of an
edge would incur a negative cost, thereby ensuring that the final cost of the
flow would always be equal to the sum of costs incurred by each update. Our
model is in between these models, where we charge flow increases with positive
cost but omit charging flow decrements. However, note that our results also
translate to results for the [GKS14] cost model within a factor of $2$ in the
cost — indeed, for any decrement in the flow value on an edge, we must have
paid the cost when we increase the flow value.
We prove the following main theorem in this section, which is a more formal
description of Theorem 1.4:
###### Theorem 2.1.
Given any $\epsilon>0$, there is an efficient deterministic algorithm for
$\mathsf{OGNF}$, such that the following holds. The cost incurred by the
algorithm at any time is at most
$\frac{1+\epsilon}{\epsilon}=O\big{(}\frac{1}{\epsilon}\big{)}$ times the cost
of the optimum flow for the general network flow instance at the time, with
all edge capacities scaled by $\frac{1}{1+\epsilon}$.
For convenience, we assume there are dummy edges from every vertex in
$V\setminus\\{\tau\\}$ to $\tau$ with infinite capacity, cost $B$ for a
sufficiently large $B$, and gain $1$, to always ensure feasibility in our
definitions. When the original instance is feasible and $B$ is large enough,
our algorithm will not use the dummy edges.
### 2.1 Augmenting Paths for General Network Flow
Our algorithm for online generalized network flow can be defined in a
straightforward manner: When a source $s_{t}$ arrives, we repeatedly find the
cheapest “generalized augmenting path” from $s_{t}$ and augment the flow along
the path as much as possible. The procedure terminates when $s_{t}$ sent 1
units of flow. Unlike the ordinary network flow problem, for which we can
define an augmenting path as a simple path from $s_{t}$ to $\tau$, an
augmenting path for the general network flow problem can be slightly more
complex.
###### Definition 2.2 (Residual Graph).
Let $(G,S,\tau,\mu,c,\gamma)$ be a generalized network flow instance. Let
${\mathbf{x}}\in\mathbb{R}_{\geq 0}^{E}$ be a vector such that
$x_{e}\in[0,\mu_{e}]$ for every $e\in E$ (it is not required that
${\mathbf{x}}$ is a valid flow for the instance). Then the _residual graph_
$G^{\mathbf{x}}=(V,E^{\mathbf{x}})$ for ${\mathbf{x}}$ is defined as the graph
containing vertices $V$ and the set $E^{\mathbf{x}}$ of edges, each $e\in
E^{\mathbf{x}}$ with parameters $\mu^{\mathbf{x}}_{e},c^{\mathbf{x}}_{e}$ and
$\gamma^{\mathbf{x}}_{e}$. They are defined as follows.
* •
For every $uv\in E$ with $x_{uv}<\mu_{uv}$, we have a _forward_ edge $uv\in
E^{\mathbf{x}}$ with $\mu^{\mathbf{x}}_{uv}=\mu_{uv}-x_{uv}$,
$c^{\mathbf{x}}_{uv}=c_{uv}$ and $\gamma^{\mathbf{x}}_{uv}=\gamma_{uv}$.
* •
For every $uv\in E$ with $x_{uv}>0$, we have a _backward_ edge $vu\in
E^{\mathbf{x}}$ with $\mu^{\mathbf{x}}_{vu}=\gamma_{uv}x_{uv}$,
$c^{\mathbf{x}}_{vu}=0$ and $\gamma^{\mathbf{x}}_{vu}=1/\gamma_{uv}$.
The definition can be viewed as an extension of the residual graph to the
generalized network flow problem. We make two remarks here. First, a backward
edge has cost $0$ instead of a negative cost, due to the cost defined in our
online model. Second, $\theta$ units of flow sent via $uv$ on the $u$’s side
transform into $\gamma_{uv}\theta$ units of flow on the $v$’th side. This
gives the definition of $\mu^{\mathbf{x}}_{vu}$ and $\gamma^{\mathbf{x}}_{vu}$
for a backward edge $vu\in E^{\mathbf{x}}$. As is typical, we assume $G$ does
not contain anti-parallel edges, so that $G^{\mathbf{x}}$ is a simple graph.
###### Definition 2.3 (Fractional Augmenting Paths).
Let $(G,S,\tau,\mu,c,\gamma)$, ${\mathbf{x}}$,
$G^{\mathbf{x}}=(V,E^{\mathbf{x}})$ and vectors
$\mu^{\mathbf{x}},\gamma^{\mathbf{x}}$ and $c^{\mathbf{x}}$ be defined as in
Definition 2.2. Let $s\in V\setminus\\{\tau\\}$ (it may be possible that
$s\notin S$). A fractional augmenting path from $s$ in $G^{\mathbf{x}}$ is a
vector $f\in\mathbb{R}_{\geq 0}^{E^{\mathbf{x}}}$ such that the excess flow
$\sum_{e\in\delta^{+}_{G^{\mathbf{x}}}(v)}f_{e}-\sum_{e\in\delta^{-}_{G^{\mathbf{x}}}(v)}\gamma^{\mathbf{x}}_{e}f_{e}$
equals $1$ for $v=s$, and equals $0$ for $v\in V\setminus\\{s,\tau\\}$. The
cost of such an augmenting path $f$ is defined as
${\mathrm{cost}}(f):=\sum_{e\in E^{\mathbf{x}}}c^{\mathbf{x}}_{e}f_{e}$.
Notice that the definition does not involve the capacities
$\mu^{\mathbf{x}}_{e}$ of edges $e\in E^{\mathbf{x}}$: As long as $e$ exists
in $E^{\mathbf{x}}$, $f_{e}$ can take any number in $\mathbb{R}_{\geq 0}$.
###### Definition 2.4 (Augmenting Paths).
Take all the notations in Definition 2.3 and assume $f$ is a fractional
augmenting path from $s$ in $G^{\mathbf{x}}$. We simply say $f$ is an
augmenting path (without the word “fractional”) if additionally it satisfies
one of the following conditions:
1. (2.4a)
The support of $f$ is a path from $s$ to $\tau$ in $G^{\mathbf{x}}$.
2. (2.4b)
The support of $f$ is a cycle $C$ in $G^{\mathbf{x}}$ containing $s$ but not
$\tau$.
3. (2.4c)
The support of $f$ is the union of a cycle $C$ in $G^{\mathbf{x}}$ not
containing $s$ and $\tau$, and a path in $G^{\mathbf{x}}$ from $s$ to $C$ that
is internally disjoint from $C$.
###### Lemma 2.5.
Consider a generalized network flow instance $(G,S,\tau,\mu,c,\gamma)$ and
${\mathbf{x}}\in\mathbb{R}_{\geq 0}^{E}$ satisfying that $x_{e}\in[0,\mu_{e}]$
for every $e\in E$. Let $v\in V\setminus\\{\tau\\}$. Then $f$, the minimum-
cost fractional augmenting path from $v$ in $G^{\mathbf{x}}$, assuming it
exists, can be achieved at an augmenting path.
The following claim will be useful later.
###### Claim 2.6.
Let $f$ be an augmenting path from some $s\in V\setminus\\{\tau\\}$ in
$G^{\mathbf{x}}$ for some ${\mathbf{x}}$. Let $v\notin\\{s,\tau\\}$ be some
vertex in the support graph of $f$. Then we can break $f$ into
$f=f^{\prime}+f^{\prime\prime}$ where $f^{\prime}$ is a flow path in
$G^{\mathbf{x}}$ that sends 1 unit flow from $s$ to $v$ (the flow received by
$v$ may not be $1$), and $f^{\prime\prime}$ is a scaled augmenting path from
$v$ in $G^{\mathbf{x}}$.
Now given a (fractional) augmenting path $f$ from $s$ in $G^{\mathbf{x}}$, and
a real number $\theta>0$, we define the operation of augmenting ${\mathbf{x}}$
by $\theta$ units using $f$ as follows:
* •
For every forward edge $uv\in E^{\mathbf{x}}$ with $f_{uv}>0$, we update
$x_{uv}\leftarrow x_{uv}+\theta\cdot f_{uv}$.
* •
For every backward edge $vu\in E^{\mathbf{x}}$ with $f_{vu}>0$, we update
$x_{uv}\leftarrow x_{uv}-\theta\cdot f_{vu}/\gamma_{uv}$.
###### Claim 2.7.
Let $f$ be an augmenting path from $s$ in $G^{\mathbf{x}}$. Then, augmenting
${\mathbf{x}}$ by $\theta>0$ units using $f$ does not change
$\sum_{e\in\delta^{+}(v)}x_{e}-\sum_{e\in\delta^{-}(v)}\gamma_{e}x_{e}$ for
every $v\in V\setminus\\{s,\tau\\}$, and increases
$\sum_{e\in\delta^{+}(s)}x_{e}-\sum_{e\in\delta^{-}(s)}\gamma_{e}x_{e}$ by
$\theta$.
### 2.2 The Online Algorithm
With the definition of augmenting paths, we can now formally describe our
online algorithm. The pseudo-code is given in Algorithm 1.
Algorithm 1 Online algorithm for generalized network flow
1:Let ${\mathbf{x}}^{(0)}$ be the all-0 vector over edges of the initial graph
$G$
2:for every $t\leftarrow 1$ to $T$ do
3: update the instance to include the source $s_{t}$
4: let ${\mathbf{x}}\leftarrow{\mathbf{x}}^{(t-1)}$, adding $0$-coordinates
for the incoming edges of $s_{t}$
5: while $\sum_{e\in\delta^{+}(s_{t})}x_{e}<1$ do
6: find the cheapest augmenting path $f$ from $s_{t}$ in the residual graph
$G^{\mathbf{x}}$
7: let $\theta>0$ be the biggest number such that after augmenting
${\mathbf{x}}$ using $f$ by $\theta$ units, we still have
$x_{e}\in[0,\mu_{e}]$ for every $e\in E$ and
$\sum_{e\in\delta^{+}(s_{t})}x_{e}\leq 1$
8: augment ${\mathbf{x}}$ using $f$ by $\theta$ units
9: ${\mathbf{x}}^{(t)}\leftarrow{\mathbf{x}}$
Remark on Running Time Notice that the running time of Algorithm 1 may be
exponentially large, as we can not bound the number of iterations of the while
loop by polynomial. However, at time $t$, the algorithm is simply trying to
send 1 unit flow from $s_{t}$ using the minimum cost in the residual graph,
and this can be simply done using an LP. Moreover sending the 1 unit flow at
once can only incur a smaller cost. Therefore we can make the algorithm
efficient. For analysis purposes, we chose to present Algorithm 1, since the
augmenting path at each step has a good structure as stated in Definition 2.4.
### 2.3 Analysis of Online Algorithm
[GKS14] define the concept of height of any vertex to be the cost of the
shortest path to the sink $\tau$ in the residual graph at any point in time,
and then track it for bounding the competitive ratio. Analogously, we define a
height for a vertex as the _cost of the cheapest augmenting path from the
vertex in the residual graph_ – notice the lack of sink in the definition. We
show that the incremental cost incurred by the $t^{\rm th}$ source is at most
the height of $s_{t}$ at time $t$. Moreover, we also show that heights can
only increase during the course of the algorithm, implying that the cost
incurred by the algorithm is at most the total height of sources at the end.
To complete the analysis, we show that the heights (after all arrivals) define
a dual solution for the final flow instance with capacities scaled down by a
factor of $(1+\epsilon)$, and upper bound its cost by $O(1/\epsilon)$ times
the offline optimum cost of the instance.
###### Definition 2.8.
Let $t\in[0,T]$ and $v\neq\tau$ be a vertex in the network $G$ at time $t$.
The _height_ of $v$ at time $t$, denoted as $0pt_{t}(v)$, is defined as the
cost of the cheapest augmenting path from $v$ in the residual graph
$G^{{\mathbf{x}}^{(t)}}$. We define $0pt_{t}(\tau)=0$.
Monotonicity of Heights. We show that heights are non-decreasing over the
course of the algorithm. The proof the following lemma is deferred to Appendix
A.
###### Lemma 2.9.
Let $t\in[T]$ and $v$ be a vertex in the graph $G$ at time $t-1$. Then
$0pt_{t}(v)\geq 0pt_{t-1}(v)$.
###### Lemma 2.10.
The cost incurred by the whole algorithm is at most
$\sum_{t=1}^{T}0pt_{T}(s_{t})$.
###### Proof.
Notice that the cost incurred by sending 1 unit of flow from the source
$s_{t}$ in time $t$ can be upper bounded by $0pt_{t}(s_{t})$, which in turn is
at most $0pt_{T}(s_{t})$ by Lemma 2.9. ∎
Bounding total heights using duality. Let $C^{*}$ be the cost of the optimum
flow for the generalized network flow instance at time $T$, with capacities
scaled by a factor of $\frac{1}{1+\epsilon}$. We consider the dual of LP (1)
for the instance:
$\max\qquad\sum_{v\in V\setminus\\{\tau\\}}a_{v}y_{v}-\sum_{e\in
E}\frac{\mu_{e}z_{e}}{1+\epsilon}\qquad\text{s.t.}$ (2) $\displaystyle-
z_{uv}+y_{u}-\gamma_{uv}y_{v}$ $\displaystyle\leq c_{uv}$
$\displaystyle\forall uv\in E$ $\displaystyle z_{e}$ $\displaystyle\geq 0$
$\displaystyle\forall e\in E$ $\displaystyle y_{\tau}$ $\displaystyle=0$
Notice that we do not have a constraint in LP (1) for $\tau$; for convenience
we also introduce a dual variable $y_{\tau}$ and let $y_{\tau}=0$. For a fixed
$y$ vector, the optimal choice for $z_{uv}$ is
$(y_{u}-\gamma_{uv}y_{v}-c_{uv})_{+}$. Also $a_{v}=0$ for every $v\in
V\setminus S\setminus\\{\tau\\}$ and $a_{s_{t}}=1$ for every $t\in[T]$.
Therefore, the dual LP can be rewritten as
$\displaystyle\max\qquad\sum_{t\in[T]}y_{s_{t}}-\sum_{uv\in
E}\frac{\mu_{uv}\left(y_{u}-\gamma_{uv}y_{v}-c_{uv}\right)_{+}}{1+\epsilon}\qquad\text{s.t.
}\quad y_{\tau}=0.$
###### Lemma 2.11.
$\sum_{t=1}^{T}0pt_{T}(s_{t})\leq\frac{1+\epsilon}{\epsilon}C^{*}$.
###### Proof.
Now we need to bound the sum of the heights of the sources at termination. To
do this we show that $\big{(}y_{v}:=0pt_{T}(v)\big{)}_{v\in V}$ is a feasible
dual solution. Then we bound the cost of this feasible dual in relation to
$C^{*}$, therefore giving us a bound on the competitive ratio.
Let ${\mathbf{x}}={\mathbf{x}}^{(T)}\in\mathbb{R}_{\geq 0}^{E}$ be the final
flow we obtained. By breaking edges, we can assume every edge $e\in E$ has
either $x_{e}=0$ or $x_{e}=\mu_{e}$. Any edge $uv\in E$ with $x_{uv}=0$ exists
in the residual graph $G^{\mathbf{x}}$. Therefore
$y_{u}\leq\gamma_{uv}y_{v}+c_{uv}$ as $y$ corresponds to the heights, and
sending 1 unit flow from $u$ can be achieved by sending $1$ unit flow from $u$
to $v$, and then sending $\gamma_{uv}$ units flow from $v$. So for such edges
$(y_{u}-\gamma_{uv}y_{v}-c_{uv})_{+}=0$. Let $E^{\prime}$ be the set of edges
with $x_{uv}=\mu_{uv}$. So,
$\displaystyle\sum_{uv\in
E}\frac{\mu_{uv}(y_{u}-\gamma_{uv}y_{v}-c_{uv})_{+}}{1+\epsilon}\leq\sum_{uv\in
E^{\prime}}\frac{x_{uv}}{1+\epsilon}(y_{u}-\gamma_{uv}y_{v})=\frac{1}{1+\epsilon}\sum_{v\in
V}y_{v}\left(\sum_{e\in\delta^{+}(v)}x_{e}-\sum_{e\in\delta^{-}(v)}\gamma_{e}x_{e}\right)=\frac{1}{1+\epsilon}\sum_{t\in[T]}y_{s_{t}}.$
For every $uv\in E^{\prime}$, $x_{uv}=\mu_{uv}$ and
$y_{v}\leq\frac{y_{u}}{\gamma_{uv}}$ since the backward edge $vu$ exists in
$G^{\mathbf{x}}$ and it has gain $\frac{1}{\gamma_{uv}}$ and cost $0$. So we
have the inequality. The first equality is by that $x_{uv}=0$ for $uv\in
E\setminus E^{\prime}$ and rearranging the terms. The second equality is by
the balance condition for ${\mathbf{x}}$ and $y_{\tau}=0$.
So the objective value of the dual solution $y$ is at least
$(1-\frac{1}{1+\epsilon})\sum_{t\in[T]}y_{s_{t}}$, which implies
$\frac{\epsilon}{1+\epsilon}\sum_{t\in[T]}y_{s_{t}}\leq C^{*}$. Multiplying
both sides by $\frac{1+\epsilon}{\epsilon}$ proves the lemma. ∎
Thus, combining Lemmas 2.10 and 2.11 proves Theorem 2.1.
## 3 Online Unrelated Machine Load Balancing with Recourse
We now show one of our main results, that of maintaining
$(2+\epsilon)$-approximate solutions for online unrelated machine load
balancing with $O_{\epsilon}(\log n)$ amortized recourse per job, as stated in
Theorem 1.2. We restate the problem setting. There is a set $M$ of $m$
machines, and $n$ jobs indexed by $[n]$. We have a bipartite graph
$G=(M\uplus[n],E)$ between machines and jobs, where $ij\in E$ indicates that
the job $j$ can be assigned to machine $i$. When $j$ is assigned to $i$, it
incurs a load of $p_{ij}>0$ on machine $i$. The goal is to assign jobs to
machines so as to minimize the maximum load over all machines, also called
makespan in the scheduling literature. In the online version, jobs arrive one
by one: job $j\in[n]$ arrives at time $j$, along with its incident edges in
$G$ and their $p_{ij}$ values. We need to maintain a solution for the arrived
jobs at any time. We allow the algorithm to re-assign prior jobs from time to
time, and separately track the recourse of the algorithm.
Known vs Unknown $T^{*}$. We first define a useful quantity $T^{*}$, which is
the smallest value of $T$ for which the following LP is feasible: $\sum_{i\in
N(j)}x_{ij}=1$ for every $j\in[t]$, $\sum_{j\in N(i)}p_{ij}x_{ij}\leq T$ for
every $i\in M$, and $x_{ij}=0$ if $p_{ij}>T^{*}$. We refer to this as the
_optimal fractional makespan_.
Our online algorithms will assume knowledge of $T^{*}$. While we can use a
standard guess-and-double approach to eliminate this assumption, we would lose
an additional constant factor in the competitive ratio. Since we are allowed
recourse, we can do better, as follows to get the $2+O(\epsilon)$ guarantee on
competitive ratio. Suppose there is algorithm ${\mathcal{A}}$ that achieves
$C\cdot T^{*}$ makespan when $T^{*}$ is given. We now design a simple
procedure which can also achieve $(1+O(\epsilon))C$-competitive solutions with
bounded recourse, even when $T^{*}$ is not given. Indeed, we break our
procedure into _stages_ , where a new stage occurs when the optimum fractional
makespan increases by a factor of at least $1/\epsilon$. Each stage is further
partitioned into many _phases_ , where a new phase starts if the optimum
fractional makespan increases by a factor of at least $1+\epsilon$. When a new
stage $g$ starts with bound $T^{*}$ on the optimal fractional makespan, we
simply re-construct an offline solution for all the jobs in $(g-2)$-th stage
with makespan $2\epsilon T^{*}$, say using the $2$-approximation algorithm
[LST90] for offline load balancing. We then _freeze the assignment of these
jobs_ according to this offline solution, i.e., we won’t change the assignment
of these jobs in the future. Note that the total load due to all the frozen
jobs on any machine is at most $O(\epsilon)T^{*}$. On the other hand, whenever
a new phase starts with optimum fractional makespan $T^{*}$, we re-run
algorithm ${\mathcal{A}}$ with the revised estimate $T^{*}$ and reintroduce
all the unfrozen jobs (i.e., jobs of this stage as well as the previous),
thereby causing recourse for all these jobs. From the gaurantee of
${\mathcal{A}}$, the makespan induced by these jobs will be at most $C\cdot
T^{*}$, giving us the desired guarantee. As for recourse, note that each job
can be unfrozen for at most
$O(\log_{1+\epsilon}\frac{1}{\epsilon})=O\left(\frac{\log(1/\epsilon)}{\epsilon}\right)$
phases across two stages, which bounds the recourse.
Algorithm Overview. The overall algorithm comprises of two steps. We first
maintain a _fractional_ solution which is $(1+\epsilon)$-competitive and has
$O\big{(}\frac{1}{\epsilon}\big{)}$-amortized fractional recourse by reducing
the problem to the online generalized network flow problem. In the second step
we round the fractional solution into an integral one, in an online manner
with low recourse by creating an intermediate bipartite matching instance
based on the fractional solution. For clarity of presentation, we present a
conceptually simpler rounding algorithm with a weaker $O(1)$-factor
competitive ratio in Section 3.3, and defer the slightly more involved
$(2+\epsilon)$-competitive algorithm to Appendix B.
### 3.1 Producing Fractional Solutions Online using Generalized Flow Instance
In the first step, we reduce the online load balancing problem to the online
generalized network flow problem. We use $G^{\prime}=(V^{\prime},E^{\prime})$
to denote the digraph for the network flow problem. Initially,
$V^{\prime}=M\cup\\{\tau\\}$, where $\tau$ is the sink. There is an edge
$i\tau\in E^{\prime}$ with $\mu_{i\tau}=(1+\epsilon)T^{*},c_{i\tau}=0$ and
$\gamma_{i\tau}=1$. For each arriving job $j$, we add $j$ to $V^{\prime}$ and
the source set. For every $ij\in E$, we add a directed edge $ji$ to
$E^{\prime}$ with $\mu_{ji}=\infty,c_{ji}=1$ and $\gamma_{ji}=p_{ij}$. Below
we let $E$ be the final set of edges between jobs and machines.
###### Lemma 3.1.
We can maintain fractional solutions $({\mathbf{x}}^{(t)})_{t\in[n]}$ online
such that the following conditions hold:
* •
For every $t\in[n]$, ${\mathbf{x}}^{(t)}\in[0,1]^{E}$ is constructed at time
$t$, and is a fractional solution for jobs $[t]$ of makespan at most
$(1+\epsilon)T^{*}$: $\sum_{i}x^{(t)}_{ij}=1$ for all jobs $j\in[t]$,
$x^{(t)}_{ij}=0$ for every $ij\in E$ with $j>t$, and
$\sum_{j}p_{ij}x^{(t)}_{ij}\leq(1+\epsilon)T^{*}$for all $i\in M$.
111Technically, we do not know the edges incident to jobs arrived after $t$.
However the $x^{(t)}_{ij}$ values for these edges are $0$. Letting
${\mathbf{x}}^{(t)}\in[0,1]^{E}$ is only for the sake of notational
convenience.
* •
The total fractional recourse until any time $t$ is bounded, i.e.,
$\sum_{t^{\prime}=1}^{t}\sum_{i}|{\mathbf{x}}^{(t^{\prime})}-{\mathbf{x}}^{(t^{\prime}-1)}|\leq
O\big{(}\frac{1}{\epsilon}\big{)}\cdot t$.
### 3.2 Online Bipartite Matching with Vertex Updates On the Right Side
Both our rounding algorithms work by carefully constructing an online instance
of a matching-type problem, _based on the load balancing fractional solution_
, and then using good online algorithms for this instance to derive our final
assignment. We therefore describe this variant of the online bipartite
matching problem where _vertices on the right side may also change_ :
There is a bipartite graph $H=(L\uplus R,E_{H})$ with $L=E_{H}=\varnothing$
initially. $H$ is changed dynamically by the following three types of updates:
(i) a new vertex $u$ is added to $L$, along with its incident edges, (ii) a
new vertex $v$ is added to $R$, along with its incident edges, and (iii) a
vertex $v\in R$ is removed. Our goal is to maintain a matching covering $L$,
given the promise that one exists at any time. The algorithm has an amortized
recourse of $\beta$, if the number of times we re-assign vertices in $L$ is at
most $\beta$ times the number of updates.
We now show that good algorithms exist if the graph $H$ has sufficient
expansion at all times.
###### Theorem 3.2.
Consider an online bipartite-matching problem with the three types of updates
specified above. Let $\alpha>1$ be a real number such that at any time, we
have $|N_{H}(A)|\geq\alpha|A|$ for every $A\subseteq L$. Then there is an
algorithm for the instance with $O(\log_{\alpha}n)$-amortized recourse, where
$n$ is the size of $|L|$ at the end.
The algorithm is simple: We maintain a matching $F$ in $H$ covering $L$.
Whenever a new vertex in $L$ arrives, we match it by changing $F$ using the
shortest augmenting path. Whenever some vertex $v\in R$ is removed, we remove
the edge in $F$ incident to $v$ if it exists. Then for the unmatched vertex,
we match it again using the shortest augmenting path. Due to the
$\alpha$-factor slack in the neighborhood, we can use an expansion argument to
show that there will always exist a short augmenting path if there is a
unmatched vertex in $L$ (Lemma 3.3).
###### Lemma 3.3.
Let $H=(L\uplus R,E_{H})$ be a bipartite graph. Let $\alpha>1$ be a real
number such that $|N_{H}(A)|\geq\alpha|A|$ for every $A\subseteq L$. Let
$F\subseteq E_{H}$ be a partial matching where not all vertices in $L$ are
matched. Then there is an augmenting path of length at most $2D+1$ w.r.t $F$,
where $D={\left\lfloor\log_{\alpha}|L|\right\rfloor}+1>\log_{\alpha}|L|$.
This finishes the proof of Theorem 3.2, as the number of times we re-assign
vertices in $L$ is at most the total length of augmentation paths over time,
which is small by Lemma 3.3.
Online $b$-Matching with Capacity Updates. Theorem 3.2 can also be generalized
to solve the variant where the vertices on the right-side have capacities
$b(v)$ which can change over time. The goal is to maintain matchings
respecting these capacities, and an algorithm is said to have
$\beta$-amortized recourse if the total number of re-assignments is at most
$\beta$ times the total change to the capacities $\sum_{v\in
R}\sum_{t}|b^{t}(v)-b^{t-1}(v)|$. To see how, note that we can simply have
$b(v)$ copies of each vertex $v$ on the right side to reduce it to the online
bipartite problem stated above.
### 3.3 Simple Online Rounding via Online $b$-Matching with Right Vertex
Updates
We now show how to use the fractional solutions
$({\mathbf{x}}^{(t)})_{t\in[n]}$ to construct the online $b$-matching instance
with capacity updates, and apply Theorem 3.2 to construct the assignment.
We use $H=(L\uplus R,E_{H})$ and $b:R\to\mathbb{Z}_{\geq 0}$ to denote the
graph for the $b$-matching instance and the capacity values of $R$. (We let
$b$ be a function instead of a vector due to the heavy notations for vertices
in $R$.) The algorithm is formally defined in Algorithm 2. Left vertices
correspond to jobs. Right vertex $v_{ik}\in R$ associated with machine $i$ is
only catering to jobs $j$ with $p_{ij}\approx T^{*}/2^{k}$. Indeed, let
$f_{ik}$ denote the total fractional allocation of such jobs to machine $i$ in
the solution $({\mathbf{x}}^{(t)})$. Then we ideally want to define the
capacity of $v_{ik}$ to be $b({v_{ik}})=\lceil 2f_{ik}\rceil$, where the
factor of $2$ would guarantee the condition in Theorem 3.2 holds with
$\alpha=2$. However, this could lead to too many changes to ${v_{ik}}$’s
capacity over time, e.g., if the fractional allocation keeps fluctuating
around a half-integral value. To overcome this, we introduce a random offset
and bound the expected change in capacities in terms of the total fractional
change. We can also derandomize this using ideas similar to our other rounding
algorithm in Appendix B.
Algorithm 2 Online Rounding of Fractional Solutions
${\mathbf{x}}^{(1)},{\mathbf{x}}^{(2)},\cdots,{\mathbf{x}}^{(n)}$
1:Let $L\leftarrow\varnothing,R\leftarrow\\{v_{ik}:i\in M,k\in[0,K]\\}$ for
$K=\lceil 2\log n\rceil$, $H\leftarrow(L\uplus R,E_{H}=\varnothing)$
2:Choose a random threshold $\rho\in[0,1)$.
3:for $t=1,2,\ldots,n$ do
4: Add $t$ to $L$, its incident edges incident to $E_{H}$ so that
$N_{H}(t)=\\{v_{ik}\,:it\in E,\,T^{*}/2^{k+1}<p_{it}\leq T^{*}/2^{k}\\}$.
5: Update $J_{ik}\leftarrow\\{j\in[t]:T^{*}/2^{k+1}<p_{ij}\leq
T^{*}/2^{k}\\}$, $f_{ik}\leftarrow\sum_{j\in J_{ik}}x^{(t)}_{ij}$ and
$b(v_{ik})\leftarrow\left\lceil 2f_{ik}+\rho\right\rceil$ for all $i,k$
6: Follow the algorithm in Theorem 3.2 to obtain a new $b$-matching in $H$.
7: Update $\sigma:[t]\to M$ by setting $\sigma(j)=i$ if $j$ is assigned to
$v_{ik}$ for some $k$ in the $b$-matching
Throughout this section, we use the superscript $(t)$ over a notion to denote
the value of the parameter at the end of time $t$ during Algorithm 2.
#### Analysis of $O(1)$-Competitive Ratio and $O(\log n)$-Amortized Recourse
###### Lemma 3.4.
At any time of the algorithm, the $b$-matching instance $(L\cup R,E_{H})$
satisfies $b(N_{H}(A))\geq 2|A|$ for every $A\subseteq L$.
###### Proof.
The lemma holds since ${\mathbf{x}}^{(t)}$ gives a fractional matching between
$[t]$ and $R$ where every $j\in[t]$ is matched to an extent of $1$ and every
$v_{ik}$ is matched to an extent of $f^{(t)}_{ik}$, and $b^{(t)}(v_{ik})\geq
2f^{(t)}_{ik}$ is an integer. ∎
By Theorem 3.2, the algorithm achieves an $O(\log n)$-amortized recourse for
the online matching instance.
###### Claim 3.5.
Let $f$ and $f^{\prime}$ be any two real values, and let $\rho$ be a random
variable sampled uniformly from $[0,1)$. Then
$\mathbf{E}_{\rho}\left[\left|\lceil f+\rho\rceil-\lceil
f^{\prime}+\rho\rceil\right|\right]=\left|f-f^{\prime}\right|$.
###### Proof.
The proof is straightforward. Indeed, suppose without loss of generality,
$f\geq f^{\prime}$. Then, $\lceil f+\rho\rceil-\lceil f^{\prime}+\rho\rceil$
is precisely the number of integral values which lie in $f+\rho$ and
$f^{\prime}+\rho$. But because $\rho$ is a uniformly random value in $[0,1)$,
it is easy to see that this number is precisely $|f-f^{\prime}|$. ∎
###### Lemma 3.6.
The total expected change in capacities
$\mathbf{E}_{\rho}\left[\sum_{t^{\prime}=1}^{t}\sum_{ik}\left|b^{(t^{\prime})}(v_{ik})-b^{(t^{\prime}-1)}(v_{ik})\right|\right]$
by time $t$, is at most
$\sum_{t^{\prime}=1}^{t}\left|{\mathbf{x}}^{(t^{\prime})}-{\mathbf{x}}^{(t^{\prime}-1)}\right|$.
###### Proof.
The proof is a simple argument using linearity of expectation and the triangle
inequality. For any $i$ and $k$, we use $J_{ik}$ to denote the final set
$J_{ik}$, i.e, $\\{j\in[n]:T^{*}/2^{k+1}<p_{ij}\leq T^{*}/2^{k}\\}$
$\displaystyle\quad\mathbf{E}_{\rho}\left[\sum_{t^{\prime}=1}^{t}\sum_{ik}\left|b^{(t^{\prime})}(v_{ik})-b^{(t^{\prime}-1)}(v_{ik})\right|\right]\
=\
\sum_{t^{\prime}=1}^{t}\sum_{ik}\mathbf{E}_{\rho}\left[\left|b^{(t^{\prime})}(v_{ik})-b^{(t^{\prime}-1)}(v_{ik})\right|\right]$
$\displaystyle=\sum_{t^{\prime}=1}^{t}\sum_{ik}\mathbf{E}_{\rho}\left[\left|\lceil
2f^{(t^{\prime})}_{ik}+\rho\rceil-\lceil
2f^{(t^{\prime}-1)}_{ik}+\rho\rceil\right|\right]\ =\
\sum_{t^{\prime}=1}^{t}\sum_{ik}2\left|f^{(t^{\prime})}_{ik}-f^{(t^{\prime}-1)}_{ik}\right|\
\leq\
2\sum_{t^{\prime}=1}^{t}\left|{\mathbf{x}}^{(t^{\prime})}-{\mathbf{x}}^{(t^{\prime}-1)}\right|.$
Above, the third equality follows from Claim 3.5, and the next inequality
follows from the triangle inequality, and by noting that the various $J_{ik}$
sets over all $k$ for a fixed $i$ form a disjoint partition of all the jobs. ∎
###### Lemma 3.7.
After arrival of the first $j$ jobs, the load on any machine $i$ due to our
assignment $\sigma_{j}$ in 7 is at most $O(1)T^{*}$.
###### Proof.
Consider a fixed machine $i$. Below, we use $\pi^{(t)}_{jik}$ to indicate if
$j$ is assigned to $v_{ik}$ in the solution for the $b$-matching instance at
time $t$. The total load $i$ receives at time $t$ is then
$\displaystyle\sum_{k=0}^{K}\sum_{j\in J_{ik}}\pi^{(t)}_{jik}p_{ij}$
$\displaystyle\leq\sum_{k=0}^{K}\frac{T^{*}}{2^{k}}\sum_{j\in
J_{ik}}\pi^{(t)}_{jik}\ \leq\ \sum_{k=0}^{K}\frac{T^{*}}{2^{k}}\cdot
b^{(t)}(v_{ik})\ \leq\ \sum_{k=0}^{K}\frac{T^{*}}{2^{k}}\left(2\sum_{j\in
J_{ik}}x^{(t)}_{ij}+2\right)$ $\displaystyle\leq
O(T^{*})+2\sum_{k=0}^{K}\frac{T^{*}}{2^{k}}\sum_{j\in J_{ik}}x^{(t)}_{ij}\
\leq\ O(T^{*})+4\sum_{k=0}^{K}\sum_{j\in J_{ik}}x^{(t)}_{ij}p_{ij}\ \leq\
O(T^{*})+4\sum_{j\in[t]}x^{(t)}_{ij}p_{ij}\ \leq\ O(T^{*}).$
∎
The competitive ratio follows from Lemma 3.7. As for the recourse, note that
the total recourse made by the load balancing algorithm is at most the number
of reassignments made by the $b$-matching algorithm. The recourse of the
latter by time $t$ is at most $O(\log n)$ times
$t+\sum_{t^{\prime}=1}^{t}\sum_{ik}|b^{(t^{\prime})}(v_{ik})-b^{(t^{\prime}-1)}(v_{ik})|$
by Theorem 3.2. From Lemma 3.6, the expected value of the summation is at most
$\sum_{t^{\prime}=1}^{t}\left|{\mathbf{x}}^{(t^{\prime})}-{\mathbf{x}}^{(t^{\prime}-1)}\right|$.
By Lemma 3.1, this is at most $O(t)$. This shows the the amortized recourse of
the algorithm is $O(\log n)$.
## 4 An $O\left(\frac{\log\log n}{\log\log\log n}\right)$-Competitive
Algorithm with $O(1)$-Amortized Recourse
In this section, we describe the $O\left(\frac{\log\log n}{\log\log\log
n}\right)$-competitive algorithm for unrelated machine load balancing
($\mathsf{OLBwR}$) with $O(1)$-amortized recourse. Notice that we can lose an
$O(1)$-factor in the competitive ratio and thus we can simply fix
$\epsilon=1$. Using Lemma 3.1, we construct a sequence
${\mathbf{x}}^{(1)},{\mathbf{x}}^{(2)},\cdots,{\mathbf{x}}^{(n)}$ of
fractional solutions online, each ${\mathbf{x}}^{(t)}$ being a fractional
schedule for jobs $[t]$. We assume the makespans of the fractional solutions
are $T^{*}$ by scaling up $T^{*}$. For every $t\in[n]$, we have
$\sum_{t^{\prime}=1}^{t}|{\mathbf{x}}^{(t^{\prime})}-{\mathbf{x}}^{(t^{\prime}-1)}|=O(1)\cdot
t$.
We prove the following theorem in this section, which in turn proves Theorem
1.2.
###### Theorem 4.1.
There is a randomized online rounding algorithm that with high probability
produces solutions of makespan $O\left(\frac{\log\log n}{\log\log\log
n}\right)\cdot T^{*}$ and incurs recourse $O(1)\cdot t$ by time $t$, for every
$t\in[n]$.
For convenience, we add a dummy machine $i_{\bot}$ to $M$, and assign jobs
that have not arrived yet to $i_{\bot}$ both in input fractional schedules and
output integral schedules. That, is we assume $x^{(t)}_{i_{\bot}j}=1$ for
every $j>t$. After this transformation, every ${\mathbf{x}}^{(t)}$ is a
schedule for the whole job set $[n]$. We assume all jobs have processing time
$0$ on the dummy machine; this will not create an issue as our algorithm will
never assign a job $j$ to a machine $i$ at time $t$ if $x^{(t)}_{ij}=0$.
### 4.1 Main Ideas
Our algorithm uses the framework of the $O\left(\frac{\log\log m}{\log\log\log
m}\right)$-competitive online rounding algorithm of [LX21]. (Recall that $m$
is the number of machines.) In their setting, the fractional assignment of a
job $j$ never changes after its arrival. As a result, their algorithm does not
need to incur a recourse.
We give a high-level overview of the rounding algorithm in [LX21]. They
describe the algorithm in the offline setting, and one can easily make it
online. We say an edge $ij\in E$ is big if $p_{ij}>\frac{T^{*}}{\log m}$, and
small otherwise. A job $j$ is big if at least $1/2$ fraction of the job is
assigned via big edges in the fractional solution, and small otherwise. For
big (small) jobs $j$, we only consider its big (small) edges. Small jobs can
be assigned by independent rounding; with high probability they incur only an
$O(1)\cdot T^{*}$ load on every machine. Thus the bulk of the algorithm is for
the assignment of big jobs, which is done in three steps:
* •
Step (b1): The algorithm does an initial rounding to make the support of
${\mathbf{x}}$ sparse: If some $x_{ij}$ for a big job $j$ has
$x_{ij}\in(0,\frac{1}{\log m})$, then it rounds $x_{ij}$ to $0$ or
$\frac{1}{\log m}$ randomly, preserving the expectation of $x_{ij}$. After
this step, every such $x_{ij}$ is either $0$ or at least $\frac{1}{\log m}$.
This guarantees that the support graph for ${\mathbf{x}}$ restricted to big
jobs have degree $O(\log^{2}m)$.
* •
Step (b2): The algorithm attempts to assign every big job $j$ to a machine $i$
randomly, using the new $x_{ij}$ values as probabilities. The assignment fails
if the target machine is overloaded.
* •
Step (b3): The crucial theorem proved in [LX21] is that the following event
happens with high probability: In the sub-graph of the support of
${\mathbf{x}}$ induced by the failed jobs and all machines, every connected
component has size at most $\text{poly}\log m$. This allows the algorithm to
apply a deterministic $O\left(\frac{\log\log m}{\log\log\log
m}\right)$-competitive online rounding procedure for each component.
We use a similar framework as that of [LX21], with the following main
differences. First, we generate a set of global random seeds that are used in
our randomized rounding procedure for each time step. They will correlate
schedules at different time steps. Second, in step (b3), we can run the simple
offline 2-approximation algorithm for each connected component, as recourse is
allowed in our setting. Finally a small difference is that our competitive
ratio is $O\left(\frac{\log\log n}{\log\log\log n}\right)$ as we need to apply
union bounds over $n$ time steps.
### 4.2 Description of Algorithm and Proof of Competitive Ratio
We now formally describe our algorithm. We say an edge $ij\in E$ is big if
$p_{ij}\geq\frac{T^{*}}{{\log n}}$, and small otherwise. For every $j\in[n]$,
we let $M^{\mathrm{big}}_{j}$ ($M^{\mathrm{small}}_{j}$ resp.) be the set of
machines $i\in N(j)$ with $ij$ being big (small resp.).
#### Generating Global Random Seeds
We choose a threshold $\beta\in[\frac{1}{2},\frac{3}{4}]$ uniformly at random,
that will be used to define big and small jobs.
###### Definition 4.2.
Given $\beta$ and a fractional solution ${\mathbf{x}}\in[0,1]^{E}$, we say a
job $j$ scheduled in ${\mathbf{x}}$ is big if $\sum_{i\in
M^{\mathrm{big}}_{j}}x_{ij}>\beta$ and small otherwise. So, if $j$ is small,
then $\sum_{i\in M^{\mathrm{small}}_{j}}x_{ij}\geq 1-\beta$. Let
$J^{\mathrm{big}}$ and $J^{\mathrm{small}}$ be the sets of big and small jobs
respectively.
When a job $j$ arrives, for every big edge $ij\in E$, we independently choose
a threshold $\delta_{ij}\in[0,1/{\log n}]$ uniformly at random; this will be
used in the initial rounding step (step (b1)) for big jobs. We also generate
an infinite sequence of pairs
$(h^{j}_{1},\theta^{j}_{1}),(h^{j}_{2},\theta^{j}_{2}),(h^{j}_{3},\theta^{j}_{3}),\cdots$,
where for every $o\in\mathbb{Z}_{>0}$, $h^{j}_{o}$ is a random machine in
$N(j)$, and $\theta^{j}_{o}$ is a random real number in $[0,1]$; all the
parameters are independently generated. The sequence will serve as the random
seeds for the acceptance-rejection sampling method, to assign small jobs, and
to assign big jobs in step (b2).
Once we generated the global random seeds, it is convenient to describe the
algorithm _in the offline setting_ : For every time step $t$, we round the
fractional solution ${\mathbf{x}}^{(t)}$ to obtain an integral assignment,
using the global random seeds. A recourse occurs when the assignment of a job
$j$ at time $t$ is different from that at time $t-1$. So till the end of this
section we focus on a time step $t$, and the fractional solution
${\mathbf{x}}:={\mathbf{x}}^{(t)}\in[0,1]^{E}$. Big and small jobs are defined
w.r.t the global seed $\beta$ and this fractional solution ${\mathbf{x}}$.
#### Step (s): Assigning Small Jobs
For every $j\in J^{\mathrm{small}}$, we find the smallest
$o\in\mathbb{Z}_{>0}$ such that $h^{j}_{o}\in M^{\mathrm{small}}_{j}$ and
$\theta^{j}_{o}\leq x_{h^{j}_{o}j}$, and we assign $j$ to $h^{j}_{o}$. This is
equivalent to the following procedure. We draw a histogram for job $j$, where
there is a bar of height $x_{ij}$ for every $i\in M^{\mathrm{small}}_{j}$, and
a bar of height $0$ for every $i\in M^{\mathrm{big}}_{j}$. Then each
$(h^{j}_{o},\theta^{j}_{o})$ denotes a random point in $N(j)\times[0,1]$; we
accept a point if it falls into a bar in the histogram. Suppose
$(h^{j}_{o},\theta^{j}_{o})$ is the first point we accept; we then assign $j$
to $h^{j}_{o}$. Therefore, conditioned on $j\in J^{\mathrm{small}}$, the
probability that $j$ is assigned to a machine $i\in J^{\mathrm{small}}_{j}$ is
proportional to $x_{ij}$. Moreover, by the definition of the small jobs, the
probability is between $x_{ij}$ and $\frac{1}{1-\beta}x_{ij}\leq 4x_{ij}$.
###### Lemma 4.3.
With probability at least $1-\frac{1}{n^{3}}$, every machine $i$ gets a load
of at most $O(1)\cdot T^{*}$ from step (s).
###### Proof.
This follows from the Chernoff bound and union bound, as the assignments of
different small jobs $j$ are independent. ∎
It remains to assign big jobs; we do this in three steps.
#### Step (b1): Initial Rounding of ${\mathbf{x}}$ Values
For a job $j\in J^{\mathrm{big}}$, and every $i\in M^{\mathrm{big}}_{j}$, we
define $x^{\prime}_{ij}$ as follows:
$\displaystyle x^{\prime}_{ij}:=\begin{cases}x_{ij}&\text{if
}x_{ij}\geq\frac{1}{\log n}\\\ 0&\text{if }x_{ij}<\frac{1}{\log n}\text{ and
}\delta_{ij}>x_{ij}\\\ \frac{1}{\log n}&\text{if }x_{ij}<\frac{1}{\log
n}\text{ and }\delta_{ij}\leq x_{ij}\end{cases}$
For other $ij$ pairs we let $x^{\prime}_{ij}=0$. By the way we generate
$\delta_{ij}$ values, we have that $\mathbb{E}[x^{\prime}_{ij}]=x_{ij}$
conditioned on $j\in J^{\mathrm{big}}$ and $i\in M^{\mathrm{big}}_{j}$.
###### Lemma 4.4.
With probability at least $1-1/n^{2}$, the following events happen:
* •
For every $j\in J^{\mathrm{big}}$, we have $\sum_{i\in
M^{\mathrm{big}}_{j}}x^{\prime}_{ij}\geq\frac{1}{10}$.
* •
For every $i\in M$, we have $\sum_{j\in N(i)}p_{ij}x^{\prime}_{ij}\leq
10T^{*}$.
###### Proof.
The rounding procedure for jobs in $J^{\mathrm{big}}$ are independent. Again
we can apply Chernoff bound and Union bound to prove the lemma. ∎
From now on, we assume the above events happen. Then, every machine $i$ is
incident to at most $O(\log^{2}n)$ jobs in the support of
${\mathbf{x}}^{\prime}$, as each positive $x^{\prime}_{ij}$ has
$p_{ij}>\frac{T^{*}}{{\log n}}$ and $x^{\prime}_{ij}\geq\frac{1}{\log n}$.
#### Step (b2): Attempts to Assign Big Jobs
For every $j\in J^{\mathrm{big}}$, we find the smallest $o\in\mathbb{Z}_{>0}$
such that $h^{j}_{o}\in M^{\mathrm{big}}_{j}$ and $\theta^{j}_{o}\leq
x_{h^{j}_{o}j}$, and temporarily assign $j$ to $i:=h^{j}_{o}$. We _mark_ a
machine $i$ if it gets a load of more than $\frac{c\log\log n}{\log\log\log
n}\cdot T^{*}$ in this step (that is, we do not count the load assigned from
step (s)), for a sufficiently large constant $c$. All the jobs in
$J^{\mathrm{big}}$ assigned to marked machines are said to be _failed_ , and
we undo these assignments.222There is a slight difference between our
algorithm and that of [LX21] in this step. In [LX21], for a marked machine
$i$, the jobs assigned to $i$ before it is marked are not failed. This is
needed for their no-recourse setting. Let $J^{\mathrm{fail}}$ be the set of
failed jobs. If some $j\in J^{\mathrm{big}}$ is assigned to an unmarked
machine, i.e., $j\in J^{\mathrm{big}}\setminus J^{\mathrm{fail}}$, then the
assignment of $j$ is successful and final for time step $t$.
#### Step (b3) : Assign Failed Jobs Component by Component
We let $G^{\prime}=(M\uplus J^{\mathrm{big}},E^{\prime})$ be the support
bipartite graph for ${\mathbf{x}}^{\prime}$: for some $j\in
J^{\mathrm{big}},i\in M$, we have $ij\in E^{\prime}$ if and only if
$x^{\prime}_{ij}>0$. Then $G^{\prime}[M\cup J^{\mathrm{fail}}]$ is the sub-
graph of $G^{\prime}$ induced by $M\cup J^{\mathrm{fail}}$.
For any connected component $C$ in $G^{\prime}[M\cup J^{\mathrm{fail}}]$. We
run the deterministic $2$-approximation algorithm [LST90] to obtain an
assignment of failed jobs in $C$ to $M$. We can guarantee that any machine is
assigned a load of at most $200T^{*}$ in this step. This holds because
${\mathbf{x}}^{\prime}$ is an approximate LP solution, where by Lemma 4.4 the
two constraints are each violated by a factor of $10$.
This finishes the description of the algorithm. With high probability, the
algorithm maintains a schedule of makespan at most $O\left(\frac{\log\log
n}{\log\log\log n}\right)\cdot T^{*}$ at any time: Small jobs, successful and
failed big jobs respectively incur a load of $O(1)\cdot
T^{*},O\left(\frac{\log\log n}{\log\log\log n}\right)\cdot T^{*}$ and
$O(1)\cdot T^{*}$ on each machine.
### 4.3 Bounding the Recourse
In this section, we bound the recourse of the algorithm case by case. Below we
fix a time step $t\geq 1$, and use ${\mathbf{x}}^{\circ}$ and ${\mathbf{x}}$
to denote the fractional solution at time $t-1$ and $t$ respectively. For
every job $j$, we let ${\mathbf{x}}_{j}$ denote the vector $(x_{ij})_{i\in
N(j)}$, i.e., the fractional assignment of $j$ at time $t$. Define
${\mathbf{x}}^{\circ}_{j}$ similarly for time $t-1$. Define
${\mathbf{x}}^{\prime\circ},{\mathbf{x}}^{\prime},{\mathbf{x}}^{\prime}_{j},{\mathbf{x}}^{\prime\circ}_{j}$
similarly for the vector ${\mathbf{x}}^{\prime}\in[0,1]^{E}$ obtained in step
(b2). In all the proofs, we assume $c$ is a sufficiently large constant.
#### Type-1 recourse: recourse from switches between small and big jobs
We say $j$ incurs a type-1 recourse at time $t$ if $j$ is small at time $t-1$,
and big at time $t$, or the other way around.
###### Lemma 4.5.
Let $j\in[n]$. In expectation over the randomness of $\beta$, the probability
that $j$ incurs a type-1 recourse at time $t$ is at most
$2\cdot\big{|}{{\mathbf{x}}}_{j}-{{\mathbf{x}}}^{\circ}_{j}\big{|}$.
###### Proof.
Let $p^{\circ}_{\mathrm{big}}$ be the fraction of job $j$ assigned via big
edges, in the fractional solution ${\mathbf{x}}^{\circ}$; define
$p_{\mathrm{big}}$ similarly for ${\mathbf{x}}$. Then we have
$|p_{\mathrm{big}}-p^{\circ}_{\mathrm{big}}|\leq\frac{1}{2}\big{|}{{\mathbf{x}}}_{j}-{{\mathbf{x}}}^{\circ}_{j}\big{|}$.
If there is a switch between $j$ being small and $j$ being big, then $\beta$
must be between $p_{\mathrm{big}}$ and $p^{\circ}_{\mathrm{big}}$. This
happens with probability at most
$4\cdot|p_{\mathrm{big}}-p^{\circ}_{\mathrm{big}}|\leq
2\cdot\big{|}{{\mathbf{x}}}_{j}-{{\mathbf{x}}}^{\circ}_{j}\big{|}$ since
$\beta$ is uniformly chosen from $\left[\frac{1}{2},\frac{3}{4}\right]$. ∎
#### Type-2 recourse: recourse from small jobs
We say a recourse incurred by $j$ at time $t$ is of type-2 if $j$ is small at
both time steps $t-1$ and $t$, but assigned to different machines in the two
time steps.
###### Lemma 4.6.
For every $j\in[n]$, the probability that $j$ incurs a type-2 recourse at time
$t$ is at most $4\cdot|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|$.
###### Proof.
We fix a $\beta$ for which $j$ is small in both time $t-1$ and $t$. Consider
the point $(h^{j}_{1},\theta^{j}_{1})$ from the random sequence for $j$. We
say the point is good if it is accepted in both time step $t-1$ and $t$; we
say the point is bad if is accepted in exactly one of the two time steps. The
point is neutral if it is not accepted in either time step. The probability
that $j$ incurs a type-2 recourse at time $t$ is at most
$\frac{\Pr[(h^{j}_{1},\theta^{j}_{1})\text{ is
bad}]}{\Pr[(h^{j}_{1},\theta^{j}_{1})\text{ is good or
bad}]}\leq\frac{\big{|}{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}\big{|}/|N(j)|}{1/(4|N(j)|)}=4\cdot\big{|}{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}\big{|}$.
De-conditioning on $\beta$ gives the lemma. ∎
#### Type-3 recourse: recourse from different target machines in step (b2)
We say a recourse incurred by $j$ at time $t$ is of type-3 if $j$ is big at
both time steps $t-1$ and $t$, but it is temporarily assigned to different
machines in step (b2) of the two time steps.
###### Lemma 4.7.
For every $j\in[n]$, the probability that $j$ incurs a type-3 recourse at time
$t$ is at most $10\cdot|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|$.
###### Proof.
First we condition on the value of $\beta$ such that $j$ is big at both time
$t-1$ and $t$. Conditioned on $\delta_{ij}$ values, the probability that $j$
incurs a type-3 recourse at time $t$ is at most
$10\cdot|{\mathbf{x}}^{\prime}_{j}-{\mathbf{x}}^{\prime\circ}_{j}|$, using the
same argument from the previous lemma. De-conditioning on $\delta_{ij}$
values, the probability is at most
$10\cdot|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|$; this holds since
$E_{\delta_{ij}}[|x^{\prime}_{ij}-x^{\prime\circ}_{ij}|]=|x_{ij}-x^{\circ}_{ij}|$
for some $i\in M^{\mathrm{big}}_{j}$. Finally, de-conditioning on $\beta$
gives the lemma. ∎
#### Type-4 recourse: recourse from failed jobs
We say a recourse incurred by a job $j$ at time $t$ is of type-4, if it is not
of type-1, 2 or 3. In this case, $j$ is big in both time steps $t-1$ and $t$,
it is temporarily assigned to the same machine in step (b2), and it fails in
at least one of the two time steps. Moreover, the connected component in
$G^{\prime}[M\cup J^{\mathrm{fail}}]$ containing $j$ is not the same in the
two time steps (we assume the condition holds if $j$ does not fail in the
other time step).
Let ${\mathcal{C}}$ be the set of connected components in $G^{\prime}[M\cup
J^{\mathrm{fail}}]$ in time step $t$; define ${\mathcal{C}}^{\circ}$ similarly
for time step $t-1$. So we need to count the number of jobs in the components
in ${\mathcal{C}}^{\circ}\,\triangle\,{\mathcal{C}}$, where
${\mathcal{C}}^{\circ}\,\triangle\,{\mathcal{C}}$ denotes the symmetric
difference between ${\mathcal{C}}^{\circ}$ and ${\mathcal{C}}$, i.e., the set
of components appearing exactly one of ${\mathcal{C}}^{\circ}$ and
${\mathcal{C}}$. Due to the symmetry, it suffices to consider the components
in ${\mathcal{C}}^{\circ}\setminus{\mathcal{C}}$.
The key theorem we use is that any $C\in{\mathcal{C}}^{\circ}$ is small with
high probability. This is proved in [LX21] and used to bound their competitive
ratio; while in our case, we apply the theorem to bound the recourse. The
crucial properties we use to prove this are the facts that $G^{\prime}$ has
degree $O(\log^{2}(n))$, and that every machine is marked with probability
$1/\operatorname{poly}\log(n)$, with a sufficiently large exponent in the
$\operatorname{poly}\log(n)$ factor. We omit its proof here as it is identical
to that in [LX21], except for minor differences in parameters.
###### Theorem 4.8 (Lemma 4.5 in [LX21]).
With probability at least $1-\frac{1}{n^{3}}$, every
$C\in{\mathcal{C}}^{\circ}$ contains at most $O(\log^{7}n)$ machines.
With the theorem, we can now focus on bounding
$|{\mathcal{C}}^{\circ}\setminus{\mathcal{C}}|$. The change of components from
${\mathcal{C}}^{\circ}$ to ${\mathcal{C}}$ are caused by the following two
types of events:
* •
Some job $j$ has changed its target machine in step (b2) from time $t-1$ to
$t$, and in at least one of the two time steps, the target machine is marked;
we also say this event happens if $j$ is small in the other time step. This
event may add/remove $j$ to/from $J^{\mathrm{fail}}$, switch up to 2 machines
between marked and unmarked from time $t-1$ to $t$. We say this is a type-a
event incurred by job $j$.
* •
For some $j$ that fails in both time step $t-1$ and $t$, and some $i\in
M^{\mathrm{big}}_{j}$, we have $x^{\prime}_{ij}\neq x^{\prime\circ}_{ij}$. We
say this is a type-b event incurred by the pair $ij$. This will add/remove the
edge $ij$ to/from the graph $G^{\prime}$.
Notice that if no events of the two types occur, then
${\mathcal{C}}^{\circ}={\mathcal{C}}$, as marked machines, failed jobs, and
$x^{\prime}$ values incident to failed jobs do not change from time $t-1$ to
time $t$.
###### Lemma 4.9.
Fix a job $j\in[n]$. The probability that $j$ incurs a type-a event is at most
$O\left(\frac{1}{\log^{14}n}\right)\cdot|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|$.
###### Proof.
The probability that $j$ is assigned to two different target machines in step
(b2) in time steps $t-1$ and $t$ is
$O(1)\cdot|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|$. Conditioned on the
event, the probability that the target machine is marked in the correspondent
time step is at most $O\left(\frac{1}{\log^{14}n}\right)$, when $c$ is big
enough. ∎
###### Lemma 4.10.
Fix a big edge $ij\in E$. The probability that $ij$ incurs a type-b event is
at most $O\left(\frac{1}{\log^{14}n}\right)\cdot|x_{ij}-x^{\circ}_{ij}|$.
###### Proof.
With probability at most $O(\log n)\cdot|x_{ij}-x^{\circ}_{ij}|$, the
$x^{\prime}_{ij}$ value is different in time step $t-1$ and $t$, and $j$ is
big in both time steps. Under the condition that this happens, the probability
that $j$ fails at time step $t$ is at most
$O\left(\frac{1}{\log^{15}n}\right)$. The lemma then follows. ∎
Each type-a event can change at most $O(\log^{2}n)$ components in
${\mathcal{C}}^{\circ}$: it may change the marking status of two machines, and
a machine is incident to $O(\log^{2}n)$ jobs in the support of
$x^{\prime\circ}$. Each type-b event can change at most at most 2 components
in ${\mathcal{C}}^{\circ}$. Therefore, the expected number of components in
${\mathcal{C}}^{\circ}\setminus{\mathcal{C}}$ is at most
$\sum_{j\in[n]}O\left(\frac{1}{\log^{14}n}\right)\big{|}{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}\big{|}\cdot
O(\log^{2}n)=O\left(\frac{1}{\log^{12}n}\right)\cdot\sum_{j\in[n]}|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|$.
By Theorem 4.8, in expectation, the type-4 recourse is at most
$O\left(\frac{1}{\log^{12}n}\right)\sum_{j\in[n]}|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|\cdot
O(\log^{7}n)\cdot
O(\log^{2}n)=O(\frac{1}{\log^{3}n})\sum_{j\in[n]}|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|$.
Therefore, we have proved that the expected recourse at time $t$ is at most
$O(1)\sum_{j\in[n]}|{\mathbf{x}}_{j}-{\mathbf{x}}^{\circ}_{j}|=|{\mathbf{x}}-{\mathbf{x}}^{\circ}|$.
Summing up the bound over all time steps $t^{\prime}$ from $1$ to $t$, we
obtain that the recourse by time $t$ is at most
$\sum_{t^{\prime}=1}^{t}|{\mathbf{x}}^{(t^{\prime})}-{\mathbf{x}}^{(t^{\prime}-1)}|=O(1)\cdot
t$. This finishes the proof of Theorem 4.1.
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## Appendix A Missing Proofs
See 1.5
###### Proof.
We reduce $\mathsf{ObMwRC}$ to the online generalized network flow problem
(indeed, the ordinary network flow problem suffices as we all gains are
defined as $1$). We maintain a digraph $G^{\prime}=(L\cup
R\cup\\{\tau\\},E^{\prime})$. We have edges from $L$ to $R$ in $G^{\prime}$
that are the same as those in $G$ (except that the edges in $G^{\prime}$ are
directed), and edges from each $v\in R$ to $\tau$. An edge $uv\in E^{\prime}$
with $u\in L,v\in R$ has cost $c_{u}$, capacity $\infty$ and gain $1$. An edge
$v\tau$ for $v\in R$ has cost $0$, capacity $\lceil(1+\epsilon)b_{v}\rceil$
and gain $1$. The sources in the network are $L$. The graph $G^{\prime}$ is
constructed online: when a new vertex in $L$ arrives, we add it and its
outgoing edges to $G^{\prime}$.
Theorem 2.1 gives an online algorithm that maintains a network flow where
every $u\in L$ sends 1 units of flow. As the edges have gain parameters $1$,
$\tau$ receives $|L|$ units of flow. As the algorithm are based on augmenting
paths, the flow is integral. Thus the algorithm maintains a matching where
every $v\in R$ is matched at most $\lceil(1+\epsilon)b_{v}\rceil$ times. The
reassignment cost of the algorithm is equal to the cost incurred by the
network flow algorithm, which is at most $\frac{1+\epsilon}{\epsilon}$ times
the optimum cost $C^{*}$ of the network flow instance with capacities scaled
by $\frac{1}{1+\epsilon}$. After scaling down the capacities, every edge
$v\tau\in E^{\prime}$ has capacity at least $b_{v}$. Thus $C^{*}\leq\sum_{u\in
L}c_{u}$. ∎
See 2.5
###### Proof.
We consider any solution $f$ to the linear system defined by the constraints
in Definition 2.3, with objective ${\mathrm{cost}}(f)$. As all edge costs are
non-negative, we can assume the optimum solution is achieved when all values
in $f$ are bounded. Therefore, it must be achieved at a vertex point defined
by some tight inequalities in the linear system. Let
$G^{\prime}=(V^{\prime},E^{\prime})$ be the subgraph of $G^{\mathbf{x}}$
containing the support of $f$, and the vertices incident to these edges. We
can assume $G^{\prime}$ contains $s$ and is connected. Every $v\in
V^{\prime}\setminus\\{s,\tau\\}$ has at least one incoming edge and one
outgoing edge; $s$ has at least one outgoing edge. As $f$ is a vertex
solution, we have $|E^{\prime}|\leq|V^{\prime}\setminus\\{\tau\\}|$. We now
consider two cases depending on whether $\tau\in V^{\prime}$ or not.
First assume $\tau\in V^{\prime}$. Then we have
$|E^{\prime}|\leq|V^{\prime}|-1$. Therefore, we have
$|E^{\prime}|=|V^{\prime}|-1$ and $G^{\prime}$ is a path from $s$ to $\tau$ in
$G^{\mathbf{x}}$ (case (2.4a)). It remains to consider the case $\tau\notin
V^{\prime}$, which implies $|E^{\prime}|\leq|V^{\prime}|$. It can only happen
that $|E^{\prime}|=|V^{\prime}|$. Since every vertex in $V^{\prime}$ except
for $s$ has at least one incoming and one outgoing edge. Two sub-cases may
happen in this case. It may be that $s$ also has an incoming edge in
$G^{\prime}$, in which case $G^{\prime}$ is a cycle containing $s$ (case
(2.4b)). It may also be that $s$ does not have an incoming edge in
$G^{\prime}$, but some other vertex in $G^{\prime}$ has 2 incoming edges. In
this case, $G^{\prime}$ contains a cycle and a path connecting $s$ to the
cycle (case (2.4c)). This finishes the proof of the lemma. ∎
See 2.9
###### Proof.
Let $s=s_{t}$ be the source arrived at time $t$. Notice that adding $s$ and
its outgoing edges to $G$ does not decrease the cost of cheapest augmenting
path from $v$ in $G^{\mathbf{x}}$, since $s$ does not have incoming edges. Let
${\mathbf{x}}$ be the flow at the beginning of some iteration of the while
loop, at time $t$. Let $f$ be the cheapest augmenting path from $s$ in the
residual graph $G^{\mathbf{x}}$, as in Step 6. Let $\bar{\mathbf{x}}$ be the
${\mathbf{x}}$ obtained at the end of the iteration, i.e., after it is
augmented using $f$. Let $f^{1}$ and $f^{2}$ be the cheapest augmenting paths
from $v$ in the residual graph $G^{\mathbf{x}}$ and $G^{\bar{\mathbf{x}}}$
respectively. Let $C_{1}={\mathrm{cost}}(f^{1})$ and
$C_{2}={\mathrm{cost}}(f^{2})$. It suffices for us to prove $C_{2}\geq C_{1}$.
We first consider the case where $v$ is in the support graph $G^{\prime}$ of
the flow $f$, which falls in one of the three cases in Definition 2.4. Towards
contradiction we assume $C_{2}<C_{1}$. By Claim 2.6, we can find an augmenting
path $f^{\prime}$ from $v$ in $G_{x}$, whose support is a sub-graph of
$G^{\prime}$. As $f^{\prime}$ is an augmenting path from $v$ in $G_{x}$, we
have ${\mathrm{cost}}(f^{\prime})\geq{\mathrm{cost}}(f^{1})=C_{1}$ by the
choice of $f^{1}$.
Let $\theta>0$ be a sufficiently small real number. We define $f^{*}:=f-\theta
f^{\prime}+\theta f^{2}$, extending the domain of the three vectors by adding
0-coordinates if necessary. When $\theta$ is small enough, all entries in
$f^{*}$ are non-negative. So, the cost of $f^{*}$ is strictly smaller than
that of $f$ as ${\mathrm{cost}}(f^{\prime})\geq
C_{1}>C_{2}={\mathrm{cost}}(f^{2})$. Clearly, $f^{*}$ satisfies the balance
constraints: the net flow sent by $s$ in $f^{*}$ is 1, and the net flow sent
by any vertex in $V\setminus\\{s,\tau\\}$ is 0.
However, some edges in the domain for $f^{2}$ may be outside $E^{\mathbf{x}}$.
This issue can be handled as follows. Suppose some $e=vu$ with $f^{2}_{e}>0$
is not in $E^{\mathbf{x}}$. Then it must be the case that $uv\in
E^{\mathbf{x}}$ and $f_{uv}>0$. Then in $f^{*}$, we update
$f^{*}_{uv}\leftarrow f^{*}_{uv}-\theta f^{2}_{vu}/\gamma_{uv}$ and
$f^{*}_{vu}\leftarrow 0$, and discard the coordinate. When $\theta>0$ is
sufficiently small, we guarantee that all entries in $f^{*}$ are non-negative.
Moreover this update operation can only decrease the cost of $f^{*}$, and thus
we still have ${\mathrm{cost}}(f^{*})<{\mathrm{cost}}(f)$. As the updated
$f^{*}$ is a fractional augmenting path from $s$ in $G^{\mathbf{x}}$, this
leads to a contradiction to the choice of $f$.
It remains to consider the case where $v$ is not in the support graph
$G^{\prime}$ of $f$. We consider first vertex $v^{\prime}$ along the path
$f^{2}$ that is in $G^{\prime}$. (In case $f^{2}$ belongs to case (2.4b) or
(2.4c), the path can be obtained by starting from $v$ and following out-going
edges in the support of $f^{2}$.) Using Claim 2.6, we can break $f^{2}$ into a
flow path $\hat{f}$ sending $1$ units flow from $v$ to $v^{\prime}$, and a
scaled augmenting path $\hat{f}^{\prime}$ from $v^{\prime}$ in
$G^{\bar{\mathbf{x}}}$. We already proved that the cost of the cheapest
augmenting path from $v^{\prime}$ in $G^{\mathbf{x}}$ is at most that in
$G^{\bar{\mathbf{x}}}$. So, we can replace $\hat{f}^{\prime}$ with the
cheapest augmenting path from $v$ in $G^{\mathbf{x}}$, scaled by the same
factor. Notice that $\hat{f}$ is also a path in $G^{\mathbf{x}}$ as $f$ does
not involve any vertex before $v^{\prime}$. Therefore, we obtained an
augmenting path from $v$ in $G^{\mathbf{x}}$ with cost no larger than that of
$f^{2}$. Therefore, we have $C_{2}\geq C_{1}$. ∎
See 3.1
###### Proof.
Notice that a valid generalized flow for the instance at time $t$ corresponds
to a fractional solution ${\mathbf{x}}^{(t)}$ for jobs $[t]$, with makespan
$(1+\epsilon)T^{*}$. The recourse incurred by the fractional solutions at any
time is precisely 2 times the cost made by the online algorithm for the
generalized flow problem. Applying Theorem 2.1, the recourse by time $t$ can
be bounded by $O\left(\frac{1}{\epsilon}\right)$ times the cost of the offline
generalized flow instance at $t$, with capacities scaled by
$\frac{1}{1+\epsilon}$. As the capacities of incoming edges of $\tau$ are
$(1+\epsilon)T^{*}$, scaling by $\frac{1}{1+\epsilon}$ reverts the capacities
back to $T^{*}$. The offline generalized network flow instance at time $t$ has
a solution of cost $t$. So, the recourse incurred by the fractional solution
at time $t$ is at most $O\big{(}\frac{1}{\epsilon}\big{)}\cdot t$. ∎
See 3.3
###### Proof.
Let $\vec{H}$ be the residual graph of $H$ w.r.t $F$: $\vec{H}$ is a directed
graph over $L\cup R$, for every edge $uv\in E_{H}$, we have $uv\in\vec{H}$,
and for every $uv\in F$, we have $vu\in\vec{H}$. We say a vertex in $L$ is
free if it is unmatched in $F$. For every integer $d\in[0,D]$, define $L^{d}$
($R^{d}$ resp.) to be the set of vertices in $L$ ($R$, resp.) to which there
exists a path in $\vec{H}$ of length _at most_ $2d$ ($2d+1$, resp.) from a
free vertex. So, we have $L^{0}\subseteq L^{1}\subseteq
L^{2}\subseteq\cdots\subseteq L^{D}$ and $R^{0}\subseteq R^{1}\subseteq
R^{2}\subseteq\cdots\subseteq R^{D}$. Also notice that $R^{d}=N_{H}(L^{d})$
for every $d\in[0,D]$.
For every $d\in[0,D]$, we have $\alpha|L^{d}|\leq|R^{d}|$ by the condition of
the lemma. All vertices in $R^{D}$ are saturated by our assumption that there
are no augmenting paths of length at most $2D+1$. So for every $d\in[0,L-1]$,
we have $|R^{d}|\leq|L^{d+1}|$ as all vertices in $R^{d}$ are matched by
vertices in $L^{d+1}$.
Combining the two statements gives us $\alpha|L^{d}|\leq|L^{d+1}|$ for every
$d\in[0,L-1]$. Thus $|L^{D}|\geq\alpha^{L}|L^{0}|$, which contradicts the
definition of $D$ and that $|L^{0}|\geq 1,|L^{D}|\leq|L|$. ∎
## Appendix B Online Rounding of Fractional Solutions for Load Balancing:
$(2+O(\epsilon))$-Approximation
In this section, we describe the rounding algorithm that converts the online
fractional solutions
${\mathbf{x}}^{(1)},{\mathbf{x}}^{(2)},\cdots,{\mathbf{x}}^{(n)}$ into
integral ones, with a $(2+O(\epsilon))$-approximation ratio. This shall finish
the proof of Theorem 1.2.
To do this, we create an online bipartite matching instance $H=(L\uplus
R,E_{H})$ with vertex updates on the right side, as well as a fractional
matching ${\mathbf{y}}$ in $H$. $L$ corresponds to the jobs, and thus vertices
in $L$ arrive one by one. Due to the recourse on the fractional solutions, we
may also insert/delete vertices from $R$. Then we use the algorithm in Section
3.2 to maintain an integral matching in $H$ covering $L$, which naturally
leads to an assignment of jobs to machines. In the analysis, we show that the
assignments have small makespan, and the number of times we reassign jobs is
bounded.
For most part in this section, we fix a job $i\in M$, and describe how to
maintain the part of $H$ and the fractional matching ${\mathbf{y}}$
correspondent to the machine $i$. Many notations in the section depend on $i$,
but for simplicity we do not include $i$ in these notations.
So, we fix $i\in M$ and a time step $t$. Let ${\mathbf{x}}={\mathbf{x}}^{(t)}$
be the fractional solution at $t$. The part of $H$ and ${\mathbf{y}}$ for $i$
can be derived from a 2-level partition of an interval for the jobs. We sort
all the jobs $j\in N(i)$ in non-increasing order of $p_{ij}$ values: Let
$N(i)=\\{j_{1},j_{2},\cdots,j_{|N(i)|}\\}$ with $p_{ij_{1}}\geq
p_{ij_{2}}\geq\cdots\geq p_{ij_{|N(i)|}}$. We then map each job $j_{o}$ to a
_job interval_ $I_{o}$ of length $x_{ij_{o}}$: $j_{1}$ is mapped to
$I_{1}:=[0,x_{ij_{1}}]$, $j_{2}$ is mapped to
$I_{2}:=[x_{ij_{1}},x_{ij_{1}}+x_{ij_{2}}]$, $j_{3}$ is mapped to
$I_{3}:=[x_{ij_{1}}+x_{ij_{2}},x_{ij_{1}}+x_{ij_{2}}+x_{ij_{3}}]$, and so on.
Let $X=\sum_{j\in N(i)}x_{ij}$ be the fractional number of jobs assigned to
$i$; that is, the total length of all job intervals. When ${\mathbf{x}}$ gets
updated, $X$ and the job intervals also change accordingly.
Then we define the 2-level partition of the interval $[0,X]$. In the first
level, we partition $[0,X]$ into a set of intervals (we distinguish between a
normal interval and a job interval), that we call _buckets_. We guarantee that
all buckets have length between $\frac{1}{\epsilon}$ and $\frac{4}{\epsilon}$.
The only exception is that when $X<\frac{1}{\epsilon}$, in which case the
whole $[0,X]$ is just one bucket. In the second level, each bucket is
partitioned into intervals, that we call _segments_. All the segments except
the last one in a bucket have length between $1-3\epsilon$ and $1-\epsilon$.
The last bucket has length at most $1-\epsilon$.
We now describe how to maintain the 2-level partition of the interval $[0,X]$,
as the fractional solution ${\mathbf{x}}$ changes. Initially, $X=0$ and there
is one bucket and segment of $0$-length at $0$. We show how to handle two
operations: increasing some $x_{ij_{o}}$ value, and decreasing some
$x_{ij_{o}}$ value. The two operations are sufficient for our algorithm: when
a new job $j$ with $ij\in E$ arrives, we create a $0$-length job interval for
the job at the appropriate position, and then we increase $x_{ij}$ from $0$ to
the desired number. The changes to ${\mathbf{x}}$ can also be handled using
the two operations.
First, consider the case where we need to increase $x_{ij_{o}}$ for some $o$.
We find the first segment ${\mathsf{seg}}$ (from left to right) that
internally intersects the job interval $I_{o}$. We then increase $x_{ij_{o}}$
and the lengths of $I_{o}$ and ${\mathsf{seg}}$ continuously at the same rate.
As a result, the length of the bucket ${\mathsf{buc}}$ containing
${\mathsf{seg}}$ also increases. Also, the segments after ${\mathsf{seg}}$ and
the buckets after ${\mathsf{buc}}$ will be shifted to the right continuously.
We run the procedure until $x_{ij_{o}}$ is increased enough, or one of the
following events happens.
* •
The length of ${\mathsf{seg}}$ reaches $1-\epsilon$. In this case, we re-
divide the bucket ${\mathsf{buc}}$ into segments: all the segments except the
last segment in ${\mathsf{buc}}$ have length exactly $1-2\epsilon$, and the
last bucket have length at most $1-2\epsilon$.
* •
The length of ${\mathsf{buc}}$ reaches $\frac{4}{\epsilon}$. In this case, we
divide ${\mathsf{buc}}$ into two buckets of length $\frac{2}{\epsilon}$ each.
Then we re-divide each of the two buckets into segments of length
$1-2\epsilon$ as in the previous case.
If we have not increased $x_{ij_{o}}$ enough after handling the event, we
continue running the procedure.
Now suppose we need to decrease $x_{ij_{o}}$ by some amount. We find the first
segment ${\mathsf{seg}}$ that intersects the job interval $I_{o}$ internally.
Let ${\mathsf{buc}}$ be the bucket containing ${\mathsf{seg}}$. Similarly, we
decrease $x_{ij_{o}}$ and the lengths of $I_{o}$, ${\mathsf{seg}}$ and
${\mathsf{buc}}$ continuously at the same rate, until we decreased
$x_{ij_{o}}$ enough, or ${\mathsf{seg}}$ does not intersect internally with
$I_{o}$ anymore, or one of the following two events happen. As before, the
segments after ${\mathsf{seg}}$ and the buckets after ${\mathsf{buc}}$ will be
shifted to the left continuously.
* •
The length of ${\mathsf{seg}}$ drops to $1-3\epsilon$ and ${\mathsf{seg}}$ is
not the last segment of ${\mathsf{buc}}$. In this case, again we re-divide
${\mathsf{buc}}$ into segments of length $1-2\epsilon$ as before.
* •
The length of ${\mathsf{buc}}$ drops to $\frac{1}{\epsilon}$ and
${\mathsf{buc}}$ is not the only bucket. In this case, we merge
${\mathsf{buc}}$ with either its previous or the next bucket, depending on
which one exists. If the merged bucket has length at most
$\frac{3}{\epsilon}$, we then keep it. Otherwise, we divide the bucket into 2
equal-length buckets, each of which has length between $\frac{1.5}{\epsilon}$
and $\frac{2.5}{\epsilon}$. In any case, we divide each of the newly created
buckets into segments of length $1-2\epsilon$.
Again, if we have not decreased $x_{ij_{o}}$ enough after handling the event,
we continue the decreasing operation.
The 2-level partition of $X$ for $i$ determines the portion of the graph $H$
and the fractional matching vector ${\mathbf{y}}$. For every segment
${\mathsf{seg}}$, we have a vertex $v_{\mathsf{seg}}\in R$ for the segment.
For every job $j_{o}\in N(i)$ such that ${\mathsf{seg}}\cap
I_{o}\neq\varnothing$, we have $j_{o}v_{\mathsf{seg}}\in E_{H}$, and
$y_{j_{o}v_{\mathsf{seg}}}$ is the length of the intersection between
${\mathsf{seg}}$ and the job interval $I_{o}$. We highlight some technicality
here: All the intervals we defined are closed interval. So it is possible that
the intersection of ${\mathsf{seg}}$ and $I_{o}$ is only one point. In this
case we have $j_{o}v_{\mathsf{seg}}\in E_{H}$ and
$y_{j_{o}v_{\mathsf{seg}}}=0$. It is allowed for the bipartite matching
algorithm to match $j_{o}$ to $v_{\mathsf{seg}}$.
###### Claim B.1.
At any time $t$, for every $ij\in E$, we have $\sum_{v\text{ is segment for
}i}y_{jv}=x_{ij}$.
###### Proof.
This holds since all the segments form a partition of $[0,X]$, and the job
interval for $j$ is inside $[0,X]$. ∎
Therefore, if we consider the whole graph $H$, ${\mathbf{y}}$ defines a
fractional matching between $L=[t]$ and $R$ at time $t$: every $j\in L$ is
covered to an extent of $1$ and every $v\in R$ is matched by an extent of at
most $1-\epsilon$. This implies the following claim:
###### Claim B.2.
At any time, we have $|N_{H}(A)|\geq\frac{|A|}{1-\epsilon}$ for every
$A\subseteq L$.
We then finish the description of the rounding algorithm. We maintain the
dynamically-changing bipartite graph $H$, and use the bipartite-matching
algorithm described in Section 3.2 and Theorem 3.2 to maintain a bipartite
matching. If a job $j$ is assigned to a segment for machine $i$, we then
assign job $j$ to machine $i$ in the load-balancing instance.
### B.1 Analysis
Now we analyze the recourse and the competitive ratio of the algorithm
respectively.
#### Analysis of Recourse
First we analyze the recourse made by the rounding algorithm.
###### Lemma B.3.
For any $t\in[n]$, the number of vertex updates on segments for $i$ by any
time $t$ is at most
$O\left(\frac{1}{\epsilon^{2}}\right)\cdot\sum_{t^{\prime}=1}^{t}\sum_{j\in
N(i)}|x^{(t^{\prime})}_{ij}-x^{(t^{\prime}-1)}_{ij}|$.
###### Proof.
We fix the machine $i$ in this proof, and consider two different causes for
vertex updates for machine $i$.
First, vertex updates may happen when a new bucket is created. When a new
bucket is created, its length is between $\frac{1.5}{\epsilon}$ and
$\frac{3}{\epsilon}$: When a bucket is created due to an increasing operation,
then the new buckets have length $2/\epsilon$. Consider a decreasing
operation. The merged bucket will have length between $\frac{2}{\epsilon}$ and
$\frac{5}{\epsilon}$. If it has length at most $3\epsilon$, then no splitting
happens and its length is between $\frac{2}{\epsilon}$ and
$\frac{3}{\epsilon}$. Otherwise, the two new buckets will have length between
$\frac{1.5}{\epsilon}$ and $\frac{2.5}{\epsilon}$. So, it takes $0.5/\epsilon$
fractional recourse on $(x_{ij})_{j}$ to create a new bucket. Moreover, when a
new bucket is created, $O(1/\epsilon)$ segments will be inserted and deleted.
Therefore, creation of new buckets incur at most
$O(1)\cdot\sum_{t^{\prime}=1}^{t}\sum_{j\in
N(i)}|x^{(t^{\prime})}_{ij}-x^{(t^{\prime}-1)}_{ij}|$.
Then we consider vertex updates due to re-division of a bucket into segments.
Whenever a re-division happens, all segments have lengths exactly
$1-2\epsilon$ except for the last one, which has length at most $1-2\epsilon$.
The next re-division happens if the length of some segment increases to
$1-\epsilon$, or the length of some segment that is not the last one decreases
to $1-3\epsilon$. So, it takes $\epsilon$ fractional recourse on
$(x_{ij})_{j}$ for the next re-division to happen. When a re-division happens,
we make $O(1/\epsilon)$ vertex updates. Therefore, re-divisions incur at most
$O\big{(}\frac{1}{\epsilon^{2}}\big{)}\cdot\sum_{t^{\prime}=1}^{t}\sum_{j\in
N(i)}|x^{(t^{\prime})}_{ij}-x^{(t^{\prime}-1)}_{ij}|$ vertex updates.
Notice that these are the only case where a vertex update happens. We need to
mention that when the length of the intersection of ${\mathsf{seg}}$ and
$I_{o}$ decreases to $0$, no vertex update happens, since still there is an
edge between $j_{o}$ and $v_{\mathsf{seg}}$ in $H$. ∎
Now we can complete the analysis of the recourse of the online rounding
algorithm. Note that the total recourse made by the load balancing algorithm
is at most the number of reassignments made by the bipartite-matching
algorithm. By Theorem 3.2 and Claim B.2, the recourse of the latter by time
$t$ is at most $O\big{(}\frac{\log
n}{\epsilon}\big{)}\big{(}t+\\#(\text{vertex updates by }t)\big{)}$. By Lemma
B.3 and summing up the bounds over all machines $i$, the number of vertex
updates by time $t$ is at most
$O\big{(}\frac{1}{\epsilon^{2}}\big{)}\sum_{t^{\prime}=1}^{t}\left|{\mathbf{x}}^{(t^{\prime})}-{\mathbf{x}}^{(t^{\prime}-1)}\right|$,
which is at most $O\big{(}\frac{t}{\epsilon^{3}}\big{)}$ by Lemma 3.1.
Therefore the total recourse by time $t$ made by the load balancing algorithm
is at most $O\left(\frac{t\log n}{\epsilon^{4}}\right)$. Recall that the extra
$O\left(\frac{\log(1/\epsilon)}{\epsilon}\right)$-factor comes from making the
assumption that $T^{*}$ is known.
#### Analysis of Competitive Ratio
It remains for us to analyze the competitive ratio of the algorithm. Again we
fix a machine $i$ and prove the following:
###### Lemma B.4.
The load of machine $i$ at any time is at most $(2+O(\epsilon))T^{*}$.
###### Proof.
Fix a time $t$ and ${\mathbf{x}}={\mathbf{x}}^{(t)}$. We now use $v_{ik}$ to
denote the $k$-th segment for $i$ from left to right, over all the buckets.
Let $J_{ik}$ be the set of jobs in $N(i)$ whose job intervals internally
intersect the segment $v_{ik}$ at the time. We use $\pi_{jik}$ to indicate if
$j$ is assigned to $v_{ik}$ in the solution for bipartite matching instance at
time $t$.
Notice that each all segments except for last segments of buckets have length
between $1-3\epsilon$ and $1-\epsilon$. The last segment of a bucket has
length at most $1-\epsilon$. All buckets have length between $1/\epsilon$ and
$4/\epsilon$; the only exception is when there is only one bucket, whose
length might be smaller than $1/\epsilon$.
The total load $i$ receives at time $t$ is then
$\displaystyle\sum_{k}\sum_{j\in J_{ik}}\pi_{jik}p_{ij}$
$\displaystyle\leq\frac{1}{1-3\epsilon}\sum_{k}\sum_{j\in
J_{ik}}y_{j,v_{ik}}p_{ij}+T^{*}+\epsilon\cdot\sum_{k}\sum_{j\in
J_{ik}}y_{j,v_{ik}}p_{ij}=(1+O(\epsilon))\sum_{k}\sum_{j\in
J_{ik}}y_{j,v_{ik}}p_{ij}+T^{*}$ $\displaystyle=(1+O(\epsilon))\sum_{j\in
N(i)}x_{ij}p_{ij}+T^{*}\leq(1+O(\epsilon))T^{*}+T^{*}=(2+O(\epsilon))T^{*}.$
We need to elaborate more on the first inequality. For every $j\in J_{ik}$
with $\pi_{jik}=1$, we can try to upper bound $p_{ij}$ by
$\frac{1}{1-3\epsilon}\sum_{j^{\prime}\in
J_{i(k-1)}}y_{j^{\prime},v_{i(k-1)}}p_{ij^{\prime}}$. Notice that every
$j^{\prime}$ in the summation has $p_{ij^{\prime}}\geq p_{ij}$. If $k$ is not
the first segment of any bucket, then the upper bound holds as the total
${\mathbf{y}}$ value for the previous segment is at least $1-3\epsilon$. The
$p_{ij}$ for the first segment of the first bucket can be upper bounded by
$T^{*}$; the $p_{ij}$ for the first segment of other buckets can be bounded by
$\epsilon$ times the budget from the previous bucket, as the bucket has length
at least $1/\epsilon$. The last equality is by Claim B.1. The last inequality
comes from that the total fractional load on machine $i$ is at most
$(1+\epsilon)T^{*}$. ∎
## Appendix C Lower Bound on Recourse for Load Balancing in the Fully Dynamic
Model
In this section we consider the online load balancing problem with recourse in
the fully dynamic model. Again, we have a set $M$ of machines and a set $J$ of
$n$ jobs. Jobs can arrive and depart, and at any time, we are guaranteed that
there is a schedule of makespan at most $T^{*}$, for a given $T^{*}$. To
achieve an $\alpha$-competitive ratio, our algorithm needs to maintain a
solution of makespan at most $\alpha T^{*}$. An algorithm with an amortized
recourse of $\beta$ can make at most $\beta t$ reassignments for the first $t$
arrival/departure events, for any $t$. By making copies of jobs, we assume
every job arrives and departs exactly once; so there are $2n$ events in the
sequence. Our negative result holds even for the restricted assignment
setting, and when we only need to maintain a fractional schedule. Indeed, the
lower bound holds in the offline setting: the best algorithm that knows
everything upfront must incur a large recourse.
###### Theorem C.1.
Let $\alpha(n)=o(\log n)$ be a monotone non-decreasing function of $n$. There
is an instance for the above problem such that the following holds. Any
offline algorithm that maintains a fractional schedule of $\alpha(n)T^{*}$
needs to incur an amortized recourse of $n^{\Omega(1/\alpha(n))}$.
To see why an algorithm needs to incur a large recourse in the fully dynamic
model, consider the following simple instance. There are 2 machines, and
$n^{\prime}$ small jobs of size $1$ each arrive at the beginning of the
algorithm. Each of them can be assigned to the two machines. Big jobs have
size $n^{\prime}$ and they are 2 types of them: type-1 big jobs can only be
assigned to machine 1, and type-2 big jobs can only be assigned to machine 2.
Consider the following online updates: a type-1 big job arrives and departs, a
type-2 big job arrives and departs, then a type-1 big job arrives and departs,
and so on. Then at any time, there is a schedule of makespan $n^{\prime}$ for
active jobs. If the sequence is long enough, then any $(1.5-c)$-competitive
algorithm must incur an amortized recourse of $\Omega(cn^{\prime})$. The
constant 1.5 can be made arbitrarily close to 2, if we introduce more
machines. However, to go beyond 2, we need to use a recursive construction,
using the basic instance as a building block.
Since our algorithm only needs to maintain fractional schedules, we can make
the problem more general by allowing each job $j$ to have a reassignment cost
$c_{j}\in\mathbb{Z}_{\geq 0}$: Reassigning $x$ fraction of a job $j$ incurs a
recourse of $xc_{j}$. To make the reassignment costs uniform without changing
the instance, one can break a job of size $p_{j}$ with reassignment cost
$c_{j}$ into $c_{j}$ jobs of size $p/c_{j}$, each with reassignment cost $1$.
This will change the number of jobs in the instance, and we take care of the
issue at the end of this section. We compare the total recourse of the
algorithm against $\sum_{j\in J}c_{j}$. (Recall that every job arrives and
departs only once).
Now describe the instance with reassignment costs. Let $L\geq 1,P\geq 2^{L}$
be two integers. We construct a perfect binary tree $\mathbf{T}$ of with $L$
levels of edges; so there are $2^{L}$ leaves in the tree. The level of a
vertex is $L$ minus its distance to the root in the tree: The root has level
$L$ and the laves have level $0$. Let
$\mathbf{r},\mathbf{V},\mathbf{V}^{\circ}$ be the root, vertex set and leaf
set of the tree $\mathbf{T}$ respectively. For every
$v\in\mathbf{V}\setminus\mathbf{V}^{\circ}$, let ${\mathrm{left}}(v)$ and
${\mathrm{right}}(v)$ be the left and right child of $v$ respectively. For
every $v\in\mathbf{V}$, let $\ell(v)$ be the level of $v$ and
$\Lambda^{\circ}(v)$ be the set of leaves that are descendants of $v$.
There are 2 machines at each leaf and so there are in total $m:=2^{L+1}$ jobs.
There is a job $j_{v}$ of size $1$ for every $v\in\mathbf{V}$, and the job
$j_{v}$ can be assigned to any machine in $\Lambda^{\circ}(v)$. The
reassignment cost $c_{j_{v}}$ of the job $j_{v}$ is defined as
$\displaystyle c_{j_{v}}:=P^{\ell(v)}.$
To guarantee that every job arrives and departs once, we make copies of these
jobs so that each copy is used only once.
Algorithm 3 construct-instance$(v)$ //$v\in\mathbf{V}$
1:let $2^{\ell(v)}$ copies of the job $j_{v}$ arrive
2:if $\ell>0$ then
3: repeat $P$ times:
4: construct-instance$(\ell-1,{\mathrm{left}}(v))$
5: construct-instance$(\ell-1,{\mathrm{right}}(v))$
6:let the $2^{\ell}$ copies of the job $j_{v}$ created at step $1$ depart
The instance is constructed recursively by calling the procedure
$\text{construct-instance}(\mathbf{r})$, defined in Algorithm 3. After Step 1
of the procedure construct-instance$(v)$ for some leaf $u$, we have
$2^{L}+2^{L-1}+2^{L-2}+\cdots+2^{0}=2^{L+1}-1$ active jobs: For any ancestor
$v$ of $u$, we have $2^{\ell(v)}$ active copies of $j_{v}$. It is easy to see
that the active jobs can be scheduled on the machines so that each machine
takes at most 1 job: For every strict ancestor $v$ of $u$ at level $\ell$, we
schedule the $2^{\ell}$ copies of $j_{v}$ on $\Lambda^{\circ}(v^{\prime})$,
where $v^{\prime}$ is the child of $v$ that is not an ancestor of $u$; the 1
copy of $j_{u}$ is scheduled on one machine at $u$ (the other machine is
idle). Therefore, the optimum makespan at any time is at most $1$.
We show that any algorithm that maintains a fractional solution of makespan at
most $\frac{L+1}{3}$ need to incur a large recourse. First, we make our task
easier, and we lower bound the recourse needed for this easier task. In the
task, the algorithm is allowed to _remove_ fractions of jobs, where the
recourse for removing $x$ fraction of a job $j$ is $xc_{j}$. This means that a
job does not need to be fully scheduled. The only requirement needed is the
following: At the beginning of each iteration of Loop 3 in an execution of
construct-instance$(v)$, the $2^{\ell(v)}$ copies of $j_{v}$ created must be
brought back to machines (at $\Lambda^{\circ}(v)$). At the time, the algorithm
is allowed to distribute the $2^{\ell}$ jobs fractionally on the machines at
$\Lambda^{\circ}(v)$ in any manner, without incurring any recourse.
Clearly, the new task is not harder as any algorithm for the original task is
an algorithm for the new one with the same recourse. For the new task, the
following assumptions on the algorithm can be made.
* •
First, we only need the removing operations, not the moving operations. We can
change the operation of moving a faction of a job from machine $i$ to
$i^{\prime}$ to the operation of removing the fractional job from $i$. This
does not change the recourse of the operation, and can only decrease the loads
of machines. Also, bringing back jobs back later does not incur any recourse.
* •
Then, focus on one iteration of Loop 3 in an execution of construct-
instance$(v)$. We can assume all the removing operations for the $2^{\ell(v)}$
copies of $j_{v}$ in this operation are done immediately after they were
brought back to the schedule. Earlier removal can only decrease the machine
loads. As a result, we can assume the following: at the beginning of the
iteration, some fraction of the $2^{\ell(v)}$ jobs are put on the machines,
and the recourse incurred is $c_{j_{v}}$ times the fractional number of jobs
that are not put on the machines. No removing is allowed anymore during the
iteration.
* •
Finally, for an execution of construct-instance$(v)$, we can assume the
algorithm uses the same procedure for the $P$ iterations of Loop 3, as there
are identical.
Therefore, the algorithm for the modified task can be specified by one
fractional assignment $f_{v}:\Lambda^{\circ}(v)\to\mathbb{R}_{\geq 0}$ for
every vertex $v\in\mathbf{V}$: $f_{v}(u)$ the fractional number of copies of
$j_{v}$ we assign to the two machines on $u$ in an execution of construct-
instance$(v)$. It is guaranteed that
$\sum_{u\in\Lambda^{\circ}(v)}f_{v}(u)\leq 2^{\ell(v)}$.
Suppose at any time, the fractional schedule has makespan at most
$\frac{L+1}{3}$. Then we have
$\displaystyle\sum_{v\text{ ancestor of }u}f_{v}(u)\leq
2\times\frac{L+1}{3}=\frac{2(L+1)}{3},\forall u\in\mathbf{V}^{\circ}.$ (3)
Summing up (3) over all leaves $u$ gives
$\displaystyle\sum_{v\in\mathbf{V},u\in\Lambda^{\circ}(v)}f_{v}(u)\leq\frac{2(L+1)}{3}\cdot
2^{L}.$
The recourse of the algorithm is then
$\displaystyle\quad\sum_{v\in\mathbf{V}}P^{L-\ell(v)+1}\cdot
c_{j_{v}}\cdot\left(2^{\ell(v)}-\sum_{u\in\Lambda^{\circ}(v)}f_{v}(u)\right)=P^{L+1}\sum_{v\in\mathbf{V}}\left(2^{\ell(v)}-\sum_{u\in\Lambda^{\circ}(v)}f_{v}(u)\right)$
$\displaystyle\geq P^{L+1}\left((L+1)\cdot 2^{L}-\frac{2(L+1)}{3}\cdot
2^{L}\right)=\frac{P^{L+1}\cdot(L+1)\cdot 2^{L}}{3}.$
In the above sequence, $P^{L-\ell(v)+1}$ is the number of iterations of Loop 3
in all executions of construct-instance$(v)$, $c_{j_{v}}=P^{\ell(v)}$ is the
per-unit recourse for copies of job $j_{v}$, and
$\left(2^{\ell(v)}-\sum_{u\in\Lambda^{\circ}(v)}f_{v}(u)\right)$ is the
fractional number of copies of $j_{v}$ that we did not schedule in each
iteration of Loop 3.
The sum of $c_{j}$ over all arrived jobs is
$\displaystyle\sum_{v\in\mathbf{V}}\big{(}2^{\ell(v)}\cdot
P^{L-\ell(v)}\big{)}\cdot
P^{\ell(v)}=P^{L}\sum_{v\in\mathbf{V}}2^{\ell(v)}=P^{L}\cdot(L+1)\cdot 2^{L}.$
Above, $2^{\ell(v)}$ is the number of copies we create in each execution of
construct-instance$(v)$, $P^{L-\ell(v)}$ is the number of times we run
construct-instance$(v)$, and $P^{\ell(v)}=c_{j_{v}}$ is the recourse of a copy
of $j_{v}$.
Therefore, we proved that the total recourse is at least $\Omega(P)$ times
$\sum_{j}c_{j}$. Splitting jobs to make all reassignment costs equaling 1, we
obtain an instance with $n=(L+1)\cdot(2P)^{L}$ jobs, with the amortized
recourse $\Omega(P)$. Therefore, for any integer $\alpha$, we can set
$L=3\alpha-1$, and
$P={\left\lfloor\left({\frac{n}{L+1}}\right)^{1/L}/2\right\rfloor}=n^{\Omega(1/\alpha)}$.
This finishes the proof of Theorem C.1.
|
Van’t Hoff Institute for Molecular Sciences, University of Amsterdam,
Amsterdam 1098 XH, The Netherlands School of Physical Sciences, University of
Kent, Canterbury CT2 7NH, UK School of Electronic Engineering and Computer
Science, Queen Mary University of London, London E1 4NS, U.K.
# Energy transfer and restructuring in amorphous solid water upon consecutive
irradiation
Herma M. Cuppen Radboud University, Institute for Molecules and Materials,
Nijmegen 6525 AJ, The Netherlands<EMAIL_ADDRESS>Jennifer A. Noble
CNRS, Aix-Marseille Univ, PIIM, Marseille 13397, France jennifer.noble@univ-
amu.fr Stephane Coussan CNRS, Aix-Marseille Univ, PIIM, Marseille 13397,
France Britta Redlich FELIX Laboratory, Radboud University, Nijmegen 6525
ED, The Netherlands Sergio Ioppolo Center for Interstellar Catalysis,
Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120,
Aarhus C 8000, Denmark<EMAIL_ADDRESS>
###### Abstract
Interstellar and cometary ices play an important role in the formation of
planetary systems around young stars. Their main constituent is amorphous
solid water (ASW). Although ASW is widely studied, vibrational energy
dissipation and structural changes due to vibrational excitation are less well
understood. The hydrogen-bonding network is likely a crucial component in
this. Here we present experimental results on hydrogen-bonding changes in ASW
induced by the intense, nearly monochromatic mid-IR free-electron laser (FEL)
radiation of the FELIX-2 beamline at the HFML-FELIX facility at the Radboud
University in Nijmegen, the Netherlands. Structural changes in ASW are
monitored by reflection-absorption infrared spectroscopy and depend on the
irradiation history of the ice. The experiments show that FEL irradiation can
induce changes in the local neighborhood of the excited molecules due to
energy transfer. Molecular Dynamics simulations confirm this picture:
vibrationally excited molecules can reorient for a more optimal tetrahedral
surrounding without breaking existing hydrogen bonds. The vibrational energy
can transfer through the hydrogen-bonding network to water molecules that have
the same vibrational frequency. We hence expect a reduced energy dissipation
in amorphous material with respect to crystalline material due to the
inhomogeneity in vibrational frequencies as well as the presence of specific
hydrogen-bonding defect sites which can also hamper the energy transfer.
## 1 Introduction
Interstellar and cometary ices play an important role in the formation of
planetary systems around young stars, and hence these ices have received quite
a lot of attention in the astrochemical community. The main constituent of
interstellar ices is amorphous solid water (ASW)1, which is formed on dust
grains in dark molecular clouds from atomic and molecular oxygen reacting with
hydrogen atoms 2, 3, 4. ASW is porous when deposited at low temperatures and
pressures, but chemically formed ice is compact using the excess energy for
restructuring of the ice5. Also, the excess energy of other surface reactions,
the formation of H2 for instance, can impact the structure of the underlying
water surface 6. This means that the excess energy can be transferred to an
ice layer. Recent Molecular Dynamics simulations 7 showed that there is little
energy transfer between different types of excitation (translational,
vibrational, and rotational), but that vibrational excitation of a molecule on
the surface can efficiently dissipate to an ASW surface through the
admolecule-surface interaction. However, the efficiency of this process varies
largely from case to case. The hydrogen-bonding network of ASW is likely a
crucial component in this. The exact nature of the hydrogen bonding network in
amorphous ices is not fully understood. So far, most vibrational excitation
studies have focused on liquid water 8, 9, 10, 11, 12. In solid materials,
vibrational energy dissipation is generally investigated for crystalline
materials, often metals, and the energy transfer is treated by interaction
with a phonon bath13. It is, however, not clear to what extent this holds for
amorphous, molecular materials.
ASW is a metastable state of ice, and vibrational energy could, in principle,
lead to structural modification toward the stable crystalline structure. In
the present work, structural changes are identified by infrared (IR)
spectroscopy. As far as we are aware, only a handful of studies have focused
on low-energy IR irradiation of ASW,14, 15, 16, 17, 18 revealing wavelength-
dependent irreversible structural changes of these ices. The exact oscillator
frequencies of the O–H stretch of water molecules depend sensitively on the
specific surroundings and hydrogen bonding structure of the particular water
molecule. While this does not allow us to study long-range crystallization
effects, local restructuring towards a perfect surrounding of two hydrogen-
bond acceptors and two donors (DDAA) can be detected. We have used a similar
method in the past17. The absorption feature associated with a perfect DDAA
surrounding increased upon IR irradiation. Concurrently, a decrease in defect
sites with missing hydrogen bonds was observed. The exact changes in
absorption depend on the irradiation wavelength, but the effect was found for
irradiation at stretch, bending, and libration frequencies. Irradiation at
off-resonance frequencies did not result in observable changes. Classical
Molecular Dynamics simulations using an oscillating electric field to simulate
the IR irradiation could reproduce the effect. They showed that the changes
occur through local heating where classes of oscillators are excited.
The present paper aims to study the dissipation of vibrational energy and its
consequences for restructuring of ices in more detail. Vibrational excitation
can occur upon resonant irradiation in the IR and terahertz (THz) spectral
ranges, and upon reaction, in particular bond-formation reactions. Here we use
consecutive IR irradiation at different frequencies in the 3 $\mu$m O–H
stretch region to study history-dependent and wavelength-dependent effects.
Molecular Dynamics simulations supplement the experimental results. Two types
of simulations are performed: sequential irradiation of ASW and vibrational
excitation of individual molecules. Energy transfer is analyzed in terms of
molecular vibrations, due to the amorphous and molecular nature of ASW.
## 2 Experimental and computational methods
### 2.1 Experiments
Experiments were performed in the ultrahigh vacuum (UHV) Laboratory Ice
Surface Astrophysics (LISA) end station at the HFML-FELIX facility, Radboud
University in the Netherlands. The version of the LISA setup used in this work
was described in Noble et al. 17, i.e. a prior version of the setup compared
to the most recent setup as presented in Ioppolo et al. 19. Briefly, the LISA
setup has been designed and optimized to perform selective IR/THz irradiation
of space-relevant molecules in the solid phase when coupled to the free-
electron lasers (FELs) FELIX-1 ($\sim$30-150 $\mu$m) and FELIX-2 ($\sim$3-45
$\mu$m). At the center of the main chamber, a custom-made
30$\times$30$\times$50 mm (l$\times$w$\times$h) oxygen-free high thermal
conductivity (OFHC) copper block substrate with four optically flat gold
plated faces is in thermal contact with a closed-cycle helium cryostat system.
The substrate temperature is controlled in the range of 15-300 K using a
Kapton tape heater connected to the OFHC copper block and regulated with a
temperature controller capable of reading temperatures through an uncalibrated
silicon diode fixed at the bottom of the substrate. The OFHC copper block can
be manipulated in the $z$ and $\theta$ directions through a $z$-translator
with a stroke of 50.8 mm and a rotary platform, respectively, allowing the
exposure of its all four faces to the FEL beam at numerous different spots
(i.e. a minimum of 6 unprocessed spots per block face).
For all experiments described here, deionized water was purified via multiple
freeze-pump-thaw cycles and dosed onto the gold-coated copper substrate by
background deposition through an all-metal leak valve connected to a 6 mm tube
that faces one of the walls of the main chamber. Two ice morphologies were
studied, namely, porous ASW (pASW) and compact ASW (cASW). Porous ASW samples
were prepared in the main chamber with a base pressure better than
8$\times$10-9 mbar and a base temperature of 16.5 K. Porous ASW was deposited
via background deposition for 370 seconds at 1.1$\times$10-6 mbar. A thickness
of $\sim$0.25 $\mu$m for pASW was chosen to ensure that photons fully
penetrated the ices, while the ice had a high enough IR signal-to-noise in
absorbance to monitor subtle structural modifications via FTIR spectroscopy.
Compact ASW samples were prepared with the substrate at 105 K and water
deposited by background deposition at a pressure of 1.0$\times$10-6 mbar for
480 seconds. Compact ASW was then cooled to 20 K before exposure to FEL
radiation. During deposition, FEL irradiation, and temperature-programmed
desorption (TPD) experiments, ices were monitored by means of Fourier
transform infrared (FTIR) spectroscopy (4000-600 cm-1, 2.5-16.6 $\mu$m) at a
grazing angle of 18∘ with respect to the surface with a spot size of $\sim$3
mm in height (diameter) and at a spectral resolution of 0.5 cm-1. The
reference spectrum was measured with 512 scans, while the experimental spectra
were measured with 128 or 256 accumulated scans.
Ices were then irradiated using the FELIX-2 IRFEL source (_i.e._ macropulses
with a duration of about 8 ms at 5 Hz repetition rate and a micropulse spacing
of 1 ns with a laser energy between 5-20 mJ) at frequencies in the mid-IR
(2.7-3.25 $\mu$m). All IRFEL irradiations were carried out for 5 minutes to
ensure complete saturation of any structural change in the ice layers. At all
wavelengths, the laser fluence at the sample was approximately $\sim$ 0.2
J/cm2. The spectral FWHM of the FELIX beam is on the order of 0.8 %
$\delta\lambda/\lambda$ for all wavelengths. The FEL beam impinges the gold-
plated flat substrate at an angle of 54∘ with respect to the surface with a
spot size of $\sim$2 mm in height (diameter) that fully overlaps with the FTIR
beam. Since the FTIR beam was larger than the FEL beam, part of the ice probed
by the FTIR was not exposed to FEL irradiation. Hence, FTIR difference spectra
acquired before and after FEL irradiation were investigated to highlight
changes in the ice. In this paper, we discuss FEL irradiations in terms of
wavelength and FTIR spectra in wavenumbers to reflect the higher spectral
resolution in the FTIR data as opposed to the transform-limited bandwidth of
the FEL radiation. “Fresh”, unirradiated ice spots were exposed to single FEL
irradiations between 2.7 and 3.25 $\mu$m. The possibility of adjusting the
sample height allowed us to start new irradiation series on other unirradiated
ice spots obtained during the same single ice deposition. Results from FEL
irradiations on “fresh” spots were compared to a series of irradiations at the
same ice spot carried out from “high” to “low” and from “low” to “high”
wavenumbers (hereafter referred to as “blue to red” frequencies (from 2.7 to
3.3 $\mu$m) and “red to blue” frequencies (from 3.3 to 2.7 $\mu$m),
respectively) across the water OH stretching mode. Detailed experimental
settings for all irradiation experiments can be found in the Supplementary
Information.
### 2.2 Simulations
Classical Molecular Dynamics (MD) simulations were performed using the LAMMPS
package (version 7/08/18).20 Water molecules were treated flexibly using the
TIP4P/2005f potential.21 To confidently prove structural changes in the ice,
either large samples are needed or many trajectories of small samples.
TIP4P/2005f gives a good trade-off between computational cost and
reproducibility of the experimental spectra. Polarizable flexible potentials
would be more accurate in describing vibrations, since they also take the
many-body effects on the vibration into account as well as the changing dipole
with the vibration. Lambros et al showed in a comparison study that fq-MB-pol
and MB-pol 22 are particularly good in this respect23.
Two ASW samples of 2880 molecules were used: one mimicking porous ASW and one
compact ASW. Both were obtained by quenching a water sample to 10 K after a
simulation in the canonical ensemble ($NVT$) at 400 K for 50 ps. The non-
porous sample has a cubic simulation box with a length of 45.07 Å, resulting
in an average density of 0.94 g cm-3, in agreement with the experimental
density of ASW. The porous sample has a box of 48 Å and an average density of
0.78 g cm-3. In the latter case, the quenching simulation resulted in a sample
with a large pore, effectively creating both surface and bulk in one
simulation box. The local density is rather similar in both cases. This is the
unannealed cASW sample. The compact ice sample was further annealed by heating
it to 70 K during 50 ps to create an annealed cASW sample. We think that the
latter is more representative of the experimental cASW ice which is formed
through deposition at higher temperatures.
Irradiation was simulated by employing an oscillating electric field of the
desired frequency along the $z$-direction, with a maximum amplitude of 15
mV/Å. All simulations were done in the $NVT$ ensemble where only 16 molecules
out of 2880 were thermostated. The oscillating electric field was switched on
for 2 ps and switched off again for 18 ps. This procedure was repeated 10
times. Properties were then calculated during another 20 ps while the full
structure was thermostated. The whole procedure hence lasted 220 ps and was
then repeated at a different wavelength. The $10\times 20$ ps sequence aims to
mimic the micropulses of the FEL irradiation. One should realize that the
micropulse interval of FELIX is much longer (1 ns) than the 18 ps intervals
with an electric field in the simulation.
However, the cooling rate of the thermostat, even when applied to only 16
molecules, is much higher than the experimental cryostat and the overall
energy that is taken from the system during the light-off interval is higher
in the simulation than in the experiments (see Ref.17 for more details). The
16 molecules are randomly distributed across the ice. A setup where the
thermostated molecules are located together might be a more realistic
represention of the experimental setup where only the bottom of the ice layer
is connected to a thermostat. Simulations with such a setup resulted in
inhomogeneous results with local hot spots far away from the thermostated
region. However, this is in part due to the short time interval between pulse
in the simulations (18 ps) whereas the experimental interval is 1 ns. The
random distribution is hence a compromise and together with the high cooling
rate this likely leads to a lower limit of the effect in the simulations.
VMD24 was used for visualization of all trajectories and bond-length
calculations to determine the oscillation wavelength of individual O–H
stretches.
### 2.3 Analysis in terms of hydrogen-bonding structure
Spectra were fitted by a combination of eight Gaussian functions (G1 – G8) to
aid in the interpretation of the spectral changes observed in the experiments.
The procedure, based on an in-house python script, has been previously
described in Ref. 17. The different Gaussians account for the contribution of
different oscillator families to the O-H stretch ice feature. A combination of
five known oscillator modes in the bulk of the ice spectrum (between $\sim$
3050–3450 cm-1) plus three surface-specific modes (two dangling-H modes at
3720 and 3698 cm-1, one dangling oxygen mode at 3549 cm-1 and the tetra-
coordinated surface s4 mode at 3503 cm-1) were identified from literature
data25, 26, 27, 28, 14, 29. For all experimental difference spectra, the same
combination of these eight Gaussian functions (G1 – G8) was fitted to each
spectrum, with identical constraints placed on peak position and full width
half maximum (FWHM). Most oscillator classes have also been classified in
terms of the local hydrogen bonding structure. The attribution of the
Gaussians to the different environments in terms of the hydrogen bonding
acceptors (A) and donors (D) is as follows: DA, G1; DAA, G2; DDA, G3; DDAA
G6+G7. The identity of the other oscillator families has not yet been
unequivocally determined in the literature.
Similar hydrogen bonding information was obtained from the simulation
trajectories by using an in-house python script. This script determined the
hydrogen bonding structure for each water molecule in terms of donor and
acceptor, taking into account the periodic boundary conditions. A radical
cutoff of 3.5 Å and a radial cutoff of 30 ∘ was applied24. Averages and
standard deviations of the total number of hydrogen bonding structures were
obtained by averaging during 20 ps while the system is fully thermostated.
## 3 Results & Discussion
### 3.1 Irradiation of pristine ice
Before studying the effect of successive irradiation, the effect of
irradiation on pristine ice is shown. Individual pASW samples were irradiated
at 2.7, 3.1, and 3.25 $\mu$m. Panel a of Fig. 1 shows the spectrum before
irradiation in blue and the difference spectrum after irradiation at 3.1
$\mu$m in red. The blue spectrum is scaled by a factor of 0.2 to ensure that
both spectra can be plotted on the same scale. The difference spectrum shows
both an increase and a decrease in absorption intensity. The spectra were
analyzed in terms of hydrogen bonding structures. Figure 1b shows the relative
change in hydrogen bonding structures with respect to the non-irradiated
spectrum as a function of irradiation wavelength. The error due to constant
low-level residual water deposition in the chamber was found to be 2% on the
DA oscillator, $<$2% on DAA, and $<$1% on the other bands. This is hence
within the size of the symbols of Fig. 1b. In all cases, there is an increase
in DDAA oscillators and a decrease in the DA, DAA, and DDA oscillators. For
the thin ices used in these experiments, individual irradiations are
reproducible over the 3-micron band in the sense that reorganization dominates
over desorption in all cases, and an increase in DDAA is observed at the
expense of the other oscillator classes.
The restructuring effect is largest for 3.1 $\mu$m, which is resonant with
oscillators in the bulk of the ice. Irradiation at 3.25 and especially at 2.7
$\mu$m are resonant with surface oscillators. The overall changes are smaller
at these wavelengths, and changes in the DA oscillators, which are mostly
located at the surface, become more important.
Figure 1: Individual experimental irradiations of pASW samples. (a) Pre-
irradiated spectrum in blue and difference spectrum after individual
irradiation at 3.1 $\mu$m in red. The original spectrum has been scaled down
by a factor of 5 for easier comparison. (b) Difference in oscillator
absorption after individual irradiations in the O-H stretch band, normalized
to pre-irradiation ices.
### 3.2 Sequential irradiation in the 3 $\mu$m band
A porous ASW sample was prepared at 16.5 K and subsequently irradiated for 5
minutes at a wavelength of 2.7 $\mu$m after which a new spectrum was recorded.
The resulting difference spectrum can be observed in Fig. 2 in cyan. A very
small increase in absorption intensity can be observed at 3200 cm-1 and a
tentative decrease around 3500 cm-1. The procedure was repeated at the same
spot for irradiation at 2.8 $\mu$m. The difference spectrum with respect to
previous irradiation can be seen in light green in Fig. 2. A larger difference
can be observed and the decrease in the spectrum appears to occur at slightly
lower wavenumbers. Again the spectra are fitted with eight Gaussians
representing the different oscillator classes. The relative change in
absorption after each irradiation is plotted in Figure 3a. It can be observed
that the decrease in spectral intensity for 2.7 $\mu$m is mainly due to a
decrease of the DA feature, whereas at 2.8 $\mu$m the DAA and DDA features are
mainly responsible for the change. The DDA feature grows in importance during
subsequent irradiation at 2.9, 3.0, and 3.1 $\mu$m. For irradiation at 3.2 and
3.25 $\mu$m, saturation can be observed. As shown in Fig. 1, irradiation of
pristine ice at 3.25 $\mu$m results in much larger spectral changes than what
can be observed in Figs. 2 and 3a for pre-irradiated ice. We think that the
oscillators that normally change upon irradiation at this wavelength in
pristine ice have already realigned themselves during previous irradiations
and are hence no longer available for further restructuring.
Figure 2: Experimental spectra of pASW upon sequential irradiation in the 3
$\mu$m band. Pre-irradiated spectrum in dark blue (scaled down by a factor of
5) and the difference spectrum after sequential irradiation from 2.7 to 3.25
$\mu$m (blue-to-red series) in black. The individual contributions of each
irradiation to the total difference spectrum are given in the color series.
The spectra are shifted for better visibility.
Panel b in Fig. 3 again shows the relative change in oscillator absorption
upon sequential irradiation, but now the irradiation order is reversed. Here,
changes can be observed at 3.25 $\mu$m, since the ice is now pristine when
irradiated at this wavelength, while saturation occurs at the lower
wavelengths. The overall change after irradiation at all wavelengths is the
same. This indicates that there is a maximum number of molecules that can
restructure.
The bottom two panels, c, and d, of Fig. 3 show the results of a similar
irradiation strategy starting from a compact ASW sample. The 3 $\mu$m band for
compact ASW is more narrow than for porous ASW, especially in the blue wing,
which contains the surface features DA and DAA. Hence, only the changes in DDA
and DDAA are shown in panels c and d since the DA and DAA features are too
small in compact ASW to obtain meaningful results. This is understandable
since both hydrogen bonding patterns are associated with surface structures
and the total surface area is substantially smaller than for porous ASW
because of the absence of pores.
Again, blue-to-red and red-to-blue irradiation lead to the same overall
changes at the end of the irradiation sequences. The relative changes here
are, however, much smaller than for porous amorphous solid water. Part of this
might be due to the reduced surface area, but the main reason is likely that
the ice is grown at elevated temperature, which means that the ice has already
been annealed to some extent and that the restructuring events with a low
barrier have already occurred.
The saturation effect that is observed for all four sequences suggests that
not only the directly excited molecules are involved in the changes in the
ice. If this were the case, changes would be uniquely linked to a specific
wavelength, whereas the results show that several different irradiation
wavelengths can lead to the same changes. Dissipation of the vibrational
excitation to the local environment is likely important. This dissipation can
lead to local heating, inducing structural modifications in the ice. The fact
that not all oscillators change simultaneously and different effects for
surface and bulk can be observed, suggests that the heating occurs locally and
the energy is not dissipated homogeneously throughout the sample.
Figure 3: Relative change in Gaussian height for different hydrogen-bonding
features after FEL sequential irradiation obtained in experiments. These are
obtained by irradiating samples of (a+b) porous ASW and (c+d) compact ASW.
Sequential exposure to FELIX irradiation is either (a+c) blue-to-red or (b+d)
red-to-blue, as indicated by the arrow. The errors due to constant background
water deposition are within the size of the symbols.
### 3.3 Simulations of sequential irradiation
Molecular Dynamics simulations were performed to study the changing ice on a
molecular level and study the role of energy dissipation. Six simulations were
performed following the procedure described above. In each case, seven
irradiation events at different wavelengths of the electric field have been
simulated, each consisting of ten irradiation and subsequent cooling cycles.
In three of the six simulations, the wavelength of the electric field
increased throughout the seven irradiation events and the other two had a
decreasing wavelength. Figure 4 shows the relative changes in hydrogen bonding
motives as a function of the irradiation wavelength. Panels a and c are for
blue-to-red irradiation, and panels b, and d are for red-to-blue irradiation.
The porous sample is used for panels a and b, the annealed cASW sample for c
and d. Results for the unannealed cASW are not shown, but fall between both
results. Significant changes only occur upon irradiation between 2.9 $\mu$m
and 3.2 $\mu$m, whereas experimentally the wavelength range in which changes
occur is much larger. This is probably due to two reasons: the simulated
spectrum of the porous sample shows that the O–H stretch band is narrower than
the experimental O–H stretch band, which limits the wavelength range in which
adsorption can occur and, secondly, the irradiation wavelength during the
simulation is a single value whereas in the experiment the laser has a FWHM of
$\sim$0.02 $\mu$m. The simulated absorption spectrum is most likely too narrow
due to missing quantum effects and polarizability terms in the interaction
potential.
Several of the trends that are observed in the experimental results are
reproduced by the simulations. In both cases, the DDAA motifs increase at the
expense of the defect sites and the initial changes are large after which
saturation sets in. Again, the changes after sequential irradiation are
similar, irrespective of the irradiation order. Comparing the upper with the
lower panels, we can further observe that the changes are smaller in cASW than
in pASW. As mentioned earlier, simulations using the cASW without additional
heating to 70 K give a result intermediate between the annealed cASW and pASW
results plotted in Fig. 4. It shows that the quantitative difference in the
results between cASW and pASW is, at least in part, due to the elevated
temperature at which the sample is prepared, which means that there are fewer
molecules available that can easily restructure, _i.e._ , have a low barrier
for restructuring. During irradiation, temperatures as high as 175 K can be
reached over a short time.
The largest discrepancy between the simulated and experimental results is in
the quantitative agreement. The changes in the experimental porous ASW are
much larger than for the simulated sample. This could be due to a difference
in time scales. During irradiation – whether experimental or simulated – the
ice can be heated, depending on the number of resonant water molecules at the
irradiation wavelength. Between irradiations, the ice cools through the
cryostat in the experiments or the thermostat in the simulations. The latter
is a much faster process than the cryostat cooling in the experiments, and
hence the ice will be at elevated temperatures for much longer times in the
experiments, potentially leading to more restructuring events. As was
discussed in Ref. 17, there is indeed experimental evidence for local heating,
although the precise temperatures are hard to constrain. Experiments with more
volatile species indicate that the effect is relatively moderate since it does
not result in spot desorption.
A second argument for the limited quantitative agreement between simulations
and experiments can again be the fact that the irradiation in the simulation
occurs only at a single frequency, which is likely resonant with fewer water
molecules per volume than in the experiments. Finally, the fraction of
molecules that can rearrange with a low barrier is likely different in the
simulation ice samples w.r.t. the experimental ice since they are obtained in
very different ways (hyperquenching versus deposition). Experimental results
of single irradiations show that the results are rather sensitive to the exact
thermal relaxation history of the ice. However, the relative changes between
the oscillators –DA/DDAA, DDA/DDAA, etc.– remain the same, although the
overall relative changes –DA/DA0, DDA/DDA0, etc.– can be smaller and closer to
the simulation results.
Figure 4: Relative change in the occurrence of different hydrogen bonding
patterns after sequential irradiation in simulations. These are obtained by
simulation of (a+b) pASW and (c+d) annealed cASW. Sequential exposure to the
electric field is either (a+c) blue-to-red or (b+d) red-to-blue. Non-porous
ASW has too low a number of DA hydrogen bonds to obtain reliable results.
Figure 5 shows the oxygen-oxygen pair distribution function obtained after
each sequential irradiation simulation. The distribution function remains
largely unaffected throughout the irradiation. Only minor changes around 4.5 Å
can be observed. X-ray diffraction experiments of ASW show that this peak is
indeed the first to change upon heating30. This peak was found to narrow –
while its intensity increases – upon heating. On the onset of crystallization
(above 130 K), much stronger changes can be observed: an extra shoulder
appears around 5 Å, and the pair distribution function becomes much more
structured, also at long range which is rather flat at 130 K and below.
Neither effect is present in the simulated $g_{\text{OO}}(r)$ of Fig. 5, which
means that all restructuring effects remain within the first coordination
shell and are not long-range.
Figure 5: The simulated oxygen-oxygen pair distribution function after
sequential irradiation compared to the initial $g_{\text{OO}}(r)$. A porous
ASW sample is sequentially exposed to the electric field from blue-to-red.
Figure 6 shows the molecules that have changed their hydrogen bonding
structure upon irradiation at different irradiation wavelengths. In blue are
the unaffected water molecules and in red are the water molecules that have
gained a hydrogen bond. The pore is visible at the center of the simulation
box. Panels (a) and (b) are the combined effects of irradiation at 2.7, 2.8,
and 2.9 $\mu$m; panels (g) and (h) for 3.2 and 3.3 $\mu$m. The panels (a),
(c), (e), and (g) are for blue-to-red irradiation; the other panels for red-
to-blue. Their irradiation order is hence (h), (f), (d), and (b). It is
clearly visible that the irradiation sequence matters in terms of which
molecules are affected at a given wavelength. Some molecules can easily change
their hydrogen bonding configuration via a small rotation or translation.
These rearrangements have only small barriers and can be facilitated by the
excitation of one of the two O-H bonds, either through resonant irradiation or
through dissipation of vibrational energy of neighboring water molecules. The
latter will be discussed in the next section. Most restructuring that is
observed is limited to these small local reorientations. Once such a small
structural change has occurred, these molecules are no longer available for
further restructuring. Since these rearrangements can be triggered by
different excitations, at different wavelengths, the irradiation order
determines which defect sites can still restructure at a given wavelength.
Figure 6c shows that many different molecules have changed hydrogen bonding
structure at 3.0 $\mu$m. These are no longer available for changes when the
ice is irradiated at 3.1 $\mu$m (Fig. 6e) where fewer molecules are affected.
For red-to-blue irradiation, it is the other way around and more molecules are
affected at 3.1 $\mu$m (Fig. 6f) than at 3.0 $\mu$m (Fig. 6d), because of the
reversed irradiation order. Once all low-energy restructuring events have
occurred, saturation is reached. This confirms our earlier conclusions on the
role of local heating based on the saturation observed experimentally.
The molecules that are affected at 3.0 $\mu$m and 3.1 $\mu$m are both bulk and
surface molecules. Excitation at 2.7-2.9 and 3.2-3.3 $\mu$m affects mostly
surface molecules. This is logical, since adsorption features associated with
surface oscillators are located at these wavelengths. Again, similar to the
bulk excitation, molecules can restructure upon irradiation at both
wavelengths, depending on the irradiation order. The molecules affected in
panel (a) are very similar to those in panel (h).
2.7-2.9 (a) (b)
3.0-3.3 (c) (d)
3.1-3.3 (e) (f)
3.2-3.3 (g) (h)
Figure 6: Molecules that gain in hydrogen bonding classification upon
irradiation for different wavelengths. Panels (a,c,e,g) are for blue-to-red
irradiation; (b,d,f,h) for red-to-blue irradiation. Panels (a) and (b) show
the total effect for irradiation at 2.7, 2.8, and 2.9 $\mu$m. Likewise panels
(g) and (h) for 3.2 and 3.3 $\mu$m. Panels (c+e) and (d+f) are for 3.0 and 3.1
$\mu$m, respectively.
### 3.4 Dissipation of energy
IR irradiation excites molecules vibrationally, which ultimately leads to
heating of the ice. The experiments suggest that the energy will remain local.
In this section, we will follow the energy transfer in the ice by MD
simulation to study how fast and how far the energy is dissipated. In this
case, the full ice is not exposed to the electric field but rather only a
single molecule. The dissipation is then followed by monitoring the internal
kinetic energy of this molecule and its surrounding molecules as a function of
time. The internal kinetic energy is used as a measure of the vibrational
excitation of the individual molecules.
This is done at three different locations in the ice: at a bulk location,
close to the surface of the pore, and at a location with a high defect
density. Hereafter, we present one example from each of the three excitation
locations. Figure 7 shows the excitation of a bulk molecule. The excited
molecule is molecule 1 in panel a and is indicated in dark blue in both
panels. Molecules 2 to 12 are the eleven molecules closed to molecule 1 and
are ordered in center-of-mass distance to molecule 1. Only molecule 1 is
exposed to a chirped electric field pulse of 5 ps in which the wavelength
increases from 2.8 to 3.1 $\mu$m. This is to ensure that a resonant wavelength
for at least one of the bonds is reached and that the molecule is indeed
excited during the pulse. Analysis of the time-dependent O–H bond length in
this molecule shows that vibration wavelengths of the two bonds in molecule 1
are 2.93 and 3.07 $\mu$m (see Table 1). The latter becomes predominantly
excited upon exposure to the electric field.
Figure 7a shows the internal kinetic energy of molecule 1 as a function of
time. Without exposure to irradiation, we expect some small fluctuations to be
observable as can be seen at long timescales and in a few cases within the
first 2 ps. A clear peak in kinetic energy can be observed around 3 ps. This
excitation quickly disappears when the energy is dissipated. Since the
resonant frequency only occurred for a very brief time, there is nothing to
sustain the excitation. Panel a further shows that molecule 2 is quickly
excited as well, reaching an even higher peak value. This molecule is
indicated in panel b in a lighter blue color and is directly linked to
molecule 1 by a hydrogen bond. The other hydrogen bond acceptor of molecule 1,
molecule 5 in cyan, does not get excited. Nor do the two hydrogen bond donors,
molecules 3 and 4; not explicitly colored in panel b. Analysis of the pre-
irradiation frequencies of molecules 2 to 5 in Table 1 shows that only
molecule 2 has an oscillation wavelength close to 3.07 $\mu$m. The other three
molecules do not have O–H bonds resonant with the excited bond of molecule 1
and therefore only molecule 2 becomes excited.
Molecule 2 transfers excitation to molecule 12 in red. Again, molecule 12 has
an oscillation frequency resonant with 3.07 $\mu$m. This shows that having the
correct resonant frequency is much more important for an efficient vibrational
energy transfer than the relative orientation or hydrogen bonding structure.
In this case, the excitation-donating molecule (molecule 2) is a hydrogen-
bonding acceptor for molecule 12 and not a donor, as in the first transfer. We
observed only minor energy transfer to hydrogen bonding acceptors of molecule
2 (not shown in Fig. 7). Judging from the time evolution, energy is then
transferred from molecule 12 to molecule 7. These, again, have a hydrogen
bonding acceptor relationship. In this case, the receiving molecule is
resonant with the oscillation of the other O–H bond of the donating molecule
(3.16 $\mu$m). This indicates that a resonance match is not required for
excitation within the molecule. Indeed, the internal vibrational transfer was
found to be fast among all bonds 7.
For the excitation series of molecules 1, 2, 12, and 7, a delay in excitation
of roughly 0.3 ps can be observed. After molecule 7, dissipation appears to
stop, which is surprising since molecule 5 is resonant with molecule 7. No
bond excitation is, however, observed for this molecule. Molecule 5 is a DDAAA
defect site connected to molecules 1, 6, 7, 8, and 26. Only molecule 7 is
resonant.
In conclusion, molecules can transfer their energy to surrounding molecules
which are connected through a hydrogen-bonding network and the excitation can
persist for up to 10 ps (see molecules 7 and 12), but can also be spread over
a much shorter timescale, as can be seen in molecules 1 and 2. Energy appears
to be transferred to molecules that are hydrogen bonded to the excited
molecules and that have similar vibrational frequencies. This occurs on a 0.3
ps timescale. Defect sites might hamper dissipation.
The simulations here follow classical dynamics and show the relevance of
resonance for energy transfer. Quantum dynamics likely make this criterion
even more strict. Panman et al.31 studied resonant vibrational energy transfer
through dipolar coupling as a function of distance and angular orientation.
They showed for liquid ethanol and $N$-methylacetamide that the transfer was
most likely under angles that coincide with hydrogen bonding geometries for
the specific molecules.
Table 1: Wavelength of the oscillation frequencies of O–H bonds in Fig. 7. Molecule | $\nu_{1}$ ($\mu$m) | $\nu_{2}$ ($\mu$m) | site
---|---|---|---
1 | 2.93 | 3.07 | DDAA
2 | 3.05 | 2.99 | DDAA
3 | 3.02 | 2.98 | DDAA
4 | 3.00 | 2.98 | DDAA
5 | 3.11 | 3.15 | DDAAA
6 | 3.04 | 2.97 | DDAA
7 | 3.16 | 2.95 | DDAA
8 | 3.11 | 3.07 | DDAA
12 | 3.07 | 3.16 | DDAA
26 | 3.02 | 3.02 | DDAA
(a) (b)
Figure 7: (a) The simulated internal kinetic energy as a function of time for
different molecules. The curves are offset for better visibility. Molecule 1
is excited as explained in the main text and is in the bulk of the ice. The
remainder of the molecules is ordered in center-of-mass distance to molecule
1. Molecules discussed in the text (1, 2, 5, 7, and 12) are indicated in panel
(b), following the color coding of (a).
Figure 8 shows the vibrational excitation of another molecule; this time a
molecule at the edge of the pore. Panel (b) shows the local geometry, where we
look perpendicular to the pore surface. Molecule 1 has two hydrogen bond
acceptors, molecules 2 and 5, and two donors, molecules 3 and 4. Molecules 1,
3, 4, and 5 are all surface molecules. Panel (a) shows, again, the internal
kinetic energy, which looks very different from Fig. 7a. In Fig. 8a, many more
molecules are involved in the dissipation of the energy, but at much lower
energies as compared to the previous example. Table 2 shows that, in this
case, the molecules are much closer in oscillation wavelength. This leads to
many more dissipation routes and more molecules that are involved in the
dissipation but at lower energy per molecule.
The dissipation appears to occur at two different timings. First, molecules 4
and 7 and, to a lesser extent, 5 and 12, are excited, while, later, molecules
2 and 3 are excited which transfers to 6 and 9. The O–H bond pointing towards
molecule 4 is excited first and later the O–H bond towards molecule 2, which
has the higher oscillation wavelength. We can hence distinguish several chains
of vibrational energy release. These are initially 1 -¿ 4 -¿ 12 and 1 -¿ 5 -¿
7. All are connected through hydrogen bonds, as is indicated by the magenta
and green dashed lines, respectively, and have oscillating wavelengths that
are resonant with the initial excitation of 2.99 $\mu$m. Molecules 5 and 7
further connect to molecules 15 and 19, which are shown in gray. At a later
stage also 1 ¡-¿ 2 ¡-¿ 6 ¡-¿ 9 ¡-¿ 3 ¡-¿ 1 becomes possible, shown with the
gray dotted line. For this second example, molecules 4 and 5 play a role in
the transfer despite being defect sites. Here they have, however, missing
hydrogen bonds, whereas in the first example the defect site has more hydrogen
bonds than the perfect surrounding.
Restructuring occurs through local heating of individual molecules that can
then reorient themselves. It does not occur through excitation of the hydrogen
bonding network. The internal kinetic energy of three specific hydrogen bonds
was followed in time as a measure of excitation. We chose the hydrogen bonds
between 1 and 4, and 1 and 5, as well as one that did not play any role in the
energy dissipation. Very little time variation was observed for all three
internal kinetic energy plots. This suggests that although hydrogen bonds play
a role in the transfer of excitation, they do not become excited themselves.
(a) (b)
Figure 8: (a) The internal kinetic energy as a function of time for different molecules. Molecule 1 is excited and is sitting at the edge of the pore. The remainder of the molecules is ordered in center-of-mass distance to molecule 1. Molecules with the most significant energy increase (1 to 7, 9, and 12) are indicated in panel (b), following the color coding of (a). Molecules 15 and 19 are indicated in black since they are not in panel (a). Table 2: Wavelength of the oscillation frequencies of O–H bonds in Fig. 8. Molecule | $\nu_{1}$ ($\mu$m) | $\nu_{2}$ ($\mu$m) | site
---|---|---|---
1 | 2.98 | 3.07 | DDAA
2 | 2.99 | 3.03 | DDAA
3 | 2.98 | 2.99 | DDAA
4 | 2.95 | 3.04 | DDA
5 | 2.76 | 2.98 | DAA
6 | 2.99 | 3.16 | DDAA
7 | 2.95 | 2.98 | DDAA
8 | 2.99 | 3.08 | DDAA
9 | 2.98 | 2.99 | DDAA
10 | 2.97 | 3.03 | DDAA
11 | 2.96 | 3.07 | DDAA
12 | 2.97 | 2.99 | DDA
Figure 9 shows a third and final example. In this case, a molecule is selected
in a region with a high defect density to further investigate the role of
DDAAA sites in the dissipation of energy. Table 3 shows that this molecule is
hydrogen bonded to three defect sites, two DDAAA and one DDA site. It is
resonant with the oscillation of all three defect molecules. Panel a of Fig. 9
shows that both molecules 4 and 5 are excited by molecule 1, whereas molecule
2 is not. We saw earlier that DDA defect sites are excited similarly to non-
defect DDAA sites. This is in agreement with the excitation of molecule 4.
Molecule 4 transfers its energy to molecule 8, which is another DDA site.
Molecule 2, which did not become excited despite being resonant with molecule
1, is a DDAAA site, whereas molecule 5, another resonant DDAAA defect site,
becomes excited, which then passes to molecules 6 and 10. These results
suggest that defect sites of type DDAAA can hamper the transfer of vibrational
energy. We think that this might be because the distance and angle are less
optimal for dipolar coupling in such a crowded defect site. We ran a few more
simulations where we excited some other molecules near defect sites which
confirmed this observation: some DDAAA sites block the transfer and others do
not or only after a small reorientation (not shown here). The results
presented here are based on a rather small number of simulations. For a more
firm conclusion, a large statistical set of simulations needs be performed and
analysed which is beyond the scope of this paper.
(a) (b)
Figure 9: (a) The internal kinetic energy as a function of time for different molecules. Molecule 1 is excited. The remainder of the molecules is ordered in center-of-mass distance to molecule 1. Molecules with the most significant energy increase (1, 3 to 6, 8 and 10) are indicated in panel (b), following the color coding of (a). Table 3: Wavelength of the oscillation frequencies of O–H bonds in Fig. 9. Molecule | $\nu_{1}$ ($\mu$m) | $\nu_{2}$ ($\mu$m) | site
---|---|---|---
1 | 2.95 | 3.07 | DDAA
2 | 2.95 | 3.12 | DDAAA
3 | 2.99 | 3.11 | DDAA
4 | 2.95 | 3.03 | DDA
5 | 2.96 | 3.18 | DDAAA
6 | 2.96 | 3.06 | DDAA
7 | 2.86 | 2.98 | DDA
8 | 2.95 | 2.99 | DDA
9 | 2.98 | 2.99 | DDAA
10 | 2.97 | 3.11 | DDAA
11 | 3.01 | 3.11 | DDAA
12 | 2.99 | 3.03 | DDAA
## 4 Conclusions
In conclusion, this paper presented the effect of sequential exposure of ASW
ice to resonant IR irradiation. The experimental results were supplemented by
Molecular Dynamics simulations of sequential irradiation and a study of the
dissipation of energy by excitation of single molecules.
Specific wavelength-dependent changes occur in the ice upon sequential
irradiation. Excitation of individual O–H stretches can spread through the ice
through transfer to hydrogen-bonded molecules with resonant O–H stretches.
Within a molecule, this strict resonant criterion is not so demanding and new
dissipation channels between water molecules of deviating frequencies can open
up after internal energy transfer between vibrational modes. This leads to
local heating of the environment and structural changes. These structural
changes are not limited to the molecules that are excited at the specific
irradiation wavelength but can also include neighboring molecules. This causes
the exact changes at a given irradiation wavelength to depend on the
irradiation history of the sample, since restructuring pathways that are
kinetically accessible might have already occurred during previous irradiation
events. Most restructuring events concern translation or rotation of a water
molecule without breaking existing hydrogen bonds. For more elaborate
restructuring that can lead to crystallization, hydrogen bonds will need to be
broken. The simulations show that vibrational excitation of the O–H bond does
not lead to hydrogen bond breaking or to excitation of hydrogen bonds. For the
latter, we likely need to irradiate at frequencies between 5 and 7 THz.
Whether excitation of hydrogen bonds also results in large hydrogen bond
rearrangement is beyond the scope of the current study.
Changes due to irradiation at wavelengths resonant with surface modes are
different from those caused by IR light at wavelengths of bulk modes. This
suggests that vibrational energy remains rather local since surface and bulk
modes are geometrically separated to some extent. Surface modes also occur
near pores that are present throughout the ice. Simulations show that
molecules can transfer their energy to surrounding molecules which are
connected through a hydrogen-bonding network and have resonant vibrational
frequencies. This occurs on a 0.3 ps timescale. The excitation can persist for
up to 10 ps, but can also be spread at a much shorter timescale. The current
LISA set-up does not have the time resolution to confirm this experimentally
for ASW. Time-resolved experimental studies of the excitation of crystalline
water ice at 3310 cm-1 have shown an ultrafast heating effect at 0.18 $\pm$
0.06 ps timescale, which is faster than that for liquid water, measured at
around 0.38 $\pm$ 0.06 ps 11. The authors of that study attribute the
difference in heating lifetimes to the difference in dipolar coupling between
crystalline ice and liquid water. We expect amorphous solid water to behave
similarly to liquid water in this respect and indeed the 0.3 ps of transfer
time in our simulations corresponds to the heating lifetimes of 0.38 $\pm$
0.06 ps reported by Sudera et al.11.
Defects with missing hydrogen bonding, like DAA and DDA, do not appear to
impact the energy transfer, whereas DDAAA defects can block the transfer in
some cases. Based on our results, we expect the vibrational energy transfer in
ASW to be less efficient than in crystalline water ice for two reasons. First,
the inhomogeneity in oscillation wavelengths is much smaller in crystalline
material, as evidenced by the narrower O–H stretch band, and hence more
molecules will be in resonance with each other leading to more dissipation
channels. Secondly, DDAAA defect sites that can block energy transfer will
not, or rarely, be present in crystalline water ice. Johari and Andersson
showed that the thermal conductivity in amorphous solids is indeed generally
much lower than in crystalline solids 32. They attributed this to the lack of
long-range phonons in amorphous solids. The present work studied energy
transfer in a wavelength regime that is more suited to a molecular description
of the energy transfer, since lattice vibrations are not excited at these
wavelengths. Although we cannot exclude the role of phonons in the work by
Johari and Andersson, our work shows that the difference in thermal
conductivity between ASW and crystalline ice can also be explained in a
molecular framework.
## Credit authorship contribution statement
H.M. Cuppen led the computational part of the manuscript. S. Ioppolo initiated
and managed the laboratory aspect of the project (FELIX-2017-01-30,
FELIX-2018-1-29, and FELIX-2018-02-32) at HFML-FELIX Laboratory with support
from B. Redlich. J.A. Noble and H.M. Cuppen performed data analysis. S.
Ioppolo, J.A. Noble, and S. Coussan performed all laboratory experiments. H.M.
Cuppen wrote the manuscript with assistance from J.A. Noble and S. Ioppolo.
All authors contributed to data interpretation and commented on the paper.
Experimental settings for the different irradiation experiments
The authors thank the HFML-FELIX Laboratory team for their experimental
assistance and scientific support. The LISA end station is designed,
constructed, and managed at the HFML-FELIX Laboratory by the group of S.
Ioppolo. This work was supported by the Royal Society University Research
Fellowships Renewals 2019 (URF/R/191018); the Royal Society University
Research Fellowship (UF130409); the Royal Society Research Fellow Enhancement
Award (RGF/EA/180306); and the Royal Society Research Grant (RSG/R1/180418).
Travel support was granted by the UK Engineering and Physical Sciences
Research Council (UK EPSRC Grant EP/R007926/1 - FLUENCE: Felix Light for the
UK: Exploiting Novel Characteristics and Expertise). S.I. acknowledges further
support from the Danish National Research Foundation through the Center of
Excellence “InterCat” (Grant agreement no.: DNRF150). J.A.N. acknowledges
additional support from the French Programme National “Physique et Chimie du
Milieu Interstellaire” (PCMI) of the CNRS/INSU with the INC/INP, co-funded by
the CEA and the CNES.
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spacing=nonfrench
# Progress on the study of the Ginibre ensembles I: GinUE
Sung-Soo Byun Center for Mathematical Challenges, Korea Institute for
Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>and Peter J. Forrester School of Mathematics and
Statistics, University of Melbourne, Victoria 3010, Australia
<EMAIL_ADDRESS>
###### Abstract.
The Ginibre unitary ensemble (GinUE) consists of $N\times N$ random matrices
with independent complex standard Gaussian entries. This was introduced in
1965 by Ginbre, who showed that the eigenvalues form a determinantal point
process with an explicit correlation kernel, and after scaling they are
supported on the unit disk with constant density. For some time now it has
been appreciated that GinUE has a fundamental place within random matrix
theory, both for its applications and for the richness of its theory. Here we
review the progress on a number of themes relating to the study of GinUE.
These are eigenvalue probability density functions and correlation functions,
fluctuation formulas, sum rules and asymptotic behaviours of correlation
functions, and normal matrix models. We discuss too applications in quantum
many body physics and quantum chaos, and give an account of some statistical
properties of the eigenvectors.
###### Contents
1. 1 Introduction
2. 2 Eigenvalue PDFs and correlations
1. 2.1 Eigenvalue PDF
2. 2.2 Correlation functions
3. 2.3 Elliptic GinUE
4. 2.4 Induced GinUE
5. 2.5 Complex spherical ensemble
6. 2.6 Sub-block of a unitary matrix
7. 2.7 Products of complex Ginibre matrices
8. 2.8 Products of truncated unitary matrices
9. 2.9 The distribution of the squared eigenvalue moduli
3. 3 Fluctuation formulas
1. 3.1 Counting function in general domains
2. 3.2 Counting function in a disk
3. 3.3 Smooth linear statistics
4. 3.4 Spatial modelling and the thinned GinUE
4. 4 Sum rules and asymptotic behaviours
1. 4.1 Asymptotics associated with the configuration integral
2. 4.2 Sum rules and asymptotics for the edge density
3. 4.3 Sum rules and asymptotics for the two and higher point correlations
5. 5 Normal matrix models
1. 5.1 Eigenvalue PDF
2. 5.2 Equilibrium measure
3. 5.3 Partition functions
4. 5.4 Correlation functions and universality
6. 6 Further theory and applications
1. 6.1 Fermi gas wave function interpretation
2. 6.2 Quantum chaos applications
3. 6.3 Singular values
4. 6.4 Eigenvectors
## 1\. Introduction
In a fundamental paper on random matrix theory from the early 1960’s, Dyson
[132] isolated three ensembles of Hermitian matrices, and three ensembles of
unitary matrices. This was done by seeking the minimal requirement of a
quantum Hamiltonian $H$, respectively evolution operator $U$, to exhibit a
time reversal symmetry or not, and then imposing a probability measure. First,
for a time reversal operator $T$ — defined in general by the requirement that
it be anti-unitary — it was first shown that there are two possibilities,
either $T^{2}=\mathbb{I}$ or $T^{2}=-\mathbb{I}$, with the latter requiring
that the Hilbert space be even dimensional. For $T$ to commute with $H$ or $U$
it was then shown that in the case $T^{2}=\mathbb{I}$ both $H$ and $U$ should
be invariant under the transpose operation. In the other possible case, that
$T^{2}=-\mathbb{I}$, an invariance under the so-called quaternion dual
$M\mapsto Z_{2N}M^{T}Z_{2N}^{-1}$ was deduced. Here $Z_{2N}$ is the $2N\times
2N$ anti-symmetric tridiagonal matrix with entries all $-1$ in the leading
upper triangular diagonal, and all $1$ in the leading lower triangular
diagonal, and moreover in this case it was shown that $T$ has the realisation
$T=Z_{2N}K$ where $K$ corresponds to complex conjugation and $2N$ is the
dimension of the Hilbert space. In the case of a quantum Hamiltonian $H$, it
was then shown that $[H,T]=0$ with $T^{2}=\mathbb{I}$ implies a basis can be
chosen so that the elements are real, while with $T^{2}=-\mathbb{I}$ it
implies that the elements can be chosen to have a $2\times 2$ block structure
(1.1) $\begin{bmatrix}z&w\\\ -\bar{w}&\bar{z}\end{bmatrix}.$
The $2\times 2$ matrix (1.1) can be identified with a member of the (real)
quaternion number field. Hence for quantum Hamiltonians, Dyson was lead to the
requirement that matrices in his sought ensemble theory should have real
entries, or have a $2\times 2$ block structure corresponding to the quaternion
number field in the presence of time reversal symmetry, or to be complex
without a time reversal symmetry. This requirement is in addition to the
matrices being Hermitian and thus having real eigenvalues and a matrix of
eigenvectors which can be chosen to be unitary.
The work [132] specifies the eigenvalue probability density function (PDF) for
an ensemble of quantum Hamiltonians $H$ modelled as random matrices, and
chosen from a Gaussian distribution on the elements proportional to
(1.2) $\exp(-\beta\,{\rm Tr}\,H^{2}/2).$
The scaling factor $\beta$ is chosen for convenience, and takes on the value
of the number of independent parts of the corresponding number field — thus
$\beta=1$ for real entries (time reversal symmetry with $T^{2}=\mathbb{I}$),
$\beta=2$ for complex entries (no time reversal symmetry), and $\beta=4$ for
quaternion entries (time reversal symmetry with $T^{2}=-\mathbb{I}$). With
this specification, $\beta$ is referred to as the Dyson index. A detail is
that Hermitian matrices commuting with the quaternion dual must have doubly
degenerate eigenvalues (Kramer’s degeneracy), with the convention in (1.2)
that the trace operation relates to the independent eigenvalues only. The
result of Dyson, known earlier in the case $\beta=1$ by Wigner (see the
Introduction section of the book edited by Porter [287] for references and
moreover reprints of the original works) is that the eigenvalue PDF is given
by [132, Eq. (146)]
(1.3) $\prod_{l=1}^{N}e^{-\beta\lambda_{l}^{2}/2}\prod_{1\leq j<k\leq
N}|\lambda_{k}-\lambda_{j}|^{\beta},$
up to proportionality. If instead of the distribution on elements being chosen
as (1.2), a weighting
(1.4) $\exp(-\beta\,{\rm Tr}\,V(H)/2)$
for some real valued function $V(\lambda)$ is chosen instead, the modification
of (1.2) is that it now reads
(1.5) $\prod_{l=1}^{N}e^{-\beta V(\lambda_{l})/2}\prod_{1\leq j<k\leq
N}|\lambda_{k}-\lambda_{j}|^{\beta}.$
Here it is being assumed that $V(\lambda)$ decays sufficiently fast at
infinity for (1.5) to be normalisable. A weighting of the form (1.4) is said
to specify an invariant ensemble, since it is invariant under conjugation by a
unitary matrix $H\mapsto UHU^{\dagger}$ (and where too the elements of $U$ are
restricted to be real ($\beta=1$) and quaternion ($\beta=4$) so that $H$
remains in the same ensemble). A weighting of the form (1.2) is said to
specify a Gaussian ensemble.
In 1965 Ginibre [179], motivated by mathematical curiosity [19, §2.2, quoting
correspondence with Ginibre], initiated a study of non-Hermitian Gaussian
ensembles with either real, complex or quaternion entries. Replacing (1.2) is
the joint distribution on elements of the corresponding matrices, now to be
denoted $G=[g_{ij}]_{i,j=1}^{N}$, proportional to
(1.6) $\exp(-\beta\,{\rm
Tr}\,G^{\dagger}G/2)=\prod_{i,j=1}^{N}\exp(-\beta|g_{ij}|^{2}/2).$
Note that the second form in this expression shows that the real and imaginary
parts (there $\beta$ such parts; e.g. in the quaternion case $\beta=4$ there
is one real and three imaginary parts) are all independent, identically
distributed Gaussians. The concern of Ginibre was with the functional form of
the eigenvalue PDF, and the implied eigenvalue statistics. The most obvious
difference with the Hermitian case is that the eigenvalues are now in general
complex. Also significant is the fact that the eigenvectors no longer form an
orthonormal basis. Notwithstanding these differences, it was found in the
complex case (referred to as the complex Ginibre ensemble, or alternatively as
GinUE, where in the latter the U stands for unitary refers to the bi-unitary
invariance of (1.6) being unchanged by the mapping $G\mapsto UGV$ for $U,V$
unitary matrices) that the eigenvalue PDF is proportional to
(1.7) $\prod_{l=1}^{N}e^{-|z_{l}|^{2}}\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2},$
which is in direct correspondence with the Hermitian result (1.3) with
$\beta=2$, obtained essentially by replacing $\lambda_{j}$ by $z_{j}$.
The eigenvalue PDF in the case of real and quaternion entries does not follow
this correspondence. First, in both these cases the eigenvalues come in
complex conjugate pairs. Appreciating this point, the functional form of the
eigenvalue PDF as found by Ginibre in the quaternion case can be obtained from
(1.7) (not (1.3) with $\beta=4$) by first replacing $N$ by $2N$, then
identifying $z_{j+N}$ as $\bar{z}_{j}$ and ignoring terms which involve only
$\\{\bar{z}_{j}\\}$ (which are thought of as part of the image system [153])
to obtain
(1.8) $\prod_{l=1}^{N}e^{-2|z_{l}|^{2}}|z_{l}-\bar{z}_{l}|^{2}\prod_{1\leq
j<k\leq N}|z_{k}-z_{j}|^{2}|z_{k}-\bar{z}_{j}|^{2},\quad{\rm Im}\,z_{l}>0.$
The real case is still more complicated. First, the eigenvalue PDF is not
absolutely continuous. Rather, it decomposes into sectors depending on the
number of real eigenvalues. Its precise functional form was not obtained in
Ginibre’s original work, with a further 25 years or so elapsing before this
was achieved in a publication by Lehmann and Sommers [253].
It would seem that the first occurrence of a Ginibre ensemble in applications
(specially the real Ginibre ensemble of GinOE, where here the “O” stands for
orthogonal and refers to the bi-orthogonal invariance of the matrices) arose
in the 1972 work of May [273] on the stability of complex ecological webs.
Upon linearising about a fixed point, and the modelling of the fluctuations
away from an attractor by a real Ginibre matrix $G$, May was led to the first
order linear differential equation system for the perturbed populations — an
$n\times 1$ column vector $\mathbf{x}$ — specified by
(1.9) ${d\over dt}\mathbf{x}=(-\mathbb{I}+\alpha G)\mathbf{x},$
where $\alpha$ is a scalar parameter. The stability is then determined by the
maximum of the real part of the spectrum of $G$, the precise determination of
which has only recently become available in the literature [54, 106].
At the beginning of the 1980’s the interpretation of (1.7), written in the
Boltzmann factor form
(1.10) $e^{-\beta U(z_{1},\dots,z_{N})},\qquad U={1\over
2}\sum_{j=1}^{N}|z_{j}|^{2}-\sum_{1\leq j<k\leq
N}\log|z_{k}-z_{j}|,\quad\beta=2,$
as a model of charged particles, repelling pairwise via a logarithmic
potential, and attracted to the origin in the plane via a harmonic potential,
gained attention [23, 81]. This viewpoint was already prominent in the works
of Dyson in the context of (1.3) (see too the even earlier work of Wigner
[327] as reprinted in [287]), and was noted for (1.8) in Ginibre’s original
article [179]. In contrast to (1.3), in (1.10) the domain is two-dimensional,
and the logarithmic potential is the solution of the corresponding Poisson
equation (in one spacial dimension, the solution of the Poisson equation is
proportional to $|x|$), so (1.10) corresponds to a type of Coulomb gas. Broad
aspects of the latter have been the subject of the recent reviews [255], [90].
Also in the early 1980’s, for all positive values of $\beta/2$ odd, (1.10)
gained attention as the absolute value squared of Laughlin’s trial wave
function for the fractional quantum Hall effect [243].
Fast forward 40 years, and there are now a multitude of applications which
require knowledge of properties of Ginibre matrices, in particular their
eigenvalues and eigenvectors. This is due in no small part due to a resurgence
of interest in non-Hermitian quantum mechanics [44]. Moreover, the progression
of time as seen a much deeper understanding of the mathematical structures
associated with the Ginibre matrices, and the theoretical progress has been
considerable. It is the purpose of this article to review a number of these
advances, both in the theory and the applications. Due to space
considerations, attention will be focused here on the complex case of the
GinUE — a subsequent review article is planned relating specifically too GinOE
and GinSE. To the era up to the year 2010, accounts of the progress with
emphasis similar to the present article can be found in [149, Ch. 15, with
proofs], [224, some proofs sketched], with the latter overlapping mainly with
§2. To make the presentation self contained, this material is also part of the
present review, albeit with some reordering and additional context. And when
practical from the viewpoint of the space required, proofs of a number of the
results are presented.
There are four main themes to the review. These form sections two through to
five: eigenvalue PDFs and correlation functions, fluctuation formulas, sum
rules and asymptotic behaviours, and normal matrix models. There is also a
sixth section entitled further theory and applications. Here the topics
considered are the analogy between GinUE and the quantum many body system for
free Fermions in the plane subject to a perpendicular magnetic field, the
relevance of GinUE statistics to studies in quantum chaos, and statistical
properties of the eigenvectors of GinUE matrices.
## 2\. Eigenvalue PDFs and correlations
### 2.1. Eigenvalue PDF
In the original paper of Ginibre [179], the diagonalisation formula
$G=V\Lambda V^{-1}$, where $\Lambda$ is the diagonal matrix of eigenvalues,
and $V$ is the matrix of corresponding eigenvectors which are unique up to
normalisation, was used as the starting point to derive the eigenvalue PDF
(1.7). The matrix $V$ was then further decomposed $V=UTD$, where $U$ is
unitary, $T$ is upper triangular with all diagonal elements equal to $1$, and
$D$ is a diagonal matrix with real positive elements. As a consequence
(2.1) ${\rm Tr}\,G^{\dagger}G={\rm Tr}\,\bar{\Lambda}B\Lambda B^{-1},\qquad
B=T^{\dagger}T$
This is independent of $U$ and $D$, and $B^{-1}$ is a simpler structure than
$V^{-1}$, since it has determinant unity. It is necessary to integrate out the
variables of $B$, which was done in $N$ steps with each one consisting of
integrating out over the last remaining row and column.
A more versatile (equally applicable to the GinOE, for example) method of
derivation of (1.7) has since been found. It is due to Dyson, and first
appeared in published form in [274, Appendix 35]. Here, instead of using the
diagonalisation formula for $G$, the starting point is the Schur decomposition
(2.2) $G=UZU^{\dagger}.$
Here $U$ is a unitary matrix, unique up to the phase of each column, and $Z$
is an upper triangular matrix with elements on the diagonal equal to the
eigenvalues of $G$.
###### Proposition 2.1.
For GinUE matrices, specified by the distribution on elements proportional to
(1.6) in the case $\beta=2$ (complex elements), the eigenvalue PDF is equal to
$1/C_{N}$ times (1.7), where upon relaxing the ordering constraint on the
eigenvalues implied by (2.2), the normalisation constant $C_{N}$ is specified
by
(2.3) $C_{N}=\pi^{N}\prod_{j=1}^{N}j!.$
###### Proof.
We have from (2.2) that
(2.4) ${\rm Tr}\,G^{\dagger}G=\sum_{j=1}^{N}|z_{j}|^{2}+\sum_{1\leq j<k\leq
N}|Z_{jk}|^{2},$
where here $\\{z_{j}\\}$ denotes the diagonal elements of $Z$ (which are the
eigenvalues of $G$), and $\\{Z_{jk}\\}$ denotes the upper triangular elements.
Note the simplification relative to (2.1).
After this brisk start, there is still quite a challenge to compute the
Jacobian corresponding to (2.2). The strategy of [274, Appendix 35] is
explained in more detail in [203], and repeated in [149, Proof of Proposition
15.1.1]. Here one begins by computing the matrix of differentials
$U^{\dagger}dG\,U$, with the Jacobian corresponding to the (absolute value of)
factor which results from the corresponding wedge product. For the latter
task, proceeding in the order of the indices $(j,k)$, with $j$ decreasing from
$N$ to $1$, and $k$ increasing from $1$ to $N$, gives the factor
(2.5) $\prod_{j<k}|z_{j}-z_{k}|^{2}.$
However the product of differentials so obtained is not immediately
recognisable in the factorised form
(2.6) $\wedge_{j}dz_{j}^{\rm r}dz_{j}^{\rm
i}\wedge(U^{\dagger}dU)\wedge_{j<k}dZ_{jk}^{\rm r}dZ_{jk}^{\rm i},$
where the superscripts indicate the real and imaginary part. Further arguing
involving a count of the number of independent real variables associated with
$U$, which implies some apparent differentials contribute zero to the wedge
product, is required to make the simplification to this form. With (2.6)
established, the integration over $\\{Z_{jk}\\}$ in (2.4) is immediate.
To deduce the normalisation (2.3) from this calculation requires first that
the normalisation of (2.4) be included throughout the calculation, and second
knowledge of the integration formula (see e.g. [126, Eq, (4.4)])
(2.7) $\int_{U}(U^{\dagger}dU)={\rm
vol}\,\Big{(}U(N)/(U(1))^{N}\Big{)}=2^{N(N-1)/2}\prod_{l=1}^{N-1}{\pi^{l}\over\Gamma(l+1)}.$
Finally, an extra factor of $N!$ is required in $C_{N}$ to account for
relaxing an ordering of the eigenvalues. ∎
### 2.2. Correlation functions
With the joint eigenvalue PDF denoted $p_{N}(z_{1},\dots,z_{N})$, the
$k$-point correlation function $\rho_{(k),N}(z_{1},\dots,z_{k})$ is specified
by
(2.8)
$\rho_{(k),N}(z_{1},\dots,z_{k})=N(N-1)\cdots(N-k+1)\int_{\mathbb{C}}d^{2}z_{k+1}\cdots\int_{\mathbb{C}}d^{2}z_{N}\,p_{N}(z_{1},\dots,z_{N}),$
where, with $z:=x+iy$, $d^{2}z:=dxdy$. In the simplest case $k=1$ this
corresponds to the eigenvalue density. Ginibre [179] showed that
(2.9)
$\rho_{(k),N}(z_{1},\dots,z_{k})=\det\Big{[}K_{N}(z_{j},z_{l})\Big{]}_{j,l=1}^{k},$
for a particular function $K_{N}(w,z)$, referred to as the correlation kernel.
The structure (2.9) makes the eigenvalues of GinUE an example of a
determinantal point process [68].
###### Proposition 2.2.
The kernel function in (2.9) is specified by
(2.10)
$K_{N}(w,z)={1\over\pi}e^{-(|w|^{2}+|z|^{2})/2}\sum_{j=1}^{N}{(w\bar{z})^{j-1}\over(j-1)!}={1\over\pi}e^{-(|w|^{2}+|z|^{2})/2}e^{w\bar{z}}{\Gamma(N;w\bar{z})\over\Gamma(N)},$
where $\Gamma(j;x)=\int_{x}^{\infty}t^{j-1}e^{-t}\,dt$ denotes the (upper)
incomplete gamma function.
###### Proof.
To deduce the determinantal structure (2.9) with $K_{N}$ specified by the
first equality in (2.10) (the second equality follows from the first by the
identity
${\Gamma(N;x)\over\Gamma(N)}=e^{-x}\sum_{j=1}^{N}{x^{j-1}\over(j-1)!}$), the
first step is to rewrite the product in (1.6) according to
(2.11) $\prod_{1\leq j<k\leq N}|z_{k}-z_{j}|^{2}=\prod_{1\leq j<k\leq
N}(z_{k}-z_{j})(\overline{z}_{k}-\overline{z}_{j}),$
then to rewrite each of the product of differences on the RHS as a Vandermonde
determinant,
(2.12) $\prod_{1\leq j<k\leq N}(z_{k}-z_{j})=\det[z_{j}^{k-1}]_{j,k=1}^{N};$
see e.g. [149, Exercises 1.9 Q.1] for a derivation. Multiplying (2.12) by its
conjugate and taking the transpose of the matrix on the RHS (which leaves the
determinant unchanged) shows
(2.13) ${1\over C_{N}}\prod_{l=1}^{N}e^{-|z_{l}|^{2}}\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2}=\det\Big{[}K_{N}(z_{j},z_{k})\Big{]}_{j,k=1}^{N}.$
The significance of the form (2.13) for purposes of computing the integrations
as required by (2.8) are the reproducing and normalisation properties of
$K_{N}$,
(2.14)
$\int_{\mathbb{C}}K_{N}(w_{1},z)K_{N}(z,w_{2})\,d^{2}z=K_{N}(w_{1},w_{2}),\qquad\int_{\mathbb{C}}K_{N}(z,z)\,d^{2}z=N.$
Using these properties, a cofactor expansion along the bottom row can be used
to show [134]
(2.15)
$\int_{\mathbb{C}}\det[K_{N}(z_{j},z_{k})]_{j,k=1}^{m}\,d^{2}z_{m}=(-(m-1)+N)\det[K_{N}(z_{j},z_{k})]_{j,k=1}^{m-1};$
see also [149, Proof of Proposition 5.1.2]. Applying this inductively gives
(2.9). ∎
According to (2.9) with $k=1$ and (2.10), the eigenvalue density is given by
the rotationally invariant functional form
(2.16) $\rho_{(1),N}(z)={1\over\pi}{\Gamma(N;|z|^{2})\over\Gamma(N)}.$
Ginibre [179] identified a sharp transition for $|z|\approx\sqrt{N}$ from the
constant value ${1\over\pi}$ for $|z|$ less than this critical value, to a
value approaching zero for $|z|$ greater than this critical value. An
equivalent statement is the limit law
(2.17)
$\lim_{N\to\infty}\rho_{(1),N}(\sqrt{N}z)=\begin{cases}\displaystyle{1\over\pi},&|z|<1,\\\
0,&|z|>1.\end{cases}$
This was the first example of what now is termed the circular law, which
specifies the (global) scaled limiting eigenvalue density for a wide class of
non-Hermitian random matrices, with identically and independently distributed
elements to be constant inside a particular circle in the complex plane, and
zero outside [67].
###### Remark 2.3.
1\. In the sense of probability theory, (2.17) is a statement in the mean, due
to the ensemble average. The strong version of the circular law establishes
that for large $N$ the eigenvalues of a single GinUE matrix obey the circular
law almost surely [67].
2\. Define the numerical range of an $N\times N$ matrix $X$ by
$W(X)=\\{\overline{X\mathbf{u}}\cdot\mathbf{u}\>|\>||\mathbf{u}||=1\\}$. By
the variational characterisation of eigenvalues, for $X$ Hermitian $W(X)$ must
be contained in the interval of the real line $[\lambda_{\rm min},\lambda_{\rm
max}]$, and in fact is equal to this interval. It is proved in [259] that for
$X$ a global scaled Ginibre matrix, and thus with eigenvalue density obeying
the circular law (2.17), $W(X)$ converges to the centred disk in the complex
plane of radius $\sqrt{2}$.
The correlation kernel (2.10), and thus the correlation functions (2.9), admit
distinct scaling limits depending on the centring of the variables being in
the bulk region, where the density is constant, or the edge region, where the
density begins to decrease to zero. The first was specified in Ginibre’s
original paper [179], whereas the latter was not made explicit until some time
later [159].
###### Proposition 2.4.
Let $K_{N}(w,z)$ be specified by (2.10). We have
(2.18) $\displaystyle K_{\infty}^{\rm b}(w,z)$
$\displaystyle:=\lim_{N\to\infty}K_{N}(w,z)={1\over\pi}e^{-(|w|^{2}+|z|^{2})/2}e^{w\bar{z}},$
(2.19) $\displaystyle K_{\infty}^{\rm e}(z_{1},z_{2})$
$\displaystyle:=\lim_{N\to\infty}K_{N}(-i\sqrt{N}+z_{1},-i\sqrt{N}+z_{2})=e^{-(|z_{1}|^{2}+|z_{2}|^{2})/2}e^{z_{1}\bar{z}_{2}}h\Big{(}{1\over
2}(-iz_{1}+i\bar{z}_{2})\Big{)},$
where $h(z)={1\over 2\pi}\Big{(}1+{\rm erf}(\sqrt{2}z)\Big{)}$.
###### Proof.
The limit (2.18) is immediate from (2.10), and the fact that for fixed
$w\bar{z}$,
$\lim_{N\to\infty}{\Gamma(N;w\bar{z})\over\Gamma(N)}=1.$
The derivation of (2.19) relies on (the first term of) the asymptotic
expansion [278]
(2.20) ${\Gamma(N;N+\tau\sqrt{N})\over\Gamma(N)}={1\over 2}(1-{\rm
erf}(\tau/\sqrt{2}))+{1\over 3\sqrt{2\pi N}}e^{-\tau^{2}/2}(\tau^{2}-1)+{\rm
O}\Big{(}{1\over N}\Big{)}.$
∎
With the leading support of the eigenvalues the disk $|z|<\sqrt{N}$, it is
natural to consider as a point in the bulk any $z_{0}=s\sqrt{N}+it\sqrt{N})$
for some $|s|,|t|<1$. Making use of the fact that $\Gamma(N;u)/\Gamma(N)\to 1$
for $|u|/\sqrt{N}\to c$, $c<1$ as $N\to\infty$, as is consistent with (2.20),
it follows that
$\lim_{N\to\infty}K_{N}(z_{0}+w,z_{0}+z)=K_{\infty}^{\rm b}(w,z)$
independent of $z_{0}$; see also [69, Appendix C]. Similarly, for any
$|\nu|=1$,
(2.21)
$\lim_{N\to\infty}K_{N}(\nu(\sqrt{N}+z_{1}),\nu(\sqrt{N}+z_{2}))=e^{-(|z_{1}|^{2}+|z_{2}|^{2})/2}e^{z_{1}\bar{z}_{2}}h\Big{(}{1\over
2}(-z_{1}-\bar{z}_{2})\Big{)}$
as calculated in [69, Appendix C with $s_{k}=\nu z_{k}$].
Generally (2.9) gives for the appropriately scaled two-point correlation
$\rho_{(2),\infty}(z_{1},z_{2})=\rho_{(1),\infty}(z_{1})\rho_{(1),\infty}(z_{2})-K_{\infty}(z_{1},z_{2})K_{\infty}(z_{2},z_{1}).$
For large separation of $z_{1}$ and $z_{2}$ the leading order of the RHS is
given by the first term which is the product of the densities. The second term
$-K_{\infty}(z_{1},z_{2})K_{\infty}(z_{2},z_{1})$ must decay sufficiently
rapidly for it to be square integrable, since the limiting form of the first
integration formula in (2.14) remains valid,
(2.22)
$\int_{\mathbb{C}}K_{\infty}(w_{1},z)K_{\infty}(z,w_{2})\,d^{2}z=K_{\infty}(w_{1},w_{2}).$
This can be verified from the results of Proposition 2.4 by the evaluation of
appropriate Gaussian integrals. To separate off the product of densities, one
defines the truncated (or connected) two-point correlation
(2.23)
$\rho_{(2),\infty}^{T}(z_{1},z_{2}):=\rho_{(2),\infty}(z_{1},z_{2})-\rho_{(1),\infty}(z_{1})\rho_{(1),\infty}(z_{2})=-K_{\infty}(z_{1},z_{2})K_{\infty}(z_{2},z_{1}).$
In particular, with bulk scaling, we read off from this and (2.18) that
(2.24) $\rho_{(2),\infty}^{{\rm
b},T}(z_{1},z_{2})=-{1\over\pi^{2}}e^{-|z_{1}-z_{2}|^{2}},$
which thus exhibits a Gaussian decay. With edge scaling, (2.23) and (2.19)
give
(2.25) $\rho_{(2),\infty}^{{\rm
e},T}(z_{1},z_{2})=-e^{-(x_{1}-x_{2})^{2}-(y_{1}-y_{2})^{2}}\bigg{|}h\Big{(}{1\over
2}(y_{1}+y_{2}-i(x_{1}-x_{2}))\Big{)}\bigg{|}^{2}.$
Use of the asymptotic expansion of the error function [281, Eq. (7.12.1)]
gives to leading order
(2.26) $\rho_{(2),\infty}^{{\rm
e},T}(z_{1},z_{2})\mathop{\sim}\limits_{|z_{1}-z_{2}|\to\infty}-{1\over
2\pi^{3}}{e^{-2y_{1}^{2}-2y_{2}^{2}}\over(y_{1}+y_{2})^{2}+(x_{1}-x_{2})^{2}}.$
While this decays in all directions, parallel to the boundary of the leading
order density (i.e. in the $x$-direction) we see that the decay is algebraic,
as an inverse square.
###### Remark 2.5.
1\. It follows from (2.20) that the edge scaling of the eigenvalue density, in
the coordinates of (2.19) with $z=x+iy$, has the large $N$ expansion
(2.27) $\rho_{(1),N}^{\rm e}(y)={1\over 2\pi}\Big{(}1+{\rm
erf}(\sqrt{2}y)\Big{)}+{1\over 3\pi\sqrt{2\pi N}}e^{-2y^{2}}(y^{2}-1)+{\rm
O}\Big{(}{1\over N}\Big{)},$
and thus in particular
(2.28) $\rho_{(1),\infty}^{\rm e}(y)={1\over 2\pi}\Big{(}1+{\rm
erf}(\sqrt{2}y)\Big{)},$
where this latter expression is consistent with (2.19) upon setting
$z_{1}=z_{2}=z$. There is interest in the functional form of the $1/\sqrt{N}$
correction term in our discussion of §4.2 below; see too [250, 31].
Integrating (2.28) over $y\in(-\infty,0]$ and multiplying by $2\pi\sqrt{N}$
(the length of the bounding circle of the leading order support) shows that to
leading order the expected number of eigenvalues with modulus greater that
$\sqrt{N}$ is $\sqrt{N}/(2\pi)$ [179]. Note too that setting $y=0$ in (2.28)
gives $\rho_{(1),\infty}^{\rm e}(0)={1\over 2\pi}$, or equivalently
$\lim_{N\to\infty}\rho_{(1),N}(\sqrt{N})={1\over 2\pi}$, which is exactly
${1\over 2}$ of the limiting value inside the unit circle as given by (2.17).
2\. The two-dimensional classical Coulomb system interpretation (1.10) of the
eigenvalue PDF (1.7) allows for (2.17) to be anticipated. For this, one scales
$z_{j}\mapsto\sqrt{N}z_{j}$ and introduces a mean field energy functional
(2.29) ${N\over 2}\sum_{j=1}^{N}|z_{j}|^{2}-\sum_{1\leq j<k\leq
N}\log|z_{k}-z_{j}|\\\ \sim{N\over
2}\Big{(}\int_{\Omega}\rho_{(1)}(z)|z|^{2}\,d^{2}z-\int_{\Omega}d^{2}w\,\rho_{(1)}(w)\int_{\Omega}d^{2}z\,\rho_{(1)}(z)\log|z-w|\Big{)},$
with the hypothesis that $\rho_{(1)}(z)$ is chosen so that this functional is
minimised and furthermore integrates over $\Omega$ to unity. Characterising
the minimisation property by the vanishing of the functional upon variation
with respect to $\rho_{(1)}(z)$ gives
(2.30) $|z|^{2}-2\int_{\Omega}\rho_{(1)}(w)\log|z-w|\,d^{2}w=C,$
where $C$ is a constant, valid for $z\in\Omega$. Applying the Laplacian
operation $\nabla_{z}^{2}$ to this, using the standard fact
$-\nabla_{z}^{2}\log|z-w|=-2\pi\delta(z-w),$
it follows
$\rho_{(1)}(w)={1\over\pi}\chi_{|w|<1}.$
Here the restriction to $|w|<1$ is implied by the rotational invariance,
together with the minimisation and normalisation requirements. The notation
$\chi_{A}$ denotes the indicator function of the condition $A$, taking on the
value $1$ when $A$ is true, and $0$ otherwise. The support $\Omega$ of
$\rho_{(1)}(w)$ — which here is the unit disk — in such a Coulomb gas picture
is typically referred to as the droplet (see e.g. [4, 199]). Moreover, this
potential theoretic reasoning can rigorously be justified; see e.g. [90, §3.1]
and references therein, as well as the discussion and references in the
paragraph including (5.7) of §5 below.
3\. Let $X_{0}$ have finite rank, and $X$ be a GinUE matrix, scaled so that
the leading support of the eigenvalues is the unit disk. Assume too that the
eigenvalues of $X_{0}$ are inside of the unit disk and near the boundary. In
the limit $N\to\infty$ it has recently been shown that the edge correlation
functions centred on the eigenvalues of $X_{0}$ form a determinantal point
process with kernel involving generalisations of the error function [263].
### 2.3. Elliptic GinUE
In 1991 Lehmann and Sommers introduced a one parameter generalisation of the
non-Hermitian complex Gaussian matrices specifying GinUE. In this
generalisation, varying the parameter allows for the Hermitian ensemble of
Gaussian matrices known as the GUE (Gaussian unitary ensemble) to be obtained.
An Hermitian matrix $H$ from the GUE can be constructed from a scaled non-
Hermitian GinUE matrix $\tilde{G}=(1/\sqrt{2})G$ according to
(2.31) $H={1\over 2}\Big{(}\tilde{G}+\tilde{G}^{\dagger}\Big{)}.$
###### Proposition 2.6.
Let the parameters $0<\tau,v<1$ be related by $\tau=(1-v^{2})/(1+v^{2})$. For
$H_{1},H_{2}$ elements of the GUE, define
(2.32) $J=\sqrt{1+\tau}(H_{1}+ivH_{2}).$
The eigenvalue PDF of the ensemble of matrices $\\{J\\}$ is given by
(2.33) $\exp\Big{(}-{1\over
1-\tau^{2}}\sum_{j=1}^{N}\Big{(}|z_{j}|^{2}-{\tau\over
2}(z_{j}^{2}+\bar{z}_{j}^{2})\Big{)}\Big{)}\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2}.$
###### Proof.
Following [171] and [149, Exercises 15.1 Q.1], the starting point is to note
that the joint element PDF of $\sqrt{1+\tau}H_{1}$ and $\sqrt{1+\tau}H_{2}$ is
proportional to $\exp(-{1\over 1+\tau}{\rm Tr}(H_{1}^{2}+H_{2}^{2}))$. The
definition of $J$ gives
${\rm Tr}\,H_{1}^{2}={1\over 2}\Big{(}{\rm Tr}(JJ^{\dagger})+{\rm Re}\,{\rm
Tr}(J^{2})\Big{)},\qquad{\rm Tr}\,H_{2}^{2}={1\over 2v^{2}}\Big{(}{\rm
Tr}(JJ^{\dagger})-{\rm Re}\,{\rm Tr}(J^{2})\Big{)}.$
As a consequence, it follows that the joint element PDF of $J$ is proportional
to
(2.34) $\exp\Big{(}-{1\over 1-\tau^{2}}{\rm Tr}(JJ^{\dagger}-\tau{\rm
Re}\,J^{2})\Big{)}.$
With this knowledge, the strategy of the proof of Proposition 2.1 leads to
(2.33). ∎
The correlations for (2.33) have, for an appropriate correlation kernel
$K_{N}(w,z;\tau)$, the determinantal form (2.9). This involves the scaled
monic Hermite polynomials
(2.35) $C_{n}(z):=\Big{(}{\tau\over
2}\Big{)}^{n/2}H_{n}\Big{(}{z\over\sqrt{2\tau}}\Big{)},\qquad z\in\mathbb{C}.$
###### Proposition 2.7.
In the notation specified above, we have
(2.36) $K_{N}(w,z;\tau)={1\over\pi}{1\over\sqrt{1-\tau^{2}}}\\\
\times\exp\Big{(}-{1\over 2(1-\tau^{2})}\Big{(}|w|^{2}+|z|^{2}-\tau({\rm
Re}\,w^{2}+{\rm
Re}\,z^{2})\Big{)}\Big{)}\sum_{l=0}^{N-1}{C_{l}(w)C_{l}(\bar{z})\over l!}.$
###### Proof.
Following the same procedure as used to begin the proof of Proposition 2.2, we
modify (2.12) so that it reads
(2.37) $\prod_{1\leq j<k\leq
N}(z_{k}-z_{j})=\det[p_{k-1}(z_{j})]_{j,k=1}^{N},$
where $\\{p_{l}\\}$ is a set of monic polynomials, each $p_{l}$ of degree $l$.
Choosing these polynomials according to (2.35), with this choice being
motivated by the orthogonality [136, 125]
$\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dy\,e^{-x^{2}/(1+\tau)-y^{2}/(1-\tau)}C_{m}(z)C_{n}(\bar{z})=\pi
m!\sqrt{1-\tau^{2}}\delta_{m,n},$
the remaining working of the proof of Proposition 2.2 establishes the result.
∎
Applying the Coulomb gas argument of Remark 2.5.2, with the scaling
$z_{l}\mapsto\sqrt{N}z_{l}$, we conclude that within some domain $\Omega$ the
density is constant, taking the value $\rho_{(1)}(w)=1/(\pi(1-\tau^{2}))$.
This domain, or equivalently droplet, can be determined to be an ellipse with
semi-axes $A=1+\tau$, $B=1-\tau$ and area equal to $\pi(1-\tau^{2})$. The
shape can be verified directly, by showing that with $\Omega$ so specified,
and $|z|^{2}$ in (2.30) replaced by
(2.38) ${1\over 1-\tau^{2}}(|z|^{2}-\tau{\rm Re}\,z^{2}),$
the required minimisation equation is indeed satisfied [104], [149, Exercises
15.2 q.4]; see also [125, 161, 250, 75]. This droplet shape explains the
terminology elliptic GinUE in relation to (2.33). For complex non-Hermitian
matrices (2.32), now with the Hermitian random matrices $H_{1},H_{2}$
constructed from (2.31) with general zero mean, unit standard deviation
identically distributed entries that are not required to be Gaussian, a
constant density in an ellipse was first deduced by Girko [180] upon
additional assumptions, and without qualification by Nguyen and O’Rourke
[279].
As for the GinUE, as the boundary of the leading support of the elliptic GinUE
is approached, there is a transition from a constant density to a density
which decays to zero. Beginning with (2.36) in the case $w=z$, the analysis of
the eigenvalue density in this edge regime is more complicated than for the
deduction of (2.20). It was carried out by Lee and Riser [250]; see also [17,
276].
###### Proposition 2.8.
Consider the elliptic GinUE, with the eigenvalues scaled
$z_{j}\mapsto\sqrt{N}z_{j}$ so that for $N$ large the leading order support is
the ellipse $\Omega$. Let $z_{0}$ be a point on the boundary of $\Omega$.
Denote the unit vector corresponding to the outer normal at this point by
$\mathbf{n}$, and the corresponding curvature by $\kappa$. We have
(2.39)
$\rho_{(1),N}\Big{(}z_{0}+{(\alpha+i\beta)\mathbf{n}\over\sqrt{N}}\Big{)}={1\over
2\pi}\Big{(}1-{\rm erf}(\sqrt{2}\alpha)\Big{)}+{\kappa\over\pi\sqrt{2\pi
N}}e^{-2\alpha^{2}}\Big{(}{\alpha^{2}-1\over 3}-\beta^{2}\Big{)}+{\rm
O}\Big{(}{1\over N}\Big{)}.$
###### Remark 2.9.
1\. The leading term in (2.39) is identical to that in (2.27) with the
identification $y=-\alpha$. The universality of this functional form has been
established for a wide class of normal matrix models (see § 5 in relation to
this class), and moreover extended to the edge scaled correlation kernel
(2.19) [38, 201]. This has recently been proved too for non-Hermitian random
matrices constructed according to (2.31) with the elements of $G$ identically
distributed mean zero, finite variance random variables [105].
2\. Consider the elliptic shaped domain
$\\{z\in\mathbb{C}:\,1-{1\over 1-\tau^{2}}(|z|^{2}-\tau{\rm Re}\,z^{2})>0\\}.$
Let the function on the LHS of the inequality be denoted by $1-f(z)$. We see
that $f(z)$ coincides with (2.38). It has been shown in [21, 277] that the
polynomials $\\{p_{n}(z)\\}$ satisfying the orthogonality
$\int_{\Omega}p_{m}(z)p_{n}(\overline{z})(1-f(z))^{\alpha}\,d^{2}z\propto\delta_{m,n},$
are simply related to the Gegenbauer polynomials
$\\{C_{n}^{(1+\alpha)}(z)\\}$. After scaling, the result (2.35) can be
reclaimed by taking the limit $\alpha\to\infty$.
3\. We see from (2.32) that as $\tau\to 1^{-}$, $J$ is proportional to a GUE
matrix, and in particular its eigenvalues are then all real. It was found by
Fyodorov, Khoruzhenko and Sommers [172] that setting
$\tau=1-\pi^{2}\alpha^{2}/2N$ and scaling the eigenvalues $z_{j}\mapsto\pi
z_{j}/N=\pi(x_{j}+iy_{j})/N$ that
(2.40) $\lim_{N\to\infty}{\pi^{2}\over N^{2}}K_{N}(\pi w/N,\pi
z/N;\tau)\Big{|}_{\tau=1-\pi^{2}\alpha^{2}/2N}\\\
=\sqrt{2\over\pi\alpha^{2}}\exp\Big{(}-{y_{1}^{2}+y_{2}^{2}\over\alpha^{2}}\Big{)}{1\over
2\pi}\int_{-\pi}^{\pi}\exp\Big{(}-{\alpha^{2}u^{2}\over
2}+iu(w-\bar{z})\Big{)}\,du.$
(For a direct comparison between (2.40) and what has been reported in the
literature, see [16, $K_{\rm weak}(z_{1},z_{2})$ in Theorem 3] and [30, §1].)
This is referred to as the weakly non-Hermitian limit. Scaling of the
correlation kernel at the edge in this limit has been considered in [174, 54].
### 2.4. Induced GinUE
The fact that a complex Gaussian matrix $G$ is bi-invariant with respect to
multiplication by unitary matrices allows for the distribution of the singular
values to be related to the eigenvalue distribution [144, 225].
###### Proposition 2.10.
Let $G$ be bi-unitary invariant, and let $U$ be a Haar distributed unitary
matrix. We have that $G$ and $(G^{\dagger}G)^{1/2}U$ have the same joint
element distribution, and so in particular have the same distribution of
eigenvalues.
###### Proof.
By the singular value decomposition, $G=U_{1}\Sigma U_{2}$ for some unitary
matrices $U_{1},U_{2}$ and where $\Sigma$ is the diagonal matrix of the
singular values. We then have $(G^{\dagger}G)^{1/2}U=U_{2}^{\dagger}\Sigma
U_{2}U$. By the assumed bi-unitary invariance of $G$, this matrix and $G$ have
the same distribution. ∎
The matrix
(2.41) $A:=(G^{\dagger}G)^{1/2}U$
is well defined for $G$ rectangular, and moreover the property of $G$ being
bi-unitary invariant can be generalised to this setting. Of interest is the
relation between the joint element distributions of $G$ and $A$ [144].
###### Proposition 2.11.
Let $G$ be a bi-unitary invariant rectangular $n\times N$ random matrix with
joint element distribution of the functional form $g(G^{\dagger}G)$. The joint
element distribution of the matrix $A$ (2.41) with $U$ a Haar distributed
unitary matrix is proportional to
(2.42) $(\det A^{\dagger}A)^{(n-N)}g(A^{\dagger}A).$
As a consequence, for $G$ a rectangular complex Ginibre matrix, the eigenvalue
PDF of $A$ is
(2.43) ${1\over
C_{n,N}}\prod_{l=1}^{N}|z_{l}|^{2(n-N)}e^{-|z_{l}|^{2}}\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2},$
with normalisation $C_{n,N}=N!\pi^{N}\prod_{j=1}^{N}(n-N+j-1)!$.
###### Proof.
Write $B=G^{\dagger}G$. With the joint element distribution of $G$ of the
functional form $g(G^{\dagger}G)$, it is a standard result (see e.g. [149, Eq.
(3.23)]) that the joint element distribution of $B$ is proportional to $\det
B^{n-N}g(B)$. But $B$ and $A^{\dagger}A$ have the same joint element
distribution from the bi-unitary invariance of $G$, and the result just quoted
with $n=N$ tells us that the Jacobian for the joint element distribution of
$A$ and $B$ is a constant, which implies (2.42). In the particular case that
$G$ is a rectangular complex Ginibre matrix, the function $g$ in (2.42) is the
exponential $g(X)=e^{-{\rm Tr}\,X}$. Furthermore, with the eigenvalues of $A$
denoted $\\{z_{l}\\}$, we have $(\det
A^{\dagger}A)^{(n-N)}=\prod_{l=1}^{N}|z_{l}|^{2(n-N)}$. Taking these points
into consideration, we see (2.43) results by following the proof of
Proposition 2.1. Moreover, with
$\omega(z)=\omega(|z|^{2})=|z|^{2(n-N)}e^{-|z|^{2}}$ the weight function, that
the analogue of (2.13) in that proof have the properties (2.14) requires that
the normalisation equal
$N!\prod_{j=0}^{N-1}2\pi\int_{0}^{\infty}r^{2j+1}\omega(r)\,dr$. This implies
the stated value of $C_{n,N}$. ∎
The correlations for (2.43) are of the determinantal form (2.9). Denoting the
corresponding correlation by $K^{\rm iG}(w,z)$, the derivation of (2.10) shows
[7, 144]
$\displaystyle K_{N}^{\rm iG}(w,z)$
$\displaystyle={1\over\pi}e^{-(|w|^{2}+|z|^{2})/2}\sum_{j=1}^{N}{(w\bar{z})^{n-N+j-1}\over(n-N+j-1)!}$
(2.44)
$\displaystyle={1\over\pi}e^{-(|w|^{2}+|z|^{2})/2}e^{w\bar{z}}\Big{(}{\Gamma(n;w\bar{z})\over\Gamma(n)}-{\Gamma(n-N;w\bar{z})\over\Gamma(n-N)}\Big{)}.$
We know that the eigenvalue density, $\rho_{(1),N}^{\rm iG}(z)$ say, results
by setting $w=z$ in $K_{N}^{\rm iG}(w,z)$. Knowing from (2.20) that for $N$
large $\Gamma(N;xN)/\Gamma(N)$ exhibits a transition from the value $1$ for
$0<x<1$ to the value $0$ for $x>1$, we see from (2.4) that
(2.45) $\displaystyle\lim_{N\to\infty}\rho_{(1),N}^{\rm
iG}(\sqrt{N}z)\Big{|}_{n/N=\alpha+1}={1\over\pi}\Big{(}\chi_{|z|<\sqrt{\alpha+1}}-\chi_{|z|<\sqrt{\alpha}}\Big{)}.$
Here $(\chi_{|z|<\sqrt{\alpha+1}}-\chi_{|z|<\sqrt{\alpha}})$ corresponds to an
annulus of inner radius $\sqrt{\alpha}$ and outer radius $\sqrt{\alpha+1}$.
Generally, for random matrices of the form $A=UTV$, where $U,V$ are Haar
distributed and the singular values of $T$ converge to a compactly supported
probability measure, it is known that the eigenvalue PDF in the complex plane
is either a disk or an annulus. This is referred to as the single ring theorem
[139, 190]. Scaling of (2.4) in the bulk of the annulus, or at the boundary,
is straightforward and leads to the functional forms exhibited in Proposition
2.4 for the GinUE. Furthermore, in the double scaling regime that the spectrum
tends to form a thin annulus of width $O(1/N)$, the scaling of (2.4) in the
bulk gives rise to the weakly non-Hermitian limit (2.40) [79]. We also stress
that the induced Ginibre ensemble was introduced more generally in [7], where
the eigenvalue PDF is a mixture of (2.33) and (2.43). (The induced GinUE model
is then obtained by setting the masses to zero $m_{f}=0$ and $\tau=0$.) In
[7], the finite-$N$ expression of the correlation functions as well as their
scaling limits both at strong and weak non-Hermiticity were obtained.
### 2.5. Complex spherical ensemble
For $G_{1},G_{2}$ matrices from the GinUE, matrices of the form
$G=G_{1}^{-1}G_{2}$ are said to form the complex spherical ensemble [230]. For
with $\alpha,\beta\in\mathbb{C}$ with $|\alpha|^{2}+|\beta|^{2}=1$, introduce
the transformed pair of matrices $(C,D)$,
$C:=-\bar{\beta}G_{2}+\bar{\alpha}G_{1},\qquad D=\alpha G_{2}+\beta G_{1}.$
Since $G_{1},G_{2}$ have independent standard complex elements, it is easy to
check that the pair $(G_{1},G_{2})$ has the same distribution as $(C,D)$. As a
consequence, the eigenvalues $\\{z_{j}\\}$ of $G$ are unchanged by the
fractional linear transformation
$z\mapsto{z\alpha-\bar{\beta}\over z\beta-\bar{\alpha}}.$
This implies that upon a stereographic projection from the complex plane to
the Riemann sphere, the eigenvalue distribution of $\\{G\\}$ is invariant
under rotation of the sphere, giving rise to the name of the ensemble and
telling us that on this surface the eigenvalue density is constant.
The explicit eigenvalue PDF for the complex spherical ensemble was calculated
by Krishnapur [230].
###### Proposition 2.12.
For the complex spherical ensemble, the eigenvalue PDF is proportional to
(2.46) $\prod_{l=1}^{N}{1\over(1+|z_{l}|^{2})^{N+1}}\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2},\qquad z_{l}\in\mathbb{C}.$
###### Proof.
The joint element distribution of $(G_{1},G_{2})$ is proportional to
$\exp(-{\rm Tr}\,G_{1}^{\dagger}G_{1}-{\rm Tr}\,G_{2}^{\dagger}G_{2})$.
Substituting $G_{2}=GG_{1}$, (this gives a Jacobian factor $|\det G_{1}|^{N}$)
then integrating over $G_{1}$ gives that the element distribution of $G$ is
proportional to
(2.47) $\det(\mathbb{I}+G^{\dagger}G)^{-2N}.$
Introducing now the Schur decomposition (2.2), and changing variables as in
the proof of Proposition 2.1, we have from this that the element distribution
of the upper triangular matrix $Z$ therein is proportional to
(2.48) $\det(\mathbb{I}+Z^{\dagger}Z)^{-2N}\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2}.$
Here $\\{z_{j}\\}$ are the eigenvalues of $G$, and also the diagonal entries
of $Z$.
It remains to integrate over the strictly upper triangular entries of $Z$. For
this, denote the leading $n\times n$ sub-block of $Z$ by $Z_{n}$ and let the
product of differentials for the strictly upper entries of $Z_{n}$ be denoted
$(d\tilde{Z}_{n})$. For $p\geq n$, define
$I_{n,p}(z_{1},\dots,z_{n}):=\int{1\over\det(\mathbb{1}_{n}+Z_{n}Z_{n}^{\dagger})^{p}}\,(d\tilde{T}_{n}).$
Writing
(2.49) $Z_{n}=\begin{bmatrix}Z_{n-1}&\mathbf{u}\\\
\mathbf{0}_{n-1}^{T}&z_{n}\end{bmatrix},\qquad\mathbf{u}=[Z_{jn}]_{j=1}^{n-1},$
it’s possible to integrate out over the elements of $\mathbf{u}$ to obtain the
recurrence [203, 163]
$I_{n,p}(z_{1},\dots,z_{n})={C_{n-1,p}\over(1+|z_{n}|^{2})^{p-n+1}}I_{n-1,p-1}(z_{1},\dots,z_{n-1}),\qquad
C_{n-1,p}=\int{(d\mathbf{v})\over(1+\mathbf{v}^{\dagger}\mathbf{v})^{p}}.$
Iterating this with $n=2N$, $p=N$ shows
$\int\det(\mathbb{I}+Z^{\dagger}Z)^{-2N}\,(d\tilde{Z}_{N})\propto\prod_{l=1}^{N}{1\over(1+|z_{l}|^{2})^{N+1}},$
which when used in (2.48) implies (2.46).
∎
The stereographic projection from the south pole of a sphere with radius
$1/2$, spherical coordinates $(\theta,\phi)$, to a plane tangent to the north
pole is specified by the equation
$z=e^{i\phi}\tan{\theta\over 2},\qquad z=x+iy.$
Making this change of variables in (2.48) gives the PDF on the sphere
proportional to
(2.50) $\prod_{1\leq j<k\leq N}|u_{k}v_{j}-u_{j}v_{k}|^{2},\qquad
u=\cos(\theta/2)e^{i\phi/2},\>v=-i\sin(\theta/2)e^{-i\phi/2}.$
Here $u,v$ are the Cayley-Klein parameters, and moreover
$|u_{k}v_{j}-u_{j}v_{k}|=|\mathbf{r}_{k}-\mathbf{r}_{j}|$ where
$\mathbf{r}_{j},\mathbf{r}_{k}$ are the vector coordinates on the sphere. It
is furthermore the case that minus the logarithm of this distance solves the
Poisson equation on the sphere, and so (2.50) has the interpretation of the
Boltzmann factor at coupling $\beta=2$ of the corresponding Coulomb gas; see
[149, §15.6] for more details.
After first removing $u_{k},u_{j}$ from the product of differences in (2.50),
the Vandermonde determinant identity (2.12) can be used to compute the
correlation functions following a strategy analogous to that used in the proof
of Proposition 2.2. This calculation was first done in the context of the
corresponding two-dimensional one-component plasma, as obtained by the rewrite
of (2.50) analogous to (1.10) [81].
###### Proposition 2.13.
The $n$-point correlations for (2.50) are given by
(2.51)
$\rho_{(n,N)}\left((\theta_{1},\phi_{1}),\dots,(\theta_{n},\phi_{n})\right)=\Big{(}{N\over\pi}\Big{)}^{n}{\rm{det}}\left[{\left(u_{j}\bar{u}_{k}+v_{j}\bar{v}_{k}\right)}^{N-1}\right]_{j,k=1,\dots,n}.$
###### Remark 2.14.
1\. Changing variables $\theta_{j}=2r_{j}\sqrt{\pi/N}$ in (2.51) and taking
$N\to\infty$, the bulk scaled correlation kernel (2.18) results. Here there is
no edge regime.
2\. Denote by $a$ ($X$) an $n\times N$ ($N\times M$), $n\geq N$ ($M\geq N$)
standard complex Gaussian matrix, and set $A=a^{\dagger}a$, $Y=A^{-1/2}X$. In
terms of $Y$ define $Z=U(YY^{\dagger})^{1/2}$, which corresponds to the
induced ensemble construction of § 2.4. It was shown in [145] that the element
PDF of $Z$ is proportional to
$(\det Z^{\dagger}Z)^{M-N}{1\over\det(\mathbb{I}+Z^{\dagger}Z)^{n+M}}.$
The proof of Proposition 2.12 now gives that the corresponding eigenvalue PDF
is proportional to
(2.52)
$\prod_{l=1}^{N}{|z_{l}|^{2(M-N)}\over(1+|z_{l}|^{2})^{n+M-N+1}}\prod_{1\leq
j<k\leq N}|z_{k}-z_{j}|^{2}.$
With $M,n$ scaled with $N$, $M/N\to\alpha_{1}\geq 1,n/N\to\alpha_{2}\geq 1$
the leading order eigenvalue support now occurs in an annulus with inner and
outer radii
(2.53) $r_{1}=\sqrt{(\alpha_{1}-1)/\alpha_{2}},\qquad
r_{2}=\sqrt{\alpha_{1}/(\alpha_{2}-1)}.$
The associated correlation kernel $K_{N}$ can be expressed in terms of the
incomplete beta function
$I_{x}(a,b)\propto\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,dt$, normalised to equal
unity for $x=1$ as
(2.54) $\displaystyle K_{N}(w,z)$
$\displaystyle=\frac{1}{\pi}\frac{|zw|^{M-N}}{((1+|z|^{2})(1+|w|^{2}))^{(n+M-N+1)/2}}$
$\displaystyle\quad\times(M+n-N)(1+w\bar{z})^{n+M-N-1}\Big{(}I_{\zeta}(M-N,n)-I_{\zeta}(M,n-N)\Big{)},$
where $\zeta=(w\bar{z})/(1+w\bar{z})$, see e.g. [77]. Scaling of the
correlation functions at either of these boundaries gives the edge kernel
(2.19). Setting $w=z$ and with $\alpha_{1},\alpha_{2}$ as in (2.53) allows for
the computation of the global density [145]
(2.55) $\lim_{N\to\infty}{1\over
N}\rho_{(1),N}(z)={\alpha_{1}+\alpha_{2}-1\over\pi(1+r^{2})^{2}}\chi_{r_{1}\leq
r\leq r_{2}}.$
### 2.6. Sub-block of a unitary matrix
Let $U$ be an $(n+N)\times(n+N)$ unitary matrix, and let $A$ be the top
$N\times N$ sub-block. The non-zero eigenvalues of $A$ are then the nonzero
eigenvalues of $DUD$, where $D$ is the diagonal matrix with the first $N$
diagonal entries $1$, and the last $n$ diagonal entries $0$. The eigenvalues
of this matrix are the same as that for $UD$, since in general for square
matrices the eigenvalues of $AB$ and $BA$ coincide, and furthermore $D^{2}=D$.
For $\mathbf{u}$ a normalised eigenvector of $UD$ with eigenvalue $\lambda$,
computing the length squared of $UD\mathbf{u}$ gives
$\mathbf{u}^{\dagger}D\mathbf{u}=|\lambda|^{2}$, where use has been made of
the fact $U^{\dagger}U=\mathbb{I}$. But with the entries of $\mathbf{u}$ all
nonzero, the action of $D$ reduces the length, implying $|\lambda|<1$. With
$U$ chosen with Haar measure, it was shown by Zyczkowski and Sommers [331]
that the exact eigenvalue PDF of $A$ can be calculated.
###### Proposition 2.15.
For $A$ the top $N\times N$ sub-block of an $(n+N)\times(n+N)$ unitary matrix
chosen with Haar measure, the eigenvalue PDF is proportional to
(2.56) $\prod_{l=1}^{N}(1-|z_{l}|^{2})^{n-1}\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2},\qquad|z_{l}|<1.$
###### Proof.
Let $C$ be the block of $U$ in the first $N$ columns directly below $A$. Then
by the unitarity of $U$, $A^{\dagger}A+C^{\dagger}C=\mathbb{I}_{N}$. This
implies a joint distribution in the space of general $N\times N$ and $n\times
N$ complex matrices $A,C$ proportional to the matrix delta function
(2.57) $\delta(A^{\dagger}A+C^{\dagger}C-\mathbb{I}_{N})\propto\int e^{{\rm
Tr}((iH-\mu\mathbb{I}_{N})(A^{\dagger}A+C^{\dagger}C-\mathbb{I}_{N})}\,(dH),\quad\mu>0.$
Here in the integral form of the matrix delta function $H$ is Hermitian; see
e.g. [149, Eq. (3.27)]. Beginning with this integral form the integration of
the complex matrix $C$ can be carried out according to [149, displayed
equation below (3.27)] to give that the element PDF of $A=:A_{N}$ is
proportional to
(2.58) $F_{n}(A_{N}):=\int(\det(iH_{N}-\mathbb{I}_{N}))^{-n}e^{{\rm
Tr}((iH_{N}-\mathbb{I}_{N})(A_{N}^{\dagger}A_{N}-\mathbb{I}_{N}))}\,(dH_{N}).$
This matrix integral is the starting point for the derivation of (2.57) given
in [11, Appendix B], which we follow below.
In (2.58) the integral is unchanged by conjugating $H$ with a unitary matrix
$V$ say. Choosing $V$ to be the unitary matrix in the Schur decomposition
(2.2) allows us to effectively replace $A_{N}$ by $Z_{N}$ throughout (2.58).
Doing this, and integrating too over $\tilde{Z}_{N}$ (i.e. the strictly upper
triangular entries of $Z_{N}$), we denote the matrix integral by
$\tilde{F}_{n}(z_{1},\dots,z_{N})$.
Substituting the decomposition (2.49) for $Z_{N}$ we see that the vector
$\mathbf{u}$ occurs as a quadratic form, and can be integrated over. To
progress further, $H_{N}$ too is decomposed by a bordering procedure of the
leading $(N-1)\times(N-1)$ sub-block $H_{N-1}$, with the $N-1$ column (row)
vector $\mathbf{w}$ ($\mathbf{w}^{\dagger}$) and entry $h_{N}$ in the bottom
right corner. Key now is a determinant identity for a block matrix
$\det\begin{bmatrix}A&B\\\ C&D\end{bmatrix}=\det(D)\det(A-BD^{-1}C),$
familiar from the theory of the Schur complement (see e.g. [284]), applied to
the first term in the integrand of (2.58) with $D=h_{N}$ (a scalar). A shift
of the integration domain according to the additive rank 1 perturbation
$H_{N-1}\mapsto H_{N-1}-i\mathbf{w}\mathbf{w}^{\dagger}$ gives a quadratic
form in $\mathbf{w}$, which after integration reduces (2.58) to
(2.59)
$\tilde{F}_{n}(z_{1},\dots,z_{N})\propto\int(\det(iH_{N-1}-\mathbb{I}_{N-1}))^{-n}(ih_{N}-1)^{-n}e^{(|z_{N}|^{2}-1)(ih_{N}-1)}\\\
\times e^{{\rm
Tr}(Z_{N-1}^{\dagger}Z_{N-1}(iH_{N-1}-\mathbb{I}_{N-1}))}\,(dH_{N-1})(dZ_{N-1})dh_{N}.$
Here, using the residue theorem, the integral over $h_{N}$ can be performed,
yielding the recurrence
$\tilde{F}_{n}(z_{1},\dots,z_{N})\propto(1-|z_{N}|^{2})^{n-1}\chi_{|z_{N}|^{2}<1}\tilde{F}_{n}(z_{1},\dots,z_{N-1}).$
Iterating gives the first factor in (2.56), while the product of differences
is due to the Jacobian (2.5) for the change of variables to the Schur
decomposition as computed in the proof of Proposition 2.1. ∎
###### Remark 2.16.
The matrix integral (2.58) can be evaluated to deduce that the element
distribution of $A$ is proportional to [173, 109]
$\det(\mathbb{I}-A^{\dagger}A)^{n-N}.$ Only for $n\geq N$ is the normalisable.
This is in keeping with $A^{\dagger}A$ having $N-n$ eigenvalues equal to $1$
for $n<N$. Starting with this expression, and thus the corresponding
restriction on $n,N$, a derivation of (2.46) can be given which is analogous
to the proof of Proposition 2.12 [163].
We have seen that the eigenvalue PDF (2.46) can, after a stereographic
projection, be identified as the Boltzmann factor for a Coulomb gas model on
the sphere. The concept of stereographic projection can also be applied to a
pseudosphere, which refers to the two-dimensional hyperbolic space with
constant negative Gaussian curvature. Doing so gives the metric specifying the
Poincaré disk. Consideration of the solution of the Poisson equation on the
latter allows (2.56) to be interpreted as the Boltzmann factor for a Coulomb
gas model on the Poincaré disk [163].
The eigenvalue correlations for (2.56) are determinantal with correlation
kernel, $K_{N,n}$ say, expressible in terms of the incomplete beta function
$I_{x}(a,b)$. Thus [143, § 3.2.3]
(2.60)
$K_{N,n}(w,z)={n\over\pi}{(1-|w|^{2})^{(n-1)/2}(1-|z|^{2})^{(n-1)/2}\over(1-w\bar{z})^{n+1}}\Big{(}1-I_{w\bar{z}}(N,n+1)\Big{)}.$
In the limit $n,N\to\infty$ with $(N+n)/N=\alpha<1$ it follows from this that
(2.61) $\lim_{N\to\infty}{1\over
N}\rho_{(1),N}(z)={(1-\alpha)\over\alpha\pi}{1\over(1-|z|^{2})^{2}}\chi_{|z|^{2}<1/(1+\alpha)}.$
Hence there is a bulk regime, and an edge regime. The neighbourhood of the
origin is typical of the bulk regime, and it follows from (2.60) that
(2.62) $\lim_{n,N\to\infty\atop n/N=\alpha}{1\over
n}K_{N,n}(w/\sqrt{n},w/\sqrt{n})=K_{\infty}^{\rm b}(w,z),$
where $K_{\infty}^{\rm b}$ is specified by (2.18). And after appropriate
scaling about $|z|=\sqrt{\alpha)}$, the universal edge density as given by the
first term in (2.39) is obtained [143],
(2.63) $\lim_{n,N\to\infty\atop(n+N)/n=\alpha}\alpha(1-\alpha){1\over
N}\rho_{(1),N}\Big{(}\sqrt{\alpha}+{\xi\over\sqrt{N}}\sqrt{\alpha(1-\alpha)}\Big{)}={1\over
2\pi}\Big{(}1-{\rm erf}({\sqrt{2}}\xi)\Big{)}.$
A distinct scaling regime, referred to as close to unitary [173], or more
specifically a multiplicative rank $n$ contraction of a random unitary matrix,
is obtained by taking $N\to\infty$ with $n$ fixed in (2.60). Scaling the
eigenvalues $z_{k}=(1-y_{k}/N)e^{i\phi_{k}/N}$ gives [224] [143, Eq.
(3.2.140)]
(2.64) $\lim_{N\to\infty}{1\over
N^{2}}K_{N,n}(z_{1},z_{2})={1\over\pi}{(2\sqrt{y_{1}y_{2}})^{n-1}\over(n-1)!}\int_{0}^{1}s^{n}e^{-(y_{1}+y_{2}+i(\phi_{1}-\phi_{2}))s}\,ds.$
This functional form was first obtained for the scaled correlation kernel in
the setting of a rank $n$ additive anti-Hermitian perturbation of a GUE matrix
[170, Eq. (15) after setting $\nu(x)=1/\pi$, identifying
$\tilde{z}_{k}=(\phi_{k}+iy_{k})/2$ and taking the limit $g_{m}\to 1$
($m=1,\dots,M$) with $M$ identified as $n$].
A generalisation of the setting of Proposition 2.15 is to consider the top
$(N+L)\times N$ sub-block of an $(n+N+L)\times(n+N+L)$ unitary matrix chosen
with Haar measure. Denote such a rectangular matrix by $A_{N,L}$, and define
from this $\tilde{A}_{N,L}=V(A_{N,L}^{\dagger}A_{N,L})^{1/2}$, where $V$ is an
$N\times N$ random Haar distributed unitary matrix. It was shown in [143] that
the eigenvalue PDF of $\tilde{A}_{N,L}$ is proportional to (2.46) with an
additional factor of $\prod_{l=1}^{N}|z_{l}|^{2L}$. Denote the corresponding
correlation kernel by $K_{N,n,L}$. Scaling $L$ by requiring that
$L/N=\alpha>0$ as $N\to\infty$, scaling the eigenvalues $z_{k}$ as in the
above paragraph, and keeping $n$ fixed, it was shown in [224], [143, Eq.
(3.2.122)] that
(2.65) $\lim_{N\to\infty}{1\over
N^{2}}K_{N,n,L}(z_{1},z_{2})\Big{|}_{L/N=\alpha}={1\over\pi}{(2\sqrt{y_{1}y_{2}})^{n-1}\over(n-1)!}\int_{\alpha}^{\alpha+1}s^{n}e^{-(y_{1}+y_{2}+i(\phi_{1}-\phi_{2}))s}\,ds.$
Note the consistency with (2.64) in the limit $\alpha\to 0^{+}$.
Another class of generalisation is, for $U$ an $(n+N)\times(n+N)$ Haar
distributed unitary matrix, and $A$ a fixed diagonal matrix with first $n$
diagonal entries $a_{1},\dots,a_{n}$ and the remaining entries unity, to
consider the product $UA$ [168, 173]. With each $a_{i}=0$, this corresponds to
the truncated Haar unitary model of this subsection. The simplest case is when
$n=1$, and furthermore we take $a_{1}=a$ with $|a|<1$. Notice that this
corresponds to a multiplicative rank $1$ perturbation of $U$ — see the recent
review [157] for context from this viewpoint. The eigenvalue PDF is then
proportional to
(2.66)
$(1-|a|^{2})^{N}\delta\Big{(}|a|^{2}-\prod_{l=1}^{N+1}|z_{l}|^{2}\Big{)}\prod_{1\leq
j<k\leq N+1}|z_{j}-z_{k}|^{2},\quad|z_{l}|<1,$
(cf. (2.56)) and an explicit formula for the general $k$-point correlation
function is known [168, 173]. The latter involves determinants but technically
the state is not (unless $a=0$) a determinantal point process as no
correlation kernel can be identified. Specifically, the functional form of the
density is
(2.67) $\rho_{(1),N+1}^{UA}(z)={1\over\pi}{1\over(1-|a|^{2})^{N}}{d\over
dx}\bigg{(}{x^{N}-1\over x-1}+\Big{(}1-{|a|^{2}\over
x}\Big{)}^{N}\bigg{)}\bigg{|}_{x=|z|^{2}}.$
For recent works relating to the $UA$ model see [160, 129].
### 2.7. Products of complex Ginibre matrices
Consideration of the eigenvalue PDF for the product $G_{1}G_{2}$, where
$G_{1}$ ($G_{2}$) are independent $N\times p$ $(p\times N)$ rectangular
complex Ginibre matrices was first undertaken by Osborn [283], in the context
of a study relating to quantum chromodynamics. (We also mention that the GinUE
has been directly used in the QCD bulk spectrum [269].) Later [205] it was
realised that this eigenvalue PDF must be the same as that for the product
$\tilde{G}_{1}\tilde{G}_{2}$ (or $\tilde{G}_{2}\tilde{G}_{1})$, where each
$\tilde{G}_{i}$ is an $N\times N$ complex random matrix with element
distribution proportional to
(2.68) $|\det\tilde{G}_{i}\tilde{G}_{i}^{\dagger}|^{\nu_{i}}e^{-{\rm
Tr}\,\tilde{G}_{i}\tilde{G}_{i}^{\dagger}},\quad\nu_{1}=0,\>\nu_{2}=p-N;$
note the construction (2.41) for random matrices with this element PDF. Hence
it suffices to consider the square case. Key for this is the generalised Schur
decomposition (equivalent to the so-called QZ decomposition in numerical
linear algebra)
(2.69)
$\tilde{G}_{1}=UZ_{1}V,\qquad\tilde{G}_{2}=V^{\dagger}Z_{2}U^{\dagger},$
where $Z_{1},Z_{2}$ are upper triangular matrices with diagonal entries
$\\{z^{(1)}_{j}\\},\\{z^{(2)}_{j}\\}$ such that
(2.70) $z^{(1)}_{j}z^{(2)}_{j}=z_{j}$
with $\\{z_{j}\\}$ the eigenvalues of $\tilde{G}_{1}\tilde{G}_{2}$.
For the change of variables (2.69) the wedge product strategy of the proof of
Proposition 2.1 can again be implemented [283], [149, proof of Proposition
15.11.2] to give for the Jacobian (2.5). Substituting (2.69) in (2.68) and
recalling (2.70), it follows that the eigenvalue PDF is proportional to
(2.71) $\prod_{l=1}^{N}w^{(2)}(z_{l})\prod_{j<k}|z_{j}-z_{k}|^{2},\quad
w^{(2)}(z):=\int_{\mathbb{C}}d^{2}z_{1}\,|z_{1}|^{2\nu_{1}}\int_{\mathbb{C}}d^{2}z_{2}\,|z_{2}|^{2\nu_{2}}\delta(z-z_{1}z_{2})e^{-|z_{1}|^{2}-|z_{2}|^{2}}.$
Moreover, it was shown in [283] that $w^{(2)}(z)$ can be expressed in terms of
the modified Bessel function $K_{\nu_{2}-\nu_{1}}(2|z|)$, assuming
$\nu_{2}\geq\nu_{1}$.
Akemann and Burda [9] (in the case of all matrices square), and soon after
Adhikari et al. [3] (the general rectangular case), generalised (2.71) to hold
for the product of $M$ complex Gaussian matrices. As already indicated in the
case $M=2$, following the work of Ipsen and Kieburg [205], it is now known
that the rectangular case can be reduced to the square case, where the square
matrices have distribution as in (2.68), with the $\nu_{i}$ equal to the
difference (assumed non-negative) between the number of rows in $G_{i}$ and
the number of rows in $G_{1}$ ($=N$). The role of the modified Bessel
function, as the special function evaluating the weight $w^{(2)}(z)$ in
(2.71), is now played by the Meijer $G$-function
(2.72) $G_{p,q}^{m,n}\Big{(}z\Big{|}{a_{1},\dots,a_{p}\atop
b_{1},\dots,b_{q}}\Big{)}={1\over 2\pi
i}\int_{\mathcal{C}}{\prod_{j=1}^{m}\Gamma(b_{j}-s)\prod_{j=1}^{n}\Gamma(1-a_{j}+s)\over\prod_{j=m+1}^{q}\Gamma(1-b_{j}+s)\prod_{j=n+1}^{p}\Gamma(a_{j}-s)}z^{s}\,ds,$
where $\mathcal{C}$ is an appropriate contour as occurs in the inversion
formula for the corresponding Mellin transform; see e.g. [265] for an extended
account.
###### Proposition 2.17.
Consider the product $\tilde{G}_{1}\cdots\tilde{G}_{M}$ (in any order) of
complex Gaussian matrices with element PDF as given in (2.68), with each
$\nu_{i}\geq 0$ but otherwise unrestricted. The eigenvalue PDF is proportional
to
(2.73) $\prod_{l=1}^{N}w^{(M)}(z_{l})\prod_{j<k}|z_{j}-z_{k}|^{2},\quad
w^{(M)}(z)=G^{M,0}_{0,M}\Big{(}|z|^{2}\Big{|}{\underline{\hskip
14.22636pt}\atop\nu_{1},\dots,\nu_{M}}\Big{)}.$
The functional form (2.73) remains valid for the product of rectangular
complex Ginibre matrices $G_{1}\cdots G_{M}$, with $\nu_{i}$ specified as in
the second sentence above (2.72).
###### Proof.
(Sketch) The generalised Schur decomposition (2.69) can be extended to a
general number $M$ of square matrices to give what is referred to as the
periodic Schur form [66], [266, Corollary 3.2]
(2.74) $\tilde{G}_{i}=U_{i}Z_{i}V_{i},\qquad
V_{i}=U_{i+1}^{\dagger}\>(i=1,\dots,M-1),\>V_{M}=U_{1}^{\dagger};$
in [9] this was deduced independently. The key features with respect to
computing the eigenvalue PDF of the product
$\tilde{G}_{1}\cdots\tilde{G}_{M}$, as already discussed in the case $M=2$,
again hold true. In particular the Jacobian factor for the change of variables
is given by (2.5), and the $k$-th diagonal entry of each $Z_{i}$ multiply
together to give the eigenvalue $z_{k}$ of the product. It follows that the
eigenvalue PDF is given by (2.73) with
(2.75)
$w^{(M)}(z)\propto\int_{\mathbb{C}}d^{2}z_{1}\,|z_{1}|^{2\nu_{1}}\cdots\int_{\mathbb{C}}d^{2}z_{M}\,|z_{M}|^{2\nu_{M}}\delta(z-z_{1}\cdots
z_{M})e^{-\sum_{j=1}^{M}|z_{j}|^{2}}.$
As noted in [9, 3], taking the Mellin transform of this expression leads to
the Meijer $G$-function form in (2.73). ∎
An easy consequence of (2.73) is that the eigenvalue correlations are
determinantal with kernel, $K_{N,M}$ say, given by [9, 3],
(2.76)
$K_{N,M}(w,z)=\Big{(}w^{(M)}(w)w^{(M)}({z})\Big{)}^{1/2}\sum_{j=1}^{N}{(w\bar{z})^{j-1}\over\prod_{m=1}^{M}\Gamma(j+\nu_{m})}.$
Rigorous asymptotic analysis of (2.76) can be carried out [9, 260]. For
example, with $w=z$ and $z\mapsto N^{M/2}z$, one obtains
(2.77) $\lim_{N\to\infty}N^{M-1}\rho_{(1),N}(N^{M/2}z)={|z|^{-2+2/M}\over\pi
M}\chi_{|z|<1},$
where it is assumed each $\nu_{i}$ is fixed. To interpret this result, form
the $M$-th power of a single GinUE matrix $G$. The eigenvalues are
$\\{\tilde{z}_{j}=z_{j}^{M}\\}$ with $\\{z_{j}\\}$ the the eigenvalues of $G$.
Changing variables in the circular law (2.17) $\tilde{z}=z^{M}$ gives the
density (2.77) in the variable $\tilde{z}$. Hence, as anticipated in [73], the
global eigenvalue density for the product of $M$ independent GinUE matrices is
identical to that of the $M$-th power of a single GinUE matrix. This same
global scaling limit is also well defined if some or all of the $\nu_{i}$ are
proportional to $N$. An explicit limit formula has been obtained by Liu and
Wang [260]. It is found that the density has the support of an annulus if each
$\mu_{i}:=\lim_{N\to\infty}\nu_{i}/N$ is positive, but otherwise is again
supported in a disk, albeit with a density function distinct to that in
(2.77).
One observes from (2.77) a singularity at $z=0$, not present in the circular
law global density (2.17) for GinUE. A consequence is that the correlation
kernel about this point, simply obtained by setting the upper terminal of the
sum in (2.76) equal to infinity, is distinct from the corresponding kernel
(2.18) for the bulk scaled GinUE. On the other hand, it is shown in [260] that
the bulk scaled GinUE kernel is reclaimed by choosing the origin a distance
$\alpha\sqrt{N}$ from $z=0$, for any $0<\alpha<1$. Also, from [9, 260] we know
that the scaling of (2.76) about $|z|=1$ reclaims the kernel (2.19) for the
edge scaled GinUE.
###### Remark 2.18.
1\. Required in the derivation of (2.77) from (2.76) is knowledge of the
$z\to\infty$ asymptotic expansion [265]
(2.78) $G_{0,M}^{M,0}\Big{(}z\Big{|}{\underline{\hskip
14.22636pt}\atop\nu_{1},\dots,\nu_{M}}\Big{)}\sim{1\over\sqrt{M}}\Big{(}{2\pi\over
z}\Big{)}^{(M-1)/2}e^{-Mz^{1/M}}z^{(\nu_{1}+\cdots+\nu_{M})/M}\Big{(}1+{\rm
O}(z^{-1/M})\Big{)}.$
Of particular interest is the exponential term herein, which with $z$ replaced
by $|z|^{2}$ as required in (2.76) reads $e^{-M|z|^{2/M}}$. This suggests a
modification of $U$ in (1.10) to read
(2.79) $U_{M}:={M\over 2}\sum_{j=1}^{N}|z_{j}|^{2/M}-\sum_{1\leq j<k\leq
N}\log|z_{k}-z_{j}|.$
Indeed starting from $U_{M}$ and repeating the working of Remark 2.5 reclaims
(2.77). The correlation kernel appearing in such a calculation is called the
Mittag-Leffler kernel. This terminology applies too in the more general case
that a term $-c\sum_{j=1}^{N}\log|z_{j}|$ is included in (2.79) [37, 41]. This
very case, for $c=(k-M)/M$, $(k=1,\dots,M)$, appears in the calculation of the
joint distribution of the eigenvalues of the $M$-th power ($M\leq N)$ of a
GinUE matrix [127, Th. 1.5]. We also refer to [221, 222] for the appearance of
particular Mittag-Leffler point process as a degenerate limit of so-called
elliptic determinantal point processes, characterised by the product of
differences in (1.7) being replaced by a product over Jacobi theta functions.
2\. The eigenvalue PDF for a product of GinUE matrices and inverses of GinUE
matrices has been shown to be of the form (2.73), but with the Meijer
$G$-function therein replace by a different Meijer $G$-function [3]. The
corresponding limiting eigenvalue density, in the case of equal numbers of
matrices and inverses, is [187, 330]
(2.80) $\lim_{N\to\infty}\rho_{(1),N}(z)={1\over\pi
M}{|z|^{-2+2/M}\over(1+|z|^{2/M})^{2}}.$
This functional form relates to the $M=1$ case in the same way as (2.77)
relates to (2.17), being the $M$-th power of the so-called spherical law.
3\. The case $M=2$ of Proposition 2.17 permits a generalisation. Thus for the
matrices in the product $\tilde{G}_{1}\tilde{G}_{2}$ choose
$\tilde{G}_{1}=\sqrt{1+\tau}X_{1}+\sqrt{1-\tau}X_{2},\quad\tilde{G}_{2}=\sqrt{1+\tau}X_{1}^{\dagger}+\sqrt{1-\tau}X_{2}^{\dagger},\qquad
0<\tau<1,$
where $X_{1},X_{2}$ are $N\times(N+p)$ rectangular complex Gaussian matrices.
The joint element distribution is then proportional to
$\exp\bigg{(}-{1\over 1-\tau^{2}}{\rm
Tr}\Big{(}\tilde{G}_{1}^{\dagger}\tilde{G}_{1}+\tilde{G}_{2}^{\dagger}\tilde{G}_{2}-2\tau{\rm
Re}\,\tilde{G}_{1}\tilde{G}_{2}\Big{)}\bigg{)};$
cf. (2.34). Results in [283] give that the eigenvalue PDF of
$\tilde{G}_{1}\tilde{G}_{2}$ is of the form in (2.71) but with $w^{(2)}(z)$
now dependent on $\tau$. Due to the shape of the resulting droplet, this gives
rise to the so-called shifted elliptic law [14, Th. 1].
4\. The product of $M$ random matrices in the limit $M\to\infty$ is of
interest from a dynamical systems viewpoint, as the scaled logarithm of the
singular values gives the Lyapunov spectrum. Closely related for product
matrices themselves are the stability exponents, defined as ${1\over
M}\log|z_{k}|$, ($k=1,\dots,N$). In fact for bi-unitary invariant random
matrices at the least, the Lyapunov and stability exponents are the same
[289]. Works relating to the computation of these exponents and their related
statistical properties for GinUE include [150, 220, 10, 152, 204, 262, 185,
261, 6].
### 2.8. Products of truncated unitary matrices
The eigenvalue PDF for a truncation of a Haar distributed unitary matrix has
been given in Proposition 2.15. It turns out that knowledge of the evaluation
of the matrix integral (2.59) as implied in the proof of Proposition 2.15,
used in conjunction with the periodic Schur form (2.74), is sufficient to
allow for the determination of the eigenvalue PDF of the product of $M$
truncated Haar distributed unitary matrices [11].
###### Proposition 2.19.
Consider $M$ independent Haar distributed $(n_{j}+N)\times(n_{j}+N)$ unitary
matrices, with the top $N\times N$ sub-block of each denoted $A_{j}$
($j=1,\dots,M$). The eigenvalue PDF of the product $A_{1}\cdots A_{M}$ (in any
order) is given by (2.73) with
(2.81) $w^{(M)}(z)=G_{M,M}^{M,0}\Big{(}|z|^{2}\Big{|}{n_{1},\dots,n_{M}\atop
0,\dots,0}\Big{)}\chi_{|z|<1}.$
###### Proof.
(Sketch) The method of proof of Proposition 2.17, which begins with the
periodic Schur decomposition (2.74), and then integrates out over the upper
triangular entries of the matrices $Z_{j}$ (here this latter step requires
knowledge of the evaluation of the matrix integral (2.59)) gives (2.73) with
$w^{(M)}(z)\propto\int_{\mathbb{C}}d^{2}\,z_{1}\cdots\int_{\mathbb{C}}d^{2}\,z_{M}\,\delta(z-z_{1}\cdots
z_{M})\prod_{j=1}^{M}(1-|z_{j}|^{2})^{n_{j}-1}$
Taking the Mellin transform of this expression leads to the Meijer
$G$-function form in (2.81).
∎
Analogous to (2.76), it follows from the result of Proposition 2.19 that the
eigenvalue correlations are determinantal with kernel (to be denoted
$K_{N,M}^{\rm U}$) given by [11]
(2.82) $K_{N,M}^{\rm
U}(z_{1},z_{2})=\Big{(}w^{(M)}(z_{1})w^{(M)}({z_{2}})\Big{)}^{1/2}\sum_{j=1}^{N}(z_{1}\bar{z}_{2})^{j-1}\prod_{m=1}^{M}\prod_{m=1}^{M}\frac{(n_{m}+j-1)!}{(j-1)!}.$
Note here, that in distinction to (2.76), the $w^{(M)}$ are given by (2.81).
Suppose all the $n_{i}$ are equal to $n$, and that for large $N$,
$n/N=\alpha>0$. Using (2.82) in the case $z_{1}=z_{2}$, it is derived in [11]
that
(2.83) $\rho_{(1),N}(z)\sim{\alpha N\over\pi
M}{|z|^{-2+2/M}\over(1-|z|^{2/M})^{2}}\chi_{|z|<(1+\alpha)^{-M/2}};$
cf. (2.80).
In the same limit, with the scaling $z\mapsto z/N^{M/2}$ (and similarly for
$w$), the kernel (2.82) multiplied by $N^{-M}$ has the same limit as the
kernel (2.76) for the product of $M$ GinUE matrices about the origin in the
case of each matrix square (all $\nu_{i}=0$). If instead a point $z_{0}$ with
$0<z_{0}<(1+\alpha)^{-M/2}$ is chosen before this scaling, one obtains instead
the GinUE result (2.18), while at the boundary of support, generalising (2.63)
it is found that [11]
(2.84) $\lim_{n,N\to\infty\atop
n/N=\alpha}\frac{M}{N}{\alpha\over(1+\alpha)^{M+1}}\rho_{(1),N}\bigg{(}{1\over(1+\alpha)^{M/2}}+{\xi\over\sqrt{N}}{\sqrt{M}\alpha^{1/2}\over(1+\alpha)^{(M+1)/2}}\bigg{)}={1\over
2\pi}\Big{(}1-{\rm erf}({\sqrt{2}}\xi)\Big{)},$
thus again exhibiting the universal functional form seen at the edge scaling
of the GinUE (2.28).
Also considered in [11] is a close to unity scaling, with $n_{i}=n$ all fixed
($i=1,\dots,M$) as $N\to\infty$. Scaling the eigenvalues
$z_{k}=(1-y_{k}/N)e^{i\phi_{k}/N}$, it is found that (2.64) again holds but
with each occurrence of $n$ replaced by $nM$.
### 2.9. The distribution of the squared eigenvalue moduli
All the explicit eigenvalue PDFs obtained in the above subsections of §2,
excluding the elliptic Ginibre ensemble, have the form
(2.85) $\prod_{l=1}^{N}w(|z_{l}|^{2})\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2},$
for some weight function $w$. In particular, they are invariant under
rotations about the origin. An observation of Kostlan [229] for the GinUE case
(1.7) of (2.85) is that the set of squared eigenvalue moduli
$\\{|z_{j}|^{2}\\}_{j=1}^{N}$, appropriately ordered, are independently
distributed (specifically as gamma random variables $\\{\Gamma[j;1]\\}$); see
too [94, Theorem 1.2].
###### Proposition 2.20.
Let $F_{w}(s_{1},\dots,s_{N})$ denote the PDF for the distribution of
$\\{s_{j}:=|z_{j}|^{2}\\}_{j=1}^{N}$ for the PDF (2.85). We have
(2.86) $F_{w}(s_{1},\dots,s_{N})={1\over N!}{\rm
Sym}\,\prod_{j=1}^{N}{w(s_{j})s_{l}^{l-1}\over\int_{0}^{\infty}w(s)s^{j-1}\,ds},$
where Sym denotes symmetrisation with respect to $\\{s_{j}\\}$.
###### Proof.
Starting with (2.11), then substituting (2.12) and its complex conjugate, we
see
$\prod_{l=1}^{N}w(|z_{l}|^{2})\prod_{1\leq j<k\leq
N}|z_{k}-z_{j}|^{2}=\prod_{l=1}^{N}w(|z_{l}|^{2})\det[f_{k-1}(z_{j})]_{j,k=1}^{N}\det[g_{k-1}(z_{j})]_{j,k=1}^{N},$
where $f_{l}(z)=z^{l}$, $g_{l}(z)=\bar{z}^{l}$. According to Andréief’s
identity (see e.g. [154]), it follows from this that the integral over
$z_{l}\in\mathbb{C}$ $(l=1,\dots,N)$, $I_{w,N}$ say, is itself given by a
determinant
$I_{w,N}=N!\det\Big{[}\int_{\mathbb{C}}w(|z_{l}|^{2})f_{j-1}(z)g_{k-1}(z)\,d^{2}z\Big{]}_{j,k=1}^{N}.$
Substituting the explicit form of $f_{l},g_{l}$ and changing to polar
coordinates $z_{l}=r_{l}e^{i\theta_{l}}$ shows that only the diagonal terms
are non-zero. This allows the determinant to be evaluated
(2.87) $I_{w,N}=N!\pi^{N}\prod_{j=1}^{N}\int_{0}^{\infty}w(s)s^{j-1}\,ds,$
where in each integration the change of variables $r^{2}=s$ has been made.
Forming now $I_{w\phi,N}/I_{w,N}$, where $\phi$ is a suitable test function,
and assuming that (2.85) is normalisable, the result (2.86) follows. ∎
## 3\. Fluctuation formulas
### 3.1. Counting function in general domains
The eigenvalues of a non-Hermitian matrix are examples of point processes in
the plane. Statistical quantities characterising the point process are
functions of the eigenvalues $\\{z_{j}\\}$ of the form
$\sum_{j=1}^{N}f(z_{j})$ — referred to as linear statistics — for given $f$.
Such statistics are closely related to the correlation functions. Thus
(3.1)
$\Big{\langle}\sum_{j=1}^{N}f(z_{j})\Big{\rangle}=\int_{\mathbb{C}}f(z)\rho_{(1),N}(z)\,d^{2}z,$
while
(3.2) $\displaystyle{\rm
Cov}\,\Big{(}\sum_{l=1}^{N}f(z_{l}),\sum_{l=1}^{N}g(z_{l})\Big{)}$
$\displaystyle=\int_{\mathbb{C}}d^{2}z\int_{\mathbb{C}}d^{2}z^{\prime}\,f(z)g(z^{\prime})\Big{(}\rho_{(2),N}^{T}(z,z^{\prime})+\rho_{(1),N}(z)\delta(z-z^{\prime})\Big{)}$
$\displaystyle=-{1\over
2}\int_{\mathbb{C}}d^{2}z\int_{\mathbb{C}}d^{2}z^{\prime}\,(f(z)-f(z^{\prime}))(g(z)-g(z^{\prime}))\rho_{(2),N}^{T}(z,z^{\prime}),$
see e.g. [158, §2.1]. Here
$\rho_{(2),N}^{T}(z,z^{\prime}):=\rho_{(2),N}(z,z^{\prime})-\rho_{(1),N}(z)\rho_{(1),N}(z^{\prime})$.
One of the most prominent examples of a linear statistic is the choice
$f(z)=\chi_{z\in\mathcal{D}}$, where $\mathcal{D}\in\mathbb{C}$. The linear
statistic is then the counting function for the number of eigenvalues in
$\mathcal{D}$, $N(\mathcal{D})$ say. Let $E_{N}(n;\mathcal{D})$ denote the
probability that there are exactly $n$ eigenvalues in $\mathcal{D}$, so that
$E_{N}(n;\mathcal{D})={\rm Pr}\,(\sum_{j=1}^{N}\chi_{z_{j}\in\mathcal{D}}=n)$.
Denote the corresponding generating function (in the variable $1-\xi$) by
$\tilde{E}_{N}(\xi;\mathcal{D})$ so that
(3.3)
$\tilde{E}_{N}(\xi;\mathcal{D})=\sum_{n=0}^{N}E_{N}(n;\mathcal{D})(1-\xi)^{n}.$
Note that with $1-\xi=e^{it}$ this corresponds to the characteristic function
for the probability mass functions $\\{E_{N}(n;\mathcal{D})\\}$. A
straightforward calculation (see e.g. [149, Prop. 9.1.1]) shows that
$\tilde{E}_{N}(\xi;\mathcal{D})$ can be expressed in terms of the correlation
functions according to
(3.4) $\tilde{E}_{N}(\xi;\mathcal{D})=1+\sum_{n=1}^{N}{(-\xi)^{n}\over
n!}\int_{\mathcal{D}}d^{2}z_{1}\cdots\int_{\mathcal{D}}d^{2}z_{n}\,\rho_{(n),N}(z_{1},\dots,z_{n}).$
It can readily be checked that (3.1) and (3.2) in the special case
$f(z)=g(z)=\chi_{z\in\mathcal{D}}$ are consistent with (3.4).
Specialising now to the circumstance that the PDF for the eigenvalues is of
the form (2.85), a particular product formula for
$\tilde{E}_{N}(\xi;\mathcal{D})$ can be deduced, which was known to Gaudin
[178].
###### Proposition 3.1.
Let $\\{\rho_{(n),N}\\}$ in (3.4) be given by (2.9) where the correlation
kernel $K_{N}$ corresponds to (2.85). Let $\mathbb{K}_{N,\mathcal{D}}$ denote
the integral operator supported on $z_{2}\in\mathcal{D}$ with kernel
$K_{N}(z_{1},z_{2})$. This integral operator has at most $N$ non-zero
eigenvalues $\\{\lambda_{j}(\mathcal{D})\\}_{j=1}^{N}$, where
$0\leq\lambda_{j}(\mathcal{D})\leq 1$, and
(3.5)
$\tilde{E}_{N}(\xi;\mathcal{D})=\prod_{j=1}^{N}(1-\xi\lambda_{j}(\mathcal{D})).$
###### Proof.
Let $\\{p_{s}(z)\\}_{s=0}^{\infty}$ be a set of orthogonal polynomials with
respect to the inner product $\langle
f,g\rangle:=\int_{\mathbb{C}}w(|z|^{2})f(z)g(\bar{z})\,d^{2}z$ with
corresponding normalisation denoted $h_{s}$. Taking as a basis
$\\{(w(|z|^{2})^{1/2}p_{k}(z)\\}_{k=0}^{\infty}$ it is straightforward to
check that
(3.6)
$K_{N}(z_{1},z_{2})=(w(|z_{1}|^{2})w(|z_{2}|^{2}))^{1/2}\sum_{s=0}^{N-1}{p_{s}(z_{1})\overline{p_{s}(z_{2})}\over
h_{s}},$
and so the eigenfunctions of $\mathbb{K}_{N,\mathcal{D}}$ are of the form
$(w(|z|^{2})^{1/2}\sum_{s=0}^{N-1}c_{s}p_{s}(z)$ (see e.g. [149, proof of
Prop. 5.2.2]). Hence there are at most $N$ nonzero eigenvalues, which moreover
can be related to an Hermitian matrix and so must be real. In terms of these
eigenvalues, the determinantal form (2.9) substituted in (3.4) implies (3.5) —
this is a result from the theory of Fredholm integral operators (see e.g.
[326]). Since by definition, each $n$-point correlation is non-negative, we
see from the RHS of (3.4) that $\tilde{E}_{N}(\xi;\mathcal{D})>0$ for $\xi<0$,
and so $\lambda_{j}(\mathcal{D})\geq 0$ ($j=1,\dots,N$). Also, the definition
(3.3) tells us that $\tilde{E}_{N}(\xi;\mathcal{D})>0$ for all $\xi<1$. This
would contradict (3.5) if it was to be that any $\lambda_{j}(\mathcal{D})>1$,
since in this circumstance there would be a $\xi$ is this range such that
$\tilde{E}_{N}(\xi;\mathcal{D})$ vanishes. ∎
Consider $\sum_{j=1}^{N}x_{j}$ where $x_{j}\in\\{0,1\\}$ is a Bernoulli random
variable with ${\rm Pr}\,(x_{j}=1)=\lambda_{j}(\mathcal{D})$. The
characteristic function is
$\prod_{j=1}^{N}(1-\lambda_{j}(\mathcal{D})+e^{it}\lambda_{j}(\mathcal{D}))$.
With $e^{it}=1-\xi$ this gives the RHS of (3.5). But it has already been noted
that $\tilde{E}_{N}(\xi;\mathcal{D})$ with $\xi$ related to $e^{it}$ in this
way is the characteristic function for the counting statistic
$\mathcal{N}(\mathcal{D})$, and hence the equality in distribution
$\mathcal{N}(\mathcal{D})\mathop{=}\limits^{\rm d}\sum_{j=1}^{N}{\rm
Bernoulli}(\lambda_{j}(\mathcal{D}))$ [196, 203]. From this, it follows using
the standard arguments (see [140, § XVI.5, Theorem 2] ) that a central limit
theorem holds for $\mathcal{N}(\mathcal{D}_{N})$ (here the subscript $N$ on
$\mathcal{D}_{N}$ is to indicate that the region $\mathcal{D}$ depends on
$N$),
(3.7)
$\lim_{N\to\infty}{\mathcal{N}(\mathcal{D}_{N})-\langle\mathcal{N}(\mathcal{D}_{N})\rangle\over({\rm
Var}\,\mathcal{N}(\mathcal{D}_{N}))^{1/2}}\mathop{=}\limits^{\rm d}{\rm
N}[0,1],$
valid provided ${\rm Var}\,\mathcal{N}(\mathcal{D}_{N})\to\infty$ as
$N\to\infty$; see also [111, 316, 313].
A stronger result, extending the central limit theorem (3.7), follows from the
fact that (3.5) in the variable $z=1-\xi$ has all its zeros on the negative
real axis [164].
###### Proposition 3.2.
In the setting of the applicability of Proposition 3.1, and with
$\sigma_{\mathcal{N}_{D}}:=({\rm Var}\,{\mathcal{N}}(\mathcal{D}_{N}))^{1/2}$,
we then have that $\\{E_{N}(k;\mathcal{D}_{N})\\}$ satisfy the local central
limit theorem
(3.8)
$\lim_{N\to\infty}\,\mathop{\sup}\limits_{x\in(-\infty,\infty)}\Big{|}\sigma_{\mathcal{N}_{D}}E_{N}([\sigma_{\mathcal{N}_{D}}x+\langle{\mathcal{N}}(\mathcal{D}_{N})\rangle];\mathcal{D}_{N})-{1\over\sqrt{2\pi}}e^{-x^{2}/2}\Big{|}=0.$
###### Proof.
The fact that the zeros of (3.5) with the variable $z=1-\xi$ are on the
negative real axis implies, by Newton’s theorem on log-cavity of the sequence
of elementary symmetric functions [280], that $\\{E_{N}(k;\mathcal{D})\\}$ is
log concave. It is known that log-cavity is a sufficient condition for
extending a central limit theorem to a local limit theorem [53]. ∎
For large $N$, inside the disk of radius $\sqrt{N}$, the eigenvalue density
for GinUE is constant and the full distribution is rotationally invariant. In
such circumstances, for a two-dimensional point process in general, it is
known [50] that Var$\,\mathcal{N}(\mathcal{D}_{N})$ cannot grow slower than of
order $|\partial\mathcal{D}_{N}|$, i.e. the length of the boundary of
$\mathcal{D}_{N}$. Thus both (3.4) and (3.5) are valid for any region
$\mathcal{D}_{N}$ constrained strictly inside the disk of radius $\sqrt{N}$
and with a boundary of length tending to infinity with $N$. In fact for GinUE
more precise asymptotic information is available [97, Eq. (11)], [141, Eq.
(2.7)], which gives that for any $D_{0}\subseteq\\{z:|z|\leq 1\\}$,
(3.9) ${\rm Var}\,\mathcal{N}(\sqrt{N}D_{0})=\sqrt{N}{|\partial D_{0}|\over
2\pi}\int_{-\infty}^{\infty}\Big{(}{\rm Var}\,\chi_{U\leq(1+{\rm
erf}(t/\sqrt{2}))/2}\Big{)}\,dt+{\rm O}\Big{(}{1\over N^{1/2}}\Big{)}.$
Here $U$ is a Bernoulli random variable. A direct calculation gives that the
integral evaluates to $\sqrt{1\over\pi}$ — the advantage of the form (3.9) is
that it remains true if the appearance of the variance throughout is replaced
by any even cumulant [97, 141]. In particular this shows that the growth of
the variance with respect to the region is the smallest order possible — the
corresponding point process is then referred to as being hyperuniform [322,
321, 181]. A corollary of the property of being hyperuniform, together with
the fast decay of the correlations, is that the bulk scaled GinUE exhibits
number rigidity [183, 182]. This means that conditioning on the positions of
the (infinite number of) points outside a region $\mathcal{D}$ fully
determines the number of (but not positions of) the points inside
$\mathcal{D}$, and their centre of mass.
### 3.2. Counting function in a disk
In the special case that $\mathcal{D}_{N}$ is a disk of radius $R$ centred at
the origin (we write this as $D_{R}$), the polynomials in (3.6) are simply the
monomials $p_{s}(z)=z^{s}$. The eigenfunctions of $\mathbb{K}_{N,\mathcal{D}}$
are also given in terms of the monomials as
$\\{(w(|z|^{2}))^{1/2}z^{j-1}\\}_{j=1,\dots,N}$, and hence for the
corresponding eigenvalues we have
$\lambda_{j}(D_{R})=\int_{0}^{R^{2}}r^{j-1}w(r)\,dr\Big{/}\int_{0}^{\infty}r^{j-1}w(r)\,dr,\qquad
j=1,\dots,N.$
Substituting in (3.5) and choosing $w(|z|^{2})=\exp(-|z|^{2})$ then shows that
for the GinUE [146]
(3.10)
$\tilde{E}_{N}(\xi;D_{R})=\prod_{j=1}^{N}\Big{(}1-\xi{\gamma(j;R^{2})\over\Gamma(j)}\Big{)},$
where $\gamma(j;x)$ denotes the (lower) incomplete gamma function. Note that
this remains valid for $N\to\infty$ in keeping with the discussion of the
previous paragraph. Setting $\xi=1$, asymptotic expansions for the incomplete
gamma function can be used to deduce the $N\to\infty$ asymptotic expansion of
$E_{N}(0;D_{\alpha\sqrt{N}})$,
(3.11) $E_{N}(0;D_{\alpha\sqrt{N}})=\exp\Big{(}C_{1}N^{2}+C_{2}N\log
N+C_{3}N+C_{4}\sqrt{N}+{1\over 3}\log N+{\rm O}(1)\Big{)},\quad 0<\alpha<1.$
Here the constants $C_{1},\dots,C_{4}$ depend on $\alpha$ and are known
explicitly (e.g. $C_{1}=-\alpha^{4}/4$), being first given in [146]. The first
two of these can be deduced from the large $R$ expansion of the quantity
$F_{\infty}(0;D_{R})$, defined in Remark 3.3.1 below, given in the still
earlier work [189]. The $\log N$ term was determined recently in [98], as too
was the explicit form of the next order term, a constant with respect to $N$.
Let us also mention that for any $p\geq 2$, the $p$-th cumulant
$\kappa_{p}(D_{R})$ of the number of eigenvalues in $D_{R}$ can be written as
(3.12)
$\kappa_{p}(R)=(-1)^{p+1}\sum_{j=0}^{N-1}{\rm{Li}}_{1-p}\Big{(}1-\frac{1}{\lambda_{j}(D_{R})}\Big{)},$
where ${\rm{Li}}_{s}(x)=\sum_{k=1}^{\infty}k^{-s}x^{k}$ is the polylogarithm
function. The formula (3.12) as well as its large $N$ behaviour both in the
bulk and at the edge were obtained in [239].
We turn our attention now to the circumstance that a (possibly scaled) large
$N$ limit has already been taken, and ask about the fluctuations of the number
of particles in a region $\mathcal{N}(\mathcal{D})$ for large values of
$|\mathcal{D}|$, i.e. the volume of $\mathcal{D}$. The first point to note is
that if the coefficient of $\xi^{n}$ in (3.4) tends to zero as $N\to\infty$,
then the expansion remains valid in this limit [254]. The decay is easy to
establish in the determinantal case, since then (see e.g. [149, Eq. (9.13)])
$\rho_{(n),N}(z_{1},\dots,z_{n})\leq\prod_{l=1}^{n}\rho_{(1),N}(z_{l})$. Hence
it is sufficient that $\int_{\Omega}\rho_{(1),N}(z)\,d^{2}z$ be bounded for
$N\to\infty$. With the limiting form of (3.4) valid, the theory of Fredholm
integral operators [326] tells us that the limit of (3.5) is well defined with
the RHS identified as the Fredholm determinant
$\det(\mathbb{I}-\xi\mathbb{K}_{\infty,\mathcal{D}})$. In (3.10) the limit
corresponds to simply replacing the upper terminal of the product by $\infty$.
We stipulate the further structure that the correlation kernel be Hermitian,
as holds for the appropriately scaled form of (3.6). Then the argument of the
proof of Proposition 3.1 tells us that the eigenvalues of
$\mathcal{K}_{\infty,\mathcal{D}}$ are all between $0$ and $1$, which in turn
allows the reasoning leading to (3.7) to be repeated. The conclusion is,
assuming ${\rm Var}\,\mathcal{N}(\mathcal{D})\to\infty$ as
$|\mathcal{D}|\to\infty$ which as already remarked is guaranteed by the
results of [50], that (3.7) remains valid with $\mathcal{D}_{N}$ replaced by
$\mathcal{D}$, and the limit $N\to\infty$ replaced by the limit
$|\mathcal{D}|\to\infty$ [111, 316, 203].
It is moreover the case that in the above setting and with these modifications
the local central limit theorem of Proposition 3.2 remains valid [164].
Another point of interest is that the expansion (3.11) is uniformly valid in
the variable $R=\alpha\sqrt{N}$, provided this quantity grows with $N$, and
hence also provides the large $R$ expansion of $E_{\infty}(0;D_{R})$. Finally
we consider results of [248] as they apply to number fluctuations in the
infinite GinUE. The plane is to be divided into squares $\Gamma_{j}$ of area
$L^{2}$ with centres at $L\mathbb{Z}^{2}$. Define
$\Upsilon_{j}=\mathcal{N}(\Gamma_{j})/\sqrt{{\rm
Var}\,\mathcal{N}(\Gamma_{j})}$. For large $L$, in keeping with (3.9) we have
${\rm Var}\,(\Gamma_{j})\sim 2L/\pi^{3/2}$. The question of interest is the
joint distribution of $\\{\Upsilon_{j}\\}$. It is established in [248] that
for $L\to\infty$ this distribution is Gaussian, with covariance ${1\over
4}[-\Delta]_{j,k}$, where $\Delta$ is the discrete Laplacian on
$\mathbb{Z}^{2}$. Consequently, fluctuations of $N(\Gamma_{j})$ induce
opposite fluctuations in the regions neighbouring $\Gamma_{j}$.
###### Remark 3.3.
1\. Closely related to the probability $E_{N}(N;\mathcal{D})$ is the
conditioned quantity $F_{N}(n;\mathcal{D}):={\rm
Pr}(\sum_{j=1}^{N}\chi_{z_{j}\in\mathcal{D}}=n|z_{j}=0)$. Denote the
corresponding generating function by $\tilde{F}_{N}(\xi;\mathcal{D})$.
Proceeding as in the derivation of (3.10) shows
$\tilde{F}_{N}(\xi;D_{R})=\tilde{E}_{N}(\xi;D_{R})/(1-\xi(1-e^{-R^{2}}))$.
Thus in particular ${F}_{N}(0;D_{R})=e^{R^{2}}E_{N}(0;D_{R})$ [189]. Note that
$-{d\over dR}F(0;D_{R})$ gives the spacing distribution between an eigenvalue
conditioned to be at the origin, and its nearest neighbour at a distance $R$.
The work [310] gives results relating to the PDF for the minimum of all the
nearest neighbour spacings with global scaling, obtaining a scale of
$N^{-3/4}$ and a PDF proportional to $x^{3}e^{-x^{4}}$.
2\. Let $\bar{D}_{\alpha\sqrt{N}}$ denote the region
$\\{z:|z|>\alpha\sqrt{N}\\}$, i.e. the region outside the disk of radius
$\alpha\sqrt{N}$, where it is assumed $0<\alpha<1$. Note that then
$E_{N}(0;\bar{D}_{\alpha\sqrt{N}})=E_{N}(N;D_{\alpha\sqrt{N}})$. The analogue
of (3.11) has been calculated in [98], where in particular it is found that
(3.13) $C_{1}=\alpha^{4}/4-\alpha^{2}+(1/2)\log\alpha^{2}+3/4,$
the coefficient $C_{4}$ is unchanged, while the coefficient ${1\over 3}$ for
$\log N$ seen in (3.11) is to be replaced by $-{1\over 4}$. (We also refer to
[114] for an earlier work for which the expansion (3.11) was obtained up to
$C_{3}$.) One sees from (3.13) that $C_{1}=0$ for $\alpha=1$, and the result
of [98] gives that $C_{2},C_{3}$ similarly vanish, giving that
$E_{N}(N;D_{\sqrt{N}})\sim e^{C_{4}\sqrt{N}}$. Extending $\alpha$ larger that
$1$ according to the precise $N$ dependent value
(3.14) $\alpha=1+{1\over
2\sqrt{N}}\Big{(}\sqrt{\gamma_{N}}+{x\over\sqrt{\gamma_{N}}}\Big{)},\qquad\gamma_{N}=\log{N\over
2\pi}-2\log\log N,$
gives the extreme value result
$\lim_{N\to\infty}E_{N}(N;D_{\alpha_{N}\sqrt{N}})=\exp(-\exp(-x))$ [290] (see
too [94, Th. 1.3 with $\alpha=2$] for a generalisation to the case of (2.79),
considered also in [98] for $\alpha<1$), which is the Gumbel law. Other
references on fluctuations of the spectral radius under various boundary
conditions include [91, 305, 40, 177, 79, 70]. Furthermore, an intermediate
fluctuation regime which interpolates between the Gumbel law with the large
deviation regime (3.13) was investigated in [237]. Another case considered in
[98] is when $D_{N}$ is specified as the outside of an annulus contained
inside of the disk of radius $\sqrt{N}$. Two features of the corresponding
asymptotic expansion (3.11) are: (1) the absence of a term proportional to
$\log N$, and (2) the presence of oscillations of order 1 that are described
in terms of the Jacobi theta function. We also refer to [100, 32, 33] for
further recent studies in this direction in the presence of hard edges.
3\. For general $\mathcal{D}_{N}$ with $|\mathcal{D}_{N}|\to\infty$ the
coefficient $C_{1}$ in (3.11) relates to an energy minimisation
(electrostatics) problem, and similarly for the $|\mathcal{D}|\to\infty$
expansion of $E_{\infty}(0;\mathcal{D})$ [133, 212, 114, 2, 1]. Thus for
$\mathcal{D}=D_{\alpha\sqrt{N}}$ the electrostatics problem is to compute the
potential due to a uniform charge density $1/\pi$ inside a disk of radius
$\alpha$, with a neutralising uniform surface charge density $-\alpha/2\pi$ on
the boundary. The applicability of electrostatics remains true for the
asymptotic expansion of $E_{N}(k;D_{\alpha\sqrt{N}})$ (and
$E_{\infty}(k;\mathcal{D})$) in the so-called large deviation regime, when
$k\ll N\alpha^{2}$ or $k\gg N\alpha^{2}$. For a disk the electrostatics
problem can be solved explicitly to give [26]
(3.15) $E_{N}(k;D_{\alpha\sqrt{N}})\sim
e^{-N^{2}\psi_{0}(\alpha;k/N)},\quad\psi_{0}(\alpha;x)={1\over
4}\Big{|}(\alpha^{2}-x)(\alpha^{2}-3x)-2x^{2}\log(x/\alpha^{2})\Big{|}.$
Note in particular that $\psi_{0}(\alpha;0)=\alpha^{4}/4$, which is the value
of $-C_{1}$ in (3.11), while setting $x=1$ gives the value of $-C_{1}$ noted
in the above paragraph. There is also a scaling regime, where
$|k-N\alpha^{2}|={\rm O}(N^{1/2})$, for the asymptotic value of
$E_{N}(k;D_{\alpha\sqrt{N}})$ which interpolates between (3.15) and the local
central limit theorem result (3.8) [238, 141]. In the case of
$E_{\infty}(k;D_{R})$, it makes sense to consider $k$ proportional to not only
$\alpha R^{2}$ but also to $\alpha R^{\gamma}$ with $\gamma>2$. Then [212,
141]
$E_{\infty}(\alpha R^{\gamma};D_{R})\sim e^{-{1\over
2}(\gamma-2)\alpha^{2}R^{2\gamma}\log R(1+{\rm o}(1))}.$
4\. For the infinite GinUE the exact result in terms of modified Bessel
functions
(3.16) ${\rm
Var}\,\mathcal{N}(D_{R})=R^{2}e^{-2R^{2}}\Big{(}I_{0}(2R^{2})+I_{1}(2R^{2})\Big{)}=\sum_{j=1}^{\infty}\frac{\gamma(j;R^{2})}{\Gamma(j)}\Big{(}1-\frac{\gamma(j;R^{2})}{\Gamma(j)}\Big{)}$
is known [311, Th. 1.3], [141, Appendix B]. The second expression in (3.16)
also appears in [239] as a large $N$ limit of the number variance of the
finite Ginbire ensemble in the deep bulk regime. This exhibits the leading
large $R$ form $R/\sqrt{\pi}$ which is consistent with identifying
$\sqrt{N}|\partial D_{0}|$ as $2\pi R$ on the RHS of (3.9); see also [13].
### 3.3. Smooth linear statistics
The theory of fluctuation formulas for GinUE in the case that $f(z)$ in (3.1)
is smooth has some different features to the discontinuous case
$f(z)=\chi_{z\in\mathcal{D}}$. This can be seen by considering the bulk scaled
limit, and in particular the truncated two-point correlation (2.24). From this
we can compute the structure factor
(3.17) $S_{\infty}^{\rm
GinUE}(\mathbf{k}):=\int_{\mathbb{R}^{2}}\Big{(}\rho_{(2),\infty}^{{\rm
b},T}(\mathbf{0},\mathbf{r})+{1\over\pi}\delta(\mathbf{r})\Big{)}e^{i\mathbf{k}\cdot\mathbf{r}}\,d\mathbf{r}={1\over\pi}\Big{(}1-e^{-|\mathbf{k}|^{2}/4}\Big{)}.$
Knowledge of the structure factor allows the limiting covariance (3.2) to be
computed using the Fourier transform
(3.18) ${\rm Cov}^{\rm GinUE_{\infty}}\,\Big{(}\sum f(\mathbf{r}_{l}),\sum
g(\mathbf{r}_{l})\Big{)}={1\over(2\pi)^{2}}{1\over\pi}\int_{\mathbb{R}^{2}}\hat{f}(\mathbf{k})\hat{g}(-\mathbf{k})\Big{(}1-e^{-|\mathbf{k}|^{2}/4}\Big{)}\,d\mathbf{k},$
valid provided the integral converges. Here, with $z=x+iy$, $\mathbf{r}=(x,y)$
and the Fourier transform $\hat{f}(\mathbf{k})$ is defined by integrating
$f(\mathbf{r})$ times $e^{i\mathbf{k}\cdot\mathbf{r}}$ over $\mathbb{R}^{2}$ —
thus according to (3.17) $S_{\infty}^{\rm GinUE}(\mathbf{k})$ is a particular
Fourier transform. Now introduce a scale $R$ so that $f(\mathbf{r})\mapsto
f(\mathbf{r}/R),\,g(\mathbf{r})\mapsto g(\mathbf{r}/R)$. It follows from
(3.18) that
(3.19) $\lim_{R\to\infty}{\rm Cov}^{\rm GinUE_{\infty}}\,\Big{(}\sum
f(\mathbf{r}_{l}/R),\sum g(\mathbf{r}_{l}/R)\Big{)}={1\over(2\pi)^{2}}{1\over
4\pi}\int_{\mathbb{R}^{2}}\hat{f}(\mathbf{k})\hat{g}(-\mathbf{k})|\mathbf{k}|^{2}\,d\mathbf{k},$
again provided the integral converges. Most noteworthy is that this limiting
quantity is O$(1)$. In contrast, with
$f(\mathbf{r})=g(\mathbf{r})=\chi_{|\mathbf{r}|<1}$, and then introducing $R$
as prescribed above, we know that (3.18) has the evaluation (3.16). As
previously commented, the large $R$ form of the latter is proportional to $R$.
###### Remark 3.4.
Consider the linear statistic
$A(\mathbf{x})=-\sum_{j=1}^{N}(\log|\mathbf{x}-\mathbf{r}_{j}|-\log|\mathbf{r}_{j}|)$.
In the Coulomb gas picture relating to (1.10), this corresponds to the
difference in the potential at $\mathbf{x}$ and the origin. For bulk scaled
GinUE, one has from (3.18) the exact result [24]
(3.20) ${\rm Var}^{\rm GinUE_{\infty}}\,A(\mathbf{x})={1\over
2}\Big{(}2\log|\mathbf{x}|+(|\mathbf{x}|^{2}+1)\int_{|\mathbf{x}|^{2}}^{\infty}{e^{-t}\over
t}\,dt-e^{-|\mathbf{x}|^{2}}+C+1\Big{)},$
where here $C$ denotes Euler’s constant. In particular, for large
$|\mathbf{x}|$, ${\rm Var}^{\rm
GinUE_{\infty}}\,A(\mathbf{x})\sim\log|\mathbf{x}|$. This last point shows
that the introduction of a scale $R$ as in (3.19) would give rise to a
divergence proportional to $\log R$. Such a log-correlated structure underlies
a relationship between the logarithm of the absolute value of the
characteristic polynomial for GinUE and Gaussian multiplicative chaos [241].
The covariance with test functions $f(\mathbf{r})\mapsto
f(\mathbf{r}/\sqrt{N})$, $g(\mathbf{r})\mapsto g(\mathbf{r}/\sqrt{N})$ assumed
to take on real or complex values is also an order one quantity for GinUE in
the $N\to\infty$ limit, upon the additional assumption that $f,g$ are
differentiable and don’t grow too fast at infinity [148, 292, 35, 36].
###### Proposition 3.5.
Require that $f,g$ have the properties as stated above. Let
$f(\mathbf{r})|_{\mathbf{r}=(\cos\theta,\sin\theta)}=\sum_{n=-\infty}^{\infty}f_{n}e^{in\theta}$
and similarly for the Fourier expansion of $g(\mathbf{r})$ for
$\mathbf{r}=(\cos\theta,\sin\theta)$. We have
(3.21) $\lim_{N\to\infty}{\rm Cov}^{\rm
GinUE}\Big{(}\sum_{j=1}^{N}f(\mathbf{r}_{j}/\sqrt{N}),\sum_{j=1}^{N}\bar{g}(\mathbf{r}_{j}/\sqrt{N})\Big{)}={1\over
4\pi}\int_{|\mathbf{r}|<1}\nabla f\cdot\nabla\bar{g}\,dxdy+{1\over
2}\sum_{n=-\infty}^{\infty}|n|f_{n}\bar{g}_{-n}.$
###### Proof.
(Sketch) In the method of [292], a direct calculation using (3.2), (2.18) and
(2.10) allows (3.21) to be established for $f,g$ polynomials jointly in
$z=x+iy$ and $\bar{z}=x-iy$. The required integrations can be computed exactly
using polar coordinates. To go beyond the polynomial case, the so-called dbar
(Cauchy-Pompeiu) representation is used. This gives that for any once
continuously differentiable $f$ in the unit disk $D_{1}$, and $z$ contained in
the interior of the disk,
(3.22) $f(z)=-{1\over\pi}\int_{D_{1}}{\partial_{\bar{w}}f(w)\over
w-z}\,d^{2}w+{1\over 2\pi i}\int_{\partial D_{1}}{f(w)\over w-z}\,dw,$
where with $w=\alpha+i\gamma$, $\partial_{\bar{w}}={1\over
2}({\partial\over\partial\alpha}+i{\partial\over\partial\gamma})$. The
covariance problem is thus reduced to the particular class of linear
statistics of the functional form $h(z)=1/(w-z)$. The required analysis in
this case is facilitated by the use of the corresponding Laurent expansion,
with only a finite number of terms contributing after integration. ∎
###### Remark 3.6.
1\. As predicted in [148], upon multiplying the RHS by $2/\beta$, (3.21)
remains valid for the Coulomb gas model (1.10) [247, 307, 49]. In the case of
the elliptic GinUE, a simple modification of (3.21) holds true. Thus the
domain $|\mathbf{r}|<1$ in the first term is to be replaced by the appropriate
ellipse, and the Fourier components of the second term are now in the variable
$\eta$, where $(A\cos\eta,B\sin\eta)$, $0\leq\eta\leq 2\pi$ parametrises the
boundary of the ellipse. The results of [148, 36, 247] also cover this case.
2\. In the case of an ellipse, major and minor axes $A,B$ say, there is
particular interest in the linear statistic $P_{x}:=\sum_{j=1}^{N}x_{j}$
[104]. Linear response theory gives for the $xx$ component of the
susceptibility tensor $\chi$ — relating the polarisation density to the
applied electric field — the formula $\chi_{xx}=(\beta/(\pi
AB))\lim_{N\to\infty}{\rm Var}\,P_{x}$ (and similarly for the $xy$ and $yy$
component). This same quantity can be computed by considerations of
macroscopic electrostatics, which gives $\chi_{xx}=(A+B)/(\pi B)$. Using
(3.21) modified as in the above paragraph, the consistency of these formulas
can be verified.
3\. Considering further the case of elliptic GinUE, dividing by $N$ and taking
the limit $\tau\to 1$ gives the GUE with eigenvalues supported on $(-2,2)$,
and similarly for the $\beta$ generalisation limiting to (1.10) restricted to
this interval. For this model it is known (see e.g. [158, Eq. (3.2) with the
identification $x=2\cos\theta$])
$\lim_{N\to\infty}{\rm
Cov}\Big{(}\sum_{j=1}^{N}f(x_{j}),\sum_{j=1}^{N}{g}(x_{j})\Big{)}={2\over\beta}\sum_{n=1}^{\infty}nf_{n}^{\rm
c}g_{n}^{\rm c},$
where $f(x)|_{x=2\cos\theta}=f_{0}^{\rm c}+2\sum_{n=1}^{\infty}f_{n}^{\rm
c}\cos n\theta$ and similarly for $g(x)$. We observe that this is identical to
the final term in (3.21), modified according to the specifications of point 1.
above.
4\. There has been a recent application of Proposition 3.5 in relation to the
computation of the analogue of the Page curve for a density matrix constructed
out of GinUE matrices [107]
We turn our attention now to the limiting distribution of a smooth linear
statistic. By way of introduction, consider the particular linear statistic
${1\over N}\sum_{j=1}^{N}|\mathbf{r}_{j}|^{2}$ for GinUE. An elementary
calculation gives that the corresponding characteristic function,
$\hat{P}_{N}(k)$ say, has the exact functional form
(3.23) $\hat{P}_{N}(k)=(1-ik/N)^{-N(N+1)/2}.$
It follows from this that after centring by the mean, the limiting
distribution is a Gaussian with variance given by (3.21) (which is this
specific case evaluates to one). A limiting Gaussian form holds in the general
case of the applicability of (3.21), as first proved by Rider and Virág [292].
###### Proposition 3.7.
Let $f$ be subject to the same conditions as in Proposition 3.5, and denote
the case $f=g$ of (3.21) by $\sigma_{f}^{2}$. For the GinUE, if $f$ takes on
complex values then as $N\to\infty$
$\sum_{j=1}^{N}f(\mathbf{r}_{j}/\sqrt{N})-\Big{\langle}\sum_{j=1}^{N}f(\mathbf{r}_{j}/\sqrt{N})\Big{\rangle}\mathop{\to}\limits^{\rm
d}{\rm N}[0,\sigma_{f}]+i{\rm N}[0,\sigma_{f}],$
while if $f$ is real valued the RHS of this expression is to be replaced by
${\rm N}[0,\sigma_{f}]$. Moreover this same limit formula holds for the
elliptic GinUE [36] and its $\beta$ generalisation [247] (both subject to
further technical restrictions on $f$), with the variance modified according
to Remark 3.6.1.
###### Proof.
(Comments only) The proof of [292] proceeds by establishing that the higher
order cumulants beyond the variance tend to zero as $N\to\infty$. Essential
use is made of the rotation invariance of GinUE. The method of [36] uses a
loop equation strategy, while [247] involves energy minimisers and transport
maps. For GinUE with $f$ a function of $|\mathbf{r}|$, a simple derivation
based on the proof of Proposition 2.20 together with a Laplace approximation
of the integrals [148] (see also [77, Appendix B]). ∎
###### Remark 3.8.
Other settings in which Proposition 3.7 has proved to be valid include
products of GinUE matrices [110, 228] (with the additional assumption that the
test function have support strictly inside the unit circle), and for the
complex spherical ensemble of subsection 2.5 after stereographic projection
onto the sphere [291, 56].
### 3.4. Spatial modelling and the thinned GinUE
The GinuE viewed as a point process in the plane has been used to model
geographical regions by way of the corresponding Voronoi tessellation [245],
the positions of objects, for example trees in a plantation [244] or the nests
of birds of prey [8], and the spatial distribution of base stations in modern
wireless networks [275, 122], amongst other examples. The wireless network
application has made use of the thinned GinUE, whereby each eigenvalue is
independently deleted with probability $(1-\zeta)$, $0<\zeta\leq 1$. The
effect of this is simple to describe in terms of the correlation functions,
according to the replacement
(3.24)
$\rho_{(n),N}(z_{1},\dots,z_{n})\mapsto\zeta^{N}\rho_{(n),N}(z_{1},\dots,z_{n}).$
With the bulk density of GinUE uniform and is equal to $1/\pi$, we can also
rescale the position $z_{j}\mapsto z_{j}/\zeta$ so that this remains true in
the thinned ensemble. For this (3.24) is to be updated to read
(3.25)
$\rho_{(n),N}(z_{1},\dots,z_{n})\mapsto\rho_{(n),N}(z_{1}/\sqrt{\zeta},\dots,z_{n}/\sqrt{\zeta}).$
Recalling now (2.9) and (2.18), for the bulk scaled limit of the thinned GinUE
we have in particular
$\rho_{(1),\infty}^{\rm tGinUE}(z)={1\over\pi},\qquad\rho_{(2),\infty}^{{\rm
tGinUE},T}(w,z)=-{1\over\pi^{2}}e^{-|w-z|^{2}/\zeta}.$
From these functional forms we see
$\int_{\mathbb{C}}\rho_{(2),\infty}^{{\rm
tGinUE},T}(w,z)\,d^{2}z=-{\zeta\over\pi}\neq-\rho_{(1),\infty}^{{\rm
tGinUE}}(w)\qquad{\rm unless}\>\zeta=1,$
where the superscript “tGinU” denotes the thinned GinUE. Equivalently, in
terms of the structure factor (3.17),
$S_{\infty}^{{\rm tGinUE}}(\mathbf{0})={1-\zeta\over\pi}\neq 0\qquad{\rm
unless}\>\zeta=1.$
Due to this last fact, the O$(1)$ scaled covariance for smooth linear
statistics (3.18) is no longer true, and now reads instead
(3.26) ${\rm Cov}^{\rm tGinUE}\,\Big{(}\sum f(\mathbf{r}_{l}/R),\sum
g(\mathbf{r}_{l}/R)\Big{)}\mathop{\sim}\limits_{R\to\infty}{R^{2}\over(2\pi)^{2}}{(1-\zeta)\over\pi}\int_{\mathbb{R}^{2}}\hat{f}(\mathbf{k})\hat{g}(-\mathbf{k})\,d\mathbf{k}.$
This leading dependence on $R^{2}$ holds too for the counting function
$f(z)=\chi_{|z|<1}$, since in distinction to (3.18) the integral now
converges. Hence, in the terminology of the text introduced below (3.9), the
statistical state is no longer hyperuniform. There is an analogous change to
the O$(1)$ scaled covariance (3.21), which is now proportional to $N$ and
reads
(3.27) ${\rm Cov}^{\rm
tGinUE}\Big{(}\sum_{j=1}^{N}f(\mathbf{r}_{j}/\sqrt{N}),\sum_{j=1}^{N}\bar{g}(\mathbf{r}_{j}/\sqrt{N})\Big{)}\mathop{\sim}\limits_{N\to\infty}N{(1-\zeta)\over\pi}\int_{|\mathbf{r}|<1}f\bar{g}\,dxdy.$
Notwithstanding this difference, the corresponding limiting distribution
function is still Gaussian [240]. A more subtle limit, also considered in
[240], is when $N\to\infty$ and $\zeta\to 1^{-}$ simultaneously, with
$N(1-\zeta)$ fixed. The quantity (3.21) returns to being O$(1)$, but consists
of a contribution of the form (3.21), and a term characteristic of a Poisson
process.
We turn our attention now to the probabilities $\\{E_{N}^{\rm
tGUE}(k;D_{\alpha\sqrt{N}})\\}$. Upon consideration of the thinning
prescription (3.25), the proof of Proposition 3.1, and (3.10) shows that the
corresponding generating function is given by
(3.28) $\tilde{E}_{N}^{\rm tGUE}(\xi;D_{\alpha\sqrt{\zeta
N}})=\prod_{j=1}^{N}\bigg{(}1-\xi\zeta{\gamma(j;\alpha^{2}N)\over\Gamma(j)}\bigg{)}.$
Setting $\xi=1$ in this gives the probability $E_{N}^{\rm
tGUE}(0;D_{\alpha\sqrt{\zeta N}})$. Note that the implied formula shows
$E_{N}^{\rm tGUE}(0;D_{\alpha\sqrt{\zeta N}})=\tilde{E}_{N}^{\rm
GUE}(\zeta;D_{\alpha\sqrt{N}})$. The large $N$ asymptotics of
$\tilde{E}_{N}^{\rm GUE}(\zeta;D_{\alpha\sqrt{N}})$, and various
generalisations, are available in the literature [99, 76]. Here we present a
self contained derivation of the first two terms (cf. (3.11)).
###### Proposition 3.9.
For large $N$ and with $0<\alpha,\zeta<1$ we have
(3.29) $\tilde{E}_{N}^{\rm tGUE}(0;D_{\alpha\sqrt{\zeta
N}})=\exp\Big{(}{\alpha^{2}N}\log(1-\zeta)+\sqrt{\alpha^{2}N}\,h(\zeta)+{\rm
O}(1)\Big{)},$
where
(3.30) $h(\zeta)=\int_{0}^{\infty}\log\Big{(}{1-(\zeta/2)(1+{\rm
erf}(t/\sqrt{2}))\over
1-\zeta}\Big{)}\,dt+\int_{0}^{\infty}\log\Big{(}1-(\zeta/2)(1-{\rm
erf}(t/\sqrt{2}))\Big{)}\,dt.$
###### Proof.
Our main tool is the uniform asymptotic expansion [323]
(3.31)
${\gamma(M-j+1;M)\over\Gamma(M-j+1)}\mathop{\sim}\limits_{M\to\infty}{1\over
2}\Big{(}1+{\rm erf}\Big{(}{j\over\sqrt{2M}}\Big{)}\Big{)};$
cf. the leading term in (2.20). Here it is known that the error term has the
structure $(1/\sqrt{M})g(j/\sqrt{2M})$ where $g(t)$ is integrable on
$\mathbb{R}$ and decays rapidly at infinity. This expansion suggests we
rewrite the product in (3.28) with $\xi=1$ in the form
$(1-\zeta)^{[M^{*}]}\bigg{(}\prod_{j=1}^{M^{*}}{1-\zeta\gamma(j;M^{*})/\Gamma(j)\over
1-\zeta}\bigg{)}\bigg{(}\prod_{j=M^{*}+1}^{N}(1-\zeta\gamma(j;M^{*})/\Gamma(j))\bigg{)},$
where $M^{*}=[\alpha^{2}N]$.
We see that the first term in this expression gives the leading order term in
(3.29). In the first product we change labels $j\mapsto M^{*}-j+1$
$(j=1,\dots,M^{*})$. In the second we change labels $j\mapsto M^{*}+j+1$
($j=0,\dots,N-M^{*}-1$). Now writing both these products as exponentials of
sums and applying (3.31) gives, upon recognising the sums as Riemann
integrals, the O$(\sqrt{\alpha^{2}N})$ term in (3.29). ∎
The leading term in (3.29) is consistent with the general form expected for
thinned log-gas systems, being of the form of the area of the rescaled
excluded region, times the density, times $\log(1-\zeta)$ [151, Conj. 10].
###### Remark 3.10.
The topic of spatial modelling using Ginibre eigenvalues naturally leads to
questions on the efficient simulation of the point process confined to a
compact subset of the $\mathbf{C}$. Practical algorithms for this task have
been given in [120, 121].
## 4\. Sum rules and asymptotic behaviours
Throughout this section we consider the Coulomb gas model (1.10) with general
$\beta>0$, for which we use the notation OCP, which stands for one-component
plasma. The special case $\beta=2$ coincides with GinUE.
### 4.1. Asymptotics associated with the configuration integral
The configuration integral for the Boltzmann factor (1.10) is specified by
(4.1) $Q_{N}^{\rm
OCP}(\beta)=\int_{\mathbb{C}}d^{2}z_{1}\cdots\int_{\mathbb{C}}d^{2}z_{N}\,e^{-(\beta/2)\sum_{j=1}^{N}|z_{j}|^{2}}\prod_{1\leq
j<k\leq N}|z_{k}-z_{j}|^{\beta}.$
From the OCP viewpoint, it is more natural to consider the renormalised
quantity
(4.2) $Z_{N}^{D_{R},\rm OCP}(\beta)={1\over N!}A_{N,\beta}Q_{N}^{\rm
OCP}(\beta),\qquad A_{N,\beta}=e^{-\beta N^{2}({1\over 4}\log N-{3\over 8})};$
see [149, Eq. (1.12)]. The use of the symbol $D_{R}$ denoting a disk of radius
$R$ (specifically $R=\sqrt{N}$) as a subscript follows from the derivation of
$A_{N,\beta}$. Thus this quantity has the interpretation as the Boltzmann
factor of the electrostatic self energy of a smeared out uniform background,
charge density $-{1\over\pi}$, confined to the disk $D_{R}$, and the constant
terms of its electrostatic energy when coupled to a particle inside of this
disk. The quantity $Z_{N}^{D_{R},\rm OCP}(\beta)$ is then the partition
function of the charge neutral OCP. Generally in statistical mechanics for a
stable system the dimensionless free energy, $\beta F_{N}(\beta)=-\log
Z_{N}(\beta)$, is an extensive quantity, meaning that for large $N$ it is
proportional to $N$. For the closely related model when the particles are
strictly restricted to the disk, the validity of this statement was
established in [257, 304], and has been reconsidered recently using more far
reaching techniques in [246] (which for example form a platform for the study
of fluctuation formulas in [247]). Making use of Proposition 2.1 the large $N$
form of $\beta F_{N}^{D_{R},\rm OCP}(\beta)$ can be computed for $\beta=2$
[319, Eq. (3.14)].
###### Proposition 4.1.
We have
(4.3) $\beta F_{N}^{D_{R},\rm OCP}(\beta)\Big{|}_{\beta=2}=N\beta
f(\beta)|_{\beta=2}+{1\over 12}\log N-\zeta^{\prime}(-1)-{1\over
720N^{2}}+{\rm O}\Big{(}{1\over N^{4}}\Big{)},$
where
(4.4) $\beta f(\beta)|_{\beta=2}={1\over 2}\log\Big{(}{1\over
2\pi^{3}}\Big{)}.$
###### Proof.
This relies on (2.3), identifying $\prod_{j=1}^{N-1}j!=G(N+1)$, where $G(x)$
denotes the Barnes $G$-function, and knowledge of the known asymptotic
expansion of $G(N+1)$ (see e.g. [142, Th. 1]). ∎
For the OCP on a sphere of radius $R$, $S_{R}^{2}$ say, and with $R={1\over
2}\sqrt{N}$ so that the particle density is $1/\pi$, the partition function of
the charge neutral system is (see e.g. [319, Eq. (2.1)])
$Z_{N}^{S_{R}^{2},\rm OCP}(\beta)={1\over N!}N^{-N\beta/2}e^{\beta
N^{2}/4}\int_{S^{2}_{R}}d\theta_{1}d\phi_{1}\cdots\int_{S^{2}_{R}}d\theta_{N}d\phi_{N}\,\prod_{1\leq
j<k\leq N}|u_{k}v_{j}-u_{j}v_{k}|^{\beta}.$
Here the variables $\\{u_{j},v_{k}\\}$ are the Cayley-Klein parameters as in
(2.50). From the analogue of Proposition 2.1 this can be evaluated exactly at
$\beta=2$ [81], implying that for large $N$ [214], [319, Eq. (3.6)]
(4.5) $\beta F_{N}^{S_{R}^{2},\rm OCP}(\beta)\Big{|}_{\beta=2}=N\beta
f(\beta)|_{\beta=2}+{1\over 6}\log N+{1\over 12}-2\zeta^{\prime}(-1)+{1\over
180N^{2}}+{\rm O}\Big{(}{1\over N^{4}}\Big{)},$
where $\beta f(\beta)|_{\beta=2}$ is as in (4.3).
Both expansions (4.3) and (4.5) illustrate a conjectured universal property of
the large $N$ expansion of $\beta F_{N}^{\rm OCP}$ in the case that the
droplet forms a shape with Euler index $\chi$ [214]
(4.6) $\beta F_{N}^{\rm OCP}(\beta)=N\beta
f(\beta)+a_{\beta}\sqrt{N}+{\chi\over 12}\log N+\cdots,$
valid for general $\beta>0$. To compare against the exact results for
$\beta=2$ it should be recalled that $\chi=1$ for a disk and $\chi=2$ for a
sphere. An analogous calculation in annulus geometry at $\beta=2$ gives an
expansion with a term proportional to $\log N$ absent, in keeping with
$\chi=0$ [145], [78]. A further point of interest is that the large $N$
expansions for $E_{N}(0;\mathcal{D}_{N})$ from [98] as reviewed in § 3.2, also
exhibit simple fractions for the coefficient of $\log N$, and moreover this
term is not present in the case the eigenvalues are constrained to a single
annulus.
We note that the term proportional to $\sqrt{N}$ in (4.6) has the
interpretation as a surface tension, and so is expected not to be present in
the case of a sphere, as seen in (4.5) for $\beta=2$. Also, for the disk
geometry, it has been conjectured (see [85, Eq. (3.2)]) that
(4.7) $a_{\beta}={4\log(\beta/2)\over 3\pi^{1/2}},$
which in particular vanishes for $\beta=2$, as is consistent with (4.5).
###### Remark 4.2.
1\. Multiplication of the configuration integral $Q_{N}^{\rm OCP}(\beta)$ by
$A_{N,\beta}$ as in (4.2) effectively shifts the microscopic energy $U$ in
(1.10) by a function of $N$. We denote this shifted, charge neutral, energy by
$U^{\prime}$, and similarly in relation to the sphere. For $\beta=2$ direct
calculation of the mean is possible. Thus in the case of the OCP on the plane,
one finds [309, equivalent to Eqns. (25) and (29)]
$\displaystyle\langle U^{\prime}\rangle^{D_{R}}\Big{|}_{\beta=2}=$
$\displaystyle-{1\over 2}\bigg{(}{N^{2}\over 2}(\Psi(N)-\log N)+{N+1\over
4}+{NC\over 2}-{\Gamma(N+3/2)\over\Gamma(N+2)\Gamma(3/2)}$
$\displaystyle\quad\times{}_{3}F_{2}\Big{(}{1,N-1,N+3/2\atop
N+2,N+1}\Big{|}1\Big{)}\bigg{)}=-{CN\over 4}+{2\sqrt{N}\over
3\sqrt{\pi}}-{5\over 48}+{\rm O}(N^{-1/2}),$
where here $C$ denotes Euler’s constant and $\Psi(N)$ denotes the digamma
function. The corresponding result for the sphere geometry at $\beta=2$ gives
the simpler formula [82, 25, 299]
$\langle U^{\prime}\rangle^{S_{R}^{2}}\Big{|}_{\beta=2}={N\over
4}\Big{(}-H_{N}+\log N\Big{)}=-{CN\over 4}-{1\over 8}+{\rm O}(N^{-1}),$
where here $H_{N}$ denotes the harmonic numbers. The common leading order
value of the charge neutral energy per particle, $-C/4$, was known to
Jancovici [207] through the formula
$\lim_{N\to\infty}{1\over N}\langle U^{\prime}\rangle^{\rm
OCP}\Big{|}_{\beta=2}=-{\pi\over
2}\int_{\mathbb{R}^{2}}\log|\mathbf{r}|\,\rho_{(2),\infty}^{\rm b,\rm
GinUE}(\mathbf{0},\mathbf{r})\,d\mathbf{r}=-{C\over 4},$
where the integral is evaluated from the explicit formula for
$\rho_{(2),\infty}^{\rm b,\rm GinUE}$ given by (2.24). Note as an expansion
about $\beta=2$, $\beta F_{N}(\beta)=\beta
F_{N}(\beta)|_{\beta=2}+(\beta-2)\langle U^{\prime}\rangle+{\rm
O}((\beta/2-1)^{2})$, so the above results allow (4.3) and (4.5) to be
extended to first order in $(\beta-2)$. In particular consistency is obtained
with (4.6).
2\. In the low temperature limit $\beta\to\infty$ the OCP particles are
expected to form a triangular lattice. Recent works relating to this include
[302, 294, 247, 61, 28, 87, 43, 42]. The conjectured exact value of twice the
charge neutral energy per particle in this limit, with bulk density $1/(4\pi)$
(not $1/\pi$ as is natural for GinUE in the plane, rather the geometry used
was a sphere of unit radius) is [72]
$2\log 2+{1\over 2}\log{2\over
3}+3\log{\sqrt{\pi}\over\Gamma(1/3)}=-0.0556053\dots.$
We also refer to [93, 12, 242] and references therein for recent works on the
opposite, high temperature limit $\beta\to 0$ of the two dimensional Coulomb
particles.
### 4.2. Sum rules and asymptotics for the edge density
First we note that for large $N$ the leading order support of the density is a
disk of radius $\sqrt{N}$, independent of $\beta$, as follows from the
potential theoretic argument of Remark 2.5.2. Using the vector coordinate
$\mathbf{r}=(x,y)$, as in the second equation in (2.19) we introduce edge
scaling coordinates by writing
(4.8) $\rho_{(1),N}^{\rm OCP}((x,\sqrt{N}-y))=\rho_{(1),\infty}^{\rm
e,OCP}(y)+{1\over\sqrt{N}}\mu^{\rm e,OCP}(y)+{\rm O}\Big{(}{1\over N}\Big{)}.$
Here the form of the correction terms, known to be valid at $\beta=2$
according to (2.27), are at this stage presented as an ansatz. We seek some
integral identities that must be satisfied by $\rho_{(1),\infty}^{\rm
e,OCP}(y)$ and $\mu_{(1),\infty}^{\rm e,OCP}(y)$. Identities of this type are
referred to as sum rules.
###### Proposition 4.3.
We have
(4.9) $\int_{-\infty}^{\infty}\Big{(}\rho_{(1),\infty}^{\rm
e,OCP}(y)-{1\over\pi}\chi_{y>0}\Big{)}\,dy=0.$
and
(4.10) $\int_{-\infty}^{\infty}y\Big{(}\rho_{(1),\infty}^{\rm
e,OCP}(y)-{1\over\pi}\chi_{y>0}\Big{)}\,dy=\int_{-\infty}^{\infty}\mu_{(1),\infty}^{\rm
e,OCP}(y)\,dy.$
###### Proof.
Because of the large $N$ form of the density, we have that $\rho_{(1),N}^{\rm
e,OCP}(\mathbf{r})-{1\over\pi}\chi_{|\mathbf{r}|<\sqrt{N}}$ will be
concentrated near $|\mathbf{r}|=\sqrt{N}$. Now, the normalisation condition
for the density gives, with the use of polar coordinates
$\int_{0}^{\infty}r\Big{(}\rho_{(1),N}^{\rm
e,OCP}(\mathbf{r})-{1\over\pi}\chi_{|\mathbf{r}|<\sqrt{N}}\Big{)}\,dr=\int_{-\infty}^{\sqrt{N}}(\sqrt{N}-y)\Big{(}\rho_{(1),N}^{\rm
e,OCP}(\sqrt{N}-y)-{1\over\pi}\chi_{y>0}\Big{)}\,dr=0.$
In the second term we now substitute (4.8) and equate terms of order
$\sqrt{N}$ and of order unity to zero to obtain (4.9) and (4.10). ∎
From the functional forms for $\rho_{(1),\infty}^{\rm e,GinUE}(y)$ and
$\mu_{(1),\infty}^{\rm e,GinUE}(y)$ as read off from (2.27), we verify both of
the above sum rules in this special case. Note that the integral on the LHS of
(4.10) has the interpretation of the dipole moment of the excess charge in the
edge boundary layer, using the Coulomb gas picture. In fact it is possible to
derive a further sum rule which evaluates this dipole moment explicitly [320,
Eq. (5.13)].
###### Proposition 4.4.
We have
(4.11) $\int_{-\infty}^{\infty}y\Big{(}\rho_{(1),\infty}^{\rm
e,OCP}(y)-{1\over\pi}\chi_{y>0}\Big{)}\,dy=-{1\over
2\pi\beta}\Big{(}1-{\beta\over 4}\Big{)}.$
###### Proof.
From the definition, we observe
(4.12)
$\Big{\langle}\sum_{j=1}^{N}|\mathbf{r}_{j}|^{2}\Big{\rangle}=-{\partial\over\partial
c}\log\bigg{(}\int_{\mathbb{R}^{2}}d\mathbf{r}_{1}\cdots\int_{\mathbb{R}^{2}}d\mathbf{r}_{N}\,e^{-c\sum_{j=1}^{N}|\mathbf{r}_{j}|^{2}}\prod_{1\leq
j<k\leq
N}|\mathbf{r}_{k}-\mathbf{r}_{j}|^{\beta}\,\bigg{)}\bigg{|}_{c=\beta/2}.$
The $c$ dependence of the integral can be factored by a simple change of
variables, allowing the integral to be replaced by $c^{-N-\beta N(N-1)/4}$,
and so giving
(4.13)
$\Big{\langle}\sum_{j=1}^{N}|\mathbf{r}_{j}|^{2}\Big{\rangle}={2N\over\beta}+{1\over
2}N(N-1).$
By writing the LHS as an average over the density and the use of polar
coordinates, we see that this is equivalent to the sum rule
$2\pi\int_{0}^{\infty}r^{3}\Big{(}\rho_{(1),N}^{\rm
OCP}(r)-{1\over\pi}\chi_{r<\sqrt{N}}\Big{)}\,dr={N\over
2}\Big{(}{4\over\beta}-1\Big{)}.$
Proceeding now as in the proof of Proposition 4.3, by substituting (4.8),
equating terms proportional to $N$ on both sides, and making use too of
(4.10), we deduce (4.11). ∎
One observes that the RHS of (4.11) changes sign as $\beta$ increase beyond
$4$. In fact in the work [84] it is predicted that for general $\beta>2$, the
edge density profile of the OCP exhibits an overshoot effect where it rises
before tailing off to zero. The exact evaluation of the edge density to
leading order in $\beta-2$ in [85] lends analytic evidence to this claim. This
edge density overshoot effect has been observed in the random matrix ensemble
of even dimensional random matrices $Z_{N}W$, where
$Z_{N}={\small\mathbb{I}_{N}\otimes\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}}$
and $W$ is a complex anti-symmetric Gaussian random matrix [197, 152].
Moreover, in [197], upon the assumption of large eigenvalue separation, an
analytic calculation of the joint eigenvalue PDF gives the functional form of
the OCP with $\beta=4$.
We now turn our attention to the large $N$ form of the global scaled density,
$\rho_{(1),N}^{\rm g,OCP}(\mathbf{r}):=\rho_{(1),N}^{\rm
OCP}(\sqrt{N}\mathbf{r})$. From the theory noted at the beginning of the
subsection, this will limit to the circular law (2.17). We ask about the
leading correction term. A hint is given by (4.13), which after dividing both
sides by $N^{2}$ to correspond to global coordinates tells us $\langle{1\over
N}\sum_{j=1}^{N}|\mathbf{r}_{j}|^{2}\rangle_{\rm g,OCP}={1\over 2}-{1\over
2}\Big{(}1-{4\over\beta}\Big{)}{1\over N}.$ The first term ${1\over 2}$ is the
average of the function $g(\mathbf{r})=|\mathbf{r}|^{2}$ over the disk
$D_{\sqrt{N}}$ with density ${1\over\pi}$. The second term is a ${1\over N}$
correction, so we might expect that the leading correction to the circular law
is O$(1/N)$. In fact knowledge of (4.11) is sufficient to compute the leading
correction term of the large $N$ expansion of all the moments $\langle{1\over
N}\sum_{j=1}^{N}|\mathbf{r}_{j}|^{p}\rangle_{\rm g,OCP}$ ($p=1,2,\dots)$,
allowing us to conclude [320, Eq. (5.18)]
(4.14) $\rho_{(1),N}^{\rm
g,OCP}(\mathbf{r})={1\over\pi}\chi_{|\mathbf{r}|<1}+{1\over N}\kappa(r)+{\rm
O}(N^{-3/2}),\quad\kappa(r)={1\over 2\pi\beta}\Big{(}1-{\beta\over
4}\Big{)}{1\over r}\delta^{\prime}(r-1).$
Thus the correction term is concentrated on the boundary. For GinUE, it can be
deduced from (2.10) that inside the droplet the corrections are exponentially
small [250, Th. 1.2],[206, Lemma 3.1]. This same formula can also be read off
from a more general formula relating to $\beta$ generalised normal matrix
models (see § 5.3) obtained by Zabrodin and Wiegmann [329, Eq. (5.16)].
Associated with (4.14) is the expansion [246, Eq. (1.14)]
(4.15) $\Big{\langle}\sum_{j=}^{N}g(\mathbf{r}_{j})\Big{\rangle}^{\rm
g,OCP}={1\over\pi}\int_{|\mathbf{r}|<1}g(\mathbf{r})\,dxdy+{1\over N}{1\over
2\pi\beta}\Big{(}1-{\beta\over
4}\Big{)}\int_{|\mathbf{r}|<1}\nabla^{2}g(\mathbf{r})\,dxdy+{\rm o}(N^{-1}),$
valid for sufficiently smooth test functions $g$ (note that rotational
invariance is not assumed). In the case $\beta=2$ this expansion can be found
in [36, Th. 2.1].
As our final topic under this heading, we consider the $y\to-\infty$
asymptotic form of $\rho_{(1),N}^{\rm e,OCP}(y)$. According to (2.28), at
$\beta=2$ we have
(4.16) $\rho_{(1),N}^{\rm
e,OCP}(y)\mathop{\sim}\limits_{y\to-\infty}{e^{-2y^{2}}\over(2\pi)^{3/2}|y|}.$
As a first step to extend this to general $\beta>0$, a large deviation formula
for $\rho_{(1),N}^{\rm OCP}(\mathbf{r})$ can be computed, which asks for the
asymptotic form of $\rho_{(1),N}^{\rm OCP}(\sqrt{N}r)$ (here polar coordinates
are being used), $r>1$ [85, Prop. 1].
###### Proposition 4.5.
With $\beta f(\beta)$ the dimensionless free energy per particle as in (4.6),
for $r>1$ we have
(4.17) $\rho_{(1),N}^{\rm OCP}(\sqrt{N}r)=\frac{e^{\beta
f(\beta)}}{N^{\beta/4}}e^{-(N\beta/2)(r^{2}-1)}\exp\Big{(}N\beta\log
r-\frac{\beta}{2}\log(r^{2}-1)+{\rm o}(1)\Big{)}.$
###### Proof.
(Sketch) The starting point is to manipulate the definition of
$\rho_{(1),N}^{\rm OCP}$ to obtain its form written as an average
(4.18) $\rho_{(1),N+1}^{\rm OCP}(\sqrt{N+1}\vec{r})=(N+1)N^{\beta
N/2}e^{-(N+1)\beta r^{2}/2}\frac{Q_{N}^{\rm OCP}(\beta)}{Q_{N+1}^{\rm
OCP}(\beta)}\Big{\langle}\prod_{l=1}^{N}\Big{|}\sqrt{\frac{N+1}{N}}\vec{r}-\vec{r_{l}}\Big{|}^{\beta}\Big{\rangle}_{\rm
OCP^{\rm g}}.$
Here $Q_{N}^{\rm OCP}(\beta)$ is the configuration integral (4.1), and the
average is with respect to the global scaled GUE, specified by the Boltzmann
factor (1.10) but with the factor of ${1\over 2}$ multiplying the first sum in
$U$ replace by ${N\over 2}$. The significance of this is that upon
exponentiating the product, the average can be recognised as the
characteristic function for a particular linear statistics. For this, with
$|\mathbf{r}|=r>1$ Proposition 3.7 applies, telling us the leading two terms
in its large $N$ asymptotic expansion, once the corresponding mean and
variance have been computed. After calculating these, (4.17) results. ∎
From (4.17) we compute the limit formula
(4.19) $\lim_{N\to\infty}{\rho}_{(1),N}^{\rm
OCP}(\sqrt{N}r)|_{r=1-y/\sqrt{N}}=e^{\beta f(\beta)}\frac{e^{-\beta
y^{2}}}{(2|y|)^{\beta/2}},$
which under the assumption that the large deviation formula connects to the
$y\to-\infty$ tail of $\rho_{(1),N}^{\rm e,OCP}(y)$ is the sought $\beta>0$
generalisation of the $\beta=2$ result (4.16). The consistency between (4.19)
and the latter is immediate upon substituting (4.4).
### 4.3. Sum rules and asymptotics for the two and higher point correlations
Setting $\mathbf{k}=\mathbf{0}$ in (3.17) gives
(4.20) $\int_{\mathbb{R}^{2}}\Big{(}\rho_{(2),\infty}^{{\rm
b},T}(\mathbf{0},\mathbf{r})+{1\over\pi}\delta(\mathbf{r})\Big{)}\,d\mathbf{r}=0.$
This constraint on $\rho_{(2),\infty}^{T}$ is an example of a sum rule. In
physical terms, using the Coulomb gas picture, it says that the response of
the system by the introduction of a charge (corresponding to the delta
function) is to create a screening cloud (corresponding to
$\rho_{(2),\infty}^{T}$) of opposite total charge. Consequently (4.20) is
referred to as the perfect screening sum rule. It relates to the point process
being hyperuniform or equivalently incompressible — for a state that is
compressible the RHS of (4.20) is not zero but rather is given in terms of the
second derivative of the pressure with respect to the fugacity; see e.g. [156,
Eq. (3.7)]. For states with an underlying long range potential (as in (1.10))
the perfect screening sum rule is expected to be a necessary condition for
thermodynamic stability [270]. Of similar general validity is the sum rule
which results when the integrand of (4.20) is replaced by
(4.21)
$q(\mathbf{r}_{1},\dots,\mathbf{r}_{k},\mathbf{r}):=\rho_{(k+1),\infty}(\mathbf{r}_{1},\dots,\mathbf{r}_{k},\mathbf{r})\\\
-{1\over\pi}\rho_{(k),\infty}(\mathbf{r}_{1},\dots,\mathbf{r}_{k})+\sum_{j=1}^{k}\delta(\mathbf{r}-\mathbf{r}_{j})\rho_{(k),\infty}(\mathbf{r}_{1},\dots,\mathbf{r}_{k}).$
Furthermore the fast decay of the correlations upon truncation (i.e. suitable
subtraction by combinations of lower order correlation as in the definition of
$\rho_{(2),\infty}^{T}$) implies that not only does the total charge
associated with $q$ vanish, but in fact so too does all the multipole moments
[270].
###### Proposition 4.6.
Let $q$ be as in (4.21) and set $\mathbf{r}=(x,y)$. For bulk scaled GinUE,
specified in terms of the correlation functions by (2.9) with $N\to\infty$ and
the correlation kernel (2.18), we have
(4.22)
$\int_{\mathbb{R}^{2}}(x-iy)^{p}q(\mathbf{r}_{1},\dots,\mathbf{r}_{k},\mathbf{r})\,d\mathbf{r}=0,\qquad
p=0,1,\dots$
###### Proof.
For any $k\geq 1$ integrating over the term involving the sum of delta
functions gives
$\Big{(}\sum_{j=1}^{k}(x_{j}-iy_{j})^{p}\Big{)}\rho_{(k),\infty}$. To then
integrate over the first two terms, we expand the determinant specifying
$\rho_{(k+1),\infty}$ by the final row. The term coming from the last entry is
recognised as $(1/\pi)\rho_{(k),\infty}$ and so cancels. For each of the $k$
other terms, we multiply the term coming from the $j$-th entry of the final
row $K_{\infty}^{\rm b}(\mathbf{r},\mathbf{r}_{j})$ times $(x-iy)^{p}$ down
the final column containing $[K_{\infty}^{\rm
b}(\mathbf{r}_{m},\mathbf{r})]_{m=1}^{k}$, and integrate over $\mathbf{r}$.
For the latter task, polar coordinates can be used to deduce that
$\int_{\mathbb{R}^{2}}(x-iy)^{p}K_{\infty}^{\rm
b}(\mathbf{r}_{m},\mathbf{r})K_{\infty}^{\rm
b}(\mathbf{r},\mathbf{r}_{j})\,d\mathbf{r}=(x_{j}-iy_{j})^{p}K_{\infty}^{\rm
b}(\mathbf{r}_{m},\mathbf{r}_{j});$
the case $p=0$ is (2.22) with bulk scaling. After rearranging the columns, the
result of the integration in each case can be identified as
$-(x_{j}-iy_{j})^{p}\rho_{(k),\infty}$, and thus in total cancel out with the
integration over the final term. ∎
Replacing $1/\pi$ in (4.21) by $\rho_{(1),\infty}(\mathbf{r})$ the sum rule
(4.22) with $p=0$ remains valid in the case of edge scaling as a consequence
of the validity of (2.22). For $p\geq 1$ the slow decay of the correlations
along the direction of the boundary as seen in (2.26) means that the integral
in (4.22) is not well defined. Specifically the $p=0$, $k=1$ sum rule (4.22)
at the edge reads
(4.23) $\int_{\mathbb{C}}\rho_{(2),\infty}^{{\rm
e,OCP},T}(z,z^{\prime})\,d^{2}z^{\prime}=-\rho_{(1),\infty}^{{\rm e,OCP}}(z).$
This is the edge counterpart of the bulk perfect screening sum rule (4.20).
###### Remark 4.7.
It turns out that in the GinUE case, the determinantal structure together with
(4.23) can be used to show that $\rho(z):=\rho_{(1),\infty}^{{\rm
e,GinUE}}(z)$ (this is (2.28)) satisfies a non-linear equation of infinite
order,
(4.24) $\rho(z)=\sum_{n=0}^{\infty}{|\partial_{z}^{(n)}\rho(z)|^{2}\over n!};$
see [38, § 3.6], where (4.23) is referred to as a mass-one equation. The non-
linear equation is equivalent to the special function function identity
(4.25) ${1\over 4}\Big{(}1+{\rm erf}(\sqrt{2}x)\Big{)}\Big{(}1-{\rm
erf}(\sqrt{2}x)\Big{)}={e^{-4x^{2}}\over\pi}\sum_{n=1}^{\infty}{(H_{n-1}(\sqrt{2}x))^{2}\over
2^{n}n!}\quad(x\in\mathbb{R});$
see [38, § 4.5] for the proof of (4.25). Together with the loop equation, the
identity (4.24) is used in [38] to study the universality of normal matrix
models; see §5.4.
To deduce (3.19) from (3.18) requires that
(4.26) $\lim_{R\to\infty}R^{2}S_{\infty}^{\rm
GinUE}(\mathbf{k}/R)={|\mathbf{k}|^{2}\over 4\pi}.$
In the general $\beta>0$ case of the OCP we denote the bulk scaled structure
factor by $S_{\infty}^{\rm OCP}(\mathbf{k})$. A perfect screening argument
extending the viewpoint which implies (4.20) (for this see e.g. [149,
§15.4.1], [147, §3.2]) predicts
(4.27) $\lim_{R\to\infty}R^{2}S_{\infty}^{\rm
OCP}(\mathbf{k}/R)={|\mathbf{k}|^{2}\over 2\beta\pi}.$
This result, in the slightly different guise of a mesoscopic scaling limit, is
implied by the recent work [247, Th. 1 mesoscopic case, formula for the
variance]. In the cases of the OCP applied to the anomalous quantum Hall
effect ($\beta=2M$ with $M$ odd), the sum rule (4.27) combined with the
Feynman-Bijl formula quantifies the collective mode excitation energy from the
ground state [184]. From the definition of the structure factor, this is
equivalent to the moment formula
(4.28) $\int_{\mathbf{R}^{2}}|\mathbf{r}|^{2}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r},\mathbf{0})\,d\mathbf{r}=-{2\over\pi\beta},$
known as the Stillinger-Lovett sum rule [317]; see [271] for a derivation
which makes use of (4.22) for $p=1$ and $k=1,2$. Higher order moment formulas
can also be derived,
(4.29) $\int_{\mathbf{R}^{2}}|\mathbf{r}|^{4}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r},\mathbf{0})\,d\mathbf{r}=-{16\over\pi\beta^{2}}\Big{(}1-{\beta\over
4}\Big{)},\quad\int_{\mathbf{R}^{2}}|\mathbf{r}|^{6}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r},\mathbf{0})\,d\mathbf{r}=-{18\over\pi\beta^{3}}\Big{(}\beta-6\Big{)}\Big{(}\beta-{8\over
3}\Big{)};$
in distinction to (4.27) the precise statement of these results depends on the
underlying bulk particle density, which in keeping with GinUE has been assumed
to be ${1\over\pi}$. A linear response argument can be used to derive the
first of these; see e.g. [149, §14.1.1]. It involves the thermal pressure for
the OCP, which by a simple scaling argument takes the value $\rho^{\rm
b}(1-\beta/4)$ (here $\rho^{\rm b}$ denotes the bulk density; for a discussion
of the relation between the thermal and mechanic pressure in the OCP, see
[103]) which explains the appearance of this factor. Mayer diagrammatic
expansion methods were used to first derive the sixth moment condition in
(4.29) [217]. Later a response argument involving variations to the spatial
geometry was used to give an alternative derivation [86]. In keeping with the
relationship between (4.27) and (4.28), the moment formulas (4.29) can be
related to the small $k$ expansion of $S_{\infty}^{\rm GinUE}(\mathbf{k})$,
which must therefore read
(4.30) ${2\pi\beta\over|\mathbf{k}|^{2}}S_{\infty}^{\rm
OCP}(\mathbf{k})=1+\Big{(}{\beta\over 4}-1\Big{)}{|\mathbf{k}|^{2}\over
2\beta}+\Big{(}{\beta\over 4}-{3\over 2}\Big{)}\Big{(}{\beta\over 4}-{2\over
3}\Big{)}\Big{(}{|\mathbf{k}|^{2}\over 2\beta}\Big{)}^{2}+{\rm
O}(|\mathbf{k}|^{6}).$
Evidence is given in [217] that the polynomial structure in $\beta/4$ of the
coefficients in this expansion breaks down at ${\rm O}(|\mathbf{k}|^{6})$.
This is in contrast to the power series expansion of the bulk scaled (density
$1/\pi$ for definiteness) structure factor for the log-gas on a one-
dimensional domain, $S_{\infty}^{(\beta)}(k)$ say. Thus expanding $\pi\beta
S_{\infty}^{(\beta)}(k)/k$ as a power series in $(k/\beta)$, the $j$-th
coefficient is a monic polynomial of degree $j$ in $\beta/2$ [162, 155].
The use of linear response to quantify the change of charge density upon the
introduction of a charge $q$ into the OCP, computing from this the change of
charge by integrating, and requiring by screening that this must equal $-q$
can be used [210] to deduce the Carnie-Chan sum rule [88, 89]
(4.31)
$-\beta\int_{\mathbb{R}^{2}}d\mathbf{r}\bigg{(}\int_{\mathbb{R}^{2}}d\mathbf{r}^{\prime}\,\log|\mathbf{r}^{\prime}|\Big{(}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r},\mathbf{r}^{\prime})+{1\over\pi}\delta(\mathbf{r}-\mathbf{r}^{\prime})\Big{)}\bigg{)}=1.$
Here the integral cannot be interchanged, as according to (4.20) integrating
over $\mathbf{r}$ first would give zero. The validity of (4.31) can be checked
directly for $\beta=2$ in the bulk (i.e. for GinUE in the bulk).
###### Proposition 4.8.
The Carnie-Chan sum rule (4.31) is valid for GinUE in the bulk.
###### Proof.
Denote the integral over $\mathbf{r}^{\prime}$ in (4.31) by $g(\mathbf{r})$.
Assuming only rotation and translation invariance of $\rho_{(2),\infty}^{{\rm
OCP},T}$ implies
$g(\mathbf{r})=\log|\mathbf{r}|\Big{(}\int_{|\mathbf{r}^{\prime}|<|\mathbf{r}|}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r}^{\prime},\mathbf{0})\,d\mathbf{r}^{\prime}+{1\over\pi}\Big{)}+\int_{|\mathbf{r}^{\prime}|>|\mathbf{r}|}\log|\mathbf{r}^{\prime}|\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r}^{\prime},\mathbf{0})\,d\mathbf{r}^{\prime}.$
Specialising now to $\beta=2$ by substituting (2.24), the use of polar
coordinates and integration by parts allows the integrals to be simplified,
with the result
$g(\mathbf{r})=-{1\over\pi}\int_{|\mathbf{r}|}^{\infty}{e^{-(r^{\prime})^{2}}\over
r^{\prime}}\,dr^{\prime}$
The integration over $\mathbf{r}\in\mathbb{R}^{2}$ of this can be computed by
further use of polar coordinates and integration by parts to confirm (4.31). ∎
Formal use of the convolution theorem in (4.31) shows that it reduces to
(4.27) [285]. In the case of the edge geometry, this approach can be used
[213] to derive from (4.31) in the edge case the edge dipole moment sum rule
(4.32)
$-2\pi\beta\int_{-\infty}^{\infty}dy\,\bigg{(}\int_{-\infty}^{\infty}dy^{\prime}\,(y^{\prime}-y)\int_{-\infty}^{\infty}dx^{\prime}\,\rho_{(2),\infty}^{{\rm
e},{\rm OCP},T}((0,y),(x^{\prime},y^{\prime}))\bigg{)}=1,$
first derived in [64]. In keeping with the analogous property of (4.31), it is
not possible to interchange the integrations over $y$ and $y^{\prime}$ in this
expression (if it was the LHS would vanish due to the sign change of
$y^{\prime}-y$). In the case $\beta=2$ this is readily verified using the
exact result (2.25).
###### Proposition 4.9.
The edge dipole moment sum rule (4.32) is valid for edge scaled GinUE.
###### Proof.
Substituting (2.25), integration by parts gives
$\int_{-\infty}^{\infty}dy^{\prime}\,(y^{\prime}-y)\int_{-\infty}^{\infty}dx^{\prime}\,\rho_{(2),\infty}^{{\rm
e},{\rm
GinUE},T}((0,y),(x^{\prime},y^{\prime}))=-{1\over\sqrt{8\pi^{3}}}e^{-2y^{2}}$
(this formula can be found in [208, Eq. (2.41)]). Now integrating over $y$
verifies (4.32) for $\beta=2$. ∎
The exact result (2.26) exhibits a slow decay of
$\rho_{(2),\infty}^{T}((x_{1},y_{1}),(x_{2},y_{2}))$ parallel to the edge.
This was first predicted by Jancovici [209, 211], who on the basis of linear
response argument relating to the screening an oscillatory external charge
density applied at the edge obtained for general $\beta>0$,
(4.33)
$\rho_{(2),\infty}^{T}((x_{1},y_{1}),(x_{2},y_{2}))\mathop{\sim}\limits_{|x_{1}-x_{2}|\to\infty}{f(y_{1},y_{2})\over(x_{1}-x_{2})^{2}},\qquad\int_{-\infty}^{\infty}dy_{1}\int_{-\infty}^{\infty}dy_{2}\,f(y_{1},y_{2})=-{1\over
2\beta\pi^{2}}.$
A derivation of this using the Carnie-Chan sum rule for the OCP confined to a
strip geometry is given in [210]. In keeping with this, in [215] the amplitude
$f(y,y^{\prime})$ in (4.33) is related to the dipole moment of the screening
cloud at the edge by deriving that
(4.34)
$\int_{-\infty}^{\infty}dy^{\prime}\,(y^{\prime}-y)\int_{-\infty}^{\infty}dx^{\prime}\,\rho_{(2),\infty}^{{\rm
e},{\rm
OCP},T}((0,y),(x^{\prime},y^{\prime}))=\pi\int_{-\infty}^{\infty}dy^{\prime}\,f(y,y^{\prime}).$
Then the result for the integral over $y$ on the RHS follows from the dipole
moment sum rule (4.32).
We consider now a global scaling, so that the droplet support is the unit
disk. For large $N$, define the charge-charge surface correlation by
(4.35) $\langle\sigma(\theta)\sigma(\theta^{\prime})\rangle_{N}^{T}:=\int
d\mathbf{n}\int d\mathbf{n}^{\prime}\,\rho_{(2),N}^{{\rm g},{\rm
OCP},T}(\mathbf{r},\mathbf{r}^{\prime}).$
Here $\rho_{(2),N}^{{\rm g},{\rm OCP},T}$ refers to $\rho_{(2),N}^{{\rm
OCP},T}$ computed using global scaling (as specified below (4.18)), then
considering the large $N$ form with edge coordinates (these can be taken as
$(r,\theta)$ with $r\approx 1$. The integration is over the normal direction
$\mathbf{n}$ (here the radial direction); the more general notation has been
used as this quantity can be defined in other geometries e.g. for the elliptic
GinUE [161]. The reasoning leading to (4.33) has been extended [211] to lead
to the prediction
(4.36)
$\langle\sigma(\theta)\sigma(\theta^{\prime})\rangle_{\infty}^{T}=-{1\over
8\pi^{2}\beta\sin^{2}((\theta-\theta^{\prime})/2)}.$
This has been checked at $\beta=2$ using the exact result for GinUE in [104].
###### Remark 4.10.
For the bulk scaled OCP with $\beta$ an even integer, there is a constraint on
the small $r=|\mathbf{r}|$ form of $\rho_{(2),\infty}^{{\rm
b}}(\mathbf{r},\mathbf{0})$. Thus [301]
$\rho_{(2),\infty}^{{\rm b}}(\mathbf{r},\mathbf{0})=r^{\beta}e^{-\beta
r^{2}/4}f(r^{4}),$
for $f(z)$ analytic. Note from (2.24) that for GinUE,
$f(z)={2\over\pi^{2}}{\sinh(\sqrt{z}/2)\over\sqrt{z}}$. Recently consequences
have been found in [300].
## 5\. Normal matrix models
### 5.1. Eigenvalue PDF
A generalisation of the joint element PDF (2.34) is the functional form
proportional to
(5.1) $\exp\Big{(}-{N\over t_{0}}{\rm Tr}(JJ^{\dagger}-2{\rm
Re}\,\sum_{p=2}^{M}t_{p}{\rm Tr}\,J^{p})\Big{)},\quad t_{0}>0;$
note the scaling so that there is a factor of $N$ in the exponent. Use of the
Schur decomposition (2.2) gives a separation of the eigenvalues and the
strictly upper triangular elements in the exponent analogous to (2.4). This
allows the latter to be integrated over, showing that the corresponding
eigenvalue PDF is proportional to
(5.2) $\exp\bigg{(}-{N\over t_{0}}\sum_{j=1}^{N}\Big{(}|z_{j}|^{2}-2{\rm
Re}\sum_{p=2}^{M}t_{p}z_{j}^{p}\Big{)}\bigg{)}.$
However for $t_{M}\neq 0$ this is not normalisable for $M>2$.
A simple remedy is to impose a cutoff by restricting the eigenvalues to a disk
centred about the origin of radius $R$, and restricting to small $t_{0}$ and
values of $t_{2},\dots,t_{M}$ so that the support $\Omega$ (assumed simply
connected) of the density is contained inside this disk [138]. The mean field
argument leading to (2.30) tells us that in relation (5.2), the normalised
density has the constant value $1/\pi t_{0}$ in $\Omega$, where the latter is
such that
(5.3) $W(z)={1\over\pi}\int_{\Omega}\log|z-w|\,d^{2}w,\qquad W(z)={1\over
2}\Big{(}|z|^{2}-2{\rm Re}\sum_{p=2}^{M}t_{p}z^{p}\Big{)},$
for $z$ contained inside of $\Omega$. From this equation the coupling
constants $\\{t_{p}\\}_{p=2}^{M}$ can be related to moments associated with
$\Omega$ [4, 328].
###### Proposition 5.1.
In the above setting, for $p\geq 2$ we have
(5.4) $t_{p}={1\over 2\pi
ip}\int_{\partial\Omega}\bar{z}z^{-p}\,dz=-{1\over\pi
p}\int_{\mathbb{C}\backslash\Omega}z^{-p}\,d^{2}z.$
###### Proof.
By applying $\partial_{z}$ to both sides of (5.3), then introducing $W(w)$ in
the integrand via the identity $2\partial_{w}\partial_{\bar{w}}W(w)=1$ shows
$\partial_{z}W(z)={1\over\pi}\int_{\Omega}{\partial_{w}\partial_{\bar{w}}W(w)\over
z-w}\,d^{2}w.$
Now using the Cauchy-Pompeiu formula (3.22) with $C_{1}$ replaced by $\Omega$,
we see from this that
$\int_{\partial\Omega}{\partial_{w}W(w)\over w-z}\,dw=0.$
Simple manipulation then shows
${1\over 2\pi i}\int_{\partial\Omega}{\bar{w}\over
w-z}\,dw=\sum_{p=2}^{M}pt_{p}z^{p-1}.$
The first equality of (5.4) now follows by power series expanding the LHS with
respect to $z$. The second equality can be deduced from the first by applying
the version of (3.22) valid for $z$ outside $C_{1}$ (chosen as
$\mathbb{C}\backslash\Omega$), for which the LHS is to be replaced by $0$. ∎
The simplest case of (5.2) beyond that corresponding to (2.34) is to take
$M=3$. After scaling, the choice of $t_{3}$ can be fixed. Choosing
$t_{3}={1\over 3}$, results from [234] give that the support of $\Omega$ is
contained in a disk for all $0<t_{0}\leq 1/8$, with the critical value
$t_{0}=1/8$ involving three cusp singularities. Then, the boundary
$\partial\Omega$ is a 3-cusped hypercycloid. For general $M\geq 2$ and
$\Omega$ contained in a disk and simply connected, it is shown in [138] that
$\partial\Omega$ can be parametrised by a Laurent polynomial of the form
$\alpha_{1}w+\alpha_{0}+\cdots+\alpha_{M-1}w^{-M+1}$ with $|w|=1$. Dropping
the assumption that $\Omega$ be simply connected requires the theory of
quadrature domains; see [249] and references therein.
More general than the joint eigenvalue PDF (5.1) is the functional form
(5.5) $\exp\Big{(}-\sum_{j,k=1}^{\infty}c_{jk}{\rm
Tr}\,(J^{j}(J^{\dagger})^{k})\Big{)}=:\exp\Big{(}-{\rm
Tr}\,W(J,J^{\dagger})\Big{)}.$
However unlike the former, substituting the Schur decomposition (2.2) does not
in general lead to a separation of the eigenvalues from the strictly upper
triangular variables. To overcome this, attention can be restricted to the
subset of $N\times N$ complex matrices having the further structure
$[J,J^{\dagger}]=0$, which specifies the matrices as being normal. For normal
matrices, the eigenvectors can be chosen to form an orthonormal set, and so
$J=UDU^{\dagger}$, for $U$ unitary and $D$ the diagonal matrix of eigenvalues.
Moreover, the change of variables from $J$ to $\\{U,D\\}$ gives a
decomposition of measure into separate eigenvalue and eigenvector factors,
with the Jacobian given by (2.5) [101]. Hence the eigenvalue PDF corresponding
to (5.5) in this setting is proportional to
(5.6) $\exp\Big{(}-\sum_{j=1}^{N}W(z_{j},\bar{z}_{j})\Big{)}\prod_{1\leq
j<k\leq N}|z_{k}-z_{j}|^{2}.$
Note that for the joint element PDF (5.1), the same eigenvalue PDF (5.2)
results for $J$ specified on the full space of $N\times N$ complex matrices,
as it does on the restriction to normal matrices as implied by (5.6).
### 5.2. Equilibrium measure
Analogous to (5.1), to obtain a compact eigenvalue support for large $N$, one
considers the case that the weight $e^{-W}$ is exponentially varying in a
sense that $W$ is of order $N$, say $W(z,\bar{z})=NQ(z)$ for a fixed potential
$Q:\mathbb{C}\to\mathbb{R}$. Thus the eigenvalue PDF (5.6) is written as
(5.7) $e^{-H(z_{1},\dots,z_{N})},\quad H(z_{1},\dots,z_{N}):=\sum_{1\leq
j<l\leq N}\log\frac{1}{|z_{j}-z_{l}|^{2}}+N\sum_{j=1}^{N}Q(z_{j}).$
As in Remark 2.5, the macroscopic behaviour of the system (5.7) can be
described using the two-dimensional Coulomb gas interpretation with the help
of logarithmic potential theory. Namely, it is well known in the literature
[216, 199, 92, 93, 29] that for a general $Q$ under suitable potential
theoretic assumptions, the empirical measure $\frac{1}{N}\sum\delta_{z_{j}}$
weakly converges to a unique probability measure $\mu_{Q}$ that minimises
(5.8)
$I_{Q}[\mu]:=\int_{\mathbb{C}^{2}}\log\frac{1}{|z-w|}\,d\mu(z)\,d\mu(w)+\int_{\mathbb{C}}Q\,d\mu;$
cf. (1.10). One may notice that (5.8) can be interpreted as a continuum limit
of the Hamiltonian $H_{N}$ in (5.7) after normalisation, generalising (2.29).
The probability measure $\mu_{Q}$ is called the equilibrium measure and its
support $S_{Q}:=\textup{supp}(\mu_{Q})$ is called the droplet, as previously
remarked. Furthermore, if $Q$ is $C^{2}$-smooth in a neighbourhood of $S_{Q}$,
by Frostman’s theorem [298], $\mu_{Q}$ is absolutely continuous with respect
to the Lebesgue measure and takes the form
(5.9) $d\mu_{Q}(z)=\frac{\partial_{z}\partial_{\bar{z}}Q(z)}{\pi}\chi_{z\in
S_{Q}}\,d^{2}z.$
In particular, for a rotationally symmetric potential $q(r)=Q(|z|=r)$, the
droplet is of the form $S_{Q}=\\{r_{1}\leq|z|\leq r_{2}\\}$, where
$(r_{1},r_{2})$ are the unique pair of constants satisfying
(5.10) $r_{1}q^{\prime}(r_{1})=0,\quad r_{2}q^{\prime}(r_{2})=2,$
see [298, § IV.6]. Here, we have assumed that $Q(z)$ is strictly subharmonic
in $\mathbb{C}$, which is equivalent to the requirement that $r\mapsto
rq^{\prime}(r)$ is increasing in $(0,\infty).$ Let us mention that all the
explicit macroscopic densities with rotation invariance in the above
subsections of §2 can be obtained as special cases of the formulas (5.9) and
(5.10). For instance, the RHS of (2.45) can be realised as the RHS of (5.9)
with $Q(z)=|z|^{2}-2\alpha\log|z|$. Beyond the case when $Q$ is radially
symmetric, the determination of the droplet, also known as the two-dimensional
equilibrium problem (e.g. the ellipse in § 2.3), is far from being obvious
even for some explicit potentials with simple form; see [45, 47, 232, 14, 113]
and references therein for recent works in this direction.
### 5.3. Partition functions
Continuing the discussion in §4.1, we consider the global scaled, non charge
neutral, partition function
(5.11)
$Z_{N}(\beta;Q)=\frac{1}{N!}\int_{\mathbb{C}}d^{2}z_{1}\cdots\int_{\mathbb{C}}d^{2}z_{N}\,e^{-\frac{\beta}{2}H(z_{1},\dots,z_{N})},$
where $H(z_{1},\dots,z_{N})$ is the Hamiltonian given in (5.7). An explicit
formula for the large $N$ expansion of $Z_{N}(\beta;Q)$ was predicted in
[329]. Fairly recently, it was shown by Leblé and Serfaty [246] that for
general $\beta>0$ and $Q$, $Z_{N}(\beta;Q)$ admits the large $N$ asymptotic
expansion of the form
(5.12) $\log
Z_{N}(\beta;Q)=-\frac{\beta}{2}N^{2}I_{Q}[\mu_{Q}]+\Big{(}\frac{\beta}{4}-1\Big{)}N\log
N-\bigg{(}C(\beta)+\Big{(}1-\frac{\beta}{4}\Big{)}E_{Q}[\mu_{Q}]\bigg{)}N+o(N);$
see also an earlier work [303] on (5.12) up to the $O(N\log N)$ term. The term
$I_{Q}[\mu_{Q}]$ appearing in the leading order asymptotic is the energy (5.8)
evaluated at the equilibrium measure $\mu_{Q}$. In the case of the OCP, up to
a sign it is the quantity appearing in the exponent of $A_{N,\beta}$ in (4.2)
at order $N^{2}$. To see this, we note that for a radially symmetric potential
$q(r)=Q(|z|=r)$ generally, it is evaluated as
(5.13) $I_{Q}[\mu_{Q}]=q(r_{1})-\log
r_{1}-\frac{1}{4}\int_{r_{0}}^{r_{1}}rq^{\prime}(r)^{2}\,dr,$
where $r_{1}$ and $r_{2}$ are the constants specified in (5.10). For the OCP
$Q(z)=|z|^{2}$, which gives $I_{Q}[\mu_{Q}]=3/4$, this indeed being the
coefficient of $N^{2}$ seen in $A_{N,\beta}$. The appearance of the term
$(\beta/4-1)N\log N$ in (5.12), whereas there is a term ${1\over 4}\beta
N^{2}\log N$ in $A_{N,\beta}$ of (4.2) is due to the simple scaling
$z_{j}\mapsto z_{j}/\sqrt{N}$ required to go from the OCP with global scaled
coordinates (as assumed in (5.12)) to the OCP itself (as assumed in (4.2)).
The terms appearing in the $O(N)$ term of (5.12) are the entropy
(5.14) $E_{Q}[\mu_{Q}]:=\int_{\mathbb{C}}\mu_{Q}(z)\,\log\mu_{Q}(z)\,d^{2}z$
associated with $\mu_{Q}$ and a constant $C(\beta)$ independent of the
potential $Q$. Their sum is the free energy per particle, $\beta f(\beta;Q)$.
The expansion (5.12) with quantitative error bounds is also available in the
literature [49, 308].
For the random normal matrix model when $\beta=2$, the determinantal structure
(2.9) allows an explicit expression
(5.15) $Z_{N}(2;Q)=\prod_{j=0}^{N-1}h_{j},$
where $h_{j}$ is the orthogonal norm in (5.18); cf. Proposition 2.1. In
particular, if $Q$ is rotationally symmetric, since $p_{j}(z)=z^{j}$, we have
(5.16) $h_{j}=2\pi\int_{0}^{\infty}r^{2j+1}e^{-Nq(r)}\,dr;$
cf. (2.87). Based on this knowledge together with a Laplace approximation of
the integrals, the precise asymptotic expansion of the free energy up to the
$O(1)$ term was derived in a recent work [78]. This in particular shows that
(5.17) $\log Z_{N}(2;Q)=-N^{2}I_{Q}[\mu_{Q}]-\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi^{2})}{2}-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}\,N-\frac{\chi}{12}\log
N+O(1),$
where $\chi$ is the Euler index; cf. (4.6). Let us recall here that $\chi=0$
for the annulus ($r_{0}>0$) and $\chi=1$ for the disk ($r_{0}=0$) geometry.
See [78, §4] and [145, §4] for some concrete examples of (5.17) associated
with the matrix models discussed in §2.
### 5.4. Correlation functions and universality
Turning to the correlation functions $\rho_{(k),N}$ of (5.7), the
determinantal structure (2.9) remains valid with the correlation kernel
(5.18)
$K_{N}(z,w)=e^{-\frac{N}{2}(Q(z)+Q(w))}\sum_{j=0}^{N-1}\frac{p_{j}(z)\overline{p_{j}(w)}}{h_{j}},$
where $p_{j}$ is the monic orthogonal polynomial of degree $j$ with respect to
the weighted Lebesgue measure $e^{-NQ(z)}\,d^{2}z$ and $h_{j}$ is its squared
orthogonal norm; cf. (3.6).
Let us first discuss the asymptotic behaviours of $\rho_{(k),N}$ in the micro-
scale. Given a base point $p\in S_{Q}$ for which we zoom the point process, we
denote by
(5.19) $\delta:=\frac{\partial_{z}\partial_{\bar{z}}Q(z)}{\pi}\Big{|}_{z=p}$
the mean eigenvalue density at $p$; cf.(5.9). From universality principles
(see e.g. [231]), one expects that for a general $Q$ and $p$ such that
$\delta\in(0,\infty)$ (i.e. the eigenvalue density does not vanish or diverge
at $p$), the universal scaling limit in Proposition 2.4 arises. For the bulk
case when $p\in\textup{Int}(S_{Q})$, such a universality was established by
Ameur, Hedenmalm and Makarov in [35] where they showed that for a fairly
general potential $Q$ under some mild assumptions,
(5.20)
$\displaystyle\frac{1}{(N\delta)^{k}}\rho_{(k),N}\Big{(}p+\frac{z_{1}}{\sqrt{N\delta}},\dots,p+\frac{z_{k}}{\sqrt{N\delta}}\Big{)}\to\det\Big{[}K_{\infty}^{\rm
b}(z_{j},z_{l})\Big{]}_{j,l=1}^{k},$
uniformly on compact subsets of $\mathbb{C}$ as $N\to\infty$. See also [55,
57]. In a sequential work [36] the theory of loop equations (or Ward’s
identities) was also used to show the bulk scaling limit (5.20). It says that
for a given test function $\psi$,
(5.21) $\mathbb{E}_{N}W_{N}^{+}[\psi]=0,\quad
W_{N}^{+}[\psi]:=\frac{1}{2}\sum_{j\not=k}\frac{\psi(z_{j})-\psi(z_{k})}{z_{j}-z_{k}}-N\sum_{j=1}^{N}[\partial
Q\cdot\psi](z_{j})+\sum_{j=1}^{N}\partial\psi(z_{j}).$
The functional $W_{N}^{+}$ is also called the stress energy tensor in the
context of conformal field theory [219, Appendix 6]. The identity (5.21)
easily follows from the integration by parts
(5.22)
$\mathbb{E}_{N}[\partial\psi(z_{j})]=\mathbb{E}_{N}[\partial_{z_{j}}H(z_{1},\dots,z_{N})\cdot\psi(z_{j})].$
The approach using Ward’s identities was further developed in [38] and several
related works to study various scaling limits of the random normal matrix
models (see e.g. [39, 30] for the bulk scaling limit at weak non-Hermiticity
(2.40); [40, 306] for the edge scaling limit with boundary confinements
(2.65); [37, 41] for normal matrices with Mittag-Leffler type singularities as
in §2.4 and §2.7) and we refer to [15, Remark 2.9] for an expository summary
of this strategy. In particular, in [38], the rescaled version of Ward’s
identity was introduced and used to show the edge universality: for
$p\in\partial S_{Q},$
(5.23)
$\displaystyle\frac{1}{(N\delta)^{k}}\rho_{(k),N}\Big{(}p+i\mathbf{n}\frac{z_{1}}{\sqrt{N\delta}},\dots,p+i\mathbf{n}\frac{z_{k}}{\sqrt{N\delta}}\Big{)}\to\det\Big{[}K_{\infty}^{\rm
e}(z_{j},z_{l})\Big{]}_{j,l=1}^{k},$
where $\mathbf{n}$ is the outer normal vector at $p$ as in Proposition 2.8.
However, the result in [38] has an additional assumption that the limiting
correlation function is translation invariant along the real axis. This
assumption is intuitively natural but hard to rigorously show in general
(except e.g. for the rotationally symmetric potential $Q$). The edge
universality was later then shown by Hedenmalm and Wennman for a wide class of
potentials $Q$ in [201], where they developed an asymptotic theory for general
planar orthogonal polynomials; see also [198] for an alternative approach to
derive the main result in [201] and [200] for a theory developed for the
orthogonal polynomials associated with non-exponentially varying weight. The
asymptotic results in [201] together with (5.18) then leads to (5.23) by the
Riemann sum approximation.
We now turn our attention to the asymptotic behaviours of $\rho_{(k),N}$ in
the macro-scale. Let us begin with the $1$-point function $\rho_{(1),N}$. For
the bulk case when $z$ is interior of the droplet, the asymptotic behaviour of
$\rho_{(1),N}$ is well known in the literature, see [55, 27] and references
therein. It says that under the suitable assumptions on $Q$, there are real-
analytic functions $B_{j}$ such that
(5.24) $\pi\rho_{(1),N}(z)=N\Delta Q(z)+\frac{1}{2}\Delta\log\Delta
Q(z)+N^{-1}B_{2}(z)+\cdots+N^{-k+1}B_{k}(z)+O(N^{-k}),$
where $\Delta:=\partial_{z}\partial_{\bar{z}}$ is the one quarter of the usual
Laplacian. Notice here that the leading order asymptotic of (5.24) is implied
by the Laplacian growth (5.9). As a concrete example, we consider the induced
spherical ensemble (2.52) with $M=\alpha_{1}N$ and $n=\alpha_{1}N-1$, which
can be realised as (5.7) with
$Q(z)=(\alpha_{1}+\alpha_{2}-1)\log(1+|z|^{2})-2(\alpha_{1}-1)\log|z|.$
On the other hand, it follows from (2.54) that with
$\zeta=|z|^{2}/(1+|z|^{2})$,
(5.25)
$\pi\rho_{(1),N}(z)=\frac{|z|^{2(M-N)}}{(1+|z|^{2})^{2}}(M+n-N)\Big{(}I_{\zeta}(M-N,n)-I_{\zeta}(M,n-N)\Big{)}.$
Then using the well-known asymptotic behaviours of the incomplete beta
function, one can observe that for $r_{1}<|z|<r_{2}$, where $r_{1},r_{2}$ are
given in (2.53), the asymptotic behaviour (5.24) holds with
$\Delta
Q(z)=\frac{\alpha_{1}+\alpha_{2}-1}{(1+|z|^{2})^{2}},\qquad\Delta\log\Delta
Q(z)=-\frac{2}{(1+|z|^{2})^{2}}.$
Next, we consider the off-diagonal asymptotic behaviour of the correlation
kernel. For the Ginibre ensemble, equivalently, for the random normal matrix
model (5.7) with $Q(z)=|z|^{2}$, the associated correlation kernel $K_{N}$ in
(5.18) satisfies
(5.26)
$K_{N}(z,w)=\sqrt{\frac{2N}{\pi}}(z\bar{w})^{N}e^{N-\frac{N}{2}(|z|^{2}+|w|^{2})}S(z,w)\Big{(}1+O(\frac{1}{N})\Big{)},\qquad(z\not=w)$
where $S(z,w)$ is the exterior Szegö kernel
(5.27) $S(z,w):=\frac{1}{2\pi}\frac{1}{z\bar{w}-1}.$
From a viewpoint of Proposition 2.2, the asymptotic behaviour (5.26) can be
realised as a uniform expansion of the incomplete gamma function
$z\mapsto\Gamma(N;Nz)$, which is available in the literature in some
particular domains; see e.g. [323] for $|\arg(z-1)|<3\pi/4.$ In a recent work
[34], generalising the classical results, it was shown that the asymptotic
behaviour (5.26) remains valid as long as $z\bar{w}$ is outside the Szegö
curve $\\{z\in\mathbb{C}:|z|\leq 1,|z\,e^{1-z}|=1\\}.$ Note in particular that
if $|z|=|w|=1,$ then (5.26) reads
(5.28)
$K_{N}(z,w)\overset{c}{\sim}\sqrt{\frac{2N}{\pi}}S(z,w)\Big{(}1+O(\frac{1}{N})\Big{)},$
where $\overset{c}{\sim}$ means that the asymptotic expansion holds up to a
sequence of cocycles (in this case $(z\bar{w})^{N}$), which cancel out when
forming a determinant (2.9). From a statistical physics point of view, the
behaviour (5.28) indicates that there are strong correlations among the
particles on the boundary of the droplet, which also shows the slow decay of
correlations at the boundary. For GinUE this is explicit in (4.36). For
elliptic GinUE, the phenomenon was studied in [161], as a test of the
generalisation to more general shaped droplets, when the RHS of (4.36) is
predicted to be given in terms of a certain Green’s function for an
electrostatics problem outside of the droplet, which acts as a macroscopic
conductor [211], [147, Eq. (3.29]. Furthermore, it was obtained by Ameur and
Cronvall [34] that for a general class of potentials $Q$, the associated
correlation kernel $K_{N}$ satisfies
(5.29)
$K_{N}(z,w)\sim\sqrt{2N}\Big{(}\frac{\partial_{z}\partial_{\bar{z}}Q(z)}{\pi}\Big{)}^{1/4}\Big{(}\frac{\partial_{w}\partial_{\bar{w}}Q(w)}{\pi}\Big{)}^{1/4}S(z,w)(1+o(1)).$
For this, the use of a general theory on the orthogonal polynomial due to
Hedenmalm and Wennman [201] was made. We also refer to [17, 80, 31] for more
recent studies in this direction.
###### Remark 5.2.
Let $\\{p_{j}^{(N,R)}(z)\\}_{j=0,1,\dots}$ be the orthogonal polynomials with
respect to the inner product $\langle
f|g\rangle:=\int_{C_{R}}f(z)g(\bar{z})e^{-2NW(z)/t_{0}}$, where $W(z)$ as in
(5.3) and it is assumed that the limiting eigenvalue support $\Omega$
corresponding to (5.2) is contained in $D_{R}$. Consider the probability
distribution $\mathcal{P}_{k}^{(N,R)}$ specified by the eigenvalue PDF
proportional to
(5.30) $\prod_{l=1}^{k}e^{-2NW(z_{l})/t_{0}}\prod_{1\leq j<l\leq
k}|z_{l}-z_{j}|^{2}.$
As a result of the underlying determinantal structure, one has the formula for
$p_{k}^{(N,R)}(z)$ as an expectation with respect to $\mathcal{P}_{k}^{(N,R)}$
(see e.g. [149, proof of Proposition 5.1.3]; in fact such formulae were known
to Heine [318])
$p_{k}^{(N,R)}(z)=\Big{\langle}\prod_{l=1}^{k}(z-z_{l})\Big{\rangle}_{\mathcal{P}_{k}^{(N,R)}}.$
For $k/N=x$ as $k,N\to\infty$ it has been conjectured [137] that the zeros of
$p_{k}^{(N,R)}(z)$ accumulate on certain arcs $\Sigma$ contained in $\Omega$,
with corresponding measure $\mu_{x}^{*}$. Assuming this, it follows that for
$z\in\mathbb{C}\backslash\Omega$
(5.31) ${1\over\pi
t_{0}}\int_{\Omega}\log|z-w|\,d^{2}w=\int_{\Sigma}\log|z-s|\,d\mu_{x}^{*}.$
This identifies $\Sigma$ as the so-called mother body or potential theoretic
skeleton of $\Omega$. A number of works give further developments along these
lines, especially in relation to the strong asymptotics of the planar
orthogonal polynomials based on Riemann-Hilbert analysis. In the case of (5.6)
with a cubic potential, references include [62, 234, 272, 63], while for the
induced Gaussian type weight $2W(z)/t_{0}=|z|^{2}-2c\log|z-a|$ or more
generally $2W(z)/t_{0}=|z|^{2}-2\sum_{j=1}^{M}c_{j}\log|z-a_{j}|$ in (5.30),
see [45, $c=O(1)$], [46, $c=O(1/N)$ and $|a|\not=1$], [60, $c=O(1/N)$ and
$|a|\approx 1$], [251, $c=O(1/N)$ and $|a|\not=1$], [252, $c_{j}=O(1/N)$]. We
also refer to [325, 118, 241] for applications of such strong asymptotics in
the context of the characteristic polynomials of the Ginibre matrix.
## 6\. Further theory and applications
### 6.1. Fermi gas wave function interpretation
We have seen that the rewrite of (1.7), written in the exponential form
(1.10), allows for the GinUE eigenvalue PDF to be interpreted as the Boltzmann
factor for a particular Coulomb gas. If instead of an exponential form we use
(2.12) to rewrite (1.7) as
(6.1)
$\Big{|}\prod_{j=1}^{N}e^{-|z_{j}|^{2}/2}\det[z_{j}^{k-1}]_{j,k=1,\dots,N}\Big{|}^{2},$
then we are lead to an interpretation as the absolute value squared of a
ground state free Fermi quantum many body wave function. Thus inside the
absolute value of (6.1) is a Slater determinant of single body wave functions
$\\{\phi_{l}(z)\\}_{l=0,\dots,N-1}$ with $\phi_{l}(z)=e^{-|z|^{2}/2}z^{l}$.
What remains then is to identify the corresponding one body Hamiltonian for
quantum particles in the plane which have these single body wave functions for
the lowest energy states.
The appropriate setting for this task is a quantum particle confined to the
$xy$-plane subject to a perpendicular magnetic field, $(0,0,B)$, $B>0$.
Fundamental to this setting is the vector potential $\mathbf{A}$, related to
the magnetic field by $\nabla\times\mathbf{A}=(0,0,B)$. The so-called
symmetric gauge corresponds to the particular choice
$\mathbf{A}=(-By,Bx,0)=:(A_{x},A_{y},0)$, which henceforth will be assumed.
Physical quantities in this setting are $m$ (the particle mass), $e$ (particle
charge), $\hbar$ (Planck’s constant), $c$ (speed of light), which together
with $B$ are combined to give $\omega_{c}:=eB/mc$ (cyclotron frequency) and
$\ell:=\sqrt{\hbar c/eB}$ (magnetic length).
Defining the generalised momenta and corresponding raising and lowering
operators by
$\Pi_{u}=-i\hbar{\partial\over\partial u}+{e\over c}A_{u}\>\>(u=x,y),\qquad
a^{\dagger}={\ell\over\sqrt{2}\hbar}(\Pi_{x}+i\Pi_{y}),\qquad
a=(a^{\dagger})^{\dagger},$
allows the quantum Hamiltonian to be written in the harmonic oscillator like
form $H_{B}=\hbar\omega_{c}(a^{\dagger}a+{1\over 2})$ [108]. Important too are
the quantum centre of orbit operators and associated raising and lowering
operators
$U=u-{\ell^{2}\over\hbar}\Pi_{u}\>\>(U=X,Y;\,u=x,y),\qquad
b^{\dagger}={1\over\sqrt{2}\ell}(X-iY),\qquad b=(b^{\dagger})^{\dagger},$
for which $X^{2}+Y^{2}=2\ell^{2}(b^{\dagger}b+{1\over 2})$. The operators
$\\{a,a^{\dagger}\\}$ commute with $\\{b,b^{\dagger}\\}$, implying that $H$
and $X^{2}+Y^{2}$ permit simultaneous eigenstates. A complete orthogonal set
can be constructed using the raising operators according to
(6.2)
$|n,m\rangle={(a^{\dagger})^{n}(b^{\dagger})^{m}\over\sqrt{n!m!}}|0,0\rangle,$
with eigenvalues of $H$ equal to $(n+{1\over 2})\hbar\omega_{c}$ and
eigenvalue of $X^{2}+Y^{2}$ equal to $(2m+1)\ell^{2}$. The ground state
$|0,0\rangle$ is characterised by $a|0,0\rangle=b|0,0\rangle=0$, which can be
checked to have the unique solution $|0,0\rangle\propto
e^{-(x^{2}+y^{2})/4\ell^{2}}$. From this, application of $(b^{\dagger})^{m}$
gives $|0,m\rangle\propto\bar{z}^{m}e^{-|z|^{2}/4\ell^{2}}$, $z=x+iy$. Forming
a Slater determinant with respect to the first $N$ eigenstates of this type
gives (6.1) with $\ell^{2}=1/2$. Generally states with quantum number $n=0$
and thus belonging to the ground state are said to be in the lowest Landau
level. One remarks that the largest eigenvalue of $X^{2}+Y^{2}$ is then
$N-1/2$, which is in keeping with the squared radius of the leading order
support in the circular law.
The above theory of a quantum particle in the plane subject to a perpendicular
magnetic field can be recast to apply to a rotating quantum particle in the
plane [202, 239]. It is further true that the elliptic GinUE PDF (2.33) admits
an interpretation as the absolute value squared of state in the lowest Landau
level, and furthermore the corresponding orthogonal polynomials (2.35) can be
constructed using a Bogolyubov transformation of $\\{b,b^{\dagger}\\}$ [161].
Also, the PDF on the sphere (2.50) permits an interpretation as the absolute
value squared of the ground state wave function for a free Fermi gas on the
sphere subject to a perpendicular magnetic field [193]. Another point of
interest relates to the $N$-body Fermi ground state corresponding to the
quantum Pauli Hamiltonian in the plane with a perpendicular inhomogeneous
magnetic field $B(x,y)$. The Hamiltonian $H_{B}$ defined above then is to be
multiplied by the $2\times 2$ identity matrix, and the spin coupling term
$-(g\hbar/2m)B(x,y){\rm diag}\,(1/2,-1/2)$ added. With $B(x,y)=-{1\over
2}\nabla^{2}W(x,y)$ for some real valued $W$, and with the assumption
$\Phi:=\int B(x,y)\,dxdy<\infty$, the ground state for this model (which is
spin polarised all spins up) permits an exact solution for $g=2$ [5]. The
ground state of normalisable eigenfunctions has degeneracy
$[\Phi/2\pi\hbar]=:N$, with basis of eigenfunctions
$\\{z^{j}e^{W(x,y)/2\hbar}\\}_{j=0}^{N-1}$. This implies the Fermi many body
ground state (6.1) with $e^{-|z_{j}|^{2}/2}$ replaced by
$e^{W(x_{j},y_{j})/2\hbar}$ [4].
###### Remark 6.1.
1\. Upon stereographic projection of the sphere to the plane, it is possible
to write the quantum Hamiltonian for a charge particle in a constant
perpendicular magnetic field in a form unified with the original planar case
[130]. This involves the Kähler metric and potential, and permits a viewpoint
which carries over to further generalise the space to higher dimensional
complex manifolds in $\mathbb{C}^{m}$. A point of interest is that doing so
gives, for the bulk scaling limit of the corresponding $N$ particle lowest
Landau level state, the natural higher dimensional analogue of the kernel
(2.19) [57, 58].
2\. The squared wavefunction for higher Landau levels (say the $r$-th) has
been shown to give rise to the determinantal point process with bulk scaled
kernel
$K_{\infty}^{r}(w,z)=L_{r}^{0}(|w-z|^{2})e^{w\bar{z}}e^{-(|w|^{2}+|z|^{2})/2};$
see e.g. [312, Prop. 2.5]. Allowing for mixing between Landau levels up to and
including level $r$ leads to squared wave functions giving rise to the same
determinantal point process except for the replacement of Laguerre polynomials
$L_{r}^{0}\mapsto L_{r}^{1}$ in the kernel [192]. Extending [239], the precise
mapping between the rotating fermions in the higher Landau levels and the
polyanalytic Ginibre ensemble was established in [235]. Furthermore, its full
counting statistics and generalisations to finite temperature were obtained in
[315, 236].
3\. In the theory of the fractional quantum Hall effect, constructing an anti-
symmetric state with filling fraction of the lowest Landau level $\nu=1/m$,
for $m$ an odd integer, plays a crucial role. To accomplish this, Laughlin
[243] proposed the ground state wave function proportional to
(6.3) $\prod_{l=1}^{N}e^{-|z_{l}|^{2}/4\ell^{2}}\prod_{1\leq j<k\leq
N}(\bar{z}_{k}-\bar{z}_{j})^{m};$
note that with the assumption that $m$ is odd, this is anti-symmetric as
required for fermions. Moreover it belongs to the lowest Landau level as
follows from the theory in the text below (6.2). The absolute value squared of
(6.3) coincides with the Boltzmann factor (1.10) with $\beta=2m$, and the
scaling $z_{l}\mapsto z_{l}/\sqrt{2m\ell^{2}}$. From potential theoretic/
Coulomb gas reasoning, the bulk density is therefore $1/(2m\pi\ell^{2})$. The
factor of $m$ in the denominator is in precise agreement with the requirement
that the filling fraction be equal to $1/m$.
4\. The ground state $N$-body free spinless Fermi gas in the plane, without a
magnetic field but confined by a radial harmonic potential, is also an example
of a determinantal point process for which exact calculations are possible;
see the recent review [117]. However, its statistical state is distinct from
that of GinUE. Thus with a global scaling so that the support is the unit
disk, the density profile as the $d=2$ Thomas-Fermi functional form
${2\over\pi}(1-|z|^{2})\chi_{|z|<1}$, in contrast to the circular law (2.17).
The bulk scaled two-point correlation function (bulk density $1/4\pi$) is
given in terms of the $J_{1}$ Bessel function
$\rho_{(2),\infty}^{\rm hF}(z_{1},z_{2})=\Big{(}{1\over
4\pi}\Big{)}^{2}\bigg{(}1-\Big{(}{2J_{1}(|z_{1}-z_{2}|)\over|z_{1}-z_{2}|}\Big{)}^{2}\bigg{)},$
in contrast to (2.24). This gives a decay proportional to
$1/|z_{1}-z_{2}|^{3}$ of $\rho_{(2),\infty}^{{\rm hF},T}$. Also, the edge
scaled correlation kernel now involves Airy functions [116], rather than the
error function seen in (2.19). Universality results relating to many body free
Fermi ground states in dimension $d\geq 2$ have recently been obtained [119].
We highlight in particular the macroscopic fluctuation theorem for the linear
statistic $G=\sum_{j}g(\mathbf{r}_{j}/R)$, with $g$ assumed sufficiently
smooth and absolutely integrable, in the $R\to\infty$ limit [119, Th. III.2]
(6.4)
${G-{R^{d}\omega_{d}\over(2\pi)^{d}}\int_{\mathbb{R}^{d}}g(\mathbf{r})\,d^{d}\mathbf{r}\over\sigma_{d}R^{(d-1)/2}}\to{\rm
N}[0,\Sigma(g)],\quad(\Sigma(g))^{2}=\int_{\mathbb{R}^{d}}|\hat{g}(\mathbf{r})|^{2}|\mathbf{r}|\,d^{d}\mathbf{r}.$
Here $\omega_{d}=\pi^{d/2}/\Gamma(1+d/2)$ is the volume of the Euclidean ball
in $\mathbb{R}^{d}$, $\omega_{d}/(2\pi)^{d}$ is the bulk density,
$\sigma_{d}^{2}:=\omega_{d-1}/(2\pi)^{d}$ and the Fourier transform has the
definition
$\hat{g}(\xi)={1\over(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}e^{-i\xi\cdot\mathbf{r}}g(\mathbf{r})\,d^{d}\mathbf{r}$.
Note in particular that in contrast to (3.19), the variance of $G$ now
diverges with the scale $R$.
### 6.2. Quantum chaos applications
The pioneering works of Wigner and Dyson relating to the Hermitian random
matrix ensembles was, as noted in §1, motivated by seeking a model for the
(highly excited) energy levels of a complex quantum system. Later, in the
1980’s, as a fundamental contribution to the then emerging subject of quantum
chaos, Bohigas et al. [65] identified the correct meaning of a complex quantum
system not by the number of particles but rather as one for which the
underlying classical mechanics is chaotic. To test this prediction on say the
numerically generated spectrum of a quantum billiard system, the energy levels
(beyond some threshold to qualify as being highly excited) were first unfolded
so that their local density became unity, and then their numerically
determined statistical properties were compared against random matrix
predictions for the appropriate symmetry class; see e.g. [191]. Most popular
among the statistical properties have been the variance for the number of
eigenvalues in a large interval, and the distribution of the spacing between
successive eigenvalues.
A natural extension of these advances is to inquire about the spectrum of a
dissipative chaotic quantum system, which due to the loss of energy need not
be real. This question was taken up by Grobe, Haake and Sommers [189] for the
specific model of a damped periodic kicked top. The quantum dynamics are
specified by a subunitary density operator. It is the spectrum of this
operator, which after unfolding, and considering only those eigenvalues in the
upper half plane away from the real axis (there is a symmetry which requires
that the eigenvalues come in complex conjugate pairs — see the recent paper
[20] for a discussion of this point in a random matrix context) that were
compared in [189] a statistical sense to GinUE. Following from precedents in
the Hermitian case, in the statistical quantity measured was the distribution
of the radial spacing between closest eigenvalues, to be denoted $P^{\rm
s,GUE}(r)$ with the normalisation $\int_{0}^{\infty}P^{\rm s,GUE}(r)\,dr=1$.
This is the quantity $F_{\infty}(0;D_{r})$ of Remark 3.3.1. Recalling (3.10)
we therefore have
(6.5) $P^{\rm s,GUE}(r)=-{d\over
dr}e^{r^{2}}\prod_{j=1}^{\infty}\Big{(}1-{\gamma(j;r^{2})\over\Gamma(j)}\Big{)}.$
It follows that for small $r$, $P^{\rm s,GUE}(r)\sim 2r^{3}$, while it follows
from (3.11) and the comment in the sentence immediately above Remark 3.3 that
for large $r$, $\log P^{\rm s,GUE}(r)\sim-r^{4}/4$. A numerical plot can be
obtained from the functional form (6.5). For the moments the formula $\langle
r^{p}\rangle=p\int_{0}^{\infty}r^{p-1}e^{r^{2}}E_{\infty}(0;r)\,dr$ holds
true. In particular, for the mean we calculate $\langle
r\rangle=1.142929\dots$.
A variation of the closest neighbour spacing for an eigenvalue at $z$ is the
complex ratio $(z^{\rm c}-z)/(z^{\rm nc}-z)$, where $z^{\rm c}$ is the closest
neighbour to $z$, and $z^{\rm nc}$ is the next closest neighbour [296]. An
approximation, with fast convergence properties to the large $N$ form, has
been given recently in [131].
Very recently a non-Hermitian Hamiltonian realisation of GinUE has been
obtained in the context of a proposed non-Hermitian $q$-body Sachev-Ye-Kitaev
(SYK) model, with $N$ Majorana fermions — $N$ large and tuned ${\rm mod}\,8$ —
and $q>2$ and tuned ${\rm mod}\,4$ [175]. On another front, again very
recently, the emergence of GinUE behaviours in certain model many body quantum
chaotic systems in the space direction has been demonstrated [314]. Of
interest in both these lines of study is the so-called dissipative (connected)
spectral form factor
${\rm K}_{N}^{\rm c}(t,s)={1\over N}{\rm
Cov}\,\Big{(}\sum_{j=1}^{N}e^{i(x_{j}t+y_{j}s)},\sum_{j=1}^{N}e^{-i(x_{j}t+y_{j}s)}\Big{)}.$
Making use of the first formula in (3.2) and the finite $N$ form of (2.23),
this can be evaluated in terms of the hypergeometric function ${}_{1}F_{1}$
[256, Eq. (3)] (corrected in [176, Appendix A]; note too that both those
references use global scaled variables, whereas we do not).
###### Proposition 6.2.
We have
(6.6) ${\rm K}_{N}^{\rm c}(t,s)=1\\\ -{1\over
N}\sum_{m,n=0}^{N-1}{(t^{2}+s^{2})^{|m-n|/2}\over n!m!2^{|m-n|}}\bigg{(}{{\rm
max}(m,n)!\over|m-n|!}\,{}_{1}F_{1}\Big{(}{\rm
max}(m,n)+1,|m-n|+1;-{t^{2}+s^{2}\over 4}\Big{)}\bigg{)}^{2}.$
In particular,
$\lim_{N\to\infty}{\rm K}_{N}^{\rm c}(t,s)=1-e^{-(t^{2}+s^{2})/4};$
cf. (3.17).
Also of interest in the many body quantum chaos application is the GinUE
average of $|{\rm Tr}\,X^{k}|^{2}$ for positive integer $k$ [314].
###### Proposition 6.3.
We have
$\Big{\langle}|{\rm Tr}\,X^{k}|^{2}\Big{\rangle}_{\rm
GinUE}={1\over(k+1)(N-1)!}\Big{(}(k+N)!-{N!(N-1)!\over(N-k-1)!}\Big{)}.$
In particular
$\lim_{N,k\to\infty\atop k/N=x}{1\over kN^{k}}\Big{\langle}|{\rm
Tr}\,X^{k}|^{2}\Big{\rangle}_{\rm GinUE}={2\sinh(x^{2}/2)\over x^{2}}.$
###### Proof.
(Sketch) In [314] the average is reduced to
$\int_{\mathbb{C}}dz_{1}\int_{\mathbb{C}}dz_{2}\,\rho_{(2),N}(z_{1},z_{2})z_{1}^{k}\bar{z}_{1}^{k}$.
Earlier, the evaluation of a more general quantity was given in [165,
Corollary 4]. ∎
We conclude this subsection with a brief account of the use of the GinUE in an
ensemble theory of Lindblad dynamics [83, 123, 297]. This relates to the
evolution of the density matrix $\rho_{t}$ for an $N$-level dissipative
quantum system in the so-called Markovian regime, specified by the master
equation $\dot{\rho}_{t}=\mathcal{L}(\rho_{t})$. Here the operator
$\mathcal{L}$ assumes a special structure identified by Lindblad [258], and by
Gorini, Kossakowski, and Sudarshan [186]. Specifically $\mathcal{L}$ consists
of the sum of two terms, the first corresponding to the familiar unitary von
Neumann evolution, and the second to a dissipative part, being the sum over
operators $D_{L}$ (referred to as simple dissipators), represented as
$N^{2}\times N^{2}$ matrices according to
$D_{L}=2L\otimes_{T}L^{\dagger}-L^{\dagger}L\otimes_{T}\mathbb{I}_{N}-\mathbb{I}_{N}\otimes_{T}L^{\dagger}L,$
for some $N\times N$ matrix $L$. Here $A\otimes_{T}B:=A\otimes B^{T}$, where
$\otimes$ is the usual Kronecker product. In an ensemble theory, there is
interest in $F_{N}(t):={1\over N^{2}}\langle{\rm Tr}\,e^{tD_{L}}\rangle_{L}$
[83].
###### Proposition 6.4.
Let $L$ be chosen from GinUE with global scaling. We have
$\lim_{N\to\infty}F_{N}(t)=e^{-4t}\Big{(}I_{0}(2t)+I_{1}(2t)\Big{)}^{2}.$
###### Proof.
(Sketch) Following Can [83], using a diagrammatic calculus, it is first
demonstrated that
$\lim_{N\to\infty}{1\over N^{2}}\langle
D_{L}^{k}\rangle=\lim_{N\to\infty}{(-1)^{k}\over
N^{2}}\Big{\langle}\Big{(}L^{\dagger}L\otimes_{T}\mathbb{I}_{N}+\mathbb{I}_{N}\otimes_{T}L^{\dagger}L\Big{)}^{k}\Big{\rangle}_{L\in{\rm
GinUE}}.$
The average on the RHS, in terms of the eigenvalues $\\{x_{j}\\}$ of
$L^{\dagger}L$, reads
$\langle\sum_{j,l=1}^{N}(x_{j}+x_{l})^{k}\rangle_{L^{\dagger}L}$ and
consequently
$\lim_{N\to\infty}F_{N}(t)=\lim_{N\to\infty}\Big{\langle}{1\over
N^{2}}\sum_{j,l=1}^{N}e^{-t(x_{j}+x_{l})}\Big{\rangle}_{L^{\dagger}L}=\lim_{N\to\infty}\bigg{(}\Big{\langle}{1\over
N}\sum_{j=1}^{N}e^{-tx_{j}}\Big{\rangle}_{L^{\dagger}L}\bigg{)}^{2}.$
The latter is the mean of a linear statistic in the ensemble
$\\{L^{\dagger}L\\}$ (complex Wishart matrices; see e.g. [149, §3.2]). Using
the Marchenko-Pastur law for the global density of this ensemble (see e.g.
[149, §3.4.1]), the stated result follows. ∎
###### Remark 6.5.
(Classification of non-Hermitian matrices) It was commented in the
Introduction that, in distinction to Dyson’s viewpoint based on symmetry
considerations, Ginibre’s study [179] was no similarly motivated. Nowadays
however, it is recognised that a symmetry viewpoint is fundamental to
topological driven effects in non-Hermitian quantum physics [44]. Starting
with [59, 268] and continuing in [223], a classification scheme based on
symmetries with respect to the involutions of transpose, complex conjugation
and Hermitian conjugation, and in which the (anti-)commutation relation
involves unitary matrices satisfying certain quadratic relations in terms of
these involution, has been given. For example, defining the block unitary
matrix $P={\rm diag}\,(\mathbb{I}_{N},-\mathbb{I}_{N})$, and requiring that
the matrix ensemble $\\{A\\}$ have the (anti-)symmetry $A=-PAP$, gives that
each $A$ has the form
(6.7) $A=\begin{bmatrix}0_{N\times N}&X\\\ Y&0_{N\times N}\end{bmatrix}$
for some square matrices $X,Y$. Denoting the eigenvalues of the matrix product
$XY$ as $\\{-z_{j}^{2}\\}$, one sees that the eigenvalues of $A$ are $\\{\pm
iz_{j}\\}$.
In keeping with the viewpoint of this subsection, a basic question are
signatures of the symmetry in the eigenvalue spectrum. For example, in (6.7),
with $X,Y$ GinUE matrices, are the bulk scaled eigenvalues of $A$
statistically distinct from individual GinUE matrices? We know from the
results quoted in the paragraph above Remark 2.18 that the answer in this case
is no. However the answer to this question is yes, if instead the symmetry is
that $A=A^{T}$, for the independent entries of $A$ standard complex Gaussians.
This was demonstrated in [195] by a numerical study of the nearest neighbour
spacing distribution, and the relevance to Lindblad dynamics discussed.
### 6.3. Singular values
One recalls that for a complex square matrix $X$ the squared singular values
are the eigenvalues of $X^{\dagger}X$. For a general ensemble of non-Hermitian
matrices $\\{X\\}$, motivation to study the singular values comes from various
viewpoints. For example, in Remark 2.18.4, singular values (specifically of
product matrices) appeared in the context of Lyapunov exponents. As other
example, one recalls that plus/minus of the singular values are the
eigenvalues of the $2N\times 2N$ Hermitian matrix
$H=\begin{bmatrix}0_{N\times N}&X\\\ X^{\dagger}&0_{N\times N}\end{bmatrix}.$
The importance of this in relation to the eigenvalues of $X$ is that resolvent
associated with $H$ is fundamental to the study of the circular law for the
spectral density beyond the Gaussian case; see e.g. [67, §4.1]. Another piece
of theory is that the condition number $\kappa_{N}$ associated with $X$ is
equal to the ratio of the smallest to the largest singular value [135]. And
from the identity $|\det X|=|\det X^{\dagger}X|^{1/2}$ the distribution of the
modulus of $\det X$ is determined by the singular values.
For the GinUE, the squared singular values $\\{s_{j}\\}_{j=1}^{N}$ say are
known to have for their joint distribution a PDF proportional to
(6.8) $\prod_{j=1}^{N}e^{-s_{j}}\prod_{1\leq j<k\leq N}(s_{k}-s_{j})^{2},\quad
s_{j}\in\mathbb{R}_{+};$
see e.g. [149, Prop. 3.2.2 with $\beta=2$, $n=m=N$]. After scaling by $N$,
almost surely the largest squared singular value has the limiting value $4$
[286]. However, after the same scaling, a simple change of variables in (6.8)
integrated from $(s,\infty)$ in each variable reveals that the smallest
singular value is an exponential random variable with rate parameter $N^{2}$.
Putting these facts together implies that for large $N$, $\kappa_{N}/N$ is
distributed according to the heavy tailed distribution with PDF ${8\over
x^{3}}e^{-4/x^{2}}\chi_{x>0}$ [135]. Also, for $n\times N$ ($n\geq N$)
rectangular GinUE matrices, it is proved in [102] that
$\langle\log\kappa_{N}\rangle<{N\over|n-N|+1}+2.24$, for any $N\geq 2$.
Let $P_{N}(t)$ denote the PDF for the distribution of $|\det X|^{2}$ for GinUE
matrices. Making use of knowledge of the PDF of squared singular values (6.8)
shows that the Mellin transform of $P_{N}(t)$ is equal to the multiple
integral
(6.9) ${1\over
C_{N}}\int_{0}^{\infty}ds_{1}\cdots\int_{0}^{\infty}ds_{N}\,\prod_{j=1}^{N}s_{j}^{s-1}e^{-s_{j}}\prod_{1\leq
j<k\leq N}(s_{k}-s_{j})^{2}=\prod_{j=0}^{N-1}{\Gamma(s+j)\over\Gamma(1+j)}.$
Here the normalisation $C_{N}$ is such that the expression equals unity for
$s=1$, while the evaluation of the multiple integral follows as a special case
of the Laguerre weight Selberg integral; see e.g. [149, Prop. 4.7.3]. As noted
in [166, Eq. (2.17)] (see also [293, Prop. 2.2]), it follows immediately from
this that
(6.10) $|\det X|^{2}\mathop{=}\limits^{\rm d}\prod_{l=1}^{N}{1\over
2}\chi_{2l}^{2}.$
In words this says that the absolute value squared of the determinant of GinUE
matrices is equal in distribution to the product of $N$ independent chi-
squared distributions, with degrees of freedom $2,4,\dots,2N$, each scaled by
a factor of 2. Starting from (6.10), and defining the global scaled GinUE
matrices $X^{\rm g}$ by $X^{\rm g}={1\over\sqrt{N}}X$, the distribution of
$\log|\det X^{\rm g}|^{2}$ can be shown to have leading order mean $-N$,
variance $\log N$, and after recentring and rescaling satisfy a central limit
theorem [293, Th. 3.5]. For a general linear statistic
$\sum_{j=1}^{N}f(z_{j})$ of global scaled GinUE matrices, the leading order
mean is ${N\over\pi}\int_{|z|<1}f(z)\,d^{2}z$. For $f(z)=\log|z|^{2}$, this
gives the stated value of $-N$. Also, we notice that substituting this choice
of $f(z)$ in the variance formula implied by (3.21) gives
${1\over\pi}\int_{|\mathbf{r}|<1}{1\over x^{2}+y^{2}}\,dxdy$, which is not
integrable at the origin, in keeping with the variance actually diverging as
$\log N$.
There is an alternative viewpoint on the result (6.10) which does not require
knowledge of the joint distribution of the singular values (6.8), nor the
evaluation of the multiple integral (6.9). The idea, used in both [293, 166]
and which goes back to Bartlett [48] in the case of real Gaussian matrices, is
to decompose $X$ in terms of its QR (Gram-Schmidt) decomposition. The matrix
of orthonormal vectors $Q$ constructed from the columns of $X$ will for
$X\in{\rm GinUE}$, be a Haar distributed unitary matrix, which we denote by
$U$. The matrix $R=[r_{jk}]_{j,k=1}^{N}$ is upper triangular with diagonal
elements real and positive. One notes
(6.11) $\det X^{\dagger}X=\prod_{j=1}^{N}r_{jj}^{2},$
and so it suffices to have knowledge on the distribution of
$\\{r_{jj}\\}_{j=1}^{N}$ for $X$.
###### Proposition 6.6.
Let $\\{r_{jj}\\}_{j=1}^{N}$ denote the diagonal elements in the QR
decomposition of a GinUE matrix $X$. We have
(6.12) $r_{jj}^{2}\mathop{=}\limits^{\rm d}{1\over 2}\chi_{2j}^{2}.$
###### Proof.
The QR decomposition $X=UR$ gives the corresponding decomposition of measure
(see e.g. [149, Prop. 3.2.5])
$(dX)=\prod_{j=1}^{N}r_{jj}^{2(N-j)+1}(dR)(U^{\dagger}U),$
where as anticipated $(U^{\dagger}U)$ is recognised as Haar measure on the
space of complex unitary matrices. The element distribution of GinUE matrices
is proportional to $e^{-{\rm Tr}\,X^{\dagger}X}=e^{-\sum_{1\leq j\leq k\leq
N}|r_{jk}|^{2}}$. The various factorisations implies that integrating over $U$
and the off diagonal elements of $R$ only changes the normalisation. We then
read off that each $r_{jj}$ has a distribution with PDF proportional to
$r^{2(N-j)+1}e^{-r^{2}}$, which implies (6.12). ∎
Using (6.12) in (6.11) reclaims (6.10).
###### Remark 6.7.
1\. Since with $\\{z_{j}\\}$ the eigenvalues of $X$, $|\det
X|^{2}=\prod_{j=1}^{N}|z_{j}|^{2}$, the fact that the Mellin transform of the
distribution of this quantity is given by the product of gamma functions in
(6.9) implies
(6.13) $\Big{\langle}\prod_{l=1}^{N}|z_{l}|^{2(s-1)}\Big{\rangle}_{\rm
GinUE}^{\rm g}=N^{N(s-1)}\prod_{j=0}^{N-1}{\Gamma(s+j)\over\Gamma(1+j)}.$
Here the superscript "g" indicates the use of global scaling coordinates
$z_{l}\mapsto\sqrt{N}z_{l}$. We observe that knowledge of the induced GinUE
normalisation $C_{n,N}$ in Proposition 2.11 provides a direct derivation of
(6.13). For large $N$ this ratio of gamma functions can be written in terms of
the Barnes $G$-function according to ${G(N+s)\over G(N+1)G(s)}$; see [149, Eq.
(4.183)]. Known asympotics for ratios of the Barnes $G$-function (see e.g.
[149, Eq. (4.185)] then gives that for large $N$, and with $s=\gamma/2+1$ for
convenience,
(6.14) $\Big{\langle}\prod_{l=1}^{N}|z_{l}|^{\gamma}\Big{\rangle}_{\rm
GinUE}^{\rm g}\sim N^{\gamma^{2}/8}e^{-(\gamma/2)N}{(2\pi)^{\gamma/4}\over
G(1+\gamma/2)}.$
This is the special case $z=0$ of an asymptotic formula for
$\langle\prod_{l=1}^{N}|z-z_{l}|^{\gamma}\rangle_{\rm GinUE}^{\rm g}$ given by
Webb and Wong [325, Th. 1.1].
2\. It is a standard result in random matrix theory (see e.g. [286]) that the
density of singular values in (6.8), after the global scaling $s_{j}\mapsto
s_{j}N$, as the particular Marchenko-Pastur form
$\rho^{|rmMP}_{(1),\infty}(x)={1\over 2\pi}({4-x\over x})^{1/2}\chi_{0<x<4}$.
The $k$-th moment of the density is given in terms of the particular mixed
moment of a global scaled Ginibre matrix $\tilde{G}$, $\langle{\rm
Tr}(\tilde{G}^{\dagger}G)^{k}\rangle$. The calculation of these moments for
large $N$ relates to free probability — see the recent introductory text [288]
for the main ideas — and to combinatorics as is seen from the fact that
$\int_{0}^{4}x^{k}\rho^{\rm MP}_{(1),\infty}(x)\,dx=C_{k}$, where $C_{k}$
denotes the $k$-th Catalan number. Works on mixed moments of Ginibre matrices
include [324, 115, 194, 124].
3\. The squared singular values as specified by the PDF (6.8) form a
determinantal points process, being a special case of the classical Laguerre
unitary ensemble; see e.g. [149, Chapters. 3 and 5]. This is similarly true of
the squared singular values of the various extensions of GinUE considered
above: for example in the case of the spherical model and truncated unitary
matrices, it is the classical Jacobi unitary ensemble which arises, while the
singular values of products of GinUE matrices, or of truncated unitary
matrices, gives rise to a class of determinantal point processes called Pólya
ensembles [233, 227, 226, 167]. A notable exception is the singular values of
elliptic GinUE matrices, which form a Pfaffian point process [218].
### 6.4. Eigenvectors
Associated with the set of eigenvalues $\\{\lambda_{j}\\}$ of a Ginibre matrix
$G$ are two sets of eigenvectors — the left eigenvectors $\\{\bm{\ell}_{j}\\}$
such that $\bm{\ell}_{j}^{T}G=\lambda_{j}\bm{\ell}_{j}^{T}$, and the right
eigenvectors $\\{\mathbf{r}_{j}\\}$ such that
$G\mathbf{r}_{j}=\lambda_{j}\mathbf{r}_{j}$. These are not independent, but
rather (upon suitable normalisation), form a biorthogonal set
(6.15) $\bm{\ell}_{i}^{T}\mathbf{r}_{j}=\delta_{i,j}.$
This property follows from the diagonalisation formula $G=XDX^{-1}$, where $X$
is the matrix of right eigenvectors, $D$ the diagonal matrix of eigenvectors,
and $X^{-1}$ identified as the matrix of left eigenvectors. For nonzero
scalars $\\{c_{i}\\}$ we see that (6.15) is unchanged by the rescalings
$\mathbf{r}_{j}\mapsto c_{j}\mathbf{r}_{j}$ and
$\bm{\ell}_{j}\mapsto(1/c_{j})\bm{\ell}_{j}$.
For $N\times 1$ column vectors $\mathbf{u},\mathbf{v}$, define the inner
product $\langle\mathbf{u},\mathbf{v}\rangle:=\bar{\mathbf{u}}^{T}\mathbf{v}$.
The so called overlap matrix has its elements $\mathcal{O}_{ij}$ expressed in
terms of this inner product according to
(6.16)
$\mathcal{O}_{ij}:=\langle\bm{\ell}_{i},\bm{\ell}_{j}\rangle\langle\mathbf{r}_{i},\mathbf{r}_{j}\rangle.$
Note that this is invariant under the mappings noted in the final sentence of
the above paragraph, and for fixed $i$ and summing over $j$ gives $1$. Also,
it follows from (6.16) that the diagonal entries relate to the lengths
(6.17) $\mathcal{O}_{jj}=||\bm{\ell}_{j}||^{2}||\mathbf{r}_{j}||^{2}.$
The square root of this quantity is known as the eigenvalue condition number;
see the introduction to [71] and [112, §1.1] for further context and
references. Significant too is the fact that the overlaps (6.17) appear in the
specification of a Dyson Brownian motion extension of GinUE [188, 71].
Statistical properties of $\\{O_{ij}\\}$ for GinUE were first considered by
Chalker and Mehlig [95, 96]. By the Schur decomposition (2.2), instead of a
GinUE matrix $G$, we may consider an upper triangular matrix $Z$ with the
eigenvalues $\\{z_{j}\\}$ of $G$ on the diagonal, and off diagonal entries
standard complex Gaussians. For the eigenvalue $\lambda_{1}$, the triangular
structure shows that $\bm{\ell}_{1}=(1,b_{2},\dots,b_{N})^{T}$ and
$\mathbf{r}_{1}=(1,0,\dots,0)^{T}$ where for $p>1$ and $b_{1}=1$,
$b_{p}={1\over z_{1}-z_{p}}\sum_{q=1}^{p-1}b_{q}Z_{pq}$. From this last
relation, it follows that with $\bm{\ell}_{1}^{(n)}=(1,b_{2},\dots,b_{n})^{T}$
for $n<N$ we have
(6.18)
$||\bm{\ell}_{1}^{(n+1)}||^{2}=||\bm{\ell}_{1}^{(n)}||^{2}\Big{(}1+{1\over|z_{1}-z_{n+1}|^{2}}\Big{|}\sum_{q=1}^{n}\tilde{b}_{q}Z_{(n+1)q}\Big{|}^{2}\Big{)},\qquad\tilde{b}_{q}:={b_{q}\over\sqrt{\sum_{q=1}^{n}|b_{q}|^{2}}}.$
This has immediate consequence in relation to $\mathcal{O}_{11}$ as shown by
Bourgade and Dubach [71].
###### Proposition 6.8.
Let the eigenvalues $\\{z_{j}\\}$ be given. We have
(6.19) $\mathcal{O}_{11}\mathop{=}\limits^{\rm
d}\prod_{n=2}^{N}\Big{(}1+{|X_{n}|^{2}\over|z_{1}-z_{n}|^{2}}\Big{)},$
where each $X_{n}$ is an independent complex standard Gaussian. Furthermore,
it follows from this that after averaging over $\\{z_{2},\dots,z_{N}\\}$
(6.20) $\mathcal{O}_{11}\Big{|}_{z_{1}=0}\mathop{=}\limits^{\rm d}{1\over{\rm
B}[2,N-1]},$
where ${\rm B}[\alpha,\beta]$ refers to the beta distribution.
###### Proof.
A product formula for $\mathcal{O}_{11}$ follows from (6.17), the fact that
$||\mathbf{r}_{1}||=1$, and by iterating (6.18). This product formula is
identified with the RHS of (6.19) upon noting that a vector of independent
standard complex Gaussians dotted with any unit vector (here
$(\tilde{b}_{1},\dots,\tilde{b}_{n})$) has distribution equal to a standard
complex Gaussian.
In relation to (6.20) a minor modification of the proof of Proposition 2.20
shows that conditioned on $z_{1}=0$, the ordered squared moduli
$\\{|z_{j}|^{2}\\}_{j=2}^{N}$ are independently distributed as
$\\{\Gamma[j;1]\\}_{j=2}^{N}$. Noting too that with each $X_{j}$ a standard
complex Gaussian, $|X_{j}|^{2}\mathop{=}\limits^{\rm d}\Gamma[1;1]$ it follows
$\mathcal{O}_{11}\Big{|}_{z_{1}=0}\mathop{=}\limits^{\rm
d}\prod_{n=2}^{N}\Big{(}1+{\tilde{X}_{n}\over
Y_{n}}\Big{)},\qquad\tilde{X}_{n}\mathop{=}\limits^{\rm
d}\Gamma[1;1],\>Y_{n}\mathop{=}\limits^{\rm d}\Gamma[n;1].$
Next, we require knowledge of the standard fact that
$Y_{n}/(Y_{n}+\tilde{X}_{n})\mathop{=}\limits^{\rm d}{\rm B}[n,1]$.
Furthermore (see e.g. [149, Exercises 4.3 q.1]), for $x\mathop{=}\limits^{\rm
d}{\rm B}[\alpha+\beta,\gamma]$, $y\mathop{=}\limits^{\rm d}{\rm
B}[\alpha,\beta]$, we have that $xy\mathop{=}\limits^{\rm d}{\rm
B}[\alpha,\beta+\gamma]$, which tells us that with
$b_{n}\mathop{=}\limits^{\rm d}{\rm B}[n,1]$ we have
$\prod_{n=2}^{N}b_{n}\mathop{=}\limits^{\rm d}{\rm B}[2,N-1]$.
∎
Dividing both sides of (6.20) by $N$ we see that the $N\to\infty$ is well
defined since $N{\rm B}[2,N-1]\to\Gamma[2,1]$. After scaling the GinUE matrix
$G\mapsto G/\sqrt{N}$ so that the leading eigenvalue support is the unit disk,
an analogous limit formula has been extended from $z_{1}=0$ to any $z_{1}=w$,
$|w|<1$ in [71]. Thus
(6.21) ${\mathcal{O}_{11}\Big{|}_{z_{1}=w_{1}}\over
N(1-|w_{1}|^{2})}\mathop{\to}\limits^{\rm d}{1\over\Gamma[2,1]}.$
In words, with $\mathcal{O}_{11}$ corresponding to the condition number, one
has that the instability of the spectrum is of order $N$ and is more stable
towards the edge. Another point of interest is that the PDF for
$1/\Gamma[2,1]$ is $\chi_{t>0}e^{-1/t}/t^{3}$, which is heavy tailed, telling
us that only the zeroth and first integer moments are well defined. We remark
that limit theorems of the universal form (6.21) have been proved in the case
of the complex spherical ensemble of §2.5, and for a sub-block of a Haar
distributed unitary matrix [128]; see also [282].
The $1/t^{3}$ tail implied by (6.21) has been exhibited from another viewpoint
in the work of Fyodorov [169]. There the joint PDF for the overlap non-
orthogonality $\mathcal{O}_{jj}-1$, and the eigenvalue position $z_{j}$, was
computed for finite $N$. The global scaled limit of this quantity,
$\mathcal{P}^{\rm g}(t,w)$ say, was evaluated as [169, Eq. (2.24)]
(6.22) $\mathcal{P}^{\rm g}(t,w)={(1-|w|^{2})^{2}\over\pi
t^{3}}e^{-(1-|w|^{2})/t},\quad|w|<1.$
Note that for the first moment in $t$ this gives
(6.23) $\int_{0}^{\infty}t\mathcal{P}^{\rm
g}(t,w)\,dt={1\over\pi}(1-|w|^{2}),$
in keeping with a prediction from [95, 96]. This was first proved in [324].
An explicit formula for the large $N$ form of the average of the overlap
(6.16) in the off diagonal case (say $(i,j)=(1,2)$), with the GinUE matrix
scaled $G\mapsto G/\sqrt{N}$, and conditioned on $z_{1}=w_{1}$, $z_{2}=w_{2}$
with $|w_{1}|,|w_{2}|<1$ is also known [95, 96, 71, 22, 112]
(6.24)
$\left\langle\mathcal{O}_{12}\Big{|}_{z_{1}=w_{1},z_{2}=w_{2}}\right\rangle\mathop{\sim}\limits_{N\to\infty}-{1\over
N}{1-w_{1}\bar{w}_{2}\over|w_{1}-w_{2}|^{4}}\bigg{(}{1-(1+N|w_{1}-w_{2}|^{2})e^{-N|w_{1}-w_{2}|^{2}}\over
1-e^{-N|w_{1}-w_{2}|^{2}}}\bigg{)}.$
This formula is uniformly valid down to the scale $N|w_{1}-w_{2}|={\rm O}(1)$.
The large $N$ form of the average value of the diagonal overlap for products
of $M$ global scaled GinUE matrices has been considered in [74, 51], with the
result
(6.25) $\lim_{N\to\infty}{1\over
N}\left\langle\mathcal{O}_{11}\Big{|}_{z_{1}=w}\right\rangle={1\over\pi}|z|^{-2+2/M}(1-|z|^{2/M})\chi_{|z|<1};$
cf. (2.77). We remark that the average values of $\mathcal{O}_{11}$ and
$\mathcal{O}_{12}$, conditioned on multiple eigenvalues are shown to have a
determinantal form in [22, Th. 1], thus exhibiting an integrable structure for
these eigenvector statistics.
For general complex non-Hermitian matrices with independently distributed
entries of the form $\xi_{jk}+i{\zeta}_{jk}$, where each
$\xi_{jk},\tilde{\zeta}$ is an identically distributed zero mean real random
variable of unit variance (this class is sometimes referred to as complex non-
Hermitian Wigner matrices; see e.g. [18]), a line of research in relation to
the normalised eigenvectors is to quantify the similarity with a complex
vector drawn from the sphere embedded in $\mathbb{C}^{N}$ with uniform
distribution. Recent references on this include [264, 267]. For random vectors
on the sphere, there are bounds on the size of the components which rule out
gaps is the spread of the size of the components, referred to in [295] as no-
gaps localisation. As noted in [267, §1.1], the bi-unitary invariance of GinUE
matrices implies individual eigenvectors are distributed uniformly at random
from the complex sphere, and thus with probability close to one have that the
$j$-th largest modulus of the entries is bounded above and below by a positive
constant times $\sqrt{N-j}/N$, for $j$ in the range from $N/2$ up to $N$ minus
a constant time $\log N$.
### Acknowledgements
This research is part of the program of study supported by the Australian
Research Council Discovery Project grant DP210102887. SB was partially
supported by the National Research Foundation of Korea grant
NRF-2019R1A5A1028324, Samsung Science and Technology Foundation grant SSTF-
BA1401-51, and KIAS Individual via the Center for Mathematical Challenges at
Korea Institute for Advanced Study grant SP083201. The authors gratefully
acknowledge Gernot Akemann, Christophe Charlier, Yan Fyodorov, Grégory Schehr
and Aron Wennman for helpful feedback on the first draft of this work.
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# Universal ion-transport descriptors and classes of inorganic solid-state
electrolytes
Cibrán López Departament de Física, Universitat Politècnica de Catalunya,
08034 Barcelona, Spain Barcelona Research Center in Multiscale Science and
Egineering, Universitat Politècnica de Catalunya, 08019 Barcelona, Spain
Institut de Ciència de Materials de Barcelona, ICMAB-CSIC, Campus UAB, 08193
Bellaterra, Spain Agustí Emperador Departament de Física, Universitat
Politècnica de Catalunya, 08034 Barcelona, Spain Edgardo Saucedo Barcelona
Research Center in Multiscale Science and Egineering, Universitat Politècnica
de Catalunya, 08019 Barcelona, Spain Department of Electronic Engineering,
Universitat Politècnica de Catalunya, 08034 Barcelona, Spain Riccardo Rurali
Institut de Ciència de Materials de Barcelona, ICMAB-CSIC, Campus UAB, 08193
Bellaterra, Spain Claudio Cazorla Departament de Física, Universitat
Politècnica de Catalunya, 08034 Barcelona, Spain Barcelona Research Center in
Multiscale Science and Egineering, Universitat Politècnica de Catalunya, 08019
Barcelona, Spain
Solid-state electrolytes (SSE) with high ion conductivity are pivotal for the
development and large-scale adoption of green-energy conversion and storage
technologies such as fuel cells, electrocatalysts and solid-state batteries.
Yet, SSE are extremely complex materials for which general rational design
principles remain indeterminate. Here, we unite first-principles materials
modelling, computational power and modern data analysis techniques to advance
towards the solution of such a fundamental and technologically pressing
problem. Our data-driven survey reveals that the correlations between ion
diffusivity and other materials descriptors in general are monotonic, although
not necessarily linear, and largest when the latter are of vibrational nature
and explicitly incorporate anharmonic effects. Surprisingly, principal
component and k-means clustering analysis show that elastic and vibrational
descriptors, rather than the usual ones related to chemical composition and
ion mobility, are best suited for reducing the high complexity of SSE and
classifying them into universal classes. Our findings highlight the need of
considering databases that incorporate temperature effects to improve our
understanding of SSE and point towards a generalized approach to the design of
energy materials.
Social networks use modern data analysis techniques to improve their customers
experience and increase advertising revenues sumpter18 . Each mouse click and
fingers action on the touchscreen reveal information on the users preferences
that can be employed to classify individuals into similarity groups and thus
better select the contents they are exposed to. Materials, in analogy to
humans, conform to highly diverse and complex collectives and as such advanced
data analysis techniques are being increasingly applied on them to improve
their design and recommend possible uses kalinin15 ; tshitoyan19 . A necessary
condition for the meaningful development and application of data-driven
materials design strategies is the existence of comprehensive and reliable
databases.
Solid-state electrolytes (SSE) are a class of energy materials in which
specific groups of ions may start to diffuse throughout the crystalline matrix
driven by the thermal excitations hull04 . SSE are the pillars of green-energy
conversion and storage technologies like fuel cells, electrocatalysts and
solid-state batteries, hence tuning of their ion-transport properties turns
out to be critical for the fields of Energy and Sustainability. SSE, however,
are highly complex materials that present disparate compositions, structures,
thermal behaviors and ion mobilities, thus it is difficult to ascribe them to
general and rational design principles. These difficulties have motivated
researchers to seek for easy to measure (or calculate) quantities that may
serve as good descriptors of the ion conductivity; examples of such
descriptors include structural parameters, defect formation energies, atomic
polarizabilities and lattice dynamics guin15 ; bachman16 ; muy18 ; katcho19 ;
muy20 . In recent years, pinpointing the role of phonon dynamics on ion
transport has attracted special and increasing attention. Actually, for some
specific SSE it has been demonstrated that lattice anharmonicity is one of the
most influential factors affecting their ion mobility muy20 ; niedziela19 ;
gupta22 ; cazorla19 ; gupta21 ; ding20 .
Quantum mechanics-based density functional theory (DFT) cazorla17 has proven
tremendously successful in the field of computational materials science, and
currently several databases of automated DFT calculations are being widely
employed for materials design applications aflowlib ; mp ; oqmd ; aiida .
Nevertheless, despite of their great successes, the existing DFT databases
might not be entirely adequate for progressing in the design and understanding
of SSE because they mostly contain information generated at zero temperature
(e.g., structural parameters and formation energies) and thus completely
disregard anharmonicity and $T$-induced effects kahle20 . In addition, modern
high-throughput and machine learning studies relying on such DFT databases
mainly have targeted Li and Na-based SSE families due to their predominance in
electrochemical storage applications katcho19 ; zhang19 ; he19 . To
holistically better understand the phenomena of ion transport, however, it
might be necessary to analyse in equal measure other classes of SSE, like
those involving mobile O, Cu, Ag and halide ions, which are technologically
relevant as well hu21 ; aznar17 ; islam21 .
Figure 1: Spearman correlograms and corresponding $p$-value matrices.
Correlations between pairs of materials features obtained for (a) all and (c)
exclusively the Li-based SSE contained in our DFT-AIMD database. The $p$-value
matrices corresponding to all and exclusively Li-based Spearman correlograms
are shown in (b) and (d), respectively. All the AIMD-based diffusive and
vibrational descriptors were estimated at $T=500\pm 100$ K.
Here, we present a data-driven analysis of SSE that covers aspects generally
unaddressed by previous computational studies and the existing DFT materials
databases. First, a comprehensive first-principles database was created for
prototypical families of inorganic SSE containing both sets of zero-
temperature DFT and finite-temperature ab initio molecular dynamics (AIMD)
results. Subsequently, a thorough correlation study between the ion diffusion
coefficient ($D$) and other materials features was performed to determine
universal ion-transport descriptors (as well as those specific to Li-based
SSE). By relying on this new knowledge and the introduced DFT-AIMD database,
several machine learning models were trained for the prediction of $D$ and
other $T$-dependent quantities. Finally, principal component and k-means
clustering analysis, data techniques customarily employed in the social
sciences, were applied to reduce the high complexity of the SSE landscape and
determine universal classes of fast-ion conductors.
Curated first-principles SSE database. The generated SSE DFT-AIMD database
database comprises a total of $61$ materials of which $46$% contain Li, $23$%
halide (i.e., F, Cl, Br and I), $15$% Na, $8$% O and $8$% Ag/Cu atoms as the
mobile ions. These percentages were selected in order to roughly reproduce the
relative abundances of fast-ion conductors reported in the literature
webofscience . The generated SSE DFT-AIMD database contains materials with
both stoichiometric and non-stoichiometric compositions and the AIMD results
were obtained over a broad range of temperatures (Supplementary Tables 1–3 and
database ).
To analyze the degree of similarity between all the surveyed SSE, a great
variety of descriptors were estimated for each material adding up to a total
of $54$ (the complete list of descriptors is detailed in the Methods section).
Some of these descriptors had been already proposed in the literature (e.g.,
band gap and vacancy formation energy) while some others were totally new
(e.g., harmonic phonon energy and Pugh’s modulus ratio). The descriptors were
classified into three general categories: “mechanical-elastic”, “diffusive-
vibrational” and “structural-compositional”. The value of some descriptors
were obtained from zero-temperature DFT calculations (“mechanical-elastic” and
“structural-compositional”) while the rest (“diffusive-vibrational”) were
deduced from AIMD simulations performed at temperatures above ambient (Methods
and Supplementary Tables 1–3).
It is worth noting that the results obtained from AIMD simulations explicitly
account for anharmonic effects, which constitutes one of the most important
novelties and technical advances of the present work and introduced SSE
database. Moreover, most vibrational descriptors were estimated considering
the following cases (1) all the ions, (2) only non-diffusive ions and (3) only
diffusive ions, in order to better substantiate the role of the vibrating
crystalline matrix on ion transport (Methods). The approximate computational
cost of the generated SSE DFT-AIMD database was of $50$ Million CPU hours.
Figure 2: Correlation study of the ion diffusion coefficient with other
materials descriptors. (a) Standardized representation of the ion diffusion
coefficient $D$ along with other materials descriptors. The descriptors
correlations are, to some extent, monotonic but not linear as it is shown by
the orange and blue lines therein (both simple guides to the eyes). Spearman
correlation coefficients for $D$ and the rest of materials descriptors
considered in this study, obtained by taking into account (b) all and (c)
exclusively the Li-based compounds included in our DFT-AIMD database. The
$p$-value results corresponding to the Spearman correlation coefficients are
indicated with different colours. All the AIMD-based diffusive and vibrational
descriptors were estimated at $T=500\pm 100$ K.
Correlations between pairs of SSE descriptors. The correlation for a couple of
materials descriptors, $x$ and $y$, can be quantified in several non-unique
ways correlations . In this work, we considered the Pearson ($c_{P}$) and
Spearman ($c_{S}$) correlation coefficients which are defined like:
$\displaystyle c_{P}(x,y)$ $\displaystyle=$ $\displaystyle\frac{{\rm
cov}(x,y)}{\sigma_{x}\sigma_{y}}~{}{\rm and}$ $\displaystyle c_{S}(x,y)$
$\displaystyle=$ $\displaystyle c_{P}\left[R(x),R(y)\right]~{},$ (1)
where $\sigma_{i}$ is the standard deviation of the descriptor $i$ and $R(i)$
the rank of the $i$ samples. The covariance function is expressed as:
${\rm cov}(x,y)=\langle xy\rangle-\langle x\rangle\langle y\rangle~{},$ (2)
where $\langle\cdot\rangle$ denotes expected value. The Spearman correlation
coefficient is able to detect monotonic dependencies between pairs of
descriptors while the Pearson can only identify linear correlations. Thus, the
$c_{S}$ correlation coefficients are more general and robust than $c_{P}$
(i.e., can assess monotonic relationships whether linear or not). For this
important reason, and despite the fact that linear correlations have been
assumed in most previous SSE studies muy18 ; muy20 , we will stick to the
Spearman correlation definition for the rest of our analysis.
Figure 1a shows the Spearman correlation coefficients estimated for all pairs
of materials descriptors considering the entire DFT-AIMD database (an
analogous Pearson correlogram can be found in the Supplementary Fig.1). In
view of the preeminence of Li-based SSE in electrochemical applications, the
same correlation analysis was performed for this family of materials alone
(Fig.1c). To assess the statistical significance of the estimated $c_{S}$
correlograms, we computed the corresponding $p$-value matrices (Figs.1b,d).
The $p$-value represents the probability for a particular correlation result
to arise if the null hypothesis (i.e., no correlation at all) were true, thus
the smaller the calculated $p$-value the more statistically significant
$c_{S}$ is.
In a bird’s eye view, the two correlograms obtained for all SSE and only those
containing Li ions look quite similar. Nevertheless, the $p$-value matrix
estimated for all SSE displays a noticeably higher number of statitiscally
significant cases (arbitrarily defined here as $p<0.2$), probably due to the
larger amount of samples. Reassuringly, a number of already expected high
correlation coefficients, like those estimated for couples of vibrational and
elastic quantities that are physically related (e.g., $F_{vib}$ and
$S_{vib}$), emerge from the calculated $c_{S}$ maps. For the sake of focus,
hereafter we will concentrate on the correlations involving the ion diffusion
coefficient ($D$).
Figure 2a encloses a standardized representation [that is,
$\hat{x}\equiv\left(x-\langle x\rangle\right)/\sigma_{x}$] of the pairs of
descriptors $D$–$C_{v}$ and $D$–$\langle\omega\rangle$, where $C_{v}$ stands
for the lattice heat capacity and $\langle\omega\rangle$ for the average
vibrational frequency (Methods). In these two cases, as well as in others not
shown here, it is clearly appreciated that the dependency beween $D$ and the
other quantities is far from linear although roughly monotonic. This outcome
confirms that for determining reliable relationships between SSE features the
Spearman correlation analysis is certainly more suitable than the usual
Pearson approach. Actually, there are significant discrepancies between the
Spearman and Pearson correlation maps; for instance, $c_{S}$ amounts to $-39$%
for the pair of descriptors $D$–$\langle\omega\rangle$ (Fig.1a) whereas
$c_{P}$ renders a significantly smaller value of $-23$% (Supplementary Fig.2).
Universal ion diffusion descriptors. Figure 2b shows the Spearman correlation
coefficients estimated for all pairs of descriptors involving $D$ and
considering the entire DFT-AIMD database. All the AIMD-based vibrational and
diffusive descriptors were estimated at $T=500\pm 100$ K. First, we note that
larger $|c_{S}|$ values are associated with statistically more significant
correlation results (i.e., smaller $p$-values). And second, the estimated
correlation coefficients in general are not very high: only $19$ out of the
$53$ pairs of materials descriptors present $|c_{S}|$’s larger than $20$%
while the maximum correlation value only amounts to $39$% (obviously, the
$D$–$D$ pair was excluded here). These low-correlation outcomes are consistent
with the usual difficulties encountered in the settlement of flawless ion
transport descriptors bachman16 .
Interestingly, the largest $D$ correlations are found for AIMD-based
vibrational descriptors (Methods) like the phonon band center (or average
lattice frequency), $\langle\omega\rangle$ ($-39$%), lattice heat capacity,
$C_{V}$ ($+39$%), vibrational free energy, $F_{vib}$ ($-37$%), and vibrational
entropy, $S_{vib}$ ($+33$%). These results indicate that insulator materials
with small average phonon frequencies, large heat capacities and large
vibrational entropies should be good ion conductors. It is worth noticing that
strongly anharmonic materials perfectly fit into this description, thus our
data-driven results generalize the conclusions of recent experimental SSE
studies revealing that low-energy phonon modes can actively influence ion
diffusion in some specific materials muy20 ; niedziela19 ; gupta22 ; cazorla19
; gupta21 ; ding20 .
Figure 3: Machine learning (ML) models trained in our DFT-AIMD database for
prediction of different SSE $T$-dependent quantities. The ML models were
trained by considering and neglecting AIMD-based vibrational descriptors that
explicitly incorporate anharmonic effects, labelled as “anharmonic” and
“harmonic”, respectively. (a) First momentum of the vibrational density of
states obtained from AIMD simulations, $\left<\omega\right>$. (b) Constant
volume heat capacity obtained from AIMD simulations, $C_{V}$. (c) Ionic
diffusion coefficient obtained from AIMD simulations, $D$. “MAPE” stands for
the mean absolute percentage error of the ML predictions.
Our correlation analysis provides further valuable insights. First, when the
vibrational descriptors were estimated considering either non-diffusive or
diffusive ions alone (superscripts “nd” and “d” in Fig.2b, respectively) the
value of the $D$ correlation coefficients slightly decreased in the first case
($|c_{S}|=30$%) and practically vanished in the second (except that
corresponding to $\langle\omega_{30}\rangle^{(d)}$). This outcome highlights
the existence of a strong and general interplay between the vibrating
crystalline matrix and mobile ions. And second, when considering vibrational
descriptors that do not explicitly take into account anharmonic effects, like
the lowest-energy optical phonon mode calculated at $T=0$ K ($\Gamma$ in
Fig.2b), the resulting $D$ correlation coefficient ($-11$%) significantly
drops in comparison to those obtained for anharmonic quantities (besides, the
corresponding $p$-value increases). Thus, scrutinity of anharmonicity appears
to be indispensable for the evaluation of reliable and statistically
meaningful $D$ correlation coefficients.
Few descriptors belonging to the “structural-compositional” category also
correlate appreciably high with $D$. Of special mention are the vacancy
formation energy of the mobile ions ($E_{vac}$, $-22$%), the crystal
polarizability ($\alpha_{C}$, $+25$% –calculated with the Clausius-Mossotti
relation–) and the symmetry of the perfect lattice ($SO$, $+27$%) wang15 . On
the other hand, intrinsically electronic properties like the energy band gap
($E_{g}$) and dielectric constant ($\epsilon$) have virtually no correlation
with the ion diffusivity ($|c_{S}|\leq 5$%). As a word of caution, we note
that when the correlations between $D$ and other materials descriptors are
assumed to be linear (i.e., Pearson’s approach) the resulting conclusions
significantly differ from those just explained (Supplementary Fig.2). In
particular, most $D$ correlation coefficients turn out to be smaller than the
corresponding Spearman values and the materials descriptors belonging to the
“mechanical-elastic” category (e.g., the Young and shear moduli –$E$ and $G$–)
become equally relevant than the vibrational features.
Figure 2c shows the Spearman $D$ correlation coefficients estimated
exclusively for Li-based SSE. Intriguingly, the resulting $c_{S}$ chart
differs appreciably from that estimated considering the entire DFT-AIMD
database (Fig.2b). First, the $D$ correlation coefficients in general present
larger values with a total of $11$ pairs of materials descriptors scoring
above $40$%. Some of the largest $|c_{S}|$’s correspond to the AIMD-based
vibrational descriptors $F_{vib}$ ($-42$%), $S_{vib}$ ($+42$%) and
$\langle\omega_{30}\rangle^{(d)}$ ($-63$%). However, in contrast to the all-
SSE case, now $\Gamma$, which is estimated at $T=0$ K and does not explicitly
account for anharmonicity, is strongly correlated with $D$ as well ($-47$%).
Moreover, several descriptors belonging to the “mechanical-elastic” category
that, to the best of our knowledge, have not been previously proposed in the
literature like the Vickers’ hardness, $H_{V}$ ($-43$%), Pugh’s modulus ratio,
$\kappa$ ($-56$%), Poisson’s ratio, $\nu$ ($+55$%), Cauchy’s pressure, $P_{C}$
($+48$%), and velocity ratio, $v_{r}$ ($+56$%), now also render very high
$|c_{S}|$ values. Therefore, in terms of key $D$ descriptors, Li-based
compounds are plainly different from the average SSE, a finding that
fundamentally justifies the large number of studies focusing on the ion
transport properties of this family of materials.
Figure 4: Principal component analysis results obtained for the SSE DFT-AIMD
database. (a) Eigenvalues corresponding to the diagonalization of the Spearman
correlation matrix obtained by considering the entire DFT-AIMD database. (b)
Eigenvector components of the first three principal components obtained from
the diagonalization of the Spearman correlation matrix obtained by considering
the entire DFT-AIMD database.
Machine learning models for prediction of $T$-dependent properties. In view of
the complex relationships between $D$ and other materials descriptors
(Fig.2a), several machine learning (ML) models based on artificial neural
networks were trained in our SSE DFT-AIMD database with the aim of predicting
the ion diffusion coefficient and other relevant $T$-dependent properties of
SSE such as $\langle\omega\rangle$ and $C_{V}$ (Methods). We considered two
different ML training schemes: (1) considering all the materials descriptors
(denoted as “anharmonic”) and (2) excluding the AIMD-based vibrational
descriptors (“harmonic”). The predictions of our trained ML models for a
validation set of $12$ compounds are shown in Fig.3. Therein, it is
appreciated that the two trained ML models can predict the finite-temperature
values of $\langle\omega\rangle$ and $C_{V}$ with high accuracy. In
particular, the mean absolute percentage error (MAPE) of the “anharmonic”
(“harmonic”) ML model amounts to $2.5$% ($7.5$%) and only $0.5$% ($1.9$%) for
$\langle\omega\rangle$ and $C_{V}$, respectively. In stark contrast, the ML
predictions for the ion diffusion coefficient are much less accurate and there
is a huge difference in the level of precision achieved with the “anharmonic”
(MAPE of $69$%) and “harmonic” ($290$%) ML models.
Several conclusions follow from the ML results enclosed in Fig.3. First, the
SSE DFT-AIMD database introduced in this work appears to be comprehensive
enough to ensure proper training of ML models able to make accurate
predictions of certain $T$-dependent materials properties. And second, ML-
based prediction of the ion diffusivity appears to be a particularly
challenging task. In this latter case, however, a big improvement is achieved
when AIMD-based anharmonic vibrational descriptors are explicitly incorporated
into the ML model (also in the $\langle\omega\rangle$ and $C_{V}$ cases). This
outcome indirectly corroborates our previous finding that anharmonicity is a
key general factor influencing ion transport. Nonetheless, to improve the
“anharmonic” ML prediction of $D$ probably it is necessary to increase the
number of SSE materials and descriptors in our DFT-AIMD database and/or resort
to alternative and more advanced ML approaches (e.g., graph neural networks
fung21 ).
Figure 5: K-means clustering analysis results obtained for the SSE DFT-AIMD
database. (a) Classification of the analyzed materials in the orthogonal
bidimensional space PC1–PC2. (b) Materials population of each group identified
in the PC1–PC2 space expressed in terms of the mobile ion species. (c)
Classification of the analyzed materials in the orthogonal tridimensional
space PC1–PC2–PC3. (d) Materials population of each group identified in the
PC1–PC2–PC3 space expressed in terms of the mobile ion species. To improve
visual clarity, some points have been removed from the plots without affecting
the main conclusions.
Complexity reduction in the SSE landscape. Principal component analysis (PCA)
is a statistical technique widely employed for analyzing large datasets
containing a high number of features. PCA increases the interpretability of a
dataset by reducing its dimensionality and simultaneously preserving the
maximum amount of information. Complexity reduction is accomplished by
linearly transforming the data into a new coordinate system where most of its
variation can be described with fewer dimensions. The principal components are
the eigenvectors of the dataset correlation matrix, which are expressed as
linear combinations of the initial descriptors. The first principal component,
the one with the largest eigenvalue, maximizes the variance of the projected
data. The $i$-th principal component corresponds to a direction that is
orthogonal to the previous $i-1$ principal components and along which the
variance of the projected data is maximized as well.
Figure 4 shows the results of diagonalizing the Spearman correlation matrix
obtained for the entire SSE DFT-AIMD database. The first three principal
components (PC) account for about two thirds of the total variance in the
original $54$-dimensional dataset (as quantified by the sum of their
normalized eigenvalues, $\approx 62$%) hence its complexity can be greatly
reduced by considering data projections on the orthogonal three-dimensional
space PC1–PC2–PC3. PC1 presents a dominant “mechanical-elastic” character, PC2
“vibrational” and PC3 “structural” (Fig.4b). Intriguingly, the contribution of
the ion diffusivity to each of these PC’s is practically zero, namely, $0.2$%
to PC1, $0.8$% to PC2 and $1.3$% to PC3. This data-driven outcome indicates
that when it comes to characterize the great disparity of SSE, with the aim of
fundamentally better understanding them and to establish general SSE
categories, the ubiquitous $D$ descriptor is actually irrelevant. Likewise,
the compound stoichiometry ($Stc$) and dielectric constant ($\epsilon$) hardly
contribute to the first three PC’s hence they neither can be regarded as
universally distinctive SSE features. By contrast, elastic and vibrational
descriptors like $E$, $H_{V}$, $\langle\omega\rangle$ and $C_{V}$ become most
pertinent for the evaluation of SSE similarities and general classification
purposes.
K-means clustering analysis. Figure 5 encloses the results of our k-means
clustering analysis performed for the entire SSE DFT-AIMD database. K-means
clustering is an unsupervised learning algorithm that classify sets of objects
in such a way that objects within the same group, called “cluster”, are more
similar to each other in a broad sense than to the objects in other clusters.
We selected a subminimal number of $7$ clusters to account for the SSE
database variance based on the outcomes of the elbow and silhouette methods
(Supplementary Figs.3–4). This number of clusters is already larger than the
number of $A$-based SSE families considered in this study (i.e., $6$ with $A=$
Li, Na, halide, Ag, Cu and O). Thus, it straightforwardly follows that the
materials composition, despite of its obvious utility in naming compounds,
should not be regarded as a fine descriptor of SSE diversity since, at least,
one SSE family will spread over more than one k-means cluster.
Figures 5a–b show the results of our k-means clustering analysis performed in
the simplified PC1–PC2 space. It is noted that Li-based SSE are present in $5$
out of the total $7$ clusters. From those $5$ clusters, Li-based SSE are the
most abundant in $80$% of the cases and overall they share similarities with
other Na, halide and O-based SSE (although not necessarily in terms of ion
conductivity). In clusters number $1$ and $2$, which are respectively
characterized by dominant PC1 (“elastic”) and PC2 (“vibrational”) components,
Li-based SSE actually conform the entire population. From these outcomes, we
may readily conclude that (1) Li-based SSE are intrinsically different from
Ag- and Cu-based SSE, which in turn are highly similar because inhabit the
same cluster, and (2) Li-based SSE can be partitioned into several similarity
subgroups attending to their elastic and vibrational properties. Likewise,
halide-based SSE appear in $4$ different clusters, Na-based in $3$ and O-based
in $2$. Thus, as it was mentioned above, chemical composition is not a good
descriptor for grouping SSE into similarity categories.
Figures 5c–d enclose the k-means clustering results obtained in the expanded
PC1–PC2–PC3 space. In this case, the main findings are very similar to those
just explained for the reduced P1–P2 space, namely, Li-based SSE are present
in $5$ out of the total $7$ clusters and they are particularly numerous in the
majority of those groups (e.g., $100$% in cluster number $7$ and $67$% in
cluster number $2$). Likewise, halide-based SSE spread over $4$ different
clusters, Na-based over $4$, O-based over $2$ and Cu/Ag-based only appear in
$1$. The Li-based SSE family overall shares similarities with other Na, halide
and O-based SSE (not so with Cu- and Ag-based SSE), and most subgroup
differences (i.e., relative distances between clusters centroids) are
contained in the P1–P2 plane. Thus, the PC3 (“structural”) dimension does not
appear to add sensible information on SSE diversity and for grouping purposes
is practically expendable (in accordance with its relatively small eigenvalue
of $\approx 4$%, Fig.4a).
The presented k-means clustering analysis enlightens the difficulties
encountered in the rational design of SSE with specific ion mobility. The bulk
of the variation in the SSE family is encoded in the materials elastic and
vibrational properties, neither in the ion mobility nor their chemical
composition. This finding implies that materials which can be rigorously
considered as overall highly similar (because they belong to a same k-means
cluster) in practice may exhibit very different ion diffusion and chemical
features (e.g., Li-based and halide-based SSE). Conversely, materials which
render very similar ion mobilities and chemical compositions (e.g., Li-based
SSE inhabiting groups $7$ and $3$ in Fig.5d) may behave radically different in
terms of other measurable quantities. These conclusions are consistent with
the $D$ correlation results enclosed in Fig.2, which show that Li-based SSE
can significantly depart from the general trends averaged over all SSE.
In summary, we have presented an original and comprehensive SSE data-driven
study on the correlations of the ion diffusion with other materials
descriptors as well as a rigorous examination of universal SSE categories,
based on a new and thorough DFT-AIMD database comprising both zero-temperature
and finite-$T$ first-principles results. It has been demonstrated that ion
diffusion correlates strongly and monotonically, not necessarily linearly,
with vibrational descriptors that explicitly incorporate anharmonic effects
(i.e., are estimated from AIMD simulations). In the particular case of Li-
based SSE, the ion mobility also correlates significantly with elastic
quantities like the Vickers’ hardness, Pugh’s modulus ratio, Poisson’s ratio
and Cauchy’s pressure, pertinent ion-diffusion descriptors that previously
have been overlooked in the literature. Furthermore, most of the variation in
the generated SSE $54$-fold dimensional space can be resolved in terms of
elastic and vibrational descriptors; ion mobility and chemical composition are
very much irrelevant when it comes to quantify the SSE diversity, a fact that
complicates the rational design of SSE with targeted ion conductivities. The
present data-driven study highlights the necessity to consider finite-
temperature effects in a high-throughput fashion to better understand SSE and
improve the predictions of machine learning models in them; it also provides
new theoretical guidelines for analyzing materials that in analogy to SSE are
highly anharmonic and technologically relevant (e.g., thermoelectrics and
superconductors).
## Methods
First-principles calculations outline. Ab initio calculations based on density
functional theory (DFT) were performed to analyse the physico-chemical
properties of bulk SSE. We performed these calculations with the VASP code
vasp by following the generalized gradient approximation to the exchange-
correlation energy due to Perdew _et al._ pbe96 . (For some halide compounds,
possible dispersion interactions were captured with the D3 correction scheme
developed by Grimme and co-workers grimmed3 .) The projector augmented-wave
method was used to represent the ionic cores bloch94 and for each element the
maximum possible number of valence electronic states was considered. Wave
functions were represented in a plane-wave basis typically truncated at $750$
eV. By using these parameters and dense ${\bf k}$-point grids for Brillouin
zone integration, the resulting zero-temperature energies were converged to
within $1$ meV per formula unit. In the geometry relaxations, a tolerance of
$0.005$ eV$\cdot$Å-1 was imposed in the atomic forces.
First-principles molecular dynamics simulations. _Ab initio_ molecular
dynamics (AIMD) simulations based on DFT were performed in the canonical
$(N,V,T)$ ensemble (i.e., constant number of particles, volume, and
temperature) for all the considered bulk materials. The selected volumes were
those determined at zero temperature hence thermal expansion effects were
neglected; nevertheless, based on previously reported molecular dynamics tests
cazorla19 , thermal expansion effects are not expected to affect significantly
the estimation of the ion-transport properties of SSE at moderate temperatures
(i.e., $T=500\pm 100$ K). The concentration of ion vacancies in the non-
stoichiometric compounds was also considered independent of the temperature
and equal to $\sim 1$–$2$%. The temperature in the AIMD simulations was kept
fluctuating around a set-point value by using Nose-Hoover thermostats. Large
simulation boxes containing $N_{ion}\sim 200$–$300$ atoms were employed in all
the cases and periodic boundary conditions were applied along the three
Cartesian directions. Newton’s equations of motion were integrated by using
the customary Verlet’s algorithm and a time-step length of $\delta t=1.5\cdot
10^{-3}$ ps. $\Gamma$-point sampling for integration within the first
Brillouin zone was employed in all the AIMD simulations. The finite-
temperature simulations typically comprised long simulation times of
$t_{total}\sim 100$–$200$ ps. For each material, we ran an average of $3$ AIMD
simulations at different temperatures and considering both stoichiometric and
non-stoichiometric compositions (Supplementary Tables 1–3 and database ).
Previous tests performed on the numerical bias stemming from the finite size
of the simulation cell and duration of the molecular dynamics runs reported in
work cazorla19 indicate that the adopted $N_{ion}$ and $t_{total}$ values
should provide reasonably well converged results for the ion diffusivity and
vibrational density of states of SSE.
Estimation of key diffusive and vibrational properties. The mean-squared
displacement (MSD) was estimated like:
$\displaystyle{\rm MSD}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{ion}\left(N_{step}-n_{\tau}\right)}\times$
$\displaystyle\sum_{i=1}^{N_{ion}}\sum_{j=1}^{N_{step}-n_{\tau}}|{\bf
r}_{i}(t_{j}+\tau)-{\bf r}_{i}(t_{j})|^{2}~{},$
where ${\bf r}_{i}(t_{j})$ is the position of the migrating ion $i$ at time
$t_{j}$ ($=j\cdot\delta t$), $\tau$ represents a lag time,
$n_{\tau}=\tau/\delta t$, $N_{ion}$ is the total number of mobile ions, and
$N_{step}$ the total number of time steps. The maximum $n_{\tau}$ was chosen
equal to $N_{step}/2$ in order to accumulate enough statistics to reduce
significantly the fluctuations in ${\rm MSD}(\tau)$ at large $\tau$’s. The
diffusion coefficient then was obtained by using the Einstein relation:
$D=\lim_{\tau\to\infty}\frac{{\rm MSD}(\tau)}{6\tau}~{}.$ (4)
In practice, we performed linear fits over the averaged ${\rm MSD}(\tau)$
values calculated within the lag time interval
$\tau_{max}/2\leq\tau\leq\tau_{max}$.
To estimate the vibrational density of states (VDOS) of bulk SSE considering
anharmonic effects, $g(\omega)$, we calculated the Fourier transform of the
corresponding velocity-velocity autocorrelation function as obtained directly
from the AIMD simulations, namely:
$g(\omega)=\frac{1}{N_{ion}}\sum_{i}^{N_{ion}}\int_{0}^{\infty}\langle{\bf
v}_{i}(\tau)\cdot{\bf v}_{i}(0)\rangle e^{i\omega\tau}d\tau~{},$ (5)
where ${\bf v}_{i}(t)$ represents the velocity of the $i^{\rm th}$ atom at
time $t$, and $\langle\cdots\rangle$ denotes statistical average in the
$(N,V,T)$ ensemble. Once the density of vibrational states was determined, it
was straightforward to calculate the corresponding phonon band center (or
average lattice frequency), $\langle\omega\rangle$, defined like:
$\langle\omega\rangle=\frac{\int_{0}^{\infty}\omega~{}g(\omega)~{}d\omega}{\int_{0}^{\infty}g(\omega)~{}d\omega}~{},$
(6)
which also depends on $T$. Likewise, the contribution of a particular group of
ions to the full VDOS was estimated by considering those ions alone in the
summation appearing in Eq.(5). In order to determine a characteristic low-
energy phonon frequency for bulk SSE, we defined the quantity:
$\langle\omega_{30}\rangle=\frac{\int_{0}^{\omega_{max}}\omega~{}g(\omega)~{}d\omega}{\int_{0}^{\omega_{max}}g(\omega)~{}d\omega}~{},$
(7)
for which we imposed an arbitrary cut-off frequency of $\omega_{max}=30$ meV.
The analytical expression for other vibrational descriptors (e.g., $F_{vib}$,
$E_{vib}$ and $C_{V}$) can be found in work phonopy .
Machine learning models. The Scikit-learn package in Python scikit was used
to encode the non-numeric descriptors as well as to implement the Artificial
Neural Network (ANN) conforming our machine learning model. For the generation
of the input data, the simulations involving all compounds, compositions and
temperatures in our SSE DFT-AIMD database were taken into consideration (i.e.,
a total of $174$ samples, Supplementary Tables 1–3 and database ). The non-
numeric descriptors (i.e., the diffusive chemical element, stoichiometricity,
chemical composition of the compound and symmetry of the relaxed structure)
were encoded with the one-hot encoding approach, and all input data was
normalized using a standard scaler. Specifically, a Multi-Layer Perceptron
Regressor (MLPR) was implemented, consisting on input, hidden and output
layers. As output layer, the algorithm was defined in such a way that any of
the considered descriptors could be used as dependent variable. Consequently,
the input layer was constructed as the set of all the other descriptors.
Optionally, anharmonic descriptors could be removed from the input layer if
desired. Finally, $6$ hidden layers of $150$, $500$, $50$, $150$, $70$ and
$100$ neurons, respectively, showed the best performance.
Attending to the extraction of metrics, K-fold validation was implemented: on
each interation, the model was required to predict the output for one element
using the rest as training set. Therefore, given that each element consists of
a different number of simulations (the original dataset presents a variable
number of simulated temperatures and stoichiometricities for each element),
the computed metrics were weighted with the number of predicted outputs and
then divided by the total amount of simulations. The optimization of the model
was monitored by using the mean absolute percentage error (MAPE) defined like:
$MAPE=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{x^{0}_{i}-x_{i}}{x^{0}_{i}}\right|~{},$
(8)
where $N$ is the total number of samples in the set, $\\{x\\}$ the predicted
outputs and $\\{x^{0}\\}$ the actual values in the DFT-AIMD database. Note
that these metrics can be extracted from both the training and test sets. As
optimal hyperparameters, Adam optimizer with the square error as loss function
and constant learning rate of $0.001$, rectified linear unit (ReLU) activation
function, and $\alpha=0.05$ strength for the $L^{2}$ regularization term of
the loss function were used.
| | |
---|---|---
Symbol | Descriptor (M-E) | Estimation approach
| |
| |
$\lambda$ | $1^{st}$ Lamé parameter | DFT
$B$ | Bulk modulus | DFT
$E$ | Young modulus | DFT
$G$ | Shear modulus | DFT
$\nu$ | Poisson’s ratio | DFT
$\sigma$ | P-wave modulus | DFT
$H_{V}$ | Vickers’ hardness | DFT
$\kappa$ | Pugh’s modulus ratio | DFT
$P_{C}$ | Cauchy’s pressure | DFT
$v_{l}$ | Longitudinal wave velocity | DFT
$v_{t}$ | Transverse wave velocity | DFT
$v_{r}$ | Velocity ratio | DFT
$\langle v\rangle$ | Average wave velocity | DFT
| |
| |
Symbol | Descriptor (D-V) | Estimation approach
| |
| |
$\gamma$ | Lindemann ratio | AIMD
$\Gamma$ | Lowest-energy optical phonon mode | DFT
$\langle\omega\rangle$ | Mean frequency | AIMD
$\langle\omega_{30}\rangle$ | Mean frequency (cut-off at $30$ meV) | AIMD
$E_{vib}$ | Vibrational phonon energy | AIMD
$C_{v}$ | Constant volume heat capacity | AIMD
$\theta_{D}$ | Debye temperature | AIMD
$F_{vib}$ | Vibrational Helmholtz free energy | AIMD
$S_{vib}$ | Vibrational entropy | AIMD
$D$ | Diffusion coefficient | AIMD
$msd$ | Mean-squared displacement | AIMD
| |
| |
Symbol | Descriptor (S-C) | Estimation approach
| |
| |
$Z_{N}$ | Nominal charge | Formula
$Z_{B}$ | Born effective charge | DFT
$\epsilon$ | Ion-clamped macroscopic dielectric constant | DFT
$M$ | Mobile ion atomic mass | Formula
$\alpha_{I}$ | Mobile ion polarizability | DFT
$\alpha_{C}$ | Crystal polarizability | DFT
$Stc$ | Stoichiometry | Formula
$Sym$ | Crystal symmetry | DFT
$a_{m}$ | Minimal lattice constant | DFT
$n$ | Number of formula units | DFT
$\Omega$ | Volume per formula unit | DFT
$\langle abc\rangle$ | Standard deviation of lattice constants | DFT
$\langle\alpha\beta\gamma\rangle$ | Standard deviation of lattice angles | DFT
$SO$ | Number of crystal symmetry operations | DFT
$N_{nn}$ | Number of nearest neighbors | DFT
$d_{nn}$ | Nearest neighbors distance | DFT
$E_{g}$ | Band gap | DFT
$E_{vac}$ | Vacancy energy of the mobile ion | DFT
| |
Table 1: Analyzed SSE descriptors and their abbreviations. The materials
descriptors were classifed into the categories (1) “mechanical and elastic”
(M-E), (2) “diffusive and vibrational” (D-V) and (3) “structural and
compositional” (S-C). The method of calculation of each descriptor, either
zero-temperature (DFT) or finite-temperature (AIMD) simulations, is indicated
in the third column. Some descriptors were directly deduced from the compounds
formula, indicated as “Formula” in the table.
SSE descriptors abbreviations. To analyze the similarities and dissimilarities
between fast-ion conductors a great variety of different physical descriptors
were estimated for each SSE, which are summarized in Table I. The descriptors
are generally classified according to the quality they refer to, in
particular: “mechanical-elastic” (M-E), “diffusive-vibrational” (D-V) and
“structural-compositional” (S-C). It may be noted that most D-V descriptors
like the mean phonon frequency (both with and without cut-off), harmonic
phonon energy, constant-volume heat capacity, Helmholtz free energy and
entropy, were calculated for the materials as a whole (i.e., considering both
diffusive and non-diffusive ions) and also exclusively considering either the
non-diffusive (denoted as “nd” in the figures) or diffusive atoms (denoted as
“d” in the figures). The total number of descriptors considered in this work
is equal to $54$. The descriptors estimated from AIMD (DFT) simulations were
obtained at $T=500\pm 100$ K ($T=0$ K).
## Data availability
The data that support the findings of this study are available upon reasonable
request from the authors C.L. and C.C. and the URL:
https://superionic.upc.edu/
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## Acknowledgements
We acknowledge financial support from the MCIN/AEI/10.13039/501100011033 under
Grant No. PID2020-119777GB-I00, the “Ramón y Cajal” fellowship
RYC2018-024947-I, the Severo Ochoa Centres of Excellence Program
(CEX2019-000917-S), the Generalitat de Catalunya under Grant No.2017SGR1506,
and the CSIC under the “JAE Intro SOMdM 2021” grant program.
## Author contributions
C.C. conceived the study and planned the research, which was discussed in
depth with the rest of co-authors. C.C. and R.R. performed and analyzed the
first-principles calculations. C.L. carried out the data analysis of the
generated DFT-AIMD database as well as the training of the SSE machine
learning models. A.E. created the website that gives access to the DFT-AIMD
database. The manuscript was written by C.C. with substantial input from the
rest of co-authors.
## Additional information
Supplementary information is available in the online version of the paper.
## Competing financial interests
The authors declare no competing financial interests.
|
# Probing first-order electroweak phase transition via primordial black holes
in the effective field theory
Katsuya Hashino Department of Physics, Faculty of Science and Technology,
Tokyo University of Science, Noda, Chiba 278-8510, Japan Shinya Kanemura
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Tomo
Takahashi Department of Physics, Saga University, Saga 840-8502, Japan
Masanori Tanaka Department of Physics, Osaka University, Toyonaka, Osaka
560-0043, Japan
###### Abstract
We investigate production of primordial black holes from first-order
electroweak phase transition in the framework of the nearly aligned Higgs
effective field theory, in which non-decoupling quantum effects are properly
described. Since the mass of such primordial black holes is evaluated to be
about $10^{-5}$ of the solar mass, current and future microlensing
observations such as Subaru HSC, OGLE, PRIME and Roman Space Telescope may be
able to probe the electroweak phase transition. We study parameter regions
where primordial black holes can be produced by the first-order electroweak
phase transition, and explore their detectability at these observations.
Complementarity of primordial black hole observations, gravitational wave
observations and collider experiments is also discussed for testing the nature
of the electroweak phase transition.
††preprint: OU-HET-1159
## I Introduction
Although a Higgs boson was discovered by the Large Hadron Collider (LHC) in
2012 ATLAS:2012yve ; CMS:2012qbp , dynamics of the electroweak symmetry
breaking remains unknown and aspects of the electroweak phase transition
(EWPT) in the early Universe is still a mystery. In particular, the nature of
the EWPT is essentially important to baryon asymmetry of the Universe which is
one of the greatest problems in cosmology and particle physics. A scenario of
electroweak baryogenesis Kuzmin:1985mm requires strongly first-order EWPT to
satisfy the condition of departure from thermal equilibrium Sakharov:1967dj .
Since the strongly first-order EWPT cannot be realized in the standard model
(SM) Dine:1992vs ; Kajantie:1995kf , extensions of the SM are necessary for a
successful scenario of electroweak baryogenesis.
As the extension from the SM for the strongly first-order EWPT, effective
field theories have often been considered like the standard model effective
field theory (SMEFT) Grojean:2004xa ; Delaunay:2007wb ; Cai:2017tmh ;
Croon:2020cgk ; Hashino:2021qoq , the Higgs effective field theory (HEFT)
Banta:2022rwg and its variations such as the nearly aligned Higgs effective
field theory (naHEFT) Kanemura:2022txx etc. In addition to the effective
field theory approaches, studies in renormalizable models with an extended
scalar sector have also been performed intensively Turok:1991uc ;
Espinosa:1993bs ; Funakubo:1993jg ; Cottingham:1995cj ; Cline:1996mga ;
Moreno:1996zm ; Kanemura:2004ch ; Funakubo:2005pu ; Fromme:2006cm ;
Profumo:2007wc ; Ahriche:2007jp ; Noble:2007kk ; Aoki:2008av ; Funakubo:2009eg
; Kanemura:2011fy ; Espinosa:2011ax ; Gil:2012ya ; Fuyuto:2014yia ;
Tamarit:2014dua ; Kanemura:2014cka ; Profumo:2014opa ; Blinov:2015vma ;
Fuyuto:2015vna ; Karam:2015jta ; Basler:2016obg ; Dorsch:2017nza ;
Ghorbani:2017jls ; Bernon:2017jgv ; Chiang:2017nmu ; Ghorbani:2019itr ;
Barman:2019oda . In both approaches, it has turned out that there is a strong
correlation between physics to satisfy the condition of strongly first-order
EWPT and to make a large deviation from the SM prediction in the triple Higgs
boson coupling ($hhh$ coupling) Grojean:2004xa ; Kanemura:2004ch . For
example, the deviation larger than about 20$\%$ is predicted by the condition
of strongly first-order EWPT in the two Higgs doublet model Kanemura:2004ch ;
Enomoto:2021dkl ; Kanemura:2022ozv . Such a large deviation in the $hhh$
coupling is expected to be measured at future collider experiments, such as
High-Luminosity LHC (HL-LHC) Cepeda:2019klc and the International Linear
Collider (ILC) Asner:2013psa ; Moortgat-Picka:2015yla ; Fujii:2015jha , by
which a scenario of strongly first-order EWPT can be tested.
If the first-order phase transition occurs in the early Universe,
gravitational waves (GWs) can be produced Grojean:2006bp . The strongly first-
order EWPT predicts specific form of GW spectrum, which is sensitive to the
detailed shape of the Higgs potential of extended models from the SM. The peak
frequency of the GW spectrum from the EWPT is typically $10^{-3}$ –
$10^{-1}$Hz, which can be tested at the future space-based GW interferometers,
such as Laser Interferometer Space Antenna (LISA) Klein:2015hvg and DECi-
hertz Interferometer Gravitational Wave Observatory (DECIGO) Yagi:2011wg . In
recent years, many authors have studied complementarity of future collider
experiments and future space-based GW observations Kakizaki:2015wua ;
Hashino:2016rvx ; Kobakhidze:2016mch ; Huang:2016cjm ; Hashino:2016xoj ;
Artymowski:2016tme ; Beniwal:2017eik ; Huang:2017rzf ; Hashino:2018zsi ;
Chala:2018ari ; Huang:2018aja ; Bruggisser:2018mrt ; Alves:2018oct ;
Hashino:2018wee ; Ahriche:2018rao ; Chala:2018opy ; Alves:2018jsw ;
Alves:2019igs ; Chen:2019ebq ; Enomoto:2021dkl ; Kanemura:2022txx ;
Kanemura:2022ozv .
Recently, it has been discussed that the primordial black hole (PBH)
observation can be used as a probe of strongly first-order EWPT
Hashino:2021qoq , which is motivated by the work by Liu et. al. Liu:2021svg
where PBH production and its abundances in general first-order phase
transition have been studied. It has been known that large density contrast in
the early Universe can gravitationally collapse into the PBHs Hawking:1971ei ;
Carr:1974nx ; Carr:1975qj . Sufficiently large density contrast can be
generated by the first-order phase transition Kodama:1982sf ; Hawking:1982ga ,
which can produce PBHs. The mass of PBHs produced by the first-order EWPT is
about $10^{-5}$ of the solar mass, which can be observed by microlensing
observations such as Subaru Hyper Suprime Cam (HSC) HSC , Optical
Gravitational Lensing Experiment (OGLE) OGLE , PRime-focus Infrared
Microlensing Experiment (PRIME) PRIME and Nancy Grace Roman (Roman) Space
Telescope Roman . Therefore, exploration of such PBHs at these observations
can be used to probe models of strongly first-order EWPT.
In the previous work Hashino:2021qoq , the authors evaluated the abundance of
PBHs from first-order EWPT in the SMEFT. Although large non-decoupling quantum
effects of new particles in the fundamental model behind are important to
realize strongly first-order EWPT, such effects may cause the SMEFT expansion
to break down Postma:2020toi ; Kanemura:2022txx . In the present paper, we
consider one of the realistic effective field theories, called the naHEFT
Kanemura:2021fvp and investigate the production of PBHs in the framework.
This effective field theory can parameterize the non-decoupling quantum
effects of the new physics at the next to leading order, by which strongly
first-order EWPT can be well described. Therefore, the naHEFT is a useful
framework to discuss the phenomena beyond the SM, such as the baryon asymmetry
of the Universe. We study the parameter region in the naHEFT where PBHs can be
produced from first-order EWPT and its detectability in the future
observations. Complementarity of PBH observations, GW observations and
collider experiments to probe the EWPT is also discussed.
In the next section, we first briefly review the naHEFT, and describe the
phase transition parameters such as released latent heat and duration of phase
transition in the framework. We then clarify parameter region in which PBHs
can be produced from the first-order EWPT. In section III, we discuss the
complementarity of PBHs, GWs and collider experiments to probe the strongly
first-order EWPT. We also examine that the parameter region, which is also
related to extended Higgs models, can be widely explored by the PBH
observations. In the final section, we give conclusions and discussions.
## II Nearly aligned Higgs effective field theory
Here, we briefly introduce the naHEFT Kanemura:2021fvp ; Kanemura:2022txx . In
this framework, the BSM part of the Higgs potential is given by
$\displaystyle V_{\rm{BSM}}(\Phi)$
$\displaystyle=\frac{\xi}{4}\kappa_{0}\,[\mathcal{M}^{2}(\Phi)]^{2}\ln\frac{\mathcal{M}^{2}(\Phi)}{\mu^{2}},$
(1)
where $\Phi$ is the SM Higgs isospin doublet field, $\xi=1/(4\pi)^{2}$,
$\kappa_{0}$ is a real dimensionless parameter, and $\mu^{2}$ is a real
massive parameter. The BSM effect to the Higgs potential is described by a
Coleman-Weinberg form Coleman:1973jx . The $\mathcal{M}^{2}(\Phi)$ in this
equation is an arbitrary function of $|\Phi|^{2}$, which is assumed to take
the following form
$\displaystyle\mathcal{M}^{2}(\Phi)\,=\,M^{2}+\kappa_{\rm{p}}\,|\Phi|^{2},$
(2)
where $M^{2}$ and $\kappa_{\rm{p}}$ are real parameters. In order to discuss
the non-decoupling effect, we introduce parameters $\Lambda$ and $r$, which
are given by
$\displaystyle\Lambda^{2}$
$\displaystyle=M^{2}+\frac{\kappa_{\rm{p}}}{2}v^{2},$ (3) $\displaystyle r$
$\displaystyle=\frac{\frac{\kappa_{\rm{p}}}{2}v^{2}}{\Lambda^{2}}=1-\frac{M^{2}}{\Lambda^{2}}.$
(4)
The dimensionful parameter $\Lambda$ corresponds to the physical mass of the
integrated new particles. The parameter $r$ shows non-decoulingness of the new
physics behind. When $r\to 0$ ($r\to 1$), $\Lambda\sim\sqrt{M^{2}}$
($\Lambda\sim\sqrt{\kappa_{p}v^{2}/2}$). Thus, the quantum effects for non-
decoupling in the potential are controlled by the parameter $r$. Therefore,
the scalar sector of the naHEFT has three independent parameters:
$\displaystyle\kappa_{0},\quad\Lambda,\quad r.$ (5)
For example, if the naHEFT is realized by integrating out additional scalar
bosons, whose masses are degenerated, in the model with an extended Higgs
sector, $\kappa_{0}$ and $\Lambda$ correspond to the number of additional
scalar fields and the physical mass of additional scalar bosons, respectively.
We explore the parameter region where PBHs can be produced from first-order
EWPT.
The effective potential in the naHEFT is given at the temperature $T$ by
Kanemura:2022txx
$\displaystyle
V_{\rm{eff}}(\varphi,T)\,=\,V^{\rm{SM}}_{\rm{eff}}(\varphi,T)+V_{\rm{BSM}}(\varphi)\,+\,\Delta
V_{{\rm BSM},T}(\varphi,T)\,,$ (6)
where $V^{\rm{SM}}_{\rm{eff}}$ is the effective potential in the SM and
$\varphi$ is the classical scalar field with
$\langle\Phi\rangle=(0,\varphi/\sqrt{2})^{T}$. Here, the finite temperature
contributions coming from the BSM effects are given by
$\displaystyle\Delta V_{{\rm BSM},T}(\varphi,T)$
$\displaystyle\,=\,8\,\xi\,T^{4}\,\kappa_{0}\,J_{\rm{BSM}}\left(\frac{\mathcal{M}^{2}(\varphi)}{T^{2}}\right)\,.$
(7)
The $J_{\rm{BSM}}$ function is given by
$\displaystyle
J_{\rm{BSM}}\left(\frac{\mathcal{M}^{2}(\varphi)}{T^{2}}\right)\,=\,\int^{\infty}_{0}dk^{2}k^{2}\ln\left(1-\mbox{sign}(\kappa_{0})\,e^{-\sqrt{k^{2}+\frac{\mathcal{M}^{2}(\varphi)}{T^{2}}}}\right)\,,$
(8)
where $\mbox{sign}(\kappa_{0})$ is positive (negative) in the case of
$\kappa_{0}>0$ $(\kappa_{0}<0)$.
Figure 1: (Left) Value of $v_{n}/T_{n}$ in the naHEFT with $\kappa_{0}=4$ and
$\Lambda=500$, 600 and 700 GeV. The green, blue and red lines correspond to
$\Lambda=500$, 600 and 700 GeV, respectively. (Right) The value of
$v_{n}/T_{n}$ in the naHEFT and the SMEFT approximation with $\Lambda=$ 700
GeV. The blue solid, blue dashed, blue dotted, blue dot-dashed and blue dashed
double-dotted lines represent the results in the potential of naHEFT truncated
up to $|\Phi|^{6}$, $|\Phi|^{8}$, $|\Phi|^{10}$, $|\Phi|^{12}$ and
$|\Phi|^{14}$, respectively.
Due to the effects of $V_{\rm eff}^{\rm BSM}$, the strongly first-order EWPT,
which is represented by the condition of $v_{n}/T_{n}>1$, can be realized.
Here, $v_{n}$ is the true vacuum at the nucleation temperature $T_{n}$, at
which one bubble nucleates in the Hubble volume. $T_{n}$ is computed by
calculating the bounce solution of equation of motion Linde:1981zj . Numerical
results of $v_{n}/T_{n}$ in the naHEFT with $\kappa_{0}=4$ are shown in the
left panel of Fig. 1. The horizontal and vertical axises of this figure are
the parameter $r$ and the value of $v_{n}/T_{n}$, respectively. The green,
blue and red lines in the left panel correspond to the $v_{n}/T_{n}$ values of
the naHEFT with $\Lambda=500$, 600 and 700 GeV, respectively. Curvatures of
these lines change when the parameter $r$ is relatively large, since the cubic
term $\varphi^{3}$ coming from finite temperature effects is no longer
important in generating a sizable barrier to realize the first-order EWPT. The
right panel shows the results of $v_{n}/T_{n}$ in the potential of naHEFT and
ones truncated up to $|\Phi|^{6}$, $|\Phi|^{8}$, $|\Phi|^{10}$, $|\Phi|^{12}$
and $|\Phi|^{14}$. The red solid line in the right panel is the value of
$v_{n}/T_{n}$ in the naHEFT with $\kappa_{0}=4$ and $\Lambda=700$ GeV. The
blue solid, blue dashed, blue dotted, blue dot-dashed and blue dashed double-
dotted lines in the right panel are the value of $v_{n}/T_{n}$ in the
potential with $\kappa_{0}=4$ and $\Lambda=700$ GeV truncated up to
$|\Phi|^{6}$, $|\Phi|^{8}$, $|\Phi|^{10}$, $|\Phi|^{12}$ and $|\Phi|^{14}$,
respectively. At relatively large values of $r$, $v_{n}/T_{n}$ in the naHEFT
rapidly blows up, since the temperature for starting the EWPT gets close to
zero. In the case of relatively large $r$, deviations on $v_{n}/T_{n}$ for the
SMEFT with truncation up to $|\Phi|^{6}$ – $|\Phi|^{14}$ operators from the
prediction in the naHEFT become large. This implies that the naHEFT is better
than the SMEFT to discuss the parameter region for strongly first-order EWPT.
It is well known that in the parameter region where the strongly first-order
EWPT is realized a large deviation is predicted in the $hhh$ coupling from the
SM value, which is given by
$\displaystyle\frac{\Delta\lambda_{hhh}}{\lambda_{hhh}^{\rm
SM}}\equiv\frac{\lambda_{hhh}-\lambda_{hhh}^{\rm SM}}{\lambda_{hhh}^{\rm
SM}},\quad\lambda_{hhh}=\left.\frac{\partial^{3}V_{\rm
eff}(\varphi)}{\partial\varphi^{3}}\right|_{\varphi=v},$ (9)
where effective potential $V_{\rm eff}(\varphi,0)$ is given in Eq. (6), which
contains one-loop level quantum corrections111 One-loop corrections to the
$hhh$ coupling in extended Higgs models is discussed in Ref. Kanemura:2004mg ;
Aoki:2012jj ; Arhrib:2015hoa ; Hashino:2015nxa ; Kanemura:2016lkz . Typically,
the two-loop effects give positive contributions to the $hhh$ coupling, which
are about 20$\%$ of those at one-loop Braathen:2019pxr ; Braathen:2019zoh ;
Braathen:2020vwo .. Therefore, the strongly first-order EWPT can be tested by
precisely measuring the $hhh$ coupling at future collider experiments
Kanemura:2004ch . At the HL-LHC, the $hhh$ coupling can be measured at 50 $\%$
accuracy (at the 68$\%$ confidence level) Cepeda:2019klc , while at the future
lepton collider such as the ILC energy upgraded version with the energy 500
GeV and 1 TeV, the $hhh$ coupling is expected to be measured by 27$\%$ and
10$\%$ accuracies (at the 68$\%$ confidence level), respectively
Bambade:2019fyw .
## III First-order EWPT and PBH in naHEFT
We here introduce phase transition parameters $\alpha$ and $\beta/H$. $\alpha$
is the normalized released latent heat by radiative energy density, which is
defined as
$\displaystyle\alpha\equiv\epsilon(T_{n})/\rho_{\rm rad}(T_{n}),$ (10)
where the $\rho_{\rm rad}(T)=(\pi^{2}/30)g_{*}T^{4}$ with $g_{*}=106.75$, and
$\epsilon$ is given by
$\displaystyle\epsilon(T)=\Delta V_{\rm eff}-T\frac{\partial\Delta V_{\rm
eff}}{\partial T},\quad\Delta V_{\rm eff}=V_{\rm eff}(\varphi_{-}(T),T)-V_{\rm
eff}(\varphi_{+}(T),T),$ (11)
where $\varphi_{+}$ and $\varphi_{-}$ denote the true and false vacua,
respectively. The $\beta/H$ corresponds to the inverse of the duration of
phase transition, and is defined as
$\displaystyle\frac{\beta}{H}\equiv
T_{n}\left.\frac{d}{dT}\left(\frac{S_{3}}{T}\right)\right|_{T=T_{n}},$ (12)
where $S_{3}$ is the three-dimensional Euclidian action.
If the first-order phase transition occurs in the early Universe, GWs can be
produced due to the bubble dynamics of the true vacuum. The GWs from the
first-order EWPT have three sources: collisions of the vacuum bubbles,
compressional waves (sound waves) and magnetohydrodynamics turbulence. The
leading contribution among the three sources comes from the sound waves, whose
amplitude is given by Caprini:2015zlo
$\displaystyle\Omega_{\rm sw}(f)h^{2}=2.65\times
10^{-6}v_{w}\left(\frac{H}{\beta}\right)\left(\frac{\kappa_{v}\alpha}{1+\alpha}\right)^{2}\left(\frac{100}{g_{*}}\right)^{1/3}(f/f_{\rm
sw})^{3}\left(\frac{7}{4+3(f/f_{\rm sw})^{2}}\right)^{7/2},$ (13)
where $v_{w}$ is the wall velocity, and $\kappa_{v}$ is the fraction of the
released latent heat contributing to sound wave formation, which is given in
Ref. Espinosa:2010hh . The peak frequency $f_{\rm sw}$ is given by
Caprini:2015zlo
$\displaystyle f_{\rm sw}=1.9\times 10^{-2}{\rm
mHz}\frac{1}{v_{w}}\left(\frac{\beta}{H}\right)\left(\frac{T_{n}}{100{\rm
GeV}}\right)\left(\frac{g_{*}}{100}\right)^{1/6}.$ (14)
The prediction of the GW spectrum in the naHEFT was investigated in Ref.
Kanemura:2022txx . To examine parameter regions where GWs can be detected at
LISA and DECIGO, we use the signal-to-noise ratio for the observation of the
GW spectrum, which is discussed in Ref. Cline:2021iff . The criterion is such
that the signal-to-noise ratio for the GW spectrum is larger than ten, which
is adopted in our later analysis.
Next, it is known that sufficient large density contrast in the early Universe
results in overdensity regions which can gravitationally collapse into PBHs.
According to Ref. Liu:2021svg , the large density contrast can be generated by
delaying the vacuum bubble nucleation. Since the vacuum bubble nucleation is
probabilistic, there is a possibility that the symmetry breaking is delayed in
a whole Hubble volume. The vacuum energy density in the unbroken symmetry
region is larger than that in the broken symmetry region because the
difference of the vacuum energy density is related to the difference of the
height of the Higgs potential. This energy density difference leads to the
energy density contrast between the inside and outside of the Hubble volume in
which the symmetry breaking is delayed. The energy density contrast is defined
as
$\displaystyle\delta\equiv\frac{|\rho_{\rm in}-\rho_{\rm out}|}{\rho_{\rm
out}},$ (15)
where $\rho_{\rm in}$ and $\rho_{\rm out}$ is the total energy density inside
and outside the Hubble volume, respectively. When the energy density contrast
$\delta$ exceeds the critical value $\delta_{c}=0.45$ Musco:2004ak ;
Harada:2013epa , the inside of the Hubble volume can gravitationally collapse
into a PBH. Therefore, the mass of PBHs is roughly determined by the Hubble
horizon mass at the time when the PBHs are produced. For the EWPT, the mass of
PBHs is about $10^{-5}M_{\odot}$ ($M_{\odot}$ is the solar mass). Thus, the
fraction of the PBHs $f_{\rm PBH}$ produced from the first-order EWPT can be
probed by the current and future microlensing observations such as Subaru HSC,
OGLE, PRIME and Roman Space Telescope. The fraction $f_{\rm PBH}$ can be
determined by the phase transition parameters $\alpha$ and $\beta/H$ in Eqs.
(10) and (12). The method of calculating the fraction $f_{\rm PBH}$ is
explained in the Appendix.
Figure 2: Contours for the PBH abundance are shown in the $\alpha$ and
$\beta/H$ plane. Red, blue, orange, green and brown lines correspond to
$f_{\rm PBH}=1,10^{-2},10^{-4},10^{-6}$ and $10^{-8}$, respectively. The
region $10^{-4}<f_{\rm PBH}<1$, between red and orange lines, can be explored
by PBH observations, such as PRIME and Roman Space Telescope. In the white
region above the brown line, the abundance of the PBH cannot be produced or
otherwise is too small. In the white region below the red line, PBHs are
overproduced ($f_{\rm PBH}>1$).
Fig. 2 represents model independent numerical results of $f_{\rm PBH}$ in the
$\alpha$-$\beta/H$ plane, which were discussed in Ref. Hashino:2021qoq . The
red, blue, orange, green and brown lines correspond to $f_{\rm
PBH}=1,10^{-2},10^{-4},10^{-6}$ and $10^{-8}$, respectively. Current
microlensing experiments, such as Subaru HSC and OGLE, can explore the region
between the red and blue lines with $10^{-2}<f_{\rm PBH}<1$. On the other
hand, the region between the red and orange lines with $10^{-4}<f_{\rm PBH}<1$
can be explored by future microlensing experiments such as PRIME and Roman
Space Telescope. In the white region above the brown line, the PBH abundance
becomes too small or the PBHs cannot be produced from the first-order EWPT. In
the white region below the red line, PBHs are overproduced ($f_{\rm PBH}>1$).
According to this figure, we can discuss whether PBH observations can be used
to probe the strongly first-order EWPT.
Figure 3: Parameters $\alpha$ and $\beta/H$ with respect to $\kappa_{0}$,
$\Lambda$ and $r$. Left (right) figure represents $\alpha$ and $\beta/H$ in
the naHEFT with $r=1$ (0.5). The red, green and blue lines are $\kappa_{0}=1$,
4 and 20, respectively. The points on these lines correspond to the $\Lambda$
value. Purple dotted and solid lines respectively are $f_{\rm PBH}=10^{-4}$
and 1.
Fig. 3 represents the parameters $\alpha$ and $\beta/H$ in the naHEFT. The
parameter $r$ is assumed as $r$ =1 and 0.5 in the left and right panels,
respectively. Red, green and blue lines correspond to the parameters $\alpha$
and $\beta/H$ with $\kappa_{0}=$ 1, 4 and 20, respectively. Points on these
lines in this figure represent the value of $\Lambda$. Purple dotted and solid
lines respectively correspond to $f_{\rm PBH}=10^{-4}$ and 1. We here comment
on the behaviors of the lines in right panel of Fig. 3, which are degenerate
in the case of large $\Lambda$ value. The behavior is ascribed to the
Boltzmann suppression with respect to finite temperature effects: $\Delta
V_{{\rm BSM},T}(0,T)\propto\exp\left[-\Lambda^{2}(1-r)/T^{2}\right]$. For
small $r$ and large $\Lambda$, the potential is mainly determined by the zero
temperature BSM effects. Then the difference between the origin and the bottom
of the potential, which is related to the phase transition parameters, is
roughly given by $\kappa_{0}r\Lambda^{4}$. For example, cases with
($\kappa_{0}$, $\Lambda$ [GeV], $r$) = (1, 1039, 0.5), (4, 736, 0.5) and (20,
492, 0.5), which are depicted in the right panel of Fig. 3, have almost the
same value of $\kappa_{0}r\Lambda^{4}$, and thus these points get close to
each other actually.
Figure 4: Regions of strongly first-order EWPT, where $v_{n}/T_{n}\geq 1$,
are shown as colored regions in the $r$-$\Lambda$ plane for $\kappa_{0}=1$, 4,
8 and 16. In the red region, $f_{\rm PBH}$ can be larger than $10^{-4}$. The
EWPT has not been finished at the current Universe in top right white regions
above the red one: $\Gamma/H^{4}<1$. The orange regions represent that the
detectable GW at DECIGO experiment can be produced. The GW spectrum for the
blue and red regions can be observed by both LISA and DECIGO experiments. The
black dotted lines are the deviation in the $hhh$ coupling from the SM
prediction value $\Delta\lambda_{hhh}/\lambda_{hhh}^{\rm SM}$ = 20, 50, 100
and 200 $\%$ from the bottom, respectively.
Fig. 4 represents the region of strongly first-order EWPT in the naHEFT with
$\kappa_{0}=$ 1, 4, 8 and 16 in the $r$-$\Lambda$ plane. For example, assuming
the O(N) singlet scalar field theory as the UV theory, the parameter region
with $r<0.3$ would be prohibited by the perturbative unitarity bound
Hashino:2016rvx , and thus, we do not take into account such parameter regions
in the following numerical analysis. The black dotted lines represent
$\Delta\lambda_{hhh}/\lambda_{hhh}^{\rm SM}$ = 20, 50, 100 and 200 $\%$ from
the bottom, respectively. In the red region, the $f_{\rm PBH}$ can be larger
than $10^{-4}$. Thus, the first-order EWPT can be tested using the PBH
observations. In this region, we can also use GWs to test the EWPT. In the
blue (orange) parameter region, the first-order EWPT can be tested by using GW
observation at both LISA and DECIGO (only at DECIGO). In the green region,
although GWs cannot be detected at LISA nor DECIGO the first-order EWPT can
still be tested by the precision measurement of the $hhh$ coupling at future
collider experiments. In the top right white region above the red solid line
of these panels, the EWPT has not been completed at the current Universe, in
which $\Gamma/H^{4}<1$. In the bottom left white region below the colored
region of this figure, the strongly first-order EWPT cannot be realized
because of $\varphi_{C}/T_{C}<1$.
Figure 5: Regions of strongly first-order EWPT, where $v_{n}/T_{n}\geq 1$,
are shown as colored regions in the $\kappa_{0}$-$\Lambda$ plane for $r=0.3$,
0.5, 0.8 and 1. Otherwise the same as Fig. 4.
Fig. 5 shows the region of strongly first-order EWPT in the naHEFT with
$r=0.3$, 0.5, 0.8 and 1 in the $\kappa_{0}$ –$\Lambda$ plane. The colored
regions and black dotted contours have the same definitions in Fig. 4. For
large $\kappa_{0}$ value, the strongly first-order EWPT can be realized by
small $\Lambda$ value. According to these Figs. 4 and 5, the naHEFT can be
complementarily tested by collider experiments, GW and PBH observations.
Figure 6: The parameter region where PBHs from strongly first-order EWPT may
be able to be detected in the $\kappa_{0}$-$\Lambda$ plane ($f_{\rm
PBH}>10^{-4}$).
Fig. 6 shows the parameter region where PBHs from strongly first-order EWPT
may be able to be detected in the $\kappa_{0}$-$\Lambda$ plane. The solid and
dashed red lines of this figure correspond to the same as the red region of
Fig. 5 for $r=1$ and 0.3, respectively. In the red region, the fraction of
PBHs can be sizable with $f_{\rm PBH}>10^{-4}$ for $0.3<r<1$. The PBH
observation may be able to be used to explore the strongly first-order EWPT in
such a wide parameter region.
## IV Conclusion
We have investigated the production of PBHs from first-order EWPT in the
framework of the naHEFT, in which non-decoupling quantum effects are properly
described. Since the mass of PBHs from first-order EWPT is about $10^{-5}$ of
the solar mass, the current and future microlensing observations such as
Subaru HSC, OGLE, PRIME and Roman Space Telescope may be able to probe the
EWPT. We have examined the parameter region where PBHs can be produced from
first-order EWPT and have found that the PBH observations could probe the
strongly first-order EWPT of the naHEFT. Complementarity in testing the
strongly first-order EWPT by future collider experiments, GW and PBH
observations has also been investigated. From Fig. 6, the PBH observation may
be able to be used to explore wide parameter region. Therefore, the PBH
observation is a powerful tool to probe the strongly first-order EWPT.
###### Acknowledgements.
The work of S. K. was supported by the Grant-in-Aid on Innovative Areas, the
Ministry of Education, Culture, Sports, Science and Technology, No. 16H06492,
and by the JSPS KAKENHI Grant No. 20H00160. The work of T. T. was supported by
JSPS KAKENHI Grant Number 19K03874. The work of M. T. was supported in part by
JSPS KAKENHI Grant No. JP21J10645.
## Appendix A PBH from first-order phase transition
We briefly review the mechanism of PBH production from first-order phase
transition by the treatment in Ref. Liu:2021svg . The critical bubble
nucleation rate per unit volume per unit time is given by Linde:1981zj
$\Gamma(T)\simeq T^{4}\left(\frac{S_{3}}{2\pi
T}\right)^{\frac{3}{2}}\exp\left(-S_{3}/T\right),$ (16)
where $S_{3}$ is the three dimensional Euclidean action. From this equation,
the probability of existing the false vacuum is given by Turner:1992tz
$F(t)=\exp\left[-\frac{4\pi}{3}\int^{t}_{t_{i}}dt^{\prime}\Gamma(t^{\prime})a^{3}(t)r^{3}(t,t^{\prime})\right],$
(17)
where $t_{i}$ is nucleation time of a first bubble in the Universe, $a(t)$ is
the scale factor, and $r(t,t^{\prime})$ is the comoving radius of the true
vacuum from $t^{\prime}$ to $t$, which is given by
$r(t,t^{\prime})\equiv\int^{t}_{t^{\prime}}\frac{1}{a(\tilde{t})}d\tilde{t}.$
(18)
We here assume that the bubble wall velocity is closed to the light speed. In
this case, the bubble wall can be treated as radiation, and then the total
radiation energy density is
$\rho_{R}=\rho_{r}+\rho_{w}.$ (19)
The evolution of this energy density can be described by
$\frac{d\rho_{R}}{dt}+4H\rho_{R}=-\frac{d\rho_{v}}{dt}.$ (20)
The vacuum energy density $\rho_{v}$ is given by
$\rho_{v}(t)\equiv F(t)\Delta V,$ (21)
where the $\Delta V$ is the difference in the potential energy density between
false and true vacua. The scale factor evolution can be determined by the
Friedmann equation
$H^{2}=\left(\frac{1}{a(t)}\frac{da(t)}{dt}\right)^{2}=\frac{1}{3}(\rho_{v}+\rho_{R}),$
(22)
where we take the unit of $M_{\rm pl}=1$. The probability that the Hubble
volume collapses into a PBH is given by
$P(t_{n})=\exp\left[-\frac{4\pi}{3}\int^{t_{n}}_{t_{i}}dt\frac{a^{3}(t)}{a^{3}(t_{\rm
PBH})}\frac{1}{H^{3}(t_{\rm PBH})}\Gamma(t)\right],$ (23)
where $t_{n}$ is time of nucleation of a bubble in the Hubble volume, and
$t_{\rm PBH}$ is the time of production of the PBH. The production time
$t_{\rm PBH}$ can be obtained when the density contrast between the inside and
outside of the Hubble volume exceeds the critical value $\delta_{c}=0.45$
Musco:2004ak ; Harada:2013epa . The energy density contrast $\delta$ is given
by Eq. (15). When $\delta$ exceeds $\delta_{c}$, the inside of the Hubble
volume can gravitationally collapse into a PBH. The PBH mass is given by
$M_{\rm PBH}\sim\frac{4\pi}{3}H^{-3}(t_{\rm PBH})\rho_{c}=4\pi H^{-1}(t_{\rm
PBH}).$ (24)
In the case of the EWPT, the PBH mass is $M_{\rm PBH}^{\rm EW}\sim
10^{-5}M_{\odot}$, where $M_{\odot}$ is the solar mass. The fraction of the
PBH from the first-order EWPT in dark matter density $f_{\rm PBH}$ may be
observed by microlensing experiments. For the EWPT, the fraction $f_{\rm
PBH}^{\rm EW}$ is given by
$f_{\rm PBH}^{\rm EW}\sim 1.49\times 10^{11}\left(\frac{0.25}{\Omega_{\rm
CDM}}\right)\left(\frac{T_{\rm PBH}}{100{\rm GeV}}\right)P(t_{n}),$ (25)
where $\Omega_{\rm CDM}$ is the current energy density of cold dark matter
normalized by the total energy density and $T_{\rm PBH}$ is the temperature at
the PBHs production. Regions of $f_{\rm PBH}>10^{-2}$ are already in the reach
of current observations at Subaru HSC Niikura:2017zjd and OGLE
Niikura:2019kqi . On the other hand, future microlensing experiments, such as
Roman Space Telescope, can test the parameter region with the fraction $f_{\rm
PBH}>10^{-4}$ Roman2 .
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# Quantum correlations in a mixed spin-(1/2,1) Heisenberg dimer
S. Bhuvaneswari Centre for Nonlinear Science (CeNSc), PG & Research Department
of Physics, Government College for Women (Autonomous), Kumbakonam, Tamil Nadu,
India. R. Muthuganesan Department of Physics, Faculty of Nuclear Sciences and
Physical Engineering, Czech Technical University in Prague, Br̆ehová 7, 115 19
Praha 1-Staré Mĕsto, Czech Republic, Email<EMAIL_ADDRESS>R. Radha
111Corresponding author
###### Abstract
In this article, we consider the heterodinuclear complex [Ni(dpt)(H2
O)Cu(pba)] · 2H2 O [pba =1,3-propylenebis(oxamato) and dpt =
bis-(3-aminopropyl)amine] realized through the theoretical model of mixed
spin-(1/2,1) coupled via Heisenberg interaction. We study the behaviors of
thermal quantum correlations of the above material via Measurement-Induced
Nonlocality (MIN) based on Hilbert-Schmidt norm and fidelity. We observe that
the quantum correlation measures increase with the magnetic field in an
unconventional way. The role of system parameters is also brought out at
thermal equilibrium. The highlight of the results is that we are able to show
the existence of room temperature quantum correlation using fidelity based MIN
whereas the entanglement ceases to exist at $141K$.
Keywords: Quantum correlation, Heisenberg interaction, Measurements,
Nonlocality.
## 1 Introduction
In an era of information and communication technology, the necessity to obtain
a potential solution for classically intractable problems can be attained by
performing quantum computing. The incorporation of quantum principles such as
superposition, entanglement, and interference into computing makes quantum
computers more efficient and effective than classical counterparts. Also, this
brings in variation to our conventional notion of the principle of locality as
proposed in EPR paradox [1, 2, 3, 4]. The nonlocality that arises due to
superposition principle and entanglement is the most peculiar manifestation of
nonlocality which has no classical analogy. The presence of entanglement or
nonlocal character of quantum system is demonstrated by the violation of Bell
inequality [3, 4]. The study of entanglement in bipartite/ multipartite states
has been investigated effectively for many decades and proven that the
entanglement is not the whole indicator of the non-classical correlation of
quantum system even in the bipartite scenario. In this connection, the seminal
work of Werner [5] and the subsequent experimental demonstration of quantum
advantages using separable states [6] suggest that the entanglement is not an
ultimate resource for quantum technology. Alternatively, the separable state
is also at the root of the power of quantum computing. The above notion has
opened up new avenues for the identification of new quantum correlation
measures beyond entanglement. Over the last couple of decades, a great deal of
attention has been devoted to capture the quantumness beyond entanglement
using different measures such as quantum discord [7], measurement-induced
nonlocality [8], measurement-induced disturbance [9] and skew information
measures [10, 11]. In general, the entropic measures are quite hard to compute
even in simpler systems and is shown that the computation of discord is an
nondeterministic polynomial problem [12].
To overcome the computational complexity, the criterion of discord modified by
the distance between quantum states in the state space is known as geometric
discord [13]. Different versions of geometric discord have been identified
using various distance measures such as p-norm [13, 14], Bures metric [15] and
affinity [16]. Alternatively, Luo and Fu identified a new version of quantum
correlation measure in terms of Hilbert-Schmidt (HS) distance between the
state and locally perturbed state known as measurement-induced nonlocality
(MIN) [8]. It is one such measure which captures correlation beyond
entanglement and is dual to the the geometric quantum discord [13]. Due to
noncontractivity of HS [17], MIN is not a faithful quantifier of bipartite
quantum correlation. Till date, different versions of MIN have been introduced
to resolve the above local ancilla problem [18, 19, 20]. From a practical
perspective, MIN is considered as a more secure resource for information
processing like quantum communication and cryptography [21, 22, 23, 24]. In
addition, intrinsic decoherence [25, 26] and local noises [27] do not seem to
impact MIN. This property can also be employed to identify quantum phase
transitions [28]. The realization and implementation of the quantum algorithms
constitute an important task in the current and near-future quantum hardware
which is considered to be susceptible to thermal fluctuations. The above
challenge provides a key understanding for the development of quantum
technology. Quantum magnetic materials have the potential to be beneficial in
a variety of applications like storage, magnetic sensors, medical appliances
and quantum applications. The Heisenberg spin model is one of the simplest
systems that possesses the nonlocal aspects. In fact, in the last two decades,
the entanglement and quantum information processing in different spin models
have been studied. Among them, the study of quantum correlations in mixed spin
Heisenberg models caught wide attention within the framework of quantum
information theory. The analysis of quantum correlation in mixed
spin-$(1/2,S>1)$ has been carried out in different perspectives bringing out
the impact of uniaxial-single ion anisotropy, magnetic field and
Dzyaloshinskii–Moriya interaction [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42]. In addition, the role of different magnitudes of spin has also
been investigated. [34, 37, 40, 41, 42]. The effect of g-factor on
entanglement which arises due to the noncommutativity of the magnetic moment
operator and Hamiltonian of the system [43] has also been brought out.
Motivated by the identification of heterodinuclear complex
$[\text{Ni(dpt)}(H_{2}O)\text{Cu(pba)}]\cdot 2H_{2}O$ [pba
=1,3-propylenebis(oxamato) and dpt = bis-(3-aminopropyl)amine] which gave rise
to the experimental realization of the mixed spin-$(1/2,1)$ Heisenberg dimer,
we investigate its quantum correlations through the theoretical model of the
mixed spin-$(1/2,\leavevmode\nobreak\ 1)$ Heisenberg spin chain. Its magnetic
properties have been studied extensively [31, 44]. The impact of single-ion
anisotropy on entaglement has also been brought out in the interacting mixed
spin Heisenberg model [32]. We employ HS-MIN and fidelity-based MIN as
quantifiers of quantum correlation and compare our results with that of
entanglement obtained in [45]. Under suitable parametric condition, it is
shown that the quantum correlation measure can increase with the magnetic
field in an unconventional way. The quantum correlation between the spins
decreases monotonically with the increase of temperature and uni-axial
anisotropy strengthens quantum correlations. Choosing appropriate experimental
parameters, we also study the quantum correlation between Cu and Ni magnetic
ions. Interestingly, F-MIN survives even at room temperature $T=300K$.
Further, the role of gyromagnetic $g$ factors is also highlighted.
The paper is organized as follows. First, we provide an overview of different
measures of quantum correlations in Sec. 2. Then, we introduce the model under
investigation and its diagonalization in Sec. 3. We then discuss the quantum
correlations in mixed spin system in Sec. 4.1. In Sec. 4.2, we study the
quantum correlations in CuNi complex. Finally, in Sec. 5, we conclude the main
results of this paper.
## 2 Measurement-induced nonlocality
In this paper, we employ two different versions of measurement-induced
nonlocality such as Hilbert-Schmidt MIN and fidelity-based MIN pertaining to
bipartite quantum correlation. In the last decade, researchers have identified
different versions of quantum correlation measures that are based on metrics
and due to the symmetry property of the Hilbert space, these measures are
having merit in the computation. In this series, a new version of quantum
correlation measure has been identified from the perspective of
eigenprojective measurements and it is defined as [8]
$N_{2}(\rho)=\leavevmode\nobreak\
^{\text{max}}_{\Pi^{a}}\|\rho-\Pi^{a}(\rho)\|_{2}^{2},$ (1)
where $\|\mathcal{O}\|_{2}^{2}=\text{Tr}(\mathcal{O}\mathcal{O}^{\dagger})$ is
Hilbert-Schmidt norm of operator $\mathcal{O}$ and the maximum is taken over
the locally invariant projective measurements on subsystem $a$ which does not
change the state $\rho^{a}$. The post-measurement state is defined as
$\Pi^{a}(\rho)=\sum_{k}(\Pi^{a}_{k}\otimes\mathds{1}^{b})\rho(\Pi^{a}_{k}\otimes\mathds{1}^{b})$
and $\Pi^{a}=\\{\Pi^{a}_{k}\\}=\\{|k\rangle\langle k|\\}$ being the projective
measurements on the subsystem $a$, which do not change the marginal state
$\rho^{a}$ locally i.e., $\Pi^{a}(\rho^{a})=\rho^{a}$. If $\rho^{a}$ is a non-
degenerate state, then, the maximization is not required and the above
quantity is equal to geometric discord. An arbitrary bipartite density matrix
$\rho$ can be written as
$\displaystyle\rho=\sum_{ij}\gamma_{ij}X_{i}\otimes Y_{j}$ (2)
where $\gamma_{ij}=\text{Tr}(\rho(X_{i}\otimes Y_{j}))$. In a bipartite state
space, the orthonormal operators in respective state spaces are
$\\{X_{0},X_{1},X_{2},X_{3}\\}=\\{\mathds{1},\sigma_{1},\sigma_{2},\sigma_{3}\\}/\sqrt{2}$
and
$\\{Y_{0},Y_{1},Y_{2},Y_{3}\\}=\\{\mathds{1},\sigma_{1},\sigma_{2},\sigma_{3}\\}/\sqrt{2}$,
where $\sigma_{i}$ are the Pauli matrices. The above state can be recast as
$\rho=\frac{1}{4}\left[\mathds{1}^{a}\otimes\mathds{1}^{b}+\sum_{i=1}^{3}x_{i}(\sigma_{i}\otimes\mathds{1}^{b})+\sum_{j=1}^{3}y_{j}(\mathds{1}^{a}\otimes\sigma_{j})+\sum_{i,j=1}^{3}t_{ij}\sigma_{i}\otimes\sigma_{j}\right]$
(3)
where $x_{i}=\text{Tr}(\rho(\sigma_{i}\otimes\mathds{1}^{b})$ and
$y_{j}=\text{Tr}(\rho(\mathds{1}^{a}\otimes\sigma_{j}))$ are the components of
Bloch vector with $t_{ij}=\text{Tr}(\rho(\sigma_{i}\otimes\sigma_{j}))$ being
real matrix elements of correlation matrix $T$. MIN has a closed formula as
$N_{2}(\rho)=\begin{cases}\text{Tr}(TT^{t})-\frac{1}{\|\textbf{x}\|^{2}}\textbf{x}^{t}TT^{t}\textbf{x}&\text{if}\quad\textbf{x}\neq
0,\\\
\text{Tr}(TT^{t})-\lambda_{\text{min}}&\text{if}\quad\textbf{x}=0\end{cases}$
(4)
where $\lambda_{\text{min}}$ is the least eigenvalue of matrix $TT^{t}$, the
superscript $t$ stands for the transpose and the vector
$\textbf{x}=(x_{1},x_{2},x_{3})^{t}$.
As mentioned earlier, Hilbert-Schmidt distance is not a bonafide measure as
indicated by Piani. This issue can be circumvented by modifying the definition
of MIN using some other distance measures. One such quantity is fidelity-based
MIN (F-MIN) and is defined as [20]
$\displaystyle N_{\mathcal{F}}(\rho)=1-\leavevmode\nobreak\
^{\text{min}}_{\Pi^{a}}\mathcal{F}(\rho,\Pi^{a}(\rho)).$ (5)
where $\mathcal{F}(\rho,\sigma)$ is the fidelity between the states [46]
$\displaystyle\mathcal{F}(\rho,\sigma)=\frac{\text{Tr}(\rho\sigma)^{2}}{\text{Tr}(\rho^{2})\text{Tr}(\sigma^{2})},$
(6)
which can be computed easily compared to fidelity introduced by Josza [47] and
satisfies all the properties of a good measure of fidelity between states. In
addition, one can realize the above fidelity using quantum circuits [48]. Here
also, the minimization is taken over the locally invariant projective
measurements. Due to the multiplicative property of the fidelity, F-MIN fixes
the local ancilla problem. The closed formula of F-MIN is computed as
$N_{\mathcal{F}}(\rho)=\begin{cases}\text{Tr}(\Gamma\Gamma^{t})-TrA\Gamma\Gamma^{t}A^{t}&\text{if}\quad\textbf{x}\neq
0,\\\
\text{Tr}(\Gamma\Gamma^{t})-\tau_{\text{min}}&\text{if}\quad\textbf{x}=0\end{cases}$
(7)
where $\tau_{\text{min}}$ is the minimal eigenvalue of the matrix
$\Gamma\Gamma^{t}$ and the matrix A is given by
$\displaystyle
A=\frac{1}{\sqrt{2}}\begin{pmatrix}1&\frac{\textbf{x}}{|\textbf{x}|}\\\
1&-\frac{\textbf{x}}{|\textbf{x}|}\end{pmatrix}.$ (8)
## 3 The model and thermalization
To understand the behaviors of thermal quantum correlations, we consider the
Hamiltonian of the mixed spin - $(1/2,1)$ Heisenberg dimer and is defined as,
$\displaystyle\mathcal{H}=J\left[\Delta(\hat{S}^{x}\hat{\Sigma}^{x}+\hat{S}^{y}\hat{\Sigma}^{y})+\hat{S}^{z}\hat{\Sigma}^{z}\right]+D(\hat{\Sigma}^{z})^{2}-g_{1}\mu_{B}B\hat{S}^{z}-g_{2}\mu_{B}B\hat{\Sigma}^{z}$
(9)
where $\hat{S}^{\alpha}(\hat{\Sigma}^{\alpha})$ denotes the spatial components
of the spin-$1/2(1)$ operators with $\alpha=x,y,z$, $J$ is the coupling
constant between the spin-1/2 and spin-1 magnetic ions, the parameter $\Delta$
determines the XXZ exchange anisotropy in this exchange interaction, $D$ is a
uniaxial single-ion anisotropy acting on the spin-1 magnetic ions only, $B$
denotes a static external magnetic field, $\mu_{B}$ is the Bohr magneton and,
$g_{1}$ and $g_{2}$ are Landé $g$ factors of the spin-1/2 and spin-1 magnetic
ions respectively.
It is worth mentioning at this juncture that the above mixed spin-$(1/2,1)$
theoretical model is realizable in hetero-bimetallic complexes such as the
CuNi compound. Here, the spin-1/2 $Cu^{2+}$ and spin-1 $Ni^{2+}$ are coupled
via Heisenberg exchange coupling $J$. In the standard qubit-qutrit
computational basis
$\\{|\frac{1}{2},0\rangle,|\frac{-1}{2},0\rangle\,|\frac{1}{2},1\rangle,|\frac{1}{2},-1\rangle,|\frac{-1}{2},1\rangle,|\frac{-1}{2},-1\rangle\\}$,
the Hamiltonian has the following matrix form,
$\displaystyle\mathcal{H}=\begin{pmatrix}A_{-}&0&0&0&0&0\\\
0&B_{-}&0&\nu&0&0\\\ 0&0&C_{+}&0&\nu&0\\\ 0&\nu&0&C_{-}&0&0\\\
0&0&\nu&0&B_{+}&0\\\ 0&0&0&0&0&A_{+}\\\ \end{pmatrix},$ (10)
where the diagonal elements are
$\displaystyle A_{\pm}=\frac{1}{2}[J+2D\pm(h_{1}+2h_{2})],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ B_{\pm}=\pm\frac{h_{1}}{2}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
C_{\pm}=-\frac{1}{2}[J-2D\pm(h_{1}-2h_{2})].$
The only off-diagonal element is $\nu=J\Delta/\sqrt{2}$. The eigenvalues and
the corresponding eigenvectors of the Hamiltonian $\mathcal{H}$ are computed
as
$\displaystyle E_{1,2}=\frac{1}{2}[J+2D\mp(h_{1}+2h_{2})],$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
|\varphi_{1}\rangle=|\frac{1}{2},1\rangle,$
$\displaystyle|\varphi_{2}\rangle=|\frac{-1}{2},-1\rangle$ $\displaystyle
E_{3,4}=\frac{-1}{4}[J-2D+2h_{2}]\mp\frac{1}{4}\eta_{-},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
|\varphi_{3,4}\rangle=c_{1}^{\mp}|\frac{1}{2},0\rangle\mp
c_{1}^{\pm}|\frac{-1}{2},1\rangle$ $\displaystyle
E_{5,6}=\frac{-1}{4}[J-2D-2h_{2}]\mp\frac{1}{4}\eta_{+},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
|\varphi_{5,6}\rangle=c_{2}^{\pm}|\frac{1}{2},-1\rangle\mp
c_{2}^{\mp}|\frac{-1}{2},0\rangle.$
The normalization constants are
$\displaystyle
c_{1}^{\pm}=\frac{1}{\sqrt{2}}\sqrt{1\pm\frac{J-2D-2(h_{1}-h_{2})}{\eta_{-}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
c_{2}^{\pm}=\frac{1}{\sqrt{2}}\sqrt{1\pm\frac{J-2D+2(h_{1}-h_{2})}{\eta_{+}}}$
(11)
with the parameter $\eta_{\pm}=\sqrt{[J-2D\pm
2(h_{1}-h_{2})]^{2}+8(J\Delta)^{2}}$.
The thermal density matrix for the mixed spin-(1/2, 1) Heisenberg dimer is
$\displaystyle\varrho(T)=\frac{1}{\mathcal{Z}}\exp{\left(-\beta\mathcal{H}\right)}=\frac{1}{\mathcal{Z}}\sum_{i=1}^{6}p_{i}|\varphi_{i}\rangle\langle\varphi_{i}|,$
(12)
where $\beta=1/k_{B}T$ and it can be calculated as
$\displaystyle\varrho(T)=\frac{1}{\mathcal{Z}}\begin{pmatrix}\varrho_{11}&0&0&0&0&0\\\
0&\varrho_{22}&0&\varrho_{24}&0&0\\\ 0&0&\varrho_{33}&0&\varrho_{35}&0\\\
0&\varrho_{42}&0&\varrho_{44}&0&0\\\ 0&0&\varrho_{53}&0&\varrho_{55}&0\\\
0&0&0&0&0&\varrho_{66}\\\ \end{pmatrix},$ (13)
where the matrix elements are
$\displaystyle\varrho_{11}={\frac{1}{Z}}\mathrm{e}^{\frac{-\beta}{2}(J+2D-(h_{1}+2h_{2}))},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\varrho_{66}={\frac{1}{Z}}\mathrm{e}^{\frac{-\beta}{2}(J+2D+(h_{1}+2h_{2}))}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\varrho_{22}={\frac{1}{Z}}\mathrm{e}^{\frac{\beta}{4}(J-2D+2h_{2})}\left[\text{cosh}\left(\frac{\beta\eta_{-}}{4}\right)-\frac{(J-2D-2(h_{1}-h_{2}))}{\eta_{-}}\text{sinh}\left(\frac{\beta\eta_{-}}{4}\right)\right],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\varrho_{33}={\frac{1}{Z}}\mathrm{e}^{\frac{\beta}{4}(J-2D-2h_{2})}\left[\text{cosh}(\frac{\beta\eta_{+}}{4})+\frac{(J-2D+2(h_{1}-h_{2}))}{\eta_{+}}\text{sinh}(\frac{\beta\eta_{+}}{4})\right],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\varrho_{44}={\frac{1}{Z}}\mathrm{e}^{\frac{\beta}{4}(J-2D+2h_{2})}\left[\text{cosh}(\frac{\beta\eta_{-}}{4})+\frac{(J-2D-2(h_{1}-h_{2}))}{\eta_{-}}\text{sinh}(\frac{\beta\eta_{-}}{4})\right],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\varrho_{55}={\frac{1}{Z}}\mathrm{e}^{\frac{\beta}{4}(J-2D-2h_{2})}\left[\text{cosh}(\frac{\beta\eta_{+}}{4})-\frac{(J-2D+2(h_{1}-h_{2}))}{\eta_{+}}\text{sinh}(\frac{\beta\eta_{+}}{4})\right],\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\varrho_{24}=\varrho_{42}=\frac{-\sqrt{8}(J\Delta)}{Z\eta_{-}}\mathrm{e}^{\frac{\beta}{4}(J-2D+2h_{2})}\text{sinh}\left(\frac{\beta\eta_{-}}{4}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ and
$\displaystyle\varrho_{35}=\varrho_{53}=\frac{-\sqrt{8}J\Delta}{Z\eta_{+}}\mathrm{e}^{\frac{\beta}{4}(J-2D-2h_{2})}\text{sinh}\left(\frac{\beta\eta_{+}}{4}\right).\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
The partition function of the system is given by
$\displaystyle\mathcal{Z}=2\left[{\mathrm{e}^{\frac{-\beta(J+2D)}{2}}\text{cosh}\left(\frac{\beta(h_{1}+2h_{2})}{2}\right)+\mathrm{e}^{\frac{\beta(J-2D)}{4}}\left[\mathrm{e}^{\frac{\beta
h_{2}}{2}}\text{cosh}\left(\frac{\beta\eta_{-}}{4}\right)+\mathrm{e}^{\frac{-\beta
h_{2}}{2}}\text{cosh}\left(\frac{\beta\eta_{+}}{4}\right)\right]}\right].$
(14)
## 4 Results and Discussions
In this section, we study the bipartite thermal quantum correlations
quantified by MIN and fidelity-based MIN (F-MIN) of the mixed spin Heisenberg
dimer and compare with that of entanglement. Using Eqs. (4), (7), and elements
of the density matrix analytically, we have computed MIN and F-MIN of the
above the mixed spin system. It can be recalled that the entanglement of the
above physical system is already studied in Ref.[45].
### 4.1 Quantum correlation in thermal states
First, we study the influence of the magnetic field on quantum correlation for
different temperatures for two different values of the uniaxial single-ion
anisotropy such as $D/J=-0.5\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ 1.5$. For this purpose, we fix the values
$g_{1}=g_{2}=2$, $\Delta=1$ . In general, the magnetic field and thermal
fluctuations suppress the degree of quantumness in the interacting spin
systems. In Fig. 1, we illustrate the behaviors of quantum correlation
measures as a function of the magnetic field for a fixed single ion anisotropy
parameter. Initially, we observe that the MINs remain constant when we
increase the magnetic field at sufficiently low temperatures, then drop to
zero at a critical magnetic field. On the other hand, the entanglement
initially increases unconventionally with the magnetic field due to Zeeman’s
splitting of two energy levels [45].
Figure 1: Thermal quantum correlations quantified by (a) & (b) MIN and (c) &
(d) F-MIN as a function of external magnetic field for different temperatures.
The fixed parameters $g_{1}=g_{2}=2,\Delta=1,(a)\leavevmode\nobreak\
\&\leavevmode\nobreak\ (c)\leavevmode\nobreak\
D/J=-0.5,(b)\leavevmode\nobreak\ \&\leavevmode\nobreak\
(d)\leavevmode\nobreak\ D/J=1.5$.
To understand the role of temperature on MINs, they are plotted for different
temperatures in Fig. 1. In this connection, we observe that increase of
temperature causes monotonic decrease of quantum correlation from the maximum
value to zero in the mixed spin system. At higher temperatures, the
correlation between the spins is very small and decreases with the increase of
magnetic field. In addition, the comparison of Fig. 1(a) and Fig. 1(b) drives
home the point that the nonzero MIN region increases with the increase of the
single-ion anisotropy parameter $D$. One observes a similar functional
behavior for F-MIN.
To attain a deeper insight into the influence of uniaxial single-ion
anisotropy $D$ on the thermal quantum correlation, in Fig. 2, we plot the
densities of MIN as a function of magnetic field and temperature for a given
value of $g$-factors. Like entanglement, here also, we observe that the spins
are strongly correlated at low temperature and weak magnetic field regions. It
is pretty obvious that MINs capture more quantumness between the spins unlike
entanglement. Further, we observe that the parameter $D$ increases the quantum
correlation implying that the enhancement of $D$ can increase the threshold
values of temperature and external magnetic field. In other words, the
parameter $D$ introduces the correlation between the spins. Further, we notice
that the single-ion anisotropy induces the correlation in the parametric space
where there is no correlation between the spins and strengthens the
correlation in the parametric space if the spins are already correlated
similar to $DM$ interaction [28]. In the asymptotic limit, all eigenstates
would be given by eigenvectors with a single separable basis state vector
without any quantum superposition, i.e., without any quantum correlation. This
implies that the increase of quantum correlation by single-ion anisotropy $D$
is not a generic feature.
Figure 2: Density of MIN (upper panel) and F-MIN (lower panel) as a function
of external magnetic field and temperature (a) & (d) $D/J=-1.5$, (b) & (e)
$D/J=0$, (c) & (f) $D/J=1.5$. The fixed parameters are $g_{1}=g_{2}=2$ and
$\Delta=1$.
Next, we analyze the impact of difference of the $g-$factors on quantum
correlations. Here, we consider the difference of the g-factors
$|g_{1}-g_{2}|=0.2$ under two criteria such as $g_{1}<g_{2}$ and
$g_{1}>g_{2}$. To understand the effects of g-factor on MINs, we have plotted
the measures as a function of magnetic field for different temperatures for a
given single-ion anisotropy parameter. Even for a small difference of
g-factors, the contribution of g-factor turns out to be subtle. For
$g_{1}<g_{2}$ ( $g_{1}>g_{2}$ ), the behaviors of MIN and F-MIN are
illustrated in solid (dashed) lines in Fig. 3. When $g_{1}<g_{2}$ and
$D=-0.5$, we notice that the measures increase unconventionally with magnetic
field at low temperature $T=0.1$. For $g_{1}<g_{2}$, MINs are found to
decrease at $T=0.1$. The significant role of difference of g-factors is more
pronounced at low temperatures and the increase of temperatures negates the
impact of difference of g-factors. Nevertheless, it should be pointed out that
the negativity tends to the same asymptotic value in the zero-field limit as
well as at high magnetic fields while the most pronounced differences can be
detected at low magnetic fields. Again, it can be noticed that the single-ion
anisotropy parameter induces and strengthens the quantum correlation between
the mixed spins system.
### 4.2 Quantum Correlation in CuNi Complex
In this section, we study the quantum correlation of the heterodinuclear
complex CuNi. It is worth pointing at this juncture that the magnetic
properties of the above mixed spin (1/2,1) Heisenberg dimer have been
experimentally predicted through the CuNi compound and verified [31, 44]. In
the following, we therefore invoke the same set of model parameters to make
the relevant theoretical prediction for the bipartite entanglement of the CuNi
dimeric compound. The reported parameter values of the CuNi compound are
$J/k_{B}=141K$ and g-factors of $\text{Cu}^{2+}$ and $\text{Ni}^{2+}$ are
$g_{1}=2.20$ and $g_{2}=2.29$ respectively [31]. Similarly, the other
parametric values reported are $J/k_{B}=121K$, $g_{1}=2.09$ and $g_{2}=2.22$
[44]. Here, we use the experimental results given in [31].
Figure 3: Thermal quantum correlations quantified by MIN and F-MIN as a
function of external magnetic field (a) & (c) $D/J=-0.5$ (b) & (d) $D/J=1.5$
for different temperatures. The fixed parameters $g_{1}=g_{2}=2.2$ and
$\Delta=1$.
Figure 4: Thermal quantum correlations quantified by MIN and F-MIN as a
function of temperature and for different external magnetic field. The fixed
parameters are $g_{1}=2.2$, $g_{2}=2.29$, $J/k_{B}=141K$, and $D/k_{B}=0$.
We measure theoretically quantum correlation quantified by MIN and F-MIN in
CuNi complex through the mixed spin (1/2,1) Heisenberg dimer. Figure. 4
depicts the temperature dependence of MIN and F-MIN for different magnetic
fields. For this purpose, we have fixed the exchange interaction
$J/k_{B}=141K$, $g_{1}=2.20$ and $g_{2}=2.29$ based on the experimental
results[31]. We confirm that the correlation between Cu and Ni magnetic ions
have different variations depending on the strength of magnetic fields. At low
magnetic fields, MINs are maximum at zero temperature and decrease with
increase of temperature. The correlation vanishes at higher temperatures.
Interestingly, we observe that the F-MIN does exist even at room temperature
which may have wider ramifications from an experimental perspective. MINs are
more robust to thermal fluctuations and exist even at room temperature in
stark contrast to entanglement[45]. In addition, one understands that we can
sustain F-MIN even at higher temperatures compared to MIN. At higher magnetic
fields, say $B=150T$, both MIN and F-MIN become zero at zero temperature and
the correlation measures are unconventionally induced with the increase of
temperature which is in stark contrast to what is being observed in spin
systems [49]
Figure 5: Thermal quantum correlations of CuNi complex quantified by MIN and
F-MIN as a function of external magnetic field for different temperatures. The
fixed parameters are $g_{1}=2.2$, $g_{2}=2.29$, $J/k_{B}=141K$, and
$D/k_{B}=0$.
The correlation between the magnetic ions are also plotted as a function of
magnetic field for different temperatures for the same set of parameters used
in Fig. 5. At zero magnetic field, the thermal correlation between Cu and Ni
magnetic ions is maximum and remains constant upto a critical magnetic field
and then drops to zero. In addition, we notice that increase of temperature
also diminishes the correlations. In Fig. 6, we show the variation of
densities as a functions of magnetic field and temperature. The quantum
correlation measures exist at higher temperatures compared to entanglement.
Figure 6: Thermal quantum correlations of CuNi complex quantified by MIN and
F-MIN as a function of external magnetic field and temperature. The fixed
parameters $g_{1}=2.2$, $J/k_{B}=141K$, $g_{2}=2.29$, $\Delta=1$ and
$D/k_{B}=0$.
## 5 Conclusion
To summarize, we have studied the thermal quantum correlation quantified by
the measurement-induced nonlocality (MIN) in the mixed spin (1/2,1) Heisenberg
dimer which have been experimentallly realized in heterodinuclear complex. We
have considered two kinds of MIN such as Hilbert-Schmidt MIN and fidelity
based MIN and shown that the correlation depends on the system parameters. It
is also noticed that the MIN can be enhanced by intrinsic parameters such as
single-ion anisotropy and exchange coupling. On the other hand, the extrinsic
parameters such as magnetic field and temperature decreases the quantum
correlation. Under suitable parametric restrictions, we highlight that the
both the measures increase unconventionally with the magnetic field due to
Zeeman’s splitting.
In addition, we have also investigated the quantum correlation in
heterodinuclear complex CuNi which provides the experimental realization of
mixed spin-(1/2, 1) Heisenberg dimer. While comparing with the entanglement,
MIN can survive relatively at higher temperatures $(T=300K)$ and magnetic
fields $(B=150T)$. The robustness of MIN against thermal fluctuations and
magnetic field may offer huge potential in quantum information processing
through the CuNi heterodinuclear complex.
The density matrix elements of the physical system are realizable in terms of
spin observables and the fidelity is also measureable using quantum circuits
[48]. Hence, we believe that our results offer more insight into the
development of quantum technology with regard to information processing and
its practical applications.
## Acknowledgment
SB and RR wish to thank Council of Scientific and Industrial Research (CSIR),
Government of India for financial support under Grant No. 03(1456)/19/EMR-II.
RM acknowledges the financial support received from Czech Technical University
in Prague, Czech Republic under research grant 122-1225204D002.
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|
[a]M. Neumann
# A machine learning approach to the classification of phase transitions in
many flavor QCD
F. Karsch A. Lahiri C. Schmidt
###### Abstract
Normalizing flows are generative machine learning models which can efficiently
approximate probability distributions, using only given samples of a
distribution. This architecture is used to interpolate the chiral condensate
obtained from QCD simulations with five degenerate quark flavors in the HISQ
action. From this a model for the probability distribution of the chiral
condensate as function of lattice volume, quark mass and gauge coupling is
obtained. Using the model, first order and crossover regions can be classified
and the boundary between these regions can be marked by a critical mass. An
extension of this model to studies of phase transitions in QCD with variable
number of flavors is expected to be possible.
## 1 Introduction
Almost 40 years ago Pisarski and Wilczek argued, that the chiral phase
transition for vanishing masses and three or more flavors ($N_{f}$) should be
of first order [1]. To this day, the search for the orderness of the
transition has been quite inconclusive. No evidence for a first order
transition in three flavor QCD has been found so far in lattice QCD
calculations [2]. A recent work on this by Cuteri et al. finds that in the
continuum limit there is no first order transition for light quark masses
($m_{l}$) for all $N_{f}\leq 6$ [3], although first order transitions can be
found at non-zero $m_{l}$ on lattices with finite lattice spacing.
Figure 1: A sketch of a possible Columbia plot for mass-degenerate quarks in
the $N_{f}$-$m_{l}$ plane, assuming a critical number of flavors $N_{f\\!,c}$
between 3 and 4. Every point represents a phase boundary. The vertical line
marks the measurements done in this work.
In Fig. 1 we show a sketch of the phase diagram in the $N_{f}$-$m_{l}$ plane
for the situation corresponding to lattices of fixed temporal extent
$N_{\tau}$. We expect to find a region of first order phase transitions, which
is shown in the lower right corner. In this work, we try to find a quark mass
value in this region, and while keeping $N_{f}$ fixed increase the mass to
find the $Z(2)$ line marking the border to the crossover region.
## 2 Lattice setup and observables
The first order signal of the chiral phase transition becomes stronger with
decreasing quark mass, on larger volumes and for larger number of flavors.
Unfortunately, all of these adjustments also increase the computational cost
of numerical simulations using the Rational Hybrid Monte Carlo (RHMC)
algorithm. We thus have to be quite careful with our choice of parameters.
Our calculations have been performed in five-flavor QCD ($N_{f}=5$) using the
HISQ action with quark masses in the range $0.001\leq m_{l}\leq 0.016$ and
gauge couplings $\beta=4.5-5.4$. We used 4-dimensional lattices,
$V=N_{\sigma}^{3}N_{\tau}$, with temporal extent $N_{\tau}=6$ and spatial
volumes $N^{3}_{\sigma}=16^{3}-24^{3}$. The partition function for these
systems is given by
$\displaystyle
Z(N_{f},\beta,m_{l})=\int\mathcal{D}U_{\mu}(\text{det}\,M[U_{\mu},m_{l}])^{N_{f}/4}e^{-\mathcal{S}[U_{\mu}]}\;,$
(1)
where $M$ is the staggered fermion matrix and $S[U_{\mu}]$ denotes the gauge
action given in terms of gauge field variables $U_{\mu}$. The number of
flavors, $N_{f}$, can easily be generalized to a continuous number, losing
properties of a local quantum field theory in the process [4]. The chiral
condensate, which is the only observable we are going to discuss here, is
defined as
$\displaystyle\langle\bar{\psi}\psi\rangle=\frac{1}{4N_{\sigma}^{3}N_{\tau}}\expectationvalue{\tr
M^{-1}},$ (2)
Figure 2: Time histories and respective histograms of the chiral condensate
for two different masses at $N_{\sigma}=24$ close to $\beta_{c}$ in a first
order region. Different colors indicate different streams.
We calculate the chiral condensate using several independent RHMC streams
generated with different starting conditions. Results for our smallest quark
mass at a value of the gauge coupling $\beta$ close to the transition
temperature are shown in Figure 2 (left). The right hand figure shows the
evolution of a single RHMC stream at a larger value of the quark mass.
In case of a first order phase transition, we expect to find two distinct
phases, resulting in two peaks in the histograms of the time histories. In the
infinite volume limit ($N_{\sigma}\rightarrow\infty$) the minimum between the
two maxima will become smaller resulting in two delta-function like peaks. On
smaller volumes the peaks broaden. Frequent changes from one peak region to
another during the RHMC evolution will populate the region between the two
maxima, up to a point where there are no more double peaks visible on small
lattices.
On the other hand, a less pronounced dip in the histogram, separating the two
peaks, allows for more transitions from one phase to the other, i.e. flips in
the time histories. In Figure 2, we can see that indeed the signal for a first
order transition weakens with increasing $m_{l}$, while the rate of flips
increases.
## 3 $\beta$-reweighting
Lattice QCD calculations typically are done at a few values of the gauge
coupling $\beta$. $\beta$-reweighting [5] is a popular method to interpolate
lattice results. Given measurements of the action $S$ and any observable $O$
for $R$ different $\beta_{m}$, it yields a continuous expectation value
$\displaystyle<\\!\\!O\\!\\!>\\!(\beta)=\frac{\sum_{S}O(S)P(S,\beta)}{\sum_{S}P(S,\beta)}\;,$
(3)
obtained with reweighting weights $P(S,\beta)$,
$\displaystyle
P(S,\beta)=\frac{\sum^{R}_{n=1}N_{n}(S)\,\text{exp}[S\beta]}{\sum^{R}_{m=1}n_{m}\,\text{exp}[S\beta_{m}-f_{m}]}\;,\quad\text{where}\quad\text{exp}[f_{m}]=\sum_{S}P(S,\beta_{m})\;,$
(4)
where $n_{m}$ denotes the total number of measurements made at $\beta_{m}$ and
$N_{m}(S)$ is the total number of action-values in a bin
$[S-\epsilon,S+\epsilon]$ around $S$ at $\beta_{m}$; $f_{m}$ is the free
energy at $\beta_{m}$. The weights defined in Eq. 4 are obtained self-
consistently by iterating. This method requires a large number of
measurements, performed at a large number of $\beta$-values, since $O(S)$ is
obtained via the 2D-histogram of the action and the observable we want to
reweight. Moreover, the action histograms obtained at the different
$\beta_{m}$ need to have a sufficiently large overlap.
The method can be extended to reweight a probability distribution of any
observable by reweighting each bin of the discretized distribution
individually. This approach is thus limited to data sets discretized in a set
of bins and only interpolates in $\beta$-direction.
Figure 3: Comparison of $\beta$-reweighting (left) and ML-reweighting (right).
Data points show results obtained from RHMC calculations in 5-flavor QCD,
while the curves are obtained from the $\beta$\- and ML-reweighting,
respectively.
In Figure 3 (left) $\beta$-reweighted data for the chiral condensate are
shown. The reweighting is done for the entire set of histograms at each mass,
but only the expectation values are shown, to obtain a compact plot. While
this yields reasonable results for the lowest masses, for larger $m_{l}$,
especially $m_{l}=0.006$, the $\beta$-reweighting obviously is over-fitting.
## 4 ML model
Normalizing flows are state-of-the-art tools for modeling probability
distributions in physical systems. We use a MAF (Masked Autoregressive Flow)
[6] model with eight MADE (Masked Autoencoder for Distribution Estimation) [7]
blocks. MADE networks have been especially designed to factorize a joint
probability distribution into a product of conditional probabilities. Using
less than eight MADE blocks caused problems with fitting the double peaks,
however, for fits in the crossover region a fewer number of MADE should be
sufficient. Compared to the classical reweighting, this method has the
advantage of allowing to interpolate in any parameter. In particular, there is
no need for overlapping distributions of the action density and the method is
able to process continuous data. However, in order to visualize the learned
probability distribution, we need to draw a large number of samples from our
model to fill a two dimensional histogram.
In the end, the model learns to transform a 2D-Gaussian distribution to
“measurements” of $(\bar{\psi}\psi,S)$, conditioned on the continuous
parameters $(N_{\sigma},m_{l},\beta)$. To avoid overfitting, we have
introduced penalty terms in the loss function, based on the L1- and L2-norms
of the parameters of the network, known as regularization. The regularization
is applied on a per-layer basis and the coefficients in front of the
regularization terms have been chosen as $l_{1}=l_{2}=0.0001$. The training
took approximately 4h on a NVIDIA V100 GPU. Evaluating the model was done for
all integer $N_{\sigma}\in[16,24]$, $\beta\in[4.5,5.4]$ in steps of $0.001$
and $m_{l}\in[0.001,0.006]$ in steps of $0.001$ and for the larger masses
$m_{l}\in[0.008,0.016]$ in steps of $0.002$. Inference took approximately
30sec per 1,000,000 measurements at each parameter combination. This allows us
to fit our entire data set (as shown in Table 1) with a single function
$p(\bar{\psi}\psi,S\,|\,N_{\sigma},m_{l},\beta)$, in contrast to the
$\beta$-reweighting, where we would need to do independent reweighting for
each mass and volume.
In Figure 3 (right), the ML-reweighted data are shown. While the interpolation
appears to be slightly under fitting for $m_{l}=0.002$, in the grand picture
we achieve a good fit. Compared to the $\beta$-reweighting, it is intuitive
that we get a better fit, since now the data points support each other also in
$m_{l}$ and $N_{\sigma}$-direction and not only in $\beta$. Of course we could
have fitted only a 1D distribution to the chiral condensate. However, we
included the action as well to stabilize the fit and enable easy comparison
with the $\beta$-reweighting approach.
Figure 4: Density plot in the $\bar{\psi}\psi$-$S$ plane for 1,000,000
evaluations of the model. Shown are results for a small (left) and large
(right) quark mass, corresponding to the first order and crossover regions,
respectively.
## 5 Results
$N_{\sigma}$ | 0.001 | 0.002 | 0.003 | 0.0035 | 0.004 | 0.0045 | 0.005
---|---|---|---|---|---|---|---
16 | 17201 | 18887 | 11526 | 0 | 18866 | 0 | 0
24 | 5294 | 83177 | 149885 | 25028 | 30571 | 19332 | 19352
$N_{\sigma}$ | 0.006 | 0.008 | 0.010 | 0.012 | 0.014 | 0.016 |
16 | 61382 | 61220 | 61456 | 61456 | 61256 | 61256 |
24 | 42762 | 82061 | 65140 | 13380 | 36574 | 36499 |
Table 1: Total number of RHMC measurements for each volume and mass, summed
over all available $\beta$ values, corresponding to approximately 300,000 GPUh
on a NVIDIA V100.
In Figure 4 we show directly the model output in form of a 2D histogram
(contour plot) in the $\bar{\psi}\psi$-$S$ plane. The two distinct phases
connected by a small band are clearly visible in the left hand figure, which
shows results for a small quark mass, while only a single phase seems to be
present in the right hand figure, which is for a large quark mass value.
Since we are mainly interested in the chiral condensate, we project the 2D
histogram on the chiral condensate axis and look at 1D histograms as shown in
Figure 5. The mean values of these distributions can again be compared to the
measured points, but this time for the whole data set, as shown in Figure 6.
Again, we point out, that the entire data set is described by a single fit
(with about 10,000 parameters).
Figure 5: $p(\bar{\psi}\psi$) with 250 histogram bins and 1,000,000
evaluations of the ML-model for a light quark mass in the first order region
(left) and a heavier mass in the crossover region (right).
Figure 6: $\langle\bar{\psi}\psi\rangle$ for $N_{\sigma}=24$ (left) and
$N_{\sigma}=16$ (right). The data points represent the RHMC measurements,
while the curves are taken from the interpolation generated by the ML model.
It is also possible to extract $\langle\bar{\psi}\psi\rangle$ as well as the
1D histograms of $\bar{\psi}\psi$ in a fine sampled $m_{l}$-$\beta$ plane.
This allows to determine the quark mass dependence of the double peaks seen in
Figure 5. They signal the occurrence of a first order phase transition, with
the right hand peak corresponding to the end of the symmetry broken phase, the
left hand peak corresponding to the symmetry restored phase and the region
between the peaks being the mixed phase. The corresponding phase
Figure 7: Phase diagram of 5-flavor QCD on lattices with fixed temporal
extent, $N_{\tau}=6$ in the $m_{l}$-$\beta$ plane.
diagram in the $m_{l}$-$\beta$ plane is shown in Figure 7. It suggests that
the first order region ends in a second order end point at about
$m_{l}^{c}\simeq 0.0045$. Clearly, as the gap between the peaks at low and
high $\beta$ becomes smaller larger lattices will be needed to resolve these
two peaks and establish a gap between them. In the next section we will
discuss a ML based approach to locate this end point.
## 6 EOS-meter
Petersen et al. have introduced the idea of using an ML image recognition
approach to classify phase transitions [8]. They used a convolutional neural
network (CNN) model to classify data sets obtained in heavy-ion collision. The
resulting density plots they called an Equation-of-State-meter. Recently, the
transformer model [9], a model solely based on attention mechanisms, has been
shown to outperform recurrent or convolutional neural networks in translation
tasks. Transformers are expected to generalize well to other tasks, including
image recognition applications. Since no CNNs are used, information on pixel
positions must be added artificially via a so-called positional encoding. Here
we have used a vision transformer based approach on density plots as shown in
Figure 8. We have labeled the histograms of the smallest masses, where a clear
gap was visible as “first order” while the histograms of the largest masses
were labeled as “crossover”.
Figure 8: Probability density plots used to train the EOS-meter. Each column
of pixels corresponds to 1000 evaluations of the model. $\beta_{c}(m_{l})$
does not need to be known exactly, as long as $\beta_{c}$ is within the plot
range.
“Firstorderness” and “crossoverness” are implemented as categories in one-hot-
encoding. During training, random translation in $\beta$-direction was
applied, since it makes the trained model more independent of our estimate of
$\beta_{c}$. Since Dropout can be used as a Bayesian Approximation to the
model uncertainty [10], we can show error bars on the determined
“firstorderness”. We also tried a more traditional CNN approach, which however
resulted in less sharp transitions.
Figure 9: An EOS-meter, the “firstorderness” of the chiral phase transition
plotted versus $m_{l}$ for different volumes. The vertical bars mark the cuts
between training and testing data.
The resulting EOS-meter is shown in Figure 9. We want to remark that even
though we can see some fluctuations, $m_{c}$ should not and does not depend on
$N_{\sigma}$. The critical masses marking the borders between first order and
crossover regions were extracted via logistic fits to the “firstorderness”
$\displaystyle f(m_{l})=\frac{1}{1+e^{k(m_{l}-m_{c})}}.$ (5)
From these fits, we can extract $m_{c}=0.005(1)$.
## 7 Conclusions
Normalizing flows appear to be a performant alternative to
$\beta$-reweighting. We achieve a good model of the $(\bar{\psi}\psi,S)$
distribution in $(N_{\sigma},m_{l},\beta)$ for our entire data range. The
model can be used to extract a fine enough sampling in the parameter range to
train an EOS-meter able to extract the “firstorderness” of the chiral phase
transition for $N_{f}=5$, making it possible to identify a critical mass
$m_{c}\approx 0.005(1)$ which marks the border between the first order and
crossover regions.
In order to use this model to extract the phase diagram of QCD with $N_{f}$
flavors in the continuum limit we need to use larger $N_{\tau}$ values. An
extension of the parameter set to $(N_{f},N_{\sigma},N_{\tau},m_{l},\beta)$ is
expected to be possible, but is going to require a large amount of training
data.
## Acknowledgments
This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG)
through the grant 315477589-TRR 211 and "NFDI 39/1" for the PUNCH4NFDI
consortium and the grant EU H2020-MSCA-ITN-2018-813942 (EuroPLEx) of the
European Union. All calculations have been performed on the Bielefeld
University GPU cluster and we thank members of the HPC.NRW team for their
support. We would also like to thank the ERuM-Data-Hub workshop “Conceptual
Advances in Deep Learning for Research on Universe and Matter” for sharing
many ideas on the choice of ML models.
## Software
We used SIMULATeQCD [11] for the RHMC calculations, and Keras [12] and
Tensorflow probability [13] to implement the ML models.
## References
* [1] R. D. Pisarski and F. Wilczek, _Remarks on the chiral phase transition in chromodynamics_ , _Phys. Rev. D_ 29 (1984) 338.
* [2] L. Dini, P. Hegde, F. Karsch, A. Lahiri, C. Schmidt and S. Sharma, _Chiral phase transition in three-flavor QCD from lattice QCD_ , _Phys. Rev. D_ 105 (2022) 034510 [2111.12599].
* [3] F. Cuteri, O. Philipsen and A. Sciarra, _On the order of the QCD chiral phase transition for different numbers of quark flavours_ , _Journal of High Energy Physics_ 2021 (2021) .
* [4] F. Cuteri, O. Philipsen and A. Sciarra, _QCD chiral phase transition from noninteger numbers of flavors_ , _Physical Review D_ 97 (2018) .
* [5] A. M. Ferrenberg and R. H. Swendsen, _Optimized monte carlo data analysis_ , _Phys. Rev. Lett._ 63 (1989) 1195.
* [6] G. Papamakarios, T. Pavlakou and I. Murray, _Masked autoregressive flow for density estimation_ , 1705.07057.
* [7] M. Germain, K. Gregor, I. Murray and H. Larochelle, _MADE: Masked autoencoder for distribution estimation_ , 1502.03509.
* [8] L.-G. Pang, K. Zhou, N. Su, H. Petersen, H. Stöcker and X.-N. Wang, _An equation-of-state-meter of quantum chromodynamics transition from deep learning_ , _Nature Commun._ 9 (2018) 210 [1612.04262].
* [9] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez et al., _Attention is all you need_ , 1706.03762.
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* [11] L. Mazur, the HotQCD collaboration et al., “SIMULATeQCD.” https://github.com/LatticeQCD/SIMULATeQCD, November, 2021.
* [12] F. Chollet et al., “Keras.” https://keras.io, 2015.
* [13] J. V. Dillon, I. Langmore, D. Tran, E. Brevdo, S. Vasudevan, D. Moore et al., _TensorFlow distributions_ , 1711.10604.
|
# SimCS: Simulation for Online Domain-Incremental Continual Segmentation
Motasem Alfarra1,2 Zhipeng Cai1 Adel Bibi3 Bernard Ghanem2 Matthias Müller1
1 Intel Labs. 2 King Abdullah University of Science and Technology (KAUST). 3
University of Oxford
###### Abstract
Continual Learning is a step towards lifelong intelligence where models
continuously learn from recently collected data without forgetting previous
knowledge. Existing continual learning approaches mostly focus on image
classification in the class-incremental setup with clear task boundaries and
unlimited computational budget. This work explores Online Domain-Incremental
Continual Segmentation (ODICS), a real-world problem that arises in many
applications, _e.g_., autonomous driving. In ODICS, the model is continually
presented with batches of densely labeled images from different domains;
computation is limited and no information about the task boundaries is
available. In autonomous driving, this may correspond to the realistic
scenario of training a segmentation model over time on a sequence of cities.
We analyze several existing continual learning methods and show that they do
not perform well in this setting despite working well in class-incremental
segmentation. We propose SimCS, a parameter-free method complementary to
existing ones that leverages simulated data as a continual learning
regularizer. Extensive experiments show consistent improvements over different
types of continual learning methods that use regularizers and even
replay.111Correspondance to<EMAIL_ADDRESS>This work was done
during an internship of the first author at Intel Labs.
## 1 Introduction
Supervised learning has been the go-to solution for many computer vision
problems [16, 29]. The large scale of available labeled data has been the key
factor for its success [27]. However, in many settings the training data is
not available all at once but generated sequentially over time. Moreover, the
distribution of the training data may vary gradually over time [6, 21],
_e.g_., images taken in winter with rain and snow versus images with clear
skies taken in the summer. Naively applying supervised learning in such a
setting suffers from _catastrophic forgetting_ [18], _i.e_., training a model
on new data of a different distribution worsens its performance on old data.
Continual learning (CL) attempts to address these issues by designing
algorithms that operate on continuous data streams and efficiently adapt to
new data while retaining previous knowledge. However, in the existing CL
literature [20, 8], methods are usually evaluated only on restricted problems
such as image classification with carefully crafted data streams that assume
non-overlapping tasks, _e.g_., the class-incremental setting where each task
corresponds to a fixed set of classes.
In this work, we define a more realistic setup inspired by real-world
applications. We study the problem of Online Domain-Incremental Continual
Learning for Semantic Segmentation (ODICS). This is an essential problem for
many applications where the perception system needs to be updated over time.
In this setting, the model is trained with a _limited computation and memory
budget_ at each time step on data sampled from a varying distribution. The
variation comes from domain shifts, _e.g_., data coming from a different
environment; the model has _no information about the domain boundaries_. This
setup mimics the practical scenario where labeled data from new scenes
(weather conditions, cities, _etc_.) arrive over time, _e.g_., when developing
a segmentation system for autonomous driving.
Despite the importance of this problem, it has received little attention in
recent years. The few prior arts [14, 23] for continual semantic segmentation
study the problem under two unrealistic assumptions. First, it is assumed that
the deployed model is aware of the _domain boundaries_ [15], _i.e_. the domain
change, in both training and testing. While this assumption simplifies the
problem, domain boundaries are often not available in real applications as the
transition between different domains is usually smooth or unknown. Second, the
model is permitted to make any number of training iterations over current
domain data, _i.e_., learning with unlimited computational budget [14, 15,
23]. This means that the model can pause the stream from revealing the data
from the new domain during training, while in realistic setups, streams
continuously and uninterruptedly reveal new data and remain agnostic to the
training status of the model [6].
Figure 1: Online Domain Incremental Continual Segmentation (ODICS) with
Simulated Data (SimCS). At each step $t$, ODICS reveals a batch of labeled
images $S_{t}$ with size $B_{t}$ from a certain domain, where different
domains are presented sequentially to the model. SimCS generates on the fly a
batch with size $B_{t}$ of simulated data $S_{\text{sim}_{t}}$ and aligns its
label space to the real stream. The concatenated batch of real and simulated
data is presented to the model to aid continual training and to mitigate
forgetting previously learnt domains.
To overcome these unrealistic assumptions and to move closer to practical
scenarios, we study the problem of online, _i.e_. limited computational
learning budget, domain-incremental continual learning for semantic
segmentation. We propose a new benchmark using public datasets captured from
different cities and different weather conditions and order them based on
acquisition time. We find that the domain shift in this benchmark is severe
enough to cause models to forget previously learned domains even though no new
classes are introduced throughout training. We benchmark regularization-based
methods [18, 2, 20] that are effective in mitigating forgetting in class-
incremental continual segmentation [14] and show that they fail in this
challenging setting. Meanwhile, although we observe that replay-based methods
[9] effectively mitigate forgetting learnt domains, they may not be feasible
due to privacy concerns, _e.g_., GDPR [1]. This is particularly a concern when
data is associated with different countries in which storing it for replay is
not permissible. Thus, we propose SimCS that uses photo-realistic simulated
data with “free” dense labels that can be generated on-the-fly during ODICS
(see Figure 1) to mitigate forgetting without violating privacy constraints.
In summary, our contributions are three-fold. (1) We propose online domain-
incremental continual learning for semantic segmentation (ODICS) and construct
a corresponding new benchmark evaluating several baselines and algorithms from
the literature. (2) We propose SimCS, which leverages simulated data for
continual learning. SimCS is parameter-free and orthogonal to existing
continual learning frameworks. We demonstrate the effectiveness of SimCS by
combining it with five different continual learning strategies showing
performance improvements across the board. (3) We conduct a comprehensive
analysis, showing that SimCS is robust to the choice of simulators, hyper-
parameters, and budget constraints in continual learning.
## 2 Related Work
Continual Learning. The main challenge in learning a sequence of tasks,
classes, or domains from a continuous stream of data is catastrophic
forgetting [34, 28]. Existing methods can be roughly categorized into two
directions, using either regularization or memory. _Regularization-based
methods_ add regularization terms into the training objective without using
previous data. The goal is to maintain parameters that are important to
remember previous knowledge [18, 2, 20]. Besides explicit regularization on
model parameters [18, 2], distilling the predictions of older models is also
widely used [20]. _Memory-based methods_ leverage historical data by storing
it in a replay buffer [22, 3]. This data is often used to regularize the
gradient of the optimizer [8] or directly mixed with new training data [9].
Most existing literature focuses on the class-incremental setup on simple
tasks such as image classification. In this work, we evaluate the current
progress in continual learning for semantic segmentation in the online domain-
incremental setup.
Continual Semantic Segmentation. Recently, class-incremental learning was
extended from image classification to semantic segmentation. [24, 7]. This was
done by presenting segmentation masks of the classes belonging to a given task
while treating the remaining classes as the background [14, 23]. More closely
related to our work, multi-domain incremental learning was analyzed in the
semantic segmentation task [15]. Despite the current progress, previous
methods assume knowledge of task boundaries at test time and unlimited
computational budget for training on each task [15]. There are many practical
applications like self-driving cars, where new data is generated constantly at
a high data rate [6] and without clear distinction between different tasks
[4]. To better study these scenarios, we propose the setup of online domain
incremental continual learning for semantic segmentation (ODICS), where a new
batch of data arrives at each time step and the model is only allowed limited
computation on each batch. The domain boundaries are not provided to the
model, but there might be sharp domain shifts between two time steps.
Simulators for Semantic Segmentation. Recent works have proposed several
simulators that generate fully annotated data for “free” such as CARLA[13] and
VIPER [30, 31]. Such simulators play a key role for different applications
such as autonomous driving [5] and visual navigation [19], where collecting
and annotating data is expensive and time consuming. More recently, simulated
data was leveraged for semantic segmentation [25, 5, 17] for better
performance and generalization. Nonetheless, the use of simulated data in
continual learning remains unexplored. In this work, we propose leveraging
simulated data generated on the fly to reduce forgetting in continual
learning.
## 3 Online Domain-Incremental Continual Segmentation (ODICS)
This section formally defines the problem of _O nline Domain-Incremental
Continual Segmentation (ODICS)_. ODICS aims to train a parametrized model
$f(\cdot|\boldsymbol{\theta})$ that maps an image $\mathbf{X}\in\mathcal{X}$
to a per-pixel class prediction $\mathbf{Y}\in\mathcal{Y}$. At each time step
$t\in\\{1,2,3,...,\infty\\}$ of ODICS, a batch of densely labeled images
$S_{t}=\\{\mathbf{X}_{i_{t}},\mathbf{Y}_{i_{t}}\\}_{i_{t}=1}^{B_{t}}\sim\mathcal{D}_{t}$
is revealed. Then, the model parameters $\boldsymbol{\theta}_{t}$ are updated
using $S_{t}$ and a limited computation budget before $t+1$. Unlike supervised
learning where the domain $\mathcal{D}_{t}$ does not change, $\mathcal{D}_{t}$
in ODICS may change, even drastically, during training. The goal of ODICS is
to obtain the model parameters $\boldsymbol{\theta}_{t}$ that perform well on
all previously seen domains, _i.e_., $\mathcal{D}_{1}$ to $\mathcal{D}_{t}$.
There are two key concepts in ODICS, online and domain-incremental. _Online_
refers to the limited computation budget, _i.e_., we cannot train a model from
scratch within each $t$. This is important for applications like autonomous
driving where new data is constantly revealed over time. _Domain-incremental_
refers to the fact that the label space $\mathcal{Y}$ remains constant
throughout training, _i.e_., only the distribution of $\mathbf{X}_{i_{t}}$ and
the ratio of different classes in $\mathbf{Y}_{i_{t}}$ can change over time.
In autonomous driving, the domain shift can come from different weather
conditions, cities, and cameras.
We focus on outdoor semantic segmentation in the context of self-driving cars.
To mimic practical scenarios, we construct the stream of data
$\\{S_{t}\\}_{t=1}^{\infty}$ from multiple domains by composing four different
standard benchmarks from the literature: CityScapes (CS) [11], Indian Driving
Dataset (IDD) [33], Berkeley Driving Dataset (BDD) [35], and Adverse Weather
Condition Dataset (ACDC) [32]; we treat each dataset as a different domain.
Note that each dataset is collected in a different country. CS was collected
in Germany, IDD in India, BDD in the United States, and ACDC in Switzerland
mimicking the realistic scenario of deploying models in different locations.
This diversity introduces a notion of domains based on geographical location.
Moreover, the choice of these domains introduces domain variations in terms of
weather conditions. For example, CS contains images with clear weather
conditions while ACDC has a variety of adverse weather conditions, _e.g_., fog
and rain. This adds another realistic aspect to our setup, since deployed
models experience such adverse conditions when deployed throughout the year.
All considered domains have an aligned label space, _i.e_. an identical set of
classes, alleviating any relabeling requirements; we choose CS as reference.
We construct the stream by concatenating all domains based on the year the
dataset was published, resulting in the following order: CS (2016) - IDD
(2019) - BDD (2020) - ACDC (2021), which mimics the nature of continual
learning where data generated earlier will be seen by the model first. For
completeness, we analyze the use of different domain orders in Sec. 5. As
typical in CL, we evaluate the performance of the model trained on the stream
on a held out test set from each domain. Please refer to Sec. 5 for further
details about the benchmarks and evaluation protocols.
## 4 Methodology
This section explores different CL training strategies for ODICS and describes
the proposed SimCS, which leverages simulated data to mitigate forgetting.
### 4.1 CL Training Strategies
We first start with the scenario where at any time step $t$ the model cannot
store, hence rehearse, any data from previous time steps (1 to $t-1$). As
discussed earlier, this captures the realistic constraint where data is
subjected to privacy restrictions (_e.g_. GDPR).
The simplest baseline in this case is applying the same optimization strategy
as in supervised learning at each time step, which we call _naive training_ in
this paper. Specifically, given the training data
$S_{t}=\\{\mathbf{X}_{i_{t}},\mathbf{Y}_{i_{t}}\\}$ at time step $t$, we
update the model by optimizing the following objective:
$\displaystyle\underset{\boldsymbol{\theta}_{t}}{\text{min}}\sum_{i_{t}}{\mathcal{L}(f(\mathbf{X}_{i_{t}}|\boldsymbol{\theta}_{t}),\mathbf{Y}_{i_{t}})},$
(1)
where $\mathcal{L}(\cdot)$ is the standard loss for semantic segmentation,
_e.g_., cross entropy. In the online setting, we apply a limited number of
(stochastic) gradient descent steps on $\boldsymbol{\theta}_{t}$. While this
training strategy usually suffers the most from forgetting in the class-
incremental setup [14], it was found to be more effective in domain-
incremental classification tasks [21]. We use this strategy as the simplest
baseline to compare against different types of CL methods.
_Regularization-based methods_ [18, 20] are a family of continual learning
methods extensively studied in image-classification. Instead of optimizing
(1), these methods update the model by optimizing:
$\displaystyle\underset{\boldsymbol{\theta}_{t}}{\text{min}}\sum_{i_{t}}{\mathcal{L}(f(\mathbf{X}_{i_{t}}|\boldsymbol{\theta}_{t}),\mathbf{Y}_{i_{t}})}+\lambda\mathcal{L}_{\text{reg}}(\boldsymbol{\theta}_{t},\boldsymbol{\theta}_{t-k},S_{t}),$
(2)
where $\mathcal{L}_{\text{reg}}(\cdot)$ is the regularization term used to
mitigate forgetting, which is algorithm-specific, and $\lambda$ is a
coefficient that controls the regularization strength. Note that only data
from the current step $S_{t}$ and possibly some cached historical models
$\boldsymbol{\theta}_{t-k}$ are used for regularization, _i.e_., no historical
data is used for optimization.
We also consider relaxing the constraint on storing old data and complement
our benchmark by comparing against _replay-based methods_ [9]. In particular,
we allow the model to store a few historical training samples in a small
replay buffer. During training, the old data provides a form of
regularization, _i.e_., we optimize:
$\displaystyle\underset{\boldsymbol{\theta}_{t}}{\text{min}}\sum_{i_{t}}{\mathcal{L}(f(\mathbf{X}_{i_{t}}|\boldsymbol{\theta}_{t}),\mathbf{Y}_{i_{t}})}+\mathcal{L}_{\text{rep}}(\boldsymbol{\theta}_{t},S_{\text{rep}_{t}}),$
(3)
where $\mathcal{L}_{\text{rep}}(\boldsymbol{\theta}_{t},S_{\text{rep}_{t}})$
computes the loss on a batch of data $S_{\text{rep}_{t}}$ sampled from the
replay buffer at step $t$. In the simplest form [9],
$\mathcal{L}_{\text{rep}}(\cdot)$ is the same as $\mathcal{L}(\cdot)$ but
computed on $S_{\text{rep}_{t}}$. Despite its simplicity, this approach is
effective in mitigating forgetting for image classification [9, 26].
### 4.2 CL with Simulation
In practice, naive training or regularization-based methods are often not
effective enough for continual learning due to the strongly biased training
data. Although replay-based methods are more effective, they are less
practical under privacy or memory constraints. To address this problem, we
take an orthogonal and unexplored path, which is using simulation data for
continual learning.
Simulation techniques have achieved impressive advancements recently,
especially for computer vision. For autonomous driving, state-of-the-art
simulators [13, 30] can generate densely labeled images of simulated driving
scenes on the fly. Using simulation for continual learning has several
advantages. First, we can obtain an infinite amount of high-quality, diverse,
and densely labeled images on the fly by running the simulator; we do not need
to store a large amount of data in memory. Second, since all data is
synthetic, there is no violation of privacy constraints.
Inspired by replay-based methods, we propose to use the loss on simulation
data as a regularization for continual learning. As shown in Figure 1, at each
time step of ODICS, we first generate a batch of labeled simulation data
$S_{\text{sim}_{t}}$ on the fly. Then, we update the model by optimizing the
following objective:
$\displaystyle\underset{\boldsymbol{\theta}_{t}}{\text{min}}\sum_{i_{t}}{\mathcal{L}(f(\mathbf{X}_{i_{t}}|\boldsymbol{\theta}_{t}),\mathbf{Y}_{i_{t}})}+\mathcal{L}_{\text{sim}}(\boldsymbol{\theta}_{t},S_{\text{sim}_{t}}).$
(4)
We set
$\mathcal{L}_{\text{sim}}(\boldsymbol{\theta}_{t},S_{\text{sim}_{t}})=\sum_{{\mathbf{X}_{j},\mathbf{Y}_{j}}\in
S_{\text{sim}_{t}}}{\mathcal{L}(f(\mathbf{X}_{j}|\boldsymbol{\theta}_{t}),\mathbf{Y}_{j})}$,
_i.e_., we compute the same loss on both real and simulated data, and sum them
together. While more complex strategies can be applied, we found that this
simple approach is effective as later shown in the experiments. We call this
method Simulation for Continual Segmentation (_SimCS_). There are few
challenges that need to be addressed to make this approach general and
effective.
Data Quality. It is unclear if the image rendering quality, scene scale, and
object variety play a major role for CL performance. To address this question,
we use two different simulators, _i.e_., CARLA [13] and VIPER [30]. Empirical
results (see Sec. 5 for more details) show that both simulators can be
effectively used in continual learning to mitigate forgetting for real data
from different environments. Hence, our approach is robust in terms of the
simulator choice.
Label Space. There are many options for defining class labels leading to
misalignment between different simulators and real-world datasets. For
example, CARLA has a single vehicle class but separate classes for road line
and road. Meanwhile, the real-world datasets have separate classes for cars
and trucks but only a single class for road. Hence, we merge or relabel the
segmentation masks generated by the simulator of choice to achieve the maximal
overlap with the label space of the real data. Then, we drop all other labels
as opposed to merging them into the background class. The full details of
relabeling the simulated data can be found in the appendix. As shown in Sec.
5, though some of the real-world classes are missing in the simulated data due
to non-overlapping label spaces, our approach is still effective. We expect
that SimCS has further potential when applied to more advanced simulators.
Data Quantity. It is also not clear how much simulation data is needed for
continual learning. Intuitively, using a large amount of simulated data could
bias the model to only perform well on the simulated data, while using a small
amount of data may only improve performance marginally. At each training
iteration of SimCS, the batch of simulated data is generated by randomly
setting simulator parameters, _e.g_. camera position, weather, time and
traffic conditions. To study the impact of the amount of simulated data on
performance, we explore varying the ratio between simulated and real data
during training. In the main experiments, we set the sim-real ratio to 1,
which provides a good trade-off between computation and performance
improvement. Our empirical analysis suggests that our approach further
benefits when increasing the sim-real ratio and remains effective within a
large range of this ratio.
## 5 Experiments
This section presents our experimental evaluation of ODICS. We first outline
our setup, and then benchmark several CL training methods and compare them to
the proposed SimCS. At last, we present a detailed analysis of different
components of SimCS.
### 5.1 Experimental Setup
We construct our benchmark by concatenating four different datasets as
domains, namely CS, IDD, BDD, and ACDC, as mentioned in Section 3. Throughout,
we use the term “domain” and “dataset” interchangeably. Following common
practice in semantic segmentation [14], we use 80% of the publicly available
data from each dataset for training and evaluate on the 20% held out test set
from each domain. This results in a stream containing roughly $17$K real
images for training and roughly $4.3$K images for testing. We follow standard
practice [14, 23] in reporting the mean Intersection over Union (mIOU) on the
held out test set from each domain. During our experiments, at each time step
$t$ of ODICS, the model is presented with a batch of real images $S_{t}$ of
size 8, _i.e_. $B_{t}=8~{}\forall t$. Before the next time step $t+1$, the
model is allowed to train on the batch using a fixed computational budget,
measured by the number $N$ of forward and backward passes. Unless stated
otherwise, we set $N=4$ throughout our experiments222We found empirically that
setting $N=4$ provides a good trade-off between preventing the model from
under-fitting and significantly increasing the computation. In the appendix we
provide results for different choices of $N$ with similar conclusions.. Once
data from $S_{t+1}$ is revealed, the older batch $S_{t}$ becomes unavailable
to the model unless replay is used. We evaluate all methods using the
benchmark introduced in Section 3. In our experiments, we use the DeepLabV3
architecture [10] pre-trained on ImageNet [12] (unless otherwise stated in the
pre-training experiments in Section 5.3), following [14]. Further details are
provided in the appendix.
We analyze five different types of training strategies. The baseline is _Naive
Training (NT)_ , _i.e_., optimizing Eq. (1). We also consider regularization-
based (Eq. (2)) and replay-based (Eq. (3)) CL algorithms. For regularization-
based methods, we consider _Elastic Weight Consolidation (EWC)_ [18], _Memory
Aware Synapses (MAS)_ [2], and _Learning without Forgetting (LwF)_ [20]. We do
not provide boundaries of dataset transitions except for regularization-based
methods during training, since they need this information to achieve
reasonable performance according to our empirical results. For each considered
regularizer, we set $\lambda=0$ in Eq. (2) when training on data from the
first domain and $\lambda>0$ for the other domains. We report the best results
for each regularizer cross-validated on different values of $\lambda$ leaving
the result for all values of $\lambda$ to the appendix. For replay-based
methods, we apply Experience Replay (ER) [9] with a replay buffer size of 800
images (along with their dense labels), throughout this section and leave the
ablations to the appendix.
Regarding our SimCS approach, we explore the use of data generated from CARLA
[13] and VIPER [30]. We generate on the fly simulated data from CARLA on the
most realistic town 10 by randomly setting the location and camera parameters.
On the other hand, with VIPER, we only sample (without replacement) from a
large pool of the publicly available pre-generated simulated data, since the
code to generate data on the fly is not available. We relabel the segmentation
masks of the simulated data to align with the labels of the real world
following the procedure described in Section 4.2. This results in 11 and 15
out of 19 overlapping classes between the simulated and real data for CARLA
and VIPER, respectively.
### 5.2 Main Results
Table 1: Performance Comparison under ODICS. We report the mIOU (%) of a model
trained on our benchmark and evaluated on each domain in the benchmark. We
also report the performance of SimCS-enhanced baselines by leveraging either
CARLA or VIPER. All methods are trained with $N=4$ iterations for each
received batch. The last row “Supervised” represents the performance of a
model trained on the entire stream for 30 epochs as a surrogate to upper bound
performance. _SimCS consistently improved the performance of all baselines on
all observed domains._
Method Domain | CS | IDD | BDD | ACDC | mIOU
---|---|---|---|---|---
NT | 40.1 | 37.9 | 35.1 | 48.9 | 40.5
\+ CARLA | 44.6 | 39.6 | 38.5 | 51.0 | 43.4
\+ VIPER | 45.4 | 43.8 | 40.0 | 50.4 | 44.9
EWC [18] | 41.5 | 38.8 | 35.9 | 47.9 | 41.0
\+ CARLA | 45.3 | 40.5 | 38.8 | 51.3 | 44.0
\+ VIPER | 45.1 | 43.4 | 40.9 | 50.9 | 45.1
MAS [2] | 41.4 | 37.1 | 34.6 | 48.2 | 40.3
\+ CARLA | 46.7 | 41.3 | 38.3 | 50.2 | 44.1
\+ VIPER | 45.8 | 43.5 | 38.8 | 49.1 | 44.3
LwF [20] | 44.5 | 41.9 | 34.6 | 46.3 | 41.8
\+ CARLA | 47.1 | 44.3 | 39.0 | 48.5 | 44.7
\+ VIPER | 46.7 | 46.7 | 38.5 | 47.9 | 45.0
ER [9] | 47.4 | 47.8 | 40.9 | 48.8 | 46.2
\+ CARLA | 48.4 | 48.5 | 43.2 | 50.8 | 47.7
\+ VIPER | 48.5 | 50.0 | 42.5 | 52.0 | 48.3
Supervised | 62.7 | 63.6 | 49.6 | 62.0 | 59.5
\+ CARLA | 62.8 | 63.7 | 49.7 | 62.9 | 59.7
\+ VIPER | 63.3 | 63.9 | 49.2 | 63.1 | 59.8
We start by analyzing the performance of different CL training strategies in
ODICS. Table 1 summarizes the results of a model after being trained on our
benchmark and evaluated on each observed domain, where the last column reports
the mIOU across all domains. The last row (Supervised) reports the performance
of a model trained for 30 epochs using standard supervised learning on all
data of the stream, representing a surrogate upper-bound performance.
Unlike the class-incremental setup [14], the simple NT in ODICS enjoys an on-
par performance to all considered regularization-based methods. For example,
while MAS outperforms NT on earlier domains, _e.g_., CS, the overall
performance degrades to 40.3% compared to 40.5% mIOU for NT. The most
effective regularization-based method is LwF, which only outperforms NT by
1.3%. This suggests that further work is needed to develop regularization
techniques for this more realistic domain-incremental setup. Meanwhile,
rehearsing previously seen examples through ER consistently outperforms other
baseline methods in all domains. This conclusion is consistent with previous
results in image classification [21, 22, 9], as storing real examples in a
replay buffer provides a simple but effective regularization for continual
learning.
Next, we analyze the effectiveness of including simulated data to all
considered training strategies. To apply our approach on methods other than
NT, we simply add $l_{\text{sim}}(\boldsymbol{\theta}_{t},S_{\text{sim}_{t}})$
in Eq. 4 to the objective of each method. As shown in Table 1, across _all_
domains and _all_ considered training schemes, SimCS provides consistent and
significant performance improvements. For example, adding VIPER to the
continual learning schemes reduces forgetting of NT and LwF on CS and IDD,
respectively, by $\sim$ 5% (from 40.1 to 45.4 and from 41.9 to 46.7). Further,
leveraging simulated data improves the strongest baseline (ER) by a notable
2%. This result shows that simulation data can be leveraged as an effective
regularizer for mitigating forgetting in continual learning. Moreover,
different simulators provide different margins of improvements. For instance,
while using either simulator (CARLA or VIPER) improves performance, simulated
data generated from VIPER often produces larger gains. This observation can be
attributed to several factors. For example, different simulators vary in
photo-realism; in addition, their labels may be more or less aligned with
real-word data labels.
### 5.3 Pre-training on Simulated Data Provides Further Improvements
Table 2: Performance Comparison under VIPER Pretraining. We compare the
performance of NT when pretrained with VIPER (on top of ImageNet). We further
boost NT + VIPER pretraining with SimCS (with VIPER) during continual
learning. _VIPER pretraining boosted the performance of both NT and NT+SimCS._
Method Dataset | CS | IDD | BDD | ACDC | mIOU
---|---|---|---|---|---
NT | 40.1 | 37.9 | 35.1 | 48.9 | 40.5
\+ VIPER Pretrain | 40.2 | 40.4 | 36.5 | 51.9 | 42.3
\+ VIPER SimCS | 47.9 | 43.0 | 41.8 | 54.2 | 46.7
In addition to using simulated data as a regularizer within the ODICS setup,
we try incorporating it in pre-training, since it can be made available even
before the start of the continual learning process. To that end, and before
commencing with ODICS, we first fine-tune ImageNet pretrained models with data
generated from VIPER. We generate $17$K synthetic images, which is equal to
the total number of real images presented in the continual setup, and train
our model for 30 epochs on this simulated data. We then perform ODICS and
compare against NT and NT + VIPER, where in the latter VIPER is used as a
regularizer during ODICS. We report the results in Table 2.
We observe that pre-training on simulated data further improves the
performance of a continual learner on all observed domains. We report an
improvement of $1.7\%$ on average across all observed domains when compared to
ImageNet pre-training. Although our model observed data generated from VIPER
in the pre-training phase, including simulated data in the continual learning
process further enhances the performance by 4% on average. It is worth
mentioning that this boosted version of NT surpasses the performance of the
best baseline ER by $0.5\%$ on average, without the need to store any new
additional data from previous domains during continual learning.
### 5.4 Impact of the Amount of Simulation Data
Figure 2: Effect of Varying Sim-Real Ratio on the Performance Gain. We analyze
the effect of varying the ratio between simulation and real data from
{$\nicefrac{{1}}{{4}}$, $\nicefrac{{1}}{{2}}$, 1, 2, 4, 5, 8, 10} on the
performance gain for each observed domain. We find that SimCS provides notable
performance improvement on a wide range of ratios ($\leq$ 5). However, for
very large Sim-Real ratios, the model becomes biased towards simulated data,
thus, degrading the performance on real data.
Figure 3: Comparison under different computational budgets. We allow NT,
NT+CARLA, and NT+VIPER different computational budgets for training on each
received batch from the stream, measured by the number of training iterations.
We measure the performance on each observed domain when varying the budget to
{1, 2, 3, 4,6,8,10} training iterations. Our approach; SimCS, provides
consistent and large performance gains irrespective of the allowed number of
training iterations and the choice of simulator.
In Sections 5.2 and 5.3, we used a 1:1 ratio between simulated and real data
to form a mini-batch during continual learning. We analyze the effect of
varying this sim-real ratio on performance in Figure 2, where we report the
performance of NT, NT+CARLA, and NT+VIPER with ratios of
$\\{\nicefrac{{1}}{{4}},\nicefrac{{1}}{{2}},1,2,4,5,10\\}$. The results show
that leveraging simulated data provides consistent performance improvements
for a wide range of sim-real ratios. As long as this ratio was smaller than 5,
_i.e_., we generate 5 batches from the simulator for each received batch from
the real-world stream, our approach provides significant gains across all
observed domains. Note that however, for larger sim-real ratios, _e.g_. 10,
the training is biased towards simulated data and thus harms the performance
on the real-world stream. The is exemplified in Figure 2, where the
performance of SimCS with sim-real ratios of 8 and 10 degrades the mIOU of NT
on 3 out of 4 domains. Furthermore, VIPER outperforms CARLA across most sim-
real ratios; this is consistent with previous observations in Table 1.
### 5.5 Impact of the Computational Budget
In Section 5.2, all methods are given a fixed computational budget of $N=4$
forward and backward passes for each received batch. In this section, we
analyze the performance with different computational budgets. We conduct
experiments with $N\in\\{1,2,3,4,6,8,10\\}$ for NT, NT+CARLA, and NT+VIPER,
and report results on each observed domain in Figure 3. We observe that small
computational budgets might result in an under-fitting model while larger
budgets ($N=10$) cause the model to over-fit to the last domain, thus,
increasing forgetting on previous domains. Nonetheless, SimCS provides a
stable performance improvement across all considered computational budgets
irrespective of the choice of the simulator.
Moreover, and in contrast to prior CL literature, we perform comparisons for
when the computational budget is normalized for all methods, particularly when
comparing NT against NT+VIPER. Since NT+VIPER uses a $1:1$ sim-real ratio in
the batch, comparing it with NT using the same computational budget ($N=4$)
might not be fair for NT. Effectively, NT+VIPER with $N=4$ is equivalent to
$N=8$ due to the additional simulated data. Our results in Figure 3 show that
even when normalizing the computational budget, SimCS still outperforms the
baseline in ODICS. For example, when NT+VIPER is allowed $N=4$ steps of
computation, it achieves an mIOU of 43.8% and 40.0% on IDD and BDD,
respectively, while NT with $N=8$ achieves 39.1% and 35.0% on the same
datasets.
### 5.6 SimCS Improves Forward Transfer
Figure 4: Forward and backward transfer under different domain orders. We
analyze the forward and backward transfer during ODICS of both NT and NT+VIPER
under different domain orders. The x-axis represents the observed domain
within the stream while the y-axis shows the domain, on which we are
evaluating the model. SimCS with VIPER improves both the forward (lower
triangular) and backward (upper triangular) transfer in ODICS under different
domain orders.
In earlier sections, we analyzed the effectiveness of the use of simulated
data in mitigating forgetting previously observed domains. Here, we conduct a
more fine-grained analysis. In particular, we analyze the performance on all
previous domains after the training of every domain. Figure 4 summarizes this
analysis, where the horizontal axis represents the last observed domain within
the stream, while the vertical axis represents the domain we evaluate the
model on. The last column corresponds to the results in Table 1, where we only
report the performance of the final model after the last domain. At last and
for completeness, we extend our analysis to different domain orders to include
(ACDC-CS-IDD-BDD), (BDD-ACDC-CS-IDD), and (IDD-BDD-ACDC-CS). Note that in each
matrix, the performance difference ($r-d$) between a diagonal element $d$ and
an element $r$ to the right of it in the same row reflects the forgetting or
backward transfer (smaller is better). On the other hand, the performance
difference $d-l$ between a diagonal element $d$ and an element $l$ to the left
of it in the same row reflects the forward transfer.
First of all, including simulated data in the ODICS setup not only reduces
forgetting, but also improves the forward transfer. For example, including
simulated data from VIPER boosts the forward transfer to ACDC in the (CS-IDD-
BDD-ACDC) setup from 34.9% to 39% when trained on (CS-IDD-BDD). We note that
this result is not specific to the order at which the considered domains are
presented. For instance, our approach improves the forward transfer from 20.4%
to 29% on BDD when trained on (ACDC-CS) in the (ACDC-CS-IDD-BDD) setup.
Meanwhile, differently ordered streams result in larger variations in both
forgetting and forward transfer. For example, the performance on CS drops from
40.1% to 36.9% when changing the setup from (CS-IDD-BDD-ACDC) to (ACDC-CS-IDD-
BDD). Furthermore, the performance on all domains (except IDD) drops
significantly when IDD is the last domain. Specifically, in the (BDD-ACDC-CS-
IDD) setup, the forgetting on ACDC is a significant 10.6% mIOU. This can be
attributed to the distribution shift that IDD has compared to all other
domains. Please refer to the appendix for further analysis on different domain
orders.
Summary of Findings Our experimental results highlight the following
observations. (i) While effective in the class-incremental setup,
regularization-based CL approaches have limited impact when evaluated in ODICS
(Sec 5.2). (ii) SimCS demonstrates consistent performance improvements in
ODICS when combined with any of the five considered CL training strategies
(Sec. 5.2). (iii) SimCS can be further boosted by pre-training on simulated
data (Sec. 5.3). (iv) SimCS is robust to the choice of simulator and the
amount of simulated data used in ODICS (Sec. 5.4). (v) SimCS provides
significant performance gains even when compensating for the computational
budget needed to process the simulated data (Sec. 5.5). (vi) SimCS
consistently improves both backward and forward transfer in ODICS (Sec. 5.6).
## 6 Conclusions
In this work, we explored the problem of online domain-incremental continual
learning for semantic segmentation (ODICS), which is important for
applications such as autonomous driving. We analyzed the limitations of
several continual learning training strategies in this setup and proposed
SimCS, an orthogonal approach that leverages simulated data generated on the
fly to reduce catastrophic forgetting. Through extensive experimental
evaluation, we found that SimCS provided consistent improvements to different
continual learning algorithms and is robust to the problem setup and hyper-
parameter choices.
It is worth noting that while ODICS is a step towards analyzing realistic
scenarios for continual learning, there are additional setups with practical
applications. For instance, ODICS mimics the scenario of autonomous systems
that are sequentially deployed throughout different locations and weather
conditions. Another interesting scenario would be to consider a fleet of
autonomous systems simultaneously deployed across several locations. The data
stream would contain multiple domains at the same time with significant domain
shifts. We provide a preliminary definition and study of this setting in the
supplement. We find that SimCS also provides performance gains in this case,
but leave a detailed analysis for future work.
## References
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## Appendix A CL with Simulation
In Section 4.2, we outlined the main details on how to include simulated data
during ODICS. Here, we elaborate on the implementation details such as the
relabeling process.
### A.1 Relabeling CARLA
Table 3: Relabeling CARLA into CS label space. Index Value | CARLA Label | CS Label
---|---|---
0 | Unlabeled | -
1 | Building | Building
2 | Fence | Fence
3 | Other | -
4 | Pedestrian | Person
5 | Pole | Pole
6 | Road Line | Road
7 | Road | Road
8 | Side Walk | Side Walk
9 | Vegetation | Vegetation
10 | Vehicles | Car
11 | Wall | Wall
12 | Traffic Sign | Traffic Sign
13 | Sky | Sky
14 | Ground | -
15 | Bridge | -
16 | Rail Track | -
17 | Guard Rail | -
18 | Traffic Light | Traffic Light
19 | Static | -
20 | Dynamic | -
21 | Water | -
22 | Terrain | Terrain
We deployed the stable v: 0.9.12 from CARLA to generate the simulated data for
SimCS. We employ our procedure in Section 4.2 while we relabel the generated
data using Table 3. We note here that all dropped labels (marked as ’-’) are
not included in the loss calculation nor in model updates.
### A.2 Relabeling VIPER
Table 4: Relabeling VIPER into CS label space. Index Value | VIPER Label | CS Label
---|---|---
1 | Ambiguous | -
2 | Sky | Sky
3 | Road | Road
4 | Side Walk | Side Walk
5 | Rail Track | -
6 | Terrain | Terrain
7 | Tree | -
8 | Vegetation | Vegetation
9 | Building | Building
10 | Infrastructure | -
11 | Fence | Fence
12 | Billboard | -
13 | Traffic Light | Traffic Light
14 | Traffic Sign | Traffic Sign
15 | Mobile barrier | -
16 | Fire Hydrant | -
17 | Chair | -
18 | Trash | -
19 | Trash Can | -
20 | Person | Person
21 | Animal | -
22 | Bicycle | -
23 | Motorcycle | Motorcycle
24 | Car | Car
25 | Van | Car
26 | Bus | Bus
27 | Truck | Truck
28 | Trailer | -
29 | Train | -
30 | Plane | -
31 | Boat | -
We leveraged the available simulated data333VIPER data: http://playing-for-
benchmarks.org/ released officially. We relabel each pixel by using Table 4.
Similar to the previous section, we ignore all dropped labels from any loss
calculations.
## Appendix B Experiments
Next, we present additional experimental ablations for ODICS and SimCS. We
first outline additional details about our experimental setup. Then, we we
present results with different computational budgets for all methods followed
by ablating the effect of varying the memory size when using ER. Further and
for completeness, we report results for regularization based methods under
different values of $\lambda$.
### B.1 Experimental Setup
During our experiments we set the learning rate to $7\times 10^{-3}$
throughout all of our experiments, following [14]. For regularization based
methods, we set $\lambda=1,10,50$ for MAS, EWC, and LwF, respectively. For
Experience Replay method, we deployed First In First Out (FIFO) algorithm for
updating our replay buffer. That is, recently received examples from the
stream will replace examples stored the the begining of the buffer.
### B.2 Experimenting with Different N
Table 5: Performance Comparison under ODICS. We report the mIOU (%) of a model
trained on our benchmark and evaluated on each domain in the benchmark. We
also report the performance of SimCS-enhanced baselines by leveraging either
CARLA or VIPER. All methods are trained with $N=1$ iterations for each
received batch. _SimCS consistently improved the performance of all baselines
on all observed domains._
Method Domain | CS | IDD | BDD | ACDC | mIOU
---|---|---|---|---|---
NT | 34.2 | 35.0 | 33.2 | 37.5 | 35.0
\+ CARLA | 37.9 | 37.8 | 36.5 | 42.5 | 38.7
\+ VIPER | 37.9 | 40.5 | 37.0 | 42.4 | 39.5
EWC [18] | 34.5 | 35.7 | 33.4 | 38.0 | 35.4
\+ CARLA | 37.4 | 37.0 | 36.7 | 41.5 | 38.2
\+ VIPER | 37.9 | 41.5 | 37.5 | 42.3 | 39.8
MAS [2] | 33.9 | 34.7 | 32.5 | 37.2 | 34.6
\+ CARLA | 36.0 | 36.2 | 35.0 | 41.2 | 37.1
\+ VIPER | 37.1 | 39.9 | 35.5 | 39.3 | 38.0
LwF [20] | 34.2 | 36.2 | 31.2 | 35.0 | 34.2
\+ CARLA | 36.3 | 40.9 | 35.1 | 38.7 | 37.8
\+ VIPER | 36.7 | 42.5 | 33.5 | 38.3 | 37.8
Throughout our experiments, we reported the results with a computational
budget of $N=4$. For completeness, we report here the results with $N=1$
mimicking a fast stream setup where the learner is allowed to do only one
training iteration on each received batch. Table 5 summarizes the results.
Similar to our earlier observations, SimCS provides consistent performance
improvements in the case $N=4$. This demonstrates the effectiveness of SimCS
on different streaming scenarios. At last, we note that the performance of all
methods with $N=1$ is significantly lower than with $N=4$. This can be
attributed to the complexity of the task and the limited number of training
data.
### B.3 Memory Size in ER
Table 6: Performance Comparison under Different Replay Buffer Sizes. We report
the mIOU (%) of a model trained ER with different buffer size reported between
parenthesis. We also report the performance of SimCS-enhanced baselines by
leveraging VIPER. All methods are trained with $N=4$ iterations for each
received batch. We observe that larger buffer sizes improves the performance
under the ODICS setups. Further, _SimCS consistently improved the performance
of all baselines on all observed domains._
Method Domain | CS | IDD | BDD | ACDC | mIOU
---|---|---|---|---|---
ER(200) | 46.9 | 46.2 | 40.9 | 49.7 | 45.9
\+ VIPER | 49.2 | 48.9 | 42.2 | 51.4 | 47.9
ER(800) | 47.4 | 47.8 | 40.9 | 48.8 | 46.2
\+ VIPER | 48.5 | 50.0 | 42.5 | 52.0 | 48.3
ER(1000) | 48.6 | 50.4 | 42.2 | 48.5 | 47.4
\+ VIPER | 49.2 | 50.2 | 43.5 | 51.5 | 48.6
ER(1200) | 49.3 | 50.2 | 43.2 | 50.3 | 48.3
\+ VIPER | 50.8 | 51.7 | 44.1 | 50.3 | 49.2
Next, we analyze the effect of the size of the replay buffer when deploying ER
[9]. In particular, we experiment with a memory size of
$\\{200,800,1000,1200\\}$. We report the results on Table 6 of the performance
of ER under different buffer sizes with or without SimCS (using VIPER). For
this experiment, we set the number of training iterations to $N=4$ following
our main setup in Seciton 5.
We observe that as the buffer size increases, the performance of the learner
improves. This is consistent with the earlier observations in the literature
[9] as larger buffer sizes allow the model to rehearse more diverse examples
including several domains. We note here that this comes at the expense of
requiring large memory consumption in order to store more examples along with
their segmentation masks. Further, we find that SimCS provides a consistent
performance improvement irrespective of the buffer size. This shows another
aspect of the robustness and how versatile SimCS under different learning
algorithms for ODICS.
### B.4 Varying Regularization Importance
Table 7: Performance Comparison under Different Regularization Importance. We
report the mIOU (%) of a model trained with different values of $\lambda$. All
methods are trained with $N=4$ iterations for each received batch. _SimCS
consistently improved the performance of all baselines on all observed
domains._
Method Domain | CS | IDD | BDD | ACDC | mIOU
---|---|---|---|---|---
EWC$(\lambda=1)$ | 34.1 | 35.8 | 33.1 | 37.8 | 27.7
EWC$(\lambda=10)$ | 41.5 | 38.8 | 35.9 | 47.9 | 41.0
MAS$(\lambda=1)$ | 41.4 | 37.1 | 34.6 | 48.2 | 40.3
MAS$(\lambda=10)$ | 36.7 | 36.1 | 31.4 | 40.0 | 36.1
LwF$(\lambda=1)$ | 43.0 | 38.9 | 35.2 | 47.9 | 41.3
LwF$(\lambda=10)$ | 43.1 | 40.7 | 36.3 | 48.7 | 42.2
In Section 5, we reported the results for regularization based methods cross
validated at the best $\lambda$ value. Here, we explore the effect of varying
$\lambda$ on the performance of regularization based methods. Table 7
summarizes the results of EWC, MAS, and LwF for different regularization
importance values $\lambda$. We found that both EWC and MAS are more sensitive
to variations of $\lambda$ as they deploy the regularization directly to
network parameters. However, we observe the distillation approach; LwF, is
more robust to such variations making it more suitable to ODICS setup.
Figure 5: Online Data Incremental Continual Semantic Segmentation with SimCS.
The learner receives a mixed batch from different domains. SimCS provides
simulated data with aligned labels space to the real data. Note that different
domains might present the learner with different amounts of data.
### B.5 Analysis on Different Domain Orders
In Section 5.6, we analyzed the performance variations under different domain
orders. We also showed that SimCS improves, not only the backward transfer;
_i.e_. forgetting, but also the performance on unseen domains; _i.e_. forward
transfer. Here we attempt at providing more insights on the reasoning for this
behaviour.
Regarding improving the forward transfer, we believe that SimCS provides the
learner with extra data diversity. Meaning, simulated data could capture
experiences that do not exist in earlier domains in the stream but might
benefit the generalization to unseen domains. Further, the distribution of
labels in the generated simulation data could be closer to some real
distributions than other real domains. For example, one could match the number
of car instances in a generated image to be closer to IDD (more vehicles) or
change the weather conditions to match ACDC. Note that SimCS, while generating
the simulated data with randomly setting the simulation parameters, provided
cross the board performance improvements on unseen domains as shown in Figure
4. We leave further experiments on better utilization of simulated data for CL
for future work.
Regarding improving the backward transfer, we hypothesize the simulated data
serve as a regularizer for not forgetting previously learnt domains. That is,
the distribution of simulated data does not exactly match any of the real-
world domains, but might be bridging different real domains. This is
demonstrated on the robustness of SimCS to different domain orders in Figure 4
where performance improvements were shown in all considered scenarios. Another
evidence to this hypothesis, is the results reported in Table 2 were SimCS
improved the performance even when using VIPER in the pretraining stage. That
is, although the learner observed VIPER data, including simulated data in the
continual learning process still provides significant performance
enhancements.
## Appendix C Online Data Incremental Setup
Table 8: Performance Comparison Under Data Incremental Learning. We report the
mIOU of a model trained on a stream and evaluated on each of dataset in the
stream.
Method Dataset | CS | IDD | BDD | ACDC | mIOU
---|---|---|---|---|---
NT | 42.3 | 47.7 | 38.9 | 39.7 | 42.1
\+ CARLA | 43.4 | 49.8 | 39.2 | 41.4 | 43.4
\+ VIPER | 43.5 | 49.8 | 39.5 | 40.8 | 43.4
At last, we present another realistic setup of continual learning for
autonomous driving systems. We consider the scenario where different systems
exist at different locations and collect data simultaneously. This results in
a data incremental setup where the learner observes at each time step $t$ a
set of images belonging to different domains. However, different domains could
reveal different amounts of data. Figure 5 summarizes this setup with the
inclusion of SimCS.
We conduct preliminary experiments with Naive Training (NT) and report the
results in Table 8 for NT, and NT+SimCS. We follow similar experimental setup
to the one in Section 5.1. We observe that SimCS provides performance
improvements under this setup, similar to ODICS. We measure a performance
improvement of $1.3\%$ when including simulated data into the training
process. We leave a further detailed analysis to this setup along with more
experimental results to a future work.
|
# Closing the gap between SVRG and TD-SVRG with Gradient Splitting
Arsenii Mustafin
Department of Computer Science
Boston University
Boston, MA 02215, USA
<EMAIL_ADDRESS>
&Alex Olshevsky
Department of Electrical and Computer Engineering
Boston University, Boston, MA 02215, USA
<EMAIL_ADDRESS>
Ioannis Ch. Paschalidis
Department of Electrical and Computer Engineering
Boston University, Boston, MA 02215, USA
<EMAIL_ADDRESS>
###### Abstract
Temporal difference (TD) learning is a policy evaluation in reinforcement
learning whose performance can be enhanced by variance reduction techniques.
Recently, multiple works have sought to fuse TD learning with SVRG to obtain a
policy evaluation method with a geometric rate of convergence. However, the
resulting convergence rate is significantly weaker than what is achieved by
SVRG in the setting of convex optimization. In this work we utilize a recent
interpretation of TD-learning as the splitting of the gradient of an
appropriately chosen function, thus simplifying the algorithm and fusing TD
with SVRG. Our main result is a geometric convergence bound with predetermined
learning rate of $1/8$, which is identical to the convergence bound available
for SVRG in the convex setting. Our theoretical findings are supported by a
set of experiments.
## 1 Introduction
Reinforcement learning (RL) is a framework for solving sequential decision
making environments. Policy evaluation is one of those problems, which seeks
to determine the expected reward an agent achieves if it chooses actions
according to a specific stationary policy. Temporal Difference learning (TD
learning, [19]) is a popular algorithm with a particularly simple form which
can be performed in an online setting. TD learning uses the Bellman equation
to bootstrap the estimation process and update the value function from each
incoming sample or mini-batch. As all RL methods, tabular TD learning suffers
from the “curse of dimensionality" when the number of states is large,
motivating parametric approximations of the value function.
Despite its simple formulation, theoretical analysis of approximate TD
learning is subtle. There are few important milestones in this process, one of
which is the work in [22], where asymptotic convergence guarantees were
established. More recent advances include [2], [18] and [10]. In particular,
[10] shows that TD learning might be viewed as an example of gradient
splitting, a process analogous to gradient descent.
TD-leaning has an inherent variance problem: the variance of the update does
not go to zero as the method converges. This problem is also present in a
class of convex optimization problems where the objective function is a sum of
functions and Stochastic Gradient Descent (SGD)-type methods are applied [16].
Such methods proceed incrementally by sampling a single function, or a mini-
batch of functions, to use for stochastic gradient evaluations. Variance
reduction techniques were developed to address this problem and yield faster
convergence, including SAG [17], SVRG [8] and SAGA [6]. Their distinguishing
feature is that they converge geometrically.
Previous research has analysed the application of variance reduction technique
to TD updates in two problem settings: $(i)$ a pre-sampled trajectory of the
Markov Decision Process (MDP) (finite sample), and $(ii)$ when states are
sampled directly from the MDP (online sampling). We briefly mention the most
relevant works in both veins. In the online sampling setting, the first
attempt to adapt variance reduction to TD learning was made in [9]. Their
results were discussed by [4] and [14]; [23] provided further analysis of such
approaches and showed geometric convergence for the so-called Variance
Reduction Temporal Difference learning (VRTD) algorithm for Markovian
sampling; [12] applies the variance reduction technique to Temporal Difference
with Correction.
Table 1: Comparison of algorithmic complexities. A table with additional algorithms and details of the comparison might be found in Appendix I | | Complexity
---|---|---
Type | Algorithm | Feature case | Tabular case
Finite | PD-SVRG | $\mathcal{O}\left(\left(N+\frac{\kappa^{2}(C)L^{2}_{G}}{\lambda_{\rm min}(A^{T}C^{-1}A)^{2}}\right)\log(\frac{1}{\epsilon})\right)$ | $\mathcal{O}\left(\left(N+\frac{1}{(1-\gamma)^{2}\pi_{\rm min}^{4}}\right)\log(\frac{1}{\epsilon})\right)$
Finite | Our | $\mathcal{O}\left(\left(N+\frac{1}{\lambda_{A}}\right)\log(\frac{1}{\epsilon})\right)$ | $\mathcal{O}\left(\left(N+\frac{1}{(1-\gamma)\pi_{\rm min}}\right)\log(\frac{1}{\epsilon})\right)$
i.i.d. | TD | $\mathcal{O}\left(\frac{1}{\lambda_{A}^{2}\epsilon}\log(\frac{1}{\epsilon})\right)$ | $\mathcal{O}\left(\frac{1}{(1-\gamma)^{2}\pi_{\rm min}^{2}\epsilon}\log(\frac{1}{\epsilon})\right)$
i.i.d. | Our | $\mathcal{O}\left(\frac{1}{\lambda_{A}\epsilon}\log(\frac{1}{\epsilon})\right)$ | $\mathcal{O}\left(\frac{1}{(1-\gamma)\pi_{\rm min}\epsilon}\log(\frac{1}{\epsilon})\right)$
Markovian | VRTD | $\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}^{2}}\log(\frac{1}{\epsilon})\right)$ | $\mathcal{O}\left(\frac{1}{(1-\gamma)^{2}\pi_{\rm min}^{2}\epsilon}\log(\frac{1}{\epsilon})\right)$
Markovian | Our | $\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}}\log^{2}(\frac{1}{\epsilon})\right)$ | $\mathcal{O}\left(\frac{1}{(1-\gamma)\pi_{\rm min}\epsilon}\log^{2}(\frac{1}{\epsilon})\right)$
The finite sample setting was analysed in [7], where authors directly applied
SVRG and SAGA to a version of policy evaluation by transforming it into an
equivalent convex-concave saddle-point problem. Since their algorithm uses two
sets of parameters, in this paper we call it Primal-Dual SVRG or PD-SVRG.
Their results were improved in [15] by introducing inexact mean path update
calculation (Batched SVRG algorithm).
An unsatisfying feature of all of this literature is that the convergence
times it derives do not match what variance reduction provides in the standard
convex optimization setting. In particular, gradient descent converges
linearly with the condition number of the problem, and SVRG inherits this
property. Unfortunately, convergence times for variance reduction for temporal
difference learning not only converge quadratically with condition number, but
also include some extraneous factors related to the condition number of
certain diagonalizing matrices. Because the underlying matrices in RL are
typically ill-conditioned, they can result in very large sample complexities.
Even on simple problem such as random MDPs with random connections, previously
reported sample complexities can reach astronomical values as we discuss in
Appendix H.2.
In this paper we analyze the convergence of SVRG applied to TD (TD-SVRG) in
both finite sample and online sampling cases. Our theoretical results are
summarized in Table 1. Our key contributions are:
* •
For the finite sample case, we show that TD-SVRG has the same convergence rate
as SVRG in the convex optimization setting. In particular, we replace the
quadratic scaling with the condition number by linear scaling and remove
extraneous factors depending on the diagonalizing matrix. Notably, we use a
simple, pre-determined learning rate of $1/8$.
* •
For i.i.d. online sampling, we similarly achieve better rates with simpler
analysis. Again, we are first to show that TD-SVRG has the same convergence
rate as SVRG in the convex optimization setting with a predetermined learning
rate of $1/8$, and a linear rather than quadratic scaling with the condition
number. Similar improvement is obtained for Markovian sampling.
* •
Our algorithms, along with the parameters leading to theoretical guarantees,
are the first amenable to be implementable in practice. Previous theoretical
results required a batch size so large that they were impractical (see
Subsection H.2) and parameter search had to be used in implementation instead
of the parameters appearing in the theoretical results. In particular, we
include experiments that show our theoretically obtained batch size and
learning rate can be applied in practice: we show they achieve geometric
convergence and outperform other algorithms which use parameters chosen by
grid search.
To summarize, in every setting our key contribution is the reduce the scaling
with a condition number from quadratic to linear, as well as to remove
extraneous factors that do not appear in the analysis of SVRG in the convex
setting. As described below, the final result matches the bounds that are
known for the SVRG in the separable convex optimization setting – thus, the
“punchline” of this paper is using SVRG for policy evaluation results in a
similar kind of complexity as the classical SVRG.
## 2 Problem formulation
We consider a discounted reward Markov Decision Process (MDP) defined by the
tuple $(\mathcal{S},\mathcal{A},\mathcal{P},r,\gamma)$, where $\mathcal{S}$ is
the state space, $\mathcal{A}$ the action space,
$\mathcal{P}=\mathcal{P}(s^{\prime}|s,a)_{s,s^{\prime}\in\mathcal{S},a\in\mathcal{A}}$
the transition probabilities, $r=r(s,s^{\prime})$ the reward function, and
$\gamma\in[0,1)$ is a discount rate. The agent follows a policy
$\pi:\mathcal{S}\rightarrow\Delta_{\mathcal{A}}$ – a mapping from states to
the probability simplex over actions. A policy $\pi$ induces a joint
probability distribution $\pi(s,a)$, defined as the probability of choosing
action $a$ while being in state $s$. Given that the policy is fixed and we are
interested only in policy evaluation, for the remainder of the paper we will
consider the transition probability matrix $P$, such that:
$P(s,s^{\prime})=\sum_{a}\pi(s,a)\mathcal{P}(s^{\prime}|s,a).$ We assume, that
the Markov process produced by the transition probability matrix is
irreducible and aperiodic with stationary distribution $\mu_{\pi}$.
The policy evaluation problem is to compute $V^{\pi}$, defined as:
$V^{\pi}(s):=\operatorname{\mathbb{E}}\left[\sum_{t=0}^{\infty}\gamma^{t}r_{t+1}\right].$
Here $r_{t}$ is the reward at time $t$ and $V^{\pi}$ is the value function,
formally defined to be the unique vector which satisfies the Bellman equation
$T^{\pi}V^{\pi}=V^{\pi}$, where $T^{\pi}$ is the Bellman operator, defined as:
$T^{\pi}V^{\pi}(s)=\sum_{s^{\prime}}P(s,s^{\prime})\left(r(s,s^{\prime})+\gamma
V^{\pi}(s^{\prime})\right).$ The TD(0) method is defined as follows: one
iteration performs a fixed point update on a randomly sampled pair of states
$s,s^{\prime}$ with learning rate $\alpha$: $V(s)\leftarrow
V(s)+\alpha(r(s,s^{\prime})+\gamma V(s^{\prime})-V(s)).$ When the state space
size $|\mathcal{S}|$ is large, tabular methods which update the value function
for every state become impractical. For this reason, a linear approximation of
the value function is often used. Each state is represented by a feature
vector $\phi(s)\in\mathbb{R}^{d}$ and the state value $V^{\pi}(s)$ is
approximated by $V^{\pi}(s)\approx\phi(s)^{T}\theta$, where $\theta$ is a
tunable parameter vector. A single TD update on a randomly sampled transition
$s,s^{\prime}$ becomes:
$\displaystyle\theta$ $\displaystyle\leftarrow$ $\displaystyle\theta+\alpha
g_{s,s^{\prime}}(\theta)$ $\displaystyle=$
$\displaystyle\theta+\alpha((r(s,s^{\prime})+\gamma\phi(s^{\prime})^{T}\theta-\phi(s)^{T}\theta)\phi(s)),$
where the second equation should be viewed as a definition of
$g_{s,s^{\prime}}(\theta)$.
Our goal is to find a parameter vector $\theta^{*}$ such that the average
update vector is zero
$\operatorname{\mathbb{E}}_{s,s^{\prime}}[g_{s,s^{\prime}}(\theta^{*})]=0.$
This expectation is also called mean-path update $\bar{g}(\theta)$ and can be
written as:
$\displaystyle\begin{split}\bar{g}(\theta)&=\operatorname{\mathbb{E}}_{s,s^{\prime}}[g_{s,s^{\prime}}(\theta)]\\\
&=\operatorname{\mathbb{E}}_{s,s^{\prime}}[(\gamma\phi(s^{\prime})^{T}\theta-\phi(s)^{T}\theta)\phi(s)]\\\
&\phantom{++++}+\operatorname{\mathbb{E}}_{s,s^{\prime}}\left[r(s,s^{\prime})\phi(s)\right]\\\
&:=-A\theta+b,\end{split}$ (1)
where the last line should be taken as the definition of the matrix $A$ and
vector $b$. Finally, the minimum eigenvalue of the matrix $(A+A^{T})/2$ plays
an important role in our analysis and will be denoted as $\lambda_{A}$.
There are a few possible settings of the problem: the samples $s,s^{\prime}$
might be drawn from the MDP on-line (Markovian sampling) or independently
(i.i.d. sampling): the first state $s$ is drawn from $\mu_{\pi}$, then
$s^{\prime}$ is drawn as the next state under the policy $\pi$. Another
possible setting for analysis is the “finite sample set": first trajectory of
length $N$ is drawn from an MDP following Markovian sampling and forms dataset
$\mathcal{D}=\\{(s_{t},a_{t},r_{t},s_{t+1})\\}_{t=1}^{N}$. Then TD(0) proceeds
by drawing samples from this data set. Note that the definition of expectation
$\operatorname{\mathbb{E}}_{s,s^{\prime}}$ and, consequently, matrix $A$ will
be slightly different in two settings: in the on-line sampling cases
probability of pair of states $s,s^{\prime}$ is defined by stationary
distribution $\mu_{\pi}$, and transition matrix $P$ and
$A_{e}=\sum_{s\in\mathcal{S}}\sum_{s^{\prime}\in\mathcal{S}}\mu_{\pi}(s)P(s,s^{\prime})(\phi(s)^{T}\theta-\gamma\phi(s^{\prime})^{T}\theta)\phi(s).$
In the “finite sample" case probability of $s,s^{\prime}$ refers to
probability of getting pair of states from one particular data point $t$:
$s=s_{t},s^{\prime}=s_{t+1}$, and matrix $A$ is defined as:
$A_{d}=\frac{1}{N}\sum_{t=1}^{N}(\phi(s_{t})^{T}\theta-\gamma\phi(s_{t+1})^{T}\theta)\phi(s_{t}).$
Likewise, the definition of $\bar{g}(\theta)$ differs between the MDP and
dataset settings, since that definition involves
$\operatorname{\mathbb{E}}_{s,s^{\prime}}$ which, as discussed above, means
slightly different things in both settings.
In the sequel, we will occasionally refer to the matrix $A$. Whenever we make
such a statement, we are in fact making two statements: one for the dataset
case when $A$ should be taken to be $A_{d}$, and one in the on-line case when
$A$ should be taken to be $A_{e}$.
We make the following standard assumptions:
###### Assumption 2.1.
(Problem solvability) The matrix $A$ is non-singular.
###### Assumption 2.2.
(Bounded features) $||\phi(s)||_{2}\leq 1$ for all $s\in\mathcal{S}$.
These assumptions are widely accepted and have been utilized in previous
research within the field ([2], [7], [9], [10], [23]) Assumption 2.1 ensures
that $A^{-1}b$ exists and the problem is solvable. At the risk of being
repetitive, we note that this is really two assumptions, one that $A_{e}$ is
non-singular in the on-line case, and one that $A_{d}$ is non-singular in the
dataset case which are stated together. Assumption 2.2 is made for simplicity
and it can be satisfied by feature vector rescaling.
Figure 1: Illustration of gradient splitting. All gradient splittings of the
function $f(\theta)$ will lie on line $l$. In addition, if we have a
constraint on the 2-norm of the matrix $A$, all gradient splittings will lie
on an interval $I$.
### 2.1 Key ideas of the Analysis
While the main body of the paper will be devoted to statements of our main
results and simulations, with proofs in the supplementary information, we take
the time here to briefly discuss a perspective on TD learning which represents
the key difference between our analysis and the previous work. In [23] the
authors note: ”In [8] , the convergence proof relies on the relationship
between the gradient and the value of the objective function, but there is not
such an objective function in the TD learning problem.” We show, that viewing
TD learning as gradient splitting allows us to find such function and
establish a relationship between the gradient and the value function, which
enables a similar analysis as in [8] to achieve stronger results.
In our analysis we often use the function $f(\theta)$, defined as:
$f(\theta)=(\theta-\theta^{*})^{T}A(\theta-\theta^{*}).$ (2)
We will use $f_{d}$ and $f_{e}$ notation for dataset ($A=A_{d}$) and
environment case ($A=A_{e}$) respectively. In [10], the authors similarly
define a function $f(\theta)$ as
$f(\theta)=(1-\gamma)||V_{\theta}-V_{\theta^{*}}||_{D}^{2}+\gamma||V_{\theta}-V_{\theta^{*}}||_{\rm
Dir}^{2},$
where $V_{\theta}$ is a vector of state values induced by $\theta$,
$||V||_{D}^{2}=\sum_{s}\mu_{\pi}(s)V(s)^{2},$
is a weighted norm and
$||V||_{\rm
Dir}^{2}=\frac{1}{2}\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})(V(s)-V(s^{\prime}))^{2}$
is a Dirichlet seminorm. It is shown in [10] that the two definitions of the
function $f$ are equivalent when states are sampled directly from an MDP.
The importance of the function $f(\theta)$ comes from the notion of gradient
splitting from [10]: we omit the precise definition but the key property is
that the inner product of a gradient splitting $h(\theta)$ and the direction
$\theta^{*}-\theta$ to the minimizer is the same as the inner product $-\nabla
f(\theta)$ and $\theta^{*}-\theta$ (see Figure 1). It is shown there that the
expected TD(0) direction $-\bar{g}(\theta)$ is a gradient splitting of the
function $f(\theta)$. This interpretation of TD learning provides a tool for
its convergence analysis, since it leads to bounds on key quantity: the angle
between the expected update direction and the direction to the ultimate limit.
Our arguments build heavily on the gradient splitting interpretation of TD,
and it is this approach that differentiates our paper from previous work on
variance-reduced policy evaluation. A key difficulty to overcome is that, in
the “finite sample” case discussed earlier, the two definitions of the
function $f(\theta)$ are no longer equivalent, and as a result the TD(0)
update is no longer a gradient splitting. This complicates things considerably
and our key idea is thus to view TD updates in this case as a form of an
approximate gradient splitting.
## 3 The TD-SVRG algorithm
We next propose a modification of the TD(0) method (TD SVRG) which can attain
a geometric rate. This algorithm is given above as Algorithm 1. The algorithm
works under the “finite sample set” setting which assumes there already exists
a sampled data set ${\cal D}$. This is the same setting as in [7]. However,
the method we propose does not add regularization and does not use dual
parameters, which makes it considerably simpler.
Algorithm 1 TD-SVRG for the finite sample case
Parameters update batch size $M$ and learning rate $\alpha$.
Initialize $\tilde{\theta}_{0}$.
for $m=1,2,...$ do
$\tilde{\theta}=\tilde{\theta}_{m-1}$,
$\bar{g}_{m}(\tilde{\theta})=\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}g_{s,s^{\prime}}(\tilde{\theta})$,
where
$g_{s,s^{\prime}}(\tilde{\theta})=(r(s,s^{\prime})+\gamma\phi(s^{\prime})^{T}\tilde{\theta}-\phi(s)^{T}\tilde{\theta})\phi(s_{t})$.
$\theta_{0}=\tilde{\theta}$.
for $t=1$ to $M$ do
Sample $s,s^{\prime}$ from ${\cal D}$.
Compute
$v_{t}=g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\tilde{\theta})+\bar{g}_{m}$.
Update parameters $\theta_{t}=\theta_{t-1}+\alpha v_{t}$.
end for
Set $\tilde{\theta}_{m}=\theta_{t^{\prime}}$ for randomly chosen
$t^{\prime}\in(0,\ldots,M-1)$.
end for
Like the classic SVRG algorithm, our proposed TD-SVRG has two nested loops. We
refer one step of the outer loop as an epoch and one step of inner loop as an
iteration. TD-SVRG keeps two parameter vectors: the current parameter vector
$\theta_{t}$, which is being updated at every iteration, and the vector
$\tilde{\theta}_{t}$, which is updated at the end of each epoch. Each epoch
contains $M$ iterations, which we call update batch size (not to be confused
with the estimation batch size, which will be used in the algorithms below to
compute an estimate of the mean-path update).
## 4 Convergence analysis
In this section, we show that under rather simple assumptions, Algorithm 1
attains geometric convergence in terms of a specially chosen function
$f_{d}(\theta)$ with $\alpha$ being $\mathcal{O}(1)$ and $M$ being
$\mathcal{O}(1/\lambda_{A})$.
### 4.1 Preliminaries
In order to analyze the convergence of the presented algorithm, we define the
expected square norm of the difference between the current and optimal
parameters as $w(\theta):$
$\displaystyle w(\theta)$ $\displaystyle=$
$\displaystyle\operatorname{\mathbb{E}}[||g_{s,s^{\prime}}(\theta)-g_{s,s^{\prime}}(\theta^{*})||^{2}],$
(3)
where expectation is taken with respect to sampled pair of states
$s,s^{\prime}$. With this notation we provide a technical lemma. The next
proofs are based on variations of this lemma.
###### Lemma 4.1.
If Assumptions 2.1, 2.2 hold, epoch parameters of two consecutive epochs $m-1$
and $m$ are related by the following inequality:
$\displaystyle\begin{split}&2\alpha
M\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})]-2M\alpha^{2}\operatorname{\mathbb{E}}[w(\tilde{\theta}_{m})]\leq\\\
&\operatorname{\mathbb{E}}[||\tilde{\theta}_{m-1}-\theta^{*}||^{2}]+2\alpha^{2}M\operatorname{\mathbb{E}}[w(\tilde{\theta}_{m-1})],\end{split}$
(4)
where the expectation is taken with respect to all previous epochs and choices
of states $s,s^{\prime}$ during the epoch $m$.
###### Proof.
The proof of the lemma generally follows the analysis in [8]; it can be found
in Appendix A. ∎
Lemma 1 plays an auxiliary role in our analysis and significantly simplifies
it. It introduces a new approach to the convergence proof by carrying
iteration to iteration and epoch to epoch bounds to the earlier part of the
analysis. In particular, deriving bounds in terms of some arbitrary function
$u(\theta)$ is now reduced to deriving upper bounds on
$||\tilde{\theta}_{m-1}-\theta^{*}||^{2}$ and $w(\theta)$, and a lower bound
on $f(\theta)$ in terms of the function $u$. Three mentioned quantities are
natural choices for the function $u$. In Appendix B we show Lemma 4.1 might be
used to derive convergence in terms of
$||\tilde{\theta}_{m-1}-\theta^{*}||^{2}$ with similar bounds as in [7]. In
this paper we use $f(\theta)$ as $u$ to improve on previous results.
### 4.2 Algorithm convergence
In this section, we derive a bound in terms of $f_{d}(\theta)$. But before we
start note that in general a first state of the first pair and a second state
of the last state pair in randomly sampled dataset would not be the same
state. That leads to the effect which we call unbalanced dataset: unlike MDP,
first and second states distributions in such dataset are different. In
unbalanced dataset case mean path update is not exactly a gradient splitting
of target function $f(\theta)$ and we need to introduce a correction term in
our analysis. Following theorem covers unbalanced dataset case and balanced
dataset case is covered in the corollary.
###### Theorem 4.2.
Suppose Assumptions 2.1, 2.2 hold and the dataset $\mathcal{D}$ may be
unbalanced. Define the error term $J=\frac{4\gamma^{2}}{N\lambda_{A}}$. Then,
if we choose learning rate $\alpha=1/(8+J)$ and update batch size
$M=2/(\lambda_{A}\alpha)$, Algorithm 1 will have a convergence rate of:
$\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})]\leq\left(\frac{2}{3}\right)^{m}f_{d}(\tilde{\theta}_{0}).$
###### Corollary 4.3.
If the dataset $\mathcal{D}$ is balanced, then we may take the error term is
$J=0$ and consequently the same convergence rate might be obtained with
choices of learning rate $\alpha=1/8$ and update batch size $M=16/\lambda_{A}$
###### Proof of Theorem 4.2.
The proof is given in Appendix C. ∎
Note that $\tilde{\theta}_{m}$ refers to the iterate after $m$ iterations of
the outer loop. Thus, the total number of samples guaranteed by this theorem
until $\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})]\leq\epsilon$ is
actually $\mathcal{O}((N+16/\lambda_{A})\log(1/\epsilon))$ in balanced case
and $\mathcal{O}((N+\frac{16+2/(N\lambda_{A})}{\lambda_{A}})\log(1/\epsilon))$
in unbalanced case, which means that two complexities are identical if dataset
size $N$ is large enough so that $N\geq\lambda_{A}^{-1}$.
Even an error term introduced by unbalanced dataset is negligible in the
randomly sampled datasets, it might not be bounded better than by $J$. In
practice the issue might be tackled by sampling for a modified dataset this
issue is discussed in Appendix D. Additionally, in Appendix E we provide an
analysis for TD-SVRG with batching.
### 4.3 Similarity of SVRG and TD-SVRG
Note that the dataset case is similar to SVRG in the convex setting in the
sense that: 1) update performed in each step is selected uniformly at random,
and 2) the exact mean-path update can be computed every epoch. If the dataset
is balanced, a negative mean-path update $-\bar{g}(\theta)$ is a gradient
splitting of the function $f(\theta)$. These allow us to further demonstrate
the significance of the function $f(\theta)$ for the TD learning process and
the greater similarity between TD-learning and convex optimization. We recall
the convergence rate obtained in [8] for a sum of convex functions:
$\frac{1}{\gamma^{\prime}\alpha^{\prime}(1-2L\alpha^{\prime})m^{\prime}}+\frac{2L\alpha^{\prime}}{1-2L\alpha^{\prime}},$
where $\gamma^{\prime}$ is a strong convexity parameter and $L$ is a Lipschitz
smoothness parameter (we employ the notation from the original paper and
introduce the symbol ′ to avoid duplicates). The function
$f(\theta)=\frac{1}{2}(\theta-\theta^{*})^{T}A(\theta-\theta^{*})$ is
$\lambda_{A}$ strongly convex and 1-Lipschitz smooth, which means that the
convergence rate obtained in this paper is identical to the convergence rate
of SVRG in the convex setting. This fact further extends the analogy between
TD learning and convex optimization earlier explored by [2] and [10].
Algorithm 2 TD-SVRG for the i.i.d. sampling case
Parameters update batch size $M$ and learning rate $\alpha$.
Initialize $\tilde{\theta}_{0}$.
for $m=1,2,...$ do
$\tilde{\theta}=\tilde{\theta}_{m-1}$,
choose estimation batch size $n_{m}$,
sample batch $\mathcal{D}^{m}$ of size $n_{m}$,
compute
$g_{m}(\tilde{\theta})=\frac{1}{n_{m}}\sum_{s,s^{\prime}\in\mathcal{D}^{m}}g_{s,s^{\prime}}(\tilde{\theta})$,
where
$g_{s,s^{\prime}}(\tilde{\theta})=(r(s,s^{\prime})+\gamma\phi(s^{\prime})^{T}\tilde{\theta}-\phi(s)^{T}\tilde{\theta})\phi(s_{t})$.
$\theta_{0}=\tilde{\theta}$.
for $t=1$ to $M$ do
Sample $s,s^{\prime}$ from ${\cal D}$.
Compute
$v_{t}=g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\tilde{\theta})+g_{m}(\tilde{\theta})$.
Update parameters $\theta_{t}=\theta_{t-1}+\alpha v_{t}$.
end for
Set $\tilde{\theta}_{m}=\theta_{t^{\prime}}$ for randomly chosen
$t^{\prime}\in(0,\ldots,M-1)$.
end for
## 5 Online i.i.d. sampling from the MDP
We now apply TD learning as gradient splitting analysis to the case of online
i.i.d. sampling from the MDP each time we need to generate a new state $s$. We
show that our methods can be applied in this case to derive tighter
convergence bounds. One issue of TD-SVRG in the i.i.d. setting is that the
mean-path update may not be computed directly. Indeed, once we have a data set
of size $N$, we can simply make a pass through it; but in an MDP setting, it
is typical to assume that making a pass through all the states of the MDP is
impossible. The inexactness of mean-path update is addressed with the sampling
technique introduced in Appendix D, which makes the i.i.d. case very similar
to TD-SVRG with non-exact mean-path computation in the finite sample case.
Thus, TD-SVRG algorithm for the i.i.d. sampling case is very similar to
Algorithm E.1, with the only difference being that states $s,s^{\prime}$ are
being sampled from the MDP instead of the dataset $\mathcal{D}$. Formal
description of the algorithm is provided in Appendix F.
In this setting, geometric convergence is not attainable with variance
reduction, which always relies on a pass through the dataset. Since here one
sample is obtained from the MDP at every step, one needs to use increasing
batch sizes. Our algorithm does so, while next theorem once again improves the
scaling with the condition number from quadratic to linear compared to the
previous literature.
###### Theorem 5.1.
Suppose Assumptions 2.1, 2.2 hold. Then if the learning rate is chosen as
$\alpha=1/16$, the update batch size as $M=32/\lambda_{A}$ and the estimation
batch size as
$n_{m}=\frac{1}{c\lambda_{A}(2/3)^{m}}(4f(\theta_{m})+2\sigma^{2})$, where $c$
is some arbitrary chosen constant, Algorithm 2 will have a convergence rate
of:
$\operatorname{\mathbb{E}}[f_{e}(\tilde{\theta}_{m})]\leq\left(\frac{2}{3}\right)^{m}(f_{e}(\tilde{\theta}_{0})+C_{1}),$
where $C_{1}$ is a constant.
###### Proof.
The proof is given in Appendix F. ∎
This convergence rate will lead to total computational complexity of
$\mathcal{O}(\frac{1}{\lambda_{A}\epsilon}\log(\epsilon^{-1}))$ to achieve
accuracy $\epsilon$.
Similarly to the previous section, a quantity
$\frac{1}{c\lambda_{A}(2/3)^{m}}(2|r_{max}|^{2}+8||\tilde{\theta}_{m-1}||^{2})$
might be used for estimation batch sizes $n_{m}$ during practical
implementation of the algorithm. Note that the expression
$|r_{max}|^{2}+4||\tilde{\theta}_{m-1}||^{2}$ is common in the literature,
e.g. it is denoted as $D_{2}$ in [23].
## 6 Online Markovian sampling from the MDP
Algorithm 3 TD-SVRG for the Markovian sampling case
Parameters update batch size $M$ and learning rate $\alpha$ and projection
radius $R$.
Initialize $\tilde{\theta}_{0}$.
for $m=1,2,...$ do
$\tilde{\theta}=\tilde{\theta}_{m-1}$,
estimation batch size $n_{m}$,
sample trajectory $\mathcal{D}^{m}$ of length $n_{m}$,
compute
$g_{m}(\tilde{\theta})=\frac{1}{n_{m}}\sum_{s,s^{\prime}\in\mathcal{D}^{m}}g_{s,s^{\prime}}(\tilde{\theta})$,
where
$g_{s,s^{\prime}}(\tilde{\theta})=(r(s,s^{\prime})+\gamma\phi(s^{\prime})^{T}\tilde{\theta}-\phi(s)^{T}\tilde{\theta})\phi(s_{t})$.
$\theta_{0}=\tilde{\theta}$.
for $t=1$ to $M$ do
Sample $s,s^{\prime}$ from ${\cal D}$.
Compute
$v_{t}=g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\tilde{\theta})+g_{m}(\tilde{\theta})$.
Update parameters $\theta_{t}=\Pi_{R}(\theta_{t-1}+\alpha v_{t})$.
end for
Set $\tilde{\theta}_{m}=\theta_{t^{\prime}}$ for randomly chosen
$t^{\prime}\in(0,\ldots,M-1)$.
end for
The Markovian sampling case is the hardest to analyse due to its dependence on
the MDP properties, which makes establishing bounds on various quantities used
during the proof much harder. Leveraging the gradient splitting view still
helps us improve over existing bounds, but the derived algorithm does not have
the nice property of a constant learning rate. To deal with sample-to-sample
dependencies we introduce one more assumption often used in the literature:
###### Assumption 6.1.
For the MDP there exist constants $m>0$ and $\rho\in(0,1)$ such that
$\sup_{s\in S}d_{TV}(\mathbb{P}(s_{t}\in\cdot|s_{0}=s),\pi)\leq
m\rho^{t},\quad\forall t\geq 0,$
where $d_{TV}(P,Q)$ denotes the total-variation distance between the
probability measures P and Q.
In the Markovian setting, we also need to employ projection, which helps to
set a bound on the update vector $v$. Following [23], after each iteration we
project the parameter vector on a ball of radius $R$ (denoted as
$\Pi_{R}(\theta)=\operatorname*{arg\,min}_{\theta^{\prime}:|\theta^{\prime}|\leq
R}|\theta-\theta^{\prime}|^{2}$. We assume that $|\theta^{*}|\leq R$, where
the choice of $R$ that satisfies this bound can be found in Section 8.2 from
[2].
###### Theorem 6.2.
Suppose Assumptions 2.1, 2.2, 6.1 hold. Then, the output of Algorithm 3
satisfies:
$\displaystyle\operatorname{\mathbb{E}}[f_{e}(\tilde{\theta}_{m})]\leq$
$\displaystyle\left(\frac{3}{4}\right)^{m}f_{e}(\theta_{0})+\frac{8C_{2}}{\lambda_{A}n_{m}}+$
$\displaystyle 4\alpha(2G^{2}(4+6\tau^{\rm mix}(\alpha))+9R^{2}),$
where $C_{2}=\frac{4(1+(m-1)\rho)}{(1-\rho)}[4R^{2}+r_{\rm max}^{2}]$.
###### Proof.
The proof is given in Appendix G. ∎
Theorem 6.2 implies that if we choose $s=\mathcal{O}(\log(1/\epsilon))$,
$n_{m}=\mathcal{O}(1/(\lambda_{A}\epsilon))$,
$\alpha=\mathcal{O}(\epsilon/\log(1/\epsilon)$ and
$M=\mathcal{O}\left(\frac{\log(1/\epsilon)}{\epsilon\lambda_{A}}\right)$, the
total sample complexity is:
$\mathcal{O}\left(\frac{\log^{2}(1/\epsilon)}{\epsilon\lambda_{A}}\right).$
This has improved scaling with the condition number $\lambda_{A}^{-1}$
compared to
$\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}^{2}}\log(1/\epsilon)\right)$
from [23].
## 7 Experimental results
Figure 2: Geometric average performance of different algorithms in the finite
sample case. Columns - dataset source environments: MDP, Acrobot, CartPole and
Mountain Car. Rows - performance measurements: $\log(f(\theta))$ and
$\log(|\theta-\theta^{*}|)$.
Figure 2 shows relative performance of TD-SVRG, GTD2 [20], “vanilla" TD
learning [19], and PD-SVRG [7] in the finite sample setting. We used theory-
suggested parameters for TD-SVRG, parameters for PD-SVRG and GTD2 are selected
by grid search. Datasets of size 5,000 are generated from 4 environments:
Random MDP [5], and the Acrobot, CartPole and Mountain car OpenAI Gym
environments [3]. As the theory predicts, TD-SVRG and PD-SVRG converge
linearly, while GTD and vanilla TD converge sub-linearly. Details on the
experiments, grid search, and additional experiments can be found in Section
H. We provide the link to a github repository with the code and instructions
how to reproduce the experiments.
## 8 Conclusions
In the paper we provide improved sample complexity results for variance-
reduced policy evaluation. Our key theoretical finding is that it is possible
to reduce the scaling with the condition number of the problem from quadratic
to linear, matching what is known for SVRG in the convex optimization setting,
while simultaneously removing a number of extraneous factors. This results in
a many orders of magnitude improvements for batch size and sample complexity
for even simple problems such as random MDPs or OpenAI Gym problems. Results
of this flavor are attained in several settings, e.g., when a dataset of size
$N$ is sampled from the MDP, and when states of the MDP are sampled online
either in an i.i.d. or Markovian fashion. In simulations we find that our
method with step-sizes and batch-sizes coming from our theorems beats
algorithms from the previous literature with the same parameters selected by
grid search. The main innovation in the proofs of our results is to draw on a
view of TD learning as an approximate splitting of gradient descent. In terms
of future work, it will be interesting to apply some of these results to the
setting of epidemic control using the models in [11].
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## Appendix
## Appendix A Proof of Lemma 1
The proof follows the same logic as in [8] and is organized in four steps.
###### Step A.1.
In the original paper, the proof starts with deriving a bound on the squared
norm of the difference between the current and optimal parameter vectors. With
the introduction of $w(\theta)$ this step in our proof is trivial. We have
$\operatorname{\mathbb{E}}_{s,s^{\prime}}||g_{s,s^{\prime}}(\theta)-g_{s,s^{\prime}}(\theta^{*})||^{2}=w(\theta),$
where $\operatorname{\mathbb{E}}_{s,s^{\prime}}$ denotes the expectation taken
with respect to the choice of a random pair of states $s,s^{\prime}$. In other
words, $\operatorname{\mathbb{E}}_{s,s^{\prime}}[\cdot]$ denotes the
conditional expectation with respect to all variables that are not
$s,s^{\prime}$, which, recall, are generated at time $t$ by sampling $s$ from
the stationary distribution and letting $s^{\prime}$ be the next state. We
will slightly abuse notation to write
$\operatorname{\mathbb{E}}_{s,s^{\prime}}[\cdot]$ instead of the more rigorous
$\operatorname{\mathbb{E}}_{s_{t},s_{t+1}}$, since what time index the states
are generated at random is usually clear from context.
###### Step A.2.
During Step 2 we derive a bound on the norm of a single iteration $t$ update
$v_{t}=g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\tilde{\theta})+\bar{g}(\tilde{\theta})$,
assuming that states ${s,s^{\prime}}$ were sampled randomly during step $t$:
$\displaystyle\operatorname{\mathbb{E}}_{s,s^{\prime}}[||v_{t}||^{2}]$
$\displaystyle=\operatorname{\mathbb{E}}_{s,s^{\prime}}||g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\tilde{\theta})+\bar{g}(\tilde{\theta})||^{2}$
$\displaystyle=\operatorname{\mathbb{E}}_{s,s^{\prime}}||(g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\theta^{*}))+(g_{s,s^{\prime}}(\theta^{*})-g_{s,s^{\prime}}(\tilde{\theta})+\bar{g}(\tilde{\theta})||^{2}$
$\displaystyle\leq
2\operatorname{\mathbb{E}}_{s,s^{\prime}}||(g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\theta^{*}))||^{2}$
$\displaystyle+2\operatorname{\mathbb{E}}_{s,s^{\prime}}||g_{s,s^{\prime}}(\tilde{\theta})-g_{s,s^{\prime}}(\theta^{*})-(\bar{g}(\tilde{\theta})-\bar{g}(\theta^{*}))||^{2}$
$\displaystyle=2\operatorname{\mathbb{E}}_{s,s^{\prime}}||(g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\theta^{*}))||^{2}+2\operatorname{\mathbb{E}}_{s,s^{\prime}}||g_{s,s^{\prime}}(\tilde{\theta})-g_{s,s^{\prime}}(\theta^{*})$
$\displaystyle-\operatorname{\mathbb{E}}_{s,s^{\prime}}[g_{s,s^{\prime}}(\tilde{\theta})-g_{s,s^{\prime}}(\theta^{*})]||^{2}$
$\displaystyle\leq
2\operatorname{\mathbb{E}}_{s,s^{\prime}}||(g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\theta^{*}))||^{2}+2\operatorname{\mathbb{E}}_{s,s^{\prime}}||g_{s,s^{\prime}}(\tilde{\theta})-g_{s,s^{\prime}}(\theta^{*})||^{2}$
$\displaystyle=2w(\theta_{t-1})+2w(\tilde{\theta}).$
The first inequality uses $\operatorname{\mathbb{E}}||a+b||^{2}\leq
2\operatorname{\mathbb{E}}||a||^{2}+2\operatorname{\mathbb{E}}||b||^{2}$. The
second inequality uses the fact that the second central moment is smaller than
the second moment. The last equality uses the equality from Step 1.
###### Step A.3.
During this step we derive a bound on the expected squared norm of a distance
to the optimal parameter vector after a single update $t$:
$\displaystyle\operatorname{\mathbb{E}}_{s,s^{\prime}}||\theta_{t}-\theta^{*}||^{2}$
$\displaystyle=\operatorname{\mathbb{E}}_{s,s^{\prime}}||\theta_{t-1}-\theta^{*}+\alpha
v_{t}||^{2}$
$\displaystyle=||\theta_{t-1}-\theta^{*}||^{2}+2\alpha(\theta_{t-1}-\theta^{*})^{T}\operatorname{\mathbb{E}}_{s,s^{\prime}}v_{t}+\alpha^{2}\operatorname{\mathbb{E}}_{s,s^{\prime}}||v_{t}||^{2}$
$\displaystyle\leq||\theta_{t-1}-\theta^{*}||^{2}+2\alpha(\theta_{t-1}-\theta^{*})^{T}\bar{g}(\theta_{t-1})+2\alpha^{2}w(\theta_{t-1})+2\alpha^{2}w(\tilde{\theta})$
$\displaystyle=||\theta_{t-1}-\theta^{*}||^{2}-2\alpha
f_{d}(\theta_{t-1})+2\alpha^{2}w(\theta_{t-1})+2\alpha^{2}w(\tilde{\theta}).$
The inequality uses the bound obtained in Step 2 and equality uses gradient
splitting properties of $\bar{g}(\theta_{t-1}):$
$\displaystyle\begin{split}(\theta_{t-1}-\theta^{*})^{T}\bar{g}(\theta_{t-1})&=(\theta_{t-1}-\theta^{*})^{T}(\bar{g}(\theta_{t-1})-\bar{g}(\theta^{*}))\\\
&=(\theta_{t-1}-\theta^{*})^{T}(-A_{d}\theta_{t-1}+b+A_{d}\theta^{*}-b)\\\
&=-(\theta_{t-1}-\theta^{*})^{T}A_{d}(\theta_{t-1}-\theta^{*})=-f_{d}(\theta_{t-1}).\end{split}$
(5)
After rearranging terms it becomes:
$\operatorname{\mathbb{E}}_{s,s^{\prime}}||\theta_{t}-\theta^{*}||^{2}+2\alpha
f_{d}(\theta_{t-1})-2\alpha^{2}w(\theta_{t-1})\leq||\theta_{t-1}-\theta^{*}||^{2}+2\alpha^{2}w(\tilde{\theta}).$
###### Step A.4.
During this step we sum the inequality obtained in Step 3 over the epoch and
take another expectation to obtain:
$\sum_{t=1}^{M}\operatorname{\mathbb{E}}||\theta_{t}-\theta^{*}||^{2}+\sum_{t=1}^{M}2\alpha\operatorname{\mathbb{E}}f_{d}(\theta_{t-1})-\sum_{t=1}^{M}2\alpha^{2}\operatorname{\mathbb{E}}w(\theta_{t-1})\leq\sum_{t=1}^{M}\operatorname{\mathbb{E}}||\theta_{t-1}-\theta^{*}||^{2}+\sum_{t=1}^{M}2\alpha^{2}\operatorname{\mathbb{E}}w(\tilde{\theta}).$
(6)
We analyze this expression term-wise.
Notice that
$\sum_{t=1}^{M}\operatorname{\mathbb{E}}||\theta_{t-1}-\theta^{*}||^{2}$ and
$\sum_{t=1}^{M}\operatorname{\mathbb{E}}||\theta_{t}-\theta^{*}||^{2}$ consist
of the same terms, except the first term in the first sum and the last term in
the last sum, which are
$\operatorname{\mathbb{E}}||\theta_{0}-\theta^{*}||^{2}$ and
$\operatorname{\mathbb{E}}||\theta_{M}-\theta^{*}||^{2}$. Since
$\operatorname{\mathbb{E}}||\theta_{M}-\theta^{*}||^{2}$ is always positive
and it is on the left hand side of the inequality, we could drop it.
We denote the parameter vector $\theta$ chosen for epoch parameters at the end
of the epoch $\tilde{\theta}_{m}$. Since this vector is chosen uniformly at
random among all iteration vectors $\theta_{t}$, $t\in(0,M-1)$ we have that
$\sum_{t=1}^{M}\operatorname{\mathbb{E}}f_{d}(\theta_{t-1})=M\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m})$
and
$\sum_{t=1}^{M}\operatorname{\mathbb{E}}w(\theta_{t-1})=M\operatorname{\mathbb{E}}w(\tilde{\theta}_{m})$.
At the same time, $\tilde{\theta}$, which was chosen at the end of the
previous epoch remains the same throughout the epoch, therefore,
$\sum_{t=1}^{M}\operatorname{\mathbb{E}}w(\tilde{\theta})=M\operatorname{\mathbb{E}}w(\tilde{\theta})$.
Note, that the current epoch starts with setting $\theta_{0}=\tilde{\theta}$.
Also, to underline that $\tilde{\theta}$ during the current epoch refers to
the previous epoch, we denote it as $\tilde{\theta}_{m-1}$. Plugging these
values in (4) we have :
$2\alpha
M\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m})-2M\alpha^{2}\operatorname{\mathbb{E}}w(\tilde{\theta}_{m})\leq\operatorname{\mathbb{E}}||\tilde{\theta}_{m-1}-\theta^{*}||^{2}+2\alpha^{2}M\operatorname{\mathbb{E}}w(\tilde{\theta}_{m-1}).$
## Appendix B Convergence in terms of squared norm
At this point, we go on an aside to prove a result that is not in the main
body of the paper. We observe it is possible to derive a bound on Algorithm 1
in the squared norm. This bound is generally worse than the results we report
in the main body of the paper since it scales with the square of the condition
number.
###### Proposition B.1.
Suppose Assumptions 2.1, 2.2 hold. If we chose the learning rate as
$\alpha=\lambda_{A}/32$ and update batch size as $M=32/\lambda_{A}^{2}$, then
Algorithm 1 has a convergence rate of:
$\operatorname{\mathbb{E}}[||\tilde{\theta}_{m}-\theta^{*}||^{2}]\leq\left(\frac{5}{7}\right)^{m}||\tilde{\theta}_{0}-\theta^{*}||^{2}.$
This leads to batch size $M$ being $\mathcal{O}(1/\lambda_{A}^{2})$, which is
better than the results in [7], since their results have complexity
$\mathcal{O}(\kappa^{2}(C)\kappa_{G}^{2})$, where $\kappa(C)$ is the condition
number of matrix
$C=\operatorname{\mathbb{E}}_{s\in\mathcal{D}}[\phi(s)\phi(s)^{T}]$ and
$\kappa_{G}\propto 1/\lambda_{\rm min}(A^{T}C^{-1}A)$. Experimental comparison
of these values is given in Subsection H.2.
###### Proof.
To transform inequality (4) from Lemma 4.1 into a convergence rate guarantee,
we need to bound $w(\theta)$ and $f_{d}(\theta)$ in terms of
$||\theta-\theta^{*}||^{2}$. Both bounds are easy to show:
$\displaystyle w(\theta)$
$\displaystyle=\operatorname{\mathbb{E}}_{s,s^{\prime}}||g_{s,s^{\prime}}(\theta)-g_{s,s^{\prime}}(\theta^{*})||^{2}$
$\displaystyle=(\theta-\theta^{*})^{T}\operatorname{\mathbb{E}}_{s,s^{\prime}}[(\gamma\phi(s^{\prime})-\phi(s))\phi(s)^{T}\phi(s)(\gamma\phi(s^{\prime})-\phi(s))^{T}](\theta-\theta^{*})$
$\displaystyle\leq(\theta-\theta^{*})^{T}\operatorname{\mathbb{E}}_{s,s^{\prime}}[||(\gamma\phi(s^{\prime})-\phi(s))||\cdot||\phi(s)||\cdot||\phi(s)||\cdot||(\gamma\phi(s^{\prime})-\phi(s))||](\theta-\theta^{*})$
$\displaystyle\leq 4||\theta-\theta^{*}||^{2},$ $\displaystyle f_{d}(\theta)$
$\displaystyle=(\theta-\theta^{*})^{T}\operatorname{\mathbb{E}}_{s,s^{\prime}}[\phi(s)(\phi(s)-\gamma\phi(s^{\prime}))^{T}](\theta-\theta^{*})\geq\lambda_{A}||\theta-\theta^{*}||^{2},$
where $\operatorname{\mathbb{E}}_{s,s^{\prime}}$ denotes the expectation taken
with respect to a choice of pair of states $s,s^{\prime}$. Plugging these
bounds into Equation (4) we have:
$(2\alpha
M\lambda_{A}-8M\alpha^{2})||\tilde{\theta}_{m}-\theta^{*}||^{2}\leq(1+8M\alpha^{2})||\tilde{\theta}_{m-1}-\theta^{*}||^{2},$
which yields an epoch to epoch convergence rate of:
$\frac{1+8M\alpha^{2}}{2\alpha M\lambda_{A}-8M\alpha^{2}}.$
For this expression to be $<1$, we need that $\alpha M$ is set to
$\mathcal{O}(1/\lambda_{A})$, which means that $\alpha$ needs to be
$\mathcal{O}(\lambda_{A})$ for $M\alpha^{2}$ to be $\mathcal{O}(1)$.
Therefore, $M$ needs to be $\mathcal{O}(1/\lambda_{A}^{2})$. Setting
$\alpha=\lambda_{A}/32$ and $M=32/\lambda_{A}^{2}$ yields a convergence rate
of $5/7$. ∎
## Appendix C Proof of Theorem 4.2
An anaylsis of balanced dataset follows from unbalanced dataset but for
clarity of presentation we provide a proof for balanced dataset separately.
### C.1 Balanced dataset case
Similar to the previous section, we start with deriving bounds, but this time
we bound $||\theta-\theta^{*}||^{2}$ and $w(\theta)$ in terms of
$f_{d}(\theta)$. The first bound is straightforward:
$f_{d}(\theta)=(\theta-\theta^{*})^{T}\operatorname{\mathbb{E}}_{s,s^{\prime}}[\phi(s)(\phi(s)-\gamma\phi(s^{\prime})^{T}](\theta-\theta^{*})\implies||\theta-\theta^{*}||^{2}\leq\frac{1}{\lambda_{A}}f_{d}(\theta),$
where $\operatorname{\mathbb{E}}_{s,s^{\prime}}$ denotes the expectation taken
with respect to a choice of pair of states $s,s^{\prime}$. For $w(\theta)$ we
have:
$\displaystyle\begin{split}w(\theta)&=(\theta-\theta^{*})^{T}\operatorname{\mathbb{E}}_{s,s^{\prime}}[(\gamma\phi(s^{\prime})-\phi(s))\phi(s)^{T}\phi(s)(\gamma\phi(s^{\prime})-\phi(s))^{T}](\theta-\theta^{*})\\\
&=(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}(\gamma\phi(s^{\prime})-\phi(s))\phi^{T}(s)\phi(s)(\gamma\phi(s^{\prime})-\phi(s))^{T}\big{]}(\theta-\theta^{*})\\\
&\leq(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}(\gamma\phi(s^{\prime})-\phi(s))(\gamma\phi(s^{\prime})-\phi(s))^{T}\big{]}(\theta-\theta^{*})\\\
&=(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}\gamma^{2}\phi(s^{\prime})\phi(s^{\prime})^{T}-\gamma\phi(s^{\prime})\phi(s)^{T}\big{]}(\theta-\theta^{*})+f_{d}(\theta)\\\
&=(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}\gamma^{2}\phi(s)\phi(s)^{T}-\gamma\phi(s)\phi(s^{\prime})^{T}\big{]}(\theta-\theta^{*})+f_{d}(\theta)\\\
&\leq 2f_{d}(\theta),\end{split}$ (7)
where the first inequality uses Assumption 2.2, the third equality uses the
dataset balance property, and
$\sum_{s^{\prime}}\gamma^{2}\phi(s^{\prime})\phi(s^{\prime})^{T}=\sum_{s}\gamma^{2}\phi(s)\phi(s)^{T}$,
since $s$ and $s^{\prime}$ are the same set of states. The last inequality
uses the fact that $\gamma<1$.
Plugging these bound into Equation (4), we have:
$2\alpha
M\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m})-4M\alpha^{2}\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m})\leq\frac{1}{\lambda_{A}}\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m-1})+4\alpha^{2}M\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m-1}),$
which yields an epoch to epoch convergence rate of:
$\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m})\leq\Big{[}\frac{1}{2\lambda_{A}\alpha
M(1-2\alpha)}+\frac{2\alpha}{1-2\alpha}\Big{]}\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m-1}).$
Setting $\alpha=\frac{1}{8}$ and $M=\frac{16}{\lambda_{A}}$ we have the
desired inequality.
### C.2 Unbalanced dataset case
To prove the theorem we follow the same strategy as in C. For the
$f_{d}(\theta)$ we can use the same bound:
$f_{d}(\theta)=(\theta-\theta^{*})^{T}E_{s,s^{\prime}}[\phi(s)(\phi(s)-\gamma\phi(s^{\prime})^{T}](\theta-\theta^{*})\implies||\theta-\theta^{*}||^{2}\leq\frac{1}{\lambda_{A}}f_{d}(\theta).$
The bound for $w(\theta)$ is a little bit more difficult:
$\displaystyle w(\theta)$
$\displaystyle=(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}(\gamma\phi(s^{\prime})-\phi(s))\phi^{T}(s)\phi(s)(\gamma\phi(s^{\prime})-\phi(s))^{T}\big{]}(\theta-\theta^{*})$
$\displaystyle\leq(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}(\gamma\phi(s^{\prime})-\phi(s))(\gamma\phi(s^{\prime})-\phi(s))^{T}\big{]}(\theta-\theta^{*})$
$\displaystyle=(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}\gamma\phi(s^{\prime})(\gamma\phi(s^{\prime})-\phi(s))^{T}-\phi(s)(\gamma\phi(s^{\prime})-\phi(s))^{T}\big{]}(\theta-\theta^{*})$
$\displaystyle=(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}\gamma^{2}\phi(s^{\prime})\phi(s^{\prime})^{T}-\gamma\phi(s^{\prime})\phi(s)^{T}\big{]}(\theta-\theta^{*})+f_{d}(\theta)$
$\displaystyle=(\theta-\theta^{*})^{T}\big{[}\frac{1}{N}\sum_{s,s^{\prime}\in\mathcal{D}}\gamma^{2}\phi(s)\phi(s)^{T}-\gamma\phi(s)\phi(s^{\prime})^{T}\big{]}(\theta-\theta^{*})+f_{d}(\theta)$
$\displaystyle+\frac{\gamma^{2}}{N}(\theta-\theta^{*})^{T}(\phi(s_{N+1})\phi(s_{N+1})^{T}-\phi(s_{1})\phi(s_{1})^{T})(\theta-\theta^{*})^{T}$
$\displaystyle\leq
2f_{d}(\theta)+\frac{\gamma^{2}}{N}(\theta-\theta^{*})^{T}(\phi(s_{N+1})\phi(s_{N+1})^{T}-\phi(s_{1})\phi(s_{1})^{T})(\theta-\theta^{*})^{T}.$
The first inequality follows from Assumption 2.2. The third equality is
obtained by adding and subtracting
$\frac{\gamma^{2}}{N}(\theta-\theta^{*})^{T}\phi(s_{1})\phi(s_{1})^{T}(\theta-\theta^{*})$.
The second inequality uses the fact that $\gamma^{2}<1$. We denote the maximum
eigenvalue of the matrix
$\phi(s_{N+1})\phi(s_{N+1})^{T}-\phi(s_{1})\phi(s_{1})^{T}$ by $\mathcal{K}$
(note that $\mathcal{K}\leq 1$). Thus,
$w(\theta)\leq
2f_{d}(\theta)+\frac{\gamma^{2}\mathcal{K}}{N}||\theta-\theta^{*}||^{2}\leq
f_{d}(\theta)(2+\frac{\gamma^{2}\mathcal{K}}{N\lambda_{A}})\leq
f_{d}(\theta)(2+\frac{\gamma^{2}}{N\lambda_{A}}).$
Plugging these bounds into Equation (4) we have:
$(2\alpha
M-2M\alpha^{2}(2+\frac{\gamma^{2}}{N\lambda_{A}}))\operatorname{\mathbb{E}}f_{d}(\tilde{\theta}_{m})\leq(\frac{1}{\lambda_{A}}+2\alpha^{2}M(2+\frac{\gamma^{2}}{N\lambda_{A}}))f_{d}(\tilde{\theta}_{m-1}),$
which yields a convergence rate of:
$\frac{1}{\lambda_{A}2\alpha
M(1-\alpha(2+\frac{\gamma^{2}}{N\lambda_{A}}))}+\frac{\alpha(2+\frac{\gamma^{2}}{N\lambda_{A}})}{1-\alpha(2+\frac{\gamma^{2}}{N\lambda_{A}})}.$
To achieve constant convergence rate, for example $\frac{2}{3}$, we set up
$\alpha$ such that $\alpha(2+\frac{\gamma^{2}}{N\lambda_{A}})=0.25$, thus the
second term is equal to 1/3 and
$\alpha=\frac{1}{8+\frac{4\gamma^{2}}{N\lambda_{A}}}$. Then, to make the first
term equal to 1/3, we need to set
$M=\frac{2}{\lambda_{A}\alpha}=\frac{2}{\lambda_{A}\frac{1}{8+\frac{4\gamma^{2}}{N\lambda_{A}}}}.$
Thus, $\alpha$ is on the order of $\frac{1}{\max(1,1/(N\lambda_{A}))})$ and
$M$ is on the order of $\frac{1}{\lambda_{A}\min(1,N\lambda_{A})}$.
## Appendix D Discussion on Unbalanced dataset
If the dataset balance assumption is not satisfied, it is always possible to
modify the MDP slightly and make it satisfied. Indeed, suppose we are given an
MDP $M$ with initial state (or distribution) $s_{0}$ and discount factor
$\gamma$. We can then modify the transition probabilities by always
transitioning to $s_{0}$ with probability $p$ regardless of state and action
chosen (and doing the normal transition from the MDP $M$ with probability
$1-p$), and changing the discount factor to a new $\gamma^{\prime}$. Calling
the new MDP $M^{\prime}$, we have that:
* •
It is very easy to draw a dataset from $M^{\prime}$ such that the last state
is the same as the first one (just make sure to end on a transition to
$s_{0}$!) and the collected dataset will have the dataset balance property.
* •
Under appropriate choice of $p$ and $\gamma^{\prime}$, the value function
$V_{M}$ in the original MDP can be easily recovered from the value function of
the new MDP $V_{M^{\prime}}$.
A formal statement of this is in the comment below. Note that all we need to
be able to do is change the discount factor (which we usually set) as well as
be able to restart the MDP (which we can do in any computer simulation).
The only caveat that the size of the dataset one can draw this way will have
to be at least $(1-\gamma)^{-1}$ in expectation because to make the above
sketch work will require a choice of $p$ that is essentially proportional to
$(1-\gamma)$ (see Theorem statement in the next comment for a formal
statement). This is not a problem in practice, as typical discount factors are
usually $\approx 0.99$, whereas datasets tend to be many orders of magnitude
bigger than $\approx 100=(1-\gamma)^{-1}$. Even a discount factor of $\approx
0.999$, much closer to one than is used in practice, only forces us to draw a
dataset of size $1000$ in expectation.
###### Theorem D.1.
Choose
$\gamma^{\prime}=\frac{1+\gamma}{2},p=\frac{1-\gamma}{1+\gamma}$
and consider the pair of MDPs $M$ and $M^{\prime}$ which are defined in our
previous comment. Then the quantities $V_{M}(s)$ and $V_{M^{\prime}}(s)$
satisfy the following recursion:
$V_{M}(s)=V_{M^{\prime}}(s)+\frac{\gamma(1-\gamma)}{1+\gamma-2\gamma^{2}}V_{M^{\prime}}(s_{0})$
###### Proof.
Let $T$ denote be a time step when the first reset appears. We can condition
on $T$ to represent $V_{M^{\prime}}(s)$ as:
$V_{M^{\prime}}(s)=\sum_{t^{\prime}=1}^{\infty}P(t^{\prime}=T)E[V_{M^{\prime}}(s)|t^{\prime}=T]$
$\quad\quad\ \ \ \
=\sum_{t^{\prime}=1}^{\infty}(1-p)^{t^{\prime}-1}p((\sum_{t=1}^{t^{\prime}}\gamma^{\prime
t-1}E[r_{t}])+\gamma^{\prime t^{\prime}}V_{M^{\prime}}(s_{0})),$
where the expected rewards $E[r_{t}]$ are the same as in the original MDP. We
next change the order of summations:
$V_{M^{\prime}}(s)=\sum_{t=1}^{\infty}(\gamma^{\prime
t-1}E[r(t)]\sum_{t^{\prime}=t}^{\infty}(1-p)^{t^{\prime}-1}p)+\sum_{t^{\prime}=1}^{\infty}(1-p)^{t^{\prime}-1}p\gamma^{\prime
t^{\prime}}V_{M^{\prime}}(s_{0})$
$\quad\quad\ \ \ \ =\sum_{t^{\prime}=1}^{\infty}\gamma^{\prime
t^{\prime}-1}(1-p)^{t^{\prime}-1}E[r(t)]+(1-p)^{t^{\prime}-1}p\gamma^{\prime
t^{\prime}}V_{M^{\prime}}(s_{0})$
Now we use the fact that the chosen $\gamma^{\prime}=\gamma/(1-p)$ and perform
some algebraic manipulations:
$V_{M^{\prime}}(s)=\sum_{t^{\prime}=1}^{\infty}\gamma^{t^{\prime}-1}E[r(t)]+(1-p)^{t^{\prime}-1}p\gamma^{\prime
t^{\prime}}V_{M^{\prime}}(s_{0})$
$\quad\quad\ \ \ \
=V_{M}(s)+\sum_{t^{\prime}=1}^{\infty}(1-p)^{t^{\prime}-1}p\gamma^{\prime
t^{\prime}}V_{M^{\prime}}(s_{0})$
$\quad\quad\ \ \ \ =V_{M}(s)+\frac{\gamma p}{1-\gamma
p}V_{M^{\prime}}(s_{0}),$
which implies the claimed equality:
$V_{M}(s)=V_{M^{\prime}}(s)-\frac{\gamma(1-\gamma)}{1+\gamma-2\gamma^{2}}V_{M^{\prime}}(s_{0})$
QED. ∎
As claimed above, this theorem can be used to recover $V_{M}$ from
$V_{M^{\prime}}$. Please don’t hesitate to ask us any questions during the
author-reviewer response period if any part of this derivation is unclear.
As stated in our comment above, the artificial addition of a reset button as
above makes it possible to generate a dataset which satisfies our dataset
balance assumption from any MDP.
## Appendix E TD-SVRG with batching
In this section, we extend our results to an inexact mean-path update
computation, applying the results of [1] to the TD SVRG algorithm. We show
that the geometric convergence rate might be achieved with a smaller number of
computations by estimating the mean-path TD-update instead of performing full
computation. This approach is similar to [15], but doesn’t require dual
variables and achieves better results.
Algorithm E.1 TD-SVRG with batching for the finite sample case
Parameters update batch size $M$ and learning rate $\alpha$.
Initialize $\tilde{\theta}_{0}$.
for $m=1,2,...$ do
$\tilde{\theta}=\tilde{\theta}_{m-1}$,
choose estimation batch size $n_{m}$,
sample batch $\mathcal{D}^{m}$ of size $n_{m}$ from $\mathcal{D}$ w/o
replacement,
compute
$g_{m}(\tilde{\theta})=\frac{1}{n_{m}}\sum_{s,s^{\prime}\in\mathcal{D}^{m}}g_{s,s^{\prime}}(\tilde{\theta})$,
where
$g_{s,s^{\prime}}(\tilde{\theta})=(r(s,s^{\prime})+\gamma\phi(s^{\prime})^{T}\tilde{\theta}-\phi(s)^{T}\tilde{\theta})\phi(s_{t})$.
$\theta_{0}=\tilde{\theta}$.
for $t=1$ to $M$ do
Sample $s,s^{\prime}$ from ${\cal D}$.
Compute
$v_{t}=g_{s,s^{\prime}}(\theta_{t-1})-g_{s,s^{\prime}}(\tilde{\theta})+g_{m}(\tilde{\theta})$.
Update parameters $\theta_{t}=\theta_{t-1}+\alpha v_{t}$.
end for
Set $\tilde{\theta}_{m}=\theta_{t^{\prime}}$ for randomly chosen
$t^{\prime}\in(0,\ldots,M-1)$.
end for
Since the computation of the mean-path error is not related to the dataset
balance, in this section we assume that the dataset is balanced for
simplicity.
###### Theorem E.1.
Suppose Assumptions 2.1, 2.2 hold and the algorithm run for total of $m$
epochs. Then if the learning rate is chosen as $\alpha=1/8$, update batch size
$M=16/\lambda_{A}$ and estimation batch size during epoch $m^{\prime}$ is
$n_{m^{\prime}}=\min\left(N,\frac{N}{N-1}\frac{1}{c\lambda_{A}(2/3)^{m}}(4f(\tilde{\theta}_{m^{\prime}})+\sigma^{2}))\right)$,
where $c$ is a parameter and $\sigma^{2}=E[g_{s,s^{\prime}}(\theta^{*})]$ is
an optimal point update variance, Algorithm E.1 will converge to optimum with
a convergence rate of:
$\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})]\leq\left(\frac{2}{3}\right)^{m}(f_{d}(\tilde{\theta}_{0})+C),$
where $C$ is a constant dependent on the parameter $c$.
###### Proof.
The proof is given below E.1. ∎
This result is an improvement on [15], compared to which it improves both the
estimation and update batch sizes. In update batch size, our result is better
by at least a factor of $1/((1-\gamma)\pi_{\rm min}^{3})$, where $\pi_{\rm
min}$ is a minimum probability in the stationary distribution, see Table 1 for
theoretical results and Section H for experimental comparison. In estimation
batch size, we have given the result explicitly in terms of the iterate norm,
while in [15] authors have a bound in terms of the variance of both primal and
dual update vectors ($\Xi^{2}$ in their notation).
Note, that both quantities $f(\tilde{\theta}_{m^{\prime}})$ and $\sigma^{2}$
required to compute estimation batch size $n_{m^{\prime}}$ are not know during
the run of the algorithm. However, we provide an alternative quantity, which
might be used in practice:
$n_{m^{\prime}}=\min(N,\frac{N}{N-1}\frac{1}{c\lambda_{A}(2/3)^{m}}(2|r_{\rm
max}|^{2}+8||\tilde{\theta}_{m^{\prime}-1}||^{2})$, where $|r_{\rm max}|$ is
the maximum absolute reward.
### E.1 Proof of Theorem E.1
In the first part of the proof we derive an inequality which relates model
parameters of two consecutive epochs similar to what we achieved in previous
proofs, but now we introduce error vector to show that the mean path update is
estimated instead of being computed exactly. In this proof, we follow the same
4 steps we introduced in the proof of Lemma 4.1. In the second part of the
proof we show that there are conditions under which the error term converges
to 0.
###### Step E.1.
During the first step we use the bound obtained in inequality (7):
$w(\theta)\leq 2f_{d}(\theta).$
###### Step E.2.
During this step we derive a bound on the squared norm of a single update
$\operatorname{\mathbb{E}}[||v_{t}||^{2}]$. But now, compared to previous
case, we do not compute the exact mean-path updated $\bar{g}(\theta)$, but its
estimate, and assume our computation has error
$g_{m}(\theta)=\bar{g}(\theta)+\eta_{m}$. Thus, during iteration $t$ of epoch
$m$ the single update vector is
$v_{t}=g_{t}(\theta_{t-1})-g_{t}(\tilde{\theta})+\bar{g}(\tilde{\theta})+\eta_{m}.$
Taking expectation conditioned on all history previous to epoch $m$, which we
denote as $\mathcal{F}_{m-1}$, the bound on the single update can be derived
as:
$\displaystyle\operatorname{\mathbb{E}}[||v_{t}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||g_{t}(\theta_{t-1})-g_{t}(\tilde{\theta})+\bar{g}(\tilde{\theta})+\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-\bar{g}(\theta^{*}))+(\bar{g}(\theta^{*})-g_{t}(\tilde{\theta})+\bar{g}(\tilde{\theta})+\eta_{m})||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq
2\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-g_{t}(\theta^{*}))||^{2}|\mathcal{F}_{m-1}]+$
$\displaystyle
2\operatorname{\mathbb{E}}[||g_{t}(\tilde{\theta})-g_{t}(\theta^{*})-(\bar{g}(\tilde{\theta})-\bar{g}(\theta^{*}))-\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=2\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-g_{t}(\theta^{*}))||^{2}|\mathcal{F}_{m-1}]+$
$\displaystyle
2\operatorname{\mathbb{E}}[||g_{t}(\tilde{\theta})-g_{t}(\theta^{*})-\operatorname{\mathbb{E}}[g_{t}(\tilde{\theta})-g_{t}(\theta^{*})]-\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=2\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-g_{t}(\theta^{*}))||^{2}|\mathcal{F}_{m-1}]+$
$\displaystyle
2\operatorname{\mathbb{E}}[||g_{t}(\tilde{\theta})-g_{t}(\theta^{*})-E[g_{t}(\tilde{\theta})-g_{t}(\theta^{*})]||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle-4\operatorname{\mathbb{E}}[\langle
g_{t}(\tilde{\theta})-g_{t}(\theta^{*})-E[g_{t}(\tilde{\theta})-g_{t}(\theta^{*})],\eta_{m}\rangle|\mathcal{F}_{m-1}]+2\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq
2\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-g_{t}(\theta^{*}))||^{2}|\mathcal{F}_{m-1}]+$
$\displaystyle
2\operatorname{\mathbb{E}}[||g_{t}(\tilde{\theta})-g_{t}(\theta^{*})||^{2}|\mathcal{F}_{m-1}]+2\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=2\operatorname{\mathbb{E}}[w(\theta_{t-1})+2w(\tilde{\theta})+2||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq\operatorname{\mathbb{E}}[4f_{d}(\theta_{t-1})+4f_{d}(\tilde{\theta})+2\operatorname{\mathbb{E}}||\eta_{m}||^{2}|\mathcal{F}_{m-1}],$
where first inequality uses $\operatorname{\mathbb{E}}||A+B||^{2}\leq
2\operatorname{\mathbb{E}}||A||^{2}+2\operatorname{\mathbb{E}}||B||^{2}$, the
second inequality uses
$\operatorname{\mathbb{E}}||A-\operatorname{\mathbb{E}}[A]||^{2}\leq\operatorname{\mathbb{E}}||A||^{2}$
and the fact
$\operatorname{\mathbb{E}}[\eta_{m}|g(\tilde{\theta})-g(\theta^{*})-\operatorname{\mathbb{E}}_{s,s^{\prime}}[g(\tilde{\theta})-g(\theta^{*})],\mathcal{F}_{m-1}]=0$;
third inequality uses the result of Step E.1.
###### Step E.3.
During this step, we derive a bound on a vector norm after a single update:
$\displaystyle\operatorname{\mathbb{E}}[||\theta_{t}-\theta^{*}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}+\alpha
v_{t}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+2\alpha(\theta_{t-1}-\theta^{*})^{T}v_{t}+\alpha^{2}||v_{t}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+2\alpha(\theta_{t-1}-\theta^{*})^{T}\bar{g}(\theta_{t-1})+2\alpha(\theta_{t-1}-\theta^{*})^{T}\eta_{m}$
$\displaystyle
4\alpha^{2}f_{d}(\theta_{t-1})+4\alpha^{2}f_{d}(\tilde{\theta})+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}-2\alpha
f_{d}(\theta_{t-1})+2\alpha(\theta_{t-1}-\theta^{*})^{T}\eta_{m}$
$\displaystyle+4\alpha^{2}f_{d}(\theta_{t-1})+4\alpha^{2}f_{d}(\tilde{\theta})+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}],$
where the first inequality uses
$\displaystyle\operatorname{\mathbb{E}}[2\alpha(\theta_{t-1}-\theta^{*})^{T}v_{t}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[2\alpha(\theta_{t-1}-\theta^{*})^{T}(g_{t}(\theta_{t-1})-g_{t}(\tilde{\theta})+\bar{g}(\tilde{\theta})+\eta_{m})|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[2\alpha(\theta_{t-1}-\theta^{*})^{T}(\bar{g}(\theta_{t-1})-\bar{g}(\tilde{\theta})+\bar{g}(\tilde{\theta})+\eta_{m})|\mathcal{F}_{m-1}]$
and the last equality uses (5). Rearranging terms we obtain:
$\displaystyle\operatorname{\mathbb{E}}[||\theta_{t}-\theta^{*}||^{2}+2\alpha
f_{d}(\theta_{t-1})-4\alpha^{2}f_{d}(\theta_{t-1})|\mathcal{F}_{m-1}]$
$\displaystyle\leq\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+4\alpha^{2}f_{d}(\tilde{\theta})+2\alpha(\theta_{t-1}-\theta^{*})^{T}\eta_{m}+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+4\alpha^{2}f_{d}(\tilde{\theta})+2\alpha||\theta_{t-1}-\theta^{*}||\cdot||\eta_{m}||+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}].$
$\displaystyle\leq\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+4\alpha^{2}f_{d}(\tilde{\theta})+2\alpha(\frac{\lambda_{A}}{2}||\theta_{t-1}-\theta^{*}||^{2}+\frac{1}{2\lambda_{A}}||\eta_{m}||^{2})+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}].$
###### Step E.4.
Now derive a bound on epoch update. We use similar logic as during the proof
of Theorem 4.2. Since the error term doesn’t change over the epoch, summing
over the epoch we have:
$\displaystyle\operatorname{\mathbb{E}}[||\theta_{m}-\theta^{*}||^{2}+2\alpha
Mf_{d}(\tilde{\theta}_{m})-4\alpha^{2}Mf_{d}(\tilde{\theta}_{m})|\mathcal{F}_{m-1}]\leq$
$\displaystyle||\theta_{0}-\theta^{*}||^{2}+4\alpha^{2}Mf_{d}(\tilde{\theta}_{m-1})+\operatorname{\mathbb{E}}[2\alpha\sum_{t=1}^{M}(\frac{\lambda_{A}}{2}||\theta_{t-1}-\theta^{*}||^{2}+\frac{1}{2\lambda_{A}}||\eta_{m}||^{2})+2\alpha^{2}M||\eta_{m}||^{2}|\mathcal{F}_{m-1}]=$
$\displaystyle||\tilde{\theta}_{m-1}-\theta^{*}||^{2}+4\alpha^{2}Mf_{d}(\tilde{\theta}_{m-1})+\operatorname{\mathbb{E}}[2\alpha
M(\frac{\lambda_{A}}{2}||\tilde{\theta}_{m}-\theta^{*}||^{2}+\frac{1}{2\lambda_{A}}||\eta_{m}||^{2})+2\alpha^{2}M||\eta_{m}||^{2}|\mathcal{F}_{m-1}]\leq$
$\displaystyle\frac{1}{\lambda_{A}}f_{d}(\tilde{\theta}_{m-1})+4\alpha^{2}Mf_{d}(\tilde{\theta}_{m-1})+\operatorname{\mathbb{E}}[2\alpha
M\left(\frac{1}{2}f_{d}(\tilde{\theta}_{m})+\frac{1}{2\lambda_{A}}||\eta_{m}||^{2}\right)+2\alpha^{2}M||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
Rearranging terms, dropping
$\operatorname{\mathbb{E}}||\theta_{m}-\theta^{*}||^{2}$ and dividing by
$2\alpha M$ we further obtain:
$\displaystyle\left(\frac{1}{2}-2\alpha\right)\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})|\mathcal{F}_{m-1}]\leq$
$\displaystyle\left(\frac{1}{2\alpha
M\lambda_{A}}+2\alpha\right)f_{d}(\tilde{\theta}_{m-1})+\left(\frac{1}{2\lambda_{a}}+\alpha\right)\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
Dividing both sides of this equation to $0.5-2\alpha$ we have the epoch
convergence:
$\displaystyle\begin{split}\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})|\mathcal{F}_{m-1}]\leq&\left(\frac{1}{\lambda_{A}2\alpha
M(0.5-2\alpha)}+\frac{2\alpha}{0.5-2\alpha}\right)f_{d}(\tilde{\theta}_{m-1})+\\\
&\left(\frac{1}{2\lambda_{a}(0.5-2\alpha)}+\frac{\alpha}{0.5-2\alpha}\right)\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}].\\\
\end{split}$ (8)
To achieve convergence, we need to guarantee the linear convergence of the
first and second terms in the sum separately. The first term is dependent on
inner loop updates; its convergence is analyzed in Theorem 4.2. Here we show
how to achieve a similar geometric convergence rate of the second term. Since
the error term has 0 mean and we are in a finite sample case with replacement,
the expected squared norm can be bounded by:
$\operatorname{\mathbb{E}}||\eta_{m}||^{2}\leq\frac{N-n_{m}}{Nn_{m}}S^{2}\leq\left(1-\frac{n_{m}}{N}\right)\frac{S^{2}}{n_{m}}\leq\frac{S^{2}}{n_{m}},$
where $S^{2}$ is a bound on the update vector norm variance. If we want the
error to be bounded by $c\rho^{m}$, we need the estimation batch size $n_{m}$
to satisfy the condition:
$n_{m}\geq\frac{S^{2}}{c\rho^{m}}.$
until growing batch size reaches sample size. Satisfying this condition,
guarantees that the second term has geometric convergence:
$\left(\frac{1}{2\lambda_{a}(0.5-2\alpha)}+\frac{\alpha}{0.5-2\alpha}\right)\operatorname{\mathbb{E}}||\eta_{m}||^{2}\leq\left(\frac{1}{2\lambda_{a}(0.5-2\alpha)}+\frac{\alpha}{0.5-2\alpha}\right)c\rho^{m}.$
It remains to derive a bound $S^{2}$ for the update vector norm sample
variance:
$\displaystyle\frac{1}{N-1}\sum_{s,s^{\prime}}||g_{s,s^{\prime}}(\theta)||^{2}-||\bar{g}(\theta)||^{2}\leq$
$\displaystyle\frac{N}{N-1}\frac{1}{N}\sum_{s,s^{\prime}}||g_{s,s^{\prime}}(\theta)||^{2}=\frac{N}{N-1}\frac{1}{N}\sum_{s,s^{\prime}}||g_{s,s^{\prime}}(\theta)-g_{s,s^{\prime}}(\theta^{*})+g_{s,s^{\prime}}(\theta^{*})||^{2}\leq$
$\displaystyle\frac{N}{N-1}\frac{1}{N}\sum_{s,s^{\prime}}2||g_{s,s^{\prime}}(\theta)-g_{s,s^{\prime}}(\theta^{*})||^{2}+2||g_{s,s^{\prime}}(\theta^{*})||^{2}=$
$\displaystyle\frac{N}{N-1}(2w(\theta)+2\sigma^{2})\leq\frac{N}{N-1}(4f(\theta)+2\sigma^{2})=S^{2},$
where $\sigma^{2}$ is the variance of the updates in the optimal point similar
to [2].
Alternatively, we might derive a bound $S^{2}$ in terms of quantities known
during the algorithm execution:
$\displaystyle\frac{1}{N-1}\sum_{s,s^{\prime}}||g_{s,s^{\prime}}(\theta)||^{2}-||\bar{g}(\theta)||^{2}\leq$
$\displaystyle\frac{N}{N-1}\frac{1}{N}\sum_{s,s^{\prime}}||g_{s,s^{\prime}}(\theta)||^{2}=\frac{N}{N-1}\frac{1}{N}\sum_{s,s^{\prime}}||(r(s,s^{\prime})+\gamma\phi(s^{\prime})^{T}\theta-\phi(s)^{T}\theta)\phi(s)||^{2}\leq$
$\displaystyle\frac{N}{N-1}\frac{1}{N}\sum_{s,s^{\prime}}2||r\phi(s)||^{2}+4||\gamma\phi(s^{\prime})^{T}\theta\phi(s)||^{2}+4||\phi(s)^{T}\theta\phi(s)||^{2}\leq$
$\displaystyle\frac{N}{N-1}(2|r_{max}|^{2}+4\gamma^{2}||\theta||^{2}+4||\theta||^{2})=\frac{N}{N-1}(2|r_{max}|^{2}+8||\theta||^{2})=S^{2}.$
Having the convergence of the both terms of 8, we proceed by expanding the
equation for earlier epochs (denoting bracket terms as $\rho$ and
$\rho^{\prime}$):
$\displaystyle\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})|\mathcal{F}_{m-1}]\leq\rho
f_{d}(\tilde{\theta}_{m-1})+\rho^{\prime}\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}]\implies$
$\displaystyle\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})|\mathcal{F}_{m-2}]\leq\rho^{2}f_{d}(\tilde{\theta}_{m-2})+\rho^{\prime}(\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-2}]+\rho\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}])\implies$
$\displaystyle\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})]\leq\rho^{m}f_{d}(\tilde{\theta}_{0})+\rho^{\prime}(\sum_{i=1}^{m}\rho^{i}\operatorname{\mathbb{E}}[||\eta_{i}||^{2}|\mathcal{F}_{i}])$
Now, assuming that estimation batch sizes are large enough that all error
terms are bounded by $c\rho^{m}$:
$\displaystyle\operatorname{\mathbb{E}}[f_{d}(\tilde{\theta}_{m})|\mathcal{F}_{m-1}]\leq\rho^{m}f_{d}(\tilde{\theta}_{0})+\rho^{\prime}(\sum_{i=1}^{m}\rho^{i}c\rho^{m})\leq\rho^{m}f_{d}(\tilde{\theta}_{0})+\rho^{m}\frac{c\rho^{\prime}}{1-\rho}$
Denoting $\frac{c\rho^{\prime}}{1-\rho}$ as $C$ we have the claimed result.
## Appendix F Proof of Theorem 5.1
The proof is very similar to 8, the only difference is that now we derive an
expectation with respect to an MDP instead of a finite sample dataset.
###### Step F.1.
During the first step we use the bound obtained during the proof of Theorem
4.2:
$\displaystyle\begin{split}w(\theta)&=(\theta-\theta^{*})^{T}\operatorname{\mathbb{E}}[(\gamma\phi(s^{\prime})-\phi(s))\phi(s)^{T}\phi(s)(\gamma\phi(s^{\prime})-\phi(s))^{T}](\theta-\theta^{*})\\\
&=(\theta-\theta^{*})^{T}\big{[}\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})(\gamma\phi(s^{\prime})-\phi(s))\phi^{T}(s)\phi(s)(\gamma\phi(s^{\prime})-\phi(s))^{T}\big{]}(\theta-\theta^{*})\\\
&\leq(\theta-\theta^{*})^{T}\big{[}\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})(\gamma\phi(s^{\prime})-\phi(s))(\gamma\phi(s^{\prime})-\phi(s))^{T}\big{]}(\theta-\theta^{*})\\\
&=(\theta-\theta^{*})^{T}\big{[}\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})(\gamma^{2}\phi(s^{\prime})\phi(s^{\prime})^{T}-\gamma\phi(s^{\prime})\phi(s)^{T})\big{]}(\theta-\theta^{*})+f_{e}(\theta)\\\
&=(\theta-\theta^{*})^{T}\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})(\gamma^{2}\phi(s)\phi(s)^{T}-\gamma\phi(s)\phi(s^{\prime})^{T})\big{]}(\theta-\theta^{*})+f_{e}(\theta)\\\
&\leq 2f_{e}(\theta),\end{split}$ (9)
where the first inequality uses Assumption 2.2, the third equality uses the
fact that $\mu_{\pi}$ is a stationary distribution of $P$
($\sum_{s^{\prime}}\gamma^{2}\mu_{\pi}(s)P(s,s^{\prime})\phi(s^{\prime})\phi(s^{\prime})^{T}=\sum_{s^{\prime}}\gamma^{2}\mu_{\pi}(s^{\prime})\phi(s^{\prime})\phi(s^{\prime})^{T}=\sum_{s}\mu_{\pi}(s)\gamma^{2}\phi(s)\phi(s)^{T}$).
The last inequality uses the fact that $\gamma<1$.
###### Step F.2.
During this step we derive a bound on the squared norm of a single update
$\operatorname{\mathbb{E}}[||v_{t}||^{2}]$, which is performed during time
step $t$ of epoch $m$. Since we are aiming to derive epoch to epoch
convergence bound, we will be taking expectation conditioned on all history
previous to epoch $m$, which we denote as $\mathcal{F}_{m-1}$. Similarly with
Appendix E.1 we assume that mean path update in the end of the previous epoch
was computed inexactly and has estimation error:
$\bar{g}(\tilde{\theta}_{m-1})+\eta_{m}$. Thus the single update vector
becomes (for simplicity we denote $\tilde{\theta}_{m-1}$ as $\tilde{\theta}$):
$v_{t}=g_{t}(\theta_{t-1})-g_{t}(\tilde{\theta})+\bar{g}(\tilde{\theta})+\eta_{m}.$
The norm of this vector is bounded by:
$\displaystyle\operatorname{\mathbb{E}}[||v_{t}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||g_{t}(\theta_{t-1})-g_{t}(\tilde{\theta})+\bar{g}(\tilde{\theta})+\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-\bar{g}(\theta^{*}))+(\bar{g}(\theta^{*})-g_{t}(\tilde{\theta})+\bar{g}(\tilde{\theta})+\eta_{m})||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq
2\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-g_{t}(\theta^{*}))||^{2}|\mathcal{F}_{m-1}]+$
$\displaystyle
2\operatorname{\mathbb{E}}[||g_{t}(\tilde{\theta})-g_{t}(\theta^{*})-(\bar{g}(\tilde{\theta})-\bar{g}(\theta^{*}))-\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=2\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-g_{t}(\theta^{*}))||^{2}|\mathcal{F}_{m-1}]+$
$\displaystyle
2\operatorname{\mathbb{E}}[||g_{t}(\tilde{\theta})-g_{t}(\theta^{*})-\operatorname{\mathbb{E}}[g_{t}(\tilde{\theta})-g_{t}(\theta^{*})]-\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=2\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-g_{t}(\theta^{*}))||^{2}|\mathcal{F}_{m-1}]+$
$\displaystyle
2\operatorname{\mathbb{E}}[||g_{t}(\tilde{\theta})-g_{t}(\theta^{*})-\operatorname{\mathbb{E}}[g_{t}(\tilde{\theta})-g_{t}(\theta^{*})]||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle-\operatorname{\mathbb{E}}[\langle
g_{t}(\tilde{\theta})-g_{t}(\theta^{*})-\operatorname{\mathbb{E}}[g_{t}(\tilde{\theta})-g_{t}(\theta^{*})],\eta_{m}\rangle|\mathcal{F}_{m-1}]+2\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq
2\operatorname{\mathbb{E}}[||(g_{t}(\theta_{t-1})-g_{t}(\theta^{*}))||^{2}|\mathcal{F}_{m-1}]+$
$\displaystyle
2\operatorname{\mathbb{E}}[||g_{t}(\tilde{\theta})-g_{t}(\theta^{*})||^{2}|\mathcal{F}_{m-1}]+2\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=2\operatorname{\mathbb{E}}[w(\theta_{t-1})+2w(\tilde{\theta})+2||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq\operatorname{\mathbb{E}}[4f_{e}(\theta_{t-1})+4f_{e}(\tilde{\theta})+2||\eta_{m}||^{2}|\mathcal{F}_{m-1}],$
where first inequality uses $\operatorname{\mathbb{E}}||A+B||^{2}\leq
2\operatorname{\mathbb{E}}||A||^{2}+2\operatorname{\mathbb{E}}||B||^{2}$,
second inequality uses
$\operatorname{\mathbb{E}}||A-\operatorname{\mathbb{E}}[A]||^{2}\leq\operatorname{\mathbb{E}}||A||^{2}$
and the fact
$\operatorname{\mathbb{E}}[\eta_{m}|g(\tilde{\theta})-g(\theta^{*})-\operatorname{\mathbb{E}}_{s,s^{\prime}}[g(\tilde{\theta})-g(\theta^{*})],\mathcal{F}_{m-1}]=0$
;the third inequality uses the result of Step F.1.
###### Step F.3.
Bound on a vector norm after a single update during iteration $t$ of epoch
$m$:
$\displaystyle\operatorname{\mathbb{E}}[||\theta_{t}-\theta^{*}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}+\alpha
v_{t}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+2\alpha(\theta_{t-1}-\theta^{*})^{T}v_{t}+\alpha^{2}||v_{t}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle=\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+2\alpha(\theta_{t-1}-\theta^{*})^{T}\bar{g}(\theta_{t-1})+2\alpha(\theta_{t-1}-\theta^{*})^{T}\eta_{m}$
$\displaystyle+4\alpha^{2}f_{e}(\theta_{t-1})+4\alpha^{2}f_{e}(\tilde{\theta})+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}].$
Applying an argument similar to 5 and rearranging terms we obtain:
$\displaystyle\operatorname{\mathbb{E}}[||\theta_{t}-\theta^{*}||^{2}+2\alpha
f_{e}(\theta_{t-1})-4\alpha^{2}f_{e}(\theta_{t-1})|\mathcal{F}_{m-1}]$
$\displaystyle\leq\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+4\alpha^{2}f_{e}(\tilde{\theta})-2\alpha(\theta_{t-1}-\theta^{*})^{T}\eta+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
$\displaystyle\leq\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+4\alpha^{2}f_{e}(\tilde{\theta})+2\alpha||\theta_{t-1}-\theta^{*}||\cdot||\eta||+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}].$
$\displaystyle\leq\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||^{2}+4\alpha^{2}f_{e}(\tilde{\theta})+2\alpha(\frac{\lambda_{A}}{2}||\theta_{t-1}-\theta^{*}||^{2}+\frac{1}{2\lambda_{A}}||\eta_{m}||^{2})+2\alpha^{2}||\eta_{m}||^{2}|\mathcal{F}_{m-1}].$
###### Step F.4.
Now derive a bound on epoch update. We use the similar logic as during the
proof of Theorem 4.2. Since the error term doesn’t change over the epoch,
summing over the epoch we have:
$\displaystyle\operatorname{\mathbb{E}}[||\theta_{m}-\theta^{*}||^{2}+2\alpha
Mf_{e}(\tilde{\theta}_{m})-4\alpha^{2}Mf_{e}(\tilde{\theta}_{m})|\mathcal{F}_{m-1}]\leq$
$\displaystyle||\theta_{0}-\theta^{*}||^{2}+4\alpha^{2}Mf_{e}(\tilde{\theta}_{m-1})+\operatorname{\mathbb{E}}[2\alpha\sum_{t=1}^{M}(\frac{\lambda_{A}}{2}||\theta_{t-1}-\theta^{*}||^{2}+\frac{1}{2\lambda_{A}}||\eta_{m}||^{2})+2\alpha^{2}M||\eta_{m}||^{2}|\mathcal{F}_{m-1}]=$
$\displaystyle||\tilde{\theta}_{m-1}-\theta^{*}||^{2}+4\alpha^{2}Mf_{e}(\tilde{\theta}_{m-1})+\operatorname{\mathbb{E}}[2\alpha
M(\frac{\lambda_{A}}{2}||\tilde{\theta}_{m}-\theta^{*}||^{2}+\frac{1}{2\lambda_{A}}||\eta_{m}||^{2})+2\alpha^{2}M||\eta_{m}||^{2}|\mathcal{F}_{m-1}]\leq$
$\displaystyle\frac{1}{\lambda_{a}}f_{e}(\tilde{\theta}_{m-1})+4\alpha^{2}Mf_{e}(\tilde{\theta}_{m-1})+\operatorname{\mathbb{E}}[2\alpha
M\left(\frac{1}{2}f_{e}(\tilde{\theta}_{m})+\frac{1}{2\lambda_{A}}||\eta_{m}||^{2}\right)+2\alpha^{2}M||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
Rearranging terms, dropping
$\operatorname{\mathbb{E}}||\theta_{m}-\theta^{*}||^{2}$ and dividing by
$2\alpha M$ we further obtain:
$\displaystyle\left(\frac{1}{2}-2\alpha\right)\operatorname{\mathbb{E}}[f_{e}(\tilde{\theta}_{m})|\mathcal{F}_{m-1}]\leq$
$\displaystyle\left(\frac{1}{2\alpha
M\lambda_{A}}+2\alpha\right)f_{e}(\tilde{\theta}_{m-1})+\left(\frac{1}{2\lambda_{a}}+\alpha\right)\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}]$
Dividing both sides of this equation to $0.5-2\alpha$ we have the epoch
convergence:
$\displaystyle\operatorname{\mathbb{E}}[f_{e}(\tilde{\theta}_{m})|\mathcal{F}_{m-1}]\leq$
$\displaystyle\left(\frac{1}{\lambda_{A}2\alpha
M(0.5-2\alpha)}+\frac{2\alpha}{0.5-2\alpha}\right)f_{e}(\tilde{\theta}_{m-1})+$
$\displaystyle\left(\frac{1}{2\lambda_{a}(0.5-2\alpha)}+\frac{\alpha}{0.5-2\alpha}\right)\operatorname{\mathbb{E}}[||\eta_{m}||^{2}|\mathcal{F}_{m-1}].$
Similarly to Appendix C, convergence for the first term might be obtained by
setting the learning rate to $\alpha=1/16$ and the update batch size to
$M=32/\lambda_{A}$. To guarantee convergence of the second term, we need to
bound $\operatorname{\mathbb{E}}||\eta_{m}||^{2}$. In the infinite population
with replacement case, the norm of the error vector is bounded by:
$\operatorname{\mathbb{E}}||\eta_{m}||^{2}\leq\frac{S^{2}}{n_{m}},$
where $S^{2}$ is a bound update vector norm variance. If we want the error to
be bounded by $c\rho^{m}$, we need the estimation batch size $n_{m}$ to
satisfy the condition:
$n_{m}\geq\frac{S^{2}}{c\rho^{m}}.$
Satisfying this condition guarantees that the second term has geometric
convergence:
$\left(\frac{1}{2\lambda_{a}(0.5-2\alpha)}+\frac{\alpha}{0.5-2\alpha}\right)\operatorname{\mathbb{E}}||\eta_{m}||^{2}\leq\left(\frac{1}{2\lambda_{a}(0.5-2\alpha)}+\frac{\alpha}{0.5-2\alpha}\right)c\rho^{m}.$
Similarly to Appendix E.1, the bound on sample variance $S^{2}$ can be derived
as follows:
$\displaystyle\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})||g_{s,s^{\prime}}(\theta)||^{2}-||\bar{g}(\theta)||^{2}\leq$
$\displaystyle\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})||g_{s,s^{\prime}}(\theta)-g_{s,s^{\prime}}(\theta^{*})+g_{s,s^{\prime}}(\theta^{*})||^{2}\leq$
$\displaystyle\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})2||g_{s,s^{\prime}}(\theta)-g_{s,s^{\prime}}(\theta^{*})||^{2}+2||g_{s,s^{\prime}}(\theta^{*})||^{2}=$
$\displaystyle 2w(\theta)+2\sigma^{2}\leq 4f(\theta)+2\sigma^{2}=S^{2},$
where $\sigma^{2}$ is the variance of the updates in the optimal point similar
to [2].
An alternative bound on $S^{2}$ with known quantities for practical
implementation:
$\displaystyle\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})||g_{s,s^{\prime}}(\theta)||^{2}-||\bar{g}(\theta)||^{2}\leq$
$\displaystyle\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})||g_{s,s^{\prime}}(\theta)||^{2}=\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})(||(r(s,s^{\prime})+\gamma\phi(s^{\prime})^{T}\theta-\phi(s)^{T}\theta)\phi(s)||^{2})\leq$
$\displaystyle\sum_{s,s^{\prime}}\mu_{\pi}(s)P(s,s^{\prime})(2||r\phi(s)||^{2}+4||\gamma\phi(s^{\prime})^{T}\theta\phi(s)||^{2}+4||\phi(s)^{T}\theta\phi(s)||^{2})\leq$
$\displaystyle(2|r_{max}|^{2}+4\gamma^{2}||\theta||^{2}+4||\theta||^{2})=(2|r_{max}|^{2}+8||\theta||^{2})=S^{2}.$
Setting hyperparameters to obtained values will results in final computational
complexity of $\mathcal{O}(\frac{1}{\epsilon\lambda_{A}}\log(\epsilon^{-1}))$
## Appendix G Proof of Theorem 6.2
In the Markovian sampling case, we cannot simply apply Lemma 4.1; due to high
estimation bias the bounds on $f_{e}(\theta)$ and $w(\theta)$ will not be
derived based on the current value of $\theta$, but based on global
constraints on the updates guaranteed by applying projection.
First, we analyse a single iteration on step $t$ of epoch $m$, during which we
apply the update vector
$v_{t}=g_{t}(\theta)-g_{t}(\tilde{\theta})+g_{m}(\tilde{\theta})$. The update
takes the form:
$\displaystyle\operatorname{\mathbb{E}}||\theta_{t}-\theta^{*}||_{2}^{2}$
$\displaystyle=\operatorname{\mathbb{E}}||\Pi_{R}(\theta_{t-1}+\alpha
v_{t})-\Pi_{R}(\theta^{*})||_{2}^{2}\leq\operatorname{\mathbb{E}}||\theta_{t-1}-\theta^{*}+(-\alpha
v_{t})||_{2}^{2}=$ (10)
$\displaystyle||\theta_{t-1}-\theta^{*}||_{2}^{2}+2\alpha(\theta_{t-1}-\theta^{*})^{T}\operatorname{\mathbb{E}}[v_{t}]+\alpha^{2}E||v_{t}||_{2}^{2}=$
$\displaystyle||\theta_{t-1}-\theta^{*}||_{2}^{2}+2\alpha(\theta_{t-1}-\theta^{*})^{T}(\operatorname{\mathbb{E}}[g_{t}(\theta_{t-1})]-\operatorname{\mathbb{E}}[g_{t}(\tilde{\theta})]+g_{m}(\tilde{\theta}))+$
$\displaystyle\alpha^{2}E||v_{t}||_{2}^{2},$
where the expectation is taken with respect to $s,s^{\prime}$ sampled during
iteration $t$. Recall that under Markovian sampling,
$\operatorname{\mathbb{E}}[g_{t}(\theta_{t-1})]\neq\bar{g}(\theta_{t-1})$ and
that for the expectation of the estimated mean-path update we have
$\operatorname{\mathbb{E}}[g_{m}(\tilde{\theta})|s_{m-1}]\neq\bar{g}(\tilde{\theta})$,
where $s_{m-1}$ is the last state of epoch $m-1$. To tackle this issue, we
follow the approach introduced in [2] and [23], and rewrite the expectation as
a sum of mean-path updates and error terms. Similar to [2], we denote the
error term on a single update as $\zeta$:
$\zeta_{t}(\theta)=(\theta-\theta^{*})^{T}(g_{t}(\theta)-\bar{g}(\theta)).$
For an error term on the trajectory, we follow [23] and denote it as $\xi$:
$\xi_{m}(\theta,\tilde{\theta})=(\theta-\theta^{*})^{T}(g_{m}(\tilde{\theta})-\bar{g}(\theta)).$
Applying this notation, (10) can be rewritten as:
$\displaystyle E||\theta_{t}-\theta^{*}||_{2}^{2}\leq$
$\displaystyle||\theta_{t-1}-\theta^{*}||_{2}^{2}+$ (11) $\displaystyle
2\alpha(\theta_{t-1}-\theta^{*})^{T}(\operatorname{\mathbb{E}}[g_{t-1}(\theta_{t-1})]-\operatorname{\mathbb{E}}[g_{t}(\tilde{\theta})]+g_{m}(\tilde{\theta}))+\alpha^{2}E||v_{t}||_{2}^{2}=$
$\displaystyle||\theta_{t-1}-\theta^{*}||_{2}^{2}+2\alpha\big{[}(E[\zeta_{t}(\theta_{t-1})]+(\theta_{t-1}-\theta^{*})^{T}\bar{g}(\theta_{t-1}))-$
$\displaystyle(E[\zeta_{t}(\tilde{\theta})]-(\theta_{t-1}-\theta^{*})^{T}\bar{g}(\tilde{\theta}))+$
$\displaystyle(E[\xi(\theta_{t-1},\tilde{\theta})]-(\theta_{t-1}-\theta^{*})^{T}\bar{g}(\tilde{\theta}))\big{]}+\alpha^{2}E||v_{t}||_{2}^{2}.$
Error terms can be bounded by slightly modified lemmas from the original
papers. For $\zeta(\theta)$, we apply a bound from [2], Lemma 11:
$|E[\zeta_{t}(\theta)]|\leq G^{2}(4+6\tau^{mix}(\alpha))\alpha.$ (12)
In the original lemma, a bound on $E[\zeta_{t}(\theta)]$ is stated, however,
in the proof a bound on absolute value of the expectation is also derived.
For the mean-path estimation error term, we use a modified version of Lemma 1
[23]. The proof of this lemma in the original paper starts by applying the
inequality
$a^{T}b\leq\frac{k}{2}||a||^{2}+\frac{1}{2k}||b||^{2}$
to the expression
$(\theta_{t-1}-\theta^{*})^{T}(g_{m}(\tilde{\theta})-\bar{g}(\theta))$, with
$k=\lambda_{A}/2$ (using the notation in [23]). For the purposes of our proof
we use $k=\lambda_{A}$. Thus, we will have the expression:
$\displaystyle\operatorname{\mathbb{E}}[\xi_{m}(\theta_{t-1},\tilde{\theta})]\leq$
$\displaystyle\frac{\lambda_{A}}{2}\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||_{2}^{2}|s_{m-1}]+\frac{4(1+(m-1)\rho)}{\lambda_{A}(1-\rho)n_{m}}[4R^{2}+r_{\rm
max}^{2}]=$ (13)
$\displaystyle\frac{\lambda_{A}}{2}\operatorname{\mathbb{E}}[||\theta_{t-1}-\theta^{*}||_{2}^{2}|s_{m-1}]+\frac{C_{2}}{\lambda_{A}n_{m}}.$
Also, note, that the term $E||v_{t}||_{2}^{2}$ might be bounded as
$E||v_{t}||_{2}^{2}\leq 18R^{2}$. Plugging these bounds into (11) we obtain:
$\displaystyle E||\theta_{t}-\theta^{*}||_{2}^{2}\leq$
$\displaystyle||\theta_{t-1}-\theta^{*}||_{2}^{2}-2\alpha
f_{e}(\theta_{t-1})+4\alpha^{2}G^{2}(4+6\tau^{mix}(\alpha))+$ $\displaystyle
2\alpha\left(\frac{\lambda_{A}}{2}||\theta_{t-1}-\theta^{*}||_{2}^{2}+\frac{C_{2}}{\lambda_{A}n_{m}}\right)+18\alpha^{2}R^{2}.$
Summing the inequality over the epoch and taking expectation with respect to
all previous history, we have:
$\displaystyle 2\alpha
M\operatorname{\mathbb{E}}[f_{e}(\tilde{\theta}_{s})]\leq$
$\displaystyle||\tilde{\theta}_{s-1}-\theta^{*}||_{2}^{2}+2\alpha
M\left(\frac{\lambda_{A}}{2}||\tilde{\theta}_{s-1}-\theta^{*}||_{2}^{2}+\frac{C_{2}}{\lambda_{A}n_{m}}\right)+$
$\displaystyle\alpha^{2}M(4G^{2}(4+6\tau^{mix}(\alpha))+18R^{2}).$
Then we divide both sides by $2\alpha M$ and use
$||\tilde{\theta}_{s-1}-\theta^{*}||_{2}^{2}\leq
f_{e}(\tilde{\theta}_{s-1})/\lambda_{A}$ to obtain:
$\displaystyle E[f_{e}(\tilde{\theta}_{s})]\leq$
$\displaystyle\left(\frac{1}{2\lambda_{A}\alpha
M}+\frac{1}{2}\right)f_{e}(\tilde{\theta}_{s-1})+\frac{C_{2}}{\lambda_{A}n_{m}}+$
$\displaystyle\alpha(2G^{2}(4+6\tau^{mix}(\alpha))+9R^{2}).$
We choose $\alpha$ and $M$ such that $\alpha M\lambda_{A}=2$. We then apply
this inequality to the value of the function $f$ in the first term of the
right-hand side recursively, which yields the desired result:
$E[f_{e}(\tilde{\theta}_{s})]\leq\left(\frac{3}{4}\right)^{s}f_{e}(\theta_{0})+\frac{8C_{2}}{\lambda_{A}n_{m}}+4\alpha(2G^{2}(4+6\tau^{mix}(\alpha))+9R^{2}).$
## Appendix H Additional experiments
### H.1 Experiment details
In this section, we compare the performance of TD-SVRG with GTD2 [20],
"vanilla" TD learning [19], and PD-SVRG [7] in the finite sample setting.
Generally, our experimental set-up is similar to [15]. Datasets of size 5,000
are generated from 4 environments: Random MDP [5], and the Acrobot, CartPole
and Mountain car OpenAI Gym environments [3]. For the Random MDP, we construct
an MDP environment with $|S|=400$, 21 features (20 random and 1 constant) and
10 actions, with action selection probabilities generated from $U[0,1)$. For
OpenAI gym environments, the agent selects states uniformly at random.
Features are constructed by applying RBF kernels to discretize the original
states and then removing highly correlated features. The decay rate $\gamma$
is set to $0.95$.
We compare the performance of TD-SVRG against the performance of other
algorithms with parameters selected by grid search. Details on the grid search
might be found in Subsection H.3. Hyperparamters for the algorithms are
selected as follows: for TD-SVRG our theoretically justified parameters are
selected, the learning rate is set to $\alpha=1/8$ and the update batch size
to $M=16/\lambda_{A}$; for GTD2 the best performing parameters were:
$\alpha=0.125$ and $\beta=0.25$; for vanilla TD a decreasing learning rate is
set to $\alpha=1/\sqrt{t}$; for PD-SVRG the parameters are set to
$\sigma_{\theta}=0.1/(L_{\rho}\kappa(\hat{C}))$, $\sigma_{w}=0.1/\lambda_{\rm
max}(C)$ and the batch size is twice the size of the dataset, i.e., $M=2N$.
Each algorithm for each setting was run 10 times. The geometric average
performance is presented in Figure 2 (in $\log$-scale).
### H.2 Comparison of theoretic batchsizes
In this subsection we compare the values of update batch sizes which are
theoretically required to guarantee convergence. We compare batch sizes of
three algorithms: TD-SVRG, PDSVRG ([7]) and VRTD ([23]). Note that PDSVRG and
VRTD are algorithms for different settings, but for TD-SVRG the batch size
value is the same: $16/\lambda_{A}$, thus, we compare two algorithms in the
same table. We compare the batch sizes required by the algorithm for three
MDPs: a first MDP with 50 state, 20 actions and $\gamma=0.8$, a second MDP
with 400 states, 10 actions and $\gamma=0.95$, and a third MDP with 1000
states, 20 actions and $\gamma=0.99$, with actions selection probabilities
generated from $U[0,1)$ (similar to the settings used for the experiments in
Sections 7 and H.6). Since the batch size is dependent on the smallest
eigenvalue of the matrix $A$, which, in turn, is dependent on the
dimensionality of the feature vector, we do the comparison for different
feature vector sizes: 5, 10, 20 and 40 randomly generated features and 1
constant feature for each state. We generate 10 datasets and environments for
each feature size. Our results are summarized in Figure H.1 and tables H.1,
H.2 and H.3.
Table H.1: Comparison of theoretically suggested batch sizes for an MDP with 50 states, 20 actions and $\gamma=0.8$. Values in the first row indicate the demensionality of the feature vectors. Values in the other rows: batch size of the corresponding method. Values are averaged over 10 generated datasets and environments. Method/Features | 6 | 11 | 21 | 41
---|---|---|---|---
TD-SVRG | $2339$ | $6808$ | $21553$ | $4.51\cdot 10^{5}$
PD-SVRG | $1.52\cdot 10^{16}$ | $3.09\cdot 10^{19}$ | $1.85\cdot 10^{23}$ | $1.41\cdot 10^{36}$
VRTD | $3.07\cdot 10^{6}$ | $2.13\cdot 10^{7}$ | $3.79\cdot 10^{8}$ | $165\cdot 10^{11}$
Table H.2: Comparison of theoretically suggested batch sizes for an MDP with 400 states, 10 actions and $\gamma=0.95$. Values in the first row indicate the demensionality of the feature vectors. Values in the other rows: batch size of the corresponding method. Values are averaged over 10 generated datasets and environments. Method/Features | 6 | 11 | 21 | 41
---|---|---|---|---
TD-SVRG | $3176$ | $6942$ | $18100$ | $54688$
PD-SVRG | $1.72\cdot 10^{16}$ | $3.83\cdot 10^{18}$ | $3.06\cdot 10^{21}$ | $5.77\cdot 10^{24}$
VRTD | $5.41\cdot 10^{6}$ | $2.53\cdot 10^{7}$ | $1.63\cdot 10^{8}$ | $1.58\cdot 10^{9}$
Table H.3: Comparison of theoretically suggested batch sizes for an MDP with 1000 states, 20 actions and $\gamma=0.99$. Values in the first row indicate the demensionality of the feature vectors. Values in the other rows: batch size of the corresponding method. Values are averaged over 10 generated datasets and environments. Method/Features | 6 | 11 | 21 | 41
---|---|---|---|---
TD-SVRG | $9206$ | $16096$ | $32723$ | $79401$
PD-SVRG | $7.38\cdot 10^{18}$ | $9.64\cdot 10^{20}$ | $5.14\cdot 10^{23}$ | $4.97\cdot 10^{26}$
VRTD | $4.35\cdot 10^{7}$ | $1.34\cdot 10^{8}$ | $5.44\cdot 10^{8}$ | $1.45\cdot 10^{9}$
Figure H.1: Theoretical batch sizes of different algorithms in log-scale,
geometrical average over 10 samples. The $x$-axis plots the dimension of the
feature vector. First row: Batch sizes for random MDP environment (see Sec.
7). Left to right: Figure 1 - 50 states, 20 actions and $\gamma=0.8$; Figure
2: 400 states, 10 actions and $\gamma=0.95$, Figure 3: 1000 states, 20 actions
and $\gamma=0.99$; Figure 4: 2000 states, 50 actions and $\gamma=0.75$. Second
row: batch sizes for dataset generated from OpenAI gym classic control
environments [3]. Features generated by applying RBF kernels and then removing
highly correlated feature vectors one by one (see Sec. 7).
### H.3 Other Algorithms Grid Search Results
In this subsection we provide the results of the grid search for each
algorithm we compare against, PD-SVRG, GTD2 and VRDT. For every algorithm we
tried a set of parameters suggested by the authors in the Experiment sections
of the corresponding papers. All experiments were run on an MDP environment
with 400 states, 21 features, 10 actions and $\gamma=0.95$, identical to one
described in Section 7 of this paper. We run 5 experiments with $10^{5}$
updates for each dataset sampling case and 10 experiments with $3\times
10^{5}$ for the i.i.d. sampling setting; the average results are compared.
Figure H.2: Results of PD-SVRG grid search. Average performances of TD-SVRG
and 4 best performing PD-SVRG algorithms are shown.
For PD-SVRG, we tried the exact values suggested in [7], i.e.,
$\sigma_{\theta}\in(10^{-1},\ldots,10^{-6})\frac{1}{L_{\rho}\kappa(\hat{C})}$,
$\sigma_{w}\in(1,10^{-1},10^{-2})\frac{1}{\lambda_{\rm max}(C)}$ and the batch
size is twice the dataset size $M=2N$. Performance of the 4 best-performing
parameter sets compared to TD SVRG is shown on Figure H.2. These results
demonstrate that all PD-SVRG algorithms exhibit geometric convergence, and the
3 best performing algorithms are those with highest $\sigma_{\theta}$, while
the value of $\sigma_{w}$ doesn’t affect the performance much. In addition,
all 3 algorithms converge slower compared to TD-SVRG, probably because of a
smaller learning rate and small batch size (it reevaluates mean path too often
unnecessarily).
Figure H.3: Results of GTD2 grid search. Average performances of TD-SVRG and 4
best performing GTD2 algorithms are shown.
For GTD2 we also tried a set of values suggested in [20], which are
$\alpha\in(1/2,1/4,1/8,1/16)$ and $\beta/\alpha\in(2,1,1/2,1/4)$. Results of
the 4 best performing methods compared against TD-SVRG are shown on Figure
H.3. Similarly to the previous experiment, these results show that value of
$\beta$ doesn’t affect the performance that much, and that the best performing
value of $\alpha$ is 1/8, which, as we show in our paper, is reasonable for
this problem. All GTD methods exhibit sub-geometric convergence, while TD-SVRG
converges geometrically.
Figure H.4: Results of VRTD grid search. Average performances of TD-SVRG and 4
best performing VRTD algorithms are shown.
For VRTD we also tried a set of values similar to [23], which are $\alpha=0.1$
and batch sizes $M\in(500,1000,2000,5000)$ (we added 5000 and removed smaller
batch sizes). Results of these methods compared against TD-SVRG are shown on
Figure H.4. The results demonstrate that, as expected, a batch size of 5000 is
the best performing, but all of them converge to some level of accuracy and
oscillate near it, while TD-SVRG converges, with every step requiring larger
batch sizes. This experiment shows the disadvantage of practical VRTD: the
learning rate required to achieve the theoretically guaranteed result is too
small to be applied in practice, while practically applied values are hard to
chose, i.e., if someone wants to run the experiment for a given number of
iteration, they would not know how to choose a batch size such that it is not
too small causing VRTD to converge to its potential best accuracy too fast
(and wasting later iterations), but also it is not too big, so that VRTD will
not converge to its best accuracy in a given number of iterations.
### H.4 Datasets with DQN features
Figure H.5: Geometric average performance of different algorithms in the
finite sample case with DQN features. Columns - dataset source environments:
Acrobot, CartPole and Mountain Car. Rows - performance measurements:
$\log(f(\theta))$ and $\log(|\theta-\theta^{*}|)$.
In this set of experiments, we compare the performance of the same algorithms
as in Section 7 on datasets collected from OpenAI Acrobot, CartPOle and
MountainCar environments [3] using DQN features. To collect these features, we
trained 1 hidden layer neural network with DQN algorithm [13] for 1000 plays.
Then, the trained agent played 5000 episodes following greedy policy, while
neural network hidden states were recorded as feature representation of the
visited states. Features collected this way tend to be highly correlated,
therefore we applied PCA clearing, keeping minimum set of principal
components, corresponding to 90 % of the variance.
For the TD-SVRG algorithm we used theoretically justified parameters, for the
other algorithms parameters selected with grid search (Sec. H.3), the results
are presented in Figure H.5. In all environments TD-SVRG exhibits stable
linear convergence, GTD2 and vanilla TD algorithms converge sublinearly, while
PD-SVRG performance is unstable due to high range of condition numbers of
dataset’s characteristic matrices $A$ and $C$ (large values of $\kappa(C)$
caused PD-SVRG divergence in Acrobot dataset).
### H.5 Batched SVRG performance
In this set of experiments we compare the performance of TD-SVRG and batched
TD-SVRG in the finite-sample case. We generate 10 datasets of size 50000 from
a similar MDP as in Section 7. Algorithms also run with the same
hyperparameters. Average results over 10 runs are presented in Figure H.6 and
show that batched TD-SVRG saves a lot of computations during the earlier
epochs, which provides faster convergence.
Figure H.6: Average performance of TD-SVRG and batching TD-SVRG in the finite
sample case. Datasets sampled from MDP environments. Left figure – performance
in terms of $\log(f(\theta))$. Right figure – performance in terms of
$\log(|\theta-\theta^{*}|)$.
### H.6 Online i.i.d. sampling from the MDP
Figure H.7: Average performance of TD-SVRG, VRTD and vanilla TD in the i.i.d.
sampling case. “TD-decr" refers to vanilla TD with decreasing learning rate,
“TD-const" - to vanilla TD with constant learning rate. Left figure –
performance in terms of $\log(f(\theta))$, right figure in terms of
$\log(|\theta-\theta^{*}|)$.
In this set of experiments we compare the performance of TD-SVRG, VRTD and
Vanilla TD with fixed and decreasing learning rates in the i.i.d. sampling
case. States and rewards are sampled from the same MDP as in Section 7.
Hyperparameters are chosen as follows: for TD-SVRG – learning rate
$\alpha=1/8$, update batch size $M=16/\lambda_{A}$. VRTD – learning rate
$\alpha=0.1$ and batch size $M=2000$. For vanilla TD with constant learning
rate its value set to $0.1$ and for decreasing learning rate it is $1/t$,
where $t$ is number of the performed update. Average results over 10 runs are
shown in Figure H.7. The figure shows that TD-SVRG converges even if its
performance suffers from high variance, VRTD and vanilla TD with constant
learning rate oscillate after reaching a certain level (due to bias) and
vanilla TD with decreasing learning rate converges very slowly.
## Appendix I Algorithms comparison
In this section, we present a more detailed comparison of TD algorithms. Our
results are summarized in Table I.1, and a detailed explanation of the
quantities in the table is provided below.
Please note that while other algorithms derive convergence in terms of
$||\theta-\theta^{|}|^{2}$, our convergence is expressed in terms of the
function $f(\theta)$. The results can be compared using the inequality
$\lambda_{A}||\theta-\theta^{|}|^{2}\leq
f(\theta)\leq||\theta-\theta^{|}|^{2}$. This implies that achieving an
accuracy of $\epsilon$ in terms of one quantity can be accomplished by
achieving an accuracy of $\lambda_{A}/\epsilon$ in terms of the other
quantity. Consequently, our results for the finite sample case are strictly
superior. For environment sampling cases, our results imply previous findings,
whereas our results are not implied by previous ones. Furthermore, it is worth
noting that the inequality $\lambda_{A}||\theta-\theta^{|}|^{2}\leq f(\theta)$
is rarely strict, which means that in most cases, the convergence implied by
our results would be superior.
Table I.1: Comparison of algorithmic parameters. PD-SVRG and PD SAGA results reported from [7], VRTD and TD results from [23], GTD2 from [21]. $\lambda_{\rm min}(Q)$ and $\kappa(Q)$ are used to define, respectively, minimum eigenvalue and condition number of a matrix $Q$. $\lambda_{A}$ in this table denotes minimum eigenvalue of the matrix $1/2(A+A^{T})$, which is defined in Equation (1). Finite sample results use $N$ for the size of the dataset sampled from the MDP. Other notation is taken from original papers, and Section 1 in the supplementary information gives self-contained definitions of all the symbols appearing in this table. For simplicity $1+\gamma$ is upper bounded by $2$ throughout, where $\gamma$ is the discount factor. Method | Learning rate | Batch size | Total complexity
---|---|---|---
Finite sample case
GTD2 | $\frac{9^{2}\times 2\sigma}{8\sigma^{2}(k+2)+9^{2}\zeta}$ | 1 | $\mathcal{O}\left(\frac{\kappa(Q)^{2}\mathcal{H}}{\lambda_{\rm min}(G)\epsilon}\right)$
PD-SVRG | $\frac{\lambda_{\rm min}(A^{T}C^{-1}A)}{48\kappa(C)L^{2}_{G}}$ | $\frac{51\kappa^{2}(C)L^{2}_{G}}{\lambda_{\rm min}(A^{T}C^{-1}A)^{2}}$ | $\mathcal{O}\left(\left(N+(\frac{\kappa^{2}(C)L^{2}_{G}}{\lambda_{\rm min}(A^{T}C^{-1}A)^{2}}\right)\log(\frac{1}{\epsilon})\right)$
PD SAGA | $\frac{\lambda_{\rm min}(A^{T}C^{-1}A)}{3(8\kappa^{2}(C)L^{2}_{G}+n\mu_{\rho})}$ | 1 | $\mathcal{O}\left(\left(N+\frac{\kappa^{2}(C)L^{2}_{G}}{\lambda_{\rm min}(A^{T}C^{-1}A)^{2}}\right)\log(\frac{1}{\epsilon})\right)$
This paper | 1/8 | $16/\lambda_{A}$ | $\mathcal{O}\left(\left(N+\frac{1}{\lambda_{A}}\right)\log(\frac{1}{\epsilon})\right)$
i.i.d. sampling
TD | $\min(\frac{\lambda_{A}}{4},\frac{1}{2\lambda_{A}})$ | 1 | $\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}^{2}}\log(\frac{1}{\epsilon})\right)$
This paper | 1/8 | $16/\lambda_{A}$ | $\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}}\log(\frac{1}{\epsilon})\right)$
Markovian sampling
TD | $\mathcal{O}(\epsilon/\log(\frac{1}{\epsilon}))$ | 1 | $\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}^{2}}\log^{2}(\frac{1}{\epsilon})\right)$
VRDT | $\mathcal{O}(\lambda_{A})$ | $\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}^{2}}\right)$ | $\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}^{2}}\log(\frac{1}{\epsilon})\right)$
This paper | $\mathcal{O}(\epsilon/\log(\frac{1}{\epsilon}))$ | $\mathcal{O}\left(\frac{\log(\frac{1}{\epsilon})}{\epsilon\lambda_{A}}\right)$ | $\mathcal{O}\left(\frac{1}{\epsilon\lambda_{A}}\log^{2}(\frac{1}{\epsilon})\right)$
Definitions of quantities in Table I.1:
GTD2 convergence analysis resutls are taken from [21]. The learning rate
required for their guarantee to work is set to $\frac{9^{2}\times
2\sigma}{8\sigma^{2}(k+2)+9^{2}\zeta}$ and the complexity to obtain accuracy
$\epsilon$ is $\mathcal{O}(\frac{\kappa(Q)^{2}\mathcal{H}d}{\lambda_{\rm
min}(G)\epsilon})$. In this notation:
* •
$\sigma$ is the minimum eigenvalue of the matrix $A^{\prime
T}M^{-1}A^{\prime}$, where the matrix
$M=\operatorname{\mathbb{E}}[\phi(s_{k},a_{k})\phi(s_{k},a_{k})^{T}]$ and
$A^{\prime}=\operatorname{\mathbb{E}}[e_{k}(\gamma\operatorname{\mathbb{E}}_{\pi}[\phi(s_{k+1},.]-\phi(s_{k},a_{k}))^{T}]$,
where $e_{k}$ is the eligibility trace vector
$e_{k}=\lambda\gamma\kappa(s_{k},a_{k})e_{k-1}+\phi(s_{k},a_{k})$.
* •
$k$ is an iteration number.
* •
The matrix $G$ plays key role in the analysis, it is a block matrix of the
form
$G=\begin{pmatrix}0&\sqrt{\beta}A^{\prime T}\\\ -\sqrt{\beta}A^{\prime}&\beta
M_{k}\end{pmatrix},$
and $G_{k}$ is a matrix of similar form generated from quantities estimated at
time point $k$.
* •
$\zeta$ is $2\times 9^{2}c(M)^{2}\rho^{2}+32c(M)L_{G}$, where $c(M)$ is the
condition number of the matrix $M$, $\rho$ is the maximum eigenvalue of the
matrix $A^{\prime T}M^{-1}A^{\prime}$ and $L_{G}$ is the
$L_{G}=||\operatorname{\mathbb{E}}[G_{K}^{T}G_{K}|\mathcal{F}_{k-1}]||$.
$\mathcal{F}_{k-1}$ in this analysis is the $\sigma$-algebra generated by all
previous history up to moment $k-1$.
* •
The quantity $\mathcal{H}$ is equal to
$\operatorname{\mathbb{E}}||G_{K}z^{*}-g_{k}||$, where
$z^{*}=(\theta^{*},\frac{1}{\sqrt{\beta}w^{*}})$ is the optimal solution and
$g_{k}=(0,\frac{1}{\sqrt{\beta}}b)$.
* •
The last quantity left undefined is $\kappa(Q)$, which is the condition number
of the matrix $Q$, obtained by diagonalization of the matrix $G=Q^{T}\Lambda
Q$.
PD-SVRG and PD SAGA use the same quantities as GTD2, except that matrices $A$
and $C$ are defined the same way as in this paper:
$A=\operatorname{\mathbb{E}}[(\phi(s)^{T}-\gamma\phi(s^{\prime})^{T})\phi(s)]$,
$C=\operatorname{\mathbb{E}}[\phi(s)\phi^{T}(s)]$.
* •
$n$ in this notation is the size of the dataset.
* •
$\mu_{\rho}$ is the minimum eigenvalue of matrix $A^{T}C^{-1}A$.
All other quantities are defined in the paper.
|
11institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, 69117
Heidelberg, Germany<EMAIL_ADDRESS>
Laboratoire d’Astrophysique de Bordeaux, CNRS and Université de Bordeaux,
Allée Geoffroy St. Hilaire, 33165 Pessac, France
# Simultaneous gas accretion onto a pair of giant planets:
Impact on their final mass and on the protoplanetary disk structure
C. Bergez-Casalou 11 B. Bitsch 11 S. N. Raymond 221122
Several planetary systems are known to host multiple giant planets. However,
when two giant planets are accreting from the same disk, it is unclear what
effect the presence of the second planet has on the gas accretion process of
both planets. In this paper we perform long-term 2D isothermal hydrodynamical
simulations (over more than 0.5 Myrs) with the FARGO-2D1D code, considering
two non-migrating planets accreting from the same gaseous disk. We find that
the evolution of the planets’ mass ratio depends on gap formation. However, in
all cases, when the planets start accreting at the same time, they end up with
very similar masses (0.9 $<m_{p,out}/m_{p,in}<$ 1.1 after 0.5 Myrs). Delaying
the onset of accretion of one planet allows the planets’ mass ratio to reach
larger values initially, but they quickly converge to similar masses afterward
(0.8 $<m_{p,out}/m_{p,in}<$ 2 in $10^{5}$ yrs). In order to reproduce the more
diverse observed mass ratios of exoplanets, the planets must start accreting
gas at different times, and their accretion must be stopped quickly after the
beginning of runaway gas accretion (less than 0.5 Myrs), for example via disk
dispersal. The evolution of the planets’ mass ratio can have an important
impact on the dynamics of the system and may constrain the formation history
of Jupiter and Saturn.
###### Key Words.:
accretion, accretion disks – protoplanetary disks – planets and satellites:
gaseous planets – hydrodynamics – planets and satellites: physical evolution
## 1 Introduction
Recent surveys show that planetary systems that host multiple giants are
common (e.g., Lissauer et al., 2012; Fabrycky et al., 2014; Zhu, 2022). These
planets are believed to have formed in the same protoplanetary disk, where
they acquired their massive gaseous atmospheres. Previous hydrodynamical
studies investigate either the growth of single planets in the disk (e.g.,
Ayliffe & Bate, 2009; D’Angelo & Bodenheimer, 2013; Crida & Bitsch, 2017;
Schulik et al., 2019; Bergez-Casalou et al., 2020) or the evolution of already
formed multiple planets (e.g., Baruteau & Papaloizou, 2013; Lega et al., 2013;
Pierens et al., 2014; Morbidelli et al., 2018). This dichotomy originates from
the difficulty to accurately model each evolution process.
It is vital to understand exactly how gas is accreted onto giant planets. Some
studies have described how gas accretion can be approximated in 2D, with or
without planet migration (Kley, 1999; Crida & Bitsch, 2017). Other studies
used complex 3D simulations to take the various fluid and thermal processes
governing the gas accretion of an embedded planet into account (e.g., Ayliffe
& Bate, 2009; Machida et al., 2010; D’Angelo & Bodenheimer, 2013; Szulágyi &
Mordasini, 2017; Lambrechts et al., 2019; Schulik et al., 2019). Due to their
high computational cost, such simulations are often integrated over short
timescales, around 100 planetary orbits, making it impossible to investigate
the long-term growth of single accreting planets with the current computing
facilities.
Moreover, the gas distribution around embedded giant planets is impacted by
gap opening. The formation of these gaps has been observed both in the dust
(e.g., ALMA Partnership et al., 2015; Andrews et al., 2018; Benisty et al.,
2021) and in the gas (e.g., Huang et al., 2018; Smirnov-Pinchukov et al.,
2020). Several studies have investigated the characteristics of the gap (such
as its depth and width) as a function of the planet and disk characteristics
(Lin & Papaloizou, 1986; Crida et al., 2006; Fung et al., 2014; Kanagawa et
al., 2015). However, due to computational limitations, these studies neglected
gas accretion. Recent works show that gap opening and gas accretion are highly
dependent on each other through the viscosity of the disk (Bergez-Casalou et
al., 2020; Rosenthal et al., 2020).
As gap opening has a non-negligible impact on the gas structure, the growth of
a giant planet must be impacted by the presence of a second accreting planet
in the same disk. The goal of this study is to quantify this impact as a
function of the disk parameters and planet characteristics (i.e., their radial
separation and delay in accretion). We perform 2D isothermal hydrodynamical
simulations similar to Bergez-Casalou et al. (2020) to monitor the growth of
two planets accreting in the runaway accretion regime from the same disk. Our
2D isothermal setup allows us to integrate the evolution of the planet masses
for around 0.5 Myrs, which is longer than the majority of the studies that
investigate gas accretion onto planets using hydrodynamical simulations.
This paper is structured as follows: The numerical setup is described in Sect.
2. In Sect. 3 we investigate the impact of different disk viscosities on the
growth of each planet. A comparison with single accreting planets is presented
in Sect. 4. Different planet separations are investigated in Sect. 5. In Sect.
6 we show the mass evolution of planets accreting at different times. Our
findings are discussed in Sect. 7, where we elaborate on the impact on the
stellar gas accretion and on the dynamical evolution of the planetary systems.
A comparison with the exoplanet population is discussed as well before we
summarize and conclude in Sect. 8.
## 2 Numerical setup
In this paper we simulate two accreting planets on fixed circular orbits
embedded in their gaseous disk with the hydrodynamical code FARGO-2D1D (Crida
et al., 2007). This code is composed of two different kind of grids: a 2D grid
where the planets are located and which is surrounded by two 1D grids. The
first 1D inner disk ranges from 0.1 AU to 0.78 AU, the 2D grid ranging from
0.78 AU to 23.4 AU and the second outer 1D disk spans from 23.4 AU to 260 AU.
With this code, the global viscous evolution of the disk is self-consistently
modeled at a reasonable computational cost. As mentioned in Bergez-Casalou et
al. (2020), it is important to have a consistent viscous evolution of the full
disk in order to accurately describe gap formation and gas accretion.
For computational accuracy, the code uses dimensionless units. We normalized
masses with the mass of the central star, $M_{0}=M_{\odot}$ and lengths with
the position of the planet, $r_{0}=5.2$ AU and we set the gravitational
constant as $G=1$. The unit of time is therefore based on the orbital period
at $r_{0}$ with $P=2\pi t_{0}$ where $t_{0}=(r_{0}^{3}/(GM_{*}))^{-1/2}$.
In order to focus on the impact of gas accretion, dynamical interactions
between the planets and planet migration are neglected. This choice is
discussed in Sect. 7.3. The planets are fixed on circular orbits, at key
positions corresponding to different period ratios. We use the term ”period
ratio” instead of ”resonance,” as they are not dynamically locked in resonance
but are forced by the code to stay at their position. Four period ratios are
investigated: 2:1, 3:1, 4:1, and 5:1. These positions were chosen such as the
planets are far enough from each other to be considered dynamically stable
during their growth (Chambers et al., 1996; Raymond et al., 2009). In order to
compare to a single-planet case, we also simulate the growth of single planets
located at positions corresponding to the investigated period ratios. The
planets’ configurations are summarized in table 1.
Each planet starts with an initial mass of 20 $\rm M_{\oplus}$, allowing it to
directly accrete in the runaway gas accretion regime (Pollack et al., 1996).
These initial cores are slowly introduced in the disk with the following mass-
taper function:
$m_{\rm taper}=\sin^{2}{(t/(4n_{\rm orb}))},$ (1)
making the planet grow from 0 to its initial mass in $n_{orb}=3$ orbits of the
inner planet. Except for Sect. 6 where the accreting times are specified, all
planets start accreting simultaneously after $100$ orbits of the inner planet.
The planets accrete following the accretion routine described in Bergez-
Casalou et al. (2020). With this accretion routine, the amount of gas accreted
by the planets is dictated by Machida et al. (2010) ($\dot{M}_{M}$) and is
limited to what the disk can provide by the approach of Kley (1999)
($\dot{M}_{K}$). It is written as follows:
$\dot{M}_{p}=min\cases{\dot{\hfil}}{M}_{M}=<\Sigma>_{0.9r_{H}}H^{2}\Omega_{K}\times
min\Big{[}0.14;0.83(r_{H}/H)^{9/2}\Big{]}\\\
\dot{M}_{K}=\iint_{A_{disk}}f_{\rm red}(d)\;\Sigma(r,\phi,t)\;\pi f_{\rm
acc}\;dr\;d\phi{},$ (2)
where $<\Sigma>_{0.9r_{H}}$ is the averaged surface density around the planet
up to 0.9 $r_{H}$ with $r_{H}=r_{p}(m_{p}/3M_{*})^{1/3}$ being the Hill sphere
of the planet; $A_{disk}$ is the disk area; $H$ is the disk scale height;
$\Omega_{K}$ is the Keplerian orbital frequency of the planet; $d$ is the
distance from the planet; $f_{\rm acc}$ is the inverse timescale upon which
the accretion rate of Kley (1999) is occurring; and $f_{\rm red}$ is a smooth
reduction function that predicts what fraction of gas must be accreted as a
function of the distance from the planet. Values for $f_{\rm acc}$ and $f_{\rm
red}$ are chosen in order to reproduce a realistic accretion efficiency111Note
that $<\Sigma>_{0.9r_{H}}$ is averaged until 0.9 $r_{H}$ because $f_{\rm
red}=0$ for $d>0.9r_{H}$ in our case.. The detailed accretion routine is
presented in Appendix A.
Once the amount of gas is determined by the accretion routine, it is removed
from the disk and added to the planet’s mass as in Kley (1999). Unless
specified, we remain in the regime where $\rm\dot{M}_{M}<\dot{M}_{K}$
throughout, meaning that we always remove the amount of mass suggested in
Machida et al. (2010).
The resolution of the 2D grid is such that there are five cells per Hill
radius of the inner planet before it starts growing, which leads to
$N_{r}=802$ and $N_{\phi}=1158$. Considering that the resolution is fixed in
time, the Hill sphere region will become better and better resolved as the
planets grow ($\rm r_{\rm H}\propto m_{\rm p}^{1/3}$). As we enhance the
planet separation, we need to adjust the 2D-1D boundary located at the outer
edge of the 2D grid in order to properly take into account the perturbations
of the furthest planet. Therefore, in the 4:1 and 5:1 case, this boundary is
moved to 36.4 AU, enhancing the radial number of cells of the 2D grid to
$N_{r}=1262$. The azimuthal number of cells remains unchanged.
Table 1: Semimajor axis of the different planet configurations considered in this paper. Configuration | Inner planet $r_{p}$ | Outer planet $r_{p}$
---|---|---
2 planets 2:1 | 5.2 AU | 8.22 AU
2 planets 3:1 | 5.2 AU | 10.82 AU
2 planets 4:1 | 5.2 AU | 13.42 AU
2 planets 5:1 | 5.2 AU | 15.18 AU
1 planet inner | 5.2 AU | -
1 planet outer 2:1 | - | 8.22 AU
1 planet outer 3:1 | - | 10.82 AU
1 planet outer 4:1 | - | 13.42 AU
1 planet outer 5:1 | - | 15.18 AU
In FARGO-2D1D, the disk is locally isothermal. As in Bergez-Casalou et al.
(2020), the surface density profile is chosen such that the total mass of the
disk is $M_{d}=0.1M_{*}$, leading to
$\Sigma_{0}(r_{0})=93.6\;\mathrm{g/cm^{2}}$ with $r_{0}$ = 5.2 AU. Even if
this can be considered as a heavy disk (Baillié et al., 2019), its large
radial extent allows us to neglect self-gravity. The aspect ratio, $h=H/r$, of
the disk is constant. The disk is subject to an $\alpha$-viscosity as
described by Shakura & Sunyaev (1973). In the following section, different gas
kinematic viscosities $\nu=\alpha h^{2}r^{2}\Omega_{K}$ are investigated by
varying the aspect ratio as well as the $\alpha$-viscosity parameter. The
investigated values are $h=0.03,\mathbf{0.05},0.07$ and
$\alpha=10^{-4},\mathbf{10^{-3}},10^{-2}$, with our fiducial values marked in
bold.
## 3 Influence of the disk viscosity
The flow of gas in the disk is dictated by the kinematic viscosity, $\nu$,
which depends on the $\alpha$-viscosity parameter on one hand and on the
aspect ratio $h$ of the disk on the other. In this section we investigate the
influence of the disk viscosity on the accretion behavior of two planets fixed
in a 3:1 period ratio. We start by changing the $\alpha$-viscosity parameter
in Sect. 3.1, then the influence of different aspect ratios is studied in
Sect. 3.2. We focus on the influence of the viscosity on the evolution of the
planet mass-ratio in a last subsection (Sect. 3.3).
### 3.1 Influence of the turbulent viscosity
As mentioned previously, the turbulent viscosity is parametrized by the
$\alpha$-viscosity parameter. Disk turbulence increases with increasing
$\alpha$, leading to faster evolving disks. We show in Fig. 1 the planetary
accretion rates (top row) and the resulting masses (bottom row) for planets in
disks with different $\alpha$ parameters. From left to right, $\alpha$
increases from $10^{-4}$ to $10^{-2}$. The behavior of the accretion rates is
the same as in Bergez-Casalou et al. (2020): reducing the viscosity induces a
slightly lower planetary accretion due to a slower flow of gas in the vicinity
of the planet.
Figure 1: Planetary accretion rates (top row) and masses (bottom row) as a
function of time for different $\alpha$-viscosities. Here the planets are
fixed in a 3:1 period ratio: the inner planet is shown in red and the outer
one in blue. As in Bergez-Casalou et al. (2020), the accretion rates slightly
increase with increasing viscosity. The oscillations in the accretion rate at
low viscosities are due to the presence of vortices. The inner and outer
planets display similar accretion rates, leading to similar planet masses.
The difference in accretion rates between the inner (red) and the outer (blue)
planet slightly evolves as a function of time and viscosity as shown on the
top row of Fig. 1. At the beginning, the planets accrete in the first Machida
et al. (2010) accretion regime (dominated by Bondi accretion), leading to a
larger inner planet. The flip in accretion rate is due to the switch of
accretion regime, from a Bondi to a Hill dominated accretion scheme (Machida
et al., 2010; Bergez-Casalou et al., 2020). While the inner planet accretes
slightly more in the first regime of accretion, the accretion rates become
more similar when the planets accrete in the second accretion regime.
This similarity in accretion rates results in planets of similar masses
(bottom row of Fig. 1). Even if the planets are more massive in the high
viscosity case than in low viscosity disks, the differences between the inner
and outer planet masses does not seem to be highly influenced by the change of
$\alpha$-viscosity. This can be expected as the Machida accretion recipe does
not directly depend on this parameter (see Eq. 2). The influence of the
$\alpha$-viscosity is indirect, as it influences the gas flow around the
planet, changing the surface density $<\Sigma>_{0.9r_{H}}$ from which the
planets accrete their gas.
At low viscosity, instabilities are triggered (Klahr & Bodenheimer, 2003; Fu
et al., 2014). The Rossby wave instability (RWI; Lovelace et al., 1999; Li et
al., 2001) produces vortices at the locations of steep pressure gradients,
such as produced by the gaps of massive planets. The presence of vortices
modifies the flow of gas in the vicinity of the planets, creating the
oscillations observed on the top left panel of Fig. 1. In the case of two
accreting planets, vortices are produced at three different locations: at the
outer edge of the outer gap, in between the planets and even at the inner edge
of the inner gap. We show in Appendix B the 2D ($r,\phi$) surface density maps
of the disk containing the planets in the 3:1 period ratio at low viscosity
($\alpha=10^{-4}$ and $h=0.05$) at three different times. The strength of
these vortices depend on the growing timescale of the planets (Hammer et al.,
2017, 2021). However, here, due to the presence of the second planet, the
vortex between the planets quickly vanishes (in less than $4\times 10^{4}$
yrs). The strongest vortex (i.e., with the highest over-density) is the one
located at the outer edge of the gap of the outer planet and takes about
$10^{5}$ yrs to vanish.
### 3.2 Influence of the aspect ratio
Another way to study the impact of the disk viscosity is to modify the disk’s
aspect ratio. We show the impact of this parameter in Fig. 2. Here, $\alpha$
is fixed to $10^{-3}$ and $h$ varies from 0.03 (left panel) to 0.07 (right
panel). The fiducial value of 0.05 is shown in the middle panel for reference.
It should be noted that changing the constant aspect ratio from 0.03 to 0.05
accounts for a reduction of viscosity $\nu$ by a factor of 2.8 while enhancing
the aspect ratio from 0.05 to 0.07 increases the disk viscosity by a factor of
2.0.
Even if the change in the disk kinematic viscosity is weaker than when the
$\alpha$-viscosity is varied in the previous section (change of a factor of
10), the accretion rates behavior are significantly different depending on the
aspect ratio. This arises from the direct dependence of the accretion rate
recipe on the aspect ratio, while it is indirectly dependent on $\alpha$. When
the gas disk scale height $H$ increases, it requires a larger Hill sphere
radius for a planet to switch from the Bondi regulated accretion regime to the
Hill regulated one (Machida et al., 2010; Bergez-Casalou et al., 2020). This
can be seen on the top row of Fig. 2: the flip in accretion rates occurs at
later times for higher aspect ratios. At low aspect ratio (left panel), the
planets initial mass is large enough to start accreting immediately in the
second regime. In this case, the accretion rates always decrease in time,
proportionally to the local surface density.
The timing of the accretion switch between the Bondi and Hill regimes has an
important influence on the evolution of the planet masses and their mass
ratios. As the inner planet accretes significantly more than the outer one
when they are limited to the Bondi accretion regime, a later switch results in
more diverse planetary masses. Therefore, the difference between the inner and
outer planets increases with an increasing aspect ratio.
### 3.3 Impact on the planets’ mass ratio
Figure 2: Same as Fig. 1, but for different aspect ratios, increasing from
left to right, with $\alpha=10^{-3}$. We note on the top of each row the
difference in kinematic viscosity caused by the change in the disk’s aspect
ratio ($\nu=\alpha h^{2}r^{2}\Omega_{K}$). The differences in the planetary
accretion rates originate from the structure of the accretion formula itself
(see Eq. 2). At a low aspect ratio ($h=0.03$, left panel), the accretion
starts immediately in the second Machida regime. On the other hand, for
$h=0.07$, the change of accretion regime occurs at a higher mass and therefore
at a later time, leading to a very different planet mass evolution.
In order to better understand the differences in planetary mass, we analyze in
Fig. 3 and Fig. 4 the ratios of the planetary masses. We focus first in Fig. 3
on the impact of the $\alpha$-parameter on these ratios. The top row
represents the mass ratio as a function of time. We arbitrarily decided to
show the ratio of the outer planet mass divided by the inner planet mass: this
means that when the ratio is decreasing, the inner planet accretes more than
the outer one and vice versa. Depending on the viscosity, we see that the
ratio shows different flips in time. A first flip 1 occurs at each viscosity
around $2.5\times 10^{3}$ yrs. This flip originates from the accretion rate
switch discussed in the previous paragraphs: the planets become massive enough
to change their accretion regime, resulting in a higher accretion rate for the
outer planet (increasing mass ratio).
A second flip 2 is observed around $10^{4}$ yrs, at all viscosity except for
$\alpha=10^{-2}$ (right panel, which is discussed below). This flip is linked
to the formation of deep planetary gaps. In the bottom panels of Fig. 3, we
show the perturbed surface densities at the different times marked by the
vertical lines in the top panel. The perturbed surface density is defined as
the surface density of the disk in presence of the planets normalized to the
surface density of a disk without planets at the same time
($\Sigma_{perturbed}(t)=\Sigma_{planet}(t)/\Sigma_{disk}(t)$). These perturbed
surface density profiles are used to determine when a gap is opened. We use
the definition suggested by Crida et al. (2006): a gap is considered opened
when the gas surface density is depleted by $90\%$ compared to a disk without
planets. This threshold is represented by the horizontal gray dotted line at
$\Sigma_{planet}/\Sigma_{disk}=0.1$. The second flip 2 in the mass ratio
occurs when the inner planet reaches this threshold: when the inner planet
opens a deep gap, it starts to accrete more than the outer planet.
Figure 3: Mass ratio (top row) and perturbed surface density profiles (bottom
row) as a function of time for different $\alpha$-viscosities. As in Fig. 1,
the planets are fixed at the 3:1 period ratio positions, represented by the
two vertical dotted lines in the bottom row. The horizontal dotted lines in
the bottom panels mark the $\Sigma_{planet}/\Sigma_{disk}=0.1$ threshold,
where we consider that a gap is opened (Crida et al., 2006). In the top
panels, a decreasing (respectively increasing) mass ratio indicates that the
inner (respectively outer) planet is accreting more than the other planet.
Different flips are observed in the evolution of the mass ratios, marked in
circled numbers. The color of the surface density profiles shown in the bottom
row corresponds to the color of the different vertical lines in the top row
and represents different times. The last snapshot of the perturbed surface
density is taken $2.4\times 10^{5}$ yrs (20000 orbits of the inner planet)
after the last mass ratio flip, shown by the vertical dotted black line. The
first flip 1 can be explained by the accretion formula itself (see Eq. 2),
while the other two flips correspond to the evolution of the surface density.
This link between the gap opening of the inner planet and the decrease of the
mass ratio can be explained by the impact of gas accretion on gap opening
described in Bergez-Casalou et al. (2020). At low viscosity, gas accretion
does not help carve a gap because the disk reaction time is long. The gas is
therefore dominantly pushed away from the planets orbit by gap opening,
enhancing the surface density in between the planets as well as in the inner
and outer regions of the disk. This results in a perturbed surface density
larger than one (light gray line in the bottom panels of Fig. 3). When the gap
is opened, the planetary gas accretion becomes the dominant process
influencing the gas distribution. As the inner planet opens its gap first, it
starts by depleting the material present in between the planets and in the
inner region of the disk, leading to an inner planet with a higher accretion
rate than the outer one. Then, the outer planet also opens a deep gap, helping
the inner planet deplete the material located in between them. At this stage,
the amount of gas present around the inner planet (at $r<r_{p,out}$) is
dictated by three different processes. Gas is removed from this region of the
disk by (i) the accretion onto the planets and (ii) the accretion onto the
star, and is replenished by (iii) the viscous diffusion of the gas from the
outer part of the disk through both planet gaps. At low viscosity, only a
small amount of gas manages to diffuse through the gaps of both planets. This
results in a depletion in gas of the inner disk, resulting in the starvation
of the inner planet.
Figure 4: Same as Fig. 3 but for different aspect ratios, with
$\alpha=10^{-3}$. As in Fig. 2, we note at the top of each row the difference
in kinematic viscosity caused by the change in the disk’s aspect ratio. Here
the mass ratio evolutions differ in amplitude, but they present the same
behavior as in Fig. 3. The difference in amplitude originates from the direct
dependence of the accretion routine on the aspect ratio, and not on the
$\alpha$-viscosity (see Eq. 2).
The third flip 3 in the mass ratio, shown in the top panels of Fig. 3,
corresponds to the moment when the outer planet starts accreting more than the
starved inner planet, because it is supplied in gas by the outer region of the
disk. The depletion of the inner disk can be seen in the corresponding
perturbed surface density profiles of the bottom row of Fig. 3: the gray line
shows a depleted region in between the planets and in the inner disk. We
additionally show the surface density profiles at 20 000 inner planetary
orbits ($\simeq 2.4\times 10^{5}$ yrs) after the last mass ratio flip 3: after
this time, the inner disk is almost completely emptied in gas, with a
perturbed surface density of less than 0.1 within 12 AU. It is clear that
after this time, the outer planet will accrete more than the inner one, until
it becomes the most massive planet of the system. The only differences between
the $\alpha=10^{-4}$ (left panel) and $\alpha=10^{-3}$ (middle panel) cases
are the delay in time of the flips due to different viscous timescales and the
presence of vortices at $\alpha=10^{-4}$, influencing the accretion rates of
both planets, as mentioned earlier.
The behavior of the mass ratio in a high viscosity disk (right panel) is
slightly different than at lower viscosities. As shown in Bergez-Casalou et
al. (2020), at high viscosity, gas accretion helps gap formation. Therefore,
the material located in between the planets and the inner disk is immediately
depleted by gas accretion and viscous stellar accretion. As these two regions
are not enhanced in gas by gap opening, the inner planet accretion rate slowly
reduces as the inner disk is immediately depleted in gas, meaning that no
additional mass ratio flip is observed except for the very first one
originating from the accretion recipe. At this viscosity, the gas manages to
diffuse efficiently through the gaps of both planets, avoiding a complete
depletion of the inner region: this can be seen by comparing the surface
density profiles at two different times toward the end of the simulation
(marked by the gray and black lines on the right panels). Indeed, the
difference in the profiles after 2 000 orbits of the inner planet is small,
meaning that the disk gas flow is high enough to prevent the total depletion
of the inner region, unlike at lower viscosities (middle and left panels).
However, in this configuration, even if the material around the inner planet
is replenished by viscous evolution, the amount of gas diffused through both
gaps is not high enough to allow the inner planet to accrete more than the
outer planet.
These behaviors at high and low viscosity are also observed when we vary the
disk aspect ratio. In Fig. 4, we show the evolution of the mass ratios as a
function of time (top row) as well as the perturbed surface density profiles
at given times (bottom row) for the different aspect ratios presented in Fig.
2. As the behaviors described in the previous paragraphs are only dependent on
the gas kinematic viscosity $\nu$, we recover the same behaviors when we
change the aspect ratio: at low viscosity, the mass ratios are highly
influenced by gap formation whereas at high viscosity, the mass ratio does not
present more than one flip. However, due to the dependence of the accretion
recipe on the aspect ratio, the amplitudes of the mass ratios are highly
dependent on $h$. While the mass ratios reached values between 0.8 and 1.1 for
the different $\alpha$ parameters (Fig. 3), here the planets show a larger
spread in mass ratio at the beginning of the simulations in disks with larger
aspect ratios. However, once the gaps are formed in all cases and the inner
disk is depleted, all the mass ratios stay quite close to 1, with an outer
planet less than 1.2 times more massive than the inner planet.
As a conclusion, the mass ratio between the planets is highly and mainly
dependent on the disk kinematic viscosity $\nu$. The resulting mass ratios are
close to 1 even after $0.5$ Myrs of evolution, leading to planets with rather
similar masses. In all the explored cases, we expect the outer planet to
become the most massive planet of the system.
## 4 Single accreting planet compared to two accreting planets
The presence of the second planet highly influences the growth of the inner
planet. In order to quantify the effect of the presence of the second planet,
we compare the growth of the two planets to two separate simulations where the
planets are alone in their disk.
### 4.1 Accretion rate and mass comparisons
It has been shown in Sect. 3.3 that the outer planet has the capacity to
starve the inner planet once the gaps are formed and the inner disk is
depleted. Consequently, the differences between two accreting planets and
single accreting planets should increase with time. We show in Fig. 5 the
comparison between three different simulations: the first one, represented by
the blue and red colors, considers the simultaneous accretion of two planets
in the 3:1 period ratio as in the previous section; in the second simulation,
the disk hosts only a single planet located at the position of the inner
planet (purple line); in the third simulation, the disk hosts also a single
planet located this time at the position of the outer planet (cyan line). Each
planet configuration is represented in the top left corner of the figure. As
in Sect. 3.1, different $\alpha$ are shown in the different columns,
increasing from left to right.
Figure 5: Comparison between single accreting planets and two simultaneously
accreting planets. The planets are fixed at positions corresponding to the 3:1
period ratio. As in Figs. 1 and 2, the top row presents the accretion rates as
a function of time and the middle row the planet masses. In the bottom row, we
show the ratio of the masses in the single- and two-planet case: the red line
represents the ratio of the inner planets ($m_{2p,in}/m_{1p,in}$), and the
blue line represents the outer planets’ ratio ($m_{2p,out}/m_{1p,out}$). At
low viscosity (left panels), the differences with the single-planet case
originate from the additional formation of vortices in between the planets,
which enhances the accretion rates during the vortex lifetimes. At high
viscosity (right panel), while the outer planet is slightly impacted by the
presence of the inner planet, we observe the starvation of the inner planet.
We expect to see similar behavior at lower viscosities but delayed in time,
due to longer viscosity timescales.
The different planetary gas accretion rates are shown in the top row of Fig.
5. At high viscosity, in the right panel, the accretion rate between the
single planets and the two planets are very similar. As expected, only the
accretion rate of the inner planet is significantly impacted by the presence
of the second planet: after $2\times 10^{4}$ yrs, the accretion rate of the
inner planet in the two-planet case (red line) starts to be reduced compared
to the single-planet case (purple line). This is particularly visible in the
bottom right panel of Fig. 5, where we compare the masses in the two-planet
case to the single-planet case: the red line represents the mass ratio of the
inner planets ($m_{2p,in}/m_{1p,in}$) while the blue line represents the mass
ratio of the outer planets ($m_{2p,out}/m_{1p,out}$). After $2\times 10^{4}$
yrs, the red line continuously decreases, meaning that the planet at the inner
position has a reduced accretion rate in the two-planet case compared to the
single planet. This is due to its starvation, as discussed in the previous
section.
At high viscosity, the outer planet is only slightly impacted by the presence
of the inner planet. While their accretion rate seems very similar (blue and
cyan line in the top right panel), the mass ratio shows a slight reduction of
the mass in the two-planet case (blue line slightly below one in the bottom
right panel). This originates also from the depletion of the inner disk: the
gas accretion rate of the outer planet only relies on the material located
outside the planet’s orbit in the two-planet case, while the single planet
accretes material from both the outer and inner disk.
At lower viscosities, the impact of the presence of the second planet occurs
earlier and is more significant. Focusing on the intermediate viscosity
($\alpha=10^{-3}$, in the middle column), we see that the accretion rates (top
panel) in the two-planet case are enhanced after $\sim 10^{4}$ yrs for both
the inner and the outer planet compared to the single planets from this point
in time. This enhancement, absent at high viscosity, originates from the gap
opening process: as mentioned in Sect. 3.3, at this viscosity, gap opening
pushes material away from the vicinity of each planet, enhancing the surface
density around them. In this case, the planets are close enough to each other
to push material in the feeding zone of the neighboring planet. Therefore, the
inner planet pushes material toward the outer planet and vice versa. Each
accretion rate is enhanced until the depletion of the inner disk
($r<r_{p,out}$), resulting in lower accretion rates than in the single-planet
cases. This enhancement can also be seen in the evolution of the planet masses
(bottom and middle panels of Fig. 5 for $\alpha=10^{-3}$). In the long term,
the evolution of the mass ratios between the two-planet and single-planet
cases are expected to behave like at high viscosity: the inner planet will be
starved in gas, leading to a smaller inner planet in the two-planet case
compared to the single-planet case. As for the outer planet, once the inner
disk is depleted, it is fed only by the outer disk in the two-planet case
whereas the single outer planet accretes also from the inner disk. This leads
to a decreasing outer planet mass ratio $m_{2p,out}/m_{1p,out}$ until reaching
an almost constant ratio, similarly to the high viscosity case.
Regarding the lowest viscosity case ($\alpha=10^{-4}$, in the left column of
Fig. 5), the presence of additional vortices located in between the planets
significantly alters the accretion rates of the two accreting planets compared
to the single planets. Even if the single planets also produce vortices,
leading to the oscillations observed in their accretion rate too, the
additional vortex present in between the planets quickly enhances the
accretion of both planets. Except from this non-negligible influence, the
overall evolution of the planet masses follows the trend observed at
$\alpha=10^{-3}$, where we expect the different flips to occur at later times
due to the larger viscous timescale.
Figure 6: Comparison of the mass ratios in the single-planet and two-planet
cases as a function of time and for different viscosities. Again, the planets
are fixed at the 3:1 period ratio positions, as shown in the upper-left
corner. In solid lines we show the mass ratios of the outer over the inner
planet in the two-planet case, whereas the dashed lines represent the mass
ratio of the single planets. Different flips are marked by circled numbers and
can be explained by the impact that gap opening and depletion of the inner
disk have on the accretion behavior of the planets.
Finally, we also compare the masses of the outer planets with the inner
planets, as in Sect. 3.3. At intermediate viscosity ($\alpha=10^{-3}$, middle
panel of Fig. 6), the mass ratio in the single-planet cases (dashed line)
features only one flip 1 before reaching a plateau after $\sim 2\times 10^{4}$
yrs. As before, the first flip 1 is due to the accretion recipe, and it occurs
at the same time in the two-planet case as in the single-planet cases. The
second flip 2 in the case of the two planets in the same disk, due to gap
opening, is absent in the single-planet cases. This is expected because it is
the presence of the second planet creating a gap that enhances the accretion
of the inner planet, which is not the case in the single-planet simulations.
With our disk profile (a surface density power law of -1 and constant aspect
ratio), the single planet located at the outer position accretes naturally
more than the inner planet, leading to an increasing mass ratio. The plateau
observed after $\sim 2\times 10^{4}$ yrs corresponds to the moment when the
single planets open a deep gap: then, the accretion of each planet is mainly
governed by the flow of gas originating from the outer region of the disk.
With our disk profile and due to the close proximity of the planets, the gas
flow from outside of the inner planet position is similar to the flow from
outside of the outer planet position, leading to similar accretion rates. The
same behavior is observed at low viscosity ($\alpha=10^{-4}$, in the left
panel of Fig. 6), considering the perturbations and enhancement produced by
the vortices.
At high viscosity ($\alpha=10^{-2}$, right panel of Fig. 6), the behavior of
the single-planet simulations is again different compared to the simulations
with two planets. The absence of additional flips after $2\times 10^{3}$ yrs
in the two-planet case is explained by the relative absence of impact of gap
opening at this high viscosity (see Sect. 3.3): the viscosity is high enough
to cause the immediate depletion of the inner disk via both viscous accretion
toward the star and the accretion of the inner planet, immediately leading to
its starvation. However, in the mass ratio of the single planets, we observe
another flip 2 around $4\times 10^{4}$ yrs. After this time, the inner planet
accretes more than the outer one. This originates from the flow of gas
reaching the inner disk once each gap is opened: at this viscosity, the gas
can significantly flow through the gaps of the planets. However, it is easier
to flow through the gap of the single planet located in the inner region as it
is less wide than the gap of the single planet located in the outer region.
This results in a inner disk that is more depleted in the case of the outer
single planet than in the case of the inner single planet, leading to the
reduction of the accretion rate onto the outer single planet.
### 4.2 Impact on the accretion onto the star
The presence of a single gap opening planet can alter the gas accretion onto
the central star (e.g., Manara et al., 2019; Bergez-Casalou et al., 2020).
Here, we investigate the impact of the presence of a second planet on the
evolution of the stellar gas accretion. The stellar gas accretion rate is
defined by the flow of mass through the inner edge of the disk:
$\dot{M}_{*}=-2\pi\;r_{in}v_{r,in}\Sigma_{in},$ where $v_{r,in}$ and
$\Sigma_{in}$ are the radial velocity and surface density at the inner
boundary located at $r_{in}=0.2$ AU. In Fig. 7, we compare the stellar
accretion rates of disks that contain two accreting planets in the 3:1 period
ratio (solid black line), disks that contain single planets located at the
inner (purple dashed line) and outer (cyan dotted dashed line) positions of
the two-planet simulation, or disks that host no planets at all (gray dotted
dashed line). Again, the $\alpha$-viscosity increases from left to right.
Independent of the viscosity, the stellar accretion rate in the presence of
two planets features oscillations: they originate from the periodic overlap of
the planets spiral density arms, locally enhancing the surface density of the
gas.
Figure 7: Influence of a second accreting planet on the stellar accretion rate
at the inner edge of the disk (0.2 AU). As in Figs. 5 and 6, the planets are
located in the 3:1 period ratio positions, with an increasing
$\alpha$-viscosity from left to right. The comparison is made between the two-
planet case (solid black line), the single-planet cases (dashed purple line
for the single inner planet and dotted cyan dashed line for the single outer
planet) and a disk without planets (dotted dashed gray line). The oscillations
present in the case of the two accreting planets are due to the overlap of
their spiral arms, coupled with the presence of vortices at low viscosity.
This sometimes results in a negative radial velocity, explaining the dip seen
at $\alpha=10^{-4}$ due to the logarithmic scale. Compared to a disk with no
planet, the accretion onto the star is only reduced by up to a factor of 3 in
the presence of multiple giant planets, similar to the scenario with a single
planet (Bergez-Casalou et al., 2020).
At intermediate viscosity ($\alpha=10^{-3}$, middle panel of Fig. 7), the
presence of the planets has two distinct impacts. Before $10^{4}$ yrs, the
planets are not large enough to influence the accretion onto the star. Between
$10^{4}$ yrs and $10^{5}$ yrs, the disks hosting planets harbor an enhanced
stellar accretion rate compared to the disk without planets. This originates
from gap opening: material is pushed to the inner region, feeding the star.
After $10^{5}$ yrs, planetary gas accretion and gaps prevent part of the gas
from reaching the inner region, leading to the decrease in the stellar
accretion rate. It takes more time for the disk hosting the single planet
located at the outer position to reduce the stellar accretion rate due to the
larger inner disk present in this configuration. After a given time (here
after $2\times 10^{5}$ yrs), the flow of gas through the gaps reach a quasi
equilibrium state, leading to a linear evolution of the stellar accretion
rate, with a similar slope compared to the slope of the stellar accretion rate
of the disk without planets. At this stage, the stellar accretion rate is
reduced by a factor of between 4 and 5 when the disk hosts two accreting
planets compared to the disk without planets.
The enhancement produced by the two accreting planets is slightly more
pronounced than in the single-planet case as it includes the material pushed
by both planets. As one could expect, the decrease of the stellar accretion
rate in the two-planet case occurs at the same time as in the disk hosting the
single planet located at the inner position. However, at this viscosity, the
reduction of the stellar accretion rate is only barely influenced by the
presence of the second planet, resulting in a decrease of only $\sim 30\%$
compared to the single inner planet case. This means that the flow of gas
reaching the inner disk is mostly governed by the influence of the inner
planet.
At high viscosity, the viscous spreading of the gas prevents the enhancement
of the stellar accretion rate: the material pushed to the inner regions by the
gap opening planets is both less important than at lower viscosity and quickly
diffused toward the star by viscous spreading. This results in a reduction of
the stellar accretion rate after $10^{4}$ yrs only. The gas is efficiently
diffused through the planet gaps, maintaining a high stellar accretion rate,
even in presence of multiple accreting planets. Here, the stellar accretion
rate is only reduced by up to a factor of 2.5 compared to the accretion in a
disk without planets. Moreover, the presence of the second planet only
influences the stellar accretion rate by less than $20\%$ compared to the
cases with single planets. Focusing on the single-planet cases, we see that
the stellar accretion reaches a similar equilibrium in both cases, independent
of the planet location. As expected, with this high viscosity, the presence of
accreting planets barely influences the gas disk evolution.
The opposite behavior is observed at low viscosity ($\alpha=10^{-4}$, left
panel of Fig. 7). Here, the stellar accretion rate is highly influenced by the
vortices and by the gaps formed by the planets. Material is efficiently pushed
in the inner regions, enhancing the stellar accretion rate by up to a factor
of 10 in the disk hosting two planets compared to the accretion of a disk
without planets. The reduction of the accretion occurs at later time than at
higher viscosity, which is expected because the planets grow slower and gap
opening takes more time Bergez-Casalou et al. (2020). However, the presence of
the second planet in the disk highly influences the stellar accretion rate,
because the viscosity is not high enough to diffuse gas through both gaps. Due
to the long viscous timescale, we are not able to determine the final
reduction factor compared to the higher viscosity cases. At the end of the
simulation containing the two planets, the stellar accretion rate is reduced
by a factor of 3 compared to the disk without planets, giving a lower
estimation of the reduction factor. In Sect. 7.1, we discuss the impact that
this reduction factor can have on observations and compare it to other
studies.
## 5 Influence of the planet separation
Our study has focused so far on planets placed at positions corresponding to
the 3:1 period ratio. However, the separation between forming planets is not
fixed in time as they can dynamically interact with the disk and with each
other (e.g., Baruteau et al., 2014; Crida & Bitsch, 2017). While we will
investigate the impact of the radial evolution of the planets on their growth
in a future study, we study in the following section the impact of the planet
separation in their growth by placing the planets at different period ratios.
Figure 8: Mass ratio as a function of time for different $\alpha$-viscosities
and different period ratios. The darker the line, the farther away the planets
are located from each other. The separation between the planets has a small
impact on the range of their mass ratio, meaning that planets that start
accreting at the same time will have similar masses. The behavior of the mass
ratio evolution is not impacted by the planet separation: at all viscosities
the mass ratio flips are due to the same reasons described in Fig. 3. The
exception is for the 2:1 period ratio at high viscosity (right panel): here
the planets are close enough and the viscosity is sufficiently high to
maintain a significant flow toward the inner disk through the gaps, preventing
the inner planet from being starved. This results in an additional mass ratio
flip, II, similar to the high viscosity single-planet case of Fig. 6.
### 5.1 Impact on the mass ratio
As shown in Sect. 3.3, the planet mass ratio evolution reflects the accretion
history of the planets. We show in Fig. 8 the mass ratio evolutions of two
simultaneously accreting planets located at different period ratios, ranging
from 2:1 up to 5:1. These period ratios, represented to scale in the top left
corner of the figure, were chosen such that the planets can be considered
dynamically stable during their growth, as we neglect their dynamical
interactions. Their stability is monitored thanks to their mutual Hill radii
(Chambers et al., 1996). As in Fig. 3, the mass ratio presented is the ratio
of the outer planet mass divided by the inner planet mass and each panel
represents an $\alpha$-viscosity, increasing from left to right. A darker line
depicts a larger planet separation.
Independent of the viscosity, we first observe that the first mass ratio flip
1, originating from the switch of accretion regime in our accretion recipe, is
delayed in time when the planets are farther away from each other. Indeed,
with our disk profile, a planet located farther away in the disk has a
slightly lower gas accretion rate, meaning that it will need more time to
reach the mass needed to switch from the Bondi to the Hill accretion regime.
This delay affects the similarities between the planets: the farther the
planets are from each other, the more extreme are their mass ratios at the
beginning of the simulations (i.e., the deeper the first mass ratio flip 1
is).
Focusing on the high viscosity behavior ($\alpha=10^{-2}$, right panel of Fig.
8), we observe the same behavior as described in Sect. 3.3 except for the 2:1
period ratio. While the other period ratios feature only the first flip 1 due
to the rapid depletion of the inner disk via viscous stellar accretion, a
second flip II is observed at around $10^{5}$ yrs in the mass ratio of the 2:1
period ratio case. In this configuration, the planets are close enough from
each other to facilitate the diffusion of gas through both gaps compared to
the other period ratios, as they quickly form a common gap. As the flow of gas
to the inner disk is higher, more material is present around the inner planet.
In this case, the amount of gas diffusing through both gaps, is high enough to
prevent the starvation of the inner planet, leading to a decreasing mass
ratio.
The behavior of the mass ratio at $\alpha=10^{-3}$ (middle panel of Fig. 8) is
also the same for each period ratio. The second mass ratio flip 2, occurring
at around $10^{4}$ yrs, is always due to the formation of the planetary gaps.
The delay in the flip originates from the time needed for the outer planet to
also significantly enhance the surface density in between the planets via gap
opening. The amplitude of the mass ratio between the second 2 and third flip 3
(i.e., the difference between the local maximum 2 and the local minimum 3) is
decreasing with decreasing planet separation. As this moment corresponds to
the moment when the inner planet is accreting more thanks to the enhancement
of material in between the planets and in the inner disk after gaps are
formed, closer planets have less material in between them by construction,
leading to a smaller mass ratio amplitude. We note that it also takes more
time for planets located farther away from each other to empty the material
located between the planets, meaning that the inner planet accretes more than
the outer planet for a longer time compared to planets closer to each other.
Therefore, the inner planet is starved more easily the closer the planets are.
Figure 9: Gap-opening mass as a function of the viscosity for different
criteria and our simulations, as in Bergez-Casalou et al. (2020). In the left
panel, we show the gap opening masses of the inner planets in the two-planet
case compared to the single planet cases located at the inner position. The
outer planet cases are shown in the right panel. In each panel, the lines
represent the different gap-opening criteria from the literature: Crida et al.
(2006) in dashed blue, Fung et al. (2014) in solid orange, and Gyeol Yun et
al. (2019) in dashed-dotted red. As shown with the schematics above the
panels, the two-planet cases are represented by triangles (upward for the
inner planets and downward for the outer planets), and the single planets are
represented by circles. The inner single-planet case is shown in gray, and the
colored circles correspond to the colors of the outer positions in each
configuration. For clarity, in the right panel, the viscosity is divided into
discrete intervals, allowing us to compare the gap opening mass at one given
viscosity in the different configurations. The gap opening masses of the inner
planet are barely impacted by the presence of a second outer planet. On the
other hand, at low viscosity, the closer the planets are to each other, the
higher the gap opening mass is compared to the single-planet case.
At low viscosity ($\alpha=10^{-4}$, left panel of Fig. 8), the planet
separation has an additional impact on the formation of vortices. In the 2:1
period ratio simulation, the planets quickly create a common gap, preventing
the formation of a strong vortex in between them. As we mentioned in Sect.
3.1, as vortices push material toward the planets, it means that the gas
accretion of the planets in the 2:1 period ratio are less impacted by the
presence of vortices compared to planets located farther away from each other.
In the 5:1 period ratio case, the planet gaps are clearly distinct from each
other. Each planet therefore create sharp density gradients at the inner and
outer edge of their gaps, resulting in the formation of four vortices. Even
though the vortices located in the inner disk (i.e., inner to the outer
planet) dissipate quickly, they influence the evolution of the planet masses.
Except from the formation of vortices, the global behavior of the mass ratio
is due to the same processes as at intermediate viscosity.
Overall, independent of the planet separation or disk viscosity, the mass
ratios stay close to one (0.7 ¡ $m_{out}/m_{in}$ ¡ 1.25). We discuss in Sect.
7.5 how does this compare to the observed planetary systems.
### 5.2 Impact on the gap opening mass
The gap opening mass is an important parameter used both in theoretical models
to approximate when a planet switches from the fast type I migration to the
slow type II migration (e.g., Ndugu et al., 2018; Bitsch et al., 2019; Miguel
et al., 2020; Ndugu et al., 2021) and in dust observations to indirectly
derive the masses of embedded planets from the observed characteristics of
gaps (e.g., Zhang et al., 2018; Asensio-Torres et al., 2021). As we showed in
Bergez-Casalou et al. (2020), gas accretion has a non-negligible impact on the
gap opening mass. In this section we investigate the influence of the presence
of a second accreting planet on the gap opening mass of each planet.
We compare in Fig. 9 the gap opening mass of our accreting planets to
different gap opening criteria derived in previous studies (Crida et al.,
2006; Fung et al., 2014; Gyeol Yun et al., 2019). The gap opening masses of
the inner planets are shown on the left panel of the figure while the outer
planets are shown on the right panel222The simulation did not reach the gap
opening mass in the case of the outer planet of the 5:1 period ratio at low
viscosity. The presence of strong vortices prevented the correct continuation
of the simulation. However, we clearly see the expected trend.. As in the
previous section, the color represents the different period ratios, as can be
seen on the schematics of the disk configurations represented in the top of
the figure. Triangles represent the gap opening masses of the planets located
in a two-planet disk and single-planet gap opening masses are shown with
circles. The gray circles correspond to the gap opening masses of the single
planets located at the inner location (they correspond to the gap opening
masses presented in Bergez-Casalou et al. (2020) for the fiducial gas
accretion rate). The different lines represent the different gap opening
criteria compared in this study. In the right panel, the viscosity is divided
into discrete intervals for each of the studied viscosity
($\alpha=10^{-4},10^{-3},10^{-2}$) to help in visualizing the different
configurations.
In Bergez-Casalou et al. (2020), we conclude that gas accretion has a
different impact on the gap opening mass depending on the disk viscosity: at
high viscosity, gas accretion helps carve deeper gaps, resulting in a lower
gap opening mass for an accreting planet while at low viscosity, gap formation
is not helped by gas accretion, resulting in a higher gap opening mass for an
accreting planet. When a second planet is added in the disk, the accretion of
each planet is impacted by the gap opening of the neighboring planet, as shown
in Sect. 4. However, this impact depends on the viscosity.
At high viscosity, gas accretion helps carve a deeper gap. Here, with
$\alpha=10^{-2}$ and $h=0.05$, the disk is at the intersection between the
high viscosity regime and the low viscosity regime described above, meaning
that the gas accretion has no important impact on the gap opening mass. The
gap opening masses are therefore solely dependent on the amount of gas
diffusing through the gaps. As the gas diffuses efficiently in this case, gap
formation is not impacted by the presence of a second planet. Indeed, the
inner disk is more depleted by viscous accretion than by the accretion of both
planets, meaning that the material pushed away by the gap forming planets is
dissipated via viscous spreading. This can be seen in Fig. 9, where the gap
opening masses are the same in the single or two-planet case, for both the
inner and outer planets.
At lower viscosities, gas accretion does not help gap formation (Bergez-
Casalou et al., 2020). Therefore, the gap opening mass depends on the
accretion rate of the planet. As presented in Sect. 4.1, the accretion rates
of the two accreting planets are slightly enhanced compared to the single
planets due to the formation of the planetary gaps pushing material in the
feeding zones of the neighboring planet. This slight enhancement of the
accretion rate of the planets leads to slightly higher gap opening masses.
This impacts both the inner and the outer planets, as shown in Fig. 9. As the
gap opening mass then relies on the amount of material pushed toward the
neighboring planet, planets that are close enough from each other enhance
their gap opening mass more. Intuitively, when the planets are farther from
each other, they tend to behave as if they are isolated and have gap opening
masses closer to the single-planet simulations.
Overall, the gap opening masses are barely impacted by the presence of a
second planet in the disk. The small differences originate from the
differences in accretion rates as the planets push material toward each other.
Considering the conclusions of Bergez-Casalou et al. (2020), it seems that the
gas accretion rate on the planet itself has a stronger impact on the gap
opening mass than the presence of a simultaneously accreting companion.
## 6 Influence of delayed accretion
Giant planet formation models have very few constraints on the timing at which
runaway gas accretion occurs (e.g., Paardekooper & Johansen, 2018; Raymond et
al., 2020) . So far, we only considered the simultaneous accretion of both
giants. However, depending on the disk local properties and on the formation
mechanism, giant planets located in the same disk could start accreting at
different times. We investigate here the influence of the delayed accretion on
the evolution of the planetary growth. Different time delays are considered,
on the outer and on the inner planet. We base our delays on the mass of the
neighboring planet: the accretion on the second planet is allowed when the
other planet reaches 0.3 $M_{J}$, 0.5 $M_{J}$ and 1 $M_{J}$. We chose to
investigate the impact of the accretion delay on the 2:1 period ratio
configuration at high and intermediate viscosities as they reach these masses
in a reasonable computational time.
All the resulting mass ratios are shown in Fig. 10. The mass ratio of planets
simultaneously accreting is shown and corresponds to the 2:1 mass ratio
presented in Fig. 8. We note the difference in the mass ratio scale: while
before the mass ratios are shown on a linear scale, here the scale is
logarithmic for readability. The large spread in mass ratio is induced by our
initial choice for the different delays. Indeed, as we wait for the
neighboring planet to reach a given mass before accreting, this sets the
maximal and minimal mass ratio reached by the planets: the maximal value that
is reached is 16 ($318/20M_{\oplus}$) and the minimal one is 0.06
($20/318M_{\oplus}$). We therefore expect the planets to reach a final mass
ratio located in between these initial values.
The evolution of the different mass ratios is very different from the
simultaneously accreting planets. When the outer planet accretion is delayed
(green lines in Fig. 10), it allows the inner planet to accrete slightly more
gas before being starved by the growth of the outer planet. At high viscosity,
the gas diffuses efficiently through the planet gaps, depleting the whole disk
in gas and leading to a high stellar accretion rate (see Sect. 4.2).
Therefore, a longer delay results in accretion in a more depleted disk. As a
consequence, the mass ratio is lower for longer delays.
At lower viscosity, the effect of viscous spreading as described above can be
perturbed by the formation of the inner planet gap. However, the gap opening
mass at this kinematic viscosity is around 0.5 $M_{J}$ at the location of the
inner planet. For a delay of 300 orb. ($\sim 4.8\times 10^{3}$ yrs),
corresponding to an inner planet mass of 0.3 $M_{J}$, the outer planets starts
accreting while the inner planet did not create a deep gap yet. This leads to
a very similar final mass ratio evolution as in the simultaneous case.
However, we know from Sect. 3.3 that when the inner planet creates its gap, it
starts to deplete the material located in the inner disk and in between the
planets. Therefore, when the outer planet starts accreting even later (e.g.,
when the inner planet has reached its gap opening mass), the inner planet
already depleted part of the material in between them, leading to a lower gas
accretion rate of the outer planet compared to the simultaneous case. This
results in lower mass ratios for longer delays.
The mass ratio of the planets in the case of the delayed accretion of the
inner planet (purple lines of Fig. 10) highly depends on the depth of the gap
of the outer planet and on the viscosity of the disk. Indeed, at low
viscosity, if the inner planet starts accreting before the formation of the
gap of the outer planet, then the inner region is not depleted in gas yet. The
behavior of the mass ratio then quickly tends to be the same as in the
simultaneous case. However, if the outer planet already opened its gap, then a
longer delay of accretion results in a more depleted inner disk. The inner
planet has therefore less material to accrete, leading to higher mass ratios.
At high viscosity, the same behavior occurs, with the inner disk being
efficiently depleted by stellar accretion.
Figure 10: Mass ratio as a function of time for different accretion delays on
the inner (purple lines) or the outer planet (green lines). The planets are
located in the 2:1 period ratio configuration. The two panels represent two
different viscosities: $\alpha=10^{-3}$ in the left panel and $\alpha=10^{-2}$
in the right. Darker lines represent shorter delays. Note that this time the
mass ratio is displayed on a logarithmic scale (previous plots are on a linear
scale) due to the extreme mass ratios induced by our initial setup here. The
dots mark the moment when both planets have reached at least 0.3 $M_{J}$ and
can be considered gas giant planets. The gray rectangle represents the region
where the mass ratio lies between 0.25 and 0.5 and shows the region where
Jupiter and Saturn meet the required conditions to enter common outward
migration, for $\alpha\lesssim 10^{-3}$ (Masset & Snellgrove, 2001; Pierens et
al., 2014). Independent of the viscosity, all the mass ratios quickly tend
toward the $m_{out}/m_{in}$ = 1 line, leading to similar planet masses in both
cases ($0.8<m_{out}/m_{in}<2$ after $10^{5}$ yrs).
Even if different mechanisms influence the mass ratio in the case of delayed
accretion, the giant planets always end up with rather similar masses:
$0.8<m_{out}/m_{in}<2$ after $10^{5}$ yrs. We discuss in Sect. 7.5 how does
this compare to the observed mass distributions in the exoplanet population
and what kind of constrains on planet formation can be derived.
## 7 Discussion
### 7.1 Accretion onto the star
In Sect. 4.2 we investigate the influence of the presence of multiple gas
accreting planets on the stellar accretion at different viscosities. The
results are compared to the stellar accretion in disks hosting single planets
and in disks without any planet. We find that the presence of the second
planet only has a significant effect when the viscosity of the disk is low
($\alpha\lesssim 10^{-4}$). For higher viscosities, the presence of the second
planet only influences the stellar accretion rate by up to $30\%$ compared to
the case with single planets.
We compare our results with two planets to the different stellar accretion
rates obtained with different planetary accretion rate in Bergez-Casalou et
al. (2020). At high viscosity, the presence of the second planet has less
impact on the stellar accretion rate compared to a significant enhancement of
the planet accretion rates. Due to the uncertainties in gas accretion rates
(e.g., spread in the values found in the following studies: Kley, 1999;
D’Angelo et al., 2003; Tanigawa & Ikoma, 2007; Ayliffe & Bate, 2009; Machida
et al., 2010; Tanigawa & Tanaka, 2016; Crida & Bitsch, 2017; Schulik et al.,
2019; Lambrechts et al., 2019) , it is impossible to use the stellar accretion
rates to determine if the protoplanetary disk hosts a single fast accreting
planet or multiple planets accreting at a lower rate.
As discussed in Bergez-Casalou et al. (2020), our results are quite different
from the study derived by Manara et al. (2019). In their models, they find
that the stellar gas accretion rates can be reduced by over two orders of
magnitude when the disk is hosting accreting giant planets. We showed in our
previous paper that these large spreads of stellar accretion rates could only
be reached by widely changing the disk viscosity (over several orders of
magnitude). With this study, we additionally show that the presence of a
second accreting companion cannot explain such a large reduction in the
stellar accretion rate either. Indeed, while reducing the disk viscosity to
$\alpha=10^{-4}$ enhanced the impact of the two accreting planets on the
stellar accretion rate, we expect the reduction to be of a factor of 10 at
most compared to a disk without planets. These discrepancies between our study
and the work done by Manara et al. (2019) are the same as mentioned in our
previous paper: their model simulates a 1D gas disk while our study is
performed in 2D, allowing us to more accurately determine the flow of gas
through the gaps of the planets (e.g., Lubow & D’Angelo, 2006). Moreover,
their planetary gas accretion rates might be overestimated as they rely on the
unperturbed surface density (Mordasini et al., 2012), while we showed here
that the depletion of the inner disk leads to the starvation of the inner
planet and consequently a reduction of its accretion rate.
Again, our simulations indicate that planetary gas accretion might have a
smaller impact than expected on the stellar gas accretion rates, even in the
presence of multiple accreting planets.
### 7.2 Dependence on the accretion behavior
The results from our simulations rely on our accretion recipe, based on the
results from Machida et al. (2010) and on the amount of gas available around
the planets (i.e., the total mass of the disk and the gas surface density
profile). However, one can argue that our understanding of planetary gas
accretion in the runaway phase can be improved. In the current literature, the
derived accretion rates can range over several orders of magnitude, ranging
from around $10^{-8}$ $\rm M_{J}/yr$ (Tanigawa & Ikoma, 2007; Tanigawa &
Tanaka, 2016) up to around $10^{-4}$ $\rm M_{J}/yr$ (D’Angelo et al., 2003;
Schulik et al., 2019). Furthermore, some studies (e.g., Lambrechts et al.,
2019) show that the actual accretion rate has to be separated from the flux
that goes through the atmosphere as the planet is embedded in the gas disk.
With this definition, the flux through the atmosphere is on the order of
$10^{-4}$ $\rm M_{J}/yr$ but the actual accretion rate is lower, around
$10^{-6}$ to $10^{-5}$ $\rm M_{J}/yr$.
However, in our study, the global evolution of the planet’s mass ratio relies
mostly on the evolution of the global gas distribution in the disk, via gap
opening and the depletion of the disk. Therefore, the way the disk is evolving
has a larger impact than the actual accretion rate.
As mentioned in Sec. 2, we start our simulations with a relatively massive
disk with a surface density profile following $\Sigma\propto r^{-1}$.
Implanting the planets early implies that a large amount of gas is still
available for the planets to accrete. If we wait for a longer time before
introducing the planetary cores, the disk has time to be accreted onto the
star and therefore less material would be available around the planets. This
will lead to a lower planetary accretion rate on one hand, and on less
material between them on the other hand. Then it is just a question of timing
again between the formation of the gaps and the depletion of the inner disk
and the material between the planets: the global evolution of the planet mass
ratio is expected to be the same but on longer timescales as the gap opening
mass is independent of the disk’s surface density (Crida et al., 2006; Fung et
al., 2014; Kanagawa et al., 2015). Therefore, the timing of formation (and
therefore the total mass of the disk) will be of importance regarding the
total mass reached by the planets but will not have a huge impact on the
evolution of their mass ratio. Regarding the importance of the surface density
profile, the evolution of the planets’ mass ratio will also be shifted in time
but the qualitative evolution is expected to be similar.
### 7.3 Impact of planet dynamics
In order to determine the impact of the gas accretion on two planets embedded
in the same disk, we neglected both the dynamical interactions between the
planets and their migration. Regarding migration, different studies
investigate how it impacts gas accretion (e.g., Dürmann & Kley, 2015; Crida &
Bitsch, 2017; Dürmann & Kley, 2017). The main results of these studies are
that the evolution of the planet characteristics (i.e., mass and semimajor
axis) highly depends on the timescales of each process: a fast migrating
planet tends to accrete more gas as it quickly moves toward regions with high
surface densities. However, as we showed in Bergez-Casalou et al. (2020), gas
accretion has an impact also on the gap opening mass, influencing the
migration speed of the planet as it transitions from a fast type I to a slow
type II migration. This effect was observed in Crida & Bitsch (2017), where
their accreting planet slowed down its migration speed earlier compared to a
non accreting planet. Therefore, the resulting planetary systems highly depend
on the timescales of migration, gap formation and gas accretion.
Another dynamical effect can play an important role in the evolution of the
planets. Previous hydrodynamical studies show that multiple planets can be
captured in resonant chains during the gas phase of the disk (e.g., Baruteau &
Papaloizou, 2013; Pierens et al., 2014; Kanagawa & Szuszkiewicz, 2020). The
capture in resonance can have an important impact on the migration behavior of
the planets. For example, in the Grand Tack scenario (e.g., Masset &
Snellgrove, 2001; Walsh et al., 2011; Pierens et al., 2014), Jupiter and
Saturn are believed to migrate inward and then outward due do their capture in
resonance. This outward migration is occurring for precise disk parameters and
mass ratios, as shown in Pierens et al. (2014). As migration can be altered by
the capture in resonance, gas accretion and gap formation will also be
indirectly altered by the planet radial motion.
The capture in resonance can lead to a slight increase of the planets
eccentricity. As an eccentric planet will open a less deep gap for the same
mass (e.g., Hosseinbor et al., 2007; Sánchez-Salcedo et al., 2022), the planet
will end up with higher mass accretion rates than our planets on circular
orbits. Having a higher accretion rate will just result in a shift in the
different timing but this impact is negligible for low eccentricities. Since
the eccentricity of the planets is damped while the amount of gas is
significant in the disk (e.g., Moorhead & Ford, 2009; Bitsch & Kley, 2010),
and since our planets do not grow massive enough to excite their eccentricity
(more than 5 $M_{J}$; e.g., Papaloizou et al., 2001; Kley & Dirksen, 2006;
Bitsch & Kley, 2010; Bitsch et al., 2013), we expect the impact of realistic
low eccentricities on our results to not be significant.
We plan on implementing the impact of both migration and capture in resonance
on the growth of our two planets in follow-up studies. We expect that, if the
planets start accreting simultaneously, then the structure of the resulting
system highly depends on the timing at which gap opening will occur because it
will slow down the migration of the planets and determine their accretion
behavior. A potential outward migration can delay the depletion of the inner
disk, altering the mass ratio behavior discussed in Sect. 3.3. However, due to
the high interdependence of each mechanism, it is difficult to precisely
predict how the planets will behave.
### 7.4 Implications for the Grand Tack scenario
In the Grand Tack scenario, if Jupiter and Saturn have a mass ratio between
0.25 and 0.5, then they can migrate outward to their current locations (e.g.,
Masset & Snellgrove, 2001; Crida et al., 2009; Pierens et al., 2014). In Sect.
5.1, we show that if the planets start accreting simultaneously, they reach
mass ratios that are between 0.7 and 1.3. Therefore, in order to trigger
outward migration, the planets have to start accreting with a delay. It also
requires that the inner planet is more massive than the outer planet;
otherwise, the torques arising from the outer disk would be too large, leading
to inward migration. This also implies that Jupiter must have started to
accrete gas efficiently before Saturn.
Pierens et al. (2014) find that outward migration depends also on both the
period ratio of the planets and the disk parameters. In order to trigger
outward migration in a low mass disk, a capture in a 2:1 resonance is needed.
If the disk is more massive, then the planets need to reach the 3:2 resonance.
Both scenarios require a relatively low $\alpha$-viscosity ($\alpha\lesssim
10^{-3}$).
Within our current parameter study, we only investigated the impact of delayed
accretion in the 2:1 period ratio configuration. At low viscosity
($\alpha=10^{-3}$,left panel of Fig. 10), the conditions are barely met for
the outward migration to occur: in all cases, the mass ratios quickly evolve
in the disk, making the planets barely stay in the needed mass ratio range
(marked by the gray area). Therefore, with our current results, it seems that
outward migration of the two giant planets is very challenging to reach as
this occurs during a very short timescale. However, we plan to expand our
parameter space study in the near future, allowing us to better analyze if and
how a planetary system like Jupiter and Saturn could have formed via the Grand
Tack scenario.
### 7.5 Comparison to exoplanets
Considering our current parameter space study, planets accreting from the same
disk end up with very similar planet masses. Delaying the accretion of the
respective planets allowed us to slightly broaden the mass ratio range
reached; however, in $10^{5}$ yrs, the final mass ratios obtained are still
between 0.8 and 2. These mass ratios are quite different from the mass ratios
observed in different planetary systems. In Fig. 11, we compare the evolution
of our mass ratios to different exoplanetary systems. The data originate from
the NASA exoplanet
archive333https://exoplanetarchive.ipac.caltech.edu/docs/data.html. We
selected the planetary systems as follows: first of all, we are interested in
systems containing exactly two giant planets (i.e., with $m_{p}>0.3\;M_{J}$)
as we investigate the accretion of two planets in the runaway gas accretion
phase. Our simulation considers planet formation in a disk orbiting the Sun;
therefore, the selected planetary systems orbit Sun-like stars
($4700K<\mathrm{T_{eff}}<6500K$ and $\log(g_{*})>4$). Each panel represents
the ratio of the outer planet mass divided by the mass of the inner planet
like in previous figures as a function of the planet period ratio. Vertical
dashed lines represent the investigated period ratios. The colorbar shows the
sum of the planet masses in $M_{J}$.
From the top panel in the figure, it is clear that the observed planetary
systems hosting two giant planets have a broad range of mass and period
ratios. We highlight the systems hosting at least one hot Jupiter (i.e.,
planets with periods shorter than 10 days) with a thick black contour. These
planets might be formed after a very efficient inward migration or via a
scattering event, leaving them very close to their host star. This results in
a system where the period ratio of the planet is very large. As we do not
implement migration in this study, we focus on the planets located closer to
each other. Interestingly, it appears that planets located closer to each
other seem to have more similar masses (except for 3 systems with mass ratios
higher than 8). For better readability and comparison with our simulation, a
zoom on the planets placed in the gray rectangle is shown in the lower panel
of the figure.
Figure 11: Mass ratio evolution of the two accreting planets as a function of
their period ratio compared to exoplanetary systems. The investigated period
ratios from our simulations are marked by vertical dashed gray lines. The data
are taken from the NASA exoplanet archive, for which we selected the systems
as follows: the system contains exactly two detected planets, both of them
larger than 0.3 $M_{J}$. They orbit a single Sun-like star
($4700K<\mathrm{T_{eff}}<6500K$ and $\log(g_{*})>4$). The color of the dots
represents the sum of the planet masses in Jupiter masses, without error bars.
A black contour surrounds planets that are considered hot Jupiters (i.e., with
a period of less than 10 days). As we expect their formation to be highly
influenced by the dynamics of the system, which we do not model here, we focus
the comparison with the exoplanets on the planets marked by the gray area in
the top part of the figure. A zoomed-in view of this region is shown in the
second panel. The vertical red and pink lines represent the maximum and
minimum mass ratio reached in each of our simulations once both planets reach
0.3 $M_{J}$. Darker colors represent lower viscosities. For visibility, the
lines corresponding to the different $\alpha$-viscosities are slightly offset
from the period ratio line. The extent of the $\alpha=10^{-4}$ was so small at
low period ratios that we represent it with squares. The cross marks the
Jupiter and Saturn couple. Simultaneously accreting planets lead to planets
that are very similar in mass compared to the exoplanet population. Some
systems seem to be consistent with simultaneous accretion; however, another
mechanism is needed to explain the existence of the other systems. Delayed
accretion as shown in Fig. 10 coupled with different disk lifetimes could
explain the difference in planetary masses.
Due to computational constraints, we could not investigate the evolution of
the mass ratio until the end of the disk lifetime. Therefore, in order to make
the comparison with fully formed planets as the ones observed in the different
planet surveys, we show the maximal and minimal mass ratios obtained by our
simultaneously accreting planets with the pink, red and purple vertical lines.
As we selected the observed systems by considering planets with
$m_{p}>0.3\;M_{J}$, we show the mass ratio spread once both planets reached
0.3 $M_{J}$. The ratios obtained for different viscosities are slightly offset
from the exact period ratio for visibility. At low viscosity and small period
ratios, the mass ratio range was too small to be represented by a line;
therefore, the final mass ratio is shown with a square. Over the 45 observed
planetary systems (including Jupiter and Saturn and hot Jupiters), 5 have a
mass ratio lower than 0.7, 11 systems have a mass ratio lying between 0.7 and
1.3 and 29 systems have a mass ratio higher than 1.3. The majority of the
systems cannot be explained by the simultaneous accretion of the planets. We
also note that very few systems (only 5 here) feature a mass ratio as seen in
our Solar System with Jupiter and Saturn. While this might be explained by the
difficulty of our current facilities to see low mass planets, it also raises
the question of the peculiarity of the Solar System among other systems,
namely whether our planetary system common or an outlier.
In this study, such a large spread in mass ratio was only reached when the
planets accrete in the runaway gas accretion phase with different delays. This
accretion delay can be justified by the dependence of the beginning of the
runaway gas accretion phase on the disk characteristics. Here, we followed the
classical core accretion model, where it is assumed that runaway gas accretion
is triggered when the planetary core reaches a mass of 20 $M_{\oplus}$, with a
solid core of 10 $M_{\oplus}$ surrounded by a first gaseous atmosphere of 10
$M_{\oplus}$ (Pollack et al., 1996). However, more recent studies show that
the initial total mass of the core can vary depending on the local properties
of the disk. Runaway gas accretion can be triggered at different core masses
depending on the local disk temperature and opacity (Ikoma et al., 2001; Piso
& Youdin, 2014; Bitsch & Savvidou, 2021). Moreover, in the pebble accretion
scenario, the pebble isolation mass corresponds to the mass at which the core
is shielded from the pebble flux by the pressure bump created by its own gap
(Morbidelli & Nesvorny, 2012; Lambrechts et al., 2014; Bitsch et al., 2018;
Ataiee et al., 2018). Then, the atmosphere of the planetary core is not heated
anymore by the accretion of solids and cools down, entering the runaway gas
accretion phase. The pebble isolation mass depends on the disk’s aspect ratio,
$\alpha$-viscosity, pressure profile and turbulent diffusion of the particles
(Bitsch et al., 2018). Therefore, the delay of accretion time between two
giants highly depends on the local properties of the disk. Protoplanetary
disks can be flared, featuring large aspect ratio variations (e.g., Bitsch et
al., 2015; Pierens & Raymond, 2016), and different hydrodynamical properties
can lead to important radial variations of viscosity (e.g., Flock et al.,
2011; Dullemond & Penzlin, 2018; Delage et al., 2022). The disk properties can
lead to the delay of either the inner or the outer planet.
Even when we applied different accretion delays, the long-term trends could
only reproduce mass ratios lying between 0.8 and 2. With these parameters, the
only remaining way to reach large (or small) mass ratios, is to stop the
accretion at a given mass ratio. The timing of the dissipation of the gas disk
can be crucial here. For example, photoevaporation can dissipate the gas disk
from inside out by creating a inner hole separating the inner disk ($r<1$ AU)
from the outer disk and quickly depleting it (for a review, see Ercolano &
Pascucci, 2017). Such a depletion of the disk might have the capacity to
starve the giant planets, influencing the evolution of their mass ratio.
Some differences between our simulations and the observations might originate
from dynamical events that are not yet included in our study. For example,
dynamical interactions such as collisions might change the planets mass ratio
(e.g., Jurić & Tremaine, 2008; Raymond et al., 2009; Sotiriadis et al., 2017;
Bitsch et al., 2020). We also note that planets might also be ejected from the
systems after the dispersal of the gaseous disk: while the observed final
system would host two giants, the gaseous protoplanetary disk where they
formed could have hosted three giants, changing the accretion behavior
compared to what we simulate in this paper. However, accurately describing the
hydrodynamical evolution of the gas together with the dynamical evolution of
more than two planets for a long time (over several Myrs) is quasi-impossible
nowadays. Therefore, further studies should slowly including more dynamical
effects during the gas phase or slowly improving the hydrodynamical
assumptions of N-Body simulations.
To summarize, the simultaneous runaway gas accretion of fixed planets cannot
explain the distribution of exoplanetary masses observed as it leads to
planets with very similar masses. To increase the difference in planet masses,
the accretion between the planets have to be delayed and efficient disk
dispersal mechanisms are required to end the growth of the planets at given
mass ratios. These two last points highly depends on the disk local properties
of the gas. Moreover, taking dynamical interactions between the planets
themselves and the planets and the disk might improve our understanding of the
giants distribution. A future study including dynamical interactions and
migration is planned to determine how the results can be impacted.
## 8 Conclusions
In this paper we investigate the mass distribution of two accreting planets
located in the same disk. Using 2D hydrodynamical simulations, we monitor the
evolution of the planetary mass ratio for different disk viscosities,
different planet configurations, and different accretion timings. Our main
conclusions can be summarized as:
1. 1.
The evolution of multiple accreting planets is mainly governed by the
viscosity of the disk. The mass ratio evolution of simultaneously accreting
planets depends on the balance between the gas accretion and gap opening
timescales. As shown in Bergez-Casalou et al. (2020), at high viscosity, when
gas accretion acts in favor of gap formation, the inner planet is rapidly
starved by the viscous accretion onto the star and the outer planet accretes
more until becoming more massive than the inner planet.
However, at lower viscosities, when gap formation is only dependent on the
disk reaction time (i.e., when gas accretion does not help gap formation), the
evolution of the mass ratio of the planets follows a different behavior: the
outer planet accretes more gas until the inner planet forms its gap. Then, the
inner planet starts depleting the inner disk and the material present in
between the planets, resulting in a higher accretion rate for the inner planet
than for the outer one. When the amount of material located in these two
regions is significantly depleted, the inner planet becomes starved by the
outer planet.
2. 2.
Simultaneously accreting planets always end up with similar masses,
independent of the disk viscosity. In order to reach more extreme mass ratios,
we simulated a delayed accretion of the inner or outer planet in one
configuration at high and intermediate viscosities. While the initial mass
ratios are large by construction, the planets quickly tend toward similar mass
ratios ($0.8<m_{out}/m_{in}<2$ in $10^{5}$ yrs).
3. 3.
Via comparisons with the observed exoplanet population, we conclude that gas
accretion occurring at the same time can explain the characteristics of only a
few planetary systems. Delayed accretion coupled with different disk lifetimes
leads to mass ratios that are more consistent with the observations. While
core formation timescales and different disk dissipation mechanisms can
explain the possibility of a delayed accretion and depletion of gas, our study
shows that the majority of the observed systems of multiple gas giant planets
should have started to accrete at different times.
Understanding how material is distributed between multiple planets is crucial
to better understanding the dynamical evolution of the forming system. As
discussed in Sect. 7.3, the radial evolution of multiple planets is governed
by the migration and capture in resonance of the planets, themselves dependent
on the gas distribution in the disk, which in turn is governed by gap
formation and by planetary and stellar gas accretion. Future studies
investigating the growth of multiple giant planets should both consider that
the gas accretion of the planets is impacted by the presence of neighboring
planets and provide mechanisms that explain the spread in mass ratios observed
in distant exoplanetary systems as well as our own Solar System.
###### Acknowledgements.
C. Bergez-Casalou and B. Bitsch thank the European Research Council (ERC
Starting Grant 757448-PAMDORA) for their financial support. S.N. Raymond
thanks the CNRS’s PNP program.
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## Appendix A Gas accretion routine
As in Crida & Bitsch (2017) and Bergez-Casalou et al. (2020), gas accretion is
modeled using the Machida et al. (2010) and Kley (1999) principles. The
Machida et al. (2010) accretion rate is derived by fitting 3D isothermal
shearing box simulations and corresponds to the runaway gas accretion phase of
the giant planet. This accretion rate is different than in the first slow
contraction phase, as predicted by the core accretion model (Piso & Youdin
2014). It can be written as
$\dot{M}_{M}=<\Sigma>_{0.9r_{H}}H^{2}\Omega_{p}\times
min\Big{[}0.14;0.83(r_{H}/H)^{9/2}\Big{]},$ (3)
where $r_{H}=r_{p}(m_{p}/3M_{*})^{1/3}$ is the Hill sphere of the planet,
$<\Sigma>_{0.9r_{H}}$ is the averaged local surface density from the planet
location up to $0.9r_{H}$; $H$ is the disk scale height and $\Omega_{p}$ is
the Keplerian orbital frequency of the planet.
On the other hand, in order to make sure that the planetary accretion is
limited by what the disk can provide, we consider that the maximal accretion
rate of the planet is given by the Kley (1999) principle. This arbitrary
accretion rate considers that the planet can accrete a given fraction of the
gas present in its Hill sphere. This fraction of gas is dependent on the
distance to the planet and is given by $f_{\rm red}$. This accretion rate can
be written as
$\dot{M}_{K}=\iint_{A_{disk}}f_{\rm red}(d)\;\Sigma(r,\phi,t)\;\pi f_{\rm
acc}\;dr\;d\phi,$ (4)
where $A_{disk}$ is the area of the disk, $f_{\rm red}$ is a smooth reduction
function predicting the fraction of gas accreted by the planet as a function
of the distance from the planet $d$, $\Sigma(r,\phi,t)$ is the gas surface
density around the planet and $f_{\rm acc}$ is the inverse timescale upon
which the accretion is occurring. $f_{\rm red}$ is the smooth reduction
function used to predict what fraction of gas can be accreted on the planet as
a function of the distance to the planet $d$. It is defined as
$f_{\rm red}=\cases{2}/3&\text{if}\;d<0.45\;r_{H}\\\
2/3\times\cos^{4}\Big{(}\pi\Big{(}\frac{d}{r_{H}}-0.45\Big{)}\Big{)}\text{if}\;0.45\;r_{H}<d<0.9\;r_{H}{}.$
(5)
This function is based on Robert et al. (2018), where the authors assume that
close to the planet gas accretion is $100\%$ efficient ($\rm f_{\rm red}=1$).
However, a $100\%$ efficiency is not realistic in such a case, as shown by
Schulik et al. (2019). The accreted mass fraction increases closer to the
planet but it does not accrete $100\%$ of the gas in the vicinity. Thus, we
chose to limit our study to a maximum $\rm f_{\rm red}$ value of 2/3.
By combining these two methods, the final amount of gas that the planet can
accrete is determined by $\dot{M}_{p}=min(\dot{M}_{M},\dot{M}_{K})$. The gas
is removed from the disk and added to the mass of the planet. To do so, we
remove the gas in this regime with the same formalism as for the Kley (1999)
method. This means that if the planet is accreting in the regime of Machida et
al. (2010), the total amount of gas it accretes is given by Eq.3 and the
distribution for where the gas is removed is given by Eq.5.
This way, the removal scheme of the gas is the same for both principles, but
the mass that can be accreted is limited either by the derived accretion rates
of Machida et al. (2010) or by the maximum amount the disk can provide, given
by the Kley (1999) method. In this study, we remain in the regime where
$\rm\dot{M}_{M}<\dot{M}_{K}$ throughout, meaning that we always remove the
amount of mass dictated by Machida et al. (2010).
## Appendix B Surface density maps
In Sect. 3.1, we investigate the influence of the disk viscosity on the
evolution of the planets growth. At low viscosity ($\alpha=10^{-4}$ and
$h=0.05$), the RWI (Lovelace et al. 1999; Li et al. 2001) is triggered at the
edges of the different planet gaps, creating vortices. In Fig. 12, we show the
2D perturbed surface density maps of the disk hosting two accreting planets in
the 3:1 period ratio, at three different times, increasing from left to right.
Polar plots of the density maps presented in the top row are shown in the
bottom row. The vortices produce asymmetric over-densities that can be used to
trace them (in yellow in Fig. 12).
At the beginning of the simulation (at 500 inner planet orbits, left panel),
we see that vortices are produced at three different locations in this
configuration: at the outer edge of the outer planet gap, in between the
planets and interior to the inner planet gap. Their presence impacts the flow
of gas in the vicinity of the planets, creating oscillations in the planetary
accretion rates (see Sect. 3.1). Quickly, the vortices located in between the
planet and in the inner disk vanish (middle panel). The strongest vortex
(i.e., with the largest over-density) is the one located at the outer edge of
the outer planet gap. It takes longer to dissipate and is completely vanished
after $10^{5}$ yrs (right panel). The strength of the vortices depends on the
growth timescale of the planets (Hammer et al. 2017): if the planets accrete
faster, the vortices would be stronger but would also vanish faster.
Figure 12: 2D perturbed surface density maps at three different times: t = 500
(left), 3 000 (middle), and 9 000 (right) inner planet orbits. The over-
densities (yellow asymmetries) are representative of vortices.
|
# Spectral Data For Parabolic Projective Symplectic/Orthogonal Higgs Bundles
Sumit Roy Center for Geometry and Physics, Institute for Basic Science (IBS),
Pohang 37673, Korea<EMAIL_ADDRESS>
###### Abstract.
Hitchin in [Duke Math. J. 54 (1), 91-114 (1987)] introduced a proper morphism
from the moduli space of stable $G$-Higgs bundles
($G=\mathrm{GL}(n,\mathbb{C}),\mathrm{Sp}(2m,\mathbb{C})$ and
$\mathrm{SO}(n,\mathbb{C})$) over a curve to a vector space of invariant
polynomials and he described the generic fibers of that morphism. In this
paper, we first describe the generic Hitchin fibers for the moduli space of
stable parabolic projective symplectic/orthogonal Higgs bundles without fixing
the determinant. We also describe the generic fibers when the determinant is
trivial.
###### Key words and phrases:
Integrable systems; Moduli space; Parabolic bundle; Higgs bundle
###### 2020 Mathematics Subject Classification:
14H70, 14D22, 14H60
E-mail<EMAIL_ADDRESS>
Affiliation: Center for Geometry and Physics, Institute for Basic Science
(IBS), Pohang 37673, Korea
## 1\. Introduction
Let $X$ be a compact (closed and connected) Riemann surface $X$ of genus
$g\geq 2$. Higgs bundles over $X$ were introduced by Hitchin in [2]. A Higgs
bundle over $X$ is a pair $(E,\phi)$ consisting of a a holomorphic vector
bundle $E$ and a Higgs field $\phi:E\to E\otimes K$, where $K$ is the
canonical bundle over $X$. The coefficients of the characteristic polynomial
of $\phi$ defines a morphism
$h:\mathcal{M}_{\mathrm{Higgs}}(r,d)\longrightarrow\mathcal{A}\coloneqq\bigoplus_{i=1}^{r}H^{0}(X,K^{i})$,
from the moduli space of stable Higgs bundles over $X$ of fixed rank $r$ and
degree $d$ to a vector space $\mathcal{A}$, called the Hitchin map (see [3]).
Hitchin in [3], showed that the generic fibers of $h$ are abelian varieties
and this map gives the Higgs bundles moduli space a structure of an
algebraically completely integrable system. Later in [6], Markman generalized
this result for the moduli space of $L$-twisted Higgs bundles $(E,\phi_{L})$,
where $L$ is a line bundle over $X$ and $\phi_{L}:E\to E\otimes L$.
Let $D\subset X$ be a fixed finite subset. The notion of parabolic bundles
over a curve and their moduli spaces were constructed by Mehta and Seshadri in
[1]. Their motivation was to extend the Narasimhan-Seshadri correspondence in
the case of irreducible unitary representations of $\pi_{1}(X-D)$. A parabolic
bundle is a holomorphic vector bundle together with a weighted flag over each
parabolic point $p\in D$. A parabolic Higgs bundle on $X$ is a parabolic
bundle $E$ on $X$ together with a parabolic Higgs field $\phi:E\to E\otimes
K(D)$. The moduli space of parabolic Higgs bundles was constructed by Yokogawa
[11].
Symplectic (resp. orthogonal) parabolic bundles are parabolic bundles with a
suitably defined nondegenerate anti-symmetric (resp. symmetric) form taking
values in a line bundle $L$ (see [13] for more details). In [5], Bhosle and
Ramanathan described the notion of parabolic principal $G$-bundles, where $G$
is a connected reductive group, and also constructed its moduli space. When
all weights are rational, the notion of symplectic (resp. orthogonal)
parabolic bundles coincides with the notion of parabolic principal $G$-bundles
where $G$ is a symplectic (resp. orthogonal) complex group (see [13]). A
symplectic (resp. orthogonal) parabolic Higgs bundle is a symplectic (resp.
orthogonal) parabolic bundle together with a parabolic Higgs field which is
compatible (in a suitable sense) with the symplectic (resp. orthogonal)
structures.
In [3], Hitchin also showed that the moduli space of stable
symplectic/orthogonal Higgs bundles also forms an algebraically completely
integrable system, fibered over a vector space, either by a Jacobian or a Prym
variety of so-called spectral curves. In [15], Roy generalized this result for
the moduli space of stable parabolic symplectic/orthogonal Higgs bundles.
In this paper, we consider the moduli space of parabolic projective
symplectic/orthogonal Higgs bundles with fixed rank and degre and fixed
parabolic structure. We know that for odd rank the projective orthogonal group
is same as the odd orthogonal group, i.e.
$\mathrm{PSO}(2m+1,\mathbb{C})=\mathrm{SO}(2m+1,\mathbb{C})$. Therefore, we
only consider the projective symplectic group $\mathrm{PSp}(2m,\mathbb{C})$
and projective even orthogonal group $\mathrm{PSO}(2m,\mathbb{C})$. A
parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs (resp.
$\mathrm{PSO}(2m,\mathbb{C})$-Higgs) bundle lifts to a parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs (resp.
$\mathrm{GSO}(2m,\mathbb{C})$-Higgs) bundle. We gave an alternative
description of the Prym varieties and we consider an action of the Jacobian
group $\mathrm{Jac}(X)$ on the Prym varieties. We showed that the generic
Hitchin fibers are isomorpic to the quotient variety (see Theorem 3.1 and
Theorem 4.1).
Finally we consider the case when the symplectic/orthogonal form takes values
in the trivial line bundle $\mathcal{O}_{X}$ and we fix this line bundle for
the respective moduli spaces. In this case, the $2$-torsion subgroup
$\mathrm{Jac}_{2}(X)\subset\mathrm{Jac}(X)$ acts on the Prym varieties and the
generic fibers of the Hitchin map are isomorphic to the quotient variety (see
Theorem 5.1 and Theorem 5.2).
## 2\. Preliminaries
### 2.1. Parabolic bundles
Let $X$ be a compact Riemann surface of genus of genus $g\geq 2$. Fix a subset
$D=\\{p_{1},\dots,p_{n}\\}\subset X$ of $n$ distinct marked points.
###### Definition 1.
A parabolic bundle $E_{*}$ of rank $r$ over $X$ is a vector bundle $E$ of rank
$r$ over $X$ with a parabolic structure over the subset $D$, i.e. for each
point $p\in D$
1. (1)
every fiber $E_{p}$ has a filtration of subspaces, i.e.
$E_{p}\eqqcolon E_{p,1}\supsetneq E_{p,2}\supsetneq\dots\supsetneq
E_{p,r_{p}}\supsetneq E_{p,r_{p}+1}=\\{0\\},$
2. (2)
an increasing sequence of real numbers (parabolic weights) satisfying
$0\leq\alpha_{1}(p)<\alpha_{2}(p)<\dots<\alpha_{r_{p}}(p)<1,$
where $1\leq r_{p}\leq r$ is an integer.
We denote the collection of all parabolic weights by
$\alpha=\\{(\alpha_{1}(p),\alpha_{2}(p),\dots,\alpha_{r_{p}}(p))\\}_{p\in D}$
corresponding to a fixed parabolic structure. The parabolic structure $\alpha$
is said to have full flags if
$\mathrm{dim}(E_{p,i}/E_{p,i+1})=1$
for all $i\in\\{1,\dots,r_{p}\\}$ and for each $p\in D$, or equivalently
$r_{p}=r$ for each $p\in D$. In this paper, we will assume that the parabolic
structure have full flag at every parabolic point $p\in D$.
The parabolic degree of $E_{*}$ is defined by
$\operatorname{pardeg}(E_{*})\coloneqq\deg(E)+\sum\limits_{p\in
D}\sum\limits_{i=1}^{r_{p}}\alpha_{i}(p)\cdot\dim(E_{p,i}/E_{p,i+1})$
and the parabolic slope of $E_{*}$ is defined by
$\mu_{\mathrm{par}}(E_{*})\coloneqq\frac{\text{pardeg}(E_{*})}{r}.$
###### Definition 2.
A parabolic homomorphism $\phi:E_{*}\to E^{\prime}_{*}$ between two parabolic
bundles is a homomorphism between underlying vector bundles such that at each
parabolic point $p\in D$ we have
$\alpha_{i}(p)>\alpha_{j}^{\prime}(p)\implies\phi(E_{p,i})\subseteq
E_{p,j+1}^{\prime}.$
Furthermore, we call such a homomorphism strongly parabolic if
$\alpha_{i}(p)\geq\alpha_{j}^{\prime}(p)\implies\phi(E_{p,i})\subseteq
E_{p,j+1}^{\prime}$
for every $p\in D$.
We denote by $\mathrm{PEnd}(E_{*})$ and $\mathrm{SPEnd}(E_{*})$ the parabolic
and strongly parabolic endomorphisms of $E_{*}$ respectively.
The dual and tensor product of parabolic bundles can be defined in a natural
way (see [12]).
###### Definition 3.
A parabolic subbundle $F_{*}$ of a parabolic bundle $E_{*}$ is a subbundle
$F\subset E$ of the underlying vector bundle endowed with an induced parabolic
structure. An induced parabolic structure on $F$ is defined as follows. For
every parabolic point $p\in D$, the quasi-parabolic structure on $F$, i.e. the
flag in $F_{p}$ is given by
$F_{p}\eqqcolon F_{p,1}\supsetneq F_{p,2}\supsetneq\dots\supsetneq
F_{p,r^{\prime}_{p}}\supsetneq\\{0\\},$
where $F_{p,i}=F_{p}\cap E_{p,i}$, i.e. we are considering the intersection
with the already given flag in $E_{p}$, and also scrapping all the repetitions
of subspaces in the filtration. The weights
$0\leq\alpha^{\prime}_{1}(p)<\alpha^{\prime}_{2}(p)<\dots<\alpha^{\prime}_{r^{\prime}_{p}}(p)<1$
are taken to be the largest possible among the given weights which are allowed
after the intersections, i.e.
$\alpha^{\prime}_{i}(p)=\mathrm{max}_{j}\\{\alpha_{j}(p)|F_{p}\cap
E_{p,j}=F_{p,i}\\}=\mathrm{max}_{j}\\{\alpha_{j}(p)|F_{p,i}\subseteq
E_{p,j}\\}$
That is to say, the weight associated to $F_{p,i}$ is the weight
$\alpha_{j}(p)$ such that $F_{p,i}\subseteq E_{p,j}$ but $F_{p,i}\nsubseteq
E_{p,j+1}$.
###### Definition 4.
A parabolic bundle $E_{*}$ is called semistable (resp. stable) if every
nonzero proper subbundle $F_{*}\subset E_{*}$ satisfies
$\mu_{\mathrm{par}}(F_{*})\leq\mu_{\mathrm{par}}(E_{*})\hskip
5.69046pt(\mathrm{resp.}\hskip 5.69046pt<).$
The moduli space $\mathcal{M}(\alpha,r,d)$ of stable parabolic bundles over
$X$ of fixed rank $r$ and degree $d$ and parabolic structure $\alpha$ was
constructed by Mehta and Seshadri in [1]. They also showed that
$\mathcal{M}(\alpha,r,d)$ is a normal projective variety of dimension
$\dim\mathcal{M}(\alpha,r,d)=r^{2}(g-1)+1+\dfrac{n(r^{2}-r)}{2},$
where $n$ is the number of marked points and the last summand comes from the
fact that the parabolic structure have full flags over each parabolic point.
### 2.2. Parabolic Higgs bundles
Let $K$ be the canonical bundle over $X$. We write $K(D)\coloneqq
K\otimes\mathcal{O}(D)$.
###### Definition 5.
A (strongly) parabolic Higgs bundle over $X$ is a parabolic bundle $E_{*}$
over $X$ together with Higgs field $\Phi:E_{*}\to E_{*}\otimes K(D)$, such
that $\Phi$ is strongly parabolic i.e. $\Phi(E_{p,i})\subset
E_{p,i+1}\otimes\left.K(D)\right|_{p}$ for all $p\in D$.
There is also a notion of parabolic Higgs bundle where the Higgs field $\Phi$
is only assumed to be parabolic, i.e. $\Phi(E_{p,i})\subset
E_{p,i}\otimes\left.K(D)\right|_{p}$ for all $p\in D$. However in this paper
we will always assume that the Higgs field is strongly parabolic.
###### Definition 6.
A subbundle $F_{*}\subset E_{*}$ is called $\Phi$-invariant if
$\Phi(F_{*})\subset F_{*}\otimes K(D)$.
###### Definition 7.
A parabolic Higgs bundle $(E_{*},\Phi)$ is called semistable (resp. stable) if
every nonzero proper $\Phi$-invariant subbundle $F_{*}\subset E_{*}$ satisfies
$\mu_{\mathrm{par}}(F_{*})\leq\mu_{\mathrm{par}}(E_{*})\hskip
5.69046pt(\mathrm{resp.}\hskip 5.69046pt<).$
The moduli space $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ of stable
parabolic Higgs bundles of fixed rank $r$, degree $d$ and parabolic structure
$\alpha$ was constructed by Yokogawa in [11] (see [7] for more details). It is
a normal quasi-projective complex variety of dimension
$\dim\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)=2\dim\mathcal{M}(\alpha,r,d).$
The parabolic version of the Serre duality (see [7], [12]) says that
$H^{1}(\mathrm{PEnd}(E_{*}))\cong H^{0}(\mathrm{SPEnd}(E_{*})\otimes
K(D))^{*}.$
Therefore, there is an open embedding
$T^{*}\mathcal{M}(\alpha,r,d)\xhookrightarrow{}\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$
and that is why the dimension of $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ is
twice the dimension of $\mathcal{M}(\alpha,r,d)$. Thus the moduli space of
parabolic Higgs bundles $\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$ has a
symplectic structure induced from the natural symplectic structure of the
cotangent space.
Let $\mathrm{Jac}^{d}(X)$ denote the space of degree $d$ line bundles over
$X$. Consider the determinant map
$\displaystyle\mathrm{det}:\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$
$\displaystyle\longrightarrow\mathrm{Jac}^{d}(X)\times H^{0}(X,K)$
$\displaystyle(E_{*},\Phi)$
$\displaystyle\longmapsto(\wedge^{r}E_{*},\mathrm{trace}(\Phi)).$
Since $\Phi$ is strongly parabolic, $\mathrm{trace}(\Phi)\in H^{0}(X,K)\subset
H^{0}(X,K(D))$. The moduli space
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ of stable parabolic Higgs
bundles with fixed determinant $\xi$ is defined by the fiber
$\mathrm{det}^{-1}(\xi,0)$, i.e.
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)\coloneqq\mathrm{det}^{-1}(\xi,0).$
If the Higgs field is zero, then the dimension of the moduli space
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ is given by
$\dim\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)=2(g-1)(r^{2}-1)+nr(r-1).$
### 2.3. Spectral correspondence
We will give a description of the spectral correspondence for the moduli space
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$ (although a similar
description can be given for the moduli space
$\mathcal{M}_{\mathrm{Higgs}}(\alpha,r,d)$).
Let $p:\mathrm{Tot}(K(D))\to X$ be the natural projection from the total space
of $K(D)$ to $X$ and let $x\in H^{0}(\mathrm{Tot}(K(D)),p^{*}K)$ denote the
tautological section. Since the Higgs field $\Phi$ is strongly parabolic, the
residue at every point of $D$ is nilpotent. Therefore the trace of the map
$\wedge^{i}\Phi:\wedge^{i}E_{*}\to\wedge^{i}E_{*}\otimes K(D)^{i}$
lies in $K^{i}(D^{i-1})$ for each $2\leq i\leq r$, where $K^{i}(D^{j})$ denote
the tensor product of the $i$-th power of $K$ and the $j$-th power of
$\mathcal{O}(D)$. The coefficients of the characteristic polynomial of $\Phi$
are precisely given by $s_{i}=\mathrm{trace}(\wedge^{i}\Phi)$. Therefore, we
have the Hitchin map
$\displaystyle
h_{\mathrm{par}}:\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$
$\displaystyle\longrightarrow\mathcal{H}_{\mathrm{par}}\coloneqq\bigoplus_{i=2}^{r}H^{0}(X,K^{i}(D^{i-1}))$
$\displaystyle(E_{*},\Phi)$ $\displaystyle\longmapsto(s_{2},\dots,s_{r}).$
By Riemann-Roch theorem, the dimension of the base
$\mathcal{H}_{\mathrm{par}}$ is
$r^{2}(g-1)+\dfrac{nr(r-1)}{2},$
which is same as the half the dimension of the moduli space
$\mathcal{M}^{\xi}_{\mathrm{Higgs}}(\alpha,r,d)$.
Given $s=(s_{2},s_{3}\dots,s_{r})\in\mathcal{H}_{\mathrm{par}}$ with
$s_{i}\in H^{0}(X,K^{i}(D^{i-1}))\subset H^{0}(X,K(D)^{i}),$
the spectral curve $X_{s}$ in $\mathrm{Tot}(K(D))$ is defined by
$x^{r}+\tilde{s}_{2}x^{r-2}+\tilde{s}_{3}x^{r-3}\cdots+\tilde{s}_{r}=0$
where $\tilde{s}_{i}=p^{*}(s_{i})$ and $x\in H^{0}(\mathrm{Tot}(K(D)),p^{*}K)$
is the tautological section. Let
$\pi:X_{s}\to X$
be the restriction of the projection $p$. For a generic point
$s\in\mathcal{H}_{\mathrm{par}}$, the spectral curve $X_{s}$ is smooth and by
[10] the fiber $h_{\mathrm{par}}^{-1}(s)$ of the Hitchin map is isomorphic to
$\mathrm{Prym}(X_{s}/X)=\\{L\in\mathrm{Pic}(X):\det\pi_{*}L\cong\xi\\}.$
### 2.4. Parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundles
Let us consider the standard symplectic form
$J=\begin{bmatrix}0&I_{m}\\\ -I_{m}&0\end{bmatrix}$
on $\mathbb{C}^{2m}$. Then the general symplectic group defined by
$\mathrm{GSp}(2m,\mathbb{C})=\\{A\in\mathrm{GL}(2m,\mathbb{C}):AJA^{t}=\lambda_{A}J\text{
for some }\lambda_{A}\in\mathbb{C}^{*}\\}$
is an extension of $\mathbb{C}^{*}$ by the symplectic group
$\mathrm{Sp}(2m,\mathbb{C})$, i.e. there exist an exact sequence
$1\to\mathrm{Sp}(2m,\mathbb{C})\xrightarrow{}\mathrm{GSp}(2m,\mathbb{C})\xrightarrow[]{p}\mathbb{C}^{*}\xrightarrow[]{}1,$
where $p(A)=\lambda_{A}$. Therefore, $\det(A)=p(A)^{m}=\lambda_{A}^{m}$ for
every $A\in\mathrm{GSp}(2m,\mathbb{C})$.
The Lie algebra of $\mathrm{GSp}(2m,\mathbb{C})$ is given by
$\mathfrak{gsp}(2m,\mathbb{C})=\\{M\in\mathfrak{gl}(2m,\mathbb{C}):MJ+JA^{t}=\frac{\mathrm{tr}(M)}{m}J\\}\cong\mathfrak{sp}(2m,\mathbb{C})\oplus\mathbb{C}.$
The decomposition $M=N+\frac{\mathrm{tr}(M)}{m}I_{2m}$, with
$N\in\mathfrak{sp}(2m,\mathbb{C})$ produces the above isomorphism.
Let $L$ be a line bundle over $X$ of degree $l$. Let $E_{*}$ be a parabolic
bundle and let
$\varphi:E_{*}\otimes E_{*}\to L$ (1)
be a homomorphism of parabolic bundles. The trivial line bundle
$\mathcal{O}_{X}$ equipped with the trivial parabolic structure is realized as
a parabolic subbundle of $E_{*}\otimes E^{\vee}_{*}$ by sending a locally
defined function $f$ to the locally defined endomorphism of $E$ given by
pointwise multiplication with $f$. Let
$\tilde{\varphi}:E_{*}\to L\otimes E^{\vee}_{*}$ (2)
be the homomorphism defined by the composition
$E_{*}=E_{*}\otimes\mathcal{O}_{X}\xhookrightarrow{}E_{*}\otimes(E_{*}\otimes
E^{\vee}_{*})=(E_{*}\otimes E_{*})\otimes
E^{\vee}_{*}\xrightarrow{\varphi\otimes Id}L\otimes E^{\vee}_{*}.$
###### Definition 8.
A symplectic parabolic bundle is a pair $(E_{*},\varphi)$ of the above form
such that $\varphi$ is anti-symmetric and the homomorphism $\tilde{\varphi}$
is an isomorphism.
Suppose $E$ is the underlying vector bundle of a symplectic parabolic bundle
$(E_{*},\varphi)$. The tensor product $E\otimes E$ is a coherent subsheaf of
the vector bundle underlying the parabolic bundle $E_{*}\otimes E_{*}$.
Therefore, $\varphi$ induces a homomorphism
$\hat{\varphi}:E\otimes E\to L$ (3)
of vector bundles.
###### Definition 9.
A subbundle $F\subset E$ of the underlying bundle of $(E_{*},\varphi)$ is
called isotropic if $\hat{\varphi}(F\otimes F)=0$.
A parabolic Higgs field $\Phi$ on a symplectic parabolic bundle
$(E_{*},\varphi)$ is said to be compatible with $\varphi$ if $\tilde{\varphi}$
takes $\Phi$ to the induced parabolic Higgs field on $L\otimes E^{\vee}_{*}$
(we are considering the zero section as the Higgs field on $L$). We can
describe this compatibility condition locally. A strongly parabolic Higgs
field $\Phi$ on $E_{*}$ can be viewed as a holomorphic section of
$\mathrm{SPEnd}(E_{*})\otimes K(D)$. Let $s$ and $t$ be any holomorphic
sections of $E_{*}$ defined over an open subset $U\subset X$. Consider
$\hat{\varphi}_{\Phi}(s,t)\coloneqq\hat{\varphi}(\Phi(s)\otimes
t)+\hat{\varphi}(s\otimes\Phi(t))\in\Gamma(U,L\otimes K(D)),$
where $\hat{\varphi}$ is the pairing defined in 3. The Higgs field $\Phi$ is
said to be compatible with $\phi$ if and only if $\hat{\varphi}_{\Phi}(s,t)=0$
for all sections $s$ and $t$.
###### Definition 10.
A symplectic parabolic Higgs bundle $(E_{*},\varphi,\Phi)$ is a symplectic
parabolic bundle $(E_{*},\varphi)$ equipped with a parabolic Higgs field
$\Phi$ on $E_{*}$ which is compatible with $\varphi$.
When the parabolic weights are all rational, the notion of symplectic
parabolic Higgs bundle is equivalent to the notion of parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundles (see [13]).
### 2.5. Parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundles
Let $B$ be a non-degenerate symmetric bilinear form on $\mathbb{C}^{2m}$. Also
for notational convenience, we denote the corresponding symmetric matrix by
$B$. Then the general (even) orthogonal group
$\mathrm{GO}(2m,\mathbb{C})=\\{A\in\mathrm{GL}(2m,\mathbb{C}):ABA^{t}=\lambda_{A}J\text{
for some }\lambda_{A}\in\mathbb{C}^{*}\\}.$
therefore,, we have $(\det(A))^{2}=\lambda_{A}^{2m}$ for every
$A\in\mathrm{GO}(2m,\mathbb{C})$.
In this case, there is a sgn morphism
$\textit{sgn}:\mathrm{GO}(2m,\mathbb{C})\longrightarrow\\{\pm 1\\}$
sending $A$ to $\det(A)/\lambda_{A}^{m}$. The general special orthogonal group
is the kernel of this sgn morphism and it is denoted by
$\mathrm{GSO}(2m,\mathbb{C})=\mathrm{ker}(\textit{sgn})$. So we have a short
exact sequence
$1\to\mathrm{GSO}(2m,\mathbb{C})\xrightarrow{}\mathrm{GO}(2m,\mathbb{C})\xrightarrow[]{\textit{sgn}}\\{\pm
1\\}\xrightarrow[]{}1.$
###### Definition 11.
An orthogonal parabolic bundle is a pair $(E_{*},\varphi)$, where $\varphi$
(as in 1) is symmetric and the homomorphism $\tilde{\varphi}$ (as in 2) is an
isomorphism.
###### Definition 12.
An orthogonal parabolic Higgs bundle $(E_{*},\varphi,\Phi)$ is an orthogonal
parabolic bundle $(E_{*},\varphi)$ equipped with a parabolic Higgs field
$\Phi$ on $E_{*}$ which is compatible with $\varphi$.
As in the $\mathrm{GSp}$-case, for rational weights the notions of (even)
orthogonal parabolic Higgs bundles and parabolic
$\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundles coincide.
### 2.6. Moduli space
###### Definition 13.
A symplectic/orthogonal parabolic Higgs bundle $(E_{*},\varphi,\Phi)$ is said
to be semistable (resp. stable) if every nonzero isotropic subbundle $F\subset
E$ such that $\Phi(F_{*})\subset F_{*}\otimes K(D)$ satisfies
$\mu_{par}(F_{*})\leq\mu_{par}(E_{*})\hskip 11.38092pt(\text{resp.}\hskip
4.26773pt<)$
holds, where $F_{*}\subset E_{*}$ has the induced parabolic structure.
The moduli space $\mathcal{M}_{G}(\alpha)$ of stable parabolic $G$-bundles of
a fixed topological type and with a fixed parabolic structure $\alpha$ is a
normal quasi-projective variety (see [8], [5]) of dimension
$\dim\mathcal{M}_{G}(\alpha)=\dim Z(G)+(g-1)\dim(G)+n\dim(G/B),$
where $Z(G)$ denotes the the center of $G$ and $n$ is the number of parabolic
points. The last summand comes from the fact that the flags we are considering
over each point of $D$ are full flags and $B$ is the Borel subgroup of $G$
determined by $\alpha$. The moduli space
$\mathcal{M}_{G\mathrm{-Higgs}}(\alpha)$ of stable parabolic $G$-Higgs bundles
(see [14]) is a normal quasi-projective variety of dimension
$\dim\mathcal{M}_{G\mathrm{-Higgs}}(\alpha)=2\dim\mathcal{M}_{G}(\alpha).$
In particular when $G=\mathrm{GSp}(2m,\mathbb{C})$, the moduli space
$\mathcal{M}_{\mathrm{GSp-Higgs}}(\alpha,2m,d)$ of stable parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle of fixed degree $d$ has dimension
$\dim\mathcal{M}_{\mathrm{GSp-Higgs}}(\alpha,2m,d)=2m(2m+1)(g-1)+2m^{2}n.$
Similarly, the moduli space $\mathcal{M}_{\mathrm{GSO-Higgs}}(\alpha,2m,d)$ of
stable parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle of fixed degree
$d$ has dimension
$\dim\mathcal{M}_{\mathrm{GSO-Higgs}}(\alpha,2m,d)=2m(2m-1)(g-1)+2mn(m-1).$
### Notation:
From now on, for notational convenience we shall denote a parabolic bundle
$E_{*}$ by $E$.
## 3\. Parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles
In this section we will discuss the Hitchin fibration for the moduli space of
parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles. The projective
symplectic group $\mathrm{PSp}(2m,\mathbb{C})$ is given by the following exact
sequence:
$1\xrightarrow{}\mathbb{C}^{*}\xrightarrow{c\to
cI_{2m}}\mathrm{GSp}(2m,\mathbb{C})\xrightarrow{}\mathrm{PSp}(2m,\mathbb{C})\xrightarrow{}1.$
The sheaf version of this sequence induces the following exact sequence in
homology :
$H^{1}(X,\mathcal{O}_{X}^{*})\xrightarrow{}H^{1}(X,\mathrm{GSp}(2m,\mathcal{O}_{X}))\xrightarrow{q}H^{1}(X,\mathrm{PSp}(2m,\mathcal{O}_{X}))\to
0$ (4)
The surjectivity of the map $q$ implies that there is a bijective
correspondence between the parabolic $\mathrm{PSp}(2m,\mathbb{C})$-bundles and
the equivalence classes of parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundles
with respect to the action given by the tensor product of line bundles on the
associated bundles. If $V$ is a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-bundle
and $\tilde{V}$ is a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundle such that
$q(\tilde{V})=V$, then we call that $\tilde{V}$ is a lifting of $V$ to a
parabolic $\mathrm{GSp}(2m,\mathbb{C})$-bundle.
The group $\mathrm{PSp}(2m,\mathbb{C})$ can also be defined by the quotient of
the symplectic group $\mathrm{Sp}(2m,\mathbb{C})$ by the action of a finite
group by the following exact sequence :
$1\xrightarrow{}\\{\pm 1\\}\xrightarrow{1\to
I_{2m}}\mathrm{Sp}(2m,\mathbb{C})\xrightarrow{}\mathrm{PSp}(2m,\mathbb{C})\xrightarrow{}1.$
Since it is a quotient by a finite group, the Lie algebras are equal, i.e.
$\mathfrak{sp}(2m,\mathbb{C})=\mathfrak{psp}(2m,\mathbb{C}).$ (5)
Consider a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle $(V,\eta)$
which lifts to a parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle
$(\tilde{V},\tilde{\eta})$, i.e. $q(\tilde{V},\tilde{\eta})=(V,\eta)$. Let
$(E,\Phi,\varphi,L)$ be the bundle corresponding to
$(\tilde{V},\tilde{\eta})$. Then $(V,\eta)$ corresponds to the equivalence
class $[(E,\Phi,\varphi,L)]$ where the equivalence relation
$\sim_{\mathrm{Jac}(X)}$ is given by :
$(E,\Phi,\varphi,L)\sim_{\mathrm{Jac}(X)}(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},L\otimes M^{2})\hskip
28.45274pt\mathrm{for}\hskip 2.84544pt\mathrm{any}\hskip
5.69046ptM\in{\mathrm{Jac}(X)}$
where
$\varphi_{M}:(E\otimes M)\otimes(E\otimes M)\to L\otimes M^{2}$
is the induced symplectic form on $E\otimes M$ taking values in $L\otimes
M^{2}$.
Let $M_{\mathrm{GSp-
Higgs}}(\alpha,2m)=\coprod_{d\in\mathbb{Z}}\mathcal{M}_{\mathrm{GSp-
Higgs}}(\alpha,2m,d)$ denote the moduli stack of parabolic
$\mathrm{GSp}$-Higgs bundles of fixed rank $2m$ with any degree. Then the
$\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ of $(E,\Phi,\varphi,L)$ under
the action of $\mathrm{Jac}(X)$ on the stack $M_{\mathrm{GSp-
Higgs}}(\alpha,2m)$ is defined by:
$\displaystyle M_{\mathrm{GSp-Higgs}}(\alpha,2m)\times\mathrm{Jac}(X)$
$\displaystyle\longrightarrow M_{\mathrm{GSp-Higgs}}(\alpha,2m)$
$\displaystyle((E,\Phi,\varphi,L),M)$ $\displaystyle\mapsto(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},L\otimes M^{2}).$
Note that $\deg(L\otimes M^{2})=\deg(L)+2\deg(M)$, i.e. it changes the degree
of $L$ by a multiple of $2$. For a parabolic $\mathrm{PSp}(2m,\mathbb{C})$\-
Higgs bundle $[(E,\Phi,\varphi,L)]$, the isomorphism $E\cong E^{\vee}\otimes
L$ implies that $\mathrm{pardeg}(E)=m\deg(L)$. So the parabolic degree of $E$
is determined by the degree of $L$. Therefore,
$\displaystyle\mathrm{pardeg}(E\otimes M)$ $\displaystyle=m\deg(L\otimes
M^{2})$ $\displaystyle=m\deg(L)+2m\deg(M)$
$\displaystyle=\mathrm{pardeg}(E)+2m\deg(M).$
Therefore, the parabolic degree of a parabolic
$\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle can be defined as follows :
###### Definition 14.
Let $(V,\eta)$ be a parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle and
let $(\tilde{V},\tilde{\eta})$ be a lifting to a parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle. Let $(E,\Phi,\varphi,L)$ be the
datum corresponding to $(\tilde{V},\tilde{\eta})$ with parabolic degree $ml$,
where $l=\deg(L)$. Then the parabolic degree of $(V,\eta)$ is given by the
class $\overline{ml}\in\mathbb{Z}/m\mathbb{Z}$.
Threfore, we will consider the moduli space $\mathcal{M}_{\mathrm{PSp-
Higgs}}(\alpha,2m,\overline{ml})$ of stable parabolic
$\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles of fixed rank $2m$ and parabolic
degree $\overline{ml}$.
Consider a basis $\\{s_{2i}\\}_{i=1,\dots,m}$ of invariant polynomials of the
lie algebra $\mathfrak{sp}(2m,\mathbb{C})=\mathfrak{psp}(2m,\mathbb{C})$,
where $s_{2i}=\mathrm{tr}(\wedge^{2i}\eta)$. Therefore the Hitchin morphism
for the moduli space of parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles
is of the form
$\displaystyle h_{\mathrm{PSp-par}}:\mathcal{M}_{\mathrm{PSp-
Higgs}}(\alpha,2m,\overline{ml})$
$\displaystyle\longrightarrow\mathcal{H}_{\mathrm{PSp-
par}}\coloneqq\bigoplus_{i=1}^{m}H^{0}(X,K^{2i}(D^{2i-1}))$
$\displaystyle(V,\eta)$ $\displaystyle\longmapsto(s_{2},\dots,s_{2m}).$
Observe that if $(\tilde{V},\tilde{\eta})$ is a lifting of $(V,\eta)$ to a
parabolic $\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle then
$h_{\mathrm{GSp-par}}(\tilde{V},\tilde{\eta})=h_{\mathrm{PSp-par}}(V,\eta).$
Let $s=(s_{2},s_{4},\dots,s_{2m})\in\mathcal{H}_{\mathrm{PSp-par}}$ be a
generic point of the Hitchin base. The spectral curve
$\pi:X_{s}\to X$
is defined by the equation
$x^{2m}+s_{2}x^{2m-2}+s_{4}x^{2m-4}+\cdots+s_{2m}=0.$
For a generic $s\in\mathcal{H}_{\mathrm{PSp-par}}$, the corresponding spectral
curve $X_{s}$ is smooth (see [9]). Since all odd coefficients of the above
equation are zero, the spectral curve $X_{s}$ possesses an involution
$\sigma:X_{s}\to X_{s}$ defined by $\sigma(\lambda)=-\lambda$. Therefore, we
can define a $2$-fold covering map
$q:X_{s}\to X_{s}/\sigma.$
Since $\sigma$ sends a degree zero line bundle on $X_{s}$ to a degree zero
line bundle, it acts on the Jacobian $\text{Jac}(X_{s})$. The Prym variety
$P_{s,\sigma}\coloneqq\mathrm{Prym}(X_{s},X_{s}/\sigma)$ is given by
$P_{s,\sigma}\coloneqq\mathrm{Prym}(X_{s},X_{s}/\sigma)=\\{N\in\text{Jac}(X_{s}):\sigma^{*}N\cong
N^{\vee}\\}$
and it is of dimension
$\dim P_{s,\sigma}=g(X_{s})-g(X_{s}/\sigma).$
Following [15, Theorem 4.1], we can give a different description of the Prym
variety. Let $J\in P_{s,\sigma}$ be an element in the Prym variety. Consider
the line bundle
$U=J\otimes R^{\vee},$
where $R=(K_{X_{s}}\otimes\pi^{*}K^{\vee}\otimes\pi^{*}L^{\vee})^{1/2}$ is a
holomorphic square root. Then it follows that $U$ satisfies the isomorphism
$\sigma^{*}U\cong
U^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L.$ (6)
Similarly, let $U\in\text{Jac}(X_{s})$ be a line bundle satisfying the
equation (6). Then the line bundle $J=U\otimes R\in P_{s,\sigma}$ is an
element in the Prym variety. Therefore, there is a bijective correspondence
between the Prym variety $P_{s,\sigma}$ and
$\Omega_{s,\sigma}\coloneqq\\{(U,L,\tau)\hskip 2.84544pt|\hskip
2.84544ptU\in\text{Jac}(X_{s}),\tau:\sigma^{*}U\cong
U^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L\\}.$ (7)
From now on, we will refer an element in the Prym variety as an element of
$\Omega_{s,\sigma}$.
The Jacobian $\mathrm{Jac}(X)$ acts on $\Omega_{s,\sigma}$ as follows:
$\displaystyle\Omega_{s,\sigma}\times\mathrm{Jac}(X)$
$\displaystyle\longrightarrow\Omega_{s,\sigma}$ $\displaystyle((U,L,\tau),M)$
$\displaystyle\longmapsto(U\otimes\pi^{*}M,L\otimes M^{2},\tau_{M})$
where $\tau_{M}=\tau\otimes\text{Id}_{\pi^{*}M}$ is the following isomorphism
$\displaystyle\sigma^{*}(U\otimes\pi^{*}M)$
$\displaystyle\cong\sigma^{*}U\otimes\pi^{*}M$ $\displaystyle\cong
U^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L\otimes\pi^{*}M$
$\displaystyle\cong
U^{\vee}\otimes\pi^{*}M^{\vee}\otimes\pi^{*}M\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L\otimes\pi^{*}M$
$\displaystyle\cong(U\otimes\pi^{*}M)^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}(L\otimes
M^{2}).$
###### Theorem 3.1.
For a generic point $s\in\mathcal{H}_{\mathrm{PSp-par}}$, the fiber
$h_{\mathrm{PSp-par}}^{-1}(s)$ is isomorphic to the quotient
$\Omega_{s,\sigma}/\mathrm{Jac}(X)$.
###### Proof.
Let $(V,\eta)\in h_{\mathrm{PSp-par}}^{-1}(s)$ be a parabolic
$\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle in the fiber of the Hitchin
morphism and let $(\tilde{V},\tilde{\eta})$ be a lifting to a parabolic
$\mathrm{GSp}(2m,\mathbb{C})$-Higgs bundle which corresponds to the datum of
$(E,\Phi,\varphi,L)$. Then $(V,\eta)$ corresponds to the datum of the
$\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ uniquely. By [15, Theorem 4.1],
the datum of $(E,\Phi,\varphi,L)$ corresponds to an element of the Pyrm
variety $\Omega_{s,\sigma}$ via the spectral correspondence. Consider an
element $M\in\mathrm{Jac}(X)$, i.e. a degree zero line bundle over $X$. Then
by the projection formula the datum of $(E\otimes
M,\Phi\otimes\text{Id}_{M},\varphi_{M},L\otimes M^{2})$ corresponds uniquely
to an element of $\Omega_{s,\sigma}/\mathrm{Jac}(X)$. Therefore, we conclude
that the datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$
corresponds uniquely to the datum of the $\mathrm{Jac}(X)$-orbit of an element
of $\Omega_{s,\sigma}$ via the spectral correspondence. ∎
## 4\. Parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles
As in the previous section, there is a bijective correspondence between the
parabolic $\mathrm{PSO}(2m,\mathbb{C})$-bundles and the equivalence classes of
parabolic $\mathrm{GSO}(2m,\mathbb{C})$-bundles. So, for every parabolic
$\mathrm{PSO}(2m,\mathbb{C})$-bundle $V$ there is a lifting $\tilde{V}$ of $V$
to a parabolic $\mathrm{GSO}(2m,\mathbb{C})$-bundle.
The group $\mathrm{PSO}(2m,\mathbb{C})$ can be defined by the quotient of
$\mathrm{SO}(2m,\mathbb{C})$ by the action of a finite group by the following
exact sequence :
$1\xrightarrow{}\\{\pm 1\\}\xrightarrow{1\to
I_{2m}}\mathrm{SO}(2m,\mathbb{C})\xrightarrow{}\mathrm{PSO}(2m,\mathbb{C})\xrightarrow{}1.$
Therefore, we have
$\mathfrak{so}(2m,\mathbb{C})=\mathfrak{pso}(2m,\mathbb{C}).$ (8)
Let $(V,\eta)$ be a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle
lifting to the parabolic $\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle
$(\tilde{V},\tilde{\eta})$ and let $(E,\Phi,\varphi,L)$ be the bundle
corresponding to $(\tilde{V},\tilde{\eta})$. Then $(V,\eta)$ corresponds to
the equivalence class $[(E,\Phi,\varphi,L)]$ where the equivalence relation
$\sim_{\mathrm{Jac}(X)}$ is given by :
$(E,\Phi,\varphi,L)\sim_{\mathrm{Jac}(X)}(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},L\otimes M^{2})\hskip
28.45274pt\mathrm{for}\hskip 2.84544pt\mathrm{any}\hskip
5.69046ptM\in{\mathrm{Jac}(X)}$
where
$\varphi_{M}:(E\otimes M)\otimes(E\otimes M)\to L\otimes M^{2}$
is the induced symmetric bilinear nondegenerate form on $E\otimes M$ taking
values in $L\otimes M^{2}$.
As in the previous case, the parabolic degree of a parabolic
$\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle can be defined as follows:
###### Definition 15.
Let $(V,\eta)$ be a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundle and
let $(\tilde{V},\tilde{\eta})$ be a lifting to a parabolic
$\mathrm{GSO}(2m,\mathbb{C})$-Higgs bundle. Let $(E,\Phi,\varphi,L)$ be the
datum corresponding to $(\tilde{V},\tilde{\eta})$ with parabolic degree $ml$,
where $l=\deg(L)$. Then the parabolic degree of $(V,\eta)$ is given by the
class $\overline{ml}\in\mathbb{Z}/m\mathbb{Z}$.
Let $\\{s_{2i}\\}_{i=1,\dots,m}$ be a basis of invariant polynomials of the
lie algebra $\mathfrak{so}(2m,\mathbb{C})=\mathfrak{pso}(2m,\mathbb{C})$ In
this case, the coefficient $s_{2m}$ is a square of a polynomial $p_{m}\in
H^{0}(X,K(D)^{m})$, the Pfaffian, of degree $m$. A basis for the invariant
polynomials on the Lie algebra
$\mathfrak{so}(2m,\mathbb{C})=\mathfrak{pso}(2m,\mathbb{C})$ is given by the
coefficients $\\{s_{2},...,s_{2m-2},p_{m}\\}$. Therefore, the Hitchin map for
the moduli space of parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles is
given by
$\displaystyle h_{\mathrm{PSO-par}}:\mathcal{M}_{\mathrm{PSO-
Higgs}}(\alpha,2m,\overline{ml})$
$\displaystyle\longrightarrow\mathcal{H}_{\mathrm{PSO-
par}}\coloneqq\bigoplus_{i=1}^{m-1}H^{0}(X,K^{2i}(D^{2i-1}))\oplus
H^{0}(X,K(D)^{m})$ $\displaystyle(V,\eta)$
$\displaystyle\longmapsto(s_{2},\dots,s_{2m-2},p_{m}).$
For $s=(s_{2},\dots,s_{2m-2},p_{m})\in\mathcal{H}_{\mathrm{PSO-par}}$, the
corresponding spectral curve $X_{s}$ is given by the equation
$x^{2m}+s_{2}x^{2m-2}+\cdots+s_{2m-2}x^{2}+p_{m}^{2}=0.$
The zeroes of $p_{m}$ are singularities of $X_{s}$ and these are the only
singularities. Since $p_{m}\in H^{0}(X,K(D)^{m})$, there are
$K(D)^{m}=m(2g-2+n)$ many singularities. As in the previous case, $X_{s}$
possesses an involution $\sigma(\eta)=-\eta$. Then the fixed points of this
involution $\sigma$ are exactly the singularities of $X_{s}$. Let
$\hat{X_{s}}$ denote the desingularisation of $X_{s}$ with genus
$g(\hat{X_{s}})=g(X_{s})-\mathrm{number\hskip 2.84544ptof\hskip
2.84544ptsingularities}.$
Since the singularities of $X_{s}$ are double points, the involution $\sigma$
on $X_{s}$ extends to an involution $\hat{\sigma}$ on $\hat{X_{s}}$. Therefore
the Prym variety
$P_{s,\hat{\sigma}}\coloneqq\mathrm{Prym}(\hat{X_{s}},\hat{X_{s}}/\hat{\sigma})$
is given by
$P_{s,\hat{\sigma}}\coloneqq\mathrm{Prym}(\hat{X_{s}},\hat{X_{s}}/\hat{\sigma})=\\{N\in\text{Jac}(\hat{X_{s}}):\hat{\sigma}^{*}N\cong
N^{\vee}\\}$ (9)
As in the symplectic case, there is a bijective correspondence between the
Prym variety $P_{s,\hat{\sigma}}$ and
$\Omega_{s,\hat{\sigma}}\coloneqq\\{(U,L,\tau)\hskip 2.84544pt|\hskip
2.84544ptU\in\text{Jac}(\hat{X_{s}}),\tau:\hat{\sigma}^{*}U\cong
U^{\vee}\otimes(K_{\hat{X_{s}}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}L\\}.$
Again the action of the group $\mathrm{Jac}(X)$ on $\Omega_{s,\hat{\sigma}}$
is given by
$\displaystyle\Omega_{s,\hat{\sigma}}\times\mathrm{Jac}(X)$
$\displaystyle\longrightarrow\Omega_{s,\hat{\sigma}}$
$\displaystyle((U,L,\tau),M)$
$\displaystyle\longmapsto(U\otimes\pi^{*}M,L\otimes M^{2},\tau_{M})$
where $\tau_{M}=\tau\otimes\text{Id}_{\pi^{*}M}$ is the isomorphism
$\hat{\sigma}^{*}(U\otimes\pi^{*}M)\cong(U\otimes\pi^{*}M)^{\vee}\otimes(K_{\hat{X_{s}}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}(L\otimes
M^{2}).$
###### Theorem 4.1.
For a generic point $s\in\mathcal{H}_{\mathrm{PSO-par}}$, the fiber
$h_{\mathrm{PSO-par}}^{-1}(s)$ is isomorphic to
$\Omega_{s,\hat{\sigma}}/\mathrm{Jac}(X)$.
###### Proof.
Let $(V,\eta)\in h_{\mathrm{PSp-par}}^{-1}(s)$ lifts to a parabolic
$\mathrm{GSO(2m,\mathbb{C})}$-Higgs bundle $(\tilde{V},\tilde{\eta})$ whose
corresponding datum is $(E,\Phi,\varphi,L)$. Then $(V,\eta)$ corresponds to
the datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ uniquely. By
[15, Theorem 4.2], the datum of $(E,\Phi,\varphi,L)$ corresponds to an element
of the Pyrm variety $\Omega_{s,\hat{\sigma}}$. Let $M\in\mathrm{Jac}(X)$. Then
the datum of $(E\otimes M,\Phi\otimes\text{Id}_{M},\varphi_{M},L\otimes
M^{2})$ corresponds uniquely to an element of
$\Omega_{s,\hat{\sigma}}/\mathrm{Jac}(X)$. Therefore, we conclude that the
datum of the $\mathrm{Jac}(X)$-orbit $[(E,\Phi,\varphi,L)]$ corresponds
uniquely to the datum of the $\mathrm{Jac}(X)$-orbit of an element of
$\Omega_{s,\hat{\sigma}}$.
∎
## 5\. Fixed line bundle : $\mathcal{O}_{X}$
In this section, we will assume that the symplectic/orthogonal form in 1 takes
values in the trivial line bundle $\mathcal{O}_{X}$. In particular, we
consider the moduli space of parabolic symplectic/orthogonal Higgs bundles
with fixed rank, degree and fixed line bundle $\mathcal{O}_{X}$. In other
words, we are considering the moduli spaces with trivial determinant.
### 5.1. Parabolic $\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundles with fixed
line bundle $\mathcal{O}_{X}$
Let $\mathrm{Jac}_{2}(X)$ be the subgorup of the Jacobian $\mathrm{Jac}(X)$
which contains the $2$-torsion elements of $\mathrm{Jac}(X)$, i.e.
$\mathrm{Jac}_{2}(X)\coloneqq\\{M\in\mathrm{Jac}(X)\hskip 2.84544pt|\hskip
2.84544ptM^{2}\cong\mathcal{O}_{X}\\}.$
Let $(E,\Phi,\varphi,\mathcal{O}_{X})$ be a parabolic symplectic Higgs bundle
with the symplectic form $\varphi:E\otimes E\to\mathcal{O}_{X}$ taking values
in $\mathcal{O}_{X}$ and let $M\in\mathrm{Jac}_{2}(X)$. Then
$\varphi_{M}:(E\otimes M)\otimes(E\otimes
M)\longrightarrow\mathcal{O}_{X}\otimes
M^{2}\cong\mathcal{O}_{X}\otimes\mathcal{O}_{X}\cong\mathcal{O}_{X}$
defines a symplectic form on $E\otimes M$ with values in $\mathcal{O}_{X}$.
Since the symplectic form takes values in $\mathcal{O}_{X}$, a parabolic
$\mathrm{PSp}(2m,\mathbb{C})$-Higgs bundle $(V,\eta)$ lifts to a parabolic
$\mathrm{Sp}(2m,\mathbb{C})$-Higgs bundle $(\tilde{V},\tilde{\eta})$. Let
$(E,\Phi,\varphi,\mathcal{O}_{X})$ be the parabolic symplectic Higgs bundle
corresponding to $(\tilde{V},\tilde{\eta})$. Then $(V,\eta)$ corresponds to
the equivalence class $[(E,\Phi,\varphi,\mathcal{O}_{X})]$ where the
equivalence relation $\sim_{\mathrm{Jac}_{2}(X)}$ is given by :
$(E,\Phi,\varphi,\mathcal{O}_{X})\sim_{\mathrm{Jac}_{2}(X)}(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},\mathcal{O}_{X})\hskip
28.45274pt\mathrm{for}\hskip 2.84544pt\mathrm{any}\hskip
5.69046ptM\in{\mathrm{Jac}_{2}(X)}.$
As in 7, the Prym variety is given by
$\Omega_{s,\sigma}\coloneqq\\{(U,\mathcal{O}_{X},\tau)\hskip 2.84544pt|\hskip
2.84544ptU\in\text{Jac}(X_{s}),\tau:\sigma^{*}U\cong
U^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}\mathcal{O}_{X}\\}.$
Also, the group $\mathrm{Jac}_{2}(X)$ acts on $\Omega_{s,\sigma}$ by
$\displaystyle\Omega_{s,\sigma}\times\mathrm{Jac}_{2}(X)$
$\displaystyle\longrightarrow\Omega_{s,\sigma}$
$\displaystyle((U,\mathcal{O}_{X},\tau),M)$
$\displaystyle\longmapsto(U\otimes\pi^{*}M,\mathcal{O}_{X}\otimes
M^{2},\tau_{M})\cong(U\otimes\pi^{*}M,\mathcal{O}_{X},\tau_{M})$
where $\tau_{M}=\tau\otimes\text{Id}_{\pi^{*}M}$ is the isomorphism
$\sigma^{*}(U\otimes\pi^{*}M)\cong(U\otimes\pi^{*}M)^{\vee}\otimes(K_{X_{s}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}\mathcal{O}_{X}.$
###### Theorem 5.1.
For a generic point $s\in\mathcal{H}_{\mathrm{PSp-par},\mathcal{O}_{X}}$ of
the parabolic $\mathrm{PSp}$-Hitchin map with the fixed line bundle
$\mathcal{O}_{X}$, the fiber $h_{\mathrm{PSp-par},\mathcal{O}_{X}}^{-1}(s)$ is
isomorphic to the quotient $\Omega_{s,\sigma}/\mathrm{Jac}_{2}(X)$.
###### Proof.
The proof is similar to the proof of the Theorem 3.1. ∎
### 5.2. Parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs bundles with fixed
line bundle $\mathcal{O}_{X}$
As in the symplectic case, a parabolic $\mathrm{PSO}(2m,\mathbb{C})$-Higgs
bundle $(V,\eta)$ with the orthogonal form taking values in $\mathcal{O}_{X}$
lifts to a parabolic $\mathrm{SO}(2m,\mathbb{C})$-Higgs bundle
$(\tilde{V},\tilde{\eta})$. Let $(E,\Phi,\varphi,\mathcal{O}_{X})$ be the
parabolic (even) orthogonal Higgs bundle corresponding to
$(\tilde{V},\tilde{\eta})$. Then $(V,\eta)$ corresponds to the equivalence
class $[(E,\Phi,\varphi,\mathcal{O}_{X})]$ where the equivalence relation
$\sim_{\mathrm{Jac}_{2}(X)}$ is given by:
$(E,\Phi,\varphi,\mathcal{O}_{X})\sim_{\mathrm{Jac}_{2}(X)}(E\otimes
M,\Phi\otimes\mathrm{Id}_{M},\varphi_{M},\mathcal{O}_{X})\hskip
28.45274pt\mathrm{for}\hskip 2.84544pt\mathrm{any}\hskip
5.69046ptM\in{\mathrm{Jac}_{2}(X)}.$
Following the above section 4, the alternative description of the Prym variety
(9) is given by
$\Omega_{s,\hat{\sigma}}\coloneqq\\{(U,\mathcal{O}_{X},\tau)\hskip
2.84544pt|\hskip
2.84544ptU\in\text{Jac}(\hat{X_{s}}),\tau:\hat{\sigma}^{*}U\cong
U^{\vee}\otimes(K_{\hat{X_{s}}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}\mathcal{O}_{X}\\}.$
Also, the action of the subgroup $\mathrm{Jac}_{2}(X)$ on
$\Omega_{s,\hat{\sigma}}$ is given by
$\displaystyle\Omega_{s,\hat{\sigma}}\times\mathrm{Jac}_{2}(X)$
$\displaystyle\longrightarrow\Omega_{s,\hat{\sigma}}$
$\displaystyle((U,\mathcal{O}_{X},\tau),M)$
$\displaystyle\longmapsto(U\otimes\pi^{*}M,\mathcal{O}_{X},\tau_{M})$
where $\tau_{M}=\tau\otimes\text{Id}_{\pi^{*}M}$ is the isomorphism
$\hat{\sigma}^{*}(U\otimes\pi^{*}M)\cong(U\otimes\pi^{*}M)^{\vee}\otimes(K_{\hat{X_{s}}}\otimes\pi^{*}K^{\vee})^{-1}\otimes\pi^{*}\mathcal{O}_{X}.$
###### Theorem 5.2.
For a generic point $s\in\mathcal{H}_{\mathrm{PSO-par},\mathcal{O}_{X}}$ of
the parabolic $\mathrm{PSO}$-Hitchin map with the fixed line bundle
$\mathcal{O}_{X}$, the fiber $h_{\mathrm{PSO-par},\mathcal{O}_{X}}^{-1}(s)$ is
isomorphic to the quotient $\Omega_{s,\hat{\sigma}}/\mathrm{Jac}_{2}(X)$.
###### Proof.
The proof is similar to the proof of the Theorem 4.1. ∎
###### Remark 5.3.
We can actually consider any degree zero line bundle in place of
$\mathcal{O}_{X}$.
## Acknowledgement
This work was supported by the Institute for Basic Science (IBS-R003-D1).
## Data Availability
Data sharing is not applicable to this article as no new data were created or
analyzed in this study.
## References
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* [2] N.J. Hitchin, _The self-duality equations on a Riemann surface_ , Proc. LMS 55, 3 (1987), 59-126.
* [3] N.J. Hitchin, _Stable bundles and integrable systems_ , Duke Math. J., Volume 54, Number 1 (1987), 91-114.
* [4] N.J. Hitchin, _Langlands duality and G2 spectral curves_ , Q.J. Math., 58 (2007), 319-344
* [5] U. Bhosle, A. Ramanathan, _Moduli of parabolic $G$-bundles on curves_, Math. Z. 202, no. 2 (1989), 161–180.
* [6] E. Markman, _Spectral curves and integrable systems_ , Compositio Mathematica, tome 93, no 3 (1994), 255-290.
* [7] H. Boden and K. Yokogawa, _Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves_. I, Internat. J. Math. 7 (1996), no. 5, 573–598.
* [8] V. Balaji, I. Biswas and D.S. Nagaraj, _Principal bundles over projective manifolds with parabolic structure over a divisor_ , Tohoku Math. Jour. 53 (2001), 337–367.
* [9] A. Beauville, M.S. Narasimhan, S. Ramanan, _Spectral curves and the generalised theta divisor_. J. Reine Angew. Math. 398,169–179 (1989)
* [10] T. Gómez and M. Logares, _A Torelli theorem for the moduli space of parabolic Higgs bundles_ , Adv. Geom. 11 (2011) 429–444
* [11] K. Yokogawa, _Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves_ , J. Math. Kyoto Univ. 33 (1993), no. 2, 451–504.
* [12] K. Yokogawa, _Infinitesimal deformation of parabolic Higgs sheaves_ , Internat. J. Math. 6 (1995), 125–148.
* [13] I. Biswas, S. Majumder, M.L. Wong, _Orthogonal and symplectic parabolic bundles_ , J. Geom. Phys. 61 (2011), 1462–1475.
* [14] P. Reisert, _Moduli spaces of parabolic twisted generalized Higgs bundles (Doctoral dissertation, LMU München)_ (2016). Retrieved from https://edoc.ub.uni-muenchen.de/19890/
* [15] S. Roy, _Hitchin Fibration on Moduli of Symplectic and Orthogonal Parabolic Higgs Bundles_ , Math Phys Anal Geom 23, 41 (2020)
|
# Molecular rotors for in situ local viscosity mapping in microfluidic chips
Dharshana Nalatamby Florence Gibouin Univ. Bordeaux, CNRS, Solvay, LOF,
Pessac, F-33600, France Javier Ordóñez-Hernández Facultad de Química,
Departamento de Química Orgánica, Universidad Nacional Autónoma de México,
04510, Ciudad de México, México Julien Renaudeau Gérald Clisson Univ.
Bordeaux, CNRS, Solvay, LOF, Pessac, F-33600, France Norberto Farfán
Facultad de Química, Departamento de Química Orgánica, Universidad Nacional
Autónoma de México, 04510, Ciudad de México, México Pierre Lidon
<EMAIL_ADDRESS>Yaocihuatl Medina-González yaocihuatl.medina-
<EMAIL_ADDRESS>Univ. Bordeaux, CNRS, Solvay, LOF, Pessac, F-33600,
France
###### Abstract
In numerous industrial processes involving fluids, viscosity is a determinant
factor for reaction rates, flows, drying, mixing, etc. Its importance is even
more determinant for phenomena observed are at the micro- and nano- scales as
in nanopores or in micro and nanochannels for instance. However, despite
notable progresses of the techniques used in microrheology in recent years,
the quantification, mapping and study of viscosity at small scales remains
challenging. Fluorescent molecular rotors are molecules whose fluorescence
properties are sensitive to local viscosity: they thus allow to obtain
viscosity maps by using fluorescence microscopes. While they are well-known as
contrast agents in bioimaging, their use for quantitative measurements remains
scarce. This paper is devoted to the use of such molecules to perform
quantitative, in situ and local measurements of viscosity in heterogeneous
microfluidic flows. The technique is first validated in the well-controlled
situation of a microfluidic co-flow, where two streams mix through transverse
diffusion. Then, a more complex situation of mixing in passive micromixers is
considered and mixing efficiency is characterized and quantified. The
methodology developed in this study thus opens a new path for flow
characterization in confined, heterogeneous and complex systems.he methodology
developed in this study thus opens a new path for flow characterization in
confined, heterogeneous micro- and nano- systems.
Fluorescent Molecular Rotors, Fluorescence Lifetime Imaging Microscopy,
Microfluidics, Viscosity mapping
## I Introduction
In any process involving fluid flows, viscosity is a key control parameter.
Viscosimeters are the most common tool to measure this quantity, but their use
requires great care to avoid artifacts [1]. More importantly, they only
measure a viscosity averaged over macroscopic volumes and are difficult to
implement inline, which strongly limits their application in industrial
processes or in confined systems, or available in very limited volume. The
impossibility of obtaining local data also impedes their use with complex
flows involving spatial heterogeneity (e.g., inpaint manufacturing, food
processing or biomedical applications) and at small scales (e.g., in nano-,
micro- or millifluidics multi-phase flows or in porous media, with
applications in enhanced oil recovery, and catalysis among others). Designing
tools for small-scale viscosity measurements is thus an important stake for
fundamental studies as well as industrial and medical applications [2].
Microfluidics has been widely used as an essential tool in numerous
applications such as high-throughput screening in the research and development
domain [3] and chemical reactions analysis [4] and give interesting
opportunities for viscosity measurements. For instance, measuring pressure
drop along a microchannel at an applied flow rate allows the determination of
viscosity averaged on tiny volumes, below
$1\text{\,}\mathrm{\SIUnitSymbolMicro L}$ [5, 6]. More local approaches
require the characterization of the flow profile by introducing fluorescent
tracers in the fluid [7, 8], but are limited to simple configurations and
remain averaged over mesoscopic scales.
Fluorescent molecular rotors (FMR) offer a direct path for local viscosity
measurements. These are fluorescent molecules whose conventional fluorescent
relaxation after photoexcitation is in competition with a non-radiative
mechanism involving the rotation of a molecular bond [9]. This motion is
hindered by the local micro-viscosity of the environment [10, 11], with higher
micro-viscosity leading to increased quantum yield, thus more intense
fluorescence with longer lifetime [12]. While the precise relationship between
micro-viscosity and the usual viscosity remains unclear, they are directly
related in molecular fluids and after preliminary calibration, fluorescent
measurements can be used to retrieve local viscosity.
FMR have been acknowledged as excellent local viscosity probes with real-time
response and high spatial resolution [13, 14, 15] and are, for instance, used
as contrast agents in cells bioimaging [16, 17]. However, their use for
quantitative characterization in other contexts remains scarce [18, 19]. More
particularly in microfluidic context, regular and confocal fluorescence
lifetime imaging (of FMR and other fluorophores) have been proved to be
powerful tools for mapping complex, three-dimensional flows [20, 21, 22, 23]
but they were performed with a sophisticated lifetime measurement setup
operating directly in time domain, which requires a pulsed laser source, and
are thus difficult to be applied by non-specialists.
In this work, a FMR was synthesized and used for fluorescence lifetime imaging
in the frequency domain, using a commercial apparatus. The measurements were
performed in well controlled experiments of purely diffusive mixing of two
miscible streams co-flowing in a simple Y-mixer microfluidic channel; the
obtained results were satisfactorily compared with existing models and results
from the literature [24, 25]. After the validation of the method in this
simple case, it was finally used to assess the mixing efficiency of a passive
micromixer. These results validate the potential of the use of FMR for
quantitative and local measurements of viscosity in microfluidic flows and
open perspectives for fluid flow characterization.
## II Material and Methods
### II.1 BODIPY-2-OH synthesis and characterization
The viscosity-sensitive fluorescent boron-dipyrromethene (BODIPY)-based probe,
BODIPY-2-OH (see Figure 1), was developed specifically for this study. Based
on previously reported rotors, [26] BODIPY-2-OH was designed to ensure its
viscosity-sensitive characteristics. Details on its synthesis are provided in
the Supporting Information. In this structure, the phenyl unit is an electron
donor group in conjugation with the BODIPY core, which is an electron acceptor
group. After photoexcitation, relaxation of the molecule to its ground state
can occur either through conventional fluorescent photoemission or via a non-
radiative relaxation from the twisted intramolecular charge transfer (TICT)
state, accompanied by a rotation around the single bond linking phenyl and
BODIPY groups [27]. The radiative relaxation rate is affected by the
refractive index of the surrounding medium, while the non-radiative relaxation
rate depends on the free volume of the micro-environment, related to the
viscosity [10, 11].
Figure 1: Chemical structure of BODIPY-2-OH synthesized in this work.
Mixtures of DMSO and glycerol were chosen as working fluids in this article,
as their viscosity significantly vary with their composition [28] and because
BODIPY-2-OH is very soluble in these solvents. The volume fraction of glycerol
in analyzed solutions was controlled, and the concentration of BODIPY-2-OH was
kept constant at ${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{l}}^{-1}$ in
all the experiments.
#### II.1.1 Synthesis of BODIPY-2-OH
All starting materials were purchased from Sigma-Aldrich and used without
further purification. Solvents were dried by standard methods or distilled
prior to use. Reactions were monitored by thin-layer chromatography on pre-
coated silica gel plates (ALUGRAM SIL G/UV254) and revealed by exposure to a
UV lamp (254 nm). Infrared spectra were obtained using a Perkin-Elmer Spectrum
400 FT-IR/FT-FIR spectrophotometer, wavelength is reported in
$\text{\,}{\mathrm{cm}}^{-1}$. 1H, 13C, 11B and 19F NMR spectra were recorded
using Varian Unity Inova 300, JEOL ECA 400 spectrometers, chemical shifts
($\delta$/ppm) are reported relative to $\mathrm{Si(CH_{3})_{4}}$,
$\mathrm{CDCl_{3}}$, $\mathrm{BF_{3}OEt_{2}}$, and $\mathrm{CDCl_{3}}$. High-
resolution mass spectrometry (HR-MS) spectra were acquired with an Agilent
Technologies ESI TOF spectrometer.
The meso-substituted BODIPY-2-OH was prepared by the synthetic route shown in
Figure 2. Compound (1), a dipyrromethane ($51\text{\,}\mathrm{\char 37\relax}$
yield) derivative was synthesized from the condensation reaction between the
corresponding p-hydroxibenzaldehyde and ten equivalents of pyrrole in the
presence of a catalytic amount of $\mathrm{CF_{3}COOH}$. [29] The oxidation of
compound (1) with 2,3-dichloro-5,6-dicyano-1,4- benzoquinone (DDQ) followed by
a complexation reaction $\mathrm{BF_{3}.OEt_{2}}$ to give Compound (2)
($35\text{\,}\mathrm{\char 37\relax}$ yield). Compound (2) was dissolved in
tetrahydrofuran (THF) and sodium hydride was added. After 30 minutes,
3-bromo-1-propanol was added to the reaction mixture to obtain BODIPY-2-OH
($80\text{\,}\mathrm{\char 37\relax}$ yield). The final product was
characterized by spectroscopic techniques such as 1H, 13C, 11B, 19F NMR, IR
and HR-MS. Detailed information on the synthesis and NMR spectra of
BODIPY-2-OH can be found in the Supporting Information (see Figures 14 to 17).
Figure 2: Synthesis of BODIPY-2-OH.
#### II.1.2 Photophysical characterization of BODIPY-2-OH
The absorption and emission spectra of BODIPY-2-OH were measured respectively
using an Agilent Technologies Cary UV-Visible Spectrophotometer and an Agilent
Technologies Cary Eclipse Fluorescence Spectrophotometer. BODIPY molecules are
susceptible to photodegradation after constant irradiation [30]: this process
is slow enough to be negligible over the duration of our experiments, but
samples were stored in amber vials to limit bleaching by ambient light.
Single-use cuvettes (BRAND GMBH $70\text{\,}\mathrm{\SIUnitSymbolMicro L}$ UV-
Cuvette micro) were used for excitation and emission spectra measurements.
Figure 3: Absorption (blue line) and emission spectra (green line) of
BODIPY-2-OH in glycerol at a concentration
$c=${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{l}}^{-1}$$. Maximum
excitation and emission wavelengths of BODIPY-2-OH in glycerol are
respectively $500\text{\,}\mathrm{nm}$ and $515\text{\,}\mathrm{nm}$.
The obtained spectra for solutions of BODIPY-2-OH in glycerol at a
concentration $c=${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{l}}^{-1}$$
are depicted in Figure 3. The maximum excitation and emission wavelengths were
respectively found to be $500\text{\,}\mathrm{nm}$ and
$515\text{\,}\mathrm{nm}$. These spectra are similar to those previously
reported for BODIPY-based molecular rotors, which proves that the fluorescence
characteristics are not significantly affected by the different groups
attached to the BODIPY base [17, 26].
### II.2 Rotor response to viscosity
The fluorescence response of a given FMR to viscosity depends on the nature of
the solvent environment and has to be carefully calibrated in order to map
viscosity quantitatively [31, 32]. Hence, calibration curves were established
by measuring BODIPY-2-OH response in mixtures of known viscosity prior to the
use in microdevices. More precisely, two parameters were characterized:
emission intensity under steady illumination and fluorescence lifetime, which
quantifies the average lifetime of the excited state [33]. Intensity is a
convenient parameter to acquire with spectrophotometers or fluorescence
microscopes. However, fluorescence intensity does not depend only on the
solvent viscosity but also on local dye concentration and excitation
intensity, which is affected by the whole optical path before reaching the
sample. Careful and tedious calibrations would thus be required to go beyond
simple qualitative observations by using fluorescence intensity. On the
contrary, while more subtle to measure, fluorescence lifetime is only
determined by the microviscosity of the dye and allows for a more
straightforward interpretation: this parameter was thus preferred in this
study [15, 16, 33].
In order to calibrate the response of BODIPY-2-OH to viscosity, solutions of
the rotor in DMSO/glycerol mixtures were prepared from an initial dye stock
solution in DMSO. Viscosity of the mixtures were tuned by changing the ratio
of DMSO and glycerol and the concentration of BODIPY-2-OH was kept constant at
$c=${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{l}}^{-1}$$ to avoid any
change in fluorescence intensity due to dye concentration.
#### II.2.1 Bulk viscosity measurements
The viscosity $\eta$ of the different DMSO/glycerol mixtures were first
determined by rheometry. Tests were performed using a Kinexus Ultra+ rheometer
(Netzsch) with a Double-Gap geometry (DG24/27 SS CUP) for samples of low
viscosity (below $15\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$) and
$1\text{\,}\mathrm{\SIUnitSymbolDegree}$, $60\text{\,}\mathrm{mm}$-diameter,
cone-plate geometry for viscous samples (above
$15\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$). The experimental temperature
was controlled by a Peltier module and set to
$T=$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. The shear viscosity of every
sample was measured by successively applying shear rates of $1$, $10$, $100$,
and $1000\text{\,}{\mathrm{s}}^{-1}$. Measurements were taken for
$60\text{\,}\mathrm{s}$ for each shear rate with an acquisition rate of 1
point/s and were repeated thrice to ensure good repeatability. All samples
displayed Newtonian rheology, and the final viscosity value was taken as an
average over all the applied shear rates. Calibration curves relating glycerol
volume fraction, glycerol molar concentration, and viscosity of the mixtures
were established and fitted by exponential evolutions. Results are provided in
Supporting Information (see Figures 18 and 19).
#### II.2.2 Fluorescence intensity measurements
Absorption and emission spectra of BODIPY-2-OH in the different DMSO/glycerol
mixtures were recorded using the previously described protocol. No changes of
maximum of absorption ($\lambda_{\text{abs}}=$500\text{\,}\mathrm{nm}$$) and
of emission ($\lambda_{\text{abs}}=$515\text{\,}\mathrm{nm}$$) were observed.
Maximum of fluorescence emission spectra were recorded in order to quantify
the fluorescence intensity.
#### II.2.3 Fluorescence lifetime measurements
Fluorescence lifetimes of BODIPY-2-OH in the different DMSO/glycerol mixtures
were measured using a Fluorescence Lifetime Imaging Microscope (FLIM). FLIM
technique is a specific case of fluorescence microscopy, enabling the
measurement of spatially resolved fluorescence lifetime in heterogeneous
samples. It has been, in particular, used with FMR to map qualitative changes
of viscosity in bioimaging [16, 17, 34]. It is a wide-field method operating
in the frequency domain based on a regular setup of fluorescence microscopy
with a continuously modulated excitation source. By modifying the phase of the
exciting light, the fluorescence lifetime is then calculated for every pixel
from the local phase shift of the fluorescence emission [35, 36]. Provided an
initial reference has been acquired to account for instrumental behavior, FLIM
allows to map fluorescence lifetime over the field of view of the microscope
within a few seconds [37].
The FLIM experiments presented in this paper were carried out with a LIFA
(Lambert Instruments FLIM Attachment) device mounted on an Olympus IX71
inverted microscope. The image acquisitions were carried out at LED modulation
frequency of $40\text{\,}\mathrm{MHz}$ with $12$ acquisition phases and
$1\times$ CCD gain. A $10\text{\,}\mathrm{\SIUnitSymbolMicro
mol}\text{\,}{\mathrm{L}}^{-1}$ fluorescein solution at buffered
$\mathrm{p}H=10$ with a tabulated lifetime of $4.02\text{\,}\mathrm{ns}$ was
taken as reference. For lifetime calibration, DMSO-glycerol solutions
containing FMR at a fixed concentration of
${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$ were held in cavity
slides with glass coverslips used to obtain a flat layer of liquid with even
thickness. The samples were illuminated with a LED beam at
$451\text{\,}\mathrm{nm}$ through the dry microscope objectives lenses (10X or
20X Olympus), using a dichroic mirror. The fluorescence emission of the
molecular rotor was collected by the same objective and transmitted to the
cooled detector after passing through an Olympus
440x-$490\text{\,}\mathrm{nm}$ long pass filter. The retrieved data were then
analyzed using a Matlab application developed by our group.
Spatial resolution of the measurement is limited by the imaging setup: for
instance, in our setup, pixel size for a 20X objective lens corresponds to a
distance of $1.14\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Time resolution was
not a question in this study as we only considered steady flows. Yet, it can
be specified that acquisition of the full field of the microscope takes
typically a few seconds for maximum resolution on lifetime, and can be reduced
by acquiring smaller portion of the field of view or decreasing the accuracy.
The employed FLIM is adapted for measurement of lifetimes above
$1\text{\,}\mathrm{ns}$ with maximum accuracy of about
$0.1\text{\,}\mathrm{ns}$.
### II.3 Microfluidics
As a proof of principle of quantitative measurements of viscosity using FMR
with the proposed setup, two microfluidic configurations involving
heterogeneous flows were studied. First, the situation of diffusive mixing of
two streams in a simple microfluidic Y-junction was considered as it is well
controlled and characterized in the literature. Then, the more complex
situation of the mixing of two incoming streams in a Y-junction with
staggered-herringbone passive micromixers (SHM) was considered, as being
qualitatively understood and of practical interest in microfluidic
applications [38]. The corresponding chip designs are depicted in Figure 4.
Figure 4: (a) Schematics of a microfluidic Y-junction of height
$h=$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and width
$w=$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The mixing of two fluids
flowing side by side in the microchannel occurs through transverse molecular
diffusion. (b) Schematics of the microfluidic Y-junction with a staggered
herringbone passive mixer (SHM) used in this work.
#### II.3.1 Microfabrication
For fabrication of the simple Y-mixer chip, a mixture of
$95\text{\,}\mathrm{\char 37\relax}$ Polyethylene Glycol Diacrylate
(PEGDA)-250 and $5\text{\,}\mathrm{\char 37\relax}$ of photoinitiator,
2-hydroxy-2-methylpropiophenone was prepared in advance before being injected
by capillarity into the interstitial space between a glass slide and a 2-level
negative photoresist mold (SU-8 3050, MicroChem). This configuration was later
exposed under the UV lamp of an aligner for $1.2\text{\,}\mathrm{s}$ (power of
UV mercury vapor lamp is $35\text{\,}\mathrm{mW}\text{\,}{\mathrm{cm}}^{-2}$
at $365\text{\,}\mathrm{nm}$). The polymerized PEGDA film was then attached to
a silanized glass slide to seal the microchannel. [39, 40] PEGDA chip was
chosen for this experiment because of its fast microfabrication method, which
only takes around 5 minutes to make a functional chip[40]. The dimensions of
the main microchannel in Figure 4(a) were measured after photolithography of
the SU-8 mold using a Sensofar Non-contact 3D Optical Profiler. Height and
width of the channel were respectively
$h=$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and the width of the channel,
$w=$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
For fabrication of the chip including SHM, A 2-level PDMS microfluidic chip
was made using a negative photoresist mold (SU-8 3050, MicroChem) with classic
soft lithography techniques (Figure 4(b)). A glass slide was sealed to the
PDMS microfluidic chip after undergoing plasma treatment for 2 minutes. This
step was necessary to ensure covalent anchoring of the PDMS block to the glass
slide [41]. The chip was then placed in an oven at
$65\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 10 minutes to strengthen the
seal. The height of the channel without the micromixers is
$31\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and the part with micromixers is
$37\text{\,}\mathrm{\SIUnitSymbolMicro m}$ whereas the width of the channel is
$650\text{\,}\mathrm{\SIUnitSymbolMicro m}$. These dimensions were measured
with the Sensofar Non-contact 3D Optical Profiler with an interferometry
acquisition setting (objective lens: 10X Nikon DI). Groups of three SHM
grooves occupying a length of $1.95\text{\,}\mathrm{mm}$ were separated by
free intervals of length $9.40\text{\,}\mathrm{mm}$. The distance from the
beginning of the Y-junction to the first group of SHM was about
$7620\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
#### II.3.2 Flow control
In all experiments (Y-mixer and SHM), mixtures of glycerol and DMSO of
different viscosities were injected at flow rates imposed with a neMESYS
syringe pump into the two entrance sleeves of the chip. The concentration of
BODIPY-2-OH in all solutions was kept constant at
${10}^{-5}\text{\,}\mathrm{mol}\text{\,}{\mathrm{l}}^{-1}$. Before starting
microfluidic experiments, an aqueous solution of BODIPY-2-OH was injected into
chips to verify the absence of adsorption or permeation of the molecule into
the PDMS matrix. Calibration of the dye response to viscosity was also
performed in situ by measuring lifetimes in the entrance sleeves, where
mixtures have a known composition prior to any mixing. In order to image the
chip, a $x-y$ microactuator was used to move the chip above the microscope
objective of the microscope, and images were taken along the main
microchannel. The mapping of fluorescence lifetime was subsequently carried
out using the LIFA-FLIM, and fluorescence data post-processing led to the
mapping of viscosity in the microchannel.
## III Results
### III.1 Response of BODIPY-2-OH to viscosity
The emission spectra of solutions of BODIPY-2-OH in DMSO/glycerol mixtures of
different concentrations, and thus of varying viscosity, are reported in
Figure 5(a). It is first interesting to note that no changes of absorption
($\lambda_{\text{abs}}=$500\text{\,}\mathrm{nm}$$) and emission
($\lambda_{\text{abs}}=$515\text{\,}\mathrm{nm}$$) maxima wavelengths were
observed, which suggests that fluorescence response is not affected by
polarity changes in the investigated DMSO/glycerol mixtures [42].
Figure 5: (a) Emission spectra of BODIPY-2-OH in different DMSO-glycerol
mixtures of varying viscosities. The black vertical line at
$515\text{\,}\mathrm{nm}$ shows that the emission maximum wavelength is
independent of the viscosity of the samples. (b) Calibration curves in
logarithmic scale of BODIPY-2-OH fluorescence intensities, $I_{\text{F}}$
(blue points) and fluorescence lifetimes, $\tau_{\text{F}}$ (green points)
versus DMSO-glycerol mixture viscosity, $\eta$. The straight lines correspond
to fits with the Förster-Hoffmann model (Equation 1) with similar exponent
$\alpha=0.6$ for both fluorescence lifetime and intensity measurements, and
prefactor $C_{I}=1.3$ for fluorescence intensity and $C_{\tau}=-1.1$ for
lifetime measurement when $\eta$ is expressed in
$\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$ and $\tau$ in
$\text{\,}\mathrm{ns}$. Error bars for $I_{\text{F}}$ and $\tau_{\text{F}}$
are smaller than the size of data points.
The variations of fluorescence intensity $I_{\text{F}}$ (acquired with a
fluorescence spectrometer) and lifetime $\tau_{\text{F}}$ (acquired with FLIM)
with viscosity are displayed in Figure 5(b). Both parameters increase with
viscosity, which qualitatively agrees with the general mechanism of FMR. This
confirms that BODIPY-2-OH can be used as a local viscosity probe. Lifetime
values of the probe measured for
$\eta>$9\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ are within the detection
limit of the LIFA-FLIM used for this work.
More quantitatively, both parameters evolve with viscosity as a power-law with
similar exponent $\alpha=0.6$ over about three orders of magnitude in
viscosity. Such an observation is consistent with previous measurements in the
literature for other BODIPY-based FMR [17, 26, 43].
A power-law increase of fluorescence properties with ambient viscosity is
usual with FMR and derives from Förster-Hoffmann equation [44]. Quantum yield,
$\phi_{\text{F}}$ of FMR has indeed been proposed to follow a power-law
relationship with viscosity along:
$\log\phi_{F}=\alpha\log{\eta}+C$ (1)
where $\phi_{F}$ and $\eta$ represent the fluorescence quantum yield and the
local viscosity respectively; whereas $\alpha$ is a dye-dependent constant and
$C$ is a proportionality constant [45]. As the steady-state intensity and
fluorescence lifetime are proportional to the quantum yield, they follow a
similar Förster-Hoffmann relationship as expressed below[32, 37] :
$\log I_{F}=\alpha\log{\eta}+C_{I_{F}}$ (2)
and:
$\log\tau_{F}=\alpha\log{\eta}+C_{\tau_{F}}$ (3)
Both $I_{\text{F}}$ and $\tau_{\text{F}}$ are thus expected to follow a power-
law relationship with viscosity, sharing a similar exponent $\alpha$ and
possibly different pre-factors $C$, which is in agreement with experimental
results.
It is known that the environment can affect the fluorescence response of FMR
[14, 32] and it is thus essential to perform calibrations in situ. The
calibration procedure for fluorescence lifetime was thus repeated directly
within microfluidic channels, that will be used in the remaining of the paper.
Channels were filled with various DMSO-glycerol mixtures of known viscosity
containing BODIPY-2-OH, and the flow was left to stabilize for 1 minute before
measurement. Different images were taken along the channel for each mixture
and showed no significant lifetime variations. Average lifetime was then used
to construct the calibration curves in Figure 6 for two different materials of
the microfluidic chip.
Figure 6: Logarithmic-scale calibration curve of BODIPY-2-OH fluorescence
lifetimes, $\tau_{\text{F}}$ versus DMSO-glycerol mixture viscosity, $\eta$ in
a PDMS microchannel (green dots) and in a PEGDA microchannel (purple dots).
Straight lines correspond to a fit with the Förster-Hoffmann model (Equation
1) with exponent $\alpha$ = 0.7 for a PDMS microchannel and $\alpha$ = 0.5 for
a PEGDA microchannel. Error bars for $\tau_{\text{F}}$ values are smaller than
the size of data points.
Förster-Hoffman relation (3) thus remains valid for both materials, and
measured lifetimes are of a similar order of magnitude to those measured in
solution. However, there was a slight difference in the value of the exponent
$\alpha$: while this does not compromise the use of the technique, it showed
that in situ calibrations are preferable in order to retrieve quantitative
values of viscosity [18]. In the following, results of this in situ
calibration are used to obtain viscosity from measured lifetimes.
### III.2 Viscosity mapping of diffusive mixing in a simple Y-mixer
In order to prove that FMR can be used for viscosity mapping in heterogeneous
systems, experiments were performed in a simple Y-mixer, as depicted on Figure
4(a). In this situation, two miscible DMSO/glycerol mixtures of different
compositions (hence viscosity) were injected in the entrance sleeves of the
microchannel. There, streams follow a laminar flow and mixing occurs only by
transverse diffusion, perpendicular to the flow direction [46]. This is a
well-known configuration in microfluidics and the evolution of the fluid
composition along the channel has been thoroughly studied. Studies of similar
cases using FLIM can be found in literature but employed fluorescent probes
whose lifetime evolved through quenching by a diffusing ion [47] or by changes
of solvent polarity [48]. Other measurements using molecular rotors have also
been proposed before [13] but exploited a more complex setup, characterizing
decay of fluorescence anisotropy or measuring lifetime in time domain (thus
requiring a pulsed laser source), and no quantitative comparison with models
was presented. The most quantitative work that was found proposes to determine
directly the composition through in situ Raman spectroscopy [24, 25, 46] and
was taken as a reference for comparison with our work: a similar chip design
was thus used. In particular, the channel aspect ratio was kept identical to
ensure a unidimensional flow profile, and flow rates were adapted to have
small Reynolds numbers. However, for practical reasons, narrower channels of a
factor of about 2 were used.
For two immiscible fluids of different viscosity co-flowing along a channel,
the fraction of the channel occupied by a given stream is proportional to the
product of its viscosity and average flow rate [49]. For this experiment, the
viscosities of the two incoming DMSO/glycerol mixtures were
$\eta_{1}\approx$9\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ and
$\eta_{2}\approx$74\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ with a ratio of
about $1:8$. Experiments were conducted with flow rates adjusted to keep the
average interdiffusion zone visible along the entire width of the channel.
Figure 7: Viscosity mapping during of two liquid streams (denoted by S1, upper
stream, and S2, lower stream) flowing in a simple Y-mixer. Injected DMSO-
glycerol mixtures have initial viscosity
$\eta_{1}=$9\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ and
$\eta_{2}=$74\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$. Images are taken at
different positions $x$ along the microchannel, $x=0$ corresponding to the
first point of contact of the two liquids. The white arrow indicates the flow
direction and corresponds to a length of
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Flow rates of the two streams are
respectively $(Q_{\text{S1}},Q_{\text{S2}})=$ (a)
$(11,1)\,$\text{\,}\mathrm{\SIUnitSymbolMicro L}\text{\,}{\mathrm{min}}^{-1}$$
and (b) $(7,1)\,$\text{\,}\mathrm{\SIUnitSymbolMicro
L}\text{\,}{\mathrm{min}}^{-1}$$.
FLIM technique allowed the measurement of local fluorescence lifetime at every
pixel on different pictures taken along the microchannel at steady state.
Using the calibration equation in Figure 6, viscosity maps could be obtained
as displayed in Figure 7 for two different sets of flow rates. It is to note
that a few outliers were observed on pictures, corresponding to inconsistent
computed lifetimes far below the detection limit of LIFA
($<$0.2\text{\,}\mathrm{ns}$$): these are, however, in limited number and were
not taken into account for further data analysis.
The two streams of initially distinct viscosity progressively mix across an
interdiffusion layer that widens along the channel. As can be seen by
comparing Figure 7(a) and (b), a slower flow leads to better mixing at the end
of the channel, as it corresponds to a longer time of contact between the
solutions. Finally, the interdiffusion layer is closer to the center of the
channel for a ratio of flow rate close to the ratio of viscosity, as would be
the case for immiscible liquids. A progressive drift of the interdiffusion
zone towards the higher-viscosity fluid can also be observed. This latter fact
results from the coupling between hydrodynamics and the mixing through the
dependence of the viscosity with the glycerol concentration along the mixing
channel [25].
### III.3 Micromixing experiments
Finally, the more complex situation of mixing of two liquid streams in a SHM
channel was studied, following a similar protocol. The design of asymmetrical
herringbone grooves that was used helped to develop laminar chaotic flows in
the microchannel by stretching and folding the two incoming steady streams,
into alternate thin liquid sheets in order to enhance diffusion efficiency
[38, 50]. In order to observe the evolution of the mixing along the channel,
zones with SHM were separated by zones of free diffusion, where flow returns
to its laminar state and the degree of mixing can be measured.
In this experiment, the mixing of streams of initial viscosity
$\eta_{1}=$9\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ and
$\eta_{2}=$206\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ in such a system was
studied. The lifetimes measured by LIFA-FLIM were converted into viscosity by
using the calibration equation for the PDMS microchannel displayed in Figure
6.
Figure 8: (Top) Schematics of a microfluidic Y-mixer with staggered
herringbone passive micromixers (SHM). (Bottom) Viscosity mapping in a Y-mixer
with SHM during a co-flow of DMSO-glycerol mixture of initial viscosity
$\eta_{1}=$9\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ (S1) and
$\eta_{2}=$206\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ (S2). Applied flow
rates are
$(Q_{\text{S1}},Q_{\text{S2}})=(35,0.25)\,$\text{\,}\mathrm{\SIUnitSymbolMicro
L}\text{\,}{\mathrm{min}}^{-1}$$. The arrow represents the flow direction and
a scale length of $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Images were
taken at different positions along the length of the microchannel, labeled on
the top schematics. Indications E, M, and S, respectively stand for inlet,
middle zone, and outlet.
The obtained viscosity maps at different positions along the channel are
depicted in Figure 8. The different positions are represented on the
schematics on top of the figure: in particular, pictures designated by E
correspond to the entrance of SHM zones, where the flow is laminar. In these
pictures, after every SHM, the mixing degree increased, which was
characterized by the widening interdiffusion layer and the homogenization of
the viscosities of the two streams. Pictures designated by M and S are taken
in and after SHM zones, respectively, where we can observe a destabilization
of the interdiffusion layer, leading to enhanced mixing.
## IV Discussion
In this last part, viscosity mappings presented in the Results section are
analyzed in more detail, confirming the relevance of FMR and of the FLIM setup
for quantitative measurements.
### IV.1 Effective diffusion coefficient in the simple Y-mixer
In order to assess the validity of the viscosity measurements, it is possible
to analyze viscosity maps in further details to compare with a model from the
literature. [24, 25, 46] From experimental viscosity maps at different
positions $x$ in the channel, transverse viscosity profiles $\eta(y)$ can
first be extracted, with $y\in[0,w]$ corresponding to the coordinate
perpendicular to the flow direction. As diffusion is slow enough, profiles on
a single picture do not significantly evolve with $x$: profiles were thus
averaged in all pictures to minimize noise level. The results are displayed in
Figure 9.
Figure 9: Viscosity profiles $\eta$ versus $y$, for different positions $x$
along the microchannel (color-coded: darker to lighter shades represent
increasing $x$ direction, downstream), in a simple Y-mixer with flow rates
$(Q_{\text{S1}},Q_{\text{S2}})=(7,1)\,$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}\mathrm{/}\min$$.
The arrow represents downstream evolution, with growing $x$.
With these profiles, the evolution of viscosity across the channel can be
observed more clearly, and in particular the widening of the interdiffusion
layer along the channel. This situation of diffusive mixing in a microfluidic
co-flow can be modeled by an advection-diffusion problem. In the case of an
infinitely wide channel, analytical solutions have been obtained for the
volume fraction $\phi$ of glycerol [46]:
$\centering{\phi}\left(x,y\right)=\frac{\phi_{0}}{2}\operatorname{erf}\left(\frac{y-y_{m}(x)}{2\sigma_{d}(x)}\right)+\phi_{\text{m}}.\@add@centering$
(4)
At a given position $x$ in the channel, $\phi$ varies from
$\phi_{\text{min}}=\phi_{\text{m}}-\phi_{0}/2$ to
$\phi_{\text{max}}=\phi_{\text{m}}+\phi_{0}/2$, with a transition described by
an error function, of center $y_{m}(x)$ and characteristic width
$\sigma_{d}(x)$.
Such an expression has been experimentally validated [25] and was used as a
reference in our work. In particular, flow rates were selected to maintain a
low Reynolds number, typically between ${10}^{-2}\text{\,}$ and
${10}^{-1}\text{\,}$, in order to obtain laminar flow in the microchannels.
Also, the entrance length $L_{\text{e}}$, of about $0.4\text{\,}\mathrm{mm}$,
was small enough to consider that the flow reached a fully developed
Poiseuille profile in the analyzed pictures [49, 51].
In order to use Equation (4), the measured viscosity $\eta$ was first
converted into glycerol volume fraction $\phi$ by using a pre-established
calibration curve (see Figure 6 in Supporting Information): corresponding
profiles at different positions $x$ are displayed in Figure 10(a) and were
fitted using Equation (4). As also observable on the viscosity profile, the
minimum and maximum volume fractions at both sides are not constant along the
microchannel. This was not observed in previous experiments from the
literature and comes from the smaller width $w$ of the channel, which induces
side effects not considered in the model and become observable in our
experiment. Parameters $\phi_{0}$, $\phi_{m}$, $\sigma_{d}$, and $y_{m}$ were
thus taken as free fit parameters at every position $x$. The resulting fits
are superimposed to experimental data on Figure 10(a).
Figure 10: (a) Profiles of glycerol volume fraction ${\phi_{\text{exp}}}(y)$
at different positions $x$ along the micro-channel (color-coded: darker to
lighter shades represent increasing $x$ direction, downstream) obtained from
viscosity profiles obtained in Figure 9. The continuous lines represent the
fits according to Equation 4. (b) Normalized glycerol volume fraction
$\tilde{\phi}$ versus normalized position $\tilde{y}$ as defined in Equation
(5). Data obtained for different flow rates
$(Q_{\text{S1}},Q_{\text{S2}})=(7,1)$ (brown), $(3,1)$ (blue) and
$(11,1)\,$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}\mathrm{/}\min$$
(blue). The black dashed line represents the theoretical master curve from
Equation (6).
First, in order to assess the accuracy of the proposed model, the normalized
volume fraction:
$\tilde{\phi}=\frac{\phi-\phi_{\text{min}}}{\phi_{\text{max}}-\phi_{\text{min}}}$
(5)
can be plotted as a function of a normalized coordinate normal to the channel
$\tilde{y}=(y-y_{m}(x))/2\sigma_{d}(x)$. According to the model (4), this
should collapse all data on a single master curve described by:
$\tilde{\phi}(\tilde{y})=\frac{1+\operatorname{erf}\tilde{y}}{2}$ (6)
This is indeed observed in Figure 10(b), where datasets corresponding to
different values of the flow rates $Q_{S1}$ and $Q_{S2}$ are represented. This
confirms the agreement of the model with the measurements.
The variations of the fitting parameters with the position $x$ along the
microchannel can be characterized in more details. As commonly done in
microfluidics, the average time $\tau$ spent by the fluid in the channel at a
given position $x$ is used to describe the evolution of the system:
$\centering\tau=\frac{2xhw}{Q_{S1}+Q_{S2}}=\frac{2x}{v_{1}+v_{2}}\@add@centering$
(7)
with $v_{1}$ and $v_{2}$ representing the velocities of the incoming streams.
The evolution of the amplitude $\phi_{0}$ of the profile and the volume
fraction at the middle of the interdiffusion layer $\phi_{m}$ are given in
Figure 11(b) and Figure 11(c). These parameters are difficult to interpret,
but they are found to be roughly constant within a $10\text{\,}\mathrm{\char
37\relax}$ variation. This may seem contradictory with the clear evolution of
the viscosity on the sides of the channel: it is however easily explained by
the strongly non-linear evolution of viscosity of glycerol-DMSO mixture with
glycerol concentration. Due to the finite width of the channel, the volume
fractions at the walls of the channel slightly evolve, which induces a strong
viscosity variation for the stream highly concentrated in glycerol.
Figure 11: Variation of fitting parameters (a) $y_{m}$, (b) ${\phi_{0}}$, and
(c) ${\phi_{min}}$ versus characteristic time $\tau$ obtained during the co-
flow experiments in a simple Y-mixer with ($Q_{\text{S1}},~{}Q_{\text{S2}}$) =
$(7,1)$ (brown); $(3,1)$ (blue);
$(11,1)\,$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}\mathrm{/}\min$$
(green).
The position of the center of the interdiffusion layer $y_{m}$ is also given
in Figure 11(a). Again, this parameter is roughly constant, showing a tendency
to increase, which becomes clearer when the flow rate ratio differs from the
viscosity ratio of the two streams. This is consistent with observations by
Dambrine et al. [25].
Figure 12: Evolution of the squared interdiffusion layer width,
$\sigma_{d}^{2}$, against the timescale, $\tau$, defined in Equation (7), with
($Q_{\text{S1}},~{}Q_{\text{S2}}$) = $(7,1)$ (brown); $(3,1)$ (blue);
$(11,1)\,$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}\mathrm{/}\min$$
(green). Continuous line illustrates fit by Equation (8) with effective
diffusion coefficient (averaged for all flow rates),
$D_{\text{eff}}=$2\text{\times}{10}^{-10}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$.
More interestingly, the evolution of width $\sigma_{d}$ of the interdiffusion
layer, which is displayed in Figure 12, is predicted by the model. As the
spreading of this layer is driven by transverse diffusion, the squared width
$\sigma^{2}_{d}(x)$ at a given position $x$ is expected to evolve linearly
with the time of diffusion $\tau(x)$ along:
$\sigma_{d}^{2}(x)=D_{\text{eff}}\,\tau(x).$ (8)
Experimental observations are in agreement with this relation, and allow to
retrieve an effective diffusion coefficient
$D_{\text{eff}}=$2\text{\times}{10}^{-10}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$,
to be compared with the reference value
$D_{\text{ref}}=$7\text{\times}{10}^{-10}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$
from the literature [25]. These two values have a similar order of magnitude,
and this agreement can be considered as satisfactory. First, the reference
value obtained from literature was for water-glycerol mixtures while DMSO-
glycerol mixtures were used in this study. Also, no systematic determination
of experimental uncertainties was performed, but considering the number of
steps to retrieve this diffusion coefficient, uncertainties are likely to be
significant. Finally, the existence of a finite size effect due to the
narrower channels employed in this study can also slightly modify the
effective diffusion coefficient.
As a conclusion, these results show that the technique allows mapping of
viscosity profile in a Y-mixer configuration which agrees with experiments and
models proposed in the literature. This validated the use of such setup for
quantitative measurements in microfluidic chips, and more generally in
confined flows.
### IV.2 Mixing in passive micromixers
#### IV.2.1 Final homogeneous state
After crossing a few SHM, the two miscible streams of different viscosities
eventually mix, leading to a homogeneous mixture. Using the principle of mass
conservation, the final glycerol concentration of the homogeneous mixture
$\mathrm{[Gly_{calc}]}$ was related to the flow rates $Q_{\text{Si}}$ and
concentrations $\mathrm{[Gly_{Si}]}$ of the two incoming streams through:
$\mathrm{[Gly_{calc}]}=\frac{\mathrm{[Gly_{S1}]}.Q_{\text{S1}}+\mathrm{[Gly_{S2}]}.Q_{\text{S2}}}{Q_{\text{S1}}+Q_{\text{S2}}}.$
(9)
The final concentration of glycerol $\mathrm{[Gly_{exp}]}$ was measured by
converting the measured viscosity into glycerol concentration with a pre-
established viscosity-concentration curve (see Figure 19) in Supporting
Information). Results are gathered in Table 1 for different sets of incoming
flow rates. The measured and calculated values are in good agreement, as can
be seen, more quantitatively through their relative discrepancy
$\sigma_{\text{rel}}$ remaining below $10\text{\,}\mathrm{\char 37\relax}$;
this again validates the quantitative nature of the measurement method.
$Q_{\text{S1}}$ ($\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}\mathrm{/}\mathrm{min}\mathrm{)}$ | $Q_{\text{S2}}$ ($\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}\mathrm{/}\mathrm{min}\mathrm{)}$ | $\mathrm{[Gly_{calc}]}$ ($\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$) | $\mathrm{[Gly_{exp}]}$ ($\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$) | $\sigma_{\text{rel}}(\%)$
---|---|---|---|---
7.0 | 0.5 | 4.28 | 4.21 | 1.5
16.0 | 0.5 | 4.02 | 4.10 | 1.9
4.0 | 0.5 | 4.59 | 4.19 | 8.6
Table 1: Comparison of calculated glycerol concentration
$\mathrm{[Gly_{calc}]}$ and experimental values $\mathrm{[Gly_{exp}]}$ after
full mixing of the two streams in a SHM, for different flow rates.
#### IV.2.2 Assessment of mixing efficiency
The obtained viscosity maps could be exploited more quantitatively to assess
the efficiency of the micromixer. It is important to note that the objective
of this study was not to provide an efficient micromixer design, but rather to
propose and test a methodology that can be used to test the efficiency of
micromixers while viscosity is mapped during mixing. As already mentioned, the
flow in the SHM parts of the channel is three-dimensional and difficult to
analyze: it is thus more relevant to focus on images taken before each of the
SHM groups, where flow returned to its laminar state. In the following
discussion, the mixer design was kept similar and the effect of injection flow
rates is discussed.
In order to characterize the mixing efficiency, following previous approaches
in the literature [38, 52], the standard deviation of lifetime $\sigma$ was
measured over the different pictures. In an ideal situation, $\sigma$ should
be maximal at the entrance of the microchannel (and associated with a stepwise
profile of lifetime), and decay to zero when the two fluids are homogeneously
mixed. In practice, due to experimental noise, $\sigma$ reaches a plateau
value at the end of the chip. In order to better compare the situations
obtained with different flow rates, it is convenient to study the normalized
parameter $\tilde{\sigma}$ defined by:
$\centering\tilde{\sigma}=\frac{\sigma-\sigma_{\text{min}}}{\sigma_{\text{max}}-\sigma_{\text{min}}}\@add@centering$
(10)
where $\sigma_{\text{min}}$ is the minimum standard deviation measured at the
outlet of the chip, and $\sigma_{\text{max}}$ is its maximum value measured at
the entrance of the chip. Values of these extrema were similar within a
$10\text{\,}\mathrm{\char 37\relax}$ variation for the different flow rates
considered here.
The obtained evolution of $\tilde{\sigma}$ after the different groups of SHM
is represented in Figure 13. For all studied flow rates, the mixing remained
minimal before the entry of the first group of SHM (MM1), as also observed in
Figure 8. Then, mixing significantly improved after every group of SHM, and a
homogeneous state is eventually reached.
The dashed line represents the evolution that would be observed without
micromixers. Our results clearly illustrate the efficiency of micromixers
compared to free diffusion. In SHM, the flow becomes three-dimensional and
decomposes the streams into alternated thin sheets, in which diffusion becomes
more efficient [53, 52]
Figure 13: Normalized standard deviation, $\tilde{\sigma}$, of the lifetime
maps such as in Fig 8 defined in Equation (10) as a function of the distance
downstream from the entrance of the channel, $x$, at the beginning of every
SHM group. Data shown in this figure were generated from a single microfluidic
device during the mixing of DMSO-glycerol solutions of different viscosity,
$\eta_{1}=$9\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ (S1) and
$\eta_{2}=$206\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ (S2). Applied flow
rates $(Q_{\text{S1}},Q_{\text{S2}})$ were $(3.5,0.25)$ ($\blacklozenge$,
average Reynolds number $\mathrm{Re}=$2\text{\times}{10}^{-2}\text{\,}$$);
$(7.0,0.5)$ ($\blacksquare$, average Reynolds number
$\mathrm{Re}=$5\text{\times}{10}^{-2}\text{\,}$$); $(14.0,1.0)$ ($\bullet$,
average Reynolds number $\mathrm{Re}=$0.1\text{\,}$$);
$(35.0,2.5)\,$\text{\,}\mathrm{\SIUnitSymbolMicro
L}\text{\,}{\mathrm{min}}^{-1}$$ ($\RHD$, average Reynolds number
$\mathrm{Re}=$0.2\text{\,}$$). Colored continuous lines are simply added for
visual clarity only. The black dashed line represents the evolution that would
be obtained for purely diffusive mixing in a simple Y-mixer.
As also demonstrated in Figure 13, stronger flow rates increased mixing
efficiency in SHM, in agreement with previous studies [38]. Care was taken to
let the flow reach a steady state prior to any measurement after changing the
flow rate. This, again, was in qualitative agreement with our understanding of
micromixers: faster flows decrease contact time between the sheets in SHM
zones, thus decreasing mixing efficiency.
The proposed method thus allows the assessment of the mixing performance of
SHM micromixers, in agreement with results from the literature. This validates
the use of molecular rotors to characterize complex microfluidic flows.
## V Conclusions
In this work, the possibility to map quantitatively local viscosity in
microfluidic flows using fluorescent molecular rotors was investigated. A
BODIPY-based molecular rotor was synthesized, and its fluorescence lifetime
response to viscosity was calibrated. In the well-controlled situation of the
transverse diffusive mixing of two liquid streams flowing in a Y-mixer
channel, the obtained viscosity maps were in quantitative agreement with
models and experiments previously proposed in the literature for a similar
system. This validates the possibility of using FMR for quantitative
measurements beyond their previously reported use as contrast agents in
bioimaging. A more complex system was then considered such as mixing in
passive micromixers; it was then proved that FMR could be used to characterize
mixing efficiency. It is important to note that they allow for measurement of
up to three decades in viscosity, nearly up to
$1\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$.
This article proves the wide opportunities offered by FMR for quantitative and
local characterization of confined flows. In particular, the technique can be
adapted to other solvents and tuned for a specific viscosity range by
synthesizing new FMR. This opens a path for the characterization of flows in
numerous contexts, ranging from industrial processes to natural flows. It is
however necessary to note the importance of performing in situ calibrations
for accurate viscosity measurements, as the Förster-Hoffmann coefficients can
be affected by the local environment of FMR.
In this work, all considered fluids were Newtonian and confined at the
microscale. Future work will consider the characterization of complex fluids
and of highly confined flows, close to the molecular scale. It will help to
determine the spatial scale over which the viscosity of the microenvironment
influences the response of FMR.
Acknowledgements
The authors thank Dr. J.B Salmon and Dr. J. Leng for scientific exchanges
concerning the fluorescence mapping in microfluidic chips. Dr. S. Harrisson is
thanked for fruitful discussion. The research presented in this article was
funded by ANR grant MicroVISCOTOR (ANR-18-CE42-0010-01). The authors also
thank Solvay and CNRS for funding. This work was also supported by CONACYT
(A1-S-7642), PAPIIT IN200422.
## Appendix A Synthesis of molecular rotor
Synthesis of the molecular rotor used in the paper follows the scheme
represented in Figure 2 of the main text, in the Material and Methods Section.
### A.1 Synthesis of Compound 1
Compound (1), 4-(di(1H-pyrrol-2-yl)methyl)phenol was synthesized following a
procedure in the literature [29]. p-hydroxybenzaldehyde
($1\text{\,}\mathrm{g}$, $8.19\text{\,}\mathrm{mmol}$) and pyrrole
($5.49\text{\,}\mathrm{g}$, $81.88\text{\,}\mathrm{mmol}$) were dissolved in
anhydrous THF ($30\text{\,}\mathrm{mL}$) under $\mathrm{N_{2}}$ atmosphere
followed by the addition of trifluoroacetic acid
($94\text{\,}\mathrm{\SIUnitSymbolMicro L}$, $1.23\text{\,}\mathrm{mmol}$).
The reaction mixture was stirred at room temperature for 15 minutes. The
progress of the reaction was monitored by TLC. After $\mathrm{CH_{2}Cl_{2}}$
and 0.1 M $\mathrm{NaOH}$ were added to the reaction mixture and the organic
phase was washed with water and filtered over $\mathrm{Na_{2}SO_{4}}$. Solvent
and pyrrole were removed under reduced pressure. The crude product was
purified by column chromatography over silica gel with hexane/ethyl acetate
(7:3) as eluent to give $1\text{\,}\mathrm{g}$ of Compound (1)
($51\text{\,}\mathrm{\char 37\relax}$ yield) as a brown solid. 1H NMR
($400\text{\,}\mathrm{MHz}$, $\mathrm{CD_{3}COCD_{3}}$, $\delta$, ppm): 9.60
(bs, 2H), 8.19 (bs, 1H), 7.02 (d, $J=$8.3\text{\,}\mathrm{Hz}$$, 2H), 6.73 (d,
$J=$8.3\text{\,}\mathrm{Hz}$$ 2H), 6.66-6.65 (m, 2H), 5.96 (q, 2H), 5.73-5.71
(m, 2H), 5.34 (s, 1H). 13C NMR ($100\text{\,}\mathrm{MHz}$,
$\mathrm{CD_{3}COCD_{3}}$, $\delta$, ppm): 155.8, 134.5, 133.7, 129.3, 116.7,
114.8, 107.1, 106.3, 43.3.
### A.2 Synthesis of Compound 2
Compound (2), 4,4-Difluoro-8-(4-hydroxyphenyl)-4-bora-3a,4a-diaza-s-indacene
was synthesized following a procedure in the literature [29]. Compound (1)
($1.0\text{\,}\mathrm{g}$, $4.20\text{\,}\mathrm{mmol}$) and DDQ
($1.14\text{\,}\mathrm{g}$, $5.04\text{\,}\mathrm{mmol}$) were dissolved in
THF ($30\text{\,}\mathrm{mL}$). The reaction mixture was stirred at room
temperature for two h. After of this time, triethylamine
($8.77\text{\,}\mathrm{mL}$, $62.95\text{\,}\mathrm{mmol}$) was added and
after 10 minutes $\mathrm{BF_{3}OEt_{2}}$ ($10.36\text{\,}\mathrm{mL}$,
$83.93\text{\,}\mathrm{mmol}$) was added dropwise. The reaction mixture was
stirred for 2 h and then washed with water and extracted with ethyl acetate.
The organic phase was filtered over $\mathrm{Na_{2}SO_{4}}$, the solvent was
removed under reduced pressure and the crude product was further purified by
column chromatography over silica gel with hexane/ethyl acetate (7:3) as
eluent to give $0.39\text{\,}\mathrm{g}$ of 2 ($35\text{\,}\mathrm{\char
37\relax}$ yield) as a red solid. 1H NMR ($400\text{\,}\mathrm{MHz}$,
$\mathrm{CD_{3}COCD_{3}}$, $\delta$, ppm): 9.28 (s, 1H), 7.97 (bs, 2H), 7.60
(d, $J=$8.72\text{\,}\mathrm{Hz}$$, 2H), 7.11-7.08 (m, 4H), 6.66 (d,
$J=$4.1\text{\,}\mathrm{Hz}$$, 2H). 13C NMR ($100\text{\,}\mathrm{MHz}$,
$\mathrm{CD_{3}COCD_{3}}$, $\delta$, ppm): 161.6, 149.0, 144.1, 135.5, 133.8,
132.2, 125.9, 119.2, 116.6.
### A.3 Synthesis of BODIPY-2-OH (Compound 3)
To solution of Compound (2) ($0.5\text{\,}\mathrm{g}$,
$1.76\text{\,}\mathrm{mmol}$) in THF anhydrum ($20\text{\,}\mathrm{mL}$) under
$\mathrm{N_{2}}$ atmosphere, was added $\mathrm{NaH}$
($46.5\text{\,}\mathrm{mg}$, $1.94\text{\,}\mathrm{mmol}$) and the reaction
mixture was stirred for 30 minutes followed by the addition of 3-bromopropanol
($318.0\text{\,}\mathrm{mg}$, $2.29\text{\,}\mathrm{mmol}$). The reaction
mixture was stirred for 4 h at room temperature and then quenched with aq.
$\mathrm{NH_{4}Cl}$. The reaction was extracted with ethyl acetate and water.
The organic phase was dried ($\mathrm{Na_{2}SO_{4}}$) and the solvent was
evaporated under reduced pressure. Purification by column chromatography
(hexane/EtOAc, 7:3) gave
4,4-Difluoro-8-(4-(3-hydroxypropoxy)phenyl)-4-bora-3a,4a-diaza-s-indacene or
BODIPY-2-OH, Compound (3) ($80\text{\,}\mathrm{\char 37\relax}$ yield) as a
red solid. FTIR-ATR ($\nu\,$\text{\,}{\mathrm{cm}}^{-1}$$): 3330,3145, 3120,
2960, 2938, 1729, 1541, 1384, 1249, 1184, 1119, 1071, 1054, 971, 841, 741,
707. 1H NMR ($300\text{\,}\mathrm{MHz}$, $\mathrm{CDCl_{3}}$, $\delta$, ppm):
7.91 (bs, 2H), 7.52 (d, $J=$8.7\text{\,}\mathrm{Hz}$$, 2H), 7.04 (d,
$J=$8.7\text{\,}\mathrm{Hz}$$, 2H), 6.96 (d, $J=$3.8\text{\,}\mathrm{Hz}$$,
2H), 6.54 (d, $J=$3.8\text{\,}\mathrm{Hz}$$, 2H), 4.22 (t,
$J=$5.9\text{\,}\mathrm{Hz}$$, 2H), 3.90 (t, $J=$5.9\text{\,}\mathrm{Hz}$$,
2H), 2.10 (q, $J=$5.9\text{\,}\mathrm{Hz}$$, 2H). 13C
($75\text{\,}\mathrm{MHz}$, $\mathrm{CDCl_{3}}$, $\delta$, ppm): 161.5, 147.5,
143.5, 134.9, 132.6, 131.5, 126.5, 118.4, 114.7, 66.7, 60.0, 32.0. 11B
($160.4\text{\,}\mathrm{MHz}$, $\mathrm{CDCl_{3}}$, $\delta$, ppm): -0.67 (t,
$J_{\mathrm{B-F}}$= 28.9 Hz). 19F ($282.4\text{\,}\mathrm{MHz}$, CDCl3,
$\delta$, ppm): -144.8 (q, $J_{\text{B-F}}=$28.9\text{\,}\mathrm{Hz}$$). HRMS
(DART) m/z Calcd. for
$\mathrm{C_{18}H_{17}BF_{2}N_{2}O_{2}}+\mathrm{H^{+}}=343.14294$ found
343.14391 (2.82 ppm).
$\mathrm{{}^{1}H}$, $\mathrm{{}^{13}C}$, $\mathrm{{}^{19}F}$ and
$\mathrm{{}^{11}B}$ NMR spectra of compound (3) are respectively displayed on
Figures 14, 15, 16 and 17.
Figure 14: $\mathrm{{}^{1}H}$ NMR spectrum of compound 3 at
$75\text{\,}\mathrm{MHz}$ in $\mathrm{CDCl_{3}}$. Figure 15:
$\mathrm{{}^{13}C}$ NMR spectrum of compound 3 at $75\text{\,}\mathrm{MHz}$ in
$\mathrm{CDCl_{3}}$. Figure 16: $\mathrm{{}^{19}F}$ NMR spectrum of compound 3
at $282.4\text{\,}\mathrm{MHz}$ in $\mathrm{CDCl_{3}}$. Figure 17:
$\mathrm{{}^{11}B}$ NMR spectrum of compound 3 at $160.4\text{\,}\mathrm{MHz}$
in $\mathrm{CDCl_{3}}$.
.
## Appendix B Viscosity-Composition calibration curves
### B.1 Viscosity-Glycerol Volume Fraction Calibration Curve
As described in the main text, the viscosity $\eta$ of mixtures of various
glycerol volume fraction $\phi$ was measured. These parameters can be related
by a fitting equation:
$\phi=a\ln\left(\frac{\eta}{\eta_{0}}\right)$ (11)
with $a=0.16$ and $\eta_{0}=$1.87\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$.
Experimental measurements and fitting curve are displayed in Figure 18.
Figure 18: Glycerol volume fraction $\phi$ as a function of the viscosity
$\eta$ of the DMSO-glycerol mixture. The continuous line corresponds to Eq.
(11) with $a=0.16$ and
$\eta_{0}=$1.87\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$.
### B.2 Glycerol Concentration-Viscosity Calibration Curve
Instead of volume fraction, it can also be useful to measure glycerol
concentration $[\mathrm{Glycerol}]$. It can be related to viscosity through:
$[\mathrm{Glycerol}]=a^{\prime}\ln\left(\frac{\eta}{\eta^{\prime}_{0}}\right)$
(12)
with $a^{\prime}=$2.17\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$$ and
$\eta_{0}^{\prime}=$1.41\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$.
Experimental measurements and fitting curve are displayed in Figure 19.
Figure 19: Glycerol concentration $[\mathrm{Glycerol}]$ as a function of the
viscosity $\eta$ of the DMSO-glycerol mixture. The continuous line corresponds
to Eq. (12) with
$a^{\prime}=$2.17\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$$ and
$\eta_{0}^{\prime}=$1.41\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$.
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|
# Data-driven port-Hamiltonian structured identification
for non-strictly passive systems
Charles Poussot-Vassal1,†, Denis Matignon2, Ghilslain Haine2 and Pierre
Vuillemin1 *This work has been supported by the AID (Agence de l’Innovation de
Défense) from the French Ministry of the Armed Forces (Ministère des Armées).
1Charles and Pierre are with ONERA – The French Aerospace Lab, 2 avenue
Edouard Belin, Toulouse 31400, France.2Denis and Ghislain are with ISAE-
SUPAERO, Université de Toulouse, 10, avenue Edouard Belin, Toulouse 31400,
France.† Corresponding author<EMAIL_ADDRESS>
###### Abstract
In this work, we detail a procedure to construct a reduced order model on the
basis of frequency-domain data, that preserves the non-strictly passive
property and the port-Hamiltonian structure. The proposed scheme is based on
Benner et al. contribution [1], which has been adapted (i) to handle non-
strictly passive model, and (ii) to handle numerical issues observed when
applying the Loewner framework on complex configurations. We validate the
proposed scheme on a very complex two-dimensional wave equation, for which the
discretized version preserves the port-Hamiltoninan form.
## 1 Introduction
### 1.1 Motivation and foreword
The present work is motivated by an efficient numerical representation of the
wave equation on a 2D domain $\Omega$, with actuators and sensors that are
_collocated_ at the boundary $\partial\Omega$; the PDE model is first
described as a distributed port-Hamiltonian system (pHs) (see the pioneering
work [2], and [3] for a recent overview), second discretized in a structure-
preserving manner thanks to the Partitioned Finite Element Method (PFEM) [4].
Although in 1D with physical parameters which are uniform in space, the input-
output transfer function is easy to compute, the task becomes more difficult
with varying parameters. In generic geometric 2D domain with heterogeneous and
anisotropic parameters, it is almost impossible. However, a high-fidelity full
order model (FOM), taking all these important properties into account can be
computed at the discrete level [5].
This model results in a linear pH one, embedding a very large number of
internal state variables, a quite large number of inputs and outputs. Such a
high dimension is a limiting factor for simulation, optimisation, analysis and
control. Computing simplified, easy to use dynamical models is one purpose of
the model approximation and reduction discipline. The goal is to approximate
the original system with a smaller and simpler system with the _same
structure_ and similar response characteristics as the original, the low-
complexity model, also called a reduced order model (ROM).
The Loewner framework (LF) employed in this work is a _data-driven_ model
identification and reduction technique that was originally introduced in [6].
Using only frequency-domain measured data, the LF constructs surrogate models
directly and with low computational effort. Its extension to pH model is
proposed in [1] and [7]. We refer the reader to [8] for an overview of Loewner
identification and reduction methods. Recently [9] gave an overview of
physics-based reduction methods, and [10] presents a successful attempts to
apply data-driven techniques to identification of pHs on a 1D example.
### 1.2 Notations and preliminaries
The set of real and complex numbers of dimension $n$ are denoted, respectively
by $\mathbb{R}^{n}$ and $\mathbb{C}^{n}$. The complex variable
$\imath=\sqrt{-1}$. The notation
$\mathbb{X}^{n}_{\Lambda}:\\{x\in\mathbb{X}^{n}\setminus\Lambda\\}$, where
$\Lambda$ denotes a finite number set (typically singularities) in
$\mathbb{X}^{n}$ ($\mathbb{X}^{n}=\\{\mathbb{R}^{n},\mathbb{C}^{n}\\}$). The
set of stable rational functions with bounded $\infty$-norm along
$\imath\mathbb{R}$, is denoted $\mathcal{RH}_{\infty}$. Similarly,
$\mathcal{RL}_{\infty}$ denotes the same set but for both stable and unstable
functions. Identity and null matrices of dimension $p$ read $I_{p}$ and
$0_{p}$. The Laplace variable is denoted $s\in\mathbb{C}$. Here we consider
the following multi-input multi-output (MIMO) linear time invariant (LTI)
continuous-time dynamical systems realisations (with $x(0)=0$):
$E\dot{x}(t)=Ax(t)+Bu(t),\quad y(t)=Cx(t),$ (1a) $\dot{x}(t)=Ax(t)+Bu(t),\quad
y(t)=Cx(t)+Dx(t),$ (1b)
$\left\\{\begin{array}[]{rcl}M\dot{x}(t)&=&(J-R)Qx(t)+(G-P)u(t)\\\
y(t)&=&(G+P)^{\top}Qx(t)+(N+S)u(t)\end{array}\right.,$ (1c)
where $x(t)\in\mathbb{R}^{n}$ and $u(t),y(t)\in\mathbb{R}^{m}$ are vector-
valued functions denoting the internal variables, input and output of the
system. In the standard descriptor (1a) and non-descriptor (1b) forms, we
consider constant matrices $E,A\in\mathbb{R}^{n\times n}$,
$B,C^{\top}\in\mathbb{R}^{n\times m}$ and $D\in\mathbb{R}^{m\times m}$. When
considering the port-Hamiltonian form (1c), $M,J,R,Q\in\mathbb{R}^{n\times
n}$, $G,P\in\mathbb{R}^{n\times m}$ and $N,S\in\mathbb{R}^{m\times m}$. For
brevity, (1a) and (1b) are denoted $\bm{\Sigma}:=(E,A,B,C,0_{m})$ and
$\bm{\Sigma}:=(I_{n},A,B,C,D)$ respectively. The pH form (1c) is shortly
denoted $\bm{\Sigma}_{\text{pH}}:=(M,Q,J,R,G,P,N,S)$. By introducing the co-
energy variable $e(t)=Qx(t)$, (1c) boils down to
$M\dot{x}(t)=(J-R)e(t)+(G-P)u(t)$ and $y(t)=(G+P)^{\top}e(t)+(N+S)u(t)$. The
latter is the so-called _co-energy pH form_ and is of specific meaning in the
computation of the Hamiltonian. In each case, we define the associated
transfer functions as
$\mathbf{H}:\mathbb{C}_{\Lambda}\mapsto\mathbb{C}^{m\times m}$, where
$\mathbf{H}(s)=C(sE-A)^{-1}B$ for (1a), $\mathbf{H}(s)=C(sI-A)^{-1}B+D$ for
(1b) and $\mathbf{H}(s)=(G+P)^{\top}Q(sM-(J-R)Q)^{-1}(G-P)+(N+S)$ for
(1c)111Here $\Lambda$ denotes the singularities being the eigenvalues of
$(A,E)$ pencil in (1a), of $A$ in (1b) and of $((J-R)Q,M)$ in (1c).. On the
basis of $\mathbf{H}$, let us denote the spectral density as
$\bm{\Phi}_{\mathbf{H}}(s):=\mathbf{H}(s)+\mathbf{H}^{\top}(-s)$ and let us
remind the following definitions, necessary to characterise a pH system.
###### Definition 1 (Positive realness)
For all $\omega\in\mathbb{R}$, the rational transfer $\mathbf{H}(s)$ is called
strictly positive-real if $\bm{\Phi}_{\mathbf{H}}(\imath\omega)\succ 0$ and
positive-real if $\bm{\Phi}_{\mathbf{H}}(\imath\omega)\succeq 0$.
###### Definition 2 (Stability)
The rational transfer function $\mathbf{H}(s)$ is called asymptotically stable
if its singularities $\Lambda$ are in the open left-half plane, and called
stable it its singularities $\Lambda$ are in the closed left-half plane with
any pole occurring on $\imath\mathbb{R}$ being not repeated.
###### Definition 3 (Passivity)
The rational transfer function $\mathbf{H}(s)$ is called strictly passive if
it is strictly positive real and asymptotically stable, and stable if positive
real and stable.
### 1.3 Contribution statement and paper organisation
This note is grounded on the LF extended by [1] to identify pH-ROM as defined
in (1c). The contributions are twofold: _(i)_ first, to propose both a
methodological and numerical adjustment from [1]’s algorithm to identify pH
models where the original system is passive but not strictly222One should also
point that this is also treated in [11] through a model-based approach via
what authors call the _regular_ and _singular_ cases. _(ii)_ and second, to
apply the proposed process to a highly dimensional ($n\gg 10^{4}$) and MIMO
system, embedding a rich dynamic: the 2D wave equation, which complexity is
far higher than standard benchmarks. The paper is organised as follows: in §2,
the proposed data-driven pH-ROM construction method for non-strictly passive
systems is presented, using an adaptation of the data-driven LF of [1]. The
approach is illustrated and validated through a numerical example resulting
from a complex 2D wave equation in §3. Conclusions and perspectives are drawn
in §4.
## 2 Port-Hamiltonian identification in the Loewner framework
We are interested in identifying a MIMO ROM preserving the pH structure of
$\bm{\Sigma}_{\text{pH}}$, using a data-driven framework. To do so, we follow
the approach of [1] which extends the LF originally presented in [6]. The
latter is first reminded in §2.1, while the former is presented in §2.2. The
proposed algorithm allowing to cope with non-strictly passive systems, is
detailed in §2.3.
### 2.1 Loewner framework preliminaries
The LF offers tools for the reduction, approximation and identification of
dynamical systems based on frequency-domain data. Let us denote as the _right
and left data_ the following sets (where $j=1,\dots,k$ and $i=1,\dots,q$):
$\\{\lambda_{j},\mathbf{r}_{j},\mathbf{w}_{j}\\}\text{ and
}\\{\mu_{i},\mathbf{l}_{i}^{\top},\mathbf{v}_{i}^{\top}\\},$ (2)
where $\lambda_{j}\in\mathbb{C}$ and $\mu_{i}\in\mathbb{C}$ are the right and
left interpolation points. Then, $\mathbf{r}_{j}\in\mathbb{C}^{m\times 1}$ and
$\mathbf{l}_{i}^{\top}\in\mathbb{C}^{1\times m}$ are the right and left
tangential directions. Both points and directions lead to the right
$\mathbf{H}(\lambda_{j})\mathbf{r}_{j}=\mathbf{w}_{j}\in\mathbb{C}^{m\times
1}$ and left
$\mathbf{l}_{i}^{\top}\mathbf{H}(\mu_{i})=\mathbf{v}_{i}^{\top}\in\mathbb{C}^{1\times
m}$ tangential responses. $\mathbf{H}(s_{k})$ is the evaluation of the high
dimensional pH-FOM at point $s_{k}\in\mathbb{C}$. Based on (2), the LF seeks
for $\hat{\bm{\Sigma}}:(\hat{E},\hat{A},\hat{B},\hat{C},0_{m})$, whose
transfer function $\mathbf{\hat{H}}(s)$ satisfies tangential interpolatory
conditions $\mathbf{\hat{H}}(\lambda_{j})\mathbf{r}_{j}=\mathbf{w}_{j}$ and
$\mathbf{l}_{i}^{\top}\mathbf{\hat{H}}(\mu_{i})=\mathbf{v}_{i}^{\top}$. By
using the matrix formulation, the right data read
$\left\\{\begin{array}[]{rcl}\Lambda&=&~{}\textbf{diag}~{}[\lambda_{1},\dots,\lambda_{k}]\in\mathbb{C}^{k\times
k},\\\
\mathbf{R}&=&\begin{bmatrix}\mathbf{r}_{1}&\mathbf{r}_{2}&\dots&\mathbf{r}_{k}\end{bmatrix}\in\mathbb{C}^{m\times
k}\\\
\mathbf{W}&=&\begin{bmatrix}\mathbf{w}_{1}&\mathbf{w}_{2}&\dots&\mathbf{w}_{k}\end{bmatrix}\in\mathbb{C}^{m\times
k}\end{array}\right.,$ (3)
and the left data read
$\left\\{\begin{array}[]{rcl}\mathbf{M}&=&~{}\textbf{diag}~{}[\mu_{1},\dots,\mu_{q}]\in\mathbb{C}^{q\times
q}\\\
\mathbf{L}^{\top}&=&\begin{bmatrix}\mathbf{l}_{1}&\mathbf{l}_{2}&\dots&\mathbf{l}_{q}\end{bmatrix}\in\mathbb{C}^{m\times
q}\\\
\mathbf{V}^{\top}&=&\begin{bmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\dots&\mathbf{v}_{q}\end{bmatrix}\in\mathbb{C}^{m\times
q}\end{array}\right..$ (4)
Then, by defining the $i,j$-th entry of the Loewner and shifted Loewner
matrices as
$(\mathbb{L})_{ij}=\dfrac{\mathbf{v}_{i}^{\top}\mathbf{r}_{j}-\mathbf{l}_{i}^{\top}\mathbf{w}_{j}}{\mu_{i}-\lambda_{j}}\text{
and
}(\mathbb{M})_{ij}=\dfrac{\mu_{i}\mathbf{v}_{i}^{\top}\mathbf{r}_{j}-\mathbf{l}_{i}^{\top}\mathbf{w}_{j}\lambda_{j}}{\mu_{i}-\lambda_{j}},$
(5)
the resulting system realization
$\bm{\hat{\Sigma}}:=(\hat{E},\hat{A},\hat{B},\hat{C},0_{m})=(-\mathbb{L},-\mathbb{M},\mathbf{V},\mathbf{W},0_{m})$
which transfer function
$\mathbf{\hat{H}}(s)=\mathbf{W}(\mathbb{M}-s\mathbb{L})^{-1}\mathbf{V}$
(tangentially) interpolates the _data_. It follows that Loewner matrices
satisfy the Sylvester equations
$\mathbf{M}\mathbb{L}-\mathbb{L}\bm{\Lambda}=\mathbf{V}\mathbf{R}-\mathbf{L}\mathbf{W}$
and
$\mathbf{M}\mathbb{M}-\mathbb{M}\bm{\Lambda}=\mathbf{M}\mathbf{V}\mathbf{R}-\mathbf{L}\mathbf{W}\bm{\Lambda}$.
If data have been generated by a linear rational model,
$\forall\xi\in\mathbb{C}\setminus\Lambda$, the rational order
$r=\textbf{rank}(\xi\mathbb{L}-\mathbb{M})=\textbf{rank}([\mathbb{L},\mathbb{M}])=\textbf{rank}([\mathbb{L}^{H},\mathbb{M}^{H}]^{H})$
recovers the the minimal realisation of the generating system, as well as its
McMillan degree $\nu=\textbf{rank}(\mathbb{L})$. These features make this
approach central in the realisation theory (see [7, 8] for a recent
overviews).
### 2.2 Loewner framework with strict passivity
#### 2.2.1 General ideas and assumptions
Applying the LF to _data_ generated by a passive system $\mathbf{H}$ do not
necessarily lead to a passive transfer $\mathbf{\hat{H}}$. This is solved in
[1] by through specific _right and left data_ selection. This result is
recalled here together with the main (limiting) assumptions.
###### Assumption 1 (Strictly passive)
In [1], authors assume that the system generating the data to be strictly
passive, implying proper transfer matrix $\mathbf{H}$ where singularities are
not on the imaginary axis or at infinity.
###### Assumption 2 (Stability)
In [1], authors assume that model $\mathbf{\hat{H}}$ (realisation
$\bm{\hat{\Sigma}}$) obtained after a first identification in the LF leads to
a stable pencil $(\mathbb{M},\mathbb{L})=(\hat{A},\hat{E})$, _i.e._
$\Lambda\in\mathbb{C}_{-}$.
#### 2.2.2 Procedure as given in [1]
First, let $\mathbf{\hat{H}}$ be identified by the LF, on the basis of a real
and strictly passive transfer function $\mathbf{H}$, where the $D$-term is
removed to avoid rank deflecting $\hat{E}=\mathbb{L}$ matrix. It results in
$\mathbf{\hat{H}}$ where McMillan degree $\nu$ is equal to the (minimal)
realisation order $r$, since no polynomial term appear (because of the strict
passivity and $D$-term removal). Note that $r$ may be automatically selected
by the rank revealing factorisation of the LF or be chosen smaller. This
identified model realisation $\bm{\hat{\Sigma}}$ is now used to estimate the
associated spectral zeros and directions pairs, denoted
$(\xi_{j},\mathbf{x}_{j})$ such that
$\bm{\Phi}_{\mathbf{\hat{H}}}(\xi_{j})\mathbf{x}_{j}=0$. This pair is computed
by solving the following low order generalized eigenvalue problem (see [12,
1]):
$\begin{bmatrix}0&\hat{A}&\hat{B}\\\ \hat{A}^{\top}&0&\hat{C}^{\top}\\\
\hat{B}^{\top}&\hat{C}&D+D^{\top}\end{bmatrix}\begin{bmatrix}p_{j}\\\ q_{j}\\\
\mathbf{x}_{j}\end{bmatrix}=\xi_{j}\begin{bmatrix}0&\hat{E}&0\\\
-\hat{E}^{\top}&0&0\\\ 0&0&0\end{bmatrix}\begin{bmatrix}p_{j}\\\ q_{j}\\\
\mathbf{x}_{j}\end{bmatrix}.$ (6)
According to Assumptions 1 and 2, this eigen-problem has $r$ zeros in the open
right half-plane, $r$ zeros in the open left half-plane and has no zeros on
the imaginary axis. By selecting the _right and left strictly passive data_
data as ($i,j=1,\dots,r=k=q$, $\lambda_{j}\leftarrow\xi_{j}$ and
$\mathbf{r}_{j}\leftarrow\mathbf{x}_{i}$),
$\\{\lambda_{j},\mathbf{r}_{j},\mathbf{w}_{j}\\}\text{ and
}\\{-\overline{\lambda}_{i},\mathbf{r}_{i}^{H},-\mathbf{w}_{i}^{H}\\},$ (7)
one gets, $\mathbf{M}=-\bm{\Lambda}^{H}$, $\mathbf{L}=\mathbf{R}$ and
$\mathbf{V}=-\mathbf{W}^{H}$. Therefore, by construction, one obtains an
Hermitian $\mathbb{L}\in\mathbb{C}^{r\times r}$ and a skew symmetric
$\mathbb{M}\in\mathbb{C}^{r\times r}$ matrix (5). By setting,
$\mathbf{H}(\infty)=D$ (which may be estimated by sampling in very high
frequency), one recovers an $m\times m$ real transfer function
$\mathbf{\hat{H}}$333Real matrices are obtained if data are sampled with
complex conjugate frequencies and by applying a unitary projection [7, 8].. As
$\mathbb{L}\succ 0$, one may apply the Cholesky decomposition
$\mathbb{L}=T^{\top}T$. Then the _normalized pH model_ is obtained as
$\bm{\Sigma}_{\text{n-pH}}:=(I_{n},T\hat{A}T^{-1},T\hat{B},\hat{C}T^{-1},D)$,
with form (1b). By defining [13]
$\mathbf{S}:=\left[\begin{array}[]{cc}-T\hat{A}T^{-1}&-T\hat{B}\\\
\hat{C}T^{-1}&D\end{array}\right],$ (8)
one obtains the equivalent pH-form (1c) by solving
$\left[\begin{array}[]{cc}-J&-G\\\
G^{\top}&N\end{array}\right]:=\dfrac{\mathbf{S}-\mathbf{S}^{\top}}{2}\text{
and }\left[\begin{array}[]{cc}R&P\\\
P^{\top}&S\end{array}\right]:=\dfrac{\mathbf{S}+\mathbf{S}^{\top}}{2}.$ (9)
However, this approach suffers from two limitations. The first one stands in
the assumption that the original model generating the data should be
_strictly_ passive, and thus is cannot be applied to _non-strictly_ passive
systems. The second one is more numerical: in practice Loewner pencil
$(\mathbb{M},\mathbb{L})$ of stable functions $\mathbf{H}$ may not be
necessarily stable. Next, we propose two steps to overcome these limitations.
### 2.3 Loewner framework with non-strict passivity and stability
#### 2.3.1 Proposed modified algorithm
Based on Algorithm 1 and 2 of [1], we suggest the following Algorithm 1, to
identify pH model for non-strictly passive systems.
Algorithm 1 Data-driven normalized pH model construction of non-strictly
passive system
1:$\\{\lambda_{j}^{0},\mathbf{r}_{j}^{0},\mathbf{w}_{j}^{0}\\}$,
$\\{\mu_{i}^{0},\mathbf{l}_{i}^{0\top},\mathbf{v}_{i}^{0\top}\\}$, shift
$D_{s}$ such that $D_{s}^{\top}+D_{s}\succ 0$, objective order $r$.
2:$\bm{\hat{\Sigma}}_{\text{pH}}:=(M,Q,J,R,G,P,N,S)$ as in (1c) and
$\bm{\hat{\Sigma}}_{\text{pH}}$ ensuring interpolatory conditions.
3:Shift the data (2) with $D_{s}$ as
$\mathbf{w}_{j}\leftarrow\mathbf{w}_{j}^{0}+D_{s}$ and
$\mathbf{v}_{i}\leftarrow\mathbf{v}_{i}^{0}+D_{s}$ $\triangleright$ New step
4:Construct the $r$-th order Loewner interpolant
$\bm{\hat{\Sigma}}:=(-\mathbb{L},-\mathbb{M},\mathbf{V},\mathbf{W},0_{m})$ as
in Section 2.1
5:Compute the equivalent formulation
$\bm{\hat{\Sigma}}:=(I_{r},\hat{A},\hat{B},\hat{C},D_{s})$ with transfer
$\mathbf{\hat{H}}$
6:Compute projection $P_{\infty}(\bm{\hat{\Sigma}})$ (or
$P_{\infty}(\mathbf{\hat{H}})$) $\triangleright$ New step
7:Compute spectral zeros of $P_{\infty}(\bm{\hat{\Sigma}})$ as in (6)
8:Set $\lambda_{j}\leftarrow\xi_{j}$,
$\mathbf{r}_{j}\leftarrow\mathbf{x}_{i}$,
$\mathbf{w}_{j}=\mathbf{\hat{H}}(\lambda_{j})\mathbf{r}_{j}$ and (7)
9:Construct $\mathbb{L}$ and $\mathbb{M}$ as in (5)
10:Construct $\mathbb{M}\leftarrow\mathbb{M}-\mathbf{L}D_{s}\mathbf{R}$,
$\mathbf{V}\leftarrow\mathbf{V}-\mathbf{L}D_{s}$ and
$\mathbf{W}\leftarrow\mathbf{W}-D_{s}\mathbf{R}$.
11:Compute Chloesky decomposition $\mathbb{L}=T^{\top}T$
12:Construct
$\bm{\hat{\Sigma}}_{\text{n-pH}}:=(I_{n},T\hat{A}T^{-1},T\hat{B},\hat{C}T^{-1},D_{s})$
13:Construct $\bm{\hat{\Sigma}}_{\text{pH}}:=(M,Q,J,R,G,P,N,S)$ using (9)
14:Set $S\leftarrow S-D_{s}$ $\triangleright$ New step
The main modifications from [1] are listed hereafter. _(i)_ ”Step 1”, we
suggest shifting the data with a positive scalar to translate the Nyquist
response on the right hand side to ensure positive realness. This result in a
data that are now strictly passive. _(ii)_ ”Step 4”, as the Loewner does not
ensures stability, we suggest a projection of the rational ROM
$\mathbf{\hat{H}}$ with realisation $\bm{\hat{\Sigma}}$ onto the
$\mathcal{RH}_{\infty}$ space, following [14]. This leads to a stable
$P_{\infty}(\mathbf{\hat{H}})$ model. _(iii)_ ”Step 12”, based on the
normalised realisation $\bm{\hat{\Sigma}}_{\text{n-Ph}}$, recover the original
non-strictly passive model, by simply applying $S\leftarrow S-D_{s}$ after
solving (9), leading to the pH-ROM fitting the original data.
#### 2.3.2 Comments
##### (i) and (iii)’s bullets
should the original model, and therefore associated data, be non strictly
passive, but only passive. This typically occurs when no direct feed-through
term exist. As a consequence, the resulting spectral zeros exhibit zeros on
the imaginary axis. The first and last bullets address this point. One simply
shifts the original problem to apply the strict-passive approach of [1].
##### (ii)’s bullet
the rational model obtained through the LF denoted $\mathbf{\hat{H}}$ may
present unstable singularities. Therefore we suggest a _post stabilisation_
using the procedure presented in [14]. This latter consists in projecting the
rational model $\mathbf{\hat{H}}\in\mathcal{RL}_{\infty}$ onto its closest
stable subset $\mathcal{RH}_{\infty}$, here using the
$\mathcal{H}_{\infty}$-norm, leading to a stable model. Mathematically, and as
exposed in details in [14], given a rational model
$\mathbf{\hat{H}}\in\mathcal{RL}_{\infty}$ equipped with realisation
$\bm{\hat{\Sigma}}$, one seeks
$P_{\infty}(\mathbf{\hat{H}})\in\mathcal{RH}_{\infty}$ such that,
$P_{\infty}(\mathbf{\hat{H}})=\arg\inf_{\mathbf{G}\in\mathcal{RH}_{\infty}}||\mathbf{\hat{H}}-\mathbf{G}||_{\mathcal{L}_{\infty}}.$
(10)
Proof and procedure to obtain $P_{\infty}(\mathbf{\hat{H}})$ are detailed in
[14]. The key steps consists in performing the stable and unstable part
separation, then solving two reduced order Lyapunov equations. Applying this
post-treatment to the Loewner-based approximate hopefully preserves the
accuracy and interpolatory properties. Note that this step may also be
addressed with [13].
##### Interpolatory properties
At step 3, $\mathbf{\hat{H}}$ interpolates the _shifted data_ of step 1. At
step 4, $\mathbf{\hat{H}}$ may not interpolate exactly these data due to the
projection, but should likely do so if original system is stable. Then, at
step 5 one should recover exactly $r$ strictly positive spectral zeros.
Accordingly, the Cholesky decomposition at step 9 is possible as
$\mathbb{L}\succ 0$, thus at step 10 and 11, the system interpolates the
_spectral zeros data_ of step 6, associated to the model $\mathbf{\hat{H}}$
constructed from the _shifted data_. At step 10, the associated transfer
function thus ensures
$\mathbf{\hat{H}}_{\text{n-pH}}(\lambda_{j}^{0})\mathbf{r}_{j}^{0}=\mathbf{w}^{0}_{j}+D_{s}$.
Step 12 shifts back the model so that $\mathbf{\hat{H}}_{\text{pH}}$
tangentially interpolates the original data and is passive.
## 3 Numerical use-case: the 2D wave equation
### 3.1 Model description
#### 3.1.1 Port-Hamiltonian formulation
Let us consider the vertical deflection from equilibrium $w$ of a 2D membrane
$\Omega\subset\mathbb{R}^{2}$. Denoting $\rho$ the mass density and $T$ the
Young’s modulus of the membrane, a positive-definite symmetric tensor, leads
to the damped wave equation given as [15] ($t\geq 0,\,x\in\Omega$)
$\rho(x)\frac{\partial^{2}}{\partial
t^{2}}w(t,x)+\varepsilon(x)\frac{\partial}{\partial t}w(t,x)-{\rm
div}\left(T(x)\cdot\mathbf{grad}\left(w(t,x)\right)\right)=0$
where $\varepsilon$ is a positive damping parameter, together with Neumann
boundary control
$\left(T(x)\cdot\mathbf{grad}\left(w(t,x)\right)\right)\cdot\mathbf{n}=u_{\partial}(t,x)$,
where $\mathbf{n}$ is the outward normal to $\Omega$. The Hamiltonian is the
total mechanical energy, given as the sum of potential and kinetic energies
$\mathcal{H}(t):=\frac{1}{2}\int_{\Omega}\left(\mathbf{grad}\left(w(t,x)\right)\right)^{\top}\cdot
T(x)\cdot\mathbf{grad}\left(w(t,x)\right){\rm d}x\\\
+\frac{1}{2}\int_{\Omega}\rho(x)\left(\frac{\partial}{\partial
t}w(t,x)\right)^{2}{\rm d}x.$ (11)
Taking the strain $\bm{\alpha}_{q}:=\mathbf{grad}\left(w\right)$ and the
linear momentum $\alpha_{p}:=\frac{\partial}{\partial t}w$ as energy
variables, the Hamiltonian rewrites
$\mathcal{H}(t)=\mathcal{H}(\bm{\alpha}_{q}(t,\cdot),\alpha_{p}(t,\cdot))\\\
=\frac{1}{2}\int_{\Omega}\left(\bm{\alpha}_{q}(t,x)\right)^{\top}\cdot
T(x)\cdot\bm{\alpha}_{q}(t,x){\rm
d}x+\frac{1}{2}\int_{\Omega}\frac{\alpha_{p}^{2}(t,x)}{\rho(x)}{\rm d}x.$ (12)
The co-energy variables are by definition the variational derivatives of the
Hamiltonian
$\mathbf{e}_{q}:=\delta_{\bm{\alpha}_{q}}\mathcal{H}=T\cdot\bm{\alpha}_{q}$
the stress, and
$e_{p}:=\delta_{\alpha_{p}}\mathcal{H}=\frac{1}{\rho}\alpha_{p}$, the
velocity. These equalities are the constitutive relations which close the
dynamical system. Thanks to these variables, the wave equation writes as a
port-Hamiltonian system
$\begin{pmatrix}\frac{\partial}{\partial t}\bm{\alpha}_{q}\\\
\frac{\partial}{\partial
t}\alpha_{p}\end{pmatrix}=\begin{bmatrix}0&\mathbf{grad}\\\ {\rm
div}&-\varepsilon\end{bmatrix}\begin{pmatrix}\mathbf{e}_{q}\\\
e_{p}\end{pmatrix},$
In addition we denote inputs and outputs as
$u_{\partial}=\mathbf{e}_{q}\cdot\mathbf{n}\mid_{\partial\Omega}$ and
$y_{\partial}=e_{p}\mid_{\partial\Omega}$. The power balance satisfied by the
Hamiltonian is
$\frac{\rm d}{{\rm d}t}\mathcal{H}=\langle
u_{\partial},y_{\partial}\rangle_{\partial\Omega}-\int_{\Omega}\varepsilon|e_{p}|^{2}\leq\langle
u_{\partial},y_{\partial}\rangle_{\partial\Omega}\,,$ (13)
proving passivity. To get rid of the algebraic constraints induced by the
constitutive relations, one rewrites the port-Hamiltonian system as
$\begin{bmatrix}T^{-1}&0\\\
0&\rho\end{bmatrix}\begin{pmatrix}\frac{\partial}{\partial t}\mathbf{e}_{q}\\\
\frac{\partial}{\partial
t}e_{p}\end{pmatrix}=\begin{bmatrix}0&\mathbf{grad}\\\ {\rm
div}&-\varepsilon\end{bmatrix}\begin{pmatrix}\mathbf{e}_{q}\\\
e_{p}\end{pmatrix},\;\left\\{\begin{array}[]{rcl}u_{\partial}&=&\mathbf{e}_{q}\cdot\mathbf{n},\\\
y_{\partial}&=&e_{p}\mid_{\partial\Omega},\end{array}\right.$
also known as the _co-energy formulation_.
#### 3.1.2 Structure-preserving discretization
Let $\bm{\varphi}_{q}$, $\varphi_{p}$ and $\psi$ be vector-valued, scalar-
valued and boundary scalar-valued test functions respectively. The weak
formulation reads
$\left\\{\begin{array}[]{rcl}\displaystyle\int_{\Omega}\bm{\varphi}_{q}\cdot
T^{-1}\cdot\frac{\partial}{\partial
t}\mathbf{e}_{q}&=&\displaystyle\int_{\Omega}\bm{\varphi}_{q}\cdot\mathbf{grad}\left(e_{p}\right),\\\
\displaystyle\int_{\Omega}\varphi_{p}\rho\frac{\partial}{\partial
t}e_{p}&=&\displaystyle\int_{\Omega}\varphi_{p}{\rm
div}\left(\mathbf{e}_{q}\right)-\int_{\Omega}\varphi_{p}\varepsilon e_{p},\\\
\displaystyle\int_{\partial\Omega}\psi
y_{\partial}&=&\displaystyle\int_{\partial\Omega}\psi
e_{p}.\end{array}\right.$
The integration by parts of the second leads to
($u_{\partial}=\mathbf{e}_{q}\cdot\mathbf{n}$)
$\left\\{\begin{array}[]{rcl}\displaystyle\int_{\Omega}\bm{\varphi}_{q}\cdot
T^{-1}\cdot\frac{\partial}{\partial
t}\mathbf{e}_{q}&=&\displaystyle\int_{\Omega}\bm{\varphi}_{q}\cdot\mathbf{grad}\left(e_{p}\right),\\\
\displaystyle\int_{\Omega}\varphi_{p}\rho\frac{\partial}{\partial
t}e_{p}&=&\displaystyle-\int_{\Omega}\mathbf{grad}\left(\varphi_{p}\right)\cdot\mathbf{e}_{q}+\int_{\partial\Omega}\varphi_{p}u_{\partial}\\\
&&\qquad\qquad-\int_{\Omega}\varphi_{p}\varepsilon e_{p},\\\
\displaystyle\int_{\partial\Omega}\psi
y_{\partial}&=&\displaystyle\int_{\partial\Omega}\psi
e_{p}.\end{array}\right.$
Let $(\bm{\varphi}_{q}^{i})_{1\leq i\leq N_{q}}$, $(\varphi_{p}^{j})_{1\leq
j\leq N_{p}}$ and $(\psi^{k})_{1\leq k\leq N_{\partial}}$ be finite element
families for $q$-type, $p$-type and boundary-type variables. Variables are
approximated in their respective finite element family
$\mathbf{e}_{q}^{d}(t,x):=\sum_{i=1}^{N_{q}}e_{q}^{i}(t)\bm{\varphi}_{q}^{i}(x),\qquad
e_{p}^{d}(t,x):=\sum_{j=1}^{N_{p}}e_{p}^{j}(t)\varphi_{p}^{j}(x),$
$u_{\partial}^{d}(t,x):=\sum_{k=1}^{N_{\partial}}u_{\partial}^{k}(t)\psi^{k}(x),\qquad
y_{\partial}^{d}(t,x):=\sum_{k=1}^{N_{\partial}}y_{\partial}^{k}(t)\psi^{k}(x).$
Denoting $\underline{\star}$ the (time-varying) vector of coordinates of the
discretisation $\star^{d}$ of $\star$ in its respective finite element family,
the discrete system reads
$\small\underset{M}{\underbrace{\begin{bmatrix}M_{q}&0&0\\\ 0&M_{p}&0\\\
0&0&M_{\partial}\end{bmatrix}}}\begin{pmatrix}\frac{\rm d}{{\rm
d}t}\underline{e_{q}}(t)\\\ \frac{\rm d}{{\rm d}t}\underline{e_{p}}(t)\\\ \
-\underline{y_{\partial}}(t)\end{pmatrix}=\underset{J-R}{\underbrace{\begin{bmatrix}0&G&0\\\
\ -G^{\top}&-M_{\varepsilon}&B\\\
0&-B^{\top}&0\end{bmatrix}}}\begin{pmatrix}\underline{e_{q}}(t)\\\
\underline{e_{p}}(t)\\\ \underline{u_{\partial}}(t)\end{pmatrix}\normalsize$
where $(M_{q})_{ij}:=\int_{\Omega}\bm{\varphi}_{q}^{i}\cdot
T^{-1}\cdot\bm{\varphi}_{q}^{j}$,
$(M_{p})_{ij}:=\int_{\Omega}\varphi_{p}^{i}\rho\varphi_{p}^{j}$,
$(M_{\varepsilon})_{ij}:=\int_{\Omega}\varphi_{p}^{i}\varepsilon\varphi_{p}^{j}$,
$(M_{\partial})_{ij}:=\int_{\partial\Omega}\psi^{i}\psi^{j}$, and
$(B)_{jk}:=\int_{\partial\Omega}\varphi_{p}^{j}\mid_{\partial\Omega}\,\psi^{k}$,
$(G)_{ij}:=\int_{\Omega}\bm{\varphi}_{q}^{i}\cdot\mathbf{grad}\left(\varphi_{p}^{j}\right)$.
By definition, the discrete Hamiltonian is equal to the continuous Hamiltonian
evaluated in the approximated variables. As we are working with the co-energy
formulation, a first step is to restate the Hamiltonian in terms of co-energy
variables, namely:
$\mathcal{H}=\frac{1}{2}\int_{\Omega}\mathbf{e}_{q}\cdot
T^{-1}\cdot\mathbf{e}_{q}+\frac{1}{2}\int_{\Omega}\rho(e_{p})^{2}.$
Then, the discrete Hamiltonian is defined as
$\mathcal{H}^{d}:=\frac{1}{2}\int_{\Omega}\mathbf{e}_{q}^{d}\cdot
T^{-1}\cdot\mathbf{e}_{q}^{d}+\frac{1}{2}\int_{\Omega}\rho(e_{p}^{d})^{2}.$
After straightforward computations, it comes
$\mathcal{H}^{d}(t)=\frac{1}{2}\underline{e_{q}}(t)^{\top}M_{q}\underline{e_{q}}(t)+\frac{1}{2}\underline{e_{p}}(t)^{\top}M_{p}\underline{e_{p}}(t),$
and the _discrete_ power balance follows
$\displaystyle\frac{\rm d}{{\rm d}t}\mathcal{H}^{d}(t)$ $\displaystyle=$
$\displaystyle\underline{u_{\partial}}(t)^{\top}M_{\partial}\underline{y_{\partial}}(t)-\underline{e_{p}}(t)^{\top}M_{\varepsilon}\underline{e_{p}}(t)$
$\displaystyle\leq$
$\displaystyle\underline{u_{\partial}}(t)^{\top}M_{\partial}\underline{y_{\partial}}(t)\,,$
mimicking (13) exactly at the discrete level. The pH-FOM is thus given by the
a realization $\bm{\Sigma}_{\textbf{pH}}$ in the form (1c). The convergence of
the numerical method is assessed in [16], where the optimal selection of
families of Finite Elements is proved. The objective is to use these matrices
and the associated transfer function $\mathbf{H}$ to generate _data_ , through
the dedicated SCRIMP simulator444See https://g-haine.github.io/scrimp/,
presented in [17]. These data shall serve the construction of a pH-ROM, as
explained in §2.
### 3.2 pH-ROM identification
The considered model $\mathbf{H}$ is a 2D wave equation, on an L-shaped domain
$\Omega$. The resulting discretized pH model is equipped with a realisation
$\bm{\Sigma}_{\text{pH}}$ in pH-form as in (1c). It has the following
characteristics: $n=63,409$ and $m=604$. For illustration, we will now
consider to sub-cases. First, _(i)_ the SISO case: $m=1$, where we consider
the first input / output pair only and second, _(ii)_ the MIMO case: $m=3$,
where we consider the following input / output index pairs $\\{1,2,496\\}$
only. Notice that as the system is collocated, transfers one and two are
highly close while the third one is far away and will thus be very different.
The model is stable, poorly damped, but not strictly passive, thus spectral
zeros on $\imath\mathbb{R}$ should unlikely occur, resulting in a non verified
Assumption 1. In what follows, we denote as:
* •
_Data_ , the original system presented in §3-A-B sampled along
$\imath\omega_{l}$, where $\omega_{l}$ are $l=1,\dots,300$ logarithmically
spaced values between $10^{-1}$ and $10^{3.5}$ rad/s.
* •
_Loewner_ , the dynamical model obtained with the standard LF, in the form
(1a).
* •
_pH-Loewner_ , the dynamical model obtained with the Algorithm 1 (input data
being closed conjugated from _data_ and $D_{s}=1$), in the form (1c).
In each case, we also use the denomination _shifted_ to point the data or
model shifted and with _post-stability enforcement_ , to ensure strict
dissipativity.
#### 3.2.1 SISO case: process illustration
first, Figure 1 presents the frequency response of the original _data_ ,
compared to the (non-dissipative and non-stable) _Loewner_ model and the _pH-
Loewner_ one, illustrating the nice restitution of the frequency response in
both cases. Here, the _pH-Loewner_ is passive and embeds the expected pH-
structure.
Figure 1: Frequency response of the original data, the Loewner and the pH-
Loewner models.
Figure 2 (3) show the (zoomed) spectral zeros. The _Loewner_ model has many
zeros along or close to the imaginary axis ($\bm{\times}$). Indeed, between
the real band $[-1,1]\times 10^{-10}$, we count 4 zeros and between
$[-1,1]\times 10^{-9}$, 12. This is an issue for selecting the positive
interpolation points. This problem is solved by the proposed algorithm
modification, thanks to both _post-stability enforcement_ and _data-shift_.
Indeed, the _pH-Loewner (shifted)_ shows zeros far from this limit ($\bm{+}$).
Then, after applying the shift-back, one recovers the sought _pH-Loewner_
model ($\bullet$), where spectral zeros are back on the imaginary axis. Note
that without the proposed algorithm adjustments, no solution can be found as
the $\mathbb{L}$ matrix is not positive definite and Cholesky decomposition is
impossible. A reduced order model can be obtained if at step 2 of Algorithm 1
one selects a $r<154$. In the presented figures, the order selection was
performed by the rank revealing decomposition of the Loewner matrices.
Figure 2: Spectral zeros response of the Loewner, the Loewner (applied on the
shifted data) and the pH-Loewner models. Figure 3: Spectral zeros response of
the Loewner, the Loewner (applied on the shifted data) and the pH-Loewner
models.
#### 3.2.2 MIMO case
as mentioned, one important challenge in this application in addition to the
complexity of the dynamics of the wave equation, is its large number of inputs
and outputs ($m=604$). So far, applying the above process to this large MIMO
system led to non fully satisfactory results. However, up to $m=10$,
reasonably good approximation have been observed. In Figure 4, we illustrate
the frequency magnitude response for $m=3$, where the first two inputs /
outputs are spatially close, whereas the third one is spatially far from the
others.
Figure 4: Frequency response of the original data, the Loewner and the pH-
Loewner models.
Figure 4 illustrates the fact that diagonal elements (more energetic since the
model is collocated) are well reproduced. Regarding the anti-diagonal ones, a
good restitution is observed on channels (1,2) and (2,1) but not on (1,3),
(2,3), (3,1) and (3,2). Indeed, these transfer are less energetic than the
other and thus not well approximated. This is a point for future
investigations, _e.g._ with a specific treatment of the tangential directions
in (2).
## 4 Conclusions & perspectives
We have shown promising numerical results of a _data-driven_ reduction
technique applied to a 2D wave PDE, modelled as a pHs: it is obtained using
the LF, and more specifically a modification of [1] using the frequency-
responses generated by the matrices provided by the structure-preserving PFEM.
The main modifications to [1] are _(i)_ the data-shift to handle non-strictly
passive models and _(ii)_ the post-stability enforcement, to cope with
numerical issues often encountered when applying the LF. These two steps where
essential to achieve the presented results. Indeed, what is successful is the
number of states that can be drastically reduced (from $n=63,409$ to $n=179$).
However, the collocated input-output pairs have been tried on a SISO case, or
on a MIMO case of small dimension ($m=3,\dots,10$). So far, the MIMO version
remains not fully satisfactory and will be of specific attention in future
researches. Further investigations will also consider handling a larger number
of inputs-outputs and different real world applications555See e.g.
https://algopaul.github.io/PortHamiltonianBenchmarkSystems.jl/, such as
Maxwell equations in 2D or even in 3D, see [18].
## References
* [1] P. Benner, P. Goyal, and P. Van Dooren, “Identification of port-Hamiltonian systems from frequency response data,” Systems & Control Letters, vol. 143, p. 104741, September 2020.
* [2] A. van der Schaft and B. Maschke, “Hamiltonian formulation of distributed-parameter systems with boundary energy flow,” Journal of Geometry and Physics, vol. 42, no. 1-2, pp. 166–194, 2002.
* [3] R. Rashad, F. Califano, A. van der Schaft, and S. Stramigioli, “Twenty years of distributed port-Hamiltonian systems: a literature review,” IMA Journal of Mathematical Control and Information, vol. 37, no. 4, pp. 1400–1422, 2020.
* [4] F. Cardoso-Ribeiro, D. Matignon, and L. Lefèvre, “A partitioned finite-element method for power-preserving discretization of open systems of conservation laws,” IMA J. Mathematical Control and Information, vol. 38, no. 2, pp. 493–533, 2021.
* [5] A. Serhani, D. Matignon, and G. Haine, “A partitioned finite element method for the structure-preserving discretization of damped infinite-dimensional port-Hamiltonian systems with boundary control,” in Geometric Science of Information (F. Nielsen and F. Barbaresco, eds.), vol. 11712 of Lecture Notes in Computer Science, pp. 549–558, Springer, 2019.
* [6] A. J. Mayo and A. C. Antoulas, “A framework for the solution of the generalized realization problem,” Linear Algebra and its Applications, vol. 425, no. 2, pp. 634–662, 2007. Special Issue in honor of Paul Fuhrmann.
* [7] A. C. Antoulas, S. Lefteriu, and A. C. Ionita, Model reduction and approximation theory and algorithms, ch. A tutorial introduction to the Loewner framework for model reduction. SIAM, Philadelphia, P. Benner, A. Cohen, M. Ohlberger and K. Willcox Eds, 2016.
* [8] I. V. Gosea, C. Poussot-Vassal, and A. C. Antoulas, “Data-driven modeling and control of large-scale dynamical systems in the Loewner framework,” Handbook of Numerical Analysis, vol. 23, no. Numerical Control: Part A, pp. 499–530, 2022.
* [9] O. Ghattas and K. Willcox, “Learning physics-based models from data: perspectives from inverse problems and model reduction,” Acta Numerica, pp. 445––554, 2021.
* [10] K. Cherifi and A. Brugnoli, “Application of data-driven realizations to port-Hamiltonian flexible structures,” in IFAC-PapersOnLine, vol. 54, pp. 180–185, 2021.
* [11] T. Breiten and B. Unger, “Passivity preserving model reduction via spectral factorization,” Automatica, vol. 142, p. 110368, 2022.
* [12] J. C. Willems, “Dissipative dynamical systems part II: Linear systems with quadratic supply rates,” Arch. Ration. Mech. Anal., vol. 45, pp. 352–393, 1972.
* [13] V. Mehrmann and P. Van Dooren, “Optimal robustness of port-Hamiltonian systems,” SIAM Journal on Matrix Analysis and Applications, vol. 41, no. 1, pp. 134–151, 2020.
* [14] M. Kohler, “On the closest stable descriptor system in the respective spaces $\mathcal{RH}_{2}$ and $\mathcal{RH}_{\infty}$,” Linear Algebra and its Applications, vol. 443, pp. 34–49, 2014.
* [15] M. Kurula and H. Zwart, “Linear wave systems on n-D spatial domains,” International Journal of Control, vol. 88, no. 5, pp. 1063–1077, 2015.
* [16] G. Haine, D. Matignon, and A. Serhani, “Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled ${N}$-dimensional wave equation as a port-Hamiltonian system,” Int. J. Numer. Anal. Mod., vol. 20, no. 1, pp. 92–133, 2023.
* [17] A. Brugnoli, G. Haine, A. Serhani, and X. Vasseur, “Numerical approximation of port-Hamiltonian systems for hyperbolic or parabolic PDEs with boundary control,” Journal of Applied Mathematics and Physics, vol. 9, pp. 1278–1321, June 2021.
* [18] G. Haine, D. Matignon, and F. Monteghetti, “Structure-preserving discretization of Maxwell’s equations as a port-Hamiltonian system,” in IFAC-PapersOnLine, vol. 55, pp. 424–429, 2022.
|
# Positivity-preserving and entropy-bounded discontinuous Galerkin method for
the chemically reacting, compressible Euler equations. Part I: The one-
dimensional case
Eric J. Ching, Ryan F. Johnson, and Andrew D. Kercher Laboratories for
Computational Physics and Fluid Dynamics, U.S. Naval Research Laboratory, 4555
Overlook Ave SW, Washington, DC 20375
###### Abstract
In this paper, we develop a fully conservative, positivity-preserving, and
entropy-bounded discontinuous Galerkin scheme for simulating the
multicomponent, chemically reacting, compressible Euler equations with complex
thermodynamics. The proposed formulation is an extension of the fully
conservative, high-order numerical method previously developed by Johnson and
Kercher [_J. Comput. Phys._ , 423 (2020), 109826] that maintains pressure
equilibrium between adjacent elements. In this first part of our two-part
paper, we focus on the one-dimensional case. Our methodology is rooted in the
minimum entropy principle satisfied by entropy solutions to the
multicomponent, compressible Euler equations, which was proved by Gouasmi et
al. [_ESAIM: Math. Model. Numer. Anal._ , 54 (2020), 373–389] for nonreacting
flows. We first show that the minimum entropy principle holds in the reacting
case as well. Next, we introduce the ingredients, including a simple linear-
scaling limiter, required for the discrete solution to have nonnegative
species concentrations, positive density, positive pressure, and bounded
entropy. We also discuss how to retain the aforementioned ability to preserve
pressure equilibrium between elements. Operator splitting is employed to
handle stiff chemical reactions. To guarantee discrete satisfaction of the
minimum entropy principle in the reaction step, we develop an entropy-stable
discontinuous Galerkin method based on diagonal-norm summation-by-parts
operators for solving ordinary differential equations. The developed
formulation is used to compute canonical one-dimensional test cases, namely
thermal-bubble advection, multicomponent shock-tube flow, and a moving
hydrogen-oxygen detonation wave with detailed chemistry. We demonstrate that
the developed formulation can achieve optimal high-order convergence in smooth
flows. Furthermore, we find that the enforcement of an entropy bound can
considerably reduce the large-scale nonlinear instabilities that emerge when
only the positivity property is enforced, to an even greater extent than in
the monocomponent, calorically perfect case. Finally, mass, total energy, and
atomic elements are shown to be discretely conserved.
###### keywords:
Discontinuous Galerkin method; Combustion; Detonation; Minimum entropy
principle; Positivity-preserving; Entropy stability; Summation-by-parts
††footnotetext:
DISTRIBUTION STATEMENT A. Approved for public release. Distribution is
unlimited.
## 1 Introduction
The discontinuous Galerkin (DG) method [1, 2, 3, 4, 5] has recently gained
considerable attention in the computational fluid dynamics community [6].
Several desirable properties, such as local conservation, arbitrarily high
order of accuracy on unstructured grids, and suitability for heterogeneous
computing systems, highlight its great potential to accurately and efficiently
simulate complex fluid flows. However, it is well-known that nonlinear
instabilities are easily introduced in underresolved regions and near flow-
field discontinuities. This issue is exacerbated when realistic thermodynamics
(e.g., the thermally perfect gas model) and multispecies chemical reactions
are incorporated. For example, fully conservative schemes (not just the DG
method) are known to generate spurious pressure oscillations in moving
interface problems [7, 8, 9]. To remedy this issue, quasi-conservative methods
are often employed, such as the double-flux scheme [10], in which the equation
of state is recast based on a calorically perfect gas model using frozen,
elementwise-constant auxiliary variables. The double-flux method has been
utilized in a number of studies involving DG simulations of multicomponent
flows [11, 12, 13]. While effective at eliminating the aforementioned pressure
oscillations, the double-flux approach violates energy conservation, which can
be crucial for the prediction of shock speeds and locations, as well as heat
release in combustion processes. As a compromise, Lv and Ihme [12] proposed a
hybrid double-flux strategy wherein a fully conservative method is employed at
shocks.
On the other hand, Johnson and Kercher recently proposed an explicit, fully
conservative, high-order method that does not generate spurious pressure
oscillations in smooth flow regions or across material interfaces when the
temperature is continuous [14]. This is done via (a) exact evaluation of the
thermodynamics and (b) calculation of the inviscid and viscous fluxes in a
consistent manner that maintains pressure equilibrium, which is drastically
simplified through a particular choice of nodal basis. It is worth noting that
their proposed strategy is not limited to DG schemes, but can be applied to
other numerical methods as well. Stiff chemical reactions were handled via
operator splitting. Efficient and accurate integration of the chemical source
terms was achieved via an $hp$-adaptive DG method for solving ordinary
differential equations, termed _DGODE_. Optimal high-order accuracy was
demonstrated for smooth flows, and a suite of complex multicomponent reacting
flows was computed. The high-order calculation of a three-dimensional reacting
shear flow in the presence of a splitter plate did not require any additional
stabilization. Also computed was a two-dimensional, moving detonation wave.
Artificial viscosity was used to stabilize the shock fronts present in the
solution. However, a very fine mesh was required to maintain robustness while
achieving correct prediction of the cellular structures behind the shock,
highlighting the difficulty of robustly and accurately simulating
multidimensional detonation waves on coarse meshes, especially for high-order
methods. Even if spurious pressure oscillations are sufficiently minimized,
the wide range of complex flow features characterizing detonations is
difficult to capture [15]. Such features include thin reaction zones,
traveling pressure waves, shock-shock interactions, Kelvin-Helmholtz
instabilities, vortical structures, and triple points. As previously
discussed, underresolution of flow features induces instabilities that can
cause solver divergence, and these instabilities may be amplified by the added
nonlinearity of the variable thermodynamics, multicomponent flow, and stiff
chemical reactions. Another difficulty associated with multicomponent flow is
the frequent occurrence of negative species concentrations, especially since
initial and boundary conditions often specify the mole fractions of certain
species to be zero. Many reacting-flow solvers simply “clip” negative
concentrations to zero, which violates conservation and introduces low-order
errors.
In light of the above, our primary objective in this study is to develop a
positivity-preserving and entropy-bounded DG scheme for simulating the
multicomponent, chemically reacting Euler equations with exact thermodynamics
for mixtures of thermally perfect gases. Specifically, we build upon the
aforementioned fully conservative high-order method that can maintain pressure
equilibrium [14]. In this first part of our two-part paper, we focus on the
one-dimensional case. The unique challenges posed by realistic thermodynamics
and stiff chemical source terms are discussed and addressed. _Entropy-bounded_
in this context means that the specific thermodynamic entropy of the discrete
solution is bounded from below, an idea rooted in the minimum entropy
principle satisfied by entropy solutions to the compressible, multicomponent
Euler equations. This principle was recently proved for the nonreacting case
by Gouasmi et al. [16], which is an important prerequisite of this work. The
developed formulation in general preserves order of accuracy for smooth
solutions. Our main contributions are as follows:
* 1.
A minimum entropy principle for the compressible, multicomponent, chemically
reacting Euler equations is demonstrated, which follows naturally from the
proof in [16].
* 2.
In a DG framework, we extend the fully conservative high-order discretization
in [14] to be positivity-preserving and entropy-bounded. We also discuss a
different local entropy bound that is less restrictive than that previously
introduced in [17] in the context of the monocomponent Euler equations.
* 3.
To provably guarantee satisfaction of the minimum entropy principle in the
temporal integration of stiff chemical source terms, we extend DGODE by
developing an entropy-stable DGODE based on diagonal-norm summation-by-parts
(SBP) operators. This involves deriving a new entropy-conservative two-point
numerical state function (note that similar entropy-conservative numerical
state functions were derived for the monocomponent Euler, shallow-water, and
ideal magnetohydrodynamics equations by Friedrich et al. [18] in the context
of an entropy-stable space-time DG discretization).
* 4.
We employ the proposed entropy-bounded DG method to robustly and accurately
compute a series of canonical one-dimensional test cases. Mass, energy, and
atom conservation are maintained. The proposed formulation more effectively
suppresses spurious oscillations than the positivity-preserving DG scheme. In
particular, we find that the relative benefit of enforcing an entropy bound is
significantly greater in the multicomponent, thermally perfect setting than in
monocomponent, calorically perfect setting.
In Part II [19], we extend the entropy-bounded DG scheme to multiple
dimensions on arbitrary elements. Our multidimensional extension is a further
generalization of the multidimensional positivity-preserving/entropy-bounded
schemes currently in the literature [20, 21, 17, 22]. Specifically,
restrictions on the numerical flux, physical modeling, element shape,
polynomial order of the geometric approximation, and/or quadrature rules are
relaxed. Complex detonation waves in two and three dimensions are computed. We
find that whereas the positivity-preserving DG scheme is often not
sufficiently stable (even with artificial viscosity to stabilize strong
discontinuities), enforcing an entropy bound enables robust calculations on
relatively coarse meshes.
### 1.1 Background
Various stabilization strategies for high-order DG schemes have been
introduced in the literature. Artificial viscosity is a popular approach that
very effectively suppresses oscillations and is perfectly compatible with high
polynomial orders, arbitrary elements, and general equation sets [23, 24, 25].
However, there are certain limitations that discourage an overreliance on
artificial viscosity for suppressing _all_ instabilities. First, it can
significantly pollute accuracy, especially in smooth regions of the flow. As
such, it should ideally be added only where necessary (e.g., strong shocks
that are otherwise difficult to robustly capture). Second, design of a shock
sensor that can reliably detect discontinuous flow features for general
configurations remains an open problem. Even in a single flow configuration
that involves discontinuities of varying strengths (as is the case for
detonations), it is difficult to detect all such discontinuities and add the
“right” amount of artificial viscosity. Third, there is typically a very
strong dependence on tunable parameters.
A common alternative to artificial viscosity is limiting. WENO-type [26, 27],
TVD/TVB [28, 29], and moment limiters [30] are well-known examples. However,
it can be difficult to extend these limiters to arbitrary polynomial orders
for both the solution and geometric approximations, as well as to general
equation sets. Furthermore, these limiters are not guaranteed to yield
physically admissible solutions (i.e., positive density and pressure), a
drawback of artificial viscosity as well. An alternative approach for
maintaining robustness is to align the grid with discontinuities. Recent
formulations that do so in an implicit manner by treating the grid as a
variable were developed by Corrigan et al. [31] and Zahr and Persson [32]. An
encouraging preliminary effort to apply implicit shock tracking to reacting
flow is discussed in [33], in which supersonic inviscid reacting flow over a
two-dimensional wedge with simple thermodynamics and chemistry was computed.
Recently, positivity-preserving DG schemes [20, 34, 21, 35] have seen
considerable success in solving the monocomponent, nonreacting Euler equations
with explicit time stepping. These schemes prevent the occurrence of negative
densities and pressures under a constraint on the time step size, a
positivity-preserving numerical flux, and a limiting procedure consisting of a
simple linear “squeezing” of the solution towards its cell average. The
limiting operator is conservative and maintains order of accuracy for smooth
solutions. However, the limiter is not very effective at dampening
oscillations. The positivity-preserving DG method was extended to the entropy-
bounded DG scheme by Zhang and Shu [36], in which an additional limiting step
based on a Newton search was introduced to enforce a global entropy bound..
Note that it is implicitly assumed that an entropy-bounded scheme is also
positivity-preserving. Under their invariant-region-preserving DG framework,
Jiang and Liu [36, 22] introduced a more straightforward limiter to enforce
the entropy principle in an algebraic manner, which is particularly desirable
in the multicomponent, thermally perfect case due to the cost of evaluating
complex thermodynamics. Lv and Ihme [17] further extended the entropy-bounded
DG scheme (for the monocomponent Euler equations) by relaxing restrictions on
the geometry and quadrature rules. They also introduced a local entropy bound.
Numerical tests demonstrated the superiority of the entropy-bounded DG scheme
for suppressing spurious oscillations, compared to the positivity-preserving
DG method. Lv and Ihme later applied entropy bounding to their reacting flow
DG solver [37], which utilizes the double-flux approach discussed above. The
frozen thermodynamics and relaxation towards a calorically perfect gas model
circumvent the difficulties of extending the entropy-bounded DG method to
reacting flow with exact thermodynamics and non-calorically-perfect gases.
Furthermore, the physico-mathematical validity of combining the double-flux
model with enforcement of a discrete minimum entropy principle is not
immediately clear. We also note that these positivity-preserving and entropy-
bounded DG methods are related to the recent geometric quasilinearization
framework by Wu and Shu [38], as well as the invariant-domain-preserving
schemes based on graph viscosity by, for example, Guermond et al. [39] and
Pazner [40], which employ a convex limiting procedure relying on an iterative
line search.
Before concluding this section, we note several other efforts to compute
chemically reacting flows with DG schemes in a stable manner. Gutierrez-
Jorquera and Kummer [41] computed steady-state diffusion flames using a low-
Mach pressure-based solver and a one-step reaction mechanism. May et al. [42]
simulated steady hypersonic flows with some high-enthalpy effects using a
hybridized DG solver that employs artificial viscosity for shock capturing.
Papoutsakis et al. [43] computed chemically reacting hypersonic flow over a
double cone with a TVB limiter; however, only a linear polynomial
approximation of the solution was employed, and there were discrepancies with
finite-volume and experimental results. A number of positivity-preserving DG
schemes for the reacting Euler equations have also been developed. For
example, Zhang and Shu [34] and Wang et al. [44] extended the positivity-
preserving DG scheme to handle source terms with explicit time stepping.
Complex thermodynamics and stiff source terms were not addressed. Du and Yang
[45] and Du et al. [46] presented a positivity-preserving DG method based on a
new explicit, exponential Runge-Kutta (RK)/multistep time integration scheme
that can handle stiff source terms [47]. However, the temporal order of
accuracy is strongly dependent on the initial conditions. Other positivity-
preserving time integrators compatible with stiff source terms include
implicit Patankar-type RK schemes [48, 49, 50], which have been applied to
finite difference [48, 49] and finite volume [50] discretizations of the
reacting Euler equations. These schemes are still undergoing development, and
there may be issues with extending them to DG discretizations [48]. An
additional obstacle of both exponential RK/multistep and Patankar-type RK
schemes is proper enforcement of the minimum entropy principle. For these
reasons and due to its proven past success, we elect to still employ operator
splitting to deal with stiff source terms. Nevertheless, exponential
RK/multistep and Pantankar-type schemes are indeed worthy of future
consideration.
That we use an entropy-stable DG discretization to temporally integrate the
chemical source terms may cause readers to question why an entropy-stable DG
discretization is not also employed for the transport step. Note that entropy-
stable schemes guarantee the global integral of mathematical entropy to be
nonincreasing in time (assuming periodic or entropy-stable boundary
conditions), whereas entropy-bounded schemes enforce, in a pointwise fashion,
the specific thermodynamic entropy to be greater than some lower bound.
Entropy-stable DG schemes for multicomponent (nonreacting) flows have emerged
only recently [51, 52]. For general hyperbolic conservation laws, there are
various ways to achieve entropy stability; two of the most well-known are as
follows: (a) the entropy variables, instead of the conservative variables, are
directly solved for, and (b) SBP operators are used to approximate discrete
derivatives. The first approach, in which the conservative variables depend
implicitly on the polynomial approximation of the entropy variables, is not
appropriate for explicit time stepping, which we employ in this work (for the
transport terms). Furthermore, with the typical choice of entropy, the entropy
variables are undefined when any of the partial densities vanishes [51].
Conversely, the second approach, which has surged in popularity in recent
years, is compatible with explicit time stepping and does not assume exact
integration to provably guarantee entropy stability. Belonging to this
category are the methods by Renac [52] and Peyvan et al. [53]. On unstructured
grids, particularly those with (possibly curved) simplicial elements, current
SBP-based entropy-stable methods can become suboptimal [54], require direct
use of the entropy variables (even if the conservative variables are the
unknowns) [55, 56], and/or significantly increase in complexity [55, 57, 56,
58]. Note that these issues are not present in the developed entropy-stable
DGODE (except a moderate increase in complexity), which entails a one-
dimensional discretization in time. Due to the above factors, as well as the
relative simplicity of the entropy-bounded DG scheme and its compatibility
with the pressure-equilibrium-maintaining discretization by Johnson and
Kercher [14], we choose to employ the proposed entropy-bounded method for the
transport step. Nevertheless, we emphasize that we are not ruling out entropy-
stable schemes; advancements to these formulations are rapid and their
potential is evident [59]. Finally, it should be noted that although the
construction of entropy-stable and entropy-bounded methods rely on different
techniques, entropy stability and entropy boundedness are not necessarily
mutually exclusive. Depending on the type of entropy-stable scheme, it should,
in principle, be feasible to achieve discrete satisfaction of both properties.
The remainder of this paper is organized as follows. Sections 2 and 3
summarize the governing equations and basic DG discretization, respectively.
Section 4 reviews the minimum entropy principle associated with the
compressible, multicomponent, nonreacting Euler equations and extends it to
the reacting Euler equations. The next section presents the positivity-
preserving and entropy-bounded DG formulation for the transport step. We then
discuss the entropy-stable DG discretization for the reaction step in Section
6. Results for fundamental nonreacting test cases and one-dimensional
detonation-wave simulations are given in the following section. We close the
paper with concluding remarks.
## 2 Governing equations
The compressible, multicomponent, chemically reacting Euler equations are
given as
$\frac{\partial y}{\partial
t}+\nabla\cdot\mathcal{F}\left(y\right)-\mathcal{S}\left(y\right)=0$ (2.1)
where $t\in\mathbb{R}^{+}$ is time,
$y(x,t):\mathbb{R}^{d}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{m\times d}$
is the conservative state vector (with $x=(x_{1},\ldots,x_{d})$ denoting the
physical coordinates),
$\mathcal{F}(y):\mathbb{R}^{m}\rightarrow\mathbb{R}^{m\times d}$ is the
convective flux, $\mathcal{S}(y):\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$ is
the chemical source term. The state vector is expanded as
$y=\left(\rho v_{1},\ldots,\rho v_{d},\rho
e_{t},C_{1},\ldots,C_{n_{s}}\right)^{T},$ (2.2)
where $n_{s}$ is the number of species (which yields $m=d+n_{s}+1$), $\rho$ is
density, $v=\left(v_{1},\ldots,v_{d}\right)$ is the velocity, $e_{t}$ is the
mass-specific total energy, and $C=\left(C_{1},\ldots,C_{n_{s}}\right)$ are
the species concentrations. The density is computed from the species
concentrations as
$\rho=\sum_{i=1}^{n_{s}}\rho_{i}=\sum_{i=1}^{n_{s}}W_{i}C_{i},$
where $\rho_{i}$ is the partial density and $W_{i}$ is the molecular weight of
the $i$th species. The mass fraction and mole fraction of the $i$th species
are defined as
$Y_{i}=\frac{\rho_{i}}{\rho}$
and
$X_{i}=\frac{C_{i}}{\sum_{i=1}^{n_{s}}C_{i}},$
respectively.
The $k$th spatial convective flux component is written as
$\mathcal{F}_{k}^{c}\left(y\right)=\left(\rho
v_{k}v_{1}+P\delta_{k1},\ldots,\rho v_{k}v_{d}+P\delta_{kd},v_{k}\left(\rho
e_{t}+P\right),v_{k}C_{1},\ldots,v_{k}C_{n_{s}}\right)^{T},$ (2.3)
where $P$ is the pressure. The mass-specific total energy is the sum of the
specific internal and kinetic energies, given by
$e_{t}=u+\frac{1}{2}\sum_{k=1}^{d}v_{k}v_{k},$
where the (mixture-averaged) mass-specific internal energy, $u$, is the mass-
weighted sum of the mass-specific internal energies of each species:
$u=\sum_{i=1}^{n_{s}}Y_{i}u_{i}.$
This work assumes thermally perfect gases, with $u_{i}$ given by [60]
$u_{i}=h_{i}-R_{i}T=h_{\mathrm{ref},i}+\int_{T_{\mathrm{ref}}}^{T}c_{p,i}(\tau)d\tau-
R_{i}T,$
where $h_{i}$ is the mass-specific enthalpy of the $i$th species,
$R_{i}=R^{0}/W_{i}$ (with
$R^{0}=8314.4621\,\mathrm{JKmol}^{-1}\mathrm{K}^{-1}$ denoting the universal
gas constant), $T$ is the temperature, $T_{\mathrm{ref}}$ is the reference
temperature (298.15 K), $h_{\mathrm{ref},i}$ is the reference-state species
formation enthalpy, and $c_{p,i}$ is the mass-specific heat at constant
pressure of the $i$th species. $c_{p,i}$ is computed from an $n_{p}$-order
polynomial as
$c_{p,i}=\sum_{k=0}^{n_{p}}a_{ik}T^{k},$ (2.4)
based on the NASA coefficients [61, 62]. The mass-specific thermodynamic
entropy of the mixture is defined as
$s=\sum_{i=1}^{n_{s}}Y_{i}s_{i},$
with $s_{i}$ given by
$s_{i}=s_{i}^{o}-R_{i}\log\frac{P_{i}}{P_{\mathrm{ref}}},\quad
s_{i}^{o}=s_{\mathrm{ref},i}^{o}+\int_{T_{\mathrm{ref}}}^{T}\frac{c_{p,i}(\tau)}{\tau}d\tau,$
where $s_{\mathrm{ref},i}^{o}$ is the species formation entropy at the
reference temperature and reference pressure ($P_{\mathrm{ref}}=1\text{
atm}$), $s_{i}^{o}$ denotes the species entropy at atmospheric pressure, and
$P_{i}=C_{i}R^{0}T$ is the partial pressure. $s_{i}$ can also be expressed as
[60, 16, 51]
$s_{i}=s_{\mathrm{ref},i}^{o}+\int_{T_{\mathrm{ref}}}^{T}\frac{c_{v,i}(\tau)}{\tau}d\tau-
R_{i}\log\frac{C_{i}}{C_{\mathrm{ref}}},$
where $C_{\mathrm{ref}}=P_{\mathrm{ref}}/R^{0}T_{\mathrm{ref}}$ is the
reference concentration and $c_{v,i}=c_{p,i}-R_{i}$ is the mass-specific heat
at constant volume of the $i$th species. Summing up the partial pressures
yields the equation of state for the mixture:
$P=R^{0}T\sum_{i=1}^{n_{s}}C_{i}.$ (2.5)
$u_{i}$, $h_{i}$, and $s_{i}^{o}$ are computed by integrating Equation (2.4)
and incorporating the integration constants in [61] and [62]. For example,
$u_{i}$ is calculated as
$u_{i}=b_{i0}+\sum_{k=0}^{n_{p}}\frac{a_{ik}}{k+1}T^{k+1}-R_{i}T=\sum_{k=0}^{n_{p}+1}b_{ik}T^{k},$
(2.6)
where $b_{i0}$ is the integration constant and
$b_{ik}=\begin{cases}\frac{a_{i,k-1}}{k},&k>1\\\
a_{i0}-R_{i},&k=1.\end{cases}$
### 2.1 Chemical reaction rates
The source term in Equation (2.1) is a smooth function of the state variables,
written as [63]
$\mathcal{S}\left(y\right)=\left(0,\ldots,0,0,\omega_{1},\ldots,\omega_{n_{s}}\right)^{T},$
(2.7)
where $\omega_{i}$ is the production rate of the $i$th species, which
satisfies mass conservation:
$\sum_{i=1}^{n_{s}}W_{i}\omega_{i}=0.$ (2.8)
The production rate is computed as
$\omega_{i}=\sum_{j=1}^{n_{r}}\nu_{ij}q_{j}.$
$n_{r}$ is the number of reactions, $\nu_{ij}=\nu_{ij}^{r}-\nu_{ij}^{f}$ is
the difference between the reverse ($\nu_{ij}^{r}$) and the forward
stoichiometric coefficients ($\nu_{ij}^{f}$), and $q_{j}$ is the rate of
progress of the $j$th reaction, computed as
$q_{j}=k_{j}^{f}\prod_{i=1}^{n_{s}}C_{i}^{\nu_{ij}^{f}}-k_{j}^{r}\prod_{i=1}^{n_{s}}C_{i}^{\nu_{ij}^{r}},$
(2.9)
where $k_{j}^{f}$ and $k_{j}^{r}$ are the forward and reverse rate constants,
respectively, of the $j$th reaction. The forward and reverse rate constants
are related via the equilibrium constant,
$K_{j}^{e}=\exp\left(-\frac{\Delta
G_{j}^{\prime}}{R^{0}T}\right)\left(\frac{P_{\mathrm{ref}}}{R^{0}T}\right)^{\sum_{i}\nu_{ij}},$
(2.10)
where $\Delta G_{j}^{\prime}$ is the change in reference-state Gibbs free
energy for the $j$th reaction, given as
$\Delta
G_{j}^{\prime}=\sum_{i=1}^{n_{s}}\nu_{ij}W_{i}h_{i}-T\sum_{i=1}^{n_{s}}\nu_{ij}W_{i}s_{i}^{\prime}.$
Introducing the reduced chemical potentials of the $i$th species,
$\displaystyle\mu_{i}$ $\displaystyle=\frac{g_{i}}{R^{0}T},$
$\displaystyle\mu_{i}^{\mathrm{u}}$ $\displaystyle=\mu_{i}-\frac{1}{W_{i}}\log
C_{i},$
where $g_{i}=h_{i}-Ts_{i}$ is the Gibbs function of the $i$th species, the
equilibrium constant can also be written as [60, Chapter 6.4]
$K_{j}^{e}=\exp\left(-\sum_{i=1}^{n_{s}}\nu_{ij}W_{i}\mu_{i}^{\mathrm{u}}\right),$
(2.11)
There exist various models for approximating the forward rate constants in
Equation (2.9), several of which will be briefly discussed next.
#### 2.1.1 Arrhenius reactions
The Arrhenius form is the most common model for approximating reaction rates.
The forward rate constants are computed as
$k_{j}^{f}=A_{j}T^{b_{j}}\exp\left(-\frac{E_{j}}{R^{0}T}\right),$
where $A_{j}>0$ and $b_{j}$ are parameters and $E_{j}\geq 0$ is the activation
energy [60, 63].
#### 2.1.2 Three-body reactions
These reactions require a “third body” in order to proceed. Dissociation and
recombination reactions are often of this type. The rate of progress is scaled
by a prefactor as [63]
$q_{j}=\left(\sum_{i=1}^{n_{s}}\alpha_{ij}C_{i}\right)\left(k_{j}^{f}\prod_{i=1}^{n_{s}}C_{i}^{\nu_{ij}^{f}}-k_{j}^{r}\prod_{i=1}^{n_{s}}C_{i}^{\nu_{ij}^{r}}\right),$
where $\alpha_{ij}$ are the third-body efficiencies.
#### 2.1.3 Unimolecular/recombination fall-off reactions
Unimolecular/recombination fall-off reactions incorporate a dependence on
pressure. In general, this model predicts an increase in the reaction rate
with increasing pressure. For brevity, we drop the $j$ subscript and $f$
superscript. Given Arrhenius-type low-pressure and high-pressure limits for
the rate constant ($k_{0}$ and $k_{\infty}$, respectively), $k$ is computed as
$k=k_{\infty}\left(\frac{P_{r}}{1+P_{r}}\right)F,$ (2.12)
where $P_{r}$ is the reduced pressure, defined as
$P_{r}=\frac{k_{0}}{k_{\infty}}\sum_{i=1}^{n_{s}}\alpha_{i}C_{i}.$
There are different ways to compute $F$ in Equation (2.12). With the Lindemann
[64] approach, $F$ is simply unity. In the Troe [65] form, $F$ is given by
$\log F=\frac{\log F_{\mathrm{cent}}}{1+\left[\frac{\log
P_{r}+c_{1}}{c_{2}-c_{3}\left(\log P_{r}+c_{1}\right)}\right]^{2}},$
where the definitions of $c_{1}$, $c_{2}$, $c_{3}$, and $F_{\mathrm{cent}}$
are given in [65].
#### 2.1.4 Chemically activated bimolecular reactions
Reactions of this type are also pressure-dependent, but the reaction rates
typically decrease with increasing pressure. The rate constants are computed
as [63]
$k=k_{0}\left(\frac{1}{1+P_{r}}\right)F,$
where $k_{0}$, $P_{r}$, and $F$ are calculated as in Section 2.1.3.
## 3 Discontinuous Galerkin discretization
In this section, we briefly describe the DG discretization of the governing
equations and review the techniques proposed in [14] to prevent spurious
pressure oscillations in smooth regions of the flow.
Let $\Omega\subset\mathbb{R}^{d}$ be the $d$-dimensional computational domain
partitioned by $\mathcal{T}$, which consists of non-overlapping cells $\kappa$
with boundaries $\partial\kappa$. Let $\mathcal{E}$ denote the set of
interfaces $\epsilon$, with
$\cup_{\epsilon\in\mathcal{E}}\epsilon=\cup_{\kappa\in\mathcal{T}}\partial\kappa$.
$\mathcal{E}$ consists of the interior interfaces,
$\epsilon_{\mathcal{I}}\in\mathcal{E_{I}}=\left\\{\epsilon_{\mathcal{I}}\in\mathcal{E}\,\middle|\,\epsilon_{\mathcal{I}}\cap\partial\Omega=\emptyset\right\\},$
and boundary interfaces,
$\epsilon_{\partial}\in\mathcal{E}_{\partial}=\left\\{\epsilon_{\partial}\in\mathcal{E}\,\middle|\,\epsilon_{\partial}\subset\partial\Omega\right\\},$
such that $\mathcal{E}=\mathcal{E_{I}}\cup\mathcal{E}_{\partial}$. At interior
interfaces, there exists $\kappa^{+},\kappa^{-}\in\mathcal{T}$ such that
$\epsilon_{\mathcal{I}}=\partial\kappa^{+}\cap\partial\kappa^{-}$. $n^{+}$ and
$n^{-}$ denote the outward facing normal of $\kappa^{+}$ and $\kappa^{-}$,
respectively, with $n^{+}=-n^{-}$. The discrete (finite-dimensional) subspace
$V_{h}^{p}$ over $\mathcal{T}$ is defined as
$\displaystyle V_{h}^{p}$ $\displaystyle=$
$\displaystyle\left\\{\mathfrak{v}\in\left[L^{2}\left(\Omega\right)\right]^{m}\,\middle|\,\forall\kappa\in\mathcal{T},\left.\mathfrak{v}\right|_{\kappa}\in\left[\mathcal{P}_{p}(\kappa)\right]^{m}\right\\},$
(3.1)
where, for $d=1$, $\mathcal{P}_{p}(\kappa)$ is the space of polynomial
functions of degree no greater than $p$ in $\kappa$. For $d>1$, the choice of
polynomial space typically depends on the element type [66].
The semi-discrete form of the governing equations (Equation (2.1)) is given
as: find $y\in V_{h}^{p}$ such that
$\displaystyle\sum_{\kappa\in\mathcal{T}}\left(\frac{\partial y}{\partial
t},\mathfrak{v}\right)_{\kappa}-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{F}\left(y\right),\nabla\mathfrak{v}\right)_{\kappa}+\sum_{\epsilon\in\mathcal{E}}\left(\mathcal{F}^{\dagger}\left(y^{+},y^{-},n\right),\left\llbracket\mathfrak{v}\right\rrbracket\right)_{\mathcal{E}}-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{S}\left(y\right),\mathfrak{v}\right)_{\kappa}=0\qquad\forall\>\mathfrak{v}\in
V_{h}^{p},$ (3.2)
where $\left(\cdot,\cdot\right)$ denotes the inner product,
$\mathcal{F}^{\dagger}\left(y^{+},y^{-},n\right)$ is the numerical flux, and
$\left\llbracket\cdot\right\rrbracket$ is the jump operator, given by
$\left\llbracket\mathfrak{v}\right\rrbracket=\mathfrak{v}^{+}-\mathfrak{v}^{-}$
at interior interfaces and
$\left\llbracket\mathfrak{v}\right\rrbracket=\mathfrak{v}^{+}$ at boundary
interfaces. Applying a standard, fully explicit time stepping scheme to
Equation (3.2) would yield an exceedingly small time step due to the stiff
chemical source terms. As such, operator splitting is employed to decouple the
temporal integration of the convection operator from that of the source term.
Specifically, we apply Strang splitting [67] over a given interval
$(t_{0},t_{0}+\Delta t]$ as
$\displaystyle\frac{\partial y}{\partial
t}+\nabla\cdot\mathcal{F}\left(y\right)=0$ $\displaystyle\textup{ in
}\Omega\times\left(t_{0},t_{0}+\nicefrac{{\Delta t}}{{2}}\right],$ (3.3)
$\displaystyle\frac{\partial y}{\partial t}-\mathcal{S}\left(y\right)=0$
$\displaystyle\textup{ in }\left(t_{0},t_{0}+\Delta t\right],$ (3.4)
$\displaystyle\frac{\partial y}{\partial
t}+\nabla\cdot\mathcal{F}\left(y\right)=0$ $\displaystyle\textup{ in
}\Omega\times\left(t_{0}+\nicefrac{{\Delta t}}{{2}},t_{0}+\Delta t\right],$
(3.5)
where Equations (3.3) and (3.5) are integrated in time with an explicit RK-
type scheme, while Equation (3.4) is solved using a fully implicit, temporal
DG discretization for ODEs (DGODE). Details on DGODE and its extension to
entropy-stable DGODE are given in Section 6. More sophisticated operator-
splitting schemes can be employed as well [68].
Unless otherwise specified, the volume and surface terms in Equation (3.2) are
evaluated using a quadrature-free approach [69, 70]. Throughout this work, we
employ a nodal basis, such that the element-local polynomial approximation of
the solution is expanded as
$y_{\kappa}=\sum_{j=1}^{n_{b}}y_{\kappa}(x_{j})\phi_{j},$ (3.6)
where $n_{b}$ is the number of basis functions,
$\left\\{\phi_{1},\ldots,\phi_{n_{b}}\right\\}$ are the basis functions, and
$\left\\{x_{1},\ldots,x_{n_{b}}\right\\}$ are the node coordinates. Its
average over $\kappa$ is given by
$\overline{y}_{\kappa}=\frac{1}{\left|\kappa\right|}\int_{\kappa}ydx,$ (3.7)
where $\left|\kappa\right|$ is the volume of $\kappa$. In the evaluation of
the second and third integrals in Equation (3.2), the nonlinear convective
flux can be approximated as
$\mathcal{F_{\kappa}}\approx\sum_{k=1}^{n_{c}}\mathcal{F}\left(y_{\kappa}\left(x_{k}\right)\right)\varphi_{k},$
(3.8)
where $n_{c}\geq n_{b}$ and
$\left\\{\varphi_{1},\ldots,\varphi_{n_{c}}\right\\}$ is a set of (potentially
different) polynomial basis functions. If $n_{c}=n_{b}$ and the integration
points are included in the set of solution nodes (e.g., Gauss-Lobatto points
for tensor-product elements), then pressure equilibrium is trivially
maintained [14]. However, over-integration (i.e., $n_{c}>n_{b}$) is often
necessary to minimize aliasing errors and improve stability. Unfortunately,
standard over-integration, as defined in Equation (3.8), causes a loss of
pressure equilibrium and generation of spurious pressure oscillations at
material interfaces [14]. Instead, Johnson and Kercher [14] proposed the
following approximation of the convective flux:
$\mathcal{F_{\kappa}}\approx\sum_{k=1}^{n_{c}}\mathcal{F}\left(\widetilde{y}_{\kappa}\left(x_{k}\right)\right)\varphi_{k},$
(3.9)
where $\widetilde{y}:\mathbb{R}^{m}\times\mathbb{R}\rightarrow\mathbb{R}^{m}$
is a modified state defined as
$\widetilde{y}\left(y,\widetilde{P}\right)=\left(\rho v_{1},\ldots,\rho
v_{d},\widetilde{\rho
u}\left(C_{1},\ldots,C_{n_{s}},\widetilde{P}\right)+\frac{1}{2}\sum_{k=1}^{d}\rho
v_{k}v_{k},C_{1},\ldots,C_{n_{s}}\right)^{T}.$ (3.10)
$\widetilde{P}$ is a polynomial approximation of the pressure that
interpolates onto the span of $\left\\{\phi_{1},\ldots,\phi_{n_{b}}\right\\}$
as
$\widetilde{P}_{\kappa}=\sum_{j=1}^{n_{b}}P\left(y_{\kappa}\left(x_{j}\right)\right)\phi_{j},$
and the modified internal energy, $\widetilde{\rho u}$, is evaluated from the
modified pressure and unmodified species concentrations. Interpolating the
convective flux (including the numerical flux function) in this manner
achieves pressure equilibrium both internally and between adjacent elements.
Of course, with finite resolution, slight deviations from pressure equilibrium
are inevitable; nevertheless, apart from severely underresolved computations,
these deviations remain small and do not generate large-scale pressure
oscillations that cause the solver to crash, which is not the case if standard
flux interpolation (3.8) is employed. Additional information on the basic DG
discretization, enforcement of boundary conditions, and nonlinear flux
interpolation, as well as a detailed discussion of the conditions under which
pressure oscillations are generated, can be found in [14].
## 4 Minimum entropy principle
It is well-known that in the presence of discontinuities, weak solutions to
general systems of hyperbolic conservation laws, including the multicomponent
Euler equations, are not unique [71]. As such, physical solutions are
typically identified as those that satisfy entropy conditions, written as (in
the absence of source terms)
$\frac{\partial U}{\partial t}+\nabla\cdot\mathcal{F}^{s}\leq 0,$ (4.1)
where $U(y):\mathbb{R}^{m}\rightarrow\mathbb{R}$ is a given convex
(mathematical) entropy function and
$\mathcal{F}^{s}(y):\mathbb{R}^{m}\rightarrow\mathbb{R}^{d}$ is the
corresponding spatial entropy flux satisfying
$\mathsf{v}^{T}\frac{\partial\mathcal{F}}{\partial
y}=\frac{\partial\mathcal{F}^{s}}{\partial y},$
with $\mathsf{v}$, the entropy variables, defined as
$\mathsf{v}=\left(\frac{\partial U}{\partial y}\right)^{T}.$
The mapping from the conservative variables to the entropy variables is one-
to-one and symmetrizes the system [72]. _Entropy solutions_ are weak solutions
that satisfy (4.1) for all entropy/entropy-flux pairs. We also introduce the
entropy potential and the corresponding entropy flux potential:
$\left(\mathcal{U},\mathcal{F}^{p}\right)=\left(\mathsf{v}^{T}y-U,\mathsf{v}^{T}\mathcal{F}-\mathcal{F}^{s}\right),$
(4.2)
which will be used in Section 6. For the multicomponent Euler equations,
$U=-\rho s$ and $\mathcal{F}^{s}=-\rho sv$ form a common admissible
entropy/entropy-flux pair [16, 60], assuming $C_{i}>0$ and $T>0$. Note the
distinction between the mathematical entropy and the thermodynamic entropy
(per unit volume); they are typically of opposite sign. The conservation
equation (for smooth solutions) for $U=-\rho s$ can be obtained by first
combining the Gibbs relation,
$Tds=du-\sum_{i=1}^{n_{s}}g_{i}dY_{i}-\frac{P}{\rho^{2}}d\rho,$
with the transport equations for species mass fractions, density, and internal
energy [16, 60],
$\displaystyle\frac{DY_{i}}{Dt}$ $\displaystyle=0,\;i=1,\ldots,n_{s},$
$\displaystyle\frac{D\rho}{Dt}$ $\displaystyle+\rho\nabla\cdot v=0,$
$\displaystyle\frac{Du}{Dt}$ $\displaystyle+\frac{P}{\rho}\nabla\cdot v=0,$
to yield the specific entropy transport equation,
$\frac{Ds}{Dt}=0.$ (4.3)
Then, using conservation of mass,
$\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho v\right)=0,$
the conservation equation for $U=-\rho s$ is obtained:
$\frac{\partial\rho s}{\partial t}+\nabla\cdot\left(\rho sv\right)=0.$
### 4.1 Review: Minimum entropy principle in the compressible,
multicomponent, nonreacting Euler equations
Gouasmi et al. [16] recently proved a minimum entropy principle satisfied by
entropy solutions to the multicomponent, nonreacting Euler equations, which
means that the spatial minimum of the specific thermodynamic entropy is a
nondecreasing function of time. In this subsection, we summarize the main
steps of the proof, which itself builds on the proof by Tadmor [73] of a
minimum entropy principle in the monocomponent Euler equations.
Integrating the inequality (4.1) over $\Omega$ and assuming nonnegative net
spatial entropy outflux across $\partial\Omega$ yields
$\frac{d}{dt}\int_{\Omega}U\left(y(x,t)\right)dx\leq 0.$
Integrating in time then gives
$\int_{\Omega}U\left(y(x,t)\right)dx\leq\int_{\Omega}U\left(y(x,0)\right)dx.$
Tadmor [74], however, showed that by integrating (4.1) over the truncated cone
$\mathsf{C}=\left\\{\left|x\right|\leq R+v_{\max}(t-\tau)|0\leq\tau\leq
t\right\\}$, a more local inequality can be written:
$\int_{\left|x\right|\leq R}U\left(y(x,t)\right)dx\leq\int_{\left|x\right|\leq
R+v_{\mathrm{max}}t}U\left(y(x,0)\right)dx,$ (4.4)
where $v_{\max}$ is the maximum speed in the domain at $t=0$. If we consider
entropy/entropy-flux pairs of the form
$\left(U,\mathcal{F}^{s}\right)=\left(-\rho f(s),-\rho vf(s)\right),$ (4.5)
where $f$ is a smooth function of $s$, (4.4) then becomes
$\int_{\left|x\right|\leq R}\rho(x,t)\cdot
f\left(s\left(y(x,t)\right)\right)dx\geq\int_{\left|x\right|\leq
R+v_{\mathrm{max}}t}\rho(x,0)\cdot f\left(s\left(y(x,0)\right)\right)dx.$
(4.6)
Consider the following choice for $f(s)$:
$f_{0}(s)=\min\left\\{s-s_{0},0\right\\},$
where $s_{0}$ is the essential infimum of the specific thermodynamic entropy
in the subdomain $\Omega_{R}=\left\\{\left|x\right|\leq R+v_{\max}t\right\\}$,
$s_{0}=\underset{\left|x\right|\leq R+v_{\max}t}{\text{Ess inf}}s(x,0).$
Although $f_{0}(s)$ is not a smooth function of $s$, it can be written as the
limit of a sequence of smooth functions,
$f_{0}(s)=\underset{\epsilon\rightarrow 0}{\lim}f_{\epsilon}(s)$, where
$f_{\epsilon}(s)$ is defined as [16]
$f_{\epsilon}(s)=\int_{-\infty}^{\infty}f_{0}(s-\mathfrak{s})g_{\epsilon}(\mathfrak{s})d\mathfrak{s},$
with
$g_{\epsilon}(\mathfrak{s})=\frac{1}{\epsilon}\frac{\exp\left(-\frac{\mathfrak{s}^{2}}{\epsilon^{2}}\right)}{\sqrt{\pi}},\;\epsilon>0.$
The first and second derivatives of $f_{\epsilon}(s)$ satisfy the following
conditions:
$\frac{df_{\epsilon}}{ds}>0,\frac{d^{2}f_{\epsilon}}{ds^{2}}<0.$
Gouasmi et al. [16] proved the key result that the entropy/entropy-flux pairs
$\left(U,\mathcal{F}^{s}\right)=\left(-\rho f_{\epsilon}(s),-\rho
vf_{\epsilon}(s)\right)$, with $\epsilon>0$, are admissible. In particular,
they showed that a conservation equation (for smooth solutions) for said pairs
can be obtained and that the entropy functions are convex with respect to the
conservative variables.
With $U=-\rho f_{0}(s)$, the inequality (4.6) can then be written as
$\int_{\left|x\right|\leq
R}\rho(x,t)\cdot\min\left\\{s(x,t)-s_{0},0\right\\}dx\geq\int_{\left|x\right|\leq
R+v_{\mathrm{max}}t}\rho(x,0)\cdot\min\left\\{s(x,0)-s_{0},0\right\\}dx,$
where the RHS is zero by definition of $s_{0}$ and the LHS is nonpositive,
yielding, for $\left|x\right|\leq R$,
$\min\left\\{s(x,t)-s_{0},0\right\\}=0.$ (4.7)
As a direct result, we obtain, for $\left|x\right|\leq R$,
$s(x,t)\geq s_{0}=\underset{\left|x\right|\leq R+v_{\max}t}{\text{Ess
inf}}s(x,0),$ (4.8)
which is the minimum entropy principle for the compressible, multicomponent
Euler equations. Note that only the entropy inequalities associated with
$U=-\rho f_{\epsilon}(s)$ need to be satisfied for a minimum entropy principle
to hold.
### 4.2 Minimum entropy principle in the compressible, multicomponent,
reacting Euler equations
We now extend the result in the previous subsection to the reacting Euler
equations, where the only difference is the inclusion of the chemical source
terms (Equation (2.7)). The presence of the chemical source terms, which are
smooth functions of only the state variables, modifies the entropy inequality
(4.1) satisfied by entropy solutions as [75, 76, 77]
$\frac{\partial U}{\partial
t}+\nabla\cdot\mathcal{F}^{s}\leq\mathsf{v}^{T}\mathcal{S},$ (4.9)
where the RHS represents the mathematical entropy production rate due to the
source term. If $\mathsf{v}^{T}\mathcal{S}\leq 0$ (i.e., the entropy
production is nonpositive), the entropy inequality in (4.1) for the
homogeneous system is recovered. The local inequality in (4.4) can then be
obtained by again integrating over $\mathsf{C}=\left\\{\left|x\right|\leq
R+v_{\max}(t-\tau)|0\leq\tau\leq t\right\\}$. If, in particular,
$\mathsf{v}^{T}\mathcal{S}\leq 0$ for entropy functions of the form $U=-\rho
f_{\epsilon}(s),\>\forall\epsilon>0$, the remaining arguments in Section 4.1
can be used to establish the same minimum entropy principle in Equation (4.8)
for the reacting Euler equations. As such, we focus on showing
$\mathsf{v}^{T}\mathcal{S}\leq 0$ with $U=-\rho
f_{\epsilon}(s),\>\forall\epsilon>0$.
Consider again entropy/entropy-flux pairs of the form
$\left(U,\mathcal{F}^{s}\right)=\left(-\rho f(s),-\rho vf(s)\right)$. The
corresponding entropy variables are given as
$\mathsf{v}=\begin{pmatrix}\frac{df}{ds}\frac{v_{1}}{T}\\\ \vdots\\\
\frac{df}{ds}\frac{v_{d}}{T}\\\ -\frac{df}{ds}\frac{1}{T}\\\
W_{1}\frac{df}{ds}\left(\frac{g_{1}-\frac{1}{2}\sum_{k=1}^{d}v_{k}v_{k}}{T}+s\right)-W_{1}f\\\
\vdots\\\
W_{n_{s}}\frac{df}{ds}\left(\frac{g_{n_{s}}-\frac{1}{2}\sum_{k=1}^{d}v_{k}v_{k}}{T}+s\right)-W_{n_{s}}f\end{pmatrix},$
which differ slightly from the entropy variables derived by Gouasmi et al.
[16] since they used partial densities instead of species concentrations in
the vector of state variables. The entropy production rate due to chemical
reactions, $\mathsf{v}^{T}\mathcal{S}$, is then written as
$\displaystyle\mathsf{v}^{T}\mathcal{S}$
$\displaystyle=\sum_{i=1}^{n_{s}}\left[W_{i}\frac{df}{ds}\left(\frac{g_{i}-\frac{1}{2}\sum_{k=1}^{d}v_{k}v_{k}}{T}+s\right)-W_{i}f\right]\omega_{i}$
$\displaystyle=\frac{df}{ds}\sum_{i=1}^{n_{s}}W_{i}\omega_{i}\frac{g_{i}}{T}+\left(-\frac{1}{2}\frac{\sum_{k=1}^{d}v_{k}v_{k}}{T}\frac{df}{ds}+s\frac{df}{ds}-f\right)\sum_{i=1}^{n_{s}}W_{i}\omega_{i}$
$\displaystyle=\frac{df}{ds}\sum_{i=1}^{n_{s}}W_{i}\omega_{i}\frac{g_{i}}{T},$
where the last equality is due to mass conservation, as given by Equation
(2.8). Since $\frac{df_{\epsilon}}{ds}>0,\>\forall\epsilon>0$, a minimum
entropy principle holds under the condition
$\sum_{i=1}^{n_{s}}W_{i}\omega_{i}\frac{g_{i}}{T}\leq 0.$ (4.10)
The term on the LHS, $\sum_{i=1}^{n_{s}}W_{i}\omega_{i}g_{i}/T$, is precisely
the entropy production rate for $U=-\rho s$, which Giovangigli [60, Chapter
6.4] already showed to be nonpositive. For completeness, we review the proof
here.
Let $\mu$ and $\mu^{\mathrm{u}}$ denote the following vectors of the reduced
chemical potentials:
$\displaystyle\mu$
$\displaystyle=\left(\mu_{1},\ldots,\mu_{n_{s}}\right)^{T},$
$\displaystyle\mu^{\mathrm{u}}$
$\displaystyle=\left(\mu_{1}^{\mathrm{u}},\ldots,\mu_{n_{s}}^{\mathrm{u}}\right)^{T}.$
We also define $\mathcal{W}$ as the matrix with the molecular weights along
the diagonal:
$\mathcal{W}=\mathrm{diag}\left(W_{1},\ldots,W_{n_{s}}\right).$
The equilibrium constant can then be rewritten as
$\log
K_{j}^{e}=-\left(\mathcal{W}\nu_{j}\right)^{T}\mu^{\mathrm{u}}=\left(\mathcal{W}\nu_{j}^{f}\right)^{T}\mu^{\mathrm{u}}-\left(\mathcal{W}\nu_{j}^{r}\right)^{T}\mu^{\mathrm{u}},$
(4.11)
where
$\displaystyle\nu_{j}^{f}$
$\displaystyle=\left(\nu_{1j}^{f},\ldots,\nu_{n_{s}j}^{f}\right)^{T},$
$\displaystyle\nu_{j}^{r}$
$\displaystyle=\left(\nu_{1j}^{r},\ldots,\nu_{n_{s}j}^{r}\right)^{T},$
$\displaystyle\nu_{j}$ $\displaystyle=\nu_{j}^{r}-\nu_{j}^{f}.$
Using Equation (4.11), Giovangigli [60, Chapter 6.4] introduced a new reaction
constant, given by
$\log K_{j}^{s}=\log
K_{j}^{f}-\left(\mathcal{W}\nu_{j}^{f}\right)^{T}\mu^{\mathrm{u}}=\log
K_{j}^{r}-\left(\mathcal{W}\nu_{j}^{r}\right)^{T}\mu^{\mathrm{u}}.$
The rate of progress of the $j$th reaction can then be expressed as
$\displaystyle q_{j}$
$\displaystyle=K_{j}^{s}\left[\exp\left(\left(\mathcal{W}\nu_{j}^{f}\right)^{T}\mu^{\mathrm{u}}\right)\prod_{i=1}^{n_{s}}C_{i}^{\nu_{ij}^{f}}-\exp\left(\left(\mathcal{W}\nu_{j}^{r}\right)^{T}\mu^{\mathrm{u}}\right)\prod_{i=1}^{n_{s}}C_{i}^{\nu_{ij}^{r}}\right],$
$\displaystyle=K_{j}^{s}\left[\exp\left(\left(\mathcal{W}\nu_{j}^{f}\right)^{T}\mu\right)-\exp\left(\left(\mathcal{W}\nu_{j}^{r}\right)^{T}\mu\right)\right].$
We rewrite the entropy production rate for $U=-\rho s$ as
$\displaystyle\sum_{i=1}^{n_{s}}W_{i}\omega_{i}\frac{g_{i}}{T}$
$\displaystyle=R^{0}\sum_{i=1}^{n_{s}}\sum_{j=1}^{n_{r}}W_{i}\mu_{i}\nu_{ij}q_{j}$
$\displaystyle=-R^{0}\sum_{j=1}^{n_{r}}\mu^{T}\mathcal{W}\left(\nu_{j}^{f}-\nu_{j}^{r}\right)q_{j}$
$\displaystyle=-R^{0}\sum_{j=1}^{n_{r}}K_{j}^{s}\left(\mu^{T}\mathcal{W}\nu_{j}^{f}-\mu^{T}\mathcal{W}\nu_{j}^{r}\right)\left[\exp\left(\mu^{T}\mathcal{W}\nu_{j}^{f}\right)-\exp\left(\mu^{T}\mathcal{W}\nu_{j}^{r}\right)\right],$
$\displaystyle\leq 0.$
Note that this is equivalent to a nonnegative production rate of
_thermodynamic_ entropy per unit volume. Since the condition in (4.10) is
satisfied, a minimum entropy principle in the compressible, multicomponent,
reacting Euler equations holds. Another consequence of (4.10) is that the
entropy production rate for any entropy function of the form $U=-\rho f(s)$ is
nonpositive provided that $\frac{df}{ds}\geq 0$.
## 5 Transport step: Entropy-bounded discontinuous Galerkin scheme in one
dimension
In this section, we detail the entropy-bounded DG methodology for solving
Equations (3.3) and (3.5) (i.e., the explicit time integrations without source
terms in the operator splitting strategy) while accounting for the modified
flux interpolation in Equation (3.9). We build on related entropy-bounded DG
schemes for the monocomponent Euler equations [36, 17, 22]. These schemes are
formulated as extensions of positivity-preserving DG methods [20, 34, 35]
since the thermodynamic entropy is well-defined only for positive densities
and pressures and enforcement of an entropy constraint can be
straightforwardly incorporated into the positivity-preserving framework. The
one-dimensional entropy-bounded DG scheme is presented in Section (5.2), then
extended to multiple dimensions in Part II [19].
### 5.1 Preliminaries
Let $\mathcal{G}_{\sigma}$ denote the following set:
$\mathcal{G_{\sigma}}=\left\\{y\mid C_{1}>0,\ldots,C_{n_{s}}>0,\rho
u^{*}>0,s\geq\sigma\right\\},$ (5.1)
where $\sigma\in\mathbb{R}$ and $u^{*}$ is the “shifted” internal energy [48],
computed as
$u^{*}=u-u_{0}=u-\sum_{i=1}^{n_{s}}Y_{i}b_{i0},$ (5.2)
such that $u^{*}>0$ if and only if $T>0$, provided
$c_{v,i}>0,\>i=1,\ldots,n_{s}$ [60]. Pressure is then also positive. Note that
$C_{i}>0,\;\forall i$, implies $\rho>0$. This set is similar to that in [45,
46], except with the additional entropy constraint. In A, we review two
important lemmas from [36], the first of which establishes the quasi-concavity
of $s(y)$ (which holds in the multicomponent case as well) and the second of
which states
$s\left(\overline{y}_{\kappa}\right)\geq\min_{x\in\kappa}s\left(y(x)\right)$.
In B, we show that $\rho u^{*}(y)$ is a concave function of the state. For a
given $\sigma$, $\mathcal{G}_{\sigma}$ is then a convex set [34, 35, 48, 78].
We assume that the exact solution to the classical Riemann problem with
initial data
$y\left(x,0\right)=\begin{cases}y_{1},&x<0\\\ y_{2},&x>0\end{cases}$
is an entropy solution that preserves positivity. Then, by Lemmas 10 and 11,
$\mathcal{G}_{\sigma}$ is an invariant set [39, 79], i.e.,
$y_{1}\in\mathcal{G}_{\sigma}$ and $y_{2}\in\mathcal{G}_{\sigma}$ imply that
the average of the exact Riemann solution over a domain that includes the
Riemann fan [79] is in $\mathcal{G}_{\sigma}$.
Consider the following three-point system arising from a $p=0$, element-local
DG discretization with forward Euler time stepping:
$\displaystyle y_{\kappa}^{j+1}=y_{\kappa}^{j}-\frac{\Delta
t}{h}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j},y_{\kappa_{L}}^{j},-1\right)+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j},y_{\kappa_{R}}^{j},1\right)\right],$
(5.3)
where $j$ indexes the time step, $h$ is the element size, and $\kappa_{L}$ and
$\kappa_{R}$ are the elements to the left and right of $\kappa$, respectively.
Let $\lambda$ be an upper bound on the maximum wave speed of the system. Under
the following condition,
$\frac{\Delta t\lambda}{h}\leq\frac{1}{2},$ (5.4)
$y_{\kappa}^{j},y_{\kappa_{L}}^{j},y_{\kappa_{R}}^{j}\in\mathcal{G}_{\sigma}$
implies $y_{\kappa}^{j+1}\in\mathcal{G}_{\sigma}$ if certain _invariant-
region-preserving_ numerical fluxes are employed [22]. In particular,
$y_{\kappa}^{j+1}$ satisfies [22, 78]
$s\left(y_{\kappa}^{j+1}\right)\geq\min\left\\{s\left(y_{\kappa_{L}}^{j}\right),s\left(y_{\kappa}^{j}\right),s\left(y_{\kappa_{R}}^{j}\right)\right\\}.$
(5.5)
The Godunov, Lax-Friedrichs, HLL, and HLLC fluxes qualify [22] (see also [73,
17, 16, 80] for additional proofs regarding the Godunov and/or Lax-Friedrichs
fluxes), some of which allow for less restrictive time-step-size constraints
than (5.4). The proofs often rely on the notion that $\mathcal{G}_{\sigma}$ is
an invariant set, which itself invokes the aforementioned assumption that the
exact Riemann solution is an entropy solution satisfying the positivity
property. Two exceptions are the proof by Zhang and Shu [20] of the positivity
property of the Lax-Friedrichs flux and the proof by Lax [81] that the Lax-
Friedrichs flux satisfies a discrete cell entropy inequality for all
entropy/entropy-flux pairs, from which a sharper version of the inequality
(5.5) follows [73, 16] (note that the corresponding time-step-size constraints
for these two proofs are not necessarily the same as (5.4)). Unless otherwise
specified, we employ the HLLC numerical flux [82].
The three-point system (5.3) will be crucial in the construction of a
positivity-preserving and entropy-bounded DG scheme for $p>0$. Specifically,
we will show in the following subsection that the element average of the
solution (for $p>0$) at the next time step, $\overline{y}_{\kappa}^{j+1}$, can
be expressed as a convex combination of both pointwise values of
$y_{\kappa}^{j}(x)$ and three-point systems involving pointwise values of
$y_{\kappa}^{j}(x)$. If all of said pointwise values of $y_{\kappa}^{j}(x)$
are in $\mathcal{G}_{\sigma}$, then $\overline{y}_{\kappa}^{j+1}$ will also be
in $\mathcal{G}_{\sigma}$ under a time-step-size constraint. A simple limiter,
described in Section 5.2.1, is applied to ensure that the pointwise values of
$y_{\kappa}^{j}(x)$ are in $\mathcal{G}_{\sigma}$.
### 5.2 Entropy-bounded, high-order discontinuous Galerkin method in one
dimension
Suppose $\kappa=\left[x_{L},x_{R}\right]$. Let $x_{q}$ and $w_{q}$ denote the
quadrature points and weights, respectively, of a quadrature rule with
$x_{q}\in\kappa$ , $w_{q}>0$, and $\sum_{q=1}^{n_{q}}w_{q}=1$, where $n_{q}$
is the number of quadrature points/weights. This set of quadrature points does
not need to include the endpoints, and the quadrature rule need not be
explicitly used to evaluate any integrals in Equation (3.2). For now, we
assume that standard flux interpolation, as in Equation (3.8), is employed;
Section 5.2.2 discusses how to account for the modified flux interpolation in
Equation (3.9). As in [17], provided that the quadrature rule is sufficiently
accurate, the element-averaged solution in Equation (3.7) can be expanded as
$\displaystyle\overline{y}_{\kappa}$
$\displaystyle=\sum_{q=1}^{n_{q}}w_{q}y_{\kappa}(x_{q})$
$\displaystyle=\sum_{q=1}^{n_{q}}\theta_{q}y_{\kappa}(x_{q})+\theta_{L}y_{\kappa}(x_{L})+\theta_{R}y_{\kappa}(x_{R}).$
(5.6)
If the set of quadrature points includes the endpoints, then we can simply
take
$\theta_{q}=\begin{cases}w_{q}&x_{q}\neq x_{L},x_{q}\neq x_{R}\\\
0&\mathrm{otherwise}\end{cases}$
and
$\theta_{L}=w_{L},\quad\theta_{R}=w_{R},$
where $w_{L}$ and $w_{R}$ are the quadrature weights at the left and right
endpoints, respectively. If the set of quadrature points does not include the
endpoints, then we can instead take
$\theta_{q}=w_{q}-\theta_{L}\psi_{q}\left(x_{L}\right)-\theta_{R}\psi_{q}\left(x_{R}\right),$
where $\left\\{\psi_{1},\ldots,\psi_{n_{d}}\right\\}$, with $n_{b}\leq
n_{d}\leq n_{q}$, is a set of Lagrange basis functions whose nodes are located
at a subset of the quadrature points, while $\psi_{v}=0$ for
$v=n_{d}+1,\ldots,n_{q}$, such that [17]
$\displaystyle\sum_{q=1}^{n_{q}}\theta_{q}y_{\kappa}(x_{q})$
$\displaystyle=\sum_{q=1}^{n_{q}}\left[w_{q}-\theta_{L}\psi_{q}\left(x_{L}\right)-\theta_{R}\psi_{q}\left(x_{R}\right)\right]y_{\kappa}(x_{q})$
$\displaystyle=\sum_{q=1}^{n_{q}}w_{q}y_{\kappa}(x_{q})-\theta_{L}\sum_{q=1}^{n_{q}}y_{\kappa}(x_{q})\psi_{q}\left(x_{L}\right)-\theta_{R}\sum_{q=1}^{n_{q}}y_{\kappa}(x_{q})\psi_{q}\left(x_{R}\right)$
$\displaystyle=\sum_{q=1}^{n_{q}}w_{q}y_{\kappa}(x_{q})-\theta_{L}y_{\kappa}(x_{L})+\theta_{R}y_{\kappa}(x_{R}).$
$\theta_{L}$ and $\theta_{R}$ will be related to a constraint on the time step
size later in this section. The positivity of the quadrature weights
guarantees the existence of positive $\theta_{L}$ and $\theta_{R}$ that yield
$\theta_{q}\geq 0,\;q=1,\ldots,n_{q}$ [17]. Furthermore,
$\sum_{q}\theta_{q}+\theta_{L}+\theta_{R}=1$ since $\sum_{q}\psi_{q}=1$.
Define $\partial\mathcal{D}_{\kappa}=\left\\{x_{L,}x_{R}\right\\}$, and let
$\mathcal{D}_{\kappa}$ denote the set of points at which the state is
evaluated in Equation (5.6):
$\mathcal{D_{\kappa}}=\partial\mathcal{D}_{\kappa}\bigcup\left\\{x_{q},q=1,\ldots,n_{q}\right\\}=\left\\{x_{L,}x_{R},x_{q},q=1,\ldots,n_{q}\right\\}.$
Applying forward Euler time stepping to Equation (3.2) and taking
$\mathfrak{v}$ to be a vector of ones (i.e., $\mathfrak{v}\in V_{h}^{0}$)
gives the fully discrete scheme satisfied by the element averages [36, 17]:
$\displaystyle\overline{y}_{\kappa}^{j+1}$ $\displaystyle=$
$\displaystyle\overline{y}_{\kappa}^{j}-\frac{\Delta
t}{h}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{L}),y_{\kappa_{L}}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{R}),y_{\kappa_{R}}^{j}(x_{R}),1\right)\right]$
(5.8) $\displaystyle=$
$\displaystyle\sum_{q=1}^{n_{q}}\theta_{q}y_{\kappa}^{j}(x_{q})+\theta_{L}y_{\kappa}^{j}(x_{L})-\frac{\Delta
t}{h}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{L}),y_{\kappa_{L}}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{L}),y_{\kappa}^{j}(x_{R}),1\right)\right]$
$\displaystyle+\theta_{R}y_{\kappa}^{j}(x_{R})-\frac{\Delta
t}{h}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{R}),y_{\kappa}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{R}),y_{\kappa_{R}}^{j}(x_{R}),1\right)\right],$
where the second equality is due to the conservation property of the numerical
flux:
$\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{L}),y_{\kappa}^{j}(x_{R}),1\right)=-\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{R}),y_{\kappa}^{j}(x_{L}),-1\right).$
Note that Equations (5.8) and (5.8) hold regardless of whether the integrals
in Equation (3.2) are evaluated using conventional quadrature or a quadrature-
free implementation [69, 70]. Though the forward Euler time integration scheme
is used here, strong-stability-preserving Runge-Kutta (SSPRK) methods [83,
84], which are convex combinations of forward Euler steps, are compatible as
well. Equation (5.8) then leads to the following theorem, where we use
$y_{\kappa}^{-}$ to denote the exterior state along $\partial\kappa$.
###### Theorem 1 ([20, 36, 17]).
If $y_{\kappa}^{j}(x)\in\mathcal{G}_{\sigma},\;\forall
x\in\mathcal{D_{\kappa}}$, and
$y_{\kappa}^{-,j}\in\mathcal{G}_{\sigma},\;\forall
x\in\partial\mathcal{D}_{\kappa}$, with
$\sigma\leq\min\left\\{\min\left\\{s\left(y_{\kappa}^{j}(x)\right)|x\in\mathcal{D_{\kappa}}\right\\},\min\left\\{s\left(y_{\kappa}^{-,j}(x)\right)|x\in\mathcal{\partial
D_{\kappa}}\right\\}\right\\},$ (5.9)
then $\overline{y}_{\kappa}^{j+1}$ in Equation (5.8) is also in
$\mathcal{G}_{\sigma}$ under the constraint
$\frac{\Delta
t\lambda}{h}\leq\frac{1}{2}\min\left\\{\theta_{L},\theta_{R}\right\\}$ (5.10)
and the conditions
$\theta_{L}>0,\theta_{R}>0,\theta_{q}\geq 0,q=1,\ldots,n_{q}.$ (5.11)
###### Proof.
The proof follows the same procedure as in [20], [36], [17], and related
papers, which we review here. We first rewrite Equation (5.8) as
$\overline{y}_{\kappa}^{j+1}=\sum_{q=1}^{n_{q}}\theta_{q}y_{\kappa}^{j}(x_{q})+\theta_{L}y_{\kappa,s1}^{j+1}+\theta_{R}y_{\kappa,s2}^{j+1},$
where
$\displaystyle y_{\kappa,s1}^{j+1}$
$\displaystyle=y_{\kappa}^{j}(x_{L})-\frac{\Delta
t}{\theta_{L}h}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{L}),y_{\kappa_{L}}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{L}),y_{\kappa}^{j}(x_{R}),1\right)\right],$
$\displaystyle y_{\kappa,s2}^{j+1}$
$\displaystyle=y_{\kappa}^{j}(x_{R})-\frac{\Delta
t}{\theta_{R}h}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{R}),y_{\kappa}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}(x_{R}),y_{\kappa_{R}}^{j}(x_{R}),1\right)\right].$
As such, $\overline{y}_{\kappa}^{j+1}$ is a convex combination of
$y_{\kappa}^{j}(x_{q})$ evaluated at $x_{q}$ and two three-point systems of
the type (5.3). Under the conditions (5.10) and (5.11), $y_{\kappa,s1}^{j+1}$
and $y_{\kappa,s2}^{j+1}$ are both in $\mathcal{G}_{\sigma}$. It then follows
that $\overline{y}_{\kappa}^{j+1}\in\mathcal{G}_{\sigma}$.
∎
###### Remark 2.
A direct result of Theorem 1 is that
$s\left(\overline{y}_{\kappa}^{j+1}\right)\geq\min\left\\{\min\left\\{s\left(y_{\kappa}^{j}(x)\right)|x\in\mathcal{D_{\kappa}}\right\\},\min\left\\{s\left(y_{\kappa}^{-,j}(x)\right)|x\in\mathcal{\partial
D_{\kappa}}\right\\}\right\\}.$
According to the inequality (5.10), the upper bound on the time step size is
proportional to $\min\left\\{\theta_{L},\theta_{R}\right\\}$. For Gauss-
Lobatto quadrature, $\theta_{L}$ and $\theta_{R}$ are both equal to the
quadrature weight corresponding to either endpoint. See [17] for information
about Gauss-Legendre quadrature, as well as a discussion on how to find the
maximum allowable value of $\min\left\\{\theta_{L},\theta_{R}\right\\}$ for
general quadrature rules.
To complete the construction of an entropy-bounded, high-order DG scheme, we
need to enforce not only the positivity of $y_{\kappa}^{j+1}(x),\>\forall
x\in\mathcal{D_{\kappa}}$, for all $\kappa\in\Omega$, but also
$s\left(y^{j+1}(x)\right)>s_{b},\>\forall x\in\mathcal{D_{\kappa}}$, for all
$\kappa\in\Omega$, where $s_{b}$ is a lower bound on the specific
thermodynamic entropy. $s_{b}$ can vary among elements and time steps, and the
prescription of $s_{b}$ should be motivated by the physical principles
examined in Section 4. We will discuss $s_{b}$ in more detail in Section
5.2.3. In other words, we enforce
$y_{\kappa}^{j+1}(x)\in\mathcal{G}_{s_{b}},\>\forall
x\in\mathcal{D_{\kappa}}$, which is done via a simple limiting procedure that
will be described in Section 5.2.1.
In practice, we relax some of the requirements introduced thus far. First,
$\lambda$ is computed in a local (instead of global) manner and is calculated
as the maximum value of $\left|v\right|+c$, where $c$ is the speed of sound,
over the points of interest. However, $\left|v\right|+c$ does not bound the
wave speeds arising from the interactions between states at interfaces. A
similar remark can be made for most invariant-region-preserving numerical flux
functions, which typically require wave-speed estimates. Simple algorithms for
bounding the wave speeds in the monocomponent case have been developed [85,
86]; extending these to the multicomponent Euler equations may indeed be
worthy of future investigation, with [87] as one example. Second, we introduce
$\chi_{\sigma}=\rho s-\rho\sigma$, which is concave with respect to the state
[22], and revise the definition of $\mathcal{G}_{\sigma}$ as
$\mathcal{G}_{\sigma}=\left\\{y\mid\rho>0,\rho u^{*}>0,C_{1}\geq
0,\ldots,C_{n_{s}}\geq 0,\chi_{\sigma}\geq 0\right\\},$ (5.12)
where $\chi_{\sigma}\geq 0$ is a reformulation of $s\geq\sigma$ and the
species concentrations are now allowed to be equal to zero. From a practical
standpoint, allowing $C_{i}=0$ is necessary since the concentrations are
frequently zero in many reacting flow problems of interest. Unfortunately,
entropy functions of the form $U=-\rho f_{\epsilon}(s)$ and $U=-\rho s$ are no
longer convex if any of the concentrations is zero [16, 51]. Furthermore, the
specific thermodynamic entropy becomes ill-defined. Nevertheless, by making
use of $0\log 0=0$ [60, Chapter 6], $\rho s$ and thus $\chi_{\sigma}$ remain
well-defined. The entire methodology developed here also remains well-defined,
unlike entropy-stable schemes that rely on the entropy variables associated
with $U=-\rho s$. Throughout this work, we did not encounter any major issues
associated with relaxing the two aforementioned requirements. One potential
reason is that $\overline{y}_{\kappa}^{j+1}$ can be in $\mathcal{G}_{\sigma}$
even if $\mathcal{G}_{\sigma}$ is not convex and/or some of the conditions in
Theorem 1 are not satisfied. Furthermore, in this work, we choose
$\mathrm{CFL}=0.1$ to maintain low temporal errors, which in general yields a
smaller time step size than necessary. Should issues emerge in future work,
they can likely be alleviated by adaptively decreasing the time step size
[35].
Finally, we remark that $\mathcal{D}_{\kappa}$ is simply the set of points at
which limiting should be applied. Specifically, the interior quadrature points
in $\mathcal{D}_{\kappa}$ need not be explicitly used in numerical
integrations in Equation (3.2); if they are indeed not, the actual integration
points are added to $\mathcal{D}_{\kappa}$ as well [35].
#### 5.2.1 Limiting procedure
In this subsection, we describe the limiting procedure to ensure
$y_{\kappa}^{j+1}(x)\in\mathcal{G}_{s_{b}},\>\forall
x\in\mathcal{D_{\kappa}}$. It is assumed that
$\overline{y}_{\kappa}^{j+1}(x)\in\mathcal{G}_{s_{b}}$. For brevity, we drop
the $j+1$ superscript and $\kappa$ subscript in this discussion. The limiting
operator is of the same form as in [44], [35], [22], [78], and related papers.
1. 1.
First, positivity of density is enforced. Specifically, if
$\rho(x)>\epsilon,\>\forall x\in\mathcal{D}_{\kappa}$, where $\epsilon$ is a
small positive number (e.g., $\epsilon=10^{-10}$), then set
$C_{i}^{(1)}=C_{i}=\sum_{j=1}^{n_{b}}C_{i}(x_{j})\phi_{j},i=1,\ldots,n_{s}$;
otherwise, compute
$C_{i}^{(1)}=\overline{C}_{i}+\theta^{(1)}\left(C_{i}-\overline{C}_{i}\right),\;i=1,\ldots,n_{s},$
with
$\theta^{(1)}=\frac{\rho(\overline{y})-\epsilon}{\rho(\overline{y})-\underset{x\in\mathcal{D}}{\min}\rho(y(x))}.$
2. 2.
Next, nonnegativity of the species concentrations is enforced. If
$C_{i}^{(1)}(x)\geq 0,\>\forall x\in\mathcal{D}_{\kappa}$, then set
$C_{i}^{(2)}=C_{i}^{(1)},i=1,\ldots,n_{s}$; otherwise, compute
$C_{i}^{(2)}=\overline{C}_{i}+\theta^{(2)}\left(C_{i}^{(1)}-\overline{C}_{i}\right),\;i=1,\ldots,n_{s},$
with
$\theta^{(2)}=\frac{\overline{C}_{i}}{\overline{C}_{i}-\underset{x\in\mathcal{D}}{\min}C_{i}^{(1)}(x)}.$
Let $y^{(2)}=\left(\rho v_{1},\ldots,\rho v_{d},\rho
e_{t},C_{1}^{(2)},\ldots,C_{n_{s}}^{(2)}\right)$.
3. 3.
Positivity of $\rho u^{*}(y)$ is then enforced. If $\rho
u^{*}\left(y^{(2)}(x)\right)>\epsilon,\>\forall x\in\mathcal{D}_{\kappa}$,
then set $y^{(3)}=y^{(2)}$; otherwise, compute
$y^{(3)}=\overline{y}+\theta^{(3)}\left(y^{(2)}-\overline{y}\right),$
with
$\theta^{(3)}=\frac{\rho u^{*}(\overline{y})-\epsilon}{\rho
u^{*}(\overline{y})-\underset{x\in\mathcal{D}}{\min}\rho u^{*}(y^{(2)}(x))}.$
It can be shown that $\rho u^{*}(y^{(3)}(x))>0,\>\forall
x\in\mathcal{D}_{\kappa}$ by concavity [44, 35]. The “positivity-preserving
limiter” refers to the limiting procedure up to this point. The “entropy
limiter” corresponds to the following step (in addition to the above steps).
4. 4.
Finally, the entropy constraint is enforced. If
$\chi\left(y^{(3)}(x)\right)\geq 0,\>\forall x\in\mathcal{D}_{\kappa}$, then
set $y^{(4)}=y^{(3)}$; otherwise, compute
$y^{(4)}=\overline{y}+\theta^{(4)}\left(y^{(3)}-\overline{y}\right),$
with
$\theta^{(4)}=\frac{\chi(\overline{y})}{\chi(\overline{y})-\underset{x\in\mathcal{D}}{\min}\chi(y^{(3)}(x))}.$
It can be shown that $s\left(y^{(4)}(x)\right)\geq s_{b},\>\forall
x\in\mathcal{D}_{\kappa}$, by concavity of $\chi$ [22, 78].
$y^{(4)}$ then replaces $y$ as the solution. The limiting operator is
conservative, maintains stability, and in general preserves the formal order
of accuracy for smooth solutions [20, 35, 36, 17, 22]. There is extensive
empirical evidence demonstrating preservation of accuracy (see previously
cited references, as well as [46], [78], and related papers). However, the
order of accuracy can potentially deteriorate when the element average is
close to the boundary of $\mathcal{G}_{s_{b}}$ [35, 22]. Furthermore, the
linear scaling is not expected to suppress all oscillations [20, 22, 17, 78].
The limiting procedure described here is applied at the end of every RK stage.
#### 5.2.2 Modified flux interpolation
We now discuss how to account for over-integration with the modified flux
interpolation in Equation (3.9). The scheme satisfied by the element averages
becomes
$\displaystyle\overline{y}_{\kappa}^{j+1}$ $\displaystyle=$
$\displaystyle\overline{y}_{\kappa}^{j}-\frac{\Delta
t}{h}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{L}),\widetilde{y}_{\kappa_{L}}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{R}),\widetilde{y}_{\kappa_{R}}^{j}(x_{R}),1\right)\right]$
(5.13) $\displaystyle=$
$\displaystyle\sum_{q=1}^{n_{q}}\theta_{q}y_{\kappa}^{j}(x_{q})+\theta_{L}y_{\kappa}^{j}(x_{L})-\frac{\Delta
t}{h}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{L}),\widetilde{y}_{\kappa_{L}}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{L}),\widetilde{y}_{\kappa}^{j}(x_{R}),1\right)\right]$
$\displaystyle+\theta_{R}y_{\kappa}^{j}(x_{R})-\frac{\Delta
t}{h}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{R}),\widetilde{y}_{\kappa}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{R}),\widetilde{y}_{\kappa_{R}}^{j}(x_{R}),1\right)\right].$
Equation (5.13) can be rewritten as
$\overline{y}_{\kappa}^{j+1}=\sum_{q=1}^{n_{q}}\theta_{q}y_{\kappa}^{j}(x_{q})+\theta_{L}y_{\kappa,s3}^{j+1}+\theta_{R}y_{\kappa,s4}^{j+1},$
where
$\displaystyle y_{\kappa,s3}^{j+1}$
$\displaystyle=y_{\kappa}^{j}(x_{L})-\frac{\Delta
t}{\theta_{L}h}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{L}),\widetilde{y}_{\kappa_{L}}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{L}),\widetilde{y}_{\kappa}^{j}(x_{R}),1\right)\right],$
$\displaystyle y_{\kappa,s4}^{j+1}$
$\displaystyle=y_{\kappa}^{j}(x_{R})-\frac{\Delta
t}{\theta_{R}h}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{R}),\widetilde{y}_{\kappa}^{j}(x_{L}),-1\right)+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}(x_{R}),\widetilde{y}_{\kappa_{R}}^{j}(x_{R}),1\right)\right],$
which are not necessarily of the type (5.3) since in general,
$y_{\kappa}^{j}(x_{L})\neq\widetilde{y}_{\kappa}^{j}(x_{L})$ and
$y_{\kappa}^{j}(x_{R})\neq\widetilde{y}_{\kappa}^{j}(x_{R})$. The
incompatibility is a result of expressing $\overline{y}_{\kappa}$ as a convex
combination of pointwise values of $y_{\kappa}(x)$ (as opposed to
$\widetilde{y}_{\kappa}(x)$). Unfortunately, the element average of
$\widetilde{y}_{\kappa}$, denoted $\overline{\widetilde{y}}_{\kappa}$, is not
necessarily equal to $\overline{y}_{\kappa}$; consequently,
$\overline{y}_{\kappa}$ cannot be directly written as a convex combination of
pointwise values of $\widetilde{y}_{\kappa}(x)$. However, if the set of
solution nodes includes the endpoints (e.g., equidistant or Gauss-Lobatto
points), then Equation (5.13) recovers Equation (5.8) since
$y_{\kappa}^{j}(x_{L})=\widetilde{y}_{\kappa}^{j}(x_{L})$ and
$y_{\kappa}^{j}(x_{R})=\widetilde{y}_{\kappa}^{j}(x_{R})$. The previous
analysis, including Theorem 1, then holds. As such, for a nodal set that
includes the endpoints, the modified flux interpolation in Equation (3.9) does
not introduce any additional difficulties to the discussed framework. Note
also that the second term in Equation (3.2) (i.e., the volumetric flux
integral) does not factor into the scheme satisfied by the element averages,
so the modified flux interpolation can be freely employed in said integral.
#### 5.2.3 Lower bound on specific thermodynamic entropy
We consider two options for specifying $s_{b}$. The first option is a _global_
entropy bound [36, 22, 78]:
$s_{b}(y)=\min\left\\{s\left(y(x)\right)|x\in\Omega\right\\},$ (5.14)
which can be evaluated once based on the initial condition, $y_{0}(x)$, or
updated at each time step [16]. Instead of calculating the true minimum via,
for example, Newton’s method, we either rely on user-specified information or
compute
$s_{b}(y)=\min\left\\{s\left(y(x)\right)|x\in\bigcup_{\kappa\in\mathcal{T}}\mathcal{D_{\kappa}}\right\\}.$
(5.15)
The second option is a _local_ entropy bound, which should satisfy
$s_{b,\kappa}^{j+1}(y)\leq\min\left\\{\min\left\\{s\left(y_{\kappa}^{j}(x)\right)|x\in\mathcal{D_{\kappa}}\right\\},\min\left\\{s\left(y_{\kappa}^{-,j}(x)\right)|x\in\mathcal{\partial
D_{\kappa}}\right\\}\right\\},$
in order to ensure compatibility with Theorem 1 and the limiting procedure,
which enforces
$y_{\kappa}^{j+1}(x)\in\mathcal{G}_{s_{b,\kappa}^{j+1}},\>\forall
x\in\mathcal{D_{\kappa}}$. Lv and Ihme [17] introduced the following local
entropy bound:
$s_{b,\kappa}^{j+1}(y)=\min\left\\{\min\left\\{s\left(y_{\kappa}^{j}(x)\right)|x\in\kappa\right\\},\min\left\\{s\left(y_{\kappa}^{-,j}(x)\right)|x\in\partial\kappa\right\\}\right\\}.$
(5.16)
They demonstrated that the local entropy bound (5.16) can more effectively
dampen overshoots and undershoots when the entropy varies significantly
throughout the domain (e.g., when multiple discontinuities are present).
However, we find that Equation (5.16) is often too restrictive, as will be
illustrated in Section 7.1.
Here, we employ a different local entropy bound,
$s_{b,\kappa}^{j+1}(y)=\min\left\\{s\left(y^{j}(x)\right)|x\in\kappa\cup\kappa_{L}\cup\kappa_{R}\right\\},$
(5.17)
which is based on the local minimum entropy principle satisfied by exact
entropy solutions (or discrete entropy solutions in the limit of infinite
resolution). Specifically, the inequality
$s(y(x,t))\geq\min\left\\{s\left(y\left(x,t_{0}\right)\right)|x\in\kappa\cup\kappa_{L}\cup\kappa_{R}\right\\},\quad
t\in\left[t_{0},t_{0}+\Delta t\right],$ (5.18)
can be considered the semi-discrete analog of (4.8) with $R=h/2$ and
$\kappa=\left[-h/2,h/2\right]$, under the condition
$\frac{\Delta tv_{\max}}{h}\leq 1.$ (5.19)
The RHS of Equation (5.17) is the fully discrete analog of the RHS of the
inequality (5.18). This property is imposed on the discrete solution as a
means to suppress instabilities. A similar bound was employed by Dzanic and
Witherden [88] in their entropy-based filtering framework. Note that if
$\Delta t$ satisfies (5.10), then it also satisfies (5.19) since
$\Delta
t\leq\frac{h}{2\lambda}\min\left\\{\theta_{L},\theta_{R}\right\\}\leq\frac{h}{2\lambda}\leq\frac{h}{\lambda}\leq\frac{h}{v_{\max}}.$
An alternate viewpoint draws from the generalized Riemann problem (GRP) [82],
which differs from the _classical_ Riemann problem by allowing for source
terms and piecewise smooth (as opposed to piecewise constant) initial
conditions. Suppose that $y_{\kappa_{L}}$ and $y_{\kappa}$ form the initial
conditions of a GRP centered at $x_{L}$, while $y_{\kappa}$ and
$y_{\kappa_{R}}$ form the initial conditions of a GRP centered at $x_{R}$.
Under the (potentially strong) assumption that exact solutions
($y_{L}^{\mathrm{GRP}}$ and $y_{R}^{\mathrm{GRP}}$, respectively) to the GRPs
exist, as well as the assumption that those solutions satisfy all entropy
inequalities, the exact solution in $\kappa$, $y_{\kappa}^{\mathrm{ex}}$,
arising from the GRPs satisfies [71]
$\int_{\kappa}U\left(y_{\kappa}^{\mathrm{ex}}(x,t)\right)\leq\int_{\kappa}U\left(y_{\kappa}(x,t_{0})\right)dx-\int_{t_{0}}^{t}\mathcal{F}^{s}\left(y_{R}^{\mathrm{GRP}}\left(x_{R},\tau\right)\right)d\tau+\int_{t_{0}}^{t}\mathcal{F}^{s}\left(y_{L}^{\mathrm{GRP}}\left(x_{L},\tau\right)\right)d\tau,$
at least before the local Riemann problems interact [89]. With
$\left(U,\mathcal{F}^{s}\right)=\left(-\rho f_{0}(s),-\rho
v_{1}f_{0}(s)\right)$ and
$s_{0}=\min\left\\{s\left(y\left(x,t_{0}\right)\right)|x\in\kappa\cup\kappa_{L}\cup\kappa_{R}\right\\}$,
the above inequality becomes
$\displaystyle\int_{\kappa}\rho\left(y_{\kappa}^{\mathrm{ex}}(x,t)\right)f_{0}\left(s\left(y_{\kappa}^{\mathrm{ex}}(x,t)\right)\right)dx$
$\displaystyle\geq$
$\displaystyle\int_{\kappa}\rho\left(y_{\kappa}(x,t_{0})\right)\min\left\\{s\left(y_{\kappa}(x,t_{0})\right)-s_{0},0\right\\}dx$
$\displaystyle-\int_{t_{0}}^{t}\rho
v_{1}\left(y_{R}^{\mathrm{GRP}}\left(x_{R},\tau\right)\right)\min\left\\{s\left(y_{R}^{\mathrm{GRP}}\left(x_{R},\tau\right)\right)-s_{0},0\right\\}dx$
$\displaystyle+\int_{t_{0}}^{t}\rho
v_{1}\left(y_{L}^{\mathrm{GRP}}\left(x_{L},\tau\right)\right)\min\left\\{s\left(y_{L}^{\mathrm{GRP}}\left(x_{L},\tau\right)\right)-s_{0},0\right\\}dx,$
where the first term on the RHS clearly vanishes and the second and third
terms vanish due to Equation (4.8). As such, we have
$s\left(y_{\kappa}^{\mathrm{ex}}(x,t)\right)\geq s_{0}$. This property is then
imposed on the discrete solution via the local entropy bound in Equation
(5.17).
To our knowledge, it is unclear whether the assumption that exact solutions to
the GRP exist is valid (even for calorically perfect gases, let alone mixtures
of thermally perfect gases). Semi-analytical methods for solving GRPs have
been developed; see [82, Chapter 19] for a detailed description and [90] for a
comparison of such methods. Discussions on exact solutions to the classical
Riemann problem for mixtures of thermally perfect gases can be found in [91,
92].
To avoid finding the true minimum over a given element via an iterative
procedure, we relax Equation (5.17) and instead compute
$s_{b,\kappa}^{j+1}(y)=\mathsf{c}\min\left\\{s\left(y^{j}(x)\right)|x\in\mathcal{D}_{\kappa}\cup\mathcal{D}_{\kappa_{L}}\cup\mathcal{D}_{\kappa_{R}}\right\\},$
(5.20)
where $\mathsf{c}\in(0,1]$ is a relaxation parameter. The entropy limiter in
Section 5.2.1 remains valid since
$s\left(\overline{y}_{\kappa}^{j+1}\right)\geq\min\left\\{s\left(y^{j}(x)\right)|x\in\mathcal{D_{\kappa}}\cup\mathcal{D}_{\kappa}^{-}\right\\}\geq\min\left\\{s\left(y^{j}(x)\right)|x\in\mathcal{D}_{\kappa}\cup\mathcal{D}_{\kappa_{L}}\cup\mathcal{D}_{\kappa_{R}}\right\\}\geq
s_{b,\kappa}^{j+1}(y).$
For $\mathsf{c}<1$, it can be useful to simultaneously account for a well-
defined global entropy bound, $s_{b}$, as
$s_{b,\kappa}^{j+1}(y)=\max\left\\{s_{b},\mathsf{c}\min\left\\{s\left(y^{j}(x)\right)|x\in\mathcal{D}_{\kappa}\cup\mathcal{D}_{\kappa_{L}}\cup\mathcal{D}_{\kappa_{R}}\right\\}\right\\}.$
(5.21)
In this work, we simply choose $\mathsf{c}=1$. An alternative approach for
estimating the true minimum in an algebraic manner can be found in [17]. Note
that Equation (5.20) with $\mathsf{c}=1$ yields the true minimum for $p=1$ if
the element endpoints are the solution nodes and included in
$\mathcal{D}_{\kappa}$. For $\kappa=\left[x_{L},x_{R}\right]$, the basis
functions are given by
$\phi_{1}(x)=\frac{x_{R}-x}{x_{R}-x_{L}},\phi_{2}(x)=\frac{x-x_{L}}{x_{R}-x_{L}},$
such that $\phi_{1}+\phi_{2}=1$ and $\phi_{i}(x)\in(0,1),\>\forall
x\in\left(x_{L},x_{R}\right)$. Therefore,
$y_{\kappa}(x)=y_{\kappa}\left(x_{1}\right)\phi_{1}(x)+y_{\kappa}\left(x_{2}\right)\phi_{2}(x)$
is a convex combination of the endpoint values of the solution. By Lemma 9,
$\min_{x}s\left(y_{\kappa}(x)\right)=\min\left\\{s\left(y_{\kappa}\left(x_{1}\right)\right),s\left(y_{\kappa}\left(x_{2}\right)\right)\right\\}$.
This is similarly true in two and three dimensions.
#### 5.2.4 Artificial viscosity
As will be demonstrated in Section 7.2, the proposed entropy-bounded DG method
does not completely suppress smaller-scale oscillations, especially in the
presence of flow-field discontinuities. Therefore, artificial viscosity is
employed in certain test cases in Section 7 to more effectively dampen such
oscillations. Specifically, the following dissipation term is added to the LHS
of Equation (3.2) [66]:
$-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{F}^{\mathrm{AV}}\left(y,\nabla
y\right),\nabla\mathfrak{v}\right)_{\kappa},$ (5.22)
where
$\mathcal{F}^{\mathrm{AV}}(y,\nabla y)=\nu_{\mathrm{AV}}\nabla y,$
with $\nu_{\mathrm{AV}}\geq 0$ denoting the artificial viscosity. The
artificial viscosity is computed as [14]
$\nu_{\mathrm{AV}}=\left(C_{\mathrm{AV}}+S_{\mathrm{AV}}\right)\left(\frac{h^{2}}{p+1}\left|\frac{\partial
T}{\partial y}\cdot\frac{\mathcal{R}\left(y,\nabla
y\right)}{T}\right|\right).$
$C_{\mathrm{AV}}$ is a user-defined coefficient, $S_{\mathrm{AV}}$ is a shock
sensor based on intra-element variations [25], and $\mathcal{R}\left(y,\nabla
y\right)$ is the strong form of the residual (2.1). Note that Equation (5.8)
(the scheme satisfied by the element averages) remains the same since the
dissipation term (5.22) vanishes for $\mathfrak{v}\in V_{h}^{0}$. Therefore,
Theorem 1 still holds. This type of artificial viscosity was found to
effectively suppress spurious oscillations in the vicinity of flow-field
discontinuities in multicomponent reacting flows [14]. However, we remark that
the artificial-viscosity formulation presented here is not the focus of this
paper; other types of artificial viscosity or limiters can be employed to
dampen the small-scale instabilities that the linear-scaling limiter fails to
cure, provided that the element-local averages are unmodified.
## 6 Reaction step: Entropy-stable discontinuous Galerkin method for ODE
integration
In this section, we describe the entropy-stable DG discretization of Equation
(3.4) (i.e., the ordinary differential equation (ODE) with stiff chemical
source terms). We build on DGODE, the (non-entropy-stable) DG method for ODE
integration described in [14]. The local, semi-discrete integral form of
Equation (3.4) is given by
$\displaystyle\int_{\kappa}\mathfrak{v}^{T}\frac{\partial y}{\partial
t}dx-\int_{\kappa}\mathfrak{v}^{T}\mathcal{S}(y)dx=0.$ (6.1)
Approximating $\mathcal{S}\left(y\right)$ locally as a polynomial in
$V_{h}^{p}$,
$\mathcal{S}_{\kappa}\approx\sum_{j=1}^{n_{b}}\mathcal{S}\left(y\left(x_{j}\right)\right)\phi_{j},$
we can write
$\frac{d}{dt}y_{\kappa}\left(x_{j},t\right)-\mathcal{S}\left(y_{\kappa}\left(x_{j},t\right)\right)=0,\quad
j=1,\ldots,n_{b},$
which is a spatially decoupled system of ODEs advanced at the solution nodes
from $t=t_{0}$ to $t=t_{f}$. Our goal here is to ensure
$\displaystyle y_{\kappa}\left(x_{j},t_{f}\right)$
$\displaystyle\in\mathcal{G}_{s\left(y_{\kappa}\left(x_{j},t_{0}\right)\right)},\quad
j=1,\ldots,n_{b}.$
Assuming a Gauss-Lobatto nodal set, since $\overline{y}_{\kappa}(t_{f})$ is a
convex combination of the nodal values, we have
$\overline{y}_{\kappa}(t_{f})\in\mathcal{G}_{s_{b}}$, where $s_{b}$ is now
given by
$s_{b}=\min_{j=1,\ldots,n_{b}}s\left(y_{\kappa}\left(x_{j},t_{0}\right)\right).$
The limiting procedure described in Section 5.2.1 can then be applied to
enforce $y_{\kappa}\left(x,t_{f}\right)\in\mathcal{G}_{s_{b}},\>\forall
x\in\mathcal{D}_{\kappa}$ (unless
$\mathcal{D}_{\kappa}=\left\\{x_{j},j=1,\ldots,n_{b}\right\\}$, in which case
the limiting procedure is unnecessary).
In the following, we drop the “$\kappa$” and “$j$” subscripts, yielding
$\frac{dy}{dt}-S(y)=0,$ (6.2)
which is the system of ODEs solved at each node. Note that the formulation
described here is slightly different from that in [12].
### 6.1 Review: DGODE
We first briefly review DGODE, referred to as “standard DGODE,” which deals
with the following one-dimensional DG discretization in time of Equation
(6.2):
$\displaystyle
N_{h}\left(y,\mathfrak{v}\right)=\sum_{\epsilon\in\mathcal{E}}\left(y^{\dagger}\left(y^{+},y^{-},n\right),\left\llbracket\mathfrak{v}\right\rrbracket\right)_{\mathcal{E}}-\sum_{\kappa\in\mathcal{T}}\left(y,\frac{d\mathfrak{v}}{dt}\right)_{\kappa}-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{S}\left(y\right),\mathfrak{v}\right)_{\kappa}=0\qquad\forall\mathfrak{v}\in
V_{h}^{p},$ (6.3)
where $\epsilon$, $\mathcal{E}$, $\kappa$, $\mathcal{T}$, and $V_{h}^{p}$ are
temporal analogs of the spatial counterparts defined in Section 3.
Specifically, $\mathcal{E}$ is the set of temporal interfaces $\epsilon$,
$\mathcal{T}$ is the set of cells $\kappa$ that partitions the computational
domain, $\Omega=\left(t_{0},t_{f}\right)=\left(t_{0},t_{0}+\Delta t\right)$,
and $V_{h}^{p}$ is the discrete subspace defined similarly to Equation (3.1).
Note that “cell” in this context corresponds to a sub-time-step (referred to
in this section as simply “time step”) of size $h\in(0,\Delta t]$. Equation
(6.3) is obtained by integrating Equation (6.2) over each cell, performing
integration by parts on the time-derivative terms, and summing over the
domain. On interior faces, the numerical temporal flux (henceforth referred to
as the “numerical state”), $y^{\dagger}$, is defined as the upwind flux
function,
$\displaystyle
y^{\dagger}\left(y^{+},y^{-},n\right)=\begin{cases}y^{+}&\textup{ if }n\geq
0\\\ -y^{-}&\textup{ if }n<0\end{cases},$ $\displaystyle\textup{ on
}\epsilon\qquad\forall\epsilon\in\mathcal{E_{I}}.$ (6.4)
On exterior interfaces, the numerical state is defined as
$\displaystyle
y^{\dagger}\left(y^{+},y^{-},n\right)=\begin{cases}y^{+}&\textup{ if }n\geq
0\\\ -y_{\partial}\left(y^{+}\right)&\textup{ if }n<0\end{cases},$
$\displaystyle\textup{ on
}\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\partial},$ (6.5)
where $y_{\partial}\left(y^{+}\right)$ is a prescribed boundary state. At the
inflow interface, located at $t=t_{0}$, $y_{\partial}=y_{0}$, which is the
initial condition. At the outflow interface, located at $t=t_{f}$,
$y_{\partial}=y^{+}$ (i.e., no boundary condition is imposed).
Let $m=\dim V_{h}^{p}$ and $\left(\phi_{1},\ldots,\phi_{m}\right)$ be a basis
for $V_{h}^{p}$. The discrete residual,
$\mathcal{R}=\left(\mathcal{R}_{1}\left(y\right),\ldots,\mathcal{R}_{m}\left(y\right)\right)$
is defined as
$\mathcal{R}_{i}\left(y\right)=\sum_{\epsilon\in\mathcal{E}}\left(y^{\dagger}\left(y^{+},y^{-},n\right),\left\llbracket\phi_{i}\right\rrbracket\right)_{\mathcal{E}}-\sum_{\kappa\in\mathcal{T}}\left(y,\frac{d\phi{}_{i}}{dt}\right)_{\kappa}-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{S}\left(y\right),\phi_{i}\right)_{\kappa}$
(6.6)
for $i=1,\ldots,m$. We can then recast (6.3) as
$\mathcal{R}\left(y\right)=0,$
which is solved for $y$ via Newton’s method. With initial guess $y^{0}$, the
$k$th update, $y^{k}$, is computed by solving the linear system
$\frac{d}{dy}\mathcal{R}\left(y^{k}\right)\left(y^{k+1}-y^{k}\right)=-\mathcal{R}\left(y^{k}\right)$
(6.7)
recursively until a convergence criterion is satisfied.
Johnson and Kercher [14] introduced a simple yet effective $hp$-adaptation
strategy to control the time step, $h$, and polynomial degree, $p$. They
applied the strategy to efficiently and accurately integrate chemical systems
with complex mechanisms. Here, we focus on $h$-adaptation with uniform $p$. W
define the norm of the local error estimate as
$\mathrm{err}_{h}=\left\|\frac{\mathcal{R}(y)}{\epsilon_{\mathrm{abs}}+\epsilon_{\mathrm{rel}}\left|y\right|}\right\|,$
where $\epsilon_{\mathrm{abs}}$ and $\epsilon_{\mathrm{rel}}$ are user-
specified absolute and relative tolerances, respectively. The convergence
criterion is
$\mathrm{err}_{h}<1.$ (6.8)
If (6.8) is satisfied within a user-specified number of Newton iterations, the
solution is updated and a new time step is determined using Gustafsson’s
method [93]. Otherwise, the time step is reduced by a factor of ten.
In this work, given that implicit time stepping schemes with order greater
than one are typically not unconditionally positivity-preserving [94], we
introduce an additional convergence criterion that improves stability and more
robustly maintains conservation of mass:
$C_{i}\geq 0,\quad i=1,\ldots,n_{s}.$ (6.9)
Specifically, we require the solution at the end of each time step to satisfy
(6.9). In our experience, this additional criterion does not significantly
restrict the time step size. Furthermore, the above criterion can be relaxed.
Instead of requiring pointwise __ nonnegativity of the species concentrations,
we can simply require that the spatial averages of the concentrations over the
element be nonnnegative. The positivity-preserving limiter in Section 5.2.1
can then be applied to guarantee nonnegativity of the species concentrations
in a pointwise manner. Though not pursued in this work, the conservative and
positivity-preserving projection method by Sandu [95] can also be employed
when (6.9) is violated. Another possible convergence criterion is $T>0$, but
this is almost never a concern.
### 6.2 Entropy stability: Preliminaries
Entropy stability in this ODE setting is defined as
$U\left(y\left(t_{f}\right)\right)\leq U\left(y_{0}\right).$ (6.10)
Here, we consider the entropy function $U=-\rho s$. Combined with discrete
mass conservation (i.e., $\rho$ is constant), we have
$s\left(y\left(t_{f}\right)\right)\geq s\left(y_{0}\right).$ (6.11)
In other words, entropy stability implies that the specific thermodynamic
entropy is nondecreasing in time, which is in line with the minimum entropy
principle.
Standard DGODE, as in Section 6.1, does not necessarily satisfy the inequality
(6.10). In the following, we introduce a modified DG discretization that is
guaranteed to be entropy-stable.
### 6.3 Entropy-stable DGODE with summation-by-parts property
We work with the following strong-form discretization, obtained by performing
integration by parts on the temporal-derivative term in Equation (6.3):
$\displaystyle
N_{h}\left(y,\mathfrak{v}\right)=\sum_{\kappa\in\mathcal{T}}\left[\left(y^{\dagger}\left(y^{+},y^{-},n\right)-n\cdot
y^{+},\mathfrak{v}^{+}\right)_{\partial\kappa}+\left(\frac{dy}{dt},\mathfrak{v}\right)_{\kappa}\right]-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{S}\left(y\right),\mathfrak{v}\right)_{\kappa}=0\qquad\forall\mathfrak{v}\in
V_{h}^{p}.$ (6.12)
It is well-known that a collocated DG scheme (i.e., solution nodes and
integration points are the same) with Gauss-Lobatto points possesses the
diagonal-norm summation-by-parts (SBP) property [54, 96]. Consider the
discrete mass matrix and discrete derivative matrix derived from Gauss-Lobatto
collocation, given by
$\mathsf{M}_{ij}=w_{i}\delta_{ij},\quad\mathsf{D}_{ij}=\ell^{\prime}_{j}(\xi_{i}),$
where $w_{i}$ is the $i$th Gauss-Lobatto weight, $\ell_{j}$ is the $j$th
Lagrange basis polynomial, and $\xi\in[0,1]$ is the reference coodinate.
$\mathsf{M}$ and $\mathsf{D}$ satisfy the SBP property,
$\mathsf{Q}+\mathsf{Q}^{T}=\mathsf{B},$
where $\mathsf{Q}=\mathsf{MD}$ and
$\mathsf{B}=\mathrm{diag}(-1,0,\ldots,0,1)$. The following element-local
discrete form is then obtained [18]:
$\displaystyle\left[y^{\dagger}\left(y_{\kappa}^{+},y_{\kappa}^{-},n\right)-n\cdot
y_{\kappa}^{+}\right]^{T}\mathfrak{v}_{\kappa}^{+}\biggr{|}_{\xi=0}+\left[y^{\dagger}\left(y_{\kappa}^{+},y_{\kappa}^{-},n\right)-n\cdot
y_{\kappa}^{+}\right]^{T}\mathfrak{v}_{\kappa}^{+}\biggr{|}_{\xi=1}$ (6.13)
$\displaystyle+\sum_{i=1}^{n_{b}}w_{i}\left[\sum_{j=1}^{n_{b}}\mathsf{D}_{ij}y_{\kappa}(t_{j})-h\mathcal{S}\left(y_{\kappa}(t_{i})\right)\right]^{T}\mathfrak{v}_{\kappa}(t_{i})=0.$
Note that in [14], standard DGODE was solved using a quadrature-free approach
[69, 70]. The discrete form (6.13), though similar, specifically invokes
quadrature in order to exploit the SBP property.
We replace the temporal-derivative interpolation operator in Equation (6.13)
with a specific temporal-derivative projection operator as [18]
$\sum_{j=1}^{n_{b}}\mathsf{D}_{ij}y_{\kappa}(t_{j})\rightarrow
2\sum_{j=1}^{n_{b}}\mathsf{D}_{ij}y^{\ddagger}(y_{\kappa}(t_{i}),y_{\kappa}(t_{j})),$
where $y^{\ddagger}(y_{1,}y_{2})$ is a two-point numerical state function
(distinct from $y^{\dagger}$) that is consistent and symmetric. Equation
(6.13) thus becomes
$\displaystyle\left[y^{\dagger}\left(y_{\kappa}^{+},y_{\kappa}^{-},n\right)-n\cdot
y_{\kappa}^{+}\right]^{T}\mathfrak{v}_{\kappa}^{+}\biggr{|}_{\xi=0}+\left[y^{\dagger}\left(y_{\kappa}^{+},y_{\kappa}^{-},n\right)-n\cdot
y_{\kappa}^{+}\right]^{T}\mathfrak{v}_{\kappa}^{+}\biggr{|}_{\xi=1}$ (6.14)
$\displaystyle+\sum_{i=1}^{n_{b}}w_{i}\left[2\sum_{j=1}^{n_{b}}\mathsf{D}_{ij}y^{\ddagger}(y_{\kappa}(t_{i}),y_{\kappa}(t_{j}))-h\mathcal{S}\left(y_{\kappa}(t_{i})\right)\right]^{T}\mathfrak{v}_{\kappa}(t_{i})=0.$
With the mean-value numerical state,
$y^{\ddagger}(y_{L},y_{R})=\frac{y_{L}+y_{R}}{2},$ (6.15)
Equation (6.14) recovers Equation (6.13) [96, 54].
The discretization (6.14) combined with the mean-value numerical state is not
necessarily entropy-stable. Although, as will be shown in Section 6.3.2, the
source-term discretization results in destruction of mathematical entropy, the
temporal-derivative operator may cause production of mathematical entropy.
Entropy stability is achieved only if the entropy destruction due to the
source term outweighs the entropy production due to the temporal-derivative
operator. In the following, by replacing the mean-value numerical state with a
more appropriate numerical state, we will obtain a temporal DG scheme that is
guaranteed to be entropy-stable.
#### 6.3.1 Entropy-conservative numerical state function
A key ingredient in the development of an entropy-stable DG scheme is a two-
point numerical state function that satisfies the following condition [18]:
$(\mathsf{v}_{R}-\mathsf{v}_{L})^{T}y^{\ddagger}(y_{L},y_{R})=\mathcal{U}_{R}-\mathcal{U}_{L}.$
(6.16)
A numerical state that satisfies (6.16) is _entropy-conservative_. Note that
an entropy-conservative numerical _flux_ satisfies an analogous condition:
$(\mathsf{v}_{R}-\mathsf{v}_{L})^{T}\mathcal{F}^{\ddagger}(y_{L},y_{R})=\mathcal{F}_{R}^{p}-\mathcal{F}_{L}^{p}.$
(6.17)
Gouasmi et al. [51] derived a simple, closed-form entropy-conservative
numerical flux for the multicomponent Euler equations with the entropy
function $U=-\rho s$. They built on the techniques by Roe [97] originally used
to construct an entropy-conservative numerical flux for the monocomponent
Euler equations. Said techniques rely on (6.17) as a starting point. An
analogous procedure, instead using (6.16) as a starting point, is employed
here to derive an entropy-conservative numerical state function.
With $\left\llbracket\cdot\right\rrbracket$ denoting the jump operator, the
entropy conservation condition (6.16) can be expressed as
$\left\llbracket\mathsf{v}\right\rrbracket{}^{T}y^{\ddagger}=\left\llbracket\mathcal{U}\right\rrbracket.$
(6.18)
For consistency with [51], we first work with a re-ordered state vector where
the species concentrations are replaced with partial densities:
$y=\left(\rho_{1},\ldots,\rho_{n_{s}},\rho v_{1},\ldots,\rho v_{d},\rho
e_{t}\right)^{T}.$ (6.19)
We employ the notation
$y^{\ddagger}=\left(y_{1,1}^{\ddagger},\ldots,y_{1,n_{s}}^{\ddagger},y_{2,1}^{\ddagger},\ldots,y_{2,d}^{\ddagger},y_{3}^{\ddagger}\right),$
where the first $n_{s}$ components correspond to the partial densities, the
next $d$ components correspond to the momentum, and the last component
corresponds to the total energy. Let $z$ denote the vector
$z=\left(\rho_{1},\ldots,\rho_{n_{s}},v_{1},\ldots,v_{d},1/T\right)=\left(z_{1,1},\ldots,z_{1,n_{s}},z_{2,1},\ldots,z_{2,d},z_{3}\right).$
With the entropy function $U=-\rho s$, the entropy variables and entropy
potential are given by
$\mathsf{v}=\left(\frac{g_{1}-\frac{1}{2}\sum_{k=1}^{d}v_{k}v_{k}}{T},\ldots,\frac{g_{n_{s}}-\frac{1}{2}\sum_{k=1}^{d}v_{k}v_{k}}{T},\frac{v_{1}}{T},\ldots,\frac{v_{d}}{T},-\frac{1}{T},\right)^{T},\quad\mathcal{U}=\sum_{i=1}^{n_{s}}W_{i}\rho_{i}.$
Next, we introduce the arithmetic mean, logarithmic mean, and product
operators,
$\displaystyle\left\\{\\!\\!\left\\{\alpha\right\\}\\!\\!\right\\}$
$\displaystyle=\frac{\alpha_{L}+\alpha_{R}}{2},$ $\displaystyle\alpha^{\ln}$
$\displaystyle=\begin{cases}\alpha_{L},&\text{if }\alpha_{L}=\alpha_{R}\\\
0,&\text{if }\alpha_{L}=0\text{ or }\alpha_{R}=0\\\
\frac{\alpha_{R}-\alpha_{L}}{\ln\alpha_{R}-\ln\alpha_{L}},&\text{otherwise},\end{cases}$
(6.20) $\displaystyle\alpha^{\times}$ $\displaystyle=\alpha_{L}\alpha_{R}$
which are equipped with the identities
$\displaystyle\left\\{\\!\\!\left\\{\alpha\beta\right\\}\\!\\!\right\\}$
$\displaystyle=\alpha\left\llbracket\beta\right\rrbracket+\beta\left\llbracket\alpha\right\rrbracket$
$\displaystyle\left\llbracket\ln\alpha\right\rrbracket$
$\displaystyle=\frac{\left\llbracket\alpha\right\rrbracket}{\alpha^{\ln}}$
$\displaystyle\left\llbracket\alpha\right\rrbracket$
$\displaystyle=-\alpha^{\times}\left\llbracket\frac{1}{\alpha}\right\rrbracket,\alpha\neq
0.$
Note that the three operators in (6.20) are symmetric; the first two are also
consistent (the product operator is consistent with $\alpha^{2}$). To compute
the logarithmic mean in a numerically stable manner, we employ the procedure
by Ismail and Roe [98]. The jump in the entropy potential can then be written
as
$\left\llbracket\mathcal{U}\right\rrbracket=\sum_{i=1}^{n_{s}}W_{i}\left\llbracket\rho_{i}\right\rrbracket=\sum_{i=1}^{n_{s}}W_{i}\left\llbracket
z_{1,i}\right\rrbracket,$ (6.21)
while the jump in the entropy variables, after some algebraic manipulation,
can be expressed as [51]
$\left\llbracket\mathsf{v}\right\rrbracket=\begin{pmatrix}\left\llbracket
z_{3}\right\rrbracket\mathsf{X}_{1}+\left\llbracket
z_{1,1}\right\rrbracket\frac{W_{1}}{\rho_{1}^{\ln}}-\sum_{k=1}^{d}\left\llbracket
z_{2,k}\right\rrbracket\left\\{\\!\\!\left\\{\frac{1}{T}\right\\}\\!\\!\right\\}\left\\{\\!\\!\left\\{v_{k}\right\\}\\!\\!\right\\}\\\
\vdots\\\ \left\llbracket
z_{3}\right\rrbracket\mathsf{X}_{n_{s}}+\left\llbracket
z_{1,n_{s}}\right\rrbracket\frac{W_{n_{s}}}{\rho_{n_{s}}^{\ln}}-\sum_{k=1}^{d}\left\llbracket
z_{2,k}\right\rrbracket\left\\{\\!\\!\left\\{\frac{1}{T}\right\\}\\!\\!\right\\}\left\\{\\!\\!\left\\{v_{k}\right\\}\\!\\!\right\\}\\\
\left\\{\\!\\!\left\\{z_{3}\right\\}\\!\\!\right\\}\left\llbracket
z_{2,1}\right\rrbracket+\left\\{\\!\\!\left\\{z_{2,1}\right\\}\\!\\!\right\\}\left\llbracket
z_{3}\right\rrbracket\\\ \vdots\\\
\left\\{\\!\\!\left\\{z_{3}\right\\}\\!\\!\right\\}\left\llbracket
z_{2,d}\right\rrbracket+\left\\{\\!\\!\left\\{z_{2,d}\right\\}\\!\\!\right\\}\left\llbracket
z_{3}\right\rrbracket\\\ -\left\llbracket z_{3}\right\rrbracket\end{pmatrix},$
(6.22)
where
$\mathsf{X}_{i}=b_{i0}+\frac{b_{i1}}{\left(1/T\right)^{\ln}}+\sum_{r=2}^{n_{p}+1}b_{ir}\left(\mathsf{f}_{r-1}(T)T^{\times}\right)-\frac{1}{2}\sum_{k=1}^{d}\left\\{\left\\{v_{d}^{2}\right\\}\right\\},$
with $\mathsf{f}_{r}(\alpha)$ denoting a special averaging operator,
consistent with $\alpha^{r-1}$, that satisfies
$\left\llbracket\alpha^{r}\right\rrbracket=r\mathsf{f}_{r}(\alpha)\left\llbracket\alpha\right\rrbracket$.
For $r=1,2,3,4$, $\mathsf{f}_{r}(\alpha)$ is defined as
$\displaystyle\mathsf{f}_{1}(\alpha)$ $\displaystyle=1,$
$\displaystyle\mathsf{f}_{2}(\alpha)$
$\displaystyle=\left\\{\\!\\!\left\\{\alpha\right\\}\\!\\!\right\\},$
$\displaystyle\mathsf{f}_{3}(\alpha)$
$\displaystyle=\frac{2}{3}\left\\{\\!\\!\left\\{\alpha\right\\}\\!\\!\right\\}\left\\{\\!\\!\left\\{\alpha\right\\}\\!\\!\right\\}+\frac{1}{3}\left\\{\\!\\!\left\\{\alpha^{2}\right\\}\\!\\!\right\\},$
$\displaystyle\mathsf{f}_{4}(\alpha)$
$\displaystyle=\left\\{\\!\\!\left\\{\alpha\right\\}\\!\\!\right\\}\left\\{\\!\\!\left\\{\alpha^{2}\right\\}\\!\\!\right\\}.$
$\mathsf{f}_{r}(\alpha)$ can be derived for $r>4$ as well [51]. Using
Equations (6.21) and (6.22), we rewrite the entropy conservation condition in
time (6.18) as a requirement that a linear combination of the jumps in the
components of $z$ equals zero:
$\displaystyle\sum_{i=1}^{n_{s}}\left\llbracket
z_{1,i}\right\rrbracket\left(y_{1,i}^{\ddagger}\frac{W_{i}}{\rho_{i}^{\ln}}-W_{i}\right)+\sum_{k=1}^{d}\left\llbracket
z_{2,k}\right\rrbracket\left(y_{2,k}^{\ddagger}\left\\{\\!\\!\left\\{z_{3}\right\\}\\!\\!\right\\}-\sum_{i=1}^{n_{s}}y_{1,i}^{\ddagger}\left\\{\\!\\!\left\\{\frac{1}{T}\right\\}\\!\\!\right\\}\left\\{\\!\\!\left\\{v_{k}\right\\}\\!\\!\right\\}\right)$
$\displaystyle+\left\llbracket
z_{3}\right\rrbracket\left(\sum_{i=1}^{n_{s}}y_{1,i}^{\ddagger}\mathsf{X}_{i}+\sum_{k=1}^{d}y_{2,k}^{\ddagger}\left\\{\\!\\!\left\\{z_{2,k}\right\\}\\!\\!\right\\}-y_{3}^{\ddagger}\right)=0.$
Invoking the independence of these jumps yields a system of $m$ equations:
$\displaystyle y_{1,i}^{\ddagger}\frac{W_{i}}{\rho_{i}^{\ln}}-W_{i}$
$\displaystyle=0,\quad i=1,\ldots,n_{s},$ $\displaystyle
y_{2,k}^{\ddagger}\left\\{\\!\\!\left\\{z_{3}\right\\}\\!\\!\right\\}-\sum_{i=1}^{n_{s}}y_{1,i}^{\ddagger}\left\\{\\!\\!\left\\{\frac{1}{T}\right\\}\\!\\!\right\\}\left\\{\\!\\!\left\\{v_{k}\right\\}\\!\\!\right\\}$
$\displaystyle=0,\quad k=1,\ldots,d,$
$\displaystyle\sum_{i=1}^{n_{s}}y_{1,i}^{\ddagger}\mathsf{X}_{i}+\sum_{k=1}^{d}y_{2,k}^{\ddagger}\left\\{\\!\\!\left\\{z_{2,k}\right\\}\\!\\!\right\\}-y_{3}^{\ddagger}$
$\displaystyle=0.$
Solving for the components of $y^{\ddagger}$ then gives
$y^{\ddagger}=\begin{pmatrix}\rho_{1}^{\ln}\\\ \vdots\\\ \rho_{n_{s}}^{\ln}\\\
\sum_{i=1}^{n_{s}}\rho_{i}^{\ln}\left\\{\\!\\!\left\\{v_{1}\right\\}\\!\\!\right\\}\\\
\vdots\\\
\sum_{i=1}^{n_{s}}\rho_{i}^{\ln}\left\\{\\!\\!\left\\{v_{d}\right\\}\\!\\!\right\\}\\\
\sum_{k=1}^{d}\sum_{i=1}^{n_{s}}\rho_{i}^{\ln}\left\\{\\!\\!\left\\{v_{k}\right\\}\\!\\!\right\\}\left\\{\\!\\!\left\\{v_{k}\right\\}\\!\\!\right\\}+\sum_{i=1}^{n_{s}}\rho_{i}^{\ln}\left[b_{i0}+\frac{b_{i1}}{\left(1/T\right)^{\ln}}+\sum_{r=2}^{n_{p}+1}b_{ir}\left(\mathsf{f}_{r-1}(T)T^{\times}\right)-\frac{1}{2}\sum_{k=1}^{d}\left\\{\\!\\!\left\\{v_{k}^{2}\right\\}\\!\\!\right\\}\right]\end{pmatrix}.$
(6.23)
Note that Equation (6.23) corresponds to the state vector (6.19). A simple re-
ordering and linear mapping [51] yields
$y^{\ddagger}=\begin{pmatrix}\sum_{i=1}^{n_{s}}\rho_{i}^{\ln}\left\\{\\!\\!\left\\{v_{1}\right\\}\\!\\!\right\\}\\\
\vdots\\\
\sum_{i=1}^{n_{s}}\rho_{i}^{\ln}\left\\{\\!\\!\left\\{v_{d}\right\\}\\!\\!\right\\}\\\
\sum_{k=1}^{d}\sum_{i=1}^{n_{s}}\rho_{i}^{\ln}\left\\{\\!\\!\left\\{v_{k}\right\\}\\!\\!\right\\}\left\\{\\!\\!\left\\{v_{k}\right\\}\\!\\!\right\\}+\sum_{i=1}^{n_{s}}\rho_{i}^{\ln}\left[b_{i0}+\frac{b_{i1}}{\left(1/T\right)^{\ln}}+\sum_{r=2}^{n_{p}+1}b_{ir}\left(\mathsf{f}_{r-1}(T)T^{\times}\right)-\frac{1}{2}\sum_{k=1}^{d}\left\\{\\!\\!\left\\{v_{k}^{2}\right\\}\\!\\!\right\\}\right]\\\
C_{1}^{\ln}\\\ \vdots\\\ C_{n_{s}}^{\ln}\end{pmatrix},$ (6.24)
which corresponds to the original state vector (2.2). We then have the
following theorem.
###### Theorem 3.
The two-point numerical state function in Equation (6.24) is entropy-
conservative, consistent, and symmetric.
###### Proof.
The numerical state (6.24) is entropy-conservative by construction. Since all
of the introduced operators are symmetric, the numerical state is symmetric.
Finally, recognizing that $\mathsf{f}_{r-1}(\alpha)\alpha^{\times}$ is
consistent with $\alpha^{r}$, taking the left and right states to be the same
yields
$y^{\ddagger}=\begin{pmatrix}\rho v_{1}\\\ \vdots\\\ \rho v_{d}\\\
\sum_{k=1}^{d}\rho
v_{k}v_{k}+\sum_{i=1}^{n_{s}}\rho_{i}\left[b_{i0}+b_{i1}T+\sum_{r=2}^{n_{p}+1}b_{ir}T^{r}-\frac{1}{2}\sum_{k=1}^{d}v_{k}^{2}\right]\\\
C_{1}\\\ \vdots\\\ C_{n_{s}}\end{pmatrix}=\begin{pmatrix}\rho v_{1}\\\
\vdots\\\ \rho v_{d}\\\ \rho u+\frac{1}{2}\rho\sum_{k=1}^{d}v_{k}v_{k}\\\
C_{1}\\\ \vdots\\\ C_{n_{s}}\end{pmatrix}=y.$
Therefore, it is also consistent. ∎
###### Remark 4.
The derived entropy-conservative numerical state function (6.24) can be
directly used in entropy-stable space-time DG schemes based on SBP operators
[18].
#### 6.3.2 Discrete temporal entropy analysis
We analyze the entropy stability of the proposed temporal DG discretization in
the following theorem.
###### Theorem 5.
Consider the DG discretization (6.14). Assume that $y^{\dagger}$ is the upwind
numerical state and $y^{\ddagger}$ is the entropy-conservative numerical state
in (6.24). Then, for the entropy function $U=-\rho s$, the resulting DG
discretization is entropy-stable.
###### Proof.
Take $\mathfrak{v}_{\kappa}$ in (6.14) to be the polynomial interpolant of the
entropy variables, such that
$\mathfrak{v}_{\kappa}\left(t_{j}\right)=\mathsf{v}\left(y_{\kappa}\left(t_{j}\right)\right),\quad
j=1,\ldots,n_{b},$
which results in
$\begin{split}&\sum_{\kappa\in\mathcal{T}}\left[\left[y^{\dagger}\left(y_{\kappa}^{+},y_{\kappa}^{-},n\right)-n\cdot
y_{\kappa}^{+}\right]^{T}\mathsf{v}\left(y_{\kappa}^{+}\right)\biggr{|}_{\xi=0}+\left[y^{\dagger}\left(y_{\kappa}^{+},y_{\kappa}^{-},n\right)-n\cdot
y_{\kappa}^{+}\right]^{T}\mathsf{v}\left(y_{\kappa}^{+}\right)\biggr{|}_{\xi=1}\right]\\\
&+\sum_{\kappa\in\mathcal{T}}\left[\sum_{i=1}^{n_{b}}w_{i}\left(2\sum_{j=1}^{n_{b}}\mathsf{D}_{ij}y^{\ddagger}(y_{\kappa}(t_{i}),y_{\kappa}(t_{j}))-h\mathcal{S}\left(y_{\kappa}(t_{i})\right)\right)^{T}\mathsf{v}\left(y_{\kappa}(t_{i})\right)\right]=0.\end{split}$
(6.25)
We introduce $\mathcal{A}_{\kappa}$ and $\mathcal{B}_{\kappa}$, defined as
$\displaystyle\mathcal{A}_{\kappa}=$
$\displaystyle\left[y^{\dagger}\left(y^{+},y^{-},n\right)-n\cdot
y^{+}\right]^{T}\mathsf{v}\left(y_{\kappa}^{+}\right)\biggr{|}_{\xi=0}+\left[y^{\dagger}\left(y^{+},y^{-},n\right)-n\cdot
y^{+}\right]^{T}\mathsf{v}\left(y_{\kappa}^{+}\right)\biggr{|}_{\xi=1}$
$\displaystyle+\sum_{i=1}^{n_{b}}w_{i}\left[2\sum_{j=1}^{n_{b}}\mathsf{D}_{ij}y^{\ddagger}(y(t_{i}),y(t_{j}))\right]^{T}\mathsf{v}\left(y_{\kappa}(t_{i})\right)$
$\displaystyle\mathcal{B}_{\kappa}=$
$\displaystyle-\sum_{i=1}^{n_{b}}w_{i}hS\left(y_{\kappa}(t_{i})\right)^{T}\mathsf{v}\left(y_{\kappa}(t_{i})\right),$
such that Equation (6.25) can be rewritten as
$\sum_{\kappa\in\mathcal{T}}\mathcal{A}_{\kappa}+\sum_{\kappa\in\mathcal{T}}\mathcal{B}_{\kappa}=0.$
(6.26)
By invoking the SBP property and the fact that the upwind numerical state
function is an entropy-stable numerical state (i.e., it satisfies
$\left\llbracket\mathsf{v}\right\rrbracket{}^{T}y^{\dagger}\leq\left\llbracket\mathcal{U}\right\rrbracket$)
[18], we obtain the inequality
$U\left(y\left(t_{f}\right)\right)-U\left(y_{0}\right)\leq\sum_{\kappa\in\mathcal{T}}\mathcal{A}_{\kappa},$
(6.27)
the proof of which is very similar to that in [18, Theorem 1] (just without
any spatial component) and is therefore not included here. It remains to
analyze $\sum_{\kappa\in\mathcal{T}}\mathcal{B}_{\kappa}$. The quantity
$S\left(y_{\kappa}(t_{i})\right)^{T}\mathsf{v}\left(y_{\kappa}(t_{i})\right)$
is simply the pointwise entropy production rate due to the chemical source
terms, which, as demonstrated in Section 4.2, is nonpositive. Since $w_{i}>0$
and $h>0$, we have
$\sum_{\kappa\in\mathcal{T}}\mathcal{B}_{\kappa}\geq 0.$ (6.28)
Combining Equations (6.26), (6.27), and (6.28) gives the inequality (6.10),
which completes the proof. ∎
In the following, we refer to the proposed entropy-stable DG discretization as
simply “entropy-stable DGODE.”
###### Remark 6.
Entropy-stable DGODE is well-defined for zero concentrations. However, during
early iterations of Newton’s method, negative concentrations can occur, making
the formulation ill-defined. As a simple remedy, we use the solution obtained
with the mean-value numerical state (6.15) as an initial guess, which we find
to be sufficiently robust since this initial guess is generally close to the
entropy-stable DGODE solution (recall that taking $y^{\ddagger}$ to be the
mean-value numerical state essentially recovers standard DGODE). Other, more
sophisticated approaches can be employed as well.
###### Remark 7.
In general, entropy-stable DGODE is more expensive than standard DGODE.
Furthermore, when using the latter, we find that the thermodynamic entropy
produced by the source terms typically outweighs any thermodynamic entropy
destruction caused by the non-entropy-stable mean-value numerical state, at
least for the reaction mechanisms used in this study (though it is important
to note that this observation may not hold true for significantly different
mechanisms). As such, in practice, the following approach can be employed to
maximize efficiency:
* 1.
Compute a solution with standard DGODE.
* 2.
If $s\left(y\left(t_{f}\right)\right)\geq s\left(y_{0}\right)$, then proceed.
Otherwise, compute a solution with entropy-stable DGODE.
In Section 7.3, however, in order to test the formulation, we calculate
spatially one-dimensional detonation waves with entropy-stable DGODE alone.
###### Remark 8.
To reduce computational cost, it is common practice, especially for large-
scale simulations, to approximate a given chemical reaction as two
irreversible forward reactions (as opposed to having forward and reverse
reaction rates). However, this strategy can cause appreciable entropy
violations [60, 99]. A detailed investigation of whether the gains in speed
outweigh the loss in accuracy is outside the scope of this study. If such a
strategy is employed, then reverting to standard DGODE is the natural course
of action. Though not considered in this work, an alternative approach is to
use the least-squares-based method by Ream et al. [100] to generate chemical
mechanisms involving irreversible reactions that do not cause entropy
violations.
## 7 One-dimensional results
We consider three one-dimensional test cases. The first one involves the
advection of a thermal bubble. The second case is a shock-tube problem with
multiple flow discontinuities. These first two tests comprise nonreacting
multicomponent flows. The final case explores sustained detonations formed via
an overdriven initialization. Stiff chemical reactions are present in this
test. The SSPRK2 time integration scheme with $\mathrm{CFL}=0.1$ (based on the
linear-stability constraint) is employed throughout. All simulations are
performed using a modified version of the JENRE® Multiphysics Framework [101,
14] that incorporates the developments and extensions described in this work.
### 7.1 Thermal bubble
We use this smooth flow problem to assess the grid convergence of the entropy-
bounded DG method (without artificial viscosity). The order of accuracy of the
limiting procedure in Section 5.2.1 is of particular interest. We also compare
the local entropy bound in [17] to that proposed here (Equation (5.20)). The
initial conditions are as follows:
$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle 1\textrm{ m/s},$
$\displaystyle Y_{H_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1-\tanh\left(|x|-10\right)\right],$
$\displaystyle Y_{O_{2}}$ $\displaystyle=$ $\displaystyle 1-Y_{H_{2}},$ (7.1)
$\displaystyle T$ $\displaystyle=$ $\displaystyle
1200-900\tanh\left(|x|-10\right)\textrm{ K},$ $\displaystyle P$
$\displaystyle=$ $\displaystyle 1\textrm{ bar}.$
The computational domain is $\Omega=[-25,25]\>\mathrm{m}$, with periodic
boundaries. Four element sizes are considered: $h$, $h/2$, $h/4$, and $h/8$,
where $h=2\>\mathrm{m}$. Smaller element sizes are not investigated since the
limiters are not activated for such fine resolutions and optimal order of
accuracy without limiting was already demonstrated in [14]. The $L^{2}$ error
at $t=5\>\mathrm{s}$ is calculated in terms of the following normalized state
variables:
$\widehat{\rho v}_{k}=\frac{1}{\sqrt{\rho_{r}P_{r}}}\rho
v_{k},\quad\widehat{\rho e}_{t}=\frac{1}{P_{r}}\rho
e_{t},\quad\widehat{C}_{i}=\frac{R^{0}T_{r}}{P_{r}}C_{i},$
where $T_{r}=1000\,\mathrm{K}$, $\rho_{r}=1\,\mathrm{kg\cdot}\mathrm{m}^{-3}$,
and $P_{r}=101325\,\mathrm{Pa}$ are reference values. The results of the
convergence study for $p=1$ to $p=3$ are displayed in Figure 7.1. The dashed
lines represent the theoretical convergence rates. The “$\times$” marker
indicates that the positivity-preserving limiter is activated, the “$\Circle$”
marker indicates that the entropy limiter is activated, and the “$\triangle$”
marker indicates that neither limiter is activated. If both limiters are
activated, then the corresponding markers are superimposed as “$\otimes$”. For
$h$ and $h/2$, both the positivity-preserving and entropy limiters are engaged
regardless of $p$; for $h/4$ and $p=1$, only the positivity-preserving limiter
is engaged. That the limiters are no longer activated for well-resolved
solutions is a desirable property. In general, optimal order of accuracy is
recovered. Suboptimal accuracy is observed for the coarser resolutions with
$p=1$, likely because the asymptotic regime is not yet reached.
Figure 7.1: Convergence under grid refinement, with $h=2\>\mathrm{m}$, for the
one-dimensional thermal bubble test case. The $L^{2}$ error of the normalized
state with respect to the exact solution at $t=5\>\mathrm{s}$ is computed. The
dashed lines represent the theroretical convergence rates. The “$\times$”
marker indicates that the positivity-preserving limiter is activated, the
“$\Circle$” marker indicates that the entropy limiter is activated, and the
“$\triangle$” marker indicates that neither limiter is activated. If both
limiters are activated, then the corresponding markers are superimposed as
“$\otimes$”.
Figure 7.2 compares the temperature profile at $t=50\>s$ computed with the
local entropy bound in [17] (“Old”) to that computed with the local entropy
bound in Equation (5.20) (“New”). The exact solution is the same as the
initial condition. The element size is $h/4$ and the polynomial order is
$p=1$. The reason for choosing $p=1$ is to eliminate any ambiguity associated
with algebraically estimating the spatial minimum of the specific
thermodynamic entropy, as discussed in Section 5.2.3. Though oscillations are
present in both cases, they are significantly larger with the local entropy
bound in [17], suggesting that it may be overly restrictive. The local Lax-
Friedrichs flux is employed in this comparison. With the selected parameters,
the differences between the two local entropy bounds are more pronounced for
this numerical flux than the HLLC flux. Nevertheless, such discrepancies are
observed for the latter flux function in other configurations as well,
especially with higher polynomial orders.
Figure 7.2: Temperature profiles at $t=50\>s$ computed with the local entropy
bound in [17] (“Old”) and the local entropy bound in Equation (5.20) (“New”).
The exact solution is the same as the initial condition. The element size is
$h/4$ and the polynomial order is $p=1$.
### 7.2 Shock tube
This test case was first presented by Houim and Kuo [102] and computed as well
by Johnson and Kercher [14]. The goals here are to compare the stabilization
properties of the limiters and to illustrate the benefits of the local entropy
bound. The initial conditions are given by
$\left(v_{1},T,P,Y_{N_{2}},Y_{He}\right)=\begin{cases}\left(0\text{
m/s},300\text{ K},1\text{ atm},1,0\right),&x\geq 0.4\\\ \left(0\text{
m/s},300\text{ K},10\text{ atm},0,1\right),&x<0.4\end{cases}.$ (7.2)
The computational domain is $\Omega=[0,1]\text{ m}$, with walls at both ends.
Figure 7.3 shows the mass fraction, pressure, temperature, and entropy
profiles at $t=300\;\mu\mathrm{s}$ for $p=3$ and 200 elements. We present
results for only the positivity-preserving limiter (referred to as “PPL”) and
for both the positivity-preserving and entropy limiters with the local entropy
bound in Equation (5.20) (referred to as “local EB”). Also included is a
reference solution computed with $p=2$, 2000 elements, and artificial
viscosity, which corresponds to the configuration in [14]. Artificial
viscosity is not employed in the coarser cases in order to isolate the effects
of the limiters. The mass fraction profiles are well-captured for both types
of limiting. However, although the solver does not crash, the positivity-
preserving limiter by itself fails to suppress large-scale oscillations and
significant overshoots/undershoots in the pressure and temperature
distributions. While instabilities are still present with the local entropy
limiter, they are of substantially smaller magnitude. The entropy distribution
obtained with the local entropy limiter is very similar to that of the
reference solution, whereas the positivity-preserving limiter generates a
notable overshoot and undershoot at the shock. For the given flow conditions,
the global entropy bound yields very similar results (not shown for brevity)
to the local entropy bound since the specific thermodynamic entropy in the
vicinity of the shock, which is where much of the limiting occurs, is close to
the global minimum.
The instabilities observed in the “PPL” case in Figures 7.3b and 7.3c are
considerably larger than those typically observed in shock-tube solutions
obtained with the positivity-preserving limiter in the monocomponent,
calorically perfect case [20, 36, 35]. This difference reflects the numerical
challenges associated with not only multicomponent mixtures, but also variable
thermodynamics. In a similar vein, the relative benefit of the entropy limiter
(compared to the positivity-preserving limiter) seems significantly greater in
the multicomponent, thermally perfect case than in the monocomponent,
calorically perfect case.
(a) Mass fractions. (b) Pressure. (c) Temperature. (d) Specific thermodynamic
entropy.
Figure 7.3: Results for $p=3$ solutions on 200 elements without artificial
viscosity for the one-dimensional, multicomponent shock-tube problem with
initialization in Equation (7.2). “PPL” corresponds to the positivity-
preserving limiter by itself, and “local EB” refers to both the positivity-
preserving and entropy limiters with the local entropy bound in Equation
(5.20). The reference solution [14] is computed with $p=2$, 2000 elements, and
artificial viscosity.
Figure 7.4 presents the percent error in conservation of mass, energy, and
atomic elements for the “local EB” case as a representative example,
calculated every $0.3\;\mu\mathrm{s}$ (for a total of 1000 samples).
$\mathsf{N}_{N}$ and $\mathsf{N}_{He}$ denote the total numbers of nitrogen
and helium atoms in the mixture. The error remains close to machine precision,
confirming that the proposed formulation is conservative. Also included is the
error in mass conservation (calculated every time step) for a solution
computed without the positivity-preserving and entropy limiters, but instead
with a simple clipping procedure in which negative species concentrations are
set to zero, a strategy employed by many reacting-flow solvers. The error
increases rapidly to non-negligible values until the solver diverges.
Figure 7.4: Percent error in conservation of mass, energy, and atomic elements
for the “local EB” case in Figure 7.3, computed with $p=3$ on 200 elements.
The initial conditions for this one-dimensional, multicomponent shock-tube
problem are given in Equation (7.2). Also included is the error in mass
conservation for a solution computed without the positivity-preserving and
entropy limiters, but instead with a simple clipping procedure in which
negative species concentrations are set to zero.
Next, we recompute this problem with artificial viscosity to confirm adequate
suppression of small-scale oscillations. Figure 7.5 presents the results for
$C_{\mathrm{AV}}=1$. The instabilities observed in Figure 7.3 are largely
eliminated by the artificial viscosity. As shown in Figure 7.5c, a temperature
undershoot at the shock emerges when only the positivity-preserving limiter is
used, but is suppressed by the entropy limiter. The artificial viscosity
causes some smearing of the solution at the contact. Note that without any
limiting, negative species concentrations occur for the $p=3$, 200-element
cases, even with artificial viscosity. As will be further discussed in Part II
[19], artificial viscosity alone, or even when combined with solely the
positivity-preserving limiter, does not provide sufficient stabilization in
simulations of complex detonation waves on relatively coarse meshes. In these
simulations, enforcement of the entropy principle is critical for robustness.
(a) Mass fractions. (b) Pressure. (c) Temperature. (d) Specfic thermodynamic
entropy.
Figure 7.5: Results for $p=3$ solutions on 200 elements with artificial
viscosity for the one-dimensional, multicomponent shock-tube problem with
initialization in Equation (7.2). “PPL” corresponds to the positivity-
preserving limiter by itself, and “local EB” refers to both the positivity-
preserving and entropy limiters with the local entropy bound in Equation
(5.20). The reference solution [14] is computed with $p=2$, 2000 elements, and
artificial viscosity. The difference between these results and those in Figure
7.3 is the use of artificial viscosity in the (non-reference) solutions here.
To highlight discrepancies between the local and global entropy bounds, we
consider different initial conditions, given by
$\left(v_{1},T,P,Y_{N_{2}},Y_{He}\right)=\begin{cases}\left(0\text{
m/s},300\text{ K},1\text{ atm},0,1\right),&x\geq 0.4\\\ \left(0\text{
m/s},300\text{ K},10\text{ atm},1,0\right),&x<0.4\end{cases}.$ (7.3)
The only difference with the previous initial conditions is in the mass
fractions. Displayed in Figure 7.6 are the results obtained with the global
entropy bound (referred to as “global EB”) and the local entropy bound (again
referred to as “local EB”). These $p=2$ solutions are computed on 200 elements
without any artificial viscosity. The positivity-preserving limiter by itself
yields very similar results (not shown for brevity) to the entropy limiter
with the global entropy bound. The reference solution is again computed with
$p=2$, 2000 elements, and artificial viscosity. The mass fractions are well-
captured in all cases. The differences between the “global EB” and “local EB”
solutions here are not as large as those between the “PPL” and “local EB”
solutions in Figure 7.3. Nevertheless, the benefit of the local entropy bound
is evident, specifically at the shock. Spurious artifacts in the pressure
profile and especially the temperature profile near the shock are noticeably
larger for the global entropy bound. The discrepancies between the two
solutions are attributed to the considerable difference between the specific
thermodynamic entropy in the vicinity of the shock and the global minimum, as
illustrated in Figure 7.6d. As such, the local entropy bound is particularly
beneficial in flow problems with large variations in the specific
thermodynamic entropy throughout the domain.
(a) Mass fractions. (b) Pressure. (c) Temperature. (d) Specfic thermodynamic
entropy.
Figure 7.6: Results for $p=2$ solutions on 200 elements without artificial
viscosity for the one-dimensional, multicomponent shock-tube problem with
initialization in Equation (7.3). “Global EB” refers to the entropy limiter
with the global entropy bound in Equation (5.15), and “local EB” refers to the
entropy limiter with the local entropy bound in Equation (5.20). The reference
solution is computed with $p=2$, 2000 elements, and artificial viscosity.
### 7.3 Detonation wave
The final one-dimensional test case is a hydrogen-oxygen detonation wave
diluted in Argon with initial conditions
$\begin{array}[]{cccc}\qquad\qquad\qquad\qquad v_{1}&=&0\text{ m/s},\\\
\quad\quad\quad\;X_{Ar}:X_{O_{2}}:X_{H_{2}}&=&7:1:2&\text{ }x>0.025\text{
m},\\\ X_{Ar}:X_{O_{2}}:X_{H_{2}}:X_{OH}&=&7:1:2:0.01&\text{ }0.015\text{
m}<x<0.025\text{ m,}\\\
\quad\quad\;X_{Ar}:X_{H_{2}O}:X_{OH}&=&8:2:0.0001&x<0.015\text{ m},\\\
\qquad\qquad\qquad\qquad P&=&\begin{cases}{5.50}\mathrm{e}{5}&\text{ Pa}\\\
{6.67}\mathrm{e}{3}&\text{ Pa}\end{cases}&\begin{array}[]{c}x<0.015\text{
m}\\\ x>0.015\text{ m}\end{array},\\\ \qquad\qquad\qquad\qquad
T&=&\begin{cases}300&\text{ K}\\\ 350&\text{ K}\\\ 3500\text{\hskip
10.00002pt\hskip 10.00002pt}&\text{ K}\end{cases}&\begin{array}[]{c}\text{
}x>0.025\text{ m}\\\ \text{ }0.015\text{ m}<x<0.025\text{ m}\\\ x<0.015\text{
m}\end{array}.\end{array}$ (7.4)
The domain is $\Omega=(0,0.45)$ m, with walls at the left and right
boundaries. In previous work [14], this case was computed with $p=1$ and mesh
spacing $h=9\times 10^{-5}$ m. At this resolution, artificial viscosity was
the only stabilization necessary to obtain an accurate solution while
maintaining conservation of mass and energy. Good agreement with the Shock and
Detonation Toolbox [103] was observed. Additional details can be found in
[14]. Here, our objective is to demonstrate that the proposed formulation,
specifically the entropy-bounded DG discretization of the convective operator
combined with entropy-stable DGODE for stiff chemical reactions, can compute
stable and accurate solutions with appreciably lower resolution. In light of
[14], we use a $p=1$, $h=9\times 10^{-5}$ calculation as a reference solution.
The only difference with [14] is that the reaction mechanism [104] is slightly
modified to solely contain reversible reactions.
Figure 7.7 presents $p=1$, $p=2$, and $p=3$ results at $t=235$
$\mu\mathrm{s}$. The mesh spacing in these simulations is $h=4.5\times
10^{-4}$ m, which is fives time larger than for the reference solution.
Artificial viscosity and the local entropy limiter are employed. At this mesh
spacing, artificial viscosity alone is not sufficient to stabilize the
solution. Good agreement in temperature and pressure with the reference
solution is observed. As shown in Figure 7.7b, which zooms in on the shock,
there are slight discrepancies in the predictions of the leading-shock-front
location and the induction region. However, these predictions improve with
increasing $p$. Also included in Figure 7.7b is a $p=1$ solution with standard
DGODE in order to ensure that entropy-stable DGODE is not the primary cause of
the disagreement. Figure 7.7d shows small-scale entropy oscillations behind
the leading shock front in the $p=3$ calculation. The presence of these
instabilities is not too surprising given the linear nature of the limiting
operator, coupled with the increased susceptibility to oscillations of high
polynomial orders. Nevertheless, these instabilities do not significantly
pollute the rest of the solution.
(a) Temperature. (b) Temperature, zoomed in on shock. (c) Pressure. (d)
Specific thermodynamic entropy.
Figure 7.7: $p=1$, $p=2$, and $p=3$ results at $t=235$ $\mu\mathrm{s}$ for
the one-dimensional hydrogen detonation test case. The mesh spacing in these
simulations is $h=4.5\times 10^{-4}$ m, which is fives time larger than for
the reference solution. Artificial viscosity and the local entropy limiter are
employed.
Figure 7.8 compares mass-fraction profiles of selected species obtained with
$p=3$ to those of the reference solution. Marginal oscillations can be
observed behind the shock, particularly in the mass-fraction profiles of
$\mathrm{H_{2}O}$ and $\mathrm{HO_{2}}$. In Figure 7.8c, the $p=3$ solution
predicts slightly lower peaks in the mass fractions of $\mathrm{HO_{2}}$ and
$\mathrm{H_{2}O_{2}}$. Furthermore, the aforementioned discrepancy in the
leading-shock location is reflected in these results. Nevertheless, there is
very good agreement between the $p=3$ solution and the reference solution,
illustrating the ability of the developed formulation to obtain stable and
accurate detonation results on coarser meshes.
(a) (b) (c)
Figure 7.8: Comparison of $p=3$ predictions of species mass fractions with
those of the reference solution for the one-dimensional hydrogen detonation
test case. The mesh spacing is $h=4.5\times 10^{-4}$ m, which is fives time
larger than for the reference solution.
Finally, Figure 7.9 gives the percent error in conservation of mass, energy,
and atomic elements for $p=1$ as a representative example, calculated every
$0.235\;\mu\mathrm{s}$ (for a total of 1000 samples). $\mathsf{N}_{O}$,
$\mathsf{N}_{H}$, and $\mathsf{N}_{Ar}$ denote the total numbers of oxygen,
hydrogen, and argon atoms in the mixture. The error remains negligible
throughout the simulation, confirming that the methodology is conservative.
Figure 7.9: Percent error in conservation of mass, energy, and atomic elements
for the $p=1$ calculation with $h=4.5\times 10^{-4}$ m. The initial conditions
for this one-dimensional hydrogen detonation problem are given in Equation
(7.4).
## 8 Concluding remarks
In this paper, we introduced a positivity-preserving and entropy-bounded DG
methodology for the chemically reacting, compressible Euler equations. The
methodology builds on the fully conservative, high-order DG method previously
developed by two of the authors [14], which does not generate spurious
pressure oscillations in smooth flow regions or across material interfaces
when the temperature is continuous. As a prerequisite for the proposed
formulation, we proved a minimum entropy principle for the compressible,
multicomponent, chemically reacting Euler equations, which follows from the
proof by Gouasmi et al. [16] of a minimum entropy principle for the
compressible, multicomponent, nonreacting Euler equations.
In this first part of our two-part paper, we focused on the one-dimensional
case. A simple linear-scaling limiter ensures that the solution at a given
time step is admissible (i.e., species concentrations are nonnegative, density
is positive, pressure is positive, and entropy is greater than some lower
bound). A requirement of the limiter is that the element average of the state
is itself admissible, which we showed to be true under the following
conditions: (a) a time-step-size constraint is satisfied, (b) an invariant-
region-preserving numerical flux is employed, and (c) certain pointwise values
of the solution at the previous time step are admissible. Both a global
entropy bound and a local entropy bound were discussed. Since the linear
scaling does not completely eliminate small-scale oscillations, artificial
viscosity is employed in tandem. We also detailed how to maintain
compatibility between the proposed framework and the pressure-equilibrium-
maintaining discretization in [14]. The temporal integration of the convection
operator is decoupled from that of the stiff chemical source term via Strang
splitting. To guarantee satisfaction of the minimum entropy principle in the
reaction step, we developed an entropy-stable discontinuous Galerkin method
based on diagonal-norm summation-by-parts operators for temporal integration
of the source term, which involved the derivation of an entropy-conservative
two-point numerical state function.
The methodology was applied to canonical one-dimensional test cases. The first
two entailed nonreacting flows: advection of a smooth, hydrogen-oxygen thermal
bubble and nitrogen-helium shock-tube flow. In the former, we demonstrated
optimal convergence of the methodology and sufficient preservation of pressure
equilibrium. In the latter, we observed the following:
* 1.
The positivity-preserving limiter (which does not consider an entropy bound)
prevents the solver from crashing, but, in the absence of additional
stabilization, gives rise to large-scale oscillations. Such instabilities are
substantially larger than those typically seen in the monocomponent,
calorically perfect case, illustrating the challenges of stabilizing
computations of multicomponent flows with realistic thermodynamics.
* 2.
The entropy limiter, on the other hand, considerably reduces the magnitude of
the aforementioned instabilities, suggesting that the relative benefit of the
entropy limiter is much greater in the multicomponent, thermally perfect case.
Small-scale oscillations are still present, but these can be cured with
artificial viscosity. Note that the inability to completely eliminate
oscillations is a well-known property of the linear-scaling limiter [20, 35,
17]. Furthermore, unless a very fine resolution is employed, artificial
viscosity alone is insufficient for robustness.
* 3.
Enforcing a local entropy bound can be more effective than enforcing a global
entropy bound when the entropy varies significantly throughout the domain.
In our final test case, we computed a moving hydrogen-oxygen detonation wave
diluted in Argon, demonstrating that the developed methodology can accurately
and robustly calculate a chemically reacting flow with detailed chemistry
using high-order polynomial approximations on relatively coarse meshes.
Conservation of mass, total energy, and atomic elements was confirmed.
In Part II [19], we will extend our formulation to multiple dimensions. In our
developed multidimensional framework, restrictions on the physical modeling,
geometry, numerical flux function, and quadrature rules are milder than those
currently in the literature. Complex two- and three-dimensional detonations
will be accurately computed in a stable manner using high-order polynomial
approximations.
## Acknowledgments
This work is sponsored by the Office of Naval Research through the Naval
Research Laboratory 6.1 Computational Physics Task Area.
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## Appendix A Supporting lemmas associated with specific entropy
We restate two key results established by Zhang and Shu [36]. We assume
throughout that $\rho>0$, $C_{i}>0$, and $T>0$.
###### Lemma 9 ([36]).
$s(y)$ is quasi-concave, such that
$s\left(\beta
y_{1}+(1-\beta)y_{2}\right)>\min\left\\{s(y_{1}),s(y_{2})\right\\},$ (A.1)
where $y_{1}\neq y_{2}$ and $0<\beta<1$.
###### Proof.
Since $U=-\rho s$ is strictly convex [16], $U$ satisfies Jensen’s inequality:
$U\left(\beta y_{1}+(1-\beta)y_{2}\right)<\beta U(y_{1})+(1-\beta)U(y_{2}).$
Therefore,
$\displaystyle-\rho\left(\beta y_{1}+(1-\beta)y_{2}\right)s\left(\beta
y_{1}+(1-\beta)y_{2}\right)$
$\displaystyle<-\beta\rho(y_{1})s(y_{1})-(1-\beta)\rho(y_{2})s(y_{2})$
$\displaystyle\leq-\beta\rho(y_{1})\min\left\\{s(y_{1}),s(y_{2})\right\\}-(1-\beta)\rho(y_{2})\min\left\\{s(y_{1}),s(y_{2})\right\\}$
$\displaystyle=-\left[\beta\rho(y_{1})+(1-\beta)\rho(y_{2})\right]\min\left\\{s(y_{1}),s(y_{2})\right\\}$
$\displaystyle=-\rho\left(\beta
y_{1}+(1-\beta)y_{2}\right)\min\left\\{s(y_{1}),s(y_{2})\right\\},$
where the last equality is due to linearity. We then have
$s\left(\beta
y_{1}+(1-\beta)y_{2}\right)>\min\left\\{s(y_{1}),s(y_{2})\right\\}$
∎
###### Lemma 10 ([36]).
For given $y_{\kappa}$ and $\overline{y}_{\kappa}$ as defined in Equations
(3.6) and (3.7), respectively, we have
$s\left(\overline{y}_{\kappa}\right)\geq\min_{x\in\kappa}s\left(y(x)\right).$
(A.2)
###### Proof.
Let $\overline{\rho}_{\kappa}=\frac{1}{|\kappa|}\int_{\kappa}\rho(y(x))dx$.
With $U=-\rho s$, we have
$\displaystyle\overline{\rho}_{\kappa}s\left(\overline{y}_{\kappa}\right)$
$\displaystyle=-U\left(\overline{y}_{\kappa}\right)$
$\displaystyle\geq-\frac{1}{|\kappa|}\int_{\kappa}U(y(x))dx$
$\displaystyle=\frac{1}{|\kappa|}\int_{\kappa}\rho(y(x))s(y(x))dx$
$\displaystyle\geq\overline{\rho}_{\kappa}\min_{x\in\kappa}s\left(y(x)\right),$
where the second line is due to Jensen’s inequality since $U$ is convex. ∎
## Appendix B Concavity of shifted internal energy
###### Lemma 11.
The “shifted” internal energy per unit volume,
$\rho u^{*}=\rho u-\sum_{i=1}^{n_{s}}\rho_{i}b_{i0},$ (B.1)
is a concave function of the state for $\rho>0$, $T>0$.
###### Proof.
Throughout this proof, we work with $d=3$ and, without loss of generality, a
re-ordered state vector where the species concentrations are replaced by the
partial densities:
$y=\left(\rho v_{1},\ldots,\rho v_{d},\rho_{1},\ldots,\rho_{n_{s}},\rho
e_{t}\right)^{T}.$
It is well-known that a function is concave if and only if its Hessian is
negative semidefinite. Here, $\mathcal{H}=\frac{d^{2}\left(\rho
u^{*}\right)}{dy^{2}}$ denotes the Hessian of $\rho u^{*}$, which is a
symmetric matrix of size $m$. We observe that since the second term on the RHS
of Equation (B.1) is linear with respect to the state,
$\frac{d^{2}}{dy^{2}}\left(\sum_{i=1}^{n_{s}}\rho_{i}b_{i0}\right)$ gives a
matrix of zeros; thus, $\mathcal{H}=\frac{d^{2}\left(\rho u\right)}{dy^{2}}$.
There exist various ways to show negative semidefiniteness. One approach is to
check the signs of the principal minors [105, 106], where an $l$th-order
principal minor, $\mathcal{M}_{l}$, of $\mathcal{H}$ is the determinant of a
submatrix obtained by eliminating $m-l$ rows and the corresponding $m-l$
columns from $\mathcal{H}$. Specifically, $\mathcal{H}$ is negative
semidefinite if and only if all even-order principal minors are nonnegative
and all odd-order principal minors are nonpositive.
We start with some useful relations:
$\displaystyle\frac{\partial^{2}\left(\rho u\right)}{\partial\left(\rho
v_{k}\right)\partial\left(\rho v_{k}\right)}=-\frac{1}{\rho},$
$\displaystyle\frac{\partial^{2}\left(\rho u\right)}{\partial\left(\rho
v_{k}\right)\left(\partial\rho v_{l}\right)}=0,\quad k\neq l,$
$\displaystyle\frac{\partial^{2}\left(\rho
u\right)}{\partial\rho_{i}\partial\rho_{j}}=-\frac{\left|v\right|^{2}}{\rho},$
$\displaystyle\frac{\partial^{2}\left(\rho
u\right)}{\partial\rho_{i}\partial\left(\rho
v_{k}\right)}=\frac{v_{k}}{\rho},$ $\displaystyle\frac{\partial^{2}\left(\rho
u\right)}{\partial\rho_{i}\partial\left(\rho
e_{t}\right)}=\frac{\partial^{2}\left(\rho u\right)}{\partial\left(\rho
v_{k}\right)\partial\left(\rho e_{t}\right)}=\frac{\partial^{2}\left(\rho
u\right)}{\partial\left(\rho e_{t}\right)\partial\left(\rho e_{t}\right)}=0.$
$\mathcal{H}$ can then be written as
$\mathfrak{\mathcal{H}}=\left(\begin{array}[]{ccccccc}-\frac{1}{\rho}&0&0&\frac{v_{1}}{\rho}&\ldots&\frac{v_{1}}{\rho}&0\\\
0&-\frac{1}{\rho}&0&\frac{v_{2}}{\rho}&\ldots&\frac{v_{2}}{\rho}&\vdots\\\
0&0&-\frac{1}{\rho}&\frac{v_{3}}{\rho}&\ldots&\frac{v_{3}}{\rho}&0\\\
\frac{v_{1}}{\rho}&\frac{v_{2}}{\rho}&\frac{v_{3}}{\rho}&-\frac{\left|v\right|^{2}}{\rho}&\ldots&-\frac{\left|v\right|^{2}}{\rho}&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\frac{v_{1}}{\rho}&\frac{v_{2}}{\rho}&\frac{v_{3}}{\rho}&-\frac{\left|v\right|^{2}}{\rho}&\ldots&-\frac{\left|v\right|^{2}}{\rho}&0\\\
0&\ldots&0&0&\ldots&0&0\end{array}\right).$ (B.2)
The $m$ first-order principal minors, $\mathcal{M}_{1}$, are simply the
diagonal entries, which are all nonpositive. The principal minors of order
greater than one take the following forms:
$\displaystyle\mathcal{M}_{l}^{0}$
$\displaystyle=\det\left(\begin{array}[]{cccc}a_{1,1}&\ldots&a_{1,l-1}&0\\\
\vdots&\ddots&\vdots&\vdots\\\ a_{l-1,1}&\ldots&a_{l-1,l-1}&\vdots\\\
0&\ldots&\ldots&0\end{array}\right),\;l=2,\ldots,m,$ (B.7)
$\displaystyle\mathcal{M}_{l}^{1}$
$\displaystyle=\det\left(\begin{array}[]{ccc}-\frac{1}{\rho}&0&0\\\
0&\ddots&0\\\ 0&0&-\frac{1}{\rho}\end{array}\right),\;l=2,\ldots,d,$ (B.11)
$\displaystyle\mathcal{M}_{l}^{2}$
$\displaystyle=\det\left(B_{l}\right),B_{l}=\left(\begin{array}[]{ccc}-\frac{|v|^{2}}{\rho}&\ldots&-\frac{|v|^{2}}{\rho}\\\
\vdots&\ddots&\vdots\\\
-\frac{|v|^{2}}{\rho}&\ldots&-\frac{|v|^{2}}{\rho}\end{array}\right),\;l=2,\ldots,n_{s},$
(B.15) $\displaystyle\mathcal{M}_{l}^{3}$
$\displaystyle=\det\left(C_{l}\right)=\det\left(\begin{array}[]{c|c}C_{1,1}&C_{1,2}\\\
\hline\cr C_{2,1}&B_{q}\end{array}\right)$ (B.18)
$\displaystyle=\det\left(\begin{array}[]{c|c}C_{1,1}&C_{1,2}\\\ \hline\cr
C_{2,1}&\begin{array}[]{ccc}-\frac{|v|^{2}}{\rho}&\ldots&-\frac{|v|^{2}}{\rho}\\\
\vdots&\ddots&\vdots\\\
-\frac{|v|^{2}}{\rho}&\ldots&-\frac{|v|^{2}}{\rho}\end{array}\end{array}\right),\;q=1,\ldots,n_{s};\;l=q+1,\ldots,q+d.$
(B.24)
$\mathcal{M}_{l}^{0}$ is zero due to the row of zeros (note that the $m$th-
order principal minor, $\mathcal{M}_{m}=\det\left(\mathcal{H}\right)$, takes
this form); $\mathcal{M}_{l}^{1}$ is negative for $l$ odd and positive for $l$
even; and $\mathcal{M}_{l}^{2}$ is zero due to linear dependence of the rows.
For $\mathcal{M}_{l}^{3}$, the corresponding submatrix, $C_{l}$, is written in
block-matrix form, where the lower-right block is a matrix of size $q$ of the
form in Equation (B.15). We consider two cases: $q=1$ and $q\geq 1$. In the
latter case, $\mathcal{M}_{l}^{3}=0$ since the last $q$ rows of $C_{l}$ are
repeated. In the former, we first consider $l=2$, which yields
$\mathcal{M}_{2}^{3}=\det\left(\begin{array}[]{cc}-\frac{1}{\rho}&\frac{v_{k}}{\rho}\\\
\frac{v_{k}}{\rho}&-\frac{\left|v\right|^{2}}{\rho}\end{array}\right)=\frac{\left|v\right|^{2}}{\rho^{2}}-\frac{v_{k}^{2}}{\rho^{2}},\;k=1,\ldots,d,$
which is nonnegative since $\left|v\right|^{2}=\sum_{i}v_{i}^{2}\geq
v_{k}^{2}\geq 0$. For $l=3$, we have
$\mathcal{M}_{4}^{3}=\det\left(\begin{array}[]{ccc}-\frac{1}{\rho}&0&\frac{v_{j}}{\rho}\\\
0&-\frac{1}{\rho}&\frac{v_{k}}{\rho}\\\
\frac{v_{j}}{\rho}&\frac{v_{k}}{\rho}&-\frac{\left|v\right|^{2}}{\rho}\end{array}\right)=-\frac{\left|v\right|^{2}}{\rho^{3}}+\frac{v_{j}^{2}}{\rho^{3}}+\frac{v_{k}^{2}}{\rho^{3}},\;j,k=1,\ldots,d;\;j\neq
k.$
which is nonpositive. Finally, $l=4$ gives
$\mathcal{M}_{4}^{3}=\det\left(\begin{array}[]{cccc}-\frac{1}{\rho}&0&0&\frac{v_{1}}{\rho}\\\
0&-\frac{1}{\rho}&0&\frac{v_{2}}{\rho}\\\
0&0&-\frac{1}{\rho}&\frac{v_{3}}{\rho}\\\
\frac{v_{1}}{\rho}&\frac{v_{2}}{\rho}&\frac{v_{3}}{\rho}&-\frac{\left|v\right|^{2}}{\rho}\end{array}\right)=\left(\begin{array}[]{ccc}\rule[2.15277pt]{10.76385pt}{0.5pt}&\mathsf{R}_{1}&\rule[2.15277pt]{10.76385pt}{0.5pt}\\\
\rule[2.15277pt]{10.76385pt}{0.5pt}&\mathsf{R}_{2}&\rule[2.15277pt]{10.76385pt}{0.5pt}\\\
\rule[2.15277pt]{10.76385pt}{0.5pt}&\mathsf{R}_{3}&\rule[2.15277pt]{10.76385pt}{0.5pt}\\\
\rule[2.15277pt]{10.76385pt}{0.5pt}&\mathsf{R}_{4}&\rule[2.15277pt]{10.76385pt}{0.5pt}\end{array}\right),$
where
$\left\\{\mathsf{R}_{1},\mathsf{R}_{2},\mathsf{R}_{3},\mathsf{R}_{4}\right\\}$
denotes the rows. We observe that
$\mathsf{R}_{4}=-v_{1}\mathsf{R}_{1}-v_{2}\mathsf{R}_{2}-v_{3}\mathsf{R}_{3}$;
therefore, the rows are linearly dependent and $\mathcal{M}_{4}^{3}=0$. As
such, the principal minors satisfy the nonpositive/nonnegative requirements
for negative semidefiniteness of $\mathcal{H}$. ∎
|
# A homogeneous method for summation and its application
Zhipeng Lu Shenzhen MSU-BIT University, 1 International University Park Road,
Dayun New Town, Longgang District, Shenzhen, Guangdong Province, P.R. China
<EMAIL_ADDRESS>
###### Abstract.
We introduce a homogeneous method to deal with summations with homogeneous
factors. Then we use it to compute main terms in the asymptotics of distance
energy of square lattices in circles, which relates to the conjecture of
distinct distances by Erdős.
###### Key words and phrases:
Summation, sum of squares, distinct distances
###### 2020 Mathematics Subject Classification:
40D05, 11Y35, 52C10, 11P21
## 1\. A simple homogeneous method in summation with a log factor
In this section, we mainly introduce a homogeneous method to specifically
facilitate summations with a log factor as follows
###### Lemma 1.1.
If the function $f(x,y)>0$ is homogeneous, i.e. $f(kx,ky)=f(x,y),\forall
k\in\mathbb{R}$, and integrable in $x$, then
$\sum_{n\leq N}f(n,N)\log n\sim cN\log N,$
where $c=\int_{0}^{1}f(x,1)dx$.
Here $f(x)\sim g(x)$ always means $\frac{f(x)}{g(x)}\rightarrow 1$ as $x$
tends to infinity. Results in this form seems new in the author’s view, but
they might have been used by other authors. Though clear enough by itself, we
prove it by double counting as follows
###### Proof.
To deal with the summation, we introduce a double counting method to split the
log factor out as follows. First, we partition the interval $[1,N]$ into
$[\frac{m-1}{K}N,\frac{m}{K}N)$ for $m=1,\dots,K$. On each sub-interval, since
$f$ is projective and continuous, we can easily squeeze the partial sum as
(1.1) $\displaystyle\xi_{m}\sum_{\frac{m-1}{K}N\leq n<\frac{m}{K}N}\log
n<\sum_{\frac{m-1}{K}N\leq n<\frac{m}{K}N}f(n,N)\log
n<\eta_{m}\sum_{\frac{m-1}{K}N\leq n<\frac{m}{K}N}\log n,$
where $\xi_{m}=\min_{\frac{m-1}{K}\leq\frac{n}{N}<\frac{m}{K}}\\{f(n/N,1)\\}$
and $\eta_{m}=\max_{\frac{m-1}{K}\leq\frac{n}{N}<\frac{m}{K}}\\{f(n/N,1)\\}$.
Then we can asymptotically approximate the partial sum of $\log n$ by integral
as
$\sum_{\frac{m-1}{K}N\leq n<\frac{m}{K}N}\log
n\sim\frac{N}{K}\int_{m-1}^{m}(\log x+\log\frac{N}{K})dx\sim\frac{N}{K}(\log
N-\log K+\log m)\sim\frac{N\log N}{K},$
if we set $\log K=o(\log N)$, i.e. $K=N^{o(1)}$. Thus by (1.1), the sum may be
abbreviated to
$\frac{1}{N\log N}\sum_{n=1}^{N-1}f(n,N)\log
n\sim\frac{1}{K}\sum_{m<K}\theta_{m}\sim\int_{0}^{1}f(x,1)dx,$
for some $\xi_{m}\leq\theta_{m}\leq\eta_{m}$, if $f(x,1)$ is (Riemann)
integrable. ∎
If $f(x,y)$ is not homogeneous, but with deviation, say,
$f(kx,ky)=k^{\alpha}f(x,y)$ for some $\alpha\in\mathbb{R}$, then (1.1) is just
scaled by $N^{\alpha}$ and the result becomes
###### Corollary 1.2.
If $f(x,y)$ is homogeneous of degree $\alpha\in\mathbb{R}$, i.e.
$f(kx,ky)=k^{\alpha}f(x,y)$, and integrable in $x$, then
$\sum_{n\leq N}f(n,N)\log n\sim cN^{1+\alpha}\log N,$
where $c=\int_{0}^{1}f(x,1)dx$.
For the most obvious example, let $f(x,y)=x^{1+\alpha}/y,\alpha>-2$. Then it
just tells us that $\sum_{n\leq N}n^{1+\alpha}\log
n\sim\frac{1}{2+\alpha}N^{2+\alpha}\log N$, which is seen from obvious
approximation by integral.
Moreover, the double counting method allows us to handle summation with other
factors than just the log factor, provided that the factor behaves as well as
###### Corollary 1.3.
Suppose that the function $f(x,y)$ is homogeneous of degree
$\alpha\in\mathbb{R}$ and integrable in $x$, and that $g(x)$ has the property
that $g(N)\rightarrow\infty$ and $g(Nx)=g(N)+o(g(N))$ for $0<\delta(N)<x<1$
and $\delta(N)\rightarrow 0$ as $N\rightarrow+\infty$. Then
$\sum_{n\leq N}f(n,N)g(n)\sim cNg(N),$
where $c=\int_{0}^{1}f(x,1)dx$.
###### Proof.
Following the proof of Lemma 1.1, the summation of logarithms is substituted
by that of $g(n)$. By the property of $g(x)$, we have for $K=1/\delta(N)$,
$\sum_{\frac{m-1}{K}N\leq
n\leq\frac{m}{K}N}g(n)\sim\int_{\frac{m-1}{K}N}^{\frac{m}{K}N}g(x)dx=\frac{N}{K}\int_{m-1}^{m}g(Nx/K)dx=\frac{N}{K}(g(N)+o(g(N)))\sim\frac{Ng(N)}{K}.$
Thus,
$\lim\limits_{K\rightarrow+\infty}\frac{1}{Ng(N)}\sum_{n\leq
N}f(n,N)g(n)=\lim\limits_{K\rightarrow+\infty}\frac{1}{K}\sum_{m<K}\theta_{m}=w\int_{0}^{1}f(x,1)dx.$
∎
###### Remark 1.4.
If $g(x)=(\log x)^{\beta}$ for some $\beta>0$, then $g(Nm/K)=(\log
N+\log(m/K))^{\beta}\sim(\log N)^{\beta}$ for $m\leq K$ and $\log K=o(\log
N)$, so that similar to Lemma 1.1 we have
$\sum_{n\leq N}f(n,N)(\log n)^{\beta}\sim cN(\log N)^{\beta}.$
Moreover, it might be generalized to arbitrary $g(x)$ of slow growth by
appropriate double counting. Also, it would be interesting to derive the minor
terms of the above summations.
## 2\. Application to distance energy estimate
Now we apply the above results to an explicit counting problem in discrete
geometry or number theory. Actually, we found the homogeneous phenomena during
studying the following problem. Let $P=[\sqrt{N}]\times[\sqrt{N}]$ be the
square grid of size $N$, where $[x]$ denotes the set of integers ranging from
$1$ to $\lfloor x\rfloor$. By studying the value distribution of $x^{2}+y^{2}$
on $P$, it can be estimated that $d(P):=|\\{d(p,q)\mid p,q\in P\\}|\sim
c\frac{|P|}{\sqrt{\log|P|}}$ for some $c>0$. This becomes the initiating
example for the Erdős conjecture on distinct distances in the Euclidean plane
$\mathbb{R}^{2}$, which says $d(P):=|\\{d(p,q)\mid p,q\in P\\}|\geq
c\frac{|P|}{\sqrt{\log|P|}}$ for any finite set $P\subset\mathbb{R}^{2}$ and
some absolute constant $c>0$.
Guth and Katz [1] established the nearly optimal bound $d(P)\geq
c\frac{|P|}{\log|P|}$. The essential object therein is what they call distance
quadruples, i.e. $Q(P)=:\\{(p_{1},q_{1},p_{2},p_{2})\in P^{4}\mid
d(p_{1},q_{1})=d(p_{2},q_{2})\\}$. We call $|Q(P)|$ the distance energy of
$P$, denoted by $E_{2}(P)$. Note that in the appendix of [1],
$E_{2}([\sqrt{N}]\times[\sqrt{N}])$ is estimated to be $\theta(N^{3}\log N)$
by counting line-line incidences in $\mathbb{R}^{3}$. In this section, we
establish the asymptotics of $E_{2}(P)$ for $P$ being square lattices in
circles resorting to our homogeneous method.
Denote $r(n):=|\\{(a,b)\in\mathbb{Z}^{2}\mid a^{2}+b^{2}=n\\}|$. On average,
we have the following estimate:
###### Lemma 2.1 (see (7.20) of Wilson [3]).
For any positive integer $k$ and $x\in\mathbb{R}_{+}$, we have
$\sum_{n\leq x}r^{2}(n)\sim 4x\log x+O(x).$
More precise estimate on the distance energy on square grids takes us more
effort to develop number theoretic methods. For convenience, we study lattice
grids in circles, i.e. $P=\mathbb{Z}^{2}\cap B_{\sqrt{N}}(0,0)$, where
$B_{n}(a,b)$ denotes the disk centered at $(a,b)$ with radius $n$. By results
of the Gauss circle problem (see 1.4 of [2]),
(2.2) $|P|=\pi N+o(N^{1/3}).$
Denote by $r_{a,b}(n)=\\{(x,y)\in P\mid(x-a)^{2}+(y-b)^{2}=n\\}$ so that
$r_{0,0}(n)=r(n)$ for $n\leq N$. Actually, if
$\sqrt{a^{2}+b^{2}}\leq\sqrt{N}-\sqrt{n}$, then $r_{a,b}(n)=r(n)$. For
$\sqrt{a^{2}+b^{2}}>\sqrt{N}-\sqrt{n}$, $\partial B_{\sqrt{n}}(a,b)$ is cut by
$\partial B_{\sqrt{N}}(0,0)$. By easy calculation, the cut arc has angle
$2\arccos\left(\frac{a^{2}+b^{2}+n-N}{2\sqrt{n(a^{2}+b^{2})}}\right)$. Then by
symmetry, one may expect that
(2.3) $r_{a,b}(n)\sim\tilde{r}_{a,b}(n):=\ \begin{cases}r(n),\text{ if
}\sqrt{a^{2}+b^{2}}\leq\sqrt{N}-\sqrt{n},n\leq N;\\\ 0,\text{ if
}\sqrt{a^{2}+b^{2}}\leq\sqrt{n}-\sqrt{N},n>N;\\\
\frac{r(n)}{\pi}\arccos\left(\frac{a^{2}+b^{2}+n-N}{2\sqrt{n(a^{2}+b^{2})}}\right),\text{
otherwise}.\end{cases}$
Although the estimate by $\tilde{r}_{a,b}(n)$ may deviate from the true
distribution, the summation $R(n):=\sum_{(a,b)\in P}r_{a,b}(n)$ counting all
the pairs of points $(p,q)\in P^{2}$ with $d(p,q)=n$, turns out to be valid
from the average symmetric point of view. We may use area counting to clarify
this. Define $s_{a,b}(n)=|\\{(x,y)\in\mathbb{Z}^{2}\mid(x-a)^{2}+(y-b)^{2}\leq
n\\}$ for any $(a,b)\in B_{\sqrt{N}}(0,0),0\leq n\leq 4N$. Denote by
$s(n)=s_{0,0}(n)$. Clearly by simple trigonometry
$s_{a,b}(n)-s_{a,b}(n-1)=\frac{s(n)-s(n-1)}{\pi}\arccos\left(\frac{a^{2}+b^{2}+n-N}{2\sqrt{n(a^{2}+b^{2})}}\right)+O(1).$
Hence we have
(2.4) $\displaystyle R(n)$ $\displaystyle=S(n)-S(n-1)=\sum_{a^{2}+b^{2}\leq
N}(s_{a,b}(n)-s_{a,b}(n-1))$
$\displaystyle=\sum_{\sqrt{a^{2}+b^{2}}\leq\sqrt{N}-\sqrt{n}}r(n)+\sum_{\sqrt{a^{2}+b^{2}}>\sqrt{N}-\sqrt{n}}\tilde{r}_{a,b}(n)+O(N).$
More explicitly, we show
###### Lemma 2.2.
Let $P$ be the integer points in the disk of radius $\sqrt{N}$ and $R(n)$ be
the number of pairs of points from $P$ with distance $\sqrt{n},n\leq 4N$ as
above. Then
$R(n)=\left(2\arccos\left(\frac{\sqrt{n/N}}{2}\right)-\sqrt{\frac{4Nn-n^{2}}{4N^{2}}}\right)Nr(n)+O(N).$
###### Proof.
By (2.2), (2.3) and (2.4), we have for $n\leq N$,
$\displaystyle R(n)$
$\displaystyle=r(n)\sum_{\sqrt{a^{2}+b^{2}}\leq\sqrt{N}-\sqrt{n}}1+\frac{r(n)}{\pi}\sum_{\sqrt{N}-\sqrt{n}<\sqrt{a^{2}+b^{2}}\leq\sqrt{N}}\arccos\left(\frac{a^{2}+b^{2}+n-N}{2\sqrt{n(a^{2}+b^{2})}}\right)+O(N)$
$\displaystyle=\pi
r(n)(\sqrt{N}-\sqrt{n})^{2}+\frac{r(n)}{\pi}\iint_{(\sqrt{N}-\sqrt{n})^{2}\leq
x^{2}+y^{2}\leq
N}\arccos\left(\frac{x^{2}+y^{2}+n-N}{2\sqrt{n(x^{2}+y^{2})}}\right)dxdy+O(N).$
Using the polar coordinates we may transform the double integral into
$\displaystyle
2\pi\int_{\sqrt{N}-\sqrt{n}}^{\sqrt{N}}r\arccos\left(\frac{r^{2}+n-N}{2\sqrt{n}r}\right)dr$
$\displaystyle=$ $\displaystyle\pi
r^{2}\arccos\left(\frac{r^{2}+n-N}{2\sqrt{n}r}\right)\mid_{\sqrt{N}-\sqrt{n}}^{\sqrt{N}}+\pi\int_{\sqrt{N}-\sqrt{n}}^{\sqrt{N}}r^{2}\frac{\frac{1}{2\sqrt{n}}+\frac{N-n}{2\sqrt{n}r^{2}}}{\sqrt{1-\frac{(r^{2}+n-N)^{2}}{4nr^{2}}}}dr$
$\displaystyle=$ $\displaystyle\pi
N\arccos\left(\frac{\sqrt{n/N}}{2}\right)-\pi^{2}(\sqrt{N}-\sqrt{n})^{2}+\frac{\pi}{2}\int_{\sqrt{N}-\sqrt{n}}^{\sqrt{N}}\frac{r^{2}+N-n}{\sqrt{4nr^{2}-(r^{2}+n-N)^{2}}}d(r^{2}).$
Substituting by $s=\frac{r^{2}-n-N}{2\sqrt{Nn}}$ we get
$\displaystyle\int_{-1}^{-\frac{\sqrt{n/N}}{2}}\frac{2\sqrt{Nn}s+2N}{\sqrt{1-s^{2}}}ds$
$\displaystyle=-2\sqrt{Nn}\sqrt{1-s^{2}}\mid_{-1}^{-\frac{\sqrt{n/N}}{2}}+2N\arcsin(s)\mid_{-1}^{-\frac{\sqrt{n/N}}{2}}$
$\displaystyle=-\sqrt{4Nn-n^{2}}+2N\left(\frac{\pi}{2}-\arcsin\left(\frac{\sqrt{n/N}}{2}\right)\right).$
Summing up everything provides us for $n\leq N$,
$\displaystyle R(n)=$ $\displaystyle\pi
r(n)(\sqrt{N}-\sqrt{n})^{2}+r(n)N\arccos\left(\frac{\sqrt{n/N}}{2}\right)-\pi
r(n)(\sqrt{N}-\sqrt{n})^{2}$
$\displaystyle-\frac{r(n)}{2}\sqrt{4Nn-n^{2}}+\frac{\pi
r(n)}{2}N-r(n)N\arcsin\left(\frac{\sqrt{n/N}}{2}\right)+O(N)$ $\displaystyle=$
$\displaystyle
r(n)\left(N\arccos\left(\frac{\sqrt{n/N}}{2}\right)-\sqrt{Nn-\frac{n^{2}}{4}}+\frac{\pi}{2}N-N\arcsin\left(\frac{\sqrt{n/N}}{2}\right)\right)+O(N)$
$\displaystyle=$
$\displaystyle\left(2\arccos\left(\frac{\sqrt{n/N}}{2}\right)-\sqrt{\frac{4Nn-n^{2}}{4N^{2}}}\right)Nr(n)+O(N).$
When $N<n\leq 4N$, we have by (2.3)
$\displaystyle R(n)$
$\displaystyle=\frac{r(n)}{\pi}\sum_{\sqrt{n}-\sqrt{N}<\sqrt{a^{2}+b^{2}}\leq\sqrt{N}}\arccos\left(\frac{a^{2}+b^{2}+n-N}{2\sqrt{n(a^{2}+b^{2})}}\right)+O(N)$
$\displaystyle=\frac{r(n)}{\pi}\iint_{(\sqrt{N}-\sqrt{n})^{2}\leq
x^{2}+y^{2}\leq
N}\arccos\left(\frac{x^{2}+y^{2}+n-N}{2\sqrt{n(x^{2}+y^{2})}}\right)dxdy+O(N)$
$\displaystyle=r(n)r^{2}\arccos\left(\frac{r^{2}+n-N}{2\sqrt{n}r}\right)\mid_{\sqrt{n}-\sqrt{N}}^{\sqrt{N}}+\frac{r(n)}{2}\int_{\sqrt{n}-\sqrt{N}}^{\sqrt{N}}r^{2}\frac{\frac{1}{2\sqrt{n}}+\frac{N-n}{2\sqrt{n}r^{2}}}{\sqrt{1-\frac{(r^{2}+n-N)^{2}}{4nr^{2}}}}dr+O(N)$
$\displaystyle=Nr(n)\arccos\left(\frac{\sqrt{n/N}}{2}\right)+\frac{r(n)}{2}\int_{\sqrt{n}-\sqrt{N}}^{\sqrt{N}}\frac{r^{2}+N-n}{\sqrt{4nr^{2}-(r^{2}+n-N)^{2}}}d(r^{2})+O(N)$
$\displaystyle=Nr(n)\arccos\left(\frac{\sqrt{n/N}}{2}\right)+\frac{r(n)}{2}\left(-\sqrt{4Nn-n^{2}}+2N\left(\frac{\pi}{2}-\arcsin\left(\frac{\sqrt{n/N}}{2}\right)\right)\right)+O(N)$
$\displaystyle=\left(2\arccos\left(\frac{\sqrt{n/N}}{2}\right)-\sqrt{\frac{4Nn-n^{2}}{4N^{2}}}\right)Nr(n)+O(N),$
which adopts the same form as for $n\leq N$. ∎
As asymptotics of single $R(n)$, the above result seems too weak, but it
provides us to the main term of the distance energy as follows
###### Theorem 2.3.
Let $P$ be the set of integral lattice points in a disk of radius $\sqrt{N}$,
then
$E_{2}(P)\sim(4\pi^{2}-8\pi+16)N^{3}\log N.$
###### Proof.
Let $E(x)=\sum_{n\leq x}r^{2}(n)$. Then by Lemma 2.1, Lemma 2.2, (2.1) and
Abel’s summation by parts, we get (noting that $\sum_{n\leq N}r(n)\sim\pi N$)
$\displaystyle E_{2}(P)$ $\displaystyle=\sum_{n\leq
4N}R(n)^{2}=N^{2}\sum_{n\leq
4N}r^{2}(n)\left(2\arccos\left(\frac{\sqrt{n/N}}{2}\right)-\sqrt{\frac{4Nn-n^{2}}{4N^{2}}}\right)^{2}+O(N^{3})$
$\displaystyle=4N^{2}\sum_{n=1}^{4N}r^{2}(n)\arccos^{2}\left(\sqrt{\frac{n}{4N}}\right)-2N^{2}\sum_{n=1}^{4N}r^{2}(n)\sqrt{\frac{4Nn-n^{2}}{N^{2}}}\arccos\left(\frac{\sqrt{n/N}}{2}\right)$
$\displaystyle\quad+\frac{N^{2}}{4}\sum_{n=1}^{4N}r^{2}(n)\frac{4Nn-n^{2}}{N^{2}}+O(N^{3})$
$\displaystyle=①-②+③+O(N^{3}).$
Using Abel summation and Lemma 2.1, we get
$\displaystyle①=$ $\displaystyle
4N^{2}\sum_{n=1}^{4N}(E(n)-E(n-1))\arccos^{2}\left(\sqrt{\frac{n}{4N}}\right)$
$\displaystyle=$ $\displaystyle
4N^{2}\sum_{n=1}^{4N-1}E(n)\left(\arccos^{2}\left(\sqrt{\frac{n}{4N}}\right)-\arccos^{2}\left(\sqrt{\frac{n+1}{4N}}\right)\right)+O(N^{2})$
$\displaystyle=$ $\displaystyle
8N^{2}\sum_{n=1}^{4N-1}E(n)\arccos\left(\sqrt{\frac{n}{4N}}\right)\left(\arccos\left(\sqrt{\frac{n}{4N}}\right)-\arccos\left(\sqrt{\frac{n+1}{4N}}\right)\right)+O(N^{2})$
$\displaystyle=$ $\displaystyle
4N^{2}\sum_{n=1}^{4N-1}\frac{E(n)}{\sqrt{(4N-n)n}}\arccos\left(\sqrt{\frac{n}{4N}}\right)+O(N^{2}\log
N)$ $\displaystyle=$ $\displaystyle 16N^{2}\sum_{n=1}^{4N-1}\frac{\sqrt{n}\log
n}{\sqrt{4N-n}}\arccos\left(\sqrt{\frac{n}{4N}}\right)+O(N^{3}).$
Now the above summation falls into the case of Lemma 1.1, which provides us
$①\sim 16c_{1}N^{3}\log N,\
c_{1}=\int_{0}^{1}\sqrt{\frac{t}{1-t}}\arccos(\sqrt{t})dt=\frac{\pi^{2}-4}{8}.$
Similarly,
$\displaystyle②$
$\displaystyle=2N^{2}\sum_{n=1}^{4N}(E(n)-E(n-1))\sqrt{\frac{4Nn-n^{2}}{N^{2}}}\arccos\left(\sqrt{\frac{n}{4N}}\right)$
$\displaystyle=2N^{2}\sum_{n=1}^{4N-1}E(n)\left(\sqrt{\frac{4Nn-n^{2}}{N^{2}}}\arccos\sqrt{\frac{n}{4N}}-\sqrt{\frac{4N(n+1)-(n+1)^{2}}{N^{2}}}\arccos\sqrt{\frac{n+1}{4N}}\right)$
$\displaystyle\quad+O(N^{2})$
$\displaystyle=8N^{2}\sum_{n=1}^{4N-1}\left(\frac{1-2\sqrt{\frac{n}{4N}}}{\sqrt{1-\frac{n}{4N}}}\arccos\sqrt{\frac{n}{4N}}-\sqrt{\frac{n}{4N}}\right)\log
n+O(N^{3})$ $\displaystyle\sim 8(c_{2}-\frac{2}{3})N^{3}\log N,$
where
$c_{2}=\int_{0}^{1}\frac{1-2\sqrt{t}}{\sqrt{1-t}}\arccos(\sqrt{t})dt=\pi-2-2c_{1}$.
And also,
$\displaystyle③$
$\displaystyle=\frac{N^{2}}{4}\sum_{n=1}^{4N}(E(n)-E(n-1))\frac{4Nn-n^{2}}{N^{2}}$
$\displaystyle=\frac{N^{2}}{4}\sum_{n=1}^{4N-1}E(n)\left(\frac{4Nn-n^{2}}{N^{2}}-\frac{4N(n+1)-(n+1)^{2}}{N^{2}}\right)+O(N^{2})$
$\displaystyle=\sum_{n=1}^{4N-1}(2n-4N+1)n\log n+O(N^{3})$
$\displaystyle\sim\frac{32}{3}N^{3}\log N.$
Finally, altogether we get
$\displaystyle E_{2}(P)$
$\displaystyle\sim①-②+③\sim(16c_{1}-8c_{2}+16)N^{3}\log N$
$\displaystyle=(4\pi^{2}-8\pi+16)N^{3}\log N.$
∎
###### Remark 2.4.
Notice that the above summations (divided by $N\log N$) converge extremely
slow. Say the last summation in $①$, computing until $N=10^{11}$, the second
decimal is not even stable.
## References
* [1] L. Guth, N. H. Katz, On the Erdős distinct distances problem in the plane, Annals of Mathematics 181 (2015), 155-190.
* [2] A. A. Karatsuba, Basic Analytic Number Theory, Springer-Verlag Berlin Heidelberg New York in 1993.
* [3] B. M. Wilson, Proofs of some formulae enunciated by Ramanujan, Proc. London Math. Soc. 21 (1922), 235-255.
|
∎
e1Corresponding Author<EMAIL_ADDRESS>
11institutetext: Department of Physics, The University of South Dakota,
Vermillion, SD 57069, USA 22institutetext: School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA 33institutetext:
Department of Physics and Astronomy, Texas A$\&$M University, College Station,
TX 77843, USA
# Observation of Time-Dependent Internal Charge Amplification in a Planar
Germanium Detector at Cryogenic Temperature
P. Acharyaaddr1 M. Frittsaddr2 D.-M. Meie1,addr1 V. Mandicaddr2 C.-J.
Wangaddr1 R. Mahapatraaddr3 and M. Plattaddr3
(Received: date / Accepted: date)
###### Abstract
For the first time, time-dependent internal charge amplification through
impact ionization has been observed in a planar germanium (Ge) detector
operated at cryogenic temperature. In a time period of 30 and 45 minutes after
applying a bias voltage, the charge energy corresponding to a baseline of the
59.54 keV $\gamma$ rays from a 241Am source is amplified for a short period of
time and then decreases back to the baseline. The amplification of charge
energy depends strongly on the applied positive bias voltage with drifting
holes across the detector. No such phenomenon is visible with drifting
electrons across the detector. We find that the observed charge amplification
is dictated by the impact ionization of charged states, which has a strong
correlation with impurity level and applied electric field. We analyze the
dominant physics mechanisms that are responsible for the creation and the
impact ionization of charged states. Our analysis suggests that the
appropriate level of impurity in a Ge detector can enhance charge yield
through the impact ionization of charged states to achieve extremely low-
energy detection threshold ($<$ 10 meV) for MeV-scale dark matter searches if
the charge amplification can be stabilized.
###### Keywords:
Charge Collection Efficiency (CCE); Time-Dependent Internal Charge
Amplification; Cluster Dipole States
††journal: Eur. Phys. J. C
## 1 Introduction
Dark matter (DM) is believed to be ubiquitous. It makes up 85% of the mass of
the universe fzw ; ghi ; jlf ; mwg . Many candidates including axions, low-
mass DM, and weakly interacting massive particles (WIMPs) have been postulated
axion ; lmdm ; wimp . WIMPs are expected to generate observable nuclear recoil
energy through elastic scattering off nuclei rjg . Despite great efforts made
in searching for axions and WIMPs ardm ; cd09 ; cd ; cd1 ; cd2 ; cog ; cre12 ;
cou ; bar ; dama ; dar ; dri ; ede ; kim ; lux ; pan ; pic ; cd14 ; xe11 ;
xe15 ; xe17 ; xma ; zep , DM remains undetected. Recently, low-mass DM in the
MeV-scale has become an exciting DM candidate ess2012 ; ess2016 ; ho ; ste .
To directly detect MeV-scale DM, a detector with sensitivity of measuring a
single electron-hole (e-h) pair is required, since the energy deposition
induced by MeV-scale DM through elastic scattering off electrons or nuclei is
in the range of sub-eV to 100 eV ess2012 ; mei . In 2018, Mei et al. proposed
to detect MeV-scale DM utilizing germanium internal charge amplification
(GeICA) geia for the charge created by the ionization of impurities mei .
GeICA can potentially achieve a detection energy threshold of $\sim$0.1 eV
(100 meV), allowing a large portion of both electronic recoils and nuclear
recoils in the range of sub-eV to 100 eV induced by MeV-scale DM to be
accessible mei .
GeICA amplifies internal charge through impact ionization, which is a process
first observed in Ge diodes a few decades ago imp1 ; imp . In this process, a
charge carrier, electron or hole, with sufficient kinetic energy can knock a
bound electron out of its valence state and elevate it to a state in the
conduction band, creating an electron-hole pair. Carriers gain sufficient
kinetic energy through applying a strong electric field. Impact ionization of
Ge atoms requires higher electric field than that of impurities, since the
bandgap of Ge is about 0.7 eV (700 meV) while the ionization potential of
neutral impurity atoms in Ge is around 0.01 eV (10 meV). When a Ge detector is
cooled down to below 10 K, the residual impurities in Ge start to freeze out
from the conduction or valence band to localized states mei2022evidence . At
below 6 K, the localized states become thermally stable and form electric
dipole states mei2022evidence , which are excited neutral impurity states with
a binding energy of less than 10 meV. The dipole states can trap charge to
form cluster dipole states mei2022evidence with even smaller binding energy
depending on the operational temperature.
The formation of excited dipole states and cluster dipole states in p-type Ge
is depicted in Figure 1 mei2022evidence . The phase space of an immobile
negative ion for trapping charge carriers is smaller than that of movable
bound holes, whose motion is restricted by the Onsager radius,
$R=\frac{1}{4\pi\varepsilon\varepsilon_{0}K_{B}T}$, where $\varepsilon$= 16.2
is the relative permittivity for Ge, $\varepsilon_{0}$ is the permittivity of
free space, $K_{B}$ is the Boltzmann constant, and $T$ is temperature.
Therefore, the probability of forming $A^{-^{*}}$ states is higher than
forming $A^{+^{*}}$ states in a p-type detector. This is why electrons are
trapped more severely than holes in a p-type detector.
Figure 1: Shown are the processes involved in the formation of the excited
dipole states and the cluster dipole states in a p-type Ge detector operated
at low temperatures, where $\vec{p}$ and $\vec{q}$ are the corresponding
dipole moments.
## 2 The Experimental Methods and the Observed Physical Phenomenon
The impact ionization of impurities in Ge specimens has been reported for a
range of temperatures ($\sim$ 4.2 K to 298 K) by many authors from 1950s -
1970s sclar ; pickin ; smith ; palm . The most recent impact ionization of
impurities was reported by Phipps et al. with SuperCDMS-style detectors at 40
milliKelvin (mK) phipps1 ; phipps2 , which is similar to this work with a
detector made from a USD crystal wang1 ; wang2 . The detector was fabricated
at Texas A $\&$ M University with four channels for charge readout,
geometrically similar to SuperCDMS style detectors supercdms , as shown in
Figure 2.
Figure 2: Shown is the detector studied in this work with four electrodes
read-out ($Q_{1},Q_{2},Q_{3},andQ_{4}$). Four ${}^{241}Am$ sources were
arranged in such a way that one source was over each channel. The opposite
side of the detector is grounded with uniform Al electrodes so that the
electric field points along the z-axis of the crystal. The size of the
detector is 10 cm in diameter and 3.3 cm in thickness with a mass of $\sim$1.4
kg.
The detector was wire-bonded, mounted in a dilution refrigerator and tested at
the K100 Detector Testing Facility at the University of Minnesota (UMN).
Figure 3: Shown is the orientation of the Am-241 source mover before it was
installed.
The detector was cooled down to $\sim$40 mK and then tested in two separate
refrigerator runs, Run 67 in 2018 and Run 74 in 2021. In Run 67 four 241Am
sources were mounted above each channel on the detector (see Figure 2). The
lead collimators with 0.2 mm holes allow 59.54 keV $\gamma$ rays through to
the detector. Alpha particles from the sources were blocked by the source
encapsulation. We observed 59.54 keV peaks in spectra from each channel and
performed different measurements over a course of two weeks. In Run 74 a
single 241Am source was mounted on a carriage that could be moved by a
superconducting stepper motor (see Figure 3). The source was of a different
design with a 0.5 mm collimator hole and a lower rate of $\gamma$ rays
incident on the detector (about 75% as much as the sources used in Run 67).
Figure 4: Shown is the time-dependent charge response from Q4 at positive
biases for Run 67. The baseline of 59.54 keV gamma ray from Q4 initially rose
linearly for a few minutes, then quickly transitioned to an $\sim$exponential
fall off over 10s of minutes. Figure 5: Shown is the time-dependent charge
response from Q4 at positive biases for Run 74. The baseline of 59.54 keV
gamma ray from Q4 initially rose linearly for a few minutes, then quickly
transitioned to an $\sim$exponential fall off over 10s of minutes.
## 3 Experimental Data Analysis and Results
We used the data measured at various bias voltages to characterize the charge
collection efficiency (CCE), defined as the ratio of the measured charge
energy to 59.54 keV gamma rays, as a function of bias for each channel, and
studied how the CCE varies with time under a given bias voltage. This work
presents the results of an analysis of the 59.54 keV calibration line
behavior, as shown in Figure 4 & Figure 5. The observation in particular had a
rich set of time dependent behaviors, which are the amplitude of the signals
from the 59.54 keV line to initially rise with time after bias, then fall to
some steady-state value. This phenomenon was only present with positive
biases. Since the 241Am sources are positioned at the biased side of the
crystal and the mean free path of 59.54 keV gamma rays in Ge is about 0.09 cm,
this means that this phenomenon is present for events in which electrons are
collected almost immediately and holes are drifted through the full detector
thickness. This indicates that the observed behavior has a polarity dependence
for all events.
In Run 67 the observed phenomenon was much stronger in the center channel (Q4)
and weakest in the outer ring channel (Q1), suggesting a radial dependence. In
Run 74, with only one source present with a weaker event rate, the radial
dependence is not evident, indicating an overall event rate dependence. In
this work, we only study the 59.54 keV line in the center channel (Q4) at
positive biases, since this channel outputs the clearest results.
The observed features of the time-dependent charge collection for the 59.54
keV line can be summarized as: (i) time-dependent impact ionization of
drifting holes kicks off at a positive bias of 4.5 volts (1.36 V/cm) and a
mixing chamber (MC) temperature in a range from (30-35) mK during Run 67 and
no such a phenomenon is observed with a negatively biased detector when
drifting electrons; similarly, we observed the time-dependent impact
ionization phenomenon for holes in Run 74 kicking off at a positive bias
around 7 volts (2.12 V/cm) and a MC temperature in a range from (35-40) mK
(ii) the hole impact ionization initially increases linearly with a rate
dependent on the bias and overall event rate; and (iii) after reaching a peak
the increased charge signal falls exponentially with a fall time dependent on
bias. The loss of both types of signal is easily explained as the loss of CCE
due to the breakdown of bulk field from trapped charges. The linear increase
of the hole impact ionization signal and lack of electron impact ionization
indicate that the population of possible hole impact ionization sites starts
small and is created by drifting electrons which are captured. This suggests a
possible physical mechanism involving the combination of the following three
processes: $e^{-}+A^{0^{*}}\rightarrow A^{-^{*}}$, $h^{+}+A^{-^{*}}\rightarrow
e^{-}+2h^{+}+A^{-}$, and $h^{+}+A^{-}\rightarrow A^{0^{*}}$. The first process
represents the creation of cluster dipole states ($A^{-^{*}}$) through
drifting electrons across the detector induced by background radiation as
shown in Figure 1; the second stands for hole impact ionization of cluster
dipole states; and the third is trapping of $h^{+}$, which determines the loss
of neutralization at the end of series.
There is also a possible process, $E_{ph}+A^{-^{*}}\rightarrow
e^{-}+A^{0{*}}$, which is the impact ionization of cluster dipole states
through absorbing Neganov-Luke phonons luke , where $E_{ph}$ is the energy of
phonons created by drifting holes under a given electric field. This process
is not a significant contribution in the experiment since the Neganov-Luke
phonons have energy smaller than 1 meV mei . However, the creation of cluster
dipole charge states ($e^{-}+A^{0^{*}}\rightarrow A^{-^{*}}$) and the impact
ionization of cluster dipole states ($h^{+}+A^{-^{*}}\rightarrow e^{-}$ \+
2$h^{+}+A^{-}$) inside the detector generate a dynamic electric field, which,
in turn, impacts charge transport and charge creation. As a result, in this
model the observed time-dependent impact ionization of 59.54 keV $\gamma$ rays
involves the growth of charge states as a function of time, the impact
ionization of time-dependent cluster dipole states, and the loss of CCE due to
the breakdown of bulk field from trapped charges, which is again a function of
time.
Thus, the analysis presented here attempts to quantify the observed behavior
in an empirical way. This will help lend insight into likely physical models
that dominate the observed behavior. We assume that the detected charge
energy, $E(t)$, is related to the input 59.54 keV $\gamma$ rays through the
following equation:
$E(t)=E_{\gamma}\\{p_{0}+p_{1}exp[\frac{p_{2}}{p_{3}}(1-exp(-p_{3}t))]\\}exp(-p_{4}t),$
(1)
where $E_{\gamma}$ = 59.54 keV. Other terms in Eq. 1 are explained below:
(1) $p_{0}+p_{1}$ represents the average CCE ($\epsilon_{0}$) at $t$ = 0. If
it is greater than 1, it means that charge energy is gained through impact
ionization. It is expected that the reaction, $h^{+}+A^{-^{*}}\rightarrow
2h^{+}+e^{-}+A^{-}$, dominates the hole impact ionization at $t$=0 in the
range of the applied field sund . The parameter $p_{0}+p_{1}$ is the direct
measurement of CCE at $t$ = 0 in equation 1, as depicted in Figure 4 & Figure
5.
(2) $\\{p_{0}+p_{1}exp[\frac{p_{2}}{p_{3}}(1-exp(-p_{3}t))]\\}exp(-p_{4}t)$ is
the average CCE ($\epsilon_{t})$ for the charge created by the impact
ionization at $t>0$. We can write $\epsilon_{t}$ = $\epsilon_{0}\times
M(t)\times d(t)$, where M(t) stands for the charge gained through time-
dependent impact ionization while $d(t)$ is a time-dependent charge damping
factor that describes the charge trapping due to a dynamic process described
in points (3) and (4) which creates more charge states. Note that
$\epsilon_{0}$, $d(t)$, and $M(t)$ are correlated with applied electric field.
(3) $p_{2}$ represents the rate of creating $A^{-^{*}}$ states. It depends on
the density of the dipole state, the overall event rate, the drift velocity,
and the charge trapping cross section. Since both the drift velocity and the
charge trapping cross section are field dependent, it is expected $p_{2}$ has
a correlation with applied electric field.
(4) $p_{3}$ represents the rate of decreasing $A^{-^{*}}$ states, which is
proportional to the applied field and the overall event rate.
(5) $exp(-p_{4}t)$ represents the loss of neutralization. If we let
$p_{4}=1/\tau$, the factor $exp(-t/\tau)$ describes the collection of charge
created by impact ionization falling exponentially as a function of time
depending on bias. The parameter $\tau$ measures the effective time constant
for the loss of charge signal due to the loss of CCE resulting from the
breakdown of bulk field from trapped charges.
Using this empirical model expressed in Eq. 1, we fit the data from Run 67 and
Run 74. The observed trends and the fits are shown for different applied bias
voltages in Figures 6 and 7. Since the two runs were taken at two different
periods at which the tower temperature of detector housing was slightly
different. Thus, we expect that the detector responds slightly different to
the impact ionization as it depends on the detector conditions including the
thermal effect. For accurate reading in a temperature, the mixing chamber (MC)
temperature was taken. The MC temperature in Run 67 was in a range between
(30-35) mK for the data series shown in Figure 6. Similarly in Run 74, the MC
temperature was in a range between (35-40) mK for the data series shown in
Figure 7, being higher compared to Run 67.
At a given bias the observed phenomenon is stronger in Run 67 than in Run 74.
This can be attributed to differences in the overall event rate in the
detector between the two runs. By introducing an additional external source it
was observed that the impact ionization effect seen in 59.54 keV events was
stronger when the overall event rate in the detector was increased. In Run 74
the overall rate was lower because the 241Am source used produced events at a
lower rate, only one source was used rather than four, and a lead shield was
erected around the cryostat which reduced the background rate from
radioactivity in the lab environment.
Figure 6: Shown is the detected charge energy created by impact ionization as
a function of applied bias voltage fitted by Eq. 1 using data from Run 67. The
error bars represent the vertical width of the band shown in Figure 4. Figure
7: Shown is the detected charge energy created by impact ionization as a
function of applied bias voltage from Run 74 fitted by Eq. 1. The error bars
represent the vertical width of the band shown in Figure 5.
We summarize the fitting parameters in Table 1. The fitting to the impact
ionization curve using Eq. 1 shows that all the parameters have a good
correlation with the electric field. This indicates that the parameters have
their own effect under the electric field. The variation of those parameters
with the electric field is shown in Figure 8. Since each fitting parameter is
a combination of at least two physical processes, it is difficult to figure
out the impact of the applied field on any single process.
Table 1: A summary of the fit parameters using equation 1. Note that the
values of $p_{2}$ and $p_{3}$ are omitted for 5.4 V in Run 74 due to the rate
of creating and decreasing $A^{-*}$ states was very small, leading to abnormal
values for $p_{2}$ and $p_{3}$, which do not follow the trend observed at
higher bias voltages, where the dependency on the applied bias field is
evident. Run 67
---
Bias | $p_{0}$ | $p_{1}$ | $p_{2}$ | $p_{3}$ | $p_{4}$
4.5 V | 1.05 | 0.015 | 0.67 | 0.22 | 0.0080
5.0 V | 1.07 | 0.018 | 0.97 | 0.29 | 0.013
5.5 V | 1.09 | 0.019 | 1.31 | 0.34 | 0.020
6.0 V | 1.13 | 0.020 | 1.41 | 0.35 | 0.026
7.0 V | 1.22 | 0.028 | 1.80 | 0.44 | 0.039
8.0 V | 1.28 | 0.032 | 2.79 | 0.67 | 0.046
9.0 V | 1.39 | 0.038 | 3.98 | 0.94 | 0.056
Run 74
Bias | $p_{0}$ | $p_{1}$ | $p_{2}$ | $p_{3}$ | $p_{4}$
5.4 V | 0.65 | 0.024 | - | - | 0.0058
6.4 V | 0.78 | 0.026 | 0.67 | 0.26 | 0.0098
6.9 V | 0.92 | 0.027 | 0.80 | 0.27 | 0.012
7.2 V | 0.95 | 0.029 | 1.002 | 0.32 | 0.013
7.4 V | 0.96 | 0.031 | 1.13 | 0.35 | 0.015
9.4 V | 1.22 | 0.033 | 2.50 | 0.63 | 0.032
Figure 8: Shown are the fitting parameters, $p_{0}$ to $p_{4}$ from equation 1
as a function of the applied field, $E$, and fits a linear regression model,
which demonstrates the correlation of the parameters with the electric field
using data from each run. The fitting functions for those parameters in Run 67
(solid lines fitted to the circles) are: $p_{0}$=0.25$E$+0.69,
$p_{1}$=0.017$E$-0.0087, $p_{2}$=2.27$E$-2.57, $p_{3}$=0.49$E$-0.49, and
$p_{4}$=0.036$E$-0.039. Similarly, the fitting functions for those parameters
in Run 74 (dotted lines fitted to the squares ) are: $p_{0}=0.47E-0.11$,
$p_{1}=0.0078E+0.011$, $p_{2}=2.1E-3.5$, $p_{3}=0.43E-0.61$, and
$p_{4}=0.022E-0.033$.
## 4 Discussion and Perspective of the Physics Application
As can be seen from Figure 8, although the parameters, $p_{0}$, $p_{1}$,
$p_{2}$, $p_{3}$, and $p_{4}$ obtained from Run 67 and Run 74 have similar
tendencies as a function of applied bias field, the values of these parameters
differ between the two runs from a minimum of a few percent to a maximum of
$\sim$40%, depending on the applied bias field. The cause of this difference
is likely due to the overall event rates used in the two runs. Run 67 has a
higher event rate compared to Run 74.
For bias 4.5 V and above in Run 67, the value of $p_{0}+p_{1}$ is greater than
1.0, indicating impact ionization at $t=0$. In Run 74 the bias required to
produce impact ionization at $t=0$ was greater, 9.4 V. The difference between
the runs is apparently attributable to the higher event rate in Run 67
mentioned previously.
For stable operation at which the average CCE is not a function of time for a
given electric field, the parameters $p_{3}$ and $p_{4}$ from the empirical
model (Eq.1) should be zero. Using the fitted functions obtained from Run 67,
$p_{3}$ = 0 and $p_{4}$ = 0 at electric fields less than $\sim$1 V/cm. At such
a small electric field, the measured CCE, $p_{0}$ \+ $p_{1}$ = 0.9483 and is
constant in time. This means that at this electric field, the detector is a
normal detector, with no visible impact ionization. The charge collection
efficiency is consistent with typical SuperCDMS detectors operated at a
similar electric field phipps2 and suggests that detectors made from USD-
grown crystals are suitable as SuperCDMS-style detectors in terms of charge
collection.
At a higher electric field, for example at the applied bias of 9 V, the CCE is
about 1.4 at $t=0$ from Run 67. This means a higher electric field is needed
to generate impact ionization. If one assumes the charge collection time per
event is less than 2 microseconds in this large-size detector, we can predict
the CCE as a function of applied electric field using the fitted parameters
from Run 67 and the empirical model (Eq. 1). Figure 9 shows that the CCE can
be increased by a factor of $\sim$100 when the detector is operated at a field
of $\sim$ 400 V/cm with a charge collection time of 2 microseconds. This means
that the charge can be amplified by a factor of $\sim$100 at a bias voltage of
$\sim$1300 volts. Note that this extrapolation was obtained by extending the
fitted empirical trend as a function of applied electric field by more than
two orders of magnitude beyond the data. Therefore, the discussion above needs
to be verified using experimental data in the future.
Figure 9: Shown is the predicted CCE as a function of applied electric field.
## 5 Conclusion
We conclude that the observed time-dependent behavior of charge amplification
for the 59.54 keV line is mainly due to the impact ionization of charged
states by drifting holes across the detector. We attempted to set out possible
mechanisms for the impact ionization, which is mainly due to the creation of
cluster dipole states, and impact ionization of cluster dipole states. These
two processes dominate the production of additional charge that contributes to
the amplification of charge while drifting holes across the detector. Our
results suggest that a detector with an appropriate density of charged states,
created by controlled radiation from the appropriate density of neutral
states, can be developed for searching for MeV-scale dark matter with a
detection threshold as low as $<$10 meV, which corresponds to recoil energy
induced by sub-MeV dark matter particles. The challenge is to test such a
detector at a higher bias voltage such as 1300 volts without triggering
avalanche breakdown. The detector avalanche breakdown occurs at about 12 volts
in Run 67 and about 15 volts in Run 74. This could be increased by improving
the fabrication of the electrical contacts. But the overall event rate, from
the 241Am source plus background radiation, is also relevant to breakdown. Run
74 has a somewhat lower event rate compared to Run 67, and the avalanche
breakdown voltage was somewhat higher. We expect the avalanche breakdown
voltage can be much higher if events of only a few electron-hole pairs are
generated by a well-controlled optical source with the detector being well-
shielded from background radiation. More research and development is needed.
## 6 Acknowledgement
This work was supported in part by NSF OISE 1743790, NSF PHYS 1902577, DOE
grant DE-FG02-10ER46709, DE-SC0004768, and a research center supported by the
State of South Dakota.
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|
# Measuring the Measuring Tools: An Automatic Evaluation of Semantic Metrics
for Text Corpora
George Kour , Samuel Ackerman11footnotemark: 1
Orna Raz, Eitan Farchi, Boaz Carmeli, Ateret Anaby-Tavor
IBM Research AI
{gkour<EMAIL_ADDRESS>
{ornar, farchi, boazc<EMAIL_ADDRESS>denotes equal contribution.
###### Abstract
The ability to compare the semantic similarity between text corpora is
important in a variety of natural language processing applications. However,
standard methods for evaluating these metrics have yet to be established. We
propose a set of automatic and interpretable measures for assessing the
characteristics of corpus-level semantic similarity metrics, allowing sensible
comparison of their behavior. We demonstrate the effectiveness of our
evaluation measures in capturing fundamental characteristics by evaluating
them on a collection of classical and state-of-the-art metrics. Our measures
revealed that recently-developed metrics are becoming better in identifying
semantic distributional mismatch while classical metrics are more sensitive to
perturbations in the surface text levels.
## 1 Introduction
While there has been a long-standing interest in developing semantic
similarity metrics111In the context of this paper, a metric is a measure of
difference (gap) in the general sense, and may not necessarily satisfy the
properties of a metric in mathematical terms. Rayson and Garside (2000),
measuring how close two text corpora are remains an open problem Pillutla et
al. (2021). Specifically, the recent advances in generative language models
have led to an increased interest in the study of content similarity between
human and generated language, as a mean for comparing the quality of
generative models Mille et al. (2021); Gehrmann et al. (2022).
While one can reasonably measure the semantic distance between two individual
sentences (e.g., by calculating the cosine distance between the sentence
embeddings), measuring the dissimilarity between two text corpora remains a
challenge Naeem et al. (2020). Corpus-level metrics seek to assess semantic
similarity at the group level, for instance, assessing generated text
fidelity, diversity, and coverage compared to the reference corpus Sajjadi et
al. (2018). Thus, one common approach for measuring the semantic dissimilarity
between two corpora is to compare the densities of their sentences in the
embedding space Pillutla et al. (2021).
However, there are no standard automatic procedures for evaluating the
precision and robustness of such similarity metrics. The semi-manual standard
approach is to correlate the results of these metrics for human judgement.
However, leveraging manual human judgements to construct numeric metrics has
significant weaknesses. As we explain in Section 2, human judgements are
expensive to obtain, are difficult to aggregate consistently from individual
text instances into a corpus-level metric in a way that reflects all relevant
aspects of the texts, and can be subjective and non-robust.
Therefore, in this paper, we adopt a middle ground between validating the
metric against human judgement on real data and evaluating the metric with
synthetic distributions by building "controllable-distance real data corpora"
(Section 3). By precisely controlling the content of test corpora, we devised
a unified evaluation of desired metric characteristics on real data. This
technique allows aggregation of many small-difference judgements that should
correspond to what a human would logically decide, to evaluate the distance
metric overall in terms of desirable properties. The middle ground thus
attempts to reflect human logical judgement in an inexpensive way, while
avoiding some of the weaknesses described, such as lack of consistency.
To summarize, our contributions are as follows. First, we present a text
similarity evaluation measures that allows researchers to compare the
robustness of their newly proposed metrics against existing metrics using the
same standards. Second, we evaluate classical and state-of-the-art similarity
metrics and show that our benchmark performs well in capturing their known
properties. Finally, we provide a pip-installable Python package to compute an
extensive set of text dissimilarity metrics, using a unified and simple
API222https://github.com/IBM/comparing-corpora.
## 2 Literature Review
The most widely-used method to compute the quality of text similarity metrics
investigates the correlation between the scores given by the metric and human
judgements. However, human judgement, even on the sentence level, has several
shortcomings, mainly that it is expensive and can be inconsistent and
subjective Popescu-Belis (2003); Lin and Och (2004); Graham et al. (2017).
Also, superficial aspects of the sentences, such as text length or syllables
per sentence, may influence human judgements of the semantic similarity
Novikova et al. (2017). Furthermore, though humans may be able judge the
relative similarity of a pair of sentences, they are usually limited in their
ability to make large-scale assessments of a similar type when comparing two
corpora (i.e., two distributions of sentences) consistently and reliably.
In an attempt to standardize metric evaluation, several competitions and
standard datasets containing compared data and human assessment were created
for specific tasks, such as translation Guo et al. (2018); Mathur et al.
(2020). However, there is currently a lack of benchmarks against which to
assess the semantic similarity between corpora.
Text similarity metrics can be thought of as belonging to several broad and
overlapping classes (see e.g., Wang and Dong 2020), which partially depend on
the form of the text representation (e.g., token-based or vector embedding).
Here, we investigate metrics from three of these classes, comparing corpora
based on these aspects: _lexicographical_ (statistical properties of words and
tokens), _distribution_ ( densities of sentences represented in the embedding
space), and _discriminatability_ (ability to classify sentences as belonging
to one corpus or the other). The metrics we use are summarized in Table 1.
Type | Metric | Measures
---|---|---
Lexicographical | CHI ($\chi^{2}$) Kilgarriff (2001) | Word/Token count comparison.
Statistics | ZIPF Holtzman et al. (2019) | Unigram rank-frequency statistics.
| FID Heusel et al. (2017) | Wasserstein distance between densities.
Distributional | PR Sajjadi et al. (2018) | Assessing distributional precision & recall.
| DC Naeem et al. (2020) | Estimating manifolds density and coverage.
| MAUVE Pillutla et al. (2021) | Quality & diversity via divergence frontiers.
Discriminative | CLASSIFIER (2016) | Classifiability between reference and target.
| IRPR Zhao et al. (2017) | Average distance between closest samples pairs.
Table 1: Summary of investigated text similarity metrics.
##### Lexicographical Statistics
These methods have been developed to compare various distributional properties
of target text $Q$ with respect to the reference samples $P$, based on some
statistic measures $T(P)$ and $T(Q)$, operating on the surface text level,
e.g., sentence, words, word-parts, tokens, etc. Such commonly-used measures
include resemblance in vocabulary distribution Kilgarriff (2001), likelihood
of repetition Pillutla et al. (2021), and $n$-gram matching Papineni et al.
(2002). However, these metrics tend to be overly sensitive or easily misled by
adversarial samples or text peculiarities. In general, $\chi^{2}$-based
metrics calculate distance between observed and expected frequencies of
categorical variables. The metric in Kilgarriff (2001), denoted here as CHI,
calculates $E$, the average (between $P$ and $Q$) frequencies of the $n$ most
common tokens in the combined vocabulary of $P$ and $Q$, then sums the
$\chi^{2}$ statistics comparing each of $P$ and $Q$ to the expected $E$,
across tokens. Here, for both CHI and ZIPF, below, we use the top $n=5000$
tokens.
In contrast, the ZIPF metric Holtzman et al. (2019) compares the use of
vocabulary using Zipf’s law, which suggests that the frequency of a given word
in human text is inversely-proportional to its frequency rank. The Zipfian
coefficient is fitted on a given corpus and the further it is from $1$, the
more the observed corpus differs from the ‘ideal’ theoretical distribution
Holtzman et al. (2019). We can thus use $|z_{P}-z_{Q}|$ as a distance metric
between corpora $P$ and $Q$.
##### Distributional Metrics
These metrics are based on quantifying the distributional relationship between
the reference and target corpora in the embedded vector space, thereby
capturing semantics beyond superficial token-level statistics. Here $P$ and
$Q$ denote the reference and target corpora in the embedding space. Given
samples from these, we can use the sample density estimates $\hat{P}$ and
$\hat{Q}$ to approximate the true unknown corpus population distributions $P$
and $Q$.
The Fréchet Inception Distance (FID, Heusel et al. 2017) is computed by
fitting a continuous multivariate Gaussian to the $P$ and $Q$, and then
calculating the Wasserstein-2 distance between them. However, FID is sensitive
to both the addition of spurious modes as well as to mode dropping Lucic et
al. (2018). Also, while FID is able to detect distributional distances in the
high-dimensional space, it cannot shed light upon the nature of this distance.
Due to these weaknesses of FID, we additionally consider a metric denoted PR
(precision and recall) proposed in computer vision Sajjadi et al. (2018);
Kynkäänniemi et al. (2019), which is inspired by the notion of precision and
recall. Intuitively, the precision captures the average resemblance of the
individual target samples to the reference set, while the recall measures how
well the target samples "cover" the full variability of the reference samples.
To obtain a single distance value using the method in Kynkäänniemi et al.
(2019), we calculate the $F1$ measure based on the returned precision and
recall, denoted here by PR.
Naeem et al. (2020) proposed an improved estimation of these precision and
recall notions by mitigating the overestimation of manifolds caused by
outliers and underestimating the similarity when the target and reference are
taken from the same distribution. Similarly to PR, we calculate the $F1$ to
obtain a similarity value using this method, denoted as DC333To calculate both
PR and DC, we employed the implementation provided in the _prdc_ Python
package..
MAUVE Pillutla et al. (2021) is a recently-developed metric that estimates the
gap between human and generated text by computing the area under the
information divergence frontiers in a quantized embedding space using the KL-
divergence444We used the _mauve-text_ Python package for calculating MAUVE as
well as ZIPF..
##### Discriminatability Metrics
Similar to the distributional metrics, discriminative metrics calculate the
distance using the embedding of the individual sentences in the two corpora.
However, they do not aim to specifically capture the overlap between the
distribution induced by the compared corpora. Rather, they calculate the
relationship in classification terms, i.e., to what extent can sentences in
one corpus be distinguished from the sentences in the other corpus, using a
discriminative model.
CLASSIFIER: Following (Lopez-Paz and Oquab, 2016), we measure the similarity
between corpora using a binary classifier. We used SVM (Cortes and Vapnik,
1995) trained on samples of both source corpora to predict corpus membership
in a test set of unseen samples. A higher test accuracy indicates higher
inter-corpora distance.
While CLASSIFIER is a model-based metric that uses the entire corpus
distribution, IRPR (information-retrieval precision and recall) is an example
of an instance- (individual sentence) based corpus distance metric. Inspired
by Zhao et al. (2017), we calculate the dissimilarity between corpora as
follows. For each embedded sentence in corpus $A$, we find its closest
neighbor in $B$ by cosine distance. The average of these distances is then
computed to find the "precision" value. The same procedure in reverse, from
$B$ to $A$, gives the "recall" value. We calculate the $F1$ score of the
recall and precision to obtain a single value. Note that the CLASSIFIER metric
is used to represent model-based discriminative approaches, while IRPR is used
to represent instance-based discriminative methods.
The values calculated by CHI, IRPR, PR, DC and Mauve capture the similarity
rather than the distance between two corpora (for all metrics $v\in[0,1]$). To
make these metrics represent distances, we take $1-v$.
Our model selection was based on considering the trade-off between embedding
quality and calculation time. The code as well as the scripts to reproduce the
experiments are available online.555https://github.com/IBM/meme
## 3 Known Similarity Corpora
Most of the metric quality measures we propose are primarily based on the
notion of _known-similarity corpora_ (KSC) introduced by Kilgarriff (2001).
The KSC set is created by mixing samples from two different source corpora $A$
and $B$ in gradually-changing proportions. The KSC set, denoted $KSC(A,B)$,
consists of $k$ corpora $\\{c_{1},c_{2},\dots,c_{k}\\}$, each of size $n\geq
k-1$, where corpus $c_{i},\>i=1,\dots,k$ is constructed by sampling
$n\left(\frac{k-i}{k-1}\right)$ observations from $A$, and the remaining
$n\left(\frac{i-1}{k-1}\right)$ from $B$ (see Figure 1). The sampling
resolution gradation between corpora is a fixed $\frac{1}{k-1}$.
Figure 1: Construction of a $k=6$ known similarity corpora (KSC) collection
from source corpora $A$ and $B$. The corpus $c_{i}$ is constructed by drawing
$n\left(\frac{k-i}{k-1}\right)$ and $n\left(\frac{i-1}{k-1}\right)$ samples
from $A$ and $B$, respectively. The adjacent densities denote the
distributions of the source and KSC set corpora.
We now introduce some notation on the $KSC$ set, which are used to define the
measures in Section 4. Let $[k]=\\{1,2,\dots,k\\}$. For given source corpora
$A$ and $B$, for each $\ell\in[k-1]$ we define the $\ell$-distant corpora set
as follows:
$KSC_{\ell}(A,B)=\left\\{(c_{i},c_{j})\colon\>i,j\in[k],\>j-i=\ell\right\\}$
(1)
Let $d(a,b)$ denote the distance from corpus $a$ to $b$, according to metric
$d$. Let $D_{\ell}(A,B,d)$— $D_{\ell}$ for short—be the set of values of
distance $d$ for corpora pairs in $KSC_{\ell}(A,B)$;
$\vspace{-0.5em}D_{\ell}(A,B,d)=\\{d(c_{i},c_{j})\colon\>(c_{i},c_{j})\in
KSC_{\ell}(A,B)\\}$ (2)
To pool results across $\ell$, we further define:
$D(A,B,d)=\\{D_{\ell}(A,B,d)\colon\>\ell\in[k-1]\\}$ (3)
Note that because we do not require the distance metrics considered to be
‘metrics’ in the mathematical sense, they may not be symmetric (i.e., possibly
$d(a,b)\neq d(b,a)$). However, since $KSC_{\ell}$ enforces a pairwise order on
pairs $(c_{i},c_{j})$ by requiring $j>i$, this ensures that $D_{\ell}(A,B,d)$
is properly defined.
Some of the metrics $d$ have a pre-defined range (e.g., CHI, MAUVE, DC, PR
only return values in the range [0,1]) while others have no preset scale or
operation range. Therefore, to allow sensible comparison of distance metrics
with different operation ranges and across source corpora, we obtain
$z$-scores by normalizing the metric values, pooled across all
$D_{\ell}(A,B,d)$. In the following analysis, if not specified otherwise,
$D_{\ell}$ will always be the normalized rather than raw distances.
##### Datasets Selection
The measures described in Section 4 are applicable to any pair of textual
datasets with differently-distributed textual content, allowing the corpora in
the KSC set to be distinguishable. To ensure that each pair of source corpora
were in fact different enough, in the following experiments we use pairs of
human text corpora from different domains, rather than pairing a human corpus
with a machine-generated version of itself. For our experiments we selected
four public datasets (ATIS, Hemphill et al. 1990;
yahoo666https://ciir.cs.umass.edu/downloads/nfL6/; banking77, Casanueva et al.
2020; clinc150, Larson et al. 2019) containing short user utterances from
different domains summarized in Table 2.
Name | Size | Description
---|---|---
atis | 4978 | Utterances to a flight
| | booking system.
yahoo | 20000 | Yahoo non-factoid
| | questions in 21 categories.
clinc150 | 22500 | Utterances in 10 domains
| | classified into 150 classes.
banking77 | 10000 | Online banking queries.
Table 2: Datasets used as source corpora in our benchmark. Although some of
the datasets are partitioned annotated with labels, in our experiments, if not
mentioned otherwise, we ignored those labels.
## 4 Metric Robustness Measures
We now describe our measurements of desirable properties for distance metrics,
given the normalized $D_{\ell}$ on the KSC sets. In the three following
measures (Monotonicity, Separability, and Linearity), we aim to capture three
attributes of well-behaved metrics that can be understood by considering the
top line scatter plots of Figure 2; these show the relation between the
$D_{\ell}$ sets and $\ell$. In these scatterplots, a high angle of the
regression line, low vertical variability around it, and linearity are all
desirable properties for the distance metric, and are captured in these
measures.
Figure 2: Top: Distance values (non-normalized) of corpora pairs in
$D_{\ell}$ versus $\ell$. ($n=100,k=12,|J|=6053$), pooled across 5 repetitions
of KSC samples. Blue line indicates regression and confidence interval at 95%.
Middle: Distance values calculated on increasing $s$ size corpora $a_{s}$ and
$b_{s}$ sampled from sources $A$ and $B$, correspondingly. Bottom: Distance
between imbalanced corpora $a_{s}$ and $b_{\bar{s}}$ where $|b_{\bar{s}}|=N-s$
and $N=2900$. The x-axis represents $s\in\\{50,250,450,\dots,2850\\}$
($repetitions=10$). In middle and bottom figures, green horizontal line
indicates the asymptotic distance $d(A,B)$. In all figures $A$=clinc150 and
$B$=banking77.
### 4.1 Metric Monotonicity
A well-behaved distance metric $d$ should have a natural monotonic
relationship with the separation levels $\ell$ of the KSC. We use Spearman’s
rank correlation between $\ell$ and $D_{\ell}$, which we denote $\rho(d)$, to
assess the monotonicity. Spearman’s correlation is defined as the Pearson
correlation between the order ranks of two variables, and measures the
strength of their monotonic, rather than linear, association. As can be seen
in Table 3, MAUVE and CHI achieve the best monotonicity results, followed by
DC and FID.
### 4.2 Metric Separability
It is desirable that (1) for a given $\ell$, $D_{\ell}$ has low variability,
and (2) for different $\ell_{2}>\ell_{1}$, the samples $D_{\ell_{1}}$ and
$D_{\ell_{2}}$ are distinguishable (e.g., by a two-sample difference test),
particularly as $\ell_{2}-\ell_{1}$ grows. Here, we measure how grouping by
$\ell$ explains the variability in $D_{\ell}$ across $\ell$. We perform a one-
way fixed-effects analysis of variance (ANOVA) with $\ell$ as the unordered
categorical treatment and $D_{\ell}$ as the numeric response. Often, an F-test
is performed; if its p-value is low, it means a significant amount of the
variance in the response ($D_{\ell}$) can be explained by the treatment
($\ell$). Since the F-test for any reasonable $d$ metric should be
significant, we instead use the similar $\omega^{2}$ effect-size metric (Hays,
1963), which is bounded by $\pm 1$, to better assess them. It is defined as
$\omega^{2}=\frac{SS_{\textrm{treatment}}-df_{\textrm{treatment}}\times
MS_{\textrm{residual}}}{SS_{\textrm{total}}+MS_{\textrm{residual}}}$ (4)
where $SS$ and $MS$ are the sum and mean sums of squares, and $df$ is the
degrees of freedom, on a dataset of size $n$ (here, $n=|D(A,B,d)|$). In the
following we denote this measure as $\mathcal{W}(d)$.
### 4.3 Metric Linearity
Here we examine to what extent linear changes in the corpus content ($\ell$)
are manifested in linear changes in the distance function. To do so, we
calculate the coefficient of determination ($R^{2}$), where higher values
indicate stronger linearity. This measure is denoted by $\mathcal{L}(d)$.
Looking at the results in Table 3, we see that MAUVE achieves the best results
followed by DC and FID. It appears that this measure is more affected by the
source corpora and by the resolution than other metrics.
### 4.4 Metric Time Efficiency
The time complexity of the metric is commonly perceived as less important,
thus seldom reported Sai et al. (2022). This aspect is becoming ever more
important, especially due to the growing interest in time-consuming divergence
frontier methods Djolonga et al. (2020). Such metrics perform multiple
measurements to estimate the area under the curve (similar to precision-recall
curves for binary classification), with tune-able but increasing resolution.
We measure the time performance of the metric $\mathcal{T}(d)$ in terms of
$100$ similarity measurements operations per second ($[100\emph{op}/sec]$) on
a standard CPU machine777CPU: 2.3 GHz 8-Core Intel Core i9. Memory: 64 GB DDR4
(2667 MHz). Note that the measurements reported in Table 3 do not include the
sentences’ embedding time. Predictably, methods that operate on the token
level and avoid complex density estimation tend to achieve the best time
performance. Among the distributional metrics, MAUVE achieves the best
results, followed by FID. PR and DC produce similar results since both are
based on similar manifold calculations.
| $\mathcal{A}(d)$ | $\mathcal{A}^{w}(d)$ | $\mathcal{T}(d)$ | $\rho(d)$ | $\mathcal{W}(d)$ | $\mathcal{L}(d)$ | $\mathcal{S}(d)$ | $\mathcal{I}(d)$
---|---|---|---|---|---|---|---|---
$k$ | $7$ | $12$ | $7$ | $12$ | $7$ | $12$ | $7$ | $12$ | $7$ | $12$ | $7$ | $12$ | |
CHI | .945 | .852 | .913 | .774 | 4.68 | 3.29 | .875 | .866 | .684 | .702 | .810 | .767 | .989 | .994
CLS. | .789 | .701 | .731 | .618 | 1.26 | 1.10 | .704 | .735 | .544 | .562 | .767 | .767 | .972 | .918
DC | .958 | .863 | .936 | .805 | .031 | .031 | .913 | .892 | .908 | .879 | .946 | .919 | .832 | .808
FID | .949 | .810 | .923 | .753 | .067 | .066 | .764 | .695 | .563 | .537 | .81 | .759 | .821 | .877
IRPR | .832 | .710 | .784 | .638 | 4.39 | 2.35 | .571 | .543 | .258 | .275 | .645 | .598 | .949 | .856
MUV. | .976 | .888 | .963 | .828 | .079 | .071 | .938 | .906 | .883 | .885 | .947 | .926 | .977 | .943
PR | .820 | .688 | .767 | .608 | .031 | .031 | .649 | .592 | .577 | .566 | .716 | .667 | .909 | .934
ZIPF | .886 | .726 | .851 | .657 | 4.65 | 2.668 | .751 | .633 | .514 | .413 | .785 | .667 | .852 | .913
CHI | .953 | .935 | .936 | .891 | 5.58 | 3.33 | .960 | .962 | .835 | .900 | .83 | .858 | 1.00 | 1.00
CLS. | .931 | .827 | .902 | .751 | 1.29 | 1.21 | .843 | .836 | .671 | .702 | .847 | .847 | .993 | .989
DC | .773 | .601 | .702 | .552 | .031 | .031 | .707 | .615 | .763 | .717 | .825 | .759 | .988 | .986
FID | .967 | .904 | .947 | .848 | .071 | .067 | .793 | .754 | .634 | .636 | .845 | .816 | .922 | .898
IRPR | .697 | .587 | .642 | .570 | 3.91 | 2.69 | .382 | .264 | -.02 | -.001 | .433 | .341 | .951 | .834
MUV. | .999 | .977 | .998 | .961 | .084 | .067 | .936 | .943 | .856 | .904 | .932 | .950 | .999 | .994
PR | .722 | .467 | .666 | .446 | .031 | .030 | .459 | .240 | .488 | .394 | .658 | .523 | .890 | .899
ZIPF | .854 | .783 | .817 | .736 | 4.77 | 3.02 | .660 | .635 | .309 | .352 | .687 | .661 | .735 | .904
Table 3: Summary of metrics evaluation scoring on two pairs of source
datasets in low ($k=7$) and high ($k=12$) resolution KSC ($n=100$). Best
results with differences below $.015$ are marked in bold. $\mathcal{T}(d)$
units are $[100\emph{op}/sec]$. MUV. stands for MAUVE and CLS. for CLASSIFIER.
In the top table, $A$=clinc150 and $B$=banking77. In the bottom table $A$=atis
and $B$=yahoo. The average results of $5$ repetitions are reported for all
measures except size and imbalance robustness, in which the number of
repetitions is $10$. More statistical details are provided in Figure 6 in the
Appendix.
### 4.5 Metric Accuracy
The assessment measures described earlier in Section 4 use the observed values
of the metric distances (or similarities) between the KSC corpora; however,
the actual values of the distance may not be known. Nevertheless, we still
have some partial information about the ordering of these values, which we
will use to define an accuracy measure, which requires us to define the notion
of a ‘judgement’, as follows.
#### 4.5.1 Comparing paired corpora distances
Suppose we do not know the observed values of $d$ in $D(A,B,d)$ for the paired
corpora in $KSC_{\ell}(A,B)$, pooled across $\ell$. Nevertheless, we can still
assume that certain pairwise distances are larger than others. For instance,
the proportions of observations from $A$ in $c_{2}$ and $c_{3}$ are more
similar than the respective proportions between $c_{1}$ and $c_{4}$. Moreover,
the interval of the first pair is ‘contained’888$(c_{i},c_{j})$ contains
$(c_{q},c_{r})$, i.e., $(q,r)\subset(i,j)$, if $i\leq q$ and $r\leq j$ and
$i<r$. in the second, and thus the first pair should have smaller distance.
Thus, it should be true that, say, $d(c_{2},c_{3})\leq(c_{1},c_{4})$ in
expectation (across repeated random sampling). In general, whenever the
interval of one corpus pair contains ($\subset$) the interval of another, we
expect the contained pair to have a smaller distance.
Given two pairs, $(c_{i},c_{j})$ and $(c_{q},c_{r})$, of paired corpora, we
can only reliably predict999For instance, say we compare pairwise distances
between $(c_{1},c_{6})$ and $(c_{5},c_{7})$. Even though the second interval
length ($7-5=2$) is smaller than the first ($6-1=5$), because it is not
contained in the first, we cannot necessarily say that $d(c_{5},c_{7})\leq
d(c_{1},c_{6})$ since inter-corpus distance may not be proportional to the
interval length. which of $d(c_{i},c_{j})$ or $d(c_{q},c_{r})$ is larger in
expectation (a decision we call a ‘judgement’) if the interval of one pair
contains the other’s. The set $J$ contains all and only such judgements:
$\begin{split}J=\\{((c_{q},c_{r}),(c_{i},c_{j}))\colon(q,r)\subset(i,j)\\}\end{split}$
(5)
The judgement $d(c_{q},c_{r})\leq d(c_{i},c_{j})$ is correct when the second
interval contains the first. This gives the most probabilistically-logical
partial order on the similarities between corpora in a KSC collection, that
can be obtained without knowledge of the true pairwise $d$-distances between
corpora. Figure 3 shows a tree representation of KSC-set pair containment
relations, from which the set of judgements $J$ can be extracted.
Figure 3: A tree representation of the judgements performed on the KSC
collection given a metric $d(\cdot,\cdot)$, for calculating the accuracy
($\mathcal{A}$, Section 4.5) measures. The leafs are the KSC collection and
the inner nodes (circles) represent the corpora tuples ($c_{i},c_{j})$. The
set $J$ contains all judgements such that each node $(i,j)$ is judged against
all descended nodes. Namely, if there is a path from node $a$ to node $b$,
there is a judgement between the two nodes, and the judgement is correct if
$d(b)\leq d(a)$. The size of the judgements set can be expressed as:
$|J|=\sum_{i=1}^{k}(k-i){\left(\frac{i(i-1)}{2}-1\right)}$. For instance,
$|J|=339$ if $k=7$, and 6053 if $k=12$.
#### 4.5.2 Accuracy
The metric accuracy is defined as the rate of correct judgements, formally:
$\mathcal{A}(d)=\frac{1}{|J|}\sum_{\jmath\in J}\mathds{1}(d(c_{q},c_{r})\leq
d(c_{i},c_{j}))$ (6)
where $\jmath=((c_{q},c_{r}),(c_{i},c_{j}))$ is a judgement in $J$ and
$\mathds{1}(\cdot)$ is the indicator function. Further, we propose a weighted
version of the accuracy metric that assigns more weight to harder judgements.
We define the hardness of judgement $\jmath$ as
$w(\jmath)=\frac{1}{\ell_{2}-\ell_{1}}$ where $\ell_{2}=j-i$ and
$\ell_{1}=r-q$, and $\ell_{2}>\ell_{1}$ by definition of $J$. Formally,
$\mathcal{A}^{w}(d)=C\sum_{\jmath\in
J}w(\jmath)\cdot\mathds{1}(d(c_{q},c_{r})\leq d(c_{i},c_{j})))$ (7)
where $C=\left(|J|\cdot\sum_{\jmath\in J}w(\jmath)\right)^{-1}$. While
$\mathcal{A}$ and $\mathcal{A}^{w}$ are correlated, as one may expect,
$\mathcal{A}^{w}$ typically returns lower values (see Table 3).
In our implementation, the set of samples in each $c_{i}$ is disjoint, namely,
$c_{i}\cap c_{j}=\emptyset,\forall c_{i},c_{j}\in KSC(A,B)$. This was done to
prevent perfect judgements by naively counting the number of common instances
(e.g., by defining $d(c_{i},c_{j})=|c_{i}\Delta c_{j}|$ where $\Delta$ denotes
the symmetric difference). MAUVE, followed closely by FID, CHI and DC,
achieves the highest accuracy results across resolutions and source corpora.
### 4.6 Size Robustness
We are also interested in capturing the sensitivity of a metric to sample
sizes. To accomplish this, we need to quantify the convergence pace of
$d(a_{s},b_{s})$ to the asymptotic distance $d(A,B)$, where $a_{s},b_{s}$ are
samples from corpora $A,B$ of increasing size $s$. Specifically, in our
experiments $s\in S=\\{50,250,450,\dots,2850\\}$. The middle plot in Figure 2
shows convergence patterns of the different metrics to the asymptotic
distance. The asymptotic distance is estimated by the mean of repeated ($10$)
calculations of the distance on samples of size $3000$ each from $A,B$, rather
than on the full corpora. To quantify the metric size robustness,
$\mathcal{(}S)$, we calculate the mean absolute error,
$|d(a_{s},b_{s})-d(A,B)|$, for all $s\in S$, normalized by the asymptotic
distance:
$\mathcal{S}(d)=1-\sum_{s\in S}\frac{|d(a_{s},b_{s})-d(A,B)|}{d(A,B)}$ (8)
Similar to previous measures, the normalization is performed to omit the
influence of metric scale and operation ranges.
While our results demonstrate (Figure 2) that most of the metrics examined
require around $1000$ samples to closely estimate the asymptotic distance
between the source corpora, their measured accuracy ($\mathcal{A}$(d) and
$\mathcal{A}^{w}(d)$) is still fairly high even on small corpora within the
$KSC$, and can capture relative differences in corpus content.
### 4.7 Imbalance Robustness
Similarity metrics are frequently used to compare datasets with unequal sample
size. Especially when comparing real and generated corpora, the size of a
generated corpus is usually much larger than the real corpus. The imbalance
robustness measure quantifies the effect of corpora size imbalance on the
metric’s performance (see Figure 2, bottom).
Unsurprisingly, asymmetric metrics such as PR and DC are most affected by size
imbalance. While PR, DC, and MAUVE were all originally designed to measure the
disparity between human and generated data (and thus asymmetric in the
reference$P$ and target $Q$), it seems that MAUVE overcomes the sensitivity to
datasets of very unequal sizes. Interestingly, imbalance causs some metrics
(CLASSIFIER and MAUVE) to underestimate the distance, while others (FID)
overestimate it. When we compare the convergence patterns of PR and DC, both
are similarly asymmetric, maintaining $d(P,Q)$. When we increase the reference
size, PR diverges from the true asymptotic distance, while DC converges to it.
The Imbalance Robustness score $\mathcal{I}(d)$ is calculated similarly to the
size robustness score, only that $|b_{s}|=N-|a_{s}|$.
Figure 4: Leading metrics characterization radar chart. Mean results from
Table 3 for $A$=clinc150, $B$=booking77 and for $k=12$, excluding time
efficiency to maintain scale.
Figure 5: Similarity between reference corpus and iteratively fine-tuned
corpora $g_{i}$ samples. Orange dots show the similarity between samples of
generated text in iteration $i$ and the source dataset. The blue line
indicates regression and confidence interval at 95%. The green horizontal line
specifies the mean estimation of the distance between two random samples of
the original corpus. The top figure shows iterative generation on unlabeled
news headlines dataset. The bottom shows the iterative conditional generation
using LAMBADA Anaby-Tavor et al. (2020) trained on banking77 dataset.
##### KSC Parameters
As shown in Section 4.6, most metrics require at least $n=1000$ samples to
capture the true distance between two source domain corpora; however, our
experiments use $n=100$. This is because our measures are relative, i.e., we
do not aim to calculate the true asymptotic distance between two domains, but
to measure the metrics’ robustness in detecting small changes in the compared
corpora. Furthermore, if $n$ is large, $k$ must also be large to ensure the
$k$ corpora in the KSC set have small enough absolute consecutive differences.
Note that small consecutive differences in KSC corpora are needed so that the
measures in Section 4 will have a high enough resolution and large enough
sample size of $D_{\ell}$ to properly differentiate the metric properties. In
particular, this ensures the judgements (Section 4.5.1) used in the accuracy
measures ($\mathcal{A}$ and $\mathcal{A}^{w}$) are not too ‘easy’ to make
correctly, in which case they would be less useful as a tool. For instance, a
metric with 100% accuracy makes all correct judgements, e.g., that
$d(c_{2},c_{3})\leq d(c_{1},c_{4})$. If $k=5$, the gap (in expectation)
between the pair distances compared is large, so the judgement is easy, and
thus all metrics may have full accuracy. When $k$ increases, the absolute
consecutive differences in corpora fall, and thus the difficulty of the
judgement increases. Some metrics will fail to make the judgement correctly
(in a given random KSC), decreasing their accuracy; this allows us to better
differentiate between the more and less accurate metrics. However, setting $k$
too high results in a computationally prohibitive number $|J|$ of judgements.
Therefore, we opted to use the smaller $n$ that are still sufficient to
capture the quality and robustness of the investigated metrics.
## 5 Increasingly Fine-tuned Corpora
Here, we qualitatively investigate the metrics’ ability to discriminate
between generated and human text using the following procedure: We generated a
sequence of equal-size synthetic corpora $IFC=(g_{1},g_{2},\dots,g_{n})$ by
sampling from a gradually fine-tuned language model on a specific source
corpus $A$. Namely, in each iteration, a fine-tuning step is performed by
training the language model on a single epoch containing $1000$ sentences
randomly drawn from $A$, followed by a generation process to synthesize a
corpus $g_{i}$ containing around $1000$ sentences. The name IFC, or
"Increasingly Fine-tuned Corpora", was chosen to parallel the name KSC ("Known
Similarity Corpora").
For each generated corpus $g_{i}$, we estimated the distance from $A$, i..e,
$d_{i}=d(A,g_{i}),\>\forall i\in[n]$. While the true distance between those
synthetic corpora and $A$ are unknown, an effective metric should capture the
decreasing distance between $A$ and $g_{i}$ with increasing $i$, namely
$d_{1}\prec d_{2}\prec\dots\prec d_{n}$. Due to our results, which show low
imbalance robustness of some metrics, we maintained the same-size corpora when
calculating corpora distance.
The results presented in Figure 5 show the gap between human and generated
text captured by each metric in each iteration. To calculate the average self-
distance of the reference corpus ($A$), we take the mean distance between two
randomly sampled sub-corpora $r_{1}$ and $r_{2}$ from $A$, i.e.
$d(A,A)=\mathds{E}_{r_{1},r_{2}\sim A}[d(r_{1},r_{2})])$.
In our experiments we used two datasets, the banking77 dataset, mentioned
above and the news dataset101010HuffPost (www.huffpost.com) news headlines
collected from 2012 to 2018 containing around 200k
headlines.(www.kaggle.com/rmisra/news-category-dataset
(https://www.huffpost.com), representing different domains of text corpora.
The IFC set for the banking77 dataset was generated in an iterative two-step
procedure similar to the one described in LAMBADA Anaby-Tavor et al. (2020).
This procedure first generates sentences conditioned on the label, then
filters out sentences that are out-of-domain or incorrectly labeled. However,
the IFC set for the news dataset was generated by finetuning the pre-trained
GPT-2 medium model Radford et al. (2019).
The results in Figure 5 show that CHI is less effective than the other metrics
in capturing the gradual nature of the $IFC$. Also, they show that FID and
IRPR are sensitive in discriminating between the original and generated
corpora, even after many fine-tuning iterations. Interestingly, the ZIPF
distance increases with the iteration. This indicates that the generated text,
despite becoming semantically closer to the original with the increasing
iterations, becomes less ‘natural’ in that the token frequencies deviate from
that of human text and the reference corpus. This can be explained, at least
in part, by the TTR measure. TTR is a standard word diversity measure,
calculated by dividing the number of unique words in a text by the total word
count. A high TTR indicates significant lexical variation. Indeed, in the IFC
of banking77, $g_{1}$’s TTR is 0.295 which is closer to the original dataset’s
TTR of 0.299 than $g_{40}$’s TTR of 0.322.
## 6 Conclusions
In this work, we propose a principled set of automatic measures for evaluating
the quality of text dissimilarity metrics. By testing various metrics using
our measures, we show that they do a good job of capturing their known
characteristics, hence increasing our confidence in these measures; also,
overall, recent metrics exhibit more favorable traits than their predecessors.
The radar chart in Figure 4 shows that our measure scores correlate well with
the compared distributional metrics recency $MAUVE\succ DC\succ PR\sim FID$ as
well as their known relative strengths.
## 7 Limitations
Although one of the main motivations for comparing corpora is to measure the
semantic gap between human and generated short text, we used pairs of human
text corpora from different domains to maintain controllably-distinct corpora
in the KSC set. Despite this, future efforts to develop human and machine-
generated benchmark pairs Mille et al. (2021) will allow for future work to
quantitatively measure the characteristics of semantic metrics on pairs of
human and generated corpora using the approach devised in this paper.
Also, for more straightforward comparisons, we used only a single sentence-
embedding model. However, as other studies (e.g., GPT-2 Radford et al. (2019)
in Pillutla et al. (2021) and Bert Devlin et al. (2018) in Lo (2019)) have
shown, the quality of a corpus distance metric can be affected by the
embedding choice. In future extensions of our work, we plan to allow for
multiple embeddings to obtain a more refined evaluation of the metrics.
An important limitation of this work is that it considers only English corpora
of short text samples. We examined only a limited set of metrics and datasets,
both of which we intend to extend.
In addition, we note that while our experiments calculate all KSC-based
measures using a single KSC collection (same $n$ and $k$ values), it could be
favourable to use different $n$ and $k$ for different measures. For instance,
the time performance is calculated using a single size small dataset $n=100$.
In future work, the time scalability of metrics can be more closely
investigated by comparing their time performance on increasing corpora sizes.
As indicated in Section 4, creating KSC collections with large $k$ creates an
excessive number of judgements (e.g., for $k>15$, $|J|>50000$), thus limiting
the scalability of our method to smaller $k$ and thus smaller $n$, if high
resolution is required. This would preclude comparing the robustness of
metrics that require large samples. We intend to rectify this in future work
by creating representative smaller judgement sets by carefully sampling from
the complete set.
As mentioned in Section 2, some of the investigated metrics were adapted to
return a single value summarizing the distance between two corpora (e.g.,
averaging the precision and recall by the $F1$ score). Further work is
required to build measures that can compare metrics returning multiple values.
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## Appendix A Appendix
Figure 6: Distributional information of the results shown in Table 3. The top
two figures showing the results for ($A$=clinc150 $B$=banking77), for $k=7$
and $k=12$, respectively. The bottom two figures are for ($A$=yahoo $B$=atis),
for $k=7$ and $k=12$, respectively. Colored boxes depict the interquartile
($25^{th}$ to $75^{th}$) range. The mean is indicated by a horizontal line.
All data points within 1.5 of the corresponding limits of the interquartile
range are depicted by whiskers. Data points outside this range are plotted
individually. CLS. indicates the CLASSIFIER metric.
|
# EIGER III. JWST/NIRCam observations of the ultra-luminous high-redshift
quasar J0100+2802
Anna-Christina Eilers MIT Kavli Institute for Astrophysics and Space Research,
77 Massachusetts Avenue, Cambridge, 02139, Massachusetts, USA Robert A.
Simcoe MIT Kavli Institute for Astrophysics and Space Research, 77
Massachusetts Avenue, Cambridge, 02139, Massachusetts, USA Minghao Yue MIT
Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Avenue,
Cambridge, 02139, Massachusetts, USA Ruari Mackenzie Department of Physics,
ETH Zürich, Wolfgang-Pauli-Strasse 27, Zürich, 8093, Switzerland Jorryt
Matthee Department of Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, Zürich,
8093, Switzerland Dominika Ďurovčíková MIT Kavli Institute for Astrophysics
and Space Research, 77 Massachusetts Avenue, Cambridge, 02139, Massachusetts,
USA Daichi Kashino Institute for Advanced Research, Nagoya University, Nagoya
464-8601, Japan Department of Physics, Graduate School of Science, Nagoya
University, Nagoya 464-8602, Japan Rongmon Bordoloi Department of Physics,
North Carolina State University, Raleigh, 27695, North Carolina, USA Simon J.
Lilly Department of Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, Zürich,
8093, Switzerland Anna-Christina Eilers<EMAIL_ADDRESS>
###### Abstract
We present the first rest-frame optical spectrum of a high-redshift quasar
observed with JWST/NIRCam in Wide Field Slitless (WFSS) mode. The observed
quasar, J0100+2802, is the most luminous quasar known at $z>6$. We measure the
mass of the central supermassive black hole (SMBH) by means of the rest-frame
optical H$\beta\,$ emission line, and find consistent mass measurements of the
quasar’s SMBH of $M_{\bullet}\approx 10^{10}\,M_{\sun}$ when compared to the
estimates based on the properties of rest-frame UV emission lines C IV and Mg
II, which are accessible from ground-based observatories. To this end, we also
present a newly reduced rest-frame UV spectrum of the quasar observed with
X-Shooter/VLT and FIRE/Magellan for a total of $16.8$ hours. We readdress the
question whether this ultra-luminous quasar could be effected by strong
gravitational lensing making use of the diffraction limited NIRCam images in
three different wide band filters (F115W, F200W, F356W), which improves the
achieved spatial resolution compared to previous images taken with the Hubble
Space Telescope by a factor of two. We do not find any evidence for a
foreground deflecting galaxy, nor for multiple images of the quasar, and
determine the probability for magnification due to strong gravitational
lensing with image separations below the diffraction limit of
$\Delta\theta\lesssim 0.05\arcsec$ to be $\lesssim 2.2\times 10^{-3}$. Our
observations therefore confirm that this quasar hosts a ten billion solar mass
black hole less than $1$ Gyr after the Big Bang, which is challenging to
explain with current black hole formation models.
dark ages, early universe — quasars: emission lines, supermassive black holes
— methods: data analysis — gravitational lensing: strong gravitational lensing
††software: numpy (Harris et al., 2020), scipy (Jones et al., 2001),
matplotlib (Hunter, 2007), astropy (Astropy Collaboration et al., 2018),
PypeIt (Prochaska et al., 2020), photutils (Bradley et al., 2020), psfMC
(Mechtley, 2014)††thanks: Pappalardo Fellow
## 1 Introduction
Since the discovery of the first luminous quasars nearly six decades ago
(Schmidt, 1965) astronomers have detected and observed more than a million
quasars in the universe up to redshifts of $z\gtrsim 7.5$ (e.g. Bañados et
al., 2016; Lyke et al., 2020; Wang et al., 2021). Quasars are the most
luminous, non-transient objects in the universe, powered by accretion onto a
central supermassive black hole (SMBH) of millions to several billions of
solar masses in size (e.g. Mazzucchelli et al., 2017; Yang et al., 2021; Wu &
Shen, 2022). The most luminous, high-redshift quasar ($z>6$) known to date was
discovered by Wu et al. (2015), i.e. J0100+2802 at a redshift of $z=6.3270\pm
0.0005$ (Wang et al., 2019). Its extreme luminosity of $L_{\rm bol}\sim
10^{48}\rm\,erg\,s^{-1}$ implies that the quasar hosts a highly accreting SMBH
in its center with a mass of $M_{\bullet}\sim 10^{10}\,M_{\sun}$ (Wu et al.,
2015).
Thus, J0100+2802 harbors the most massive SMBH known at $z\gtrsim 6$ – which
is among the $0.2\%$ of the most massive SMBHs at all redshifts (Wu & Shen,
2022) – at a time when the universe is only $\sim 800$ Myr old. Its existence
poses significant challenges to current models aiming to explain the rapid
growth of SMBHs in the early universe (e.g. Bañados et al., 2018). Adding to
the challenge, the quasar’s rest-frame ultraviolet (UV) spectrum exhibits a
very small proximity zone in the vicinity of the quasar, which indicates a
short UV luminous quasar lifetime of $t_{\rm Q}\sim 10^{5}$ yr, during which
accretion onto the black hole is expected to occur (e.g. Eilers et al., 2017;
Davies et al., 2020; Morey et al., 2021).
To explain the anomaly of J0100+2802’s SMBH it has been suspected that this
ultra-luminous quasar could possibly be strongly gravitationally lensed.
Arguments in favor of this hypothesis were largely based on the clumpy
morphology of observations with the Atacama Large Millimetre Array (ALMA;
Fujimoto et al., 2020). However, high-resolution images taken with the Hubble
Space Telescope (HST) did not reveal any evidence for multiple images due to
strong gravitational lensing (Fujimoto et al., 2020, Yue et al. in prep.).
Furthermore, detailed analyses of the flux transmission in the quasar’s
proximity zone (Davies et al., 2020), the expected ratios of X-ray luminosity
to rest-frame UV and IR luminosities (Connor et al., 2021), as well as the
implications for the bright-end slope of the quasar luminosity function
(Pacucci & Loeb, 2020) indicate that a significant magnification of the
quasar’s luminosity is unlikely.
The quasar’s extremely massive SMBH has also called into question the accuracy
of all black hole mass estimates of high-redshift quasars, which are obtained
using scaling relations between the width of broad emission lines observed in
single-epoch quasar spectra, the quasars’ luminosities and their black hole
masses (e.g. Vestergaard & Osmer, 2009). These scaling relations are
calibrated based on a small sample of quasars at very low redshifts, i.e.
$z\lesssim 0.2$, for which black hole masses can be precisely determined by
means of reverberation mapping (RM) measurements (e.g. Vestergaard & Peterson,
2006). However, RM measurements require long monitoring campaigns and thus for
most quasars – especially at higher redshifts where time delays are longer due
to time dilation and their generally more massive black holes – this method is
unfeasible. Thus, masses of SMBHs are inferred from single-epoch spectra using
aforementioned scaling relations, most commonly calibrated to the, at low-
redshift easily observable, rest-frame optical H$\beta\,$ emission line (e.g.
Vestergaard & Peterson, 2006; Grier et al., 2017). Since the H$\beta\,$ line
is very challenging to observe for quasars at $z>4$ with ground-based
observatories, additional uncertainty and possibly biases in the black hole
mass estimates for high-redshift quasars could be introduced by re-scaling the
H$\beta\,$ emission line properties to properties of rest-frame UV emission
lines, such as C IV and Mg II (e.g. Vestergaard & Osmer, 2009; Coatman et al.,
2017). These uncertainties have motivated recent efforts to avoid such scaling
relations altogether and derive black hole mass estimates and other quasar
properties from the spectra themselves by means of data-driven modeling of the
quasars’ spectral features (e.g. Eilers et al., 2022).
In this Letter we present the first spectrum of the ultra-luminous quasar
J0100+2802 at rest-frame optical wavelengths observed with the NIRCam grism on
JWST, as well as a newly reduced high signal-to-noise spectrum at rest-frame
UV wavelengths taken with ground-based facilities (§ 2). This spectrum covers
the wavelength region between $0.6\mu{\rm m}\leq\lambda\leq 4\mu$m (with a gap
between $2.3\mu{\rm m}\leq\lambda\leq 3.1\mu$m), which enables us to obtain
measurements of the quasar’s black hole mass based on the rest-frame UV C IV
and Mg II broad emission lines, as well as the rest-frame optical H$\beta\,$
emission (§ 3). Furthermore, we will show NIRCam Imaging observations that set
constraints on the possibility of strong gravitational lensing of the quasar
(§ 4), before summarizing our results (§ 5). Throughout this work, we adopt a
flat $\Lambda$CDM cosmology with $\Omega_{m}=0.31$ and
$H_{0}=67.7~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$ (Planck Collaboration et al.,
2018).
## 2 Data
### 2.1 Ground-based Spectroscopy
The ground-based optical and NIR spectroscopic data for J0100+2802 (RA:
$01^{\rm h}00^{\rm m}13^{\rm s}.020$; DEC:
$+28^{\circ}02{\arcmin}25{\arcsec}.840$) have been obtained using both the
X-Shooter spectrograph (Vernet et al., 2011) on the Very Large Telescope (VLT)
as well as the Folded-port InfraRed Echellette instrument (FIRE; Simcoe et
al., 2013) on the Magellan Telescope in the years of 2015 and 2016. The target
was observed for a total of $16.8$ hours, of which $11$ hours were observed
with VLT/X-Shooter (program ID: 096.A-0095; PI: Pettini) and $5.8$ hours
observed with Magellan/FIRE (PI: Simcoe).
The ground-based optical and NIR data are reduced consistently with the open-
source python-based spectroscopic data reduction pipeline
PypeIt111https://github.com/pypeit/PypeIt version 1.7.1 (Prochaska et al.,
2020). We derive the wavelength solution from the night sky OH lines, in order
to have on-sky wavelength calibrations and to reduce the overheads of our
observations. The sky subtraction is performed on the 2D images by
differencing exposures dithered along the slit in the standard A-B mode and
fitting a $b$-spline to further eliminate sky line residuals following
Bochanski et al. (2009). We then perform an optimal extraction (Horne, 1986)
to obtain the 1D spectra. The spectra are flux calibrated using sensitivity
functions of standard stars observed during the same observing run. For a few
runs no standard stars were observed and hence a sensitivity function is
created from an A0 star with known magnitude.
We co-add the flux-calibrated 1D spectra from each night and correct for
telluric absorption features by jointly fitting an atmospheric model and a
quasar model. The telluric model grids are produced using the Line-By-Line
Radiative Transfer Model (LBLRTM; Clough et al., 2005), while we fit a third
order polynomial as the quasar model for this very featureless spectrum. The
individual telluric-corrected 1D spectra observed in different observing runs
or with different telescopes and instruments are then co-added weighted by the
average square SNR of the exposure, and stacked on a common wavelength grid.
Note that this steps avoids any interpolation of the data in order to prevent
correlated noise properties.
In order to combine the optical and NIR part of the quasar spectrum observed
with X-Shooter, the two parts are stitched together by matching the median
flux at the intersection between $9900$ Å and $10100$ Å. As a last step we
apply an absolute flux calibration to the fully reduced and co-added quasar
spectrum by requiring their integrated flux within the UKIRT-J filter to match
the observed $J$-band magnitude of the quasar, i.e. $J_{\rm AB}=17.64\pm
0.02$.
Finally, the spectrum is re-binned to a wavelength grid sampled linearly in
velocity space with a chosen step size of $\Delta v\approx 50~{}\rm
km\,s^{-1}$ per pixel. We then divide all native pixels into the respective
wavelength bins, and determine the stacked flux in each bin as the mean from
all native pixels. Note that the final wavelength grid is the weighted average
of the individual wavelengths used for each exposure that fall into a given
wavelength bin in the input wavelength grid, and hence not necessarily
linearly sampled anymore.
The resulting ground-based spectrum shown in the top panel of Fig. 1 has a
${\rm SNR}\approx 205$ per $\Delta v\approx 50~{}\rm km\,s^{-1}$ pixel
measured at a rest-frame wavelength of $\lambda_{\rm rest}=1280\pm 20$ Å.
Figure 1: Quasar spectrum of J0100+2802 observed with ground-based facilities
(top) and NIRCam WFSS on JWST (bottom) with the filter F356W shown as the
orange shaded region. Grey shaded areas show masked wavelengths affected by
foreground absorption systems, regions of significant telluric absorption,
regions with strong sky line residuals, as well as regions at the very edge of
the filter transmission. The red curve shows the best fit to the quasar
spectrum, while the colored curves in the insets show the individual
components of the spectral fit (power-law continuum: blue dashed; Balmer
continuum: dark green dotted; iron-template: green; Gaussian line components:
yellow, grey or purple).
### 2.2 Observations with JWST/NIRCam WFSS
The quasar field of J0100+2802 was observed with JWST/NIRCam in Imaging and
WFSS mode on August 22, 2022, as part of the Emission-line galaxies and
Intergalactic Gas in the Epoch of Reionization (EIGER) GTO program (program
ID: 1243, PI: S. Lilly). The quasar field is observed in a mosaic containing
four individual visits such that the final field of view spans approximately
$3\arcmin\times 6\arcmin$ corresponding to $7.6\,{\rm cMpc}\times 15.3\,{\rm
cMpc}$ at $z\approx 6$. The central $40\arcsec\times 40\arcsec$ area around
the quasar is part of every visit and thus this region has the deepest
observations with a total exposure time of $8760\,$s per visit, i.e.
approximately $9.7$ hours in total on source. The NIRCam WFSS observations of
the J0100+2802 field use the grism “R” in the F356W filter. Simultaneously, we
obtain direct imaging observations in two short wavelengths filters, i.e.
F115W and F200W (see § 2.3). The observational setup as well as the data
reduction process are described in detail Kashino et al. (2022) and Matthee et
al. (2022). We will summarize the main steps briefly below.
The data is reduced using the jwst pipeline222https://jwst-
pipeline.readthedocs.io/en/latest/ version 1.8.0 provided by STScI. We first
run the Detector1Pipeline step to obtain the uncalibrated NIRCam exposures
(*_rate.fits). We then apply two additional reduction steps from the
Spec2Pipeline to transfer the pixel coordinates to astronomical coordinates
(AssignWcsStep) and afterwards apply the flat fielding to the exposures
(FlatFieldStep). We then extract the 2D spectrum of the quasar from each of
the flat-fielded exposures using customized scripts, which are making use of
the grismconf333https://github.com/npirzkal/GRISMCONF module using the latest
(V4) trace models. The spectra are normalized by the filter’s sensitivity
curve and rectified to a common observed wavelength grid, ranging from
$3.0\mu{\rm m}\leq\lambda_{\rm obs}\leq 4.2\mu$m in bins of
$\Delta\lambda=9.75$ Å. We correct the trace of the quasar in each exposure
for small curvatures and then perform an optimal extraction of the quasar
spectrum from each 2D exposure. This results in $96$ individual 1D spectra,
which are co-added using the average square SNR of each exposure as weights to
a combined 1D spectrum. We estimate the uncertainty on each pixel flux via
bootstrapping: to this end we re-sample the $96$ individual spectra with
replacement to create $1000$ co-added 1D spectra from which we take the $16$th
and $84$th percentile at each pixel as the uncertainty. The final spectrum is
shown in the bottom panel of Fig. 1 and has a ${\rm SNR}\approx 235$ per
$9.75\rm{\AA}$ pixel ($\Delta v\sim 80~{}\rm km\,s^{-1}$) measured at a rest-
frame wavelength of $\lambda_{\rm rest}=5100\pm 20$ Å .
The spectrum of the quasar from both the ground-based observatories as well as
JWST/NIRCam WFSS will be made publicly available upon publication.
### 2.3 Imaging Data from JWST/NIRCam
The NIRCam imaging data was reduced using the jwst pipeline version 1.8.2. In
order to avoid saturation in the central pixels of bright sources in the
quasar field we run the Detector1Pipeline without suppressing the computations
for saturated ramps with only one good, unsaturated sample in the ramp-fitting
step. We then run the Image2Pipeline to obtain calibrated images (*_cal.fits),
and apply the $1/f$ noise reduction, remove any snowballs and other artifacts
from the data, and subtract the sky and any wisp artifacts. We use
tweakwcs444https://github.com/spacetelescope/tweakwcs to align all exposures
relative to each other, and then stack all exposure with a global alignment
calibrated to Gaia stars. Any offsets are applied to the calibrated images
directly, before all exposures are ultimately combined to a final mosaic by
the Image3Pipeline. The exposure time of the images in the two short-
wavelengths filters F115W and F200W add up to $4380\,$s per visit, i.e.
approximately $4.9$ hours in total, while the direct images with the F356W
filter sum up to $1578\,$s per visit, i.e. $1.8$ hours total. Individual
exposures are stacked weighted by their inverse variance.
The final images (for one of the four visits) in the three broad band NIRCam
filters F115W, F200W and F356W of a $5\arcsec\times 5\arcsec$ region around
the quasar are shown in the left column of Fig. 3. Note that we do not take
the quasar images from the full mosaic for modeling the PSF (see § 4), but
only the images from one visit (i.e. visit 1), since it results in a better
PSF subtraction.
Since the noise properties in the short wavelength filters that are reported
by the jwst pipeline seem to overestimate the pixel-to-pixel variance in the
science image, we re-scale the inverse variance in the science images by a
factor of $2.0$ in F115W and $1.8$ in F200W. We do not apply any re-scaling in
the long wavelength filter F356W.
## 3 Mass estimates of the quasar’s supermassive black hole
Under the assumption that the dynamics in the quasar’s broad line region (BLR)
are dominated by the gravitational pull of the black hole and the system is in
virial equilibrium, we can estimate the mass of the SMBH using the FWHM of
broad emission lines, which gives an estimate for the velocity of the gas
clouds in the BLR orbiting the black hole, and the quasar’s luminosity, which
acts as a proxy for the radius of the BLR, i.e.
$\frac{M_{\bullet}}{M_{\sun}}=10^{\alpha}\left(\frac{\rm FWHM_{\rm
line}}{1000\,\rm km\,s^{-1}}\right)^{\beta}\left(\frac{\lambda
L_{\lambda}}{10^{44}\,\rm erg\,s^{-1}}\right)^{\gamma}.$ (1)
Traditionally, these scaling relations are calibrated based on a relatively
small sample of low-redshift quasars using the properties of the H$\beta\,$
emission line. However, as H$\beta\,$ is not observable with ground-based
observatories for quasars at $z>4$, these scaling relations have been re-
calibrated using properties of the rest-frame UV broad emission lines, such as
Mg II and C IV instead. The best-fit values for the scaling parameters
$\alpha$, $\beta$ and $\gamma$ dependent on the respective emission lines and
monochromatic luminosity $L_{\lambda}$, and are listed in Tab. 1.
Table 1: Parameters of the black hole mass scaling relations. emission line | $\alpha$ | $\beta$ | $\gamma$ | $\lambda$ | reference
---|---|---|---|---|---
C IV (corrected) | 6.71 | 2 | 0.53 | 1350 Å | Coatman et al. (2017)
Mg II | 6.86 | 2 | 0.50 | 3000 Å | Vestergaard & Osmer (2009)
H$\beta\,$ | 6.91 | 2 | 0.50 | 5100 Å | Vestergaard & Peterson (2006)
### 3.1 Spectral Fitting
To estimate the mass of the SMBH in J0100+2802, we fit the ground-based and
NIRCam/WFSS quasar spectrum in the wavelength regions between $0.95~{}\mu$m
and $2.25~{}\mu$m, as well as $3.15~{}\mu$m and $3.99~{}\mu$m, respectively.
We mask all wavelengths that are affected by foreground metal absorption
systems, regions affected by significant telluric absorption (at
$\lambda\approx 14,000$ Å and $\lambda\approx 19,000$ Å), regions with strong
sky line residuals (at $\lambda\approx 20,000$ Å), as well as regions at the
very edge of the filter transmission (at $\lambda<31,500$ Å and
$\lambda>40,000$ Å). We fit the observed ground- and space-based spectrum
separately, with a combination of (A) a power-law continuum, (B) an iron
template (provided by Vestergaard & Wilkes (2001) for the rest-frame UV iron
emission and by Park et al. (2022) in the rest-frame optical) that is
broadened to match the widths of the broad emission lines through a
convolution of the template with a corresponding Gaussian kernel, (C) a Balmer
continuum below the Balmer edge (which only applies to the ground-based
spectrum; see De Rosa et al., 2014; Schindler et al., 2020, for details), and
(D) one or multiple Gaussian components for the broad emission lines.
We use a single Gaussian component to fit the Si IV $\lambda 1304$, the
unresolved Si II $\lambda\lambda 1393,1402$ doublet, Mg II $\lambda 2799$,
H$\gamma\,$ $\lambda 4341$, [O III] $\lambda 4363$, and He II $\lambda 4868$
emission lines, while we use two Gaussian components to capture the more
complex line profile of the C IV $\lambda\lambda 1548,1550$, and H$\beta\,$
$\lambda 4861$ emission lines, as well as the [O III] $\lambda\lambda
4959,5007$ doublet. This results in $19$ and $32$ free parameters of the
spectral fit for the ground-based and NIRCam/WFSS spectrum, respectively,
which we estimate using the Markov Chain Monte Carlo (MCMC) algorithm emcee
(Foreman-Mackey et al., 2013). We apply flat priors for each parameter and
take the median of the posterior probability distribution as the best
parameter estimate. The final spectral fits are shown in red in Fig. 1.
### 3.2 Black Hole Mass Estimates
In order to estimate the mass of the quasar’s SMBH using scaling relations, we
need to estimate the FWHM of the broad emission lines as well as the
monochromatic luminosity $L_{\lambda}$ (see Eqn. 1). In order to derive the
FWHM of the emission lines with multiple Gaussian components, we first combine
both profiles to form a joint line profile and measure the width of this joint
profile at the half maximum. The estimates for the monochromatic luminosities
$L_{\lambda}$ are determined from the continuum flux $f_{\lambda}$, for which
we evaluate the power-law continuum fit at the respective wavelength $\lambda$
(we evaluate $f_{1350}$ and $f_{3000}$ from the power-law fit to the ground-
based spectrum, i.e. $f_{\lambda}\propto\lambda^{-1.6}$, and $f_{5100}$ based
on the power-law fit to the grism spectrum, i.e.
$f_{\lambda}\propto\lambda^{-2.4}$). The uncertainties on both the FWHM and
$L_{\lambda}$ are estimated from the $16$th and $84$th percentile of $10,000$
random draws from the MCMC posterior.
In order to estimate the black hole mass based on the C IV emission line we
follow the procedure suggested in Coatman et al. (2017). It is well known that
the C IV emission line in high-redshift quasars is often blue shifted due to
strong winds and outflows in the BLR (e.g. Meyer et al., 2019; Schindler et
al., 2020), and thus Coatman et al. (2017) report better consistency in their
black hole mass estimates once they “correct” the FWHM of the C IV line by its
velocity shift from the systemic redshift of $6.3270\pm 0.0005$ which was
determined from the sub-mm [C II] emission using observations from the Atacama
Large Millimetre Array (ALMA) (Wang et al., 2019). However, determining the
velocity shift of the C IV emission line is challenging for for J0100+2802
since it is affected by significant foreground and telluric absorption. Thus,
to reduce the impact of narrow absorption systems or telluric effects on the
emission-line profile we define a “pseudo continuum” within the wavelength
interval $1450-1600$ Å by applying a median filter to the quasar spectrum, as
suggested by (Coatman et al., 2016). Pixels within the wavelength range around
the C IV emission line profile that lie more than $2\sigma$ below the pseudo-
continuum are deemed to be affected by absorption and are additionally masked.
All measured spectral properties are listed in Tab. 2.
These spectral properties allow us to obtain three different estimates of the
black hole mass based on the C IV, Mg II and H$\beta\,$ emission using Eqn. 1.
Reassuringly, we find consistent results between the black hole mass
measurements derived from the rest-frame UV as well as the rest-frame optical
emission lines. All estimates point to a SMBH mass of J0100+2802 between
$9.7\leq\log_{10}(M_{\bullet}/M_{\odot})\leq 10.2$ (see Tab. 2). The
observational uncertainties on these measurements are obtained using again the
$16$th and $84$th percentile of black hole mass estimates from $10,000$ random
draws for the spectral fit parameters from their MCMC posteriors. While the
statistical errors on these measurements are small, there are large systematic
uncertainties in the applied scaling relations of approximately $0.4-0.5$ dex
(e.g. Vestergaard & Peterson, 2006), and thus all estimates of the SMBHs are
consistent within the expected uncertainties. The measurements show that this
quasar indeed hosts a ten billion solar mass black hole at $z=6.3270$, which
is the most massive SMBH known in the early universe.
Figure 2: Median rest-frame optical quasar spectra from low-redshift ($0\leq
z\leq 1$) SDSS quasars, binned by the quasars’ bolometric luminosity (left) or
EW(H$\beta\,$) (right). All spectra are normalized to unity at $5100$ Å. The
spectra show a clear decrease in the strength of the narrow [O III] emission
lines with increasing $L_{\rm bol}$ and decreasing EW. The quasar J0100+2802
is with a bolometric luminosity of $\log_{10}(L_{\rm bol}/\rm
erg\,s^{-1})\approx 48$ more luminous than any of the low-redshift SDSS
quasars. However, low-redshift SDSS quasars with a similar EW(H$\beta\,$)
compared to J0100+2802 (i.e. EW(H$\beta\,$)$=215\pm 2$ Å) exhibit clearly
narrow [O III] emission, which is not present in the spectrum of J0100+2802.
### 3.3 Equivalent Width of the Quasar’s [O III] Emission
We will now take a closer look at the spectral shape of the quasar around the
H$\beta\,$+[O III] emission line complex. We estimate an equivalent width (EW)
in the observed wavelength frame for the [O III] doublet emission of EW([O
III]$)\approx 86\pm 1$ Å. Interestingly, nearly all of this emission arises
from the broad [O III] components, since we do not find any narrow [O III]
emission (i.e. EW([O III]${}_{\rm narrow})<1$ Å) in the spectrum of J0100+2802
(see inset in Fig. 1). It is not unusual to observe weaker narrow [O III]
emission lines with increasing bolometric luminosity (e.g. Baldwin, 1977) and
decreasing equivalent width of the H$\beta\,$ emission in low-redshift quasar
spectra from the Sloan Digital Sky Survey (SDSS; Ahumada et al., 2020; Lyke et
al., 2020), as shown in the median low-redshift SDSS quasar spectra in Fig. 2.
However, these low-redshift quasars at $0\leq z\leq 1$ clearly exhibit narrow
[O III] emission lines even if the quasars are bright (i.e. $L_{\rm
bol}\gtrsim 10^{47}\rm erg\,s^{-1}$), and have a H$\beta\,$ equivalent width
comparable to J0100+2802 (i.e. EW(H$\beta\,$)$=215\pm 2$ Å). At higher-
redshifts, however, this situation changes. Vietri et al. (2018) study the
rest-frame UV and optical spectra of 18 hyper-luminous quasars at $2\lesssim
z\lesssim 4$ and show that weak [O III] emission is common in about $70\%$ of
their sample and is strongly correlated with large blueshifts in the C IV
emission line, as also seen here for J0100+2802. In a similar study using
$330$ luminous quasars at $1.5\leq z\leq 4.0$ Coatman et al. (2019) confirm
the observed anti-correlation between the [O III] EW and C IV blueshifts, and
find that a signifcant fraction of approximately $10\%$ of their quasars
exhibit very weak narrow [O III] emission (i.e. EW$<1$ Å) as seen in
J0100+2802, which is $10$ times higher than among the lower-redshift and
lower-luminosity SDSS quasars. Thus, the absence of narrow [O III] emission in
the hyper-luminous quasar J0100+2802 could point towards evolutionary effects
in the quasar spectra and might be a sign that the narrow line region (NLR)
around the quasar has already been cleared by strong quasar-driven winds on a
relatively short timescale (Vietri et al., 2018; Coatman et al., 2019).
Alternatively, it has been suggested recently that under-luminous [O III]
emission in quasar spectra might be linked to a low gas content in the NLR of
the quasar (Agostino et al., 2022). Upcoming additional rest-frame optical
spectra of high-redshift quasars observed with JWST will be required to shed
more light on the evolution of the NLR at high redshifts.
Table 2: Spectral properties of the quasar J0100+2802. spectral property | measurement
---|---
$\rm FWHM_{\rm C\,IV}~{}[\rm km\,s^{-1}]$ (corrected) | 3190±2070
$\rm FWHM_{\rm Mg\,II}~{}[\rm km\,s^{-1}]$ | 3890±40
$\rm FWHM_{\rm H\beta}~{}[\rm km\,s^{-1}]$ | 6730±170
$\Delta v_{\rm C\,IV}~{}[\rm km\,s^{-1}]$ | -8850±2970
$\Delta v_{\rm Mg\,II}~{}[\rm km\,s^{-1}]$ | -980±20
$\Delta v_{\rm H\beta}~{}[\rm km\,s^{-1}]$ | -30±130
$1350{\rm{\AA}}\,L_{1350\rm{\AA}}~{}[10^{46}\,\rm erg\,s^{-1}]$ | 40.9±0.1
$3000{\rm{\AA}}\,L_{3000\rm{\AA}}~{}[10^{46}\,\rm erg\,s^{-1}]$ | 25.7±0.1
$5100{\rm{\AA}}\,L_{5100\rm{\AA}}~{}[10^{46}\,\rm erg\,s^{-1}]$ | 20.1±0.1
$\log_{10}(M_{\bullet}/M_{\sun})$ (C IV) | 9.6±0.9
$\log_{10}(M_{\bullet}/M_{\sun})$ (Mg II) | 9.7±0.1
$\log_{10}(M_{\bullet}/M_{\sun})$ (H$\beta\,$) | 10.2±0.1
## 4 No Evidence for Strong Gravitational Lensing
Figure 3: PSF modeling of the quasar images in the NIRCam filters F115W (top),
F200W (middle) and F356W (bottom). The left columns show a $5\arcsec\times
5\arcsec$ cutouts of the quasars, the middle columns show the quasar model,
i.e. a single point source convolved with the PSF model, while the right
columns show the residuals of the quasar after subtracting the model,
normalized by the peak pixel value of the quasar. The black dashed circle
indicates the most probable image separation of $\Delta\theta\approx
0.8\arcsec$ for a lensed source at the quasar’s redshift (see Fig. 4).
All estimates of the SMBH mass rely on the assumption that the observed
luminosity of the quasar is intrinsic to the quasar itself. If the quasar’s
luminosity was, however, magnified by a factor of $\mu$ due to strong
gravitational lensing, the black hole masses would be overestimated by a
factor of $\mu^{\gamma}$ (with $\gamma\approx 0.5$; see Tab. 1).
Previous studies searched for multiple images of the quasar or deviations from
a single point spread function (PSF) due to strong gravitational lensing
effects based on high spatial resolution HST imaging, but have been
unsuccessful. The observed quasar images are consistent with a single PSF
indicating no evidence for multiple images or arcs with a spatial separation
of the diffraction limit of $\theta\approx 0.10\arcsec$, using the F850LP
filter at a central wavelength of $\lambda=0.914\,\mu$m on HST’s Advanced
Camera for Surveys (ACS; Fujimoto et al., 2020, Yue et al. in prep.).
The diameter of JWST’s mirror is, at $6.5$ meters, nearly $3$-times larger
compared to the $2.4$-meter mirror of HST, which reduces the diffraction limit
by a factor of two to $\theta\approx 0.05\arcsec$ in the F115W filter at a
central wavelength of $\lambda=1.154\,\mu$m. In the F200W (F356W) filter with
a longer central wavelength $\lambda=1.989\,\mu$m ($\lambda=3.568\,\mu$m) the
diffraction limit approaches $\theta\approx 0.08\arcsec$ ($\theta\approx
0.14\arcsec$). With the improved spatial resolution of JWST we will now
readdress the question, whether we can find any evidence for strong
gravitational lensing in this ultra-luminous quasar by detailed modeling of
the quasar’s PSF.
### 4.1 PSF modeling and subtraction
Making use of the photutils package (Bradley et al., 2020), we build an
effective PSF by means of the brightest stars in the quasar field in the
respective filter. To this end we first select stars by searching for peaks
($\geq 30~{}\rm MJy\,sr^{-1}$) in the stacked images, while excluding all
peaks that belong to any extended sources rather than stars as well as stars
too close ($\leq 2.5\arcsec$) to the edge of our field of view. This results
in $5$ ($3$, $4$) stars for the F155W (F200W, F356W) filter from which we
build an effective PSF model (Anderson & King, 2000).
Note that since the spectral energy distribution (SED) of the quasar is much
redder than the SED of stars, we expect small differences in the shape of the
PSFs. However, we nevertheless obtain better results when modeling the PSF
when building an effective PSF using stars in the field than when applying
webbpsf555https://webbpsf.readthedocs.io/en/latest/, which allows the user to
input a quasar SED to construct a PSF model.
Leveraging the capabilities of psfMC, which performs 2D surface brightness
modelling on astronomical images using a MCMC algorithm (Mechtley, 2014;
Marshall et al., 2021), we model the quasar as a single point source and find
the best fit PSF model to the quasar images. Note that psfMC requires a noise
model for the effective PSF. To this end we assume a SNR$=100$ of the
effective PSF at each pixel with a fixed noise floor for all pixels below the
$20$th percentile of pixel values.
The results of the PSF modeling and subtraction are shown in Fig. 3 for the
three different NIRCam filters F115W, F200W and F356W. The first column shows
the quasar images, the second column shows our best fit model, i.e. a single
point source convolved with the PSF model, while the last column shows the
residuals of the quasar after subtracting the PSF model. We do not find any
evidence from deviations of a single PSF in any of the three filters.
In order to further test whether we can find any evidence for a second quasar
image, we repeat the modeling procedure and fit the quasar images with two
PSFs instead of a single one. The best fit parameters for the central pixels
of the two models are less than $1$ pixel (i.e. $0.03\arcsec$) apart, when the
magnitudes of both images are approximately equal. Thus we do not find any
evidence for two quasar images and conclude that any possible effects due to
strong gravitational lensing of the source have to result in image separations
of less than a couple of pixels, i.e. below the diffraction limit of
$\theta\approx 0.05\arcsec$.
### 4.2 Searching for a foreground deflector galaxy
We also search for a possible deflector galaxy in the foreground of the
quasar, which could cause the quasar to be gravitationally lensed. To this
end, we search for any extended emission close to the quasar sightline, since
we expect any low-redshift deflector galaxy to be spatially extended given the
size distribution of low-redshift galaxies. In van der Wel et al. (2014), the
authors report a median effective radius of $R\approx 1-4$ kpc ($R\approx 3-6$
kpc) for early-type (late-type) galaxies with a stellar mass of $M_{\star}\sim
5\times 10^{10}\,M_{\odot}$ at redshifts $0<z<3$, which corresponds to spatial
extents of $\sim 0.1\arcsec-0.5\arcsec$ ($\sim 0.4\arcsec-0.8\arcsec$).
We do not see any extended sources close to the quasar sightline in the NIRCam
filters F115W, F200W and F356W down to the $5\sigma$ maximum sensitivity of
28.7, 29.2, 29.0 magnitude, respectively, in the deepest area of the mosaic
around the quasar (Kashino et al., 2022). Furthermore, HST imaging in the
filter F606W at an effective wavelength of $\lambda_{\rm eff}=5776.43$ Å,
where the quasar’s emission is heavily suppressed thanks to the intervening
intergalactic medium acting as a natural band pass filter, does not reveal any
extended sources around the quasar sightline (Yue et al. in prep.).
If a deflecting object would be right along our line-of-sight to the quasar,
we would only be able to see the object in the images after subtracting the
quasar’s PSF, since the quasar would outshine all foreground objects. However,
we do not see any significant extended emission around the quasar or nearby,
even in the PSF subtracted residual images (right panels of Fig. 3)
### 4.3 Probability for strong gravitational lensing
Figure 4: Distribution of the lensing separation $\Delta\theta$ for sources at
the quasar’s redshift $z_{\rm s}=6.327$, predicted by the analytical model
described in Yue et al. (2022) and Oguri & Marshall (2010, OM10). The
diffraction limit obtained from the JWST/NIRCam F115W quasar image is shown as
the dashed line.
We will now estimate the likelihood that the quasar is affected by strong
gravitational lensing given the aforementioned constraints on the Einstein
radius. We will use two models to describe the population of low-redshift
deflector galaxies described in detail in Oguri & Marshall (2010) and Yue et
al. (2022). Both models use singular isothermal spheres (SIS) to describe the
mass profile of the low-redshift deflector galaxies and parameterize the
velocity dispersion function (VDF) of the deflectors with a Schechter function
(Schechter, 1976). However, the VDF applied in Oguri & Marshall (2010) is
based on early-type galaxies derived from SDSS data (Choi et al., 2007), while
the VDF in Yue et al. (2022) is based on both late- and early-type galaxies in
the local universe (Hasan & Crocker, 2019) and assumes a redshift evolution in
the parameters of the Schechter function as suggested by Geng et al. (2021).
Note that the model by Oguri & Marshall (2010) predicts a lower number of less
massive galaxies compared to Yue et al. (2022), since the VDF is based on
early-type galaxies only (see Fig. $1$ in Yue et al. (2022) for a comparison
between the VDFs).
Making use of these VDFs of the low-redshift deflector galaxy population, we
can built an analytic lensing model of the high-redshift lensed quasar
population (Yue et al., 2022). We show the distribution of image separations
of all lensed objects at the quasar’s redshift of $z=6.327$ in Fig. 4. Both
deflector models peak at approximately $\Delta\theta\approx 0.8\arcsec$
indicating the most probable image separation. Based on the PSF modeling of
the quasar in the JWST/NIRCam images described in the previous section we can
constrain the maximum image separation to be below the diffraction limit.
Thus, the fraction of lensed objects with image separation
$\Delta\theta<0.05\arcsec$ among all lensed objects at $z=6.327$ — which
constitutes the probability that J0100+2802 is gravitationally lensed with an
image separation below the diffraction limit — is
$P(\Delta\theta<0.05\arcsec)\approx 2.2\times 10^{-3}$ for the model by Yue et
al. (2022), and $P(\Delta\theta<0.05\arcsec)\approx 6.4\times 10^{-4}$ for the
model by Oguri & Marshall (2010).
## 5 Summary & Discussion
In this letter we present the rest-frame UV as well as the rest-frame optical
spectrum at observed wavelengths between $0.6\mu{\rm m}\leq\lambda_{\rm
obs}\leq 2.3\mu$m and $3.1\mu{\rm m}\leq\lambda_{\rm obs}\leq 4.0\mu$m,
respectively, of the ultra-luminous quasar J0100+2802 at a redshift of
$z=6.327$. The ground-based data consist of a total of $16.8$ hours observed
with the VLT/X-Shooter spectrograph and Magellan/FIRE, while the rest-frame
optical data are observed with JWST/NIRCam in WFSS mode for a total of $9.7$
hours on source.
We measure the mass of the quasar’s SMBH based on the C IV, Mg II and
H$\beta\,$ emission line properties and find estimates that are consistent
between the different line estimators within their systematic uncertainties of
$0.5$ dex, indicating a SMBH mass of $M_{\bullet}\sim 10^{10}\,M_{\odot}$. In
the near future, larger samples of rest-frame optical spectra of high-redshift
quasars will be able to confirm this long-questioned consistency in black hole
mass measurements based on rest-frame UV and rest-frame optical emission line
properties for a wider range of black hole masses and quasar properties.
We revisit the question whether the luminosity of this ultra-luminous quasar
is magnified due to strong gravitational lensing with the increased spatial
resolution of the JWST/NIRCam images compared to previous HST images, and thus
search for evidence for multiple images of the quasars or arcs in three
different NIRCam filters, i.e. F115W, F200W and F356W. We do not find any
evidence from deviations of a single point source at the diffraction limit,
nor do we find any evidence for a foreground deflector galaxy. We thus
estimate the probability of J0100+2802 being affected by strong gravitational
lensing with image separations below the diffraction limit to be $\lesssim
2.2\times 10^{-3}$ using two different models for the foreground galaxy
population (Oguri & Marshall, 2010; Yue et al., 2022).
Our results confirm that J0100+2802 indeed hosts a SMBH with
$M_{\bullet}\approx 10^{10}\,M_{\odot}$ at $z=6.327$, when the universe is
only about $800$ Myr old, which challenges our current understanding of black
hole growth. It requires us to explore models beyond the standard black hole
formation paradigm, such as for instance massive initial black hole seeds in
excess of stellar remnants (e.g. Oh & Haiman, 2002; Begelman et al., 2006), or
radiatively inefficient black hole accretion episodes (e.g. Begelman &
Volonteri, 2017; Davies et al., 2019). JWST has ushered us into a new era for
exploring the early universe. Future observations of high-redshift quasars
will enable us to study the black hole masses for ensembles of quasars,
understand their large-scale environments, and measure their luminosity
functions to gain new insights onto the properties and origins of SMBH seeds,
and their rapid growth at early cosmic times.
The authors would like to thank Paul Schechter for insightful discussions
about gravitational lensing. Furthermore, we would like to Madeline Marshall
for helpful discussions on the PSF modeling, and Gisella De Rosa for
interesting discussions on quasar evolution. This work is based on
observations collected at the European Organisation for Astronomical Research
in the Southern Hemisphere under ESO programme 096.A-0095. This paper includes
data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas
Observatory, Chile. This work is based in part on observations made with the
NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the
Mikulski Archive for Space Telescopes at the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These
observations are associated with program ID $\\#1243$. The specific
observations analyzed can be accessed via https://doi.org/10.17909/xbs2-v060
(catalog 10.17909/xbs2-v060). DK has been supported by JSPS KAKENHI Grant
Number JP21K13956.
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# Quark-Hadron Transition and Entanglement
Berndt Müller Department of Physics, Duke University, Durham, NC 27708-0305,
USA Andreas Schäfer Institut für Theoretische Physik, Universität
Regensburg, D-Regensburg, Germany
###### Abstract
The dual holographic description has enjoyed many successes in explaining
fundamental properties of the early stages of relativistic heavy ion
collisions up to the formation of a minimal-viscosity quark-gluon fluid.
However, there have been few attempts to extend its application beyond this
stage. Here we explore the prospects for such an extension beyond the time of
hadronization. Our discussion makes use of recent insights into the duality of
entanglement properties of field theory states in the edge of Anti-de Sitter
space and non-trivial topologies of horizons in the bulk, often referred to as
ER = EPR duality. We discuss this topic from the point of view of heavy-ion
phenomenology, review several relevant concepts, and map out a path toward
combining them into a comprehensive, at least semiquantitative description of
relativistic heavy ion collisions. We outline possible next steps in this
direction.
## I Introduction
One of the fundamental questions of high energy physics is how apparently
thermal behavior emerges in relativistic heavy ion collisions. A large number
of measurements show such thermal behavior and are best described when one
assumes that the produced transient Quark-Gluon Plasma (QGP) is in a fully
thermalized state. Yet the nuclear reaction, which occurs in isolation, cannot
produce a state of high von Neumann entropy because the reaction is governed
by the laws of quantum chromodynamics (QCD) which assures unitarity of the
S-matrix. The transient QGP and all the hadrons measured in the final state
must exist in a highly entangled quantum state that mimics the properties of a
thermal ensemble that is in interaction with an external heat bath.
One is immediately led to the question whether there are any observable
differences between a thermal ensemble and such a highly entangled excited
state. The answer is not as simple as one might think as local observables
and, more generally, observables that depend on only a fraction of the
complete many-particle final state have difficulty differentiating between
these two alternatives.
Whether entropy can be generated in an isolated system in a process that is
governed by time reversal invariant laws or not, depends on the definition of
generalized, non-equilibrium “entropy”. Several proposals for such a notion
exist and, unsurprisingly, the answer depends on the entropy definition used.
In addition, as the deviation from equilibrium is time dependent, the most
suitable notion of entropy may yield an answer that depends on the time
elapsed after the start of the collision.
The complexity of this issue is illustrated by the fact that the Anti-de
Sitter space-conformal field theory (AdS/CFT) duality allows to map the
production of a QGP to the formation of a black hole in five-dimensional Anti-
de Sitter (AdS5) space. Any description of entropy production will share the
subtleties of the black hole information paradox expressed most succinctly by
the Page curve Page (1993a, 2013). For any pure quantum state of an isolated
many-particle system the von Neumann entropy is zero and remains zero under
unitary time evolution. If, however, only part of the system is observed, i.e.
if the information residing in the rest of the system is discarded, the
reduced system appears thermal with the corresponding thermal entropy.
Although the Page curve was specifically introduced to resolve the information
problem of black holes, it is understood to be the characteristic property of
any generic many-body quantum system Page (1993b). Its existence has been
confirmed experimentally in various experiments, see, e.g., Fig. 4 in Kaufman
et al. Kaufman _et al._ (2016).
Our goal here is to outline a program which aims at describing essential
information theoretical properties of high energy nuclear collisions,
including those properties resulting in a Page curve, using concepts from
AdS/CFT duality. Obviously this is an ambitious goal, and in the present
article we can only outline the first steps. We will provide a list of open
issues at the end of this work that may help define a systematic pathway
towards a more precise understanding of how the apparent thermal properties of
the final state of a relativistic heavy ion collision can be reconciled with
the unitarity of the S-matrix in QCD.
There are many quantum field theories for which holographic duals are known to
exist. In the realm of gauge theories the original prototype, the ${\cal N}=4$
supersymmetric conformal Yang-Mills theory, is by far the most widely explored
example, because for large $N_{c}$ and large ’t Hooft coupling $\lambda$ it
has a tractable holographic dual, i. e. classical supergravity in AdS space.
The exact holographic dual for QCD is not known, although models that share
important features with QCD have been constructed Witten (1998); Sakai and
Sugimoto (2005). Therefore, any attempt to apply AdS/CFT phenomenology to QCD
is problematic from the start and can only be successful when combined with
other field theory and string theory techniques which allow to correct for
some of the differences. For example, the violation of conformal symmetry can
be treated perturbatively in QCD, see, e.g. Braun _et al._ (2003, 2019);
Kumericki _et al._ (2007), the finiteness of the ’t Hooft coupling can be
corrected by string perturbation theory, see e.g. Waeber _et al._ (2015), and
the calculation of $1/N_{c}$ expansions on the field theory side has evolved
into a broad research field of its own ’t Hooft (1974).
However, QCD has also certain properties, like the confinement/deconfinement
transition, which cannot be treated perturbatively and thus require a more
incisive approach. As we are interested in the hadronization of the quark-
gluon plasma we cannot avoid studying holographic models that incorporate a
confinement/deconfinement transition into their basic framework through some
versions of the Hawking-Page transition Hawking and Page (1983). Examples of
such models include the ${\cal N}=4$ super-Yang-Mills theory on a compactified
spatial volume, such as $S^{3}$, models including a compactified additional
dimension of AdS space Aharony _et al._ (2006), and models that break
conformal invariance through the introduction of a dilaton field Gürsoy _et
al._ (2009a); Mandal and Morita (2011). Although there are reasons to believe
that such holographic models can describe many relevant properties of QCD
qualitatively, it is far from clear whether they can be refined to decribe
these relevant features with sufficient precision to allow for a quantitative
comparison with experimental data.
In view of these caveats the success of holographic methods in modeling
certain aspects of relativistic heavy ion collisions came as a pleasant
surprise. They have been used extensively to understand rapid thermalization
Lin and Shuryak (2008); Balasubramanian _et al._ (2011a, b); Shuryak (2012);
Chesler and Yaffe (2011, 2014); Hubeny _et al._ (2013), the fast transition
to hydrodynamic flow Janik and Peschanski (2006); Heller _et al._ (2012a),
the small value of the specific shear viscosity Kovtun _et al._ (2005), and
much more. In fact, the last two decades have witnessed such continuous
progress that today a much larger assortment of powerful techniques and
insights exists than ever before. For example, semiclassical quantum gravity
and applications to black hole physics have recently conceptually resolved the
information paradox Maldacena and Susskind (2013); Maldacena and Qi (2018);
Akers _et al._ (2020); Almheiri _et al._ (2020a, 2019, 2021); Penington _et
al._ (2022); Penington (2020); Anderson _et al._ (2020). All these
encouraging successes motivate us to try to understand aspects of the breakup
of the QGP into hadrons, which have only rarely been explored using
holographic methods.
Our effort to expand the use of holographic duality to a description of the
complete heavy ion collision is also motivated by the recognition that a field
theoretical description of hadronization on the quantum level is far too
complex to be tractable. Without tractable holographic descriptions that
capture salient features of this transition, even if they cannot reproduce all
aspects quantitatively, we simply lack a perspective to fully understand the
many-body quantum mechanics of a heavy ion collision.
Let us start by reminding the reader of a few salient phenomenological
properties of heavy ion collisions that any comprehensive description must
address:
* •
There exist two distinct ways in which the particles that are ultimately
detected, hadrons, are produced from the evaporating QGP. Beginning
immediately after formation of the QGP ”fireball”, hadrons are emitted from
its surface. As the expanding QGP cools down to the pseudocritical temperature
$T_{c}$, hydrodynamics ceases to be valid, and the fireball converts into
hadrons throughout the remaining volume. The thermal model Andronic _et al._
(2018) describes the production rates of all hadrons and nuclei remarkably
well with a single universal temperature parameter (the chemical freeze-out
temperature $T_{\rm CF}=156.6\pm 1.7$ MeV Andronic _et al._ (2021)) that is
determined to percent-level accuracy. Because of the closeness of this value
to the central temperature of the QCD crossover transition determined by
lattice QCD ($T_{c}=158\pm 0.6$ MeV Borsanyi _et al._ (2020) or
$T_{c}=156.5\pm 1.5$ MeV Bazavov _et al._ (2019)) $T_{\rm CF}$ is interpreted
as QCD deconfinement temperature. The precision of the coincidence between
$T_{\rm CF}$ and $T_{c}$ itself is remarkable because the QCD crossover
transition, as measured by the chiral susceptibility, has an intrinsic width
of $15\pm 1$ MeV Borsanyi _et al._ (2020).
* •
The fact that the thermal model works so well is quite astonishing in view of
the strong interactions among the constituents of the system before and after
the hadronization transition. A focus of the discussion on this point is
provided by the hypertriton He${}^{3}_{\Lambda}$ whose yield in Pb-Pb
collisions at LHC agrees with the prediction of the thermal model for the
universal temperature $T_{\rm CF}$ despite the fact that its binding energy is
only 0.4 MeV, i. e. much smaller than $T_{\rm CF}$, and its size is comparable
to the size of the collision system. Recently, however, it was found Acharya
_et al._ (2022a) that this is no longer true for a smaller collision system
(p-Pb) with earlier hadronization time, adding to the conundrum of
hadronization in heavy ion collisions.
* •
Heavy ion collisions create thermal conditions that resemble those prevailing
in the early universe, but the time scales are vastly different. In the
cosmos, different particle species fell out of thermal equilibrium at
different times due to cosmic expansion and cooling. For example, photons and
neutrinos have different cosmic background radiation temperatures. The same
could be expected for heavy ion collisions but is not observed for Pb-Pb
collisions. As we argued in Müller and Schäfer (2017) one framework that can
potentially provide an explanation for the lack of thermal differentiation is
the Eigenstate Thermalization Hypothesis (ETH) Deutsch (1991); Srednicki
(1994); D’Alessio _et al._ (2016). The ETH may explain not only the success
of the thermal model but also the difference between Pb-Pb and p-Pb collisions
as the lifetime of the QGP in p-Pb collisions may be too short to apply the
full ETH formalism, which is based on energy eigenstates (see Section III).
Our manuscript is organized as follows. Section II contains some general
remarks about entanglement entropy in the context of relativistic heavy ion
collisions, and we introduce the concept of the Page curve. Because the ETH
plays an important role in our arguments we present a brief review in Section
III of those aspects that are relevant for us. In Sections IV and VI we will
further discuss some relevant aspects of heavy ion collisions and provide more
details on the properties mentioned above. In particular, we will argue in
Section IV that the usual numerical AdS calculations in the Poincaré patch are
insufficient to obtain the Page curve. In Section V we will review these
holographic model calculations of the early stages of heavy ion collisions up
until apparent thermalization. In this time interval they contain the relevant
physics and give phenomenologically satisfactory results.
To extend the holographic description to later times we propose two distinct
hadronization mechanisms: surface radiation and bulk hadronization. In Section
VII we adapt recent ideas Aharony _et al._ (2006); Mateos _et al._ (2006);
Bigazzi _et al._ (2020) for the Hawking-Page phase transition to bulk
hadronization. To describe the quantum entanglement of hadrons emitted from
the fireball surface requires a very different approach. As hadrons cannot
exist within the QGP fireball, its surface acts as a boundary for hadron
states. In Section VIII we discuss how hadron emission can be treated as the
production of particle-hole states at the QGP surface in analogy to Hawking
radiation from the black hole horizon. This allows us to adapt the arguments
for the formation of ”islands” within black holes which are entangled with the
outgoing Hawking radiation to the QGP fireball. The increasing entanglement of
the remaining QGP with the emitted hadrons is holographically mapped onto the
black hole evaporation process. This will make it possible to transcribe the
arguments for the occurrence of a Page curve to the QGP hadronization process.
We synthesize these considerations into a schematic holographic description of
the hadronization process in Section IX.
Our arguments are admittedly speculative, but they have the advantage that
their underlying assumptions can be confirmed or refuted by detailed
calculations. In Section X we present a limited list of possible future
studies.
## II Entanglement Entropy
In high-energy heavy ion collisions one never measures the complete final
state but detects only a subset of all emitted particles. Detector constraints
typically limit the detection to hadrons in a certain rapidity interval
$\Delta y\sim 1$ that is much smaller than the full beam rapidity range
$-y_{\rm beam}\leq y\leq y_{\rm beam}$.111The acceptance of detectors usually
covers a certain pseudorapidity window rather than a rapidity window. At high
energy, however, by far the largest number of emitted particles are pions
whose mass is smaller than their average transverse momentum, the effect of
the difference between pseudorapidity and rapidity is small for most global
observables. Because the subsystem $\Phi_{\Delta y}$ within the kinematic
range $\Delta y$ is entangled with the rest of the final state
$\Phi^{\prime}_{\Delta y}=\Phi\setminus\Phi_{\Delta y}$ outside this rapidity
window, it carries a certain entanglement entropy:
$S(\Phi_{\Delta y})={\rm Tr}[\rho(\Phi_{\Delta y})\ln\rho(\Phi_{\Delta y})].$
(1)
Here
$\rho(\Phi_{\Delta y})={\rm Tr}_{\Phi^{\prime}}[\rho(\Phi)],$ (2)
denotes the reduced density matrix of the final state constrained to the
rapidity window $\Delta y$, $\Phi$ denotes the complete final state, and
$\rho(\Phi)$ is the complete final-state density matrix. Unitarity of the
S-matrix in QCD ensures that $S(\Phi)=0$, if the initial state is a pure
quantum state, which is satisfied to a high degree of precision in the
collision of two heavy ions.
For $\Delta y\ll 2y_{\rm beam}$ the entanglement entropy $S(\Phi_{\Delta y})$
grows with the size of the rapidity window. Boost invariance at midrapidity,
realized at high collision energy, dictates that $S(\Phi_{\Delta y})/\Delta y$
is approximately independent of $\Delta y$ for small rapidity windows. The
complementarity law of quantum information also dictates that $S(\Phi_{\Delta
y})=S(\Phi^{\prime}_{\Delta y})$. Accordingly, when $\Delta y$ exceeds half
the total rapidity range of the final state, $\Delta y>y_{\rm beam}$, the
entanglement entropy has to decrease again and approach zero when $\Delta y\to
2y_{\rm beam}$. The entanglement entropy associated with a rapidity window
thus follows a Page curve Page (1993b, a), as sketched by the red solid line
in Fig. 1.
Figure 1: Page curve (red solid line) for the entanglement entropy associated
with a rapidity window $\Delta y$.
Particle detectors are not capable of measuring all information contained
within $\Delta y$, e.g., they generally do not record the relative phases
among all emitted particles. Although many modern detectors are capable of
recording the full set of kinematic correlations among hundreds or thousands
of detected particles, a large part of this information is usually discarded
in the data analysis and only selected few-body correlations are retained
222There are exceptions to this statement, e.g., in the measurement of the
collective flow vector or in fully resolved jet measurements, but even these
analyses ignore the full correlation information that is, in principle,
contained in the recorded data.. The reduction to a subset of data within the
rapidity window $\Delta y$ further changes the concept of entropy into a
notion that has been aptly called “entropy of ignorance” about the full
details of the final state Duan _et al._ (2020). What is commonly called
“entropy” of the final state of a relativistic heavy ion collision is derived
from the single-particle distribution in momentum space; here we denote this
single-particle entropy as $S_{\rm sp}(\Delta y)$. The single-particle entropy
grows at a different rate than the entanglement entropy (see Duan et al. Duan
_et al._ (2020) for a detailed analysis in a slightly different context) and
continues to grow as $\Delta y$ exceeds $y_{\rm beam}$, as illustrated by the
blue dashed line in Fig. 2.
Figure 2: Schematic diagram of the final-state entropy $S_{\rm sp}(\Delta y)$
in a rapidity window $\Delta y$ derived from the single-particle momentum
distribution, shown as blue dashed line, in comparison with the entanglement
entropy in the same window, shown as red solid line.
A number of estimates of the single-particle entropy have been presented in
the literature. The estimates are based on the measured spectra of produced
particles and calculate the entropy of an equilibrated Boltzmann (or
Bose/Fermi) gas with the same momentum and mass distribution. This is usually
done per unit of rapidity (at midrapidity), resulting in a measured value of
$dS_{\rm sp}/dy=S_{\rm sp}(\Delta y)/\Delta y$, which corresponds to the slope
of the dashed line in Fig. 2. At top RHIC energy, $dS_{\rm sp}/dy\approx
5,000$ Pal and Pratt (2004); Müller and Rajagopal (2005); at top LHC energy
$dS_{\rm sp}/dy\approx 11,500$ Hanus _et al._ (2019).
## III The Eigenstate Thermalization Hypothesis
ETH Deutsch (1991); Srednicki (1994); D’Alessio _et al._ (2016) posits that
even a single energy eigenstate can appear like a thermal system for most, if
not all, observables of practical interest. It is thought to apply generally
to systems that exhibit chaos at the quantum level. The ETH represents a
generalization of Random Matrix Theory (RMT) Wigner (1967) and, like RMT, is
formulated in terms of the matrix elements of an observable ${\cal A}$ in the
energy basis:
$A_{\alpha\beta}=\langle E_{\alpha}|{\cal A}|E_{\beta}\rangle\,.$ (3)
The ETH goes beyond RMT by assuming that the off-diagonal elements of the
matrix $A_{\alpha\beta}$ can be written as normalized elements
$R_{\alpha\beta}$ a random matrix modified by factors involving the
microcanonical entropy $S(E)$ and the spectral function $f(E,\omega)$ of the
system:
$A_{\alpha\beta}=A(E)\delta_{\alpha\beta}+e^{-S(E)/2}f(E,\omega)R_{\alpha\beta}\,,$
(4)
where $E=(E_{\alpha}+E_{\beta})/2$ and $\omega=E_{\alpha}-E_{\beta}$. The
exponential factor implies that the individual off-diagonal elements are
suppressed relative to the diagonal elements for a system with a high energy
level density, i. e., with $S(E)\gg 1$. Then the diagonal matrix elements
alone determine the thermal average of the observable:
$\displaystyle\langle{\cal A}\rangle_{T}$ $\displaystyle=$ $\displaystyle
Z(T)^{-1}\int\frac{dE}{E}\,e^{S(E)-E/T}\,A(E),$ $\displaystyle Z(T)$
$\displaystyle=$ $\displaystyle\int\frac{dE}{E}\,e^{S(E)-E/T},$ (5)
with corrections that are exponentially suppressed.
On the basis of the assumption (4) it is possible to show that the system
prepared in an energy eigenstate behaves like a thermal system. A few
particularly noteworthy properties are D’Alessio _et al._ (2016):
* •
The long-time average of an observable equals the thermal average:
$\overline{A}=\langle{\cal A}\rangle_{T}$.
* •
The quantum fluctuations of ${\cal A}$ are equal to the thermal fluctuations
with corrections of $O(1/N)$, where $N$ is the number of degrees of freedom of
the system.
* •
The time correlation function of finite-time expectation values $\langle{\cal
A}\rangle_{t}$ obeys a Kubo relation with the function $f(E,\omega)$ as the
spectral density.
These properties assure that the system, when monitored through the observable
${\cal A}$, is indistinguishable from a thermal system.
The Thouless energy $E_{\rm Th}$ is defined as the energy difference $\omega$
below which the factor $f(E,\omega)$ in (4) can be replaced with a constant
for most observables, and ETH reaches the RMT limit. In a dynamically evolving
system, the relevant range of energy differences $\omega$ is inversely related
to the evolution time by the uncertainty relation. It is often assumed that
the behavior of a chaotic quantum system for times longer than the Thouless
time $t_{\rm Th}=\hbar/E_{\rm Th}$ is described by RMT. However, it was found
in numerical studies of discrete quantum systems that the time until full ETH
behavior is established, $t_{\rm ETH}$, can be much longer, depending on the
operator under investigation Dymarsky (2019); Richter _et al._ (2020). In
fact, can be parametrically longer than $t_{\rm Th}$ by a factor proportional
to the system size.
The intuitive interpretation of this latter time scale is that the
establishment of quantum entanglement requires causal connection across the
whole system, and full entanglement is only reached asymptotically. The most
important prediction of ETH for a fully entangled system is that if only a
small part of the density operator enters an observable, i.e. if the trace is
taken over more than half of the states, the observable behaves as for a
system in contact with a heat bath. It is tempting to assume that this
property explains the experimental success of the thermal model in
relativistic heavy ion collisions mentioned in the Introduction.
The operator dependence of the onset of ETH behavior might actually allow us
to understand why the measured yield of hypertritons in p+Pb collisions
differs from the prediction of the thermal model. The question is whether the
time of hadronization, $t_{\rm H}$, is larger or smaller than $t_{\rm ETH}$.
As the hypertriton wave function is much more extended than that of the proton
one should expect a strong form-factor suppression of the corresponding matrix
element, resulting in an especially large $t_{\rm ETH}$. At the same time
$t_{\rm H}$ is especially small for the proton size fireball produced in p+Pb.
Both effects suggest that ETH-behavior has not yet established itself when
hypertritons are formed in p+Pb collisions, which would explain the observed
suppression compared to the thermal model.
The current consensus is that ETH applies to most, if not all, systems that
exhibit quantum chaos. This conjecture has been confirmed in tractable model
systems where precise numerical calculations are possible. It leads to the
question whether QCD exhibits chaos at the level of the full quantum field
theory. We know that nonabelian gauge theories are chaotic at the classical
level and exhibit ergodic properties Bolte _et al._ (2000). Unfortunately,
however, we are still unable to construct highly excited energy eigenstates of
a system governed by the laws of QCD and confirm the validity of ETH for such
systems. The validity of ETH for QCD thus remains a conjecture.
## IV Time scales in the collision
The phenomenology of thermalization in heavy ion collisions is characterized
by multiple different time scales. The initial state is very far from thermal
equilibrium, but the time available to reach it is limited by the rapid
longitudinal expansion of the created quark-gluon system. Hadronization, a
confinement-deconfinement transition, takes place during this short time span
and drastically changes the properties of the system. This transition is
usually modeled as an early ”chemical freezout”, to be distinguished from
complete kinematic freeze-out, which is assumed to occur later.
The very large momenta of the colliding nuclei imply large Lorentz factors
$\gamma$ in the center-of-mass (CM) system, up to $\gamma\sim O(1000)$.
Accordingly, the transverse extent of the colliding nuclei at the initial
collision instant in the CM system is much larger (by a factor $\gamma$) than
their longitudinal thickness. In addition, transverse fluctuations are
substantial, of order 50% for the energy density Müller and Schäfer (2012), on
length scales between the inverse saturation scale $1/Q_{s}\sim 0.2$ fm and
the nucleon radius (0.7 fm). This generates, in combination with the velocity
of light $c$ for ballistic processes or the velocity of sound
$v_{s}=c/\sqrt{3}$ for diffusive processes, additional time scales. The
various time scales pose a challenge to any holographic description spanning
the entire collision.
However, upon closer inspection one realizes that all these time and length
scales can be traced back to three primary scales related to the underlying
field theory (QCD), and the initial and spatial boundary conditions:
* •
The energy density $\varepsilon_{0}$ initially deposited in the collision; in
the glasma model $\varepsilon_{0}\sim Q_{s}^{4}$, where $Q_{s}$ is the gluon
saturation scale Lappi (2006).
* •
The initial transverse extent $R_{T}$ of the collision region, which depends
on the nuclear radii $R$ and the impact parameter $b$.
* •
The QCD confinement scale $\Lambda_{\rm QCD}$ reflected in the nucleon radius
$r_{N}$ and the pseudocritical temperature $T_{c}$.
There are three analogous independent scales that can be present in the
AdS/CFT dual description:
* •
The energy density $\varepsilon_{0}$ initially deposited in the shock wave
collision.
* •
The initial transverse extension $R_{T}$ of the collision region, which
depends on the transverse width of the shock waves and the impact parameter
$b$.
* •
In holographic models, which contain an extraneous length scale, this can set
the scale for a transition of the Hawking-Page type 333We refer to any
transition between AdS space with a black brane and thermal AdS space as
“Hawking-Page type” transition, independent of the mechanism that sets the
temperature scale at which the transition occurs. that marks the boundary
between a confined and a deconfined phase. For example, in the strong coupling
limit of the simplest holographic model dual to ${\cal N}=4$ super-Yang-Mills
theory Maldacena (1998a), such a transition occurs when the dual theory is
considered in global AdS space Hawking and Page (1983). Similar transitions
exist in more realistic holographic models of QCD even when considered on the
Poincaré patch.
When the dual description is constrained to the Poincaré patch of AdS space
Bayona and Braga (2007), as it is necessary to describe a heavy ion collision,
the third scale must be introduced by some modification of the holographic
dual that breaks the conformal symmetry, if hadronization is to be described.
Examples include models with imbedded D-branes Kruczenski _et al._ (2004);
Sakai and Sugimoto (2005) and the Scherk-Schwarz compactification of a dual
theory with an additional dimension considered by Witten Witten (1998) and
used in the study of Aharony et al. Aharony _et al._ (2006).
In recognition of these primary scales we can divide the progression of a
relativistic heavy ion collision into four stages, which are summarized in
Table 1.
time | HIC phenomenology | Holographic dual model
---|---|---
Stage I | Hydrodynamization | Numerical AdS simulations
$t\leq 1$ fm/c | hydrodynamic attractors lead from | entropy production mapped by apparent horizon
| transient large amplitude fluctuations | $t\leq 0.2$ fm/c equilibration without fluctuations
| to viscous hydrodynamic expansion | $t\leq 1-2$ fm/c equilibration with fluctuations
Stage II | Expansion of the QGP fireball | Collision of localized shocks Waeber and Yaffe (2022)
$t\leq O(15)$ fm/c | hadronization at QGP surface | (Sections V and VII)
Stage III | Bulk hadronization | Smoothed HP transition
$t\approx O(15)$ fm/c | chemical freeze-out | (Sections VII and IX)
Stage IV | Expansion of HRG | Entangled hadrons
$t\geq O(15)$ fm/c | kinetic freeze-out | network of ER bridges (Section VIII)
Table 1: The different stages of a heavy ion collision (HIC) and their
proposed holographic modelling.
We now explain the reasoning behind Table 1 and describe the roles the primary
scales play in the different stages.
Stage I: At very short times neither the confinement scale nor the transverse
extent of the reaction region are important. During this stage the local
energy density far exceeds the critical density $\varepsilon_{c}$ at the
deconfinement transition, and information about the finite transverse size has
not spread widely because of causality. The initial energy density
$\varepsilon_{0}$ is the only relevant scale both on the QFT and holographic
side of the duality. Both statements may not apply to very small collision
systems, such as those produced in proton-proton or proton-nucleus collisions.
Stage II: The resulting hydrodynamic QGP expands during the second stage.
During this expansion a small fraction of the QGP hadronizes on the surface of
the expanding fireball. By construction, this surface is at the confinement-
deconfinement transition temperature $T_{c}$, which is of the order of
$\Lambda_{\rm QCD}$. Thus, the expansion dynamics of the QGP fireball depends
on the two scales $\varepsilon_{0}$ and $R_{T}$, while the hadronization
processes at its surface depend, in addition, on $\Lambda_{QCD}$. Therefore,
the latter aspect cannot be described in the standard framework of AdS
modeling on the Poincaré patch, while the hydrodynamical expansion can.
Stage III: At the end of the second stage the remaining fireball becomes
thermodynamically and hydrodynamically unstable and completely hadronizes into
a quantum entangled hadron gas. The duration of this process depends on the
nature of the phase transition. In models with a very large number of colors
and a first-order transition the conversion from a gauge plasma to a hadron
(glueball) gas takes an extended period of time because the formation rate of
color singlet hadrons is small Aharony _et al._ (2006). In QCD, where the
transition is a rapid, but smooth crossover, the process is deemed to be so
fast that it is usually described as instantaneous. Because time evolution in
QCD is unitary no thermal entropy is produced, but ETH applies to this stage,
explaining the validity of the “thermal” model. (For this to apply the time
from the start of the collision until this third stage must be long enough for
global ETH behavior to be established Dymarsky (2019), which may not apply to
very small collision systems. Numerical simulations in the AdS Poincaré patch
are also unable to describe this stage adequately.)
The transition from (slow) surface hadronization to (rapid) bulk hadronization
means that the Page curve for a heavy ion collision has an asymmetric shape.
Before the final hadronization transition only a small fraction of the QGP (in
Section VI we will argue for roughly 20%) has decayed, while the rest decays
on a much shorter time scale during Stage III. Figure 3 shows a schematic
sketch of the Page curve for this scenario.
Figure 3: Schematic representation of the Page curve for the hadronizing
quark-gluon plasma in a relativistic heavy ion collision. The growth rate of
the entanglement entropy of the emitted hadrons is low during Stage II because
hadrons are only emitted from the surface. The growth rate then increases
quickly during Stage III due to bulk hadronization before dropping equally
rapidly after half of the QGP has hadronized. At the end of the hadronization
transition, the entropy again equals the initial entropy because all hadrons
are entangled with each other.
Stage IV: The final stage of a heavy ion collision, which features an
expanding cloud of entangled hadrons, resembles the final stage of a decaying
black hole which comprises an expanding cloud of entangled photons without a
black hole remnant at the center. Again, numerical simulations limited to the
Poincaré patch of AdS space are insufficient to capture this entanglement.
While each stage of a heavy ion collision has a well motivated holographic
dual, is is not clear how well holography can model a heavy ion collision in
quantitative terms. This is not a concern for the very early stage, when the
quark-gluon system is far away from the confinement scale, and a holographic
description can be justified by its approximate conformal symmetry and the
limited sensitivity of hydrodynamics to details of the initial state. It is
unclear whether tractable holographic models can provide for a quantitative
description at later times where more time scales are relevant.
Even if the holographic model cannot mirror a real heavy ion collision
quantitatively, in can potentially provide important conceptual insights with
parametric validity. As an example, we consider the final state when the whole
system consists of hadrons that are freely streaming toward the detectors.
There are two possibilities: Either the hadrons are in thermal equilibrium and
described by a density matrix with high entropy, which is the standard
assumption made in heavy ion physics, or the hadrons form a highly entangled,
nearly pure quantum state with very little entropy, as we argue here. In the
first case there exist well established techniques that permit detailed
theoretical predictions, whereas any such calculation requires additional
assumptions in the latter case. We will argue in the next Section, building on
work by van Raamsdonk et al. Van Raamsdonk (2020), that the holographic dual
of an entangled state of many hadrons differs only in subtle ways from the
holographic dual of a QGP fireball and that, therefore, a holographic
description can extend smoothly beyond the hadronization transition.
Another important question is whether the fireball reaches ETH-type behavior
before hadronization. This cannot be true for all observables, as the
following argument shows: As already noted, transverse variations in locally
conserved quantities equilibrate by diffusion. The associated time scale
$R_{T}^{2}/D$, where $D$ is the relevant diffusion constant in the QGP, is
identical to the one introduced in the previous Section as Thouless time
$t_{\rm Th}=1/E_{\rm th}$. It is useful to estimate this time constant for a
typical volume of QGP created in a heavy ion collision. The typical value of a
diffusion coefficient in QCD is $D\sim(\pi T)^{-1}$ Ding _et al._ (2012).
With $R_{T}\approx 5$ fm and $T\approx 300$ MeV one finds $t_{\rm Th}\sim 120$
fm/c, which is much longer than the lifetime of the QGP.
The crucial question in this context is, how long it takes until few-particle
observables that are amenable to experimental measurement are well
approximated by their thermal value. The time it takes for entanglement to
propagate throughout the fireball provides for a lower bound. Quantum
information propagates with velocity $v_{E}<c$ yielding this information time
scale to be of order $R_{T}/v_{E}$. At strong coupling holographic models in
$d=4$ dimensions give $v_{E}\approx 0.62c$ Liu and Suh (2014), which means
that the speed of information transport is similar to that for energy
transport, which is given by the speed of sound $c_{s}=c/\sqrt{3}$. This means
that entanglement can spread through the entire fireball before the onset of
hadronization.
In this work we discuss certain ideas that may ultimately make it possible to
extend the successful holographic description of heavy ion collisions beyond
the Page time. The crucial feature of any such attempt is the preservation of
full quantum coherence which, in turn, requires the addition of novel concepts
to the holographic description that become relevant at later times in the
collision. As we discussed above, this implies that numerical simulations of
AdS gravity must be expanded to cover the global AdS geometry, not just the
Poincaré patch.
The underlying idea is that a heavy ion collision proceeds from the highly
entangled many-body wave functions of the colliding nuclei to a highly
entangled QGP which evolves continuously into a highly entangled multi-hadron
state but never to a truly thermal state with large von Neumann entropy. In
other words, the collision can be viewed as a unitary S-matrix mapping between
the initial and final hadron states. In this language, it is completely
irrelevant that in some intermediate state certain local observables look
thermal. It is ultimately the interaction of the emitted hadrons with the
detectors, which act as incoherent environment, that creates the thermal
entropy. This view implies that the establishment of full ETH behavior and
thus the success of the thermal model is far less of a mystery, because
$t_{\rm Th}$ is much shorter than the time that elapses until final-state
hadrons hit the first detectors.
Elucidating the microscopic processes which govern thermalization of many-
particle quantum systems is relevant for many subfields of physics. For
example, in Popescu _et al._ (2006) it was argued that thermodynamics can be
based on the assumption of a pure state wave function of the universe with
which all systems investigated in the laboratory are entangled. Clearly, such
an assumption requires experimental verification. HICs involving two nuclei
colliding in the ultra-high vacuum of the beam-pipe, may be the “cleanest”
system to obtain such verification or refutation as the evolution of the
system can be followed without environmental influence over times many orders
of magnitude longer than the intrinsic time scales and with a dynamics which
is completely defined by QCD. A huge amount of precise data has already been
accumulated and is readily available for theoretical analysis.
It is not clear whether measurements that can detect the presence of complex
entanglement features among the emitted hadrons are feasible in heavy ion
collisions, just as it is unlikely that a practical test of the entanglement
pattern of Hawking radiation could be conducted even if we had access to an
isolated evaporating black hole. Examples of observables that might be both,
sensitive to the entanglement properties of the emitting system and amenable
to measurement, are quantum correlations among hadron spins, especially in
small collision systems Gong _et al._ (2022), or among isospins, so-called
disoriented chiral condensates Anselm and Ryskin (1997); Mohanty and Serreau
(2005), for which possible indications have been recently observed Acharya
_et al._ (2022b).
There is hope that such experiments are possible with atomic physics analogues
of black holes Steinhauer (2016); Kolobov _et al._ (2021). As QED is also
time reversal invariant, QED and QCD processes should have similar information
theoretical properties and, therefore, the experimental verification of ETH
behavior and Hawking radiation for QED systems suggest that QCD systems behave
in a similar manner. However, while these insights are intellectually
satisfying, their practical utility depends on whether or not a holographic
description of entanglement and ETH behavior is feasible in practice. This
will be discussed in Section VII.
## V Holographic Simulations of Early Collision Stages
Starting with the pioneering papers Chesler and Yaffe (2009, 2011, 2014)
numerical AdS calculations have played an ever more important role to inform
our understanding of the early phase of high-energy heavy ion collisions. Over
time the numerical techniques have been steadily improved, such that today
realistic three-dimensional collisions can be studied Waeber and Yaffe (2022)
instead of somewhat schematic collisions of infinite plane shock waves. Also,
more and more details and special cases were studied Heller _et al._ (2012b);
Casalderrey-Solana _et al._ (2014); Chesler _et al._ (2015); Ecker _et al._
(2016); Casalderrey-Solana _et al._ (2016); Endrodi _et al._ (2018); Waeber
_et al._ (2019); Müller _et al._ (2020). In parallel, calculations of the
entropy density Gubser _et al._ (1998), shear viscosity Buchel (2008),
conductivity Waeber and Schäfer (2018) and inverse equilibration times Waeber
and Schäfer (2018) at NLO in the string coupling were found to be in good
agreement with QGP phenomenology. As these corrections can be encoded in
higher derivative terms of classical gravity, there is a possible path to more
quantitative holographic simulations of the early collision stages.
Numerical solutions of the classical Einstein equations with AdS boundary
conditions share the difficult problem of having to deal with the
diffeomorphism invariance of general relativity. Using a suitable metric as an
ansatz one can significantly reduce the diffeomorphism freedom. Chesler and
Yaffe used Eddington-Finkelstein coordinates and managed to rewrite the
Einstein equation as a system of nested ordinary differential equations, which
they solved using functional methods (Chebychev functions). A nice review of
early developments can be found in Chesler and van der Schee (2015). In recent
years activity has somewhat abated although there is still continuous
progress, see e.g. Waeber and Yaffe (2022). Already early on these simulations
showed that AdS dynamics leads to rapid hydrodynamization van der Schee _et
al._ (2013); Chesler _et al._ (2015) with sizeable transverse flow Chesler
and Yaffe (2015).
The prevailing strategy has been to merge these holographic simulations of the
early stages of heavy ion collisions with statistical descriptions of the
later stages of the collisions van der Schee _et al._ (2013). Early
calculations focused on highly symmetric settings in order to reduce the
computational demands, but more recently asymmetric settings have been
studied, and by now semi-realistic heavy ion collisions can be simulated
Heller _et al._ (2012b); Chesler _et al._ (2015); Ecker _et al._ (2016);
Casalderrey-Solana _et al._ (2016); Endrodi _et al._ (2018); Waeber _et
al._ (2019); Müller _et al._ (2020). In particular, these studies have shown
that holographic simulations reproduce relativistic viscous hydrodynamics,
validating the hybrid holographic-transport approach, see Schenke (2021) for a
recent review. In the present context it is relevant that hydrodynamization
occurs at a fixed proper time Waeber _et al._ (2019) and that AdS dynamics
and viscous hydrodynamics are indistinguishable for hydrodynamic observables
close to hydrodynamization such that the whole QGP fireball shows hydrodynamic
behavior at nearly the same proper time.
Holographic calculations are attractive because they allow to treat problems
that are too difficult to solve using QFT techniques. The hope is that the
successful application of holographic techniques, which has been realized for
the early stages of heavy ion collisions, can be extended to the later
collision stages. This hope is nurtured by several observations documented in
the literature, which we will discuss now.
The insight that quantum entanglement in the field theory is holographically
encoded in features of the higher dimensional geometry was most clearly
expressed by Maldacena and Susskind Maldacena and Susskind (2013) who argued,
building on ideas formulated in Israel (1976); Maldacena (2003), that the
maximally extended (eternal) AdS-BH geometry is dual to a thermofield double
state in the quantum field theory, which means that the quantum states of the
field theory in the two asymptotic regions are maximally entangled. More
generally, in this picture, the entanglement between the quantum states in two
different space-time regions is geometrically represented by an Einstein-Rosen
(ER) bridge between two regions in AdS space. We refer the reader to Almheiri
_et al._ (2020b) for a review of the many publications building on this
paradigm and as well as its relation to recent progress in understanding the
role of entanglement for BH decay by Hawking radiation.
In Maldacena and Susskind (2013) the label “ER = EPR” was coined for this
connection, where EPR stands for Einstein-Podolsky-Rosen and, more generally,
for any form of entanglement in the quantum field theory. For our discussion
Fig. 13 in Maldacena and Susskind (2013) is most stimulating as it suggests
that the emission of entangled Hawking radiation can be described as creation
of higher dimensional Einstein-Rosen bridges between the decaying BH and the
emitted particles. This picture, however, also displays an obvious problem of
this idea: In a high-energy heavy ion collison thousands of hadrons are
produced allowing for a plethora of entanglement patterns, far more than the
number of simple geometries available for the dual picture.
In this context the recent ideas advanced by van Raamsdonk et al. Van
Raamsdonk (2017, 2020, 2021); Raamsdonk and Waddell (2021); May and Van
Raamsdonk (2021) are relevant, which posit that one large connected region of
conformal field theory (CFT) and several smaller entangled domains of the same
CFT have a nearly indistinguishable holographic dual. If it were legitimate to
interpret the large region as QGP fireball and the many small domains as
individual hadrons this would imply that the crossover transition from QGP to
hadron gas could be continuous in a holographic dual description. This notion
instills hope that a holographic description of the hadronization process is
feasible and practical.444Our concept inverts van Raamdonk’s reasoning. His
aim was to show that continuous space can emerge from the entanglement of
isolated domains; we want to describe entanglement among isolated components
of the final state geometrically. The dual description would be the
disintegration of one large AdS black hole into several small ones as
discussed, e.g., in Maldacena and Susskind (2013) in the contact of black hole
evaporation.
The analogy is now clear. Quarks and gluons are confined to the interior of
hadrons. Therefore, a diluted gas of thousands of hadrons can be interpreted
in analogy to many isolated conformal field theories (CFT) on the edge. As
each of these covers only a small faction of the fireball or expanding hadron
cloud, in the bulk AdS direction this leads only to noticeable effects very
close to the asymptotic region of AdS space. (In order to distinguish this
asymptotic region from spatial boundaries of localized states, we will use the
term “AdS edge”. Such localized systems are usually modeled in the framework
of boundary conformal field theory (BCFT) and its holographic dual description
Takayanagi (2011).) Deeper inside the bulk, all hadronic regions will remain
connected via a network of ER bridges. As a result, the dual geometry will be
very similar to that of the QGP fireball which is just one large AdS-BH. This
means that from the information theoretical point of view there is no marked
difference between entangled quarks and gluons with large relative momentum in
the early fireball phase, a mixed QGP plus hadron gas state at intermediary
times, and an entangled purely hadronic state at late times and that,
therefore, many properties of this system can be calculated from the dual
description valid at early times. In Sections VII and VIII we will expand on
this from the AdS perspective.
To summarize, we argued that a successful extension of AdS calculations to the
description of hadronization requires their extension from the Poincaré patch
to the global AdS geometry and the inclusion of the scale $L$. We will suggest
in Sections VII and VIII simplified models as first steps towards this long-
term goal.
## VI Hadronization in Relativistic Heavy Ion Collisions
At very high energies, the valence quarks of two colliding nuclei effectively
pass through each other and deposit some of their energy on a time scale much
shorter than 1 fm/c, a process that can be modeled as a quantum quench.
Following this quench. the deposited energy thermalizes and forms a quark-
gluon plasma on a time scale of the order $\tau_{\rm th}\sim O(1/T)$ where $T$
denotes the temperature after thermalization. The thermalization process can
be studied in holographic models, either by energy shell collapse in
5-dimensional Anti-de Sitter space (AdS5) Balasubramanian _et al._ (2011a, b)
or by numerically solving shock front collisions in AdS5 Chesler and Yaffe
(2011, 2014).
The quark-gluon plasma then expands hydrodynamically while hadronization
occurs in regions where the temperature reaches $T_{c}\approx 150$ MeV. At
high collision energies the evolution of the matter near the center of
momentum of the two nuclei is approximately boost invariant and can be
described in terms of Milne coordinates $\tau,\eta,x,y$, where
$\tau=\sqrt{t^{2}-z^{2}}$ is the proper time and
$\eta=\frac{1}{2}\ln[(t+z)/(t-z)]$ is called space-time rapidity. Massless
particles moving along lines of constant $\eta$ also have rapidity $\eta$ in
momentum space. Boost invariance means that to a good approximation the
evolution is a function of $\tau$ and the transverse coordinates $x,y$ only
for moderate values $|\eta|\ll\eta_{\rm beam}$, where $\eta_{\rm beam}$ is the
beam rapidity in the center-of-momentum frame.
Hydrodynamics simulations show that the hypersurface where the transition from
quark-gluon plasma (deconfined matter) to hadrons (confined matter) occurs is
composed of two main domains (see Fig. 4). One domain is time-like and located
approximately at constant $\sqrt{x^{2}+y^{2}}\equiv r\approx R$, where $R$ is
the nuclear radius, and stretches from $\tau_{\rm th}$ to a time $\tau_{c}$
when the quark-gluon plasma converts to hadrons in bulk. The bulk
hadronization proceeds along a space-like hypersurface given by $r<R$ and
$\tau=\tau_{c}$, which defines the second domain.
Figure 4: Contour plot of the evolution of the energy density in a midcentral
Au+Au collision at the highest RHIC energy $\sqrt{s_{\rm NN}}=200$ GeV. The
horizontal axis shows the proper time $\tau$, the vertical axis shows one of
the transverse coordinates. The black dashed line delineates the hadronization
hypersurface $T=T_{c}=150$ MeV Gale _et al._ (2021).
The hadronization processes in these two domains are different: On the time-
like boundary of the quark-gluon plasma, hadron emission can be visualized as
surface radiation. The emission rate is given by
$dE_{\rm had}^{\rm surface}/d\tau=\int dx\,dy\,\tau
d\eta\,\delta(r-R)\,T^{0i}n_{i},$ (6)
where $n_{i}$ is the outward directed (space-like) normal vector on the
hadronization hypersurface. Introducing the notation $S_{c}=T^{0i}n_{i}$ for
the energy flow density at the hadronization temperature $T_{c}$, one finds
$dE_{\rm had}^{\rm surface}/d\eta=2\pi R\int_{\tau_{\rm
th}}^{\tau_{c}}\tau\,d\tau\,S_{c}\approx\pi R\,\tau_{c}^{2}\,S_{c},$ (7)
since $\tau_{c}\gg\tau_{\rm th}$. The energy converted into hadrons on the
space-like hadronization surface is similarly given by
$dE_{\rm had}^{\rm
bulk}/d\eta=2\pi\int_{0}^{R}r\,dr\,\tau_{c}\,\varepsilon_{c}\approx\pi
R^{2}\,\tau_{c}\,\varepsilon_{c},$ (8)
where $\varepsilon_{c}$ is the energy density of matter at temperature
$T_{c}$.
For noninteracting hadrons with mass $m$ at temperature $T=1/\beta$,
neglecting quantum statistics, one has
$\displaystyle S(T,m)$ $\displaystyle=$
$\displaystyle\frac{T^{4}}{4\pi^{2}}\left(3(1+\beta m)+(\beta
m)^{2}\right)e^{-\beta m}$ (9) $\displaystyle\varepsilon(T,m)$
$\displaystyle=$ $\displaystyle\frac{T^{4}}{2\pi^{2}}\left(3(\beta
m)^{2}K_{2}(\beta m)+(\beta m)^{3}K_{1}(\beta m)\right).$
When summed over all well established hadron species in the Particle Data Book
weighted by their statistical degeneracies $d_{i}$, one finds
$\frac{S_{c}}{\varepsilon_{c}}\equiv\frac{\sum_{i}d_{i}S(T_{c},m_{i})}{\sum_{i}d_{i}\varepsilon(T_{c},m_{i})}\approx
0.17.$ (10)
Putting everything together and considering that $\tau_{c}\approx R$, one
obtains
$\frac{dE_{\rm had}^{\rm surface}/d\eta}{dE_{\rm had}^{\rm
bulk}/d\eta}\approx\frac{\tau_{c}\,S_{c}}{R\,\varepsilon_{c}}\approx
0.15-0.20.$ (11)
This means that approximately $80-85$% of the hadrons produced by
hadronization of the quark-gluon plasma are created during the bulk
transition.555As some regions at the fringe of the nuclear fireball may never
become hot enough for deconfinement to occur, a small fraction of the final
state hadrons may be produced directly without going through an intervening
plasma phase. This phenomenon is described in so-called core-corona models
Werner (2007). The relative magnitude of the corona contribution to hadron
production shrinks with increasing size of the collision region and increasing
collision energy Petrovici _et al._ (2017).
Figure 5: Intensity plots of the emission hypersurface for hadron pairs in
Au+Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV (from Plumberg and Heinz
(2015)). Upper panel: Hadron pairs with $K_{T}=0$, selectively weighting
hadrons from bulk hadronization. Lower panel: Hadron pairs with $K_{T}=2$
GeV/c, selectively weighting hadrons from surface radiation. The emission
intensity is highest in the dark red regions and lowest in the dark blue
regions.
The two hadronization mechanisms can be experimentally separated by measuring
the emission of hadron pairs instead of single hadrons. Hadron pairs radiated
off the surface of the outward flowing fireball carry a large outward directed
total momentum $K_{T}=p_{T,1}+p_{T,2}$. On the other hand, hadron pairs formed
when the QGP hadronizes in bulk have on average $K_{T}=0$. This is nicely seen
in Fig. 5. where the emission hypersurface of hadron pairs at midrapidity for
the same Au+Au collisions at RHIC is shown in the $(x,y,t)$ space, where $x,y$
are the transverse coordinates Plumberg and Heinz (2015). Just as in Fig. 4
the hypersurface resembles a capped cylinder. The upper panel shows the
emission points of hadron pairs with $K_{T}=0$; the lower panel shows the
emission points with $K_{T}=2$ GeV/c. Red color indicates the highest rate of
emission; blue color indicates the lowest emission rate.
## VII Hadronization as an AdS phase transition
The Einstein equations with a negative cosmological constant have two
solutions with asymptotic AdS geometry. One is plain AdS space; the other is a
black hole (AdS-BH) imbedded in AdS space. The black hole geometry is
parametrized by the Schwarzschild horizon (we will use the notation $r_{h}$).
As we shall discuss, $r_{h}$ is related to the color screening distance in the
dual gauge theory. At thermal equilibrium, $r_{h}$ is uniquely determined by
the temperature $T=1/\beta$ encoded in the period of the Euclidean version of
the geometry Gibbons and Hawking (1977).
The Euclidean action plays the role of free energy in the space of geometries;
its minimum thus corresponds to the stable equilibrium state at a given
temperature. At low values of $T$, plain AdS space ($r_{h}=0$) has the lowest
Euclidean action; above a certain temperature $T_{c}$, which depends on the
details of the dual gravity theory, the AdS-BH geometry has the lowest free
energy. As mentioned in a footnote in Section IV, we here refer to such
transitions as “Hawking-Page type” transitions Hawking and Page (1983). In the
simplest holographic model, the pure $(d+1)$-dimensional Einstein action with
negative cosmological constant $\Lambda=-d/L^{2}$ on global AdS space, the
transition is discontinuous with $r_{h}(T_{c})=L$, which means that the
Hawking-Page transition is a first-order phase transition. For a detailed
derivation and the exact relation between $r_{h}$ and $T$, see e. g. Ref.
Witten (1998).
In the dual gauge theory, the Hawking-Page transition corresponds to the
confinement-deconfinement transition Witten (1998); Aharony _et al._ (2004).
This can be seen in multiple ways. The most commonly used argument is that the
free energy of the thermal plain AdS geometry is described by a Hagedorn
spectrum of gauge-singlet excitations, whereas the free energy of the AdS-BH
geometry is proportional to $N_{c}^{2}$. Maybe the most intuitive argument is
obtained by considering the potential of a heavy quark-antiquark pair, i. e.,
static objects carrying color charge in the fundamental representation of the
gauge group Maldacena (1998b). In the gauge theory, this potential is
determined by a Wilson loop connecting the world lines of the quark-antiquark
pair. In the holographic dual representation the potential is determined by
the Nambu-Goto action of a string connecting the quark-antiquark pair through
the bulk (see left panel of Fig. 6).
Figure 6: String configurations in the AdS bulk connecting a static quark-
antiquark pair separated by distance $\ell$. The left panel shows the string
in the plain AdS geometry; the right panel shows the two disconnected strings
reaching from the quark (antiquark) to the black hole horizon $r_{h}$ in the
AdS-BH geometry.
For the conformally invariant ${\cal N}=4$, large-$N_{c}$ super-Yang-Mills
theory and for the plain AdS geometry in the bulk, the potential is found to
be Maldacena (1998b)
$U(\ell)=-\frac{4\pi^{2}\sqrt{2\lambda}}{\Gamma(1/4)^{4}\,\ell},$ (12)
where $\lambda=g^{2}N_{c}$ is the ’t Hooft coupling and $\ell$ is the quark-
antiquark separation. (There is no confining potential because the theory is
conformally invariant.) For the AdS-BH geometry one finds that the potential
$U(\ell)$ vanishes for separations
$\ell>\ell_{s}\approx 0.869L^{2}/r_{h},$ (13)
which defines the screening distance of the color force in the gauge theory
Rey _et al._ (1998); Brandhuber _et al._ (1998); Liu _et al._ (2007). The
screening occurs, because for $\ell>\ell_{s}$ the lowest energy string
configuration corresponds to a disconnected pair of strings stretching
straight from the quark (antiquark) to the black hole horizon $r_{h}$ (shown
in the right panel of Fig. 6). We can thus consider $r_{h}$ as a geometric
parameter related to the color screening length in the gauge theory. Since for
$T\gg T_{c}$ the black hole radius is related to the temperature $T$ at
thermal equilibrium by $r_{h}=(4\pi/n)L^{2}T$, this translates into a thermal
color screening length $\ell_{s}\approx 0.869/(\pi T)$in $d=4$ space-time
dimensions.
It is possible to relax the firm connection between AdS-BH radius $r_{h}$ and
the temperature $T$ by allowing for geometries with a conical singularity Eune
_et al._ (2013). Such geometries do not correspond to a minimum of the free
energy but they allow for a smooth interpolation between the equilibrium AdS-
BH geometry and the thermal AdS geometry. This is illustrated in Fig. 7 for
AdS5. The solid lines show the free energy $F(r_{h},T)$ for several different
values of $T$, and the dashed line traces the location of the minima. The
uppermost solid (red) curve is for the lowest temperature $T_{\rm min}$ for
which an AdS-BH solution of Einstein’s equations exists, $T_{\rm
min}=\sqrt{2}/(\pi L)$, the middle solid (blue) curve represents the free
energy at the temperature of the Hawking-Page phase transition, $T_{c}=3/(2\pi
L)$, and the lowest solid (black) curve shows the free energy for
$T=1.2T_{c}$.
Figure 7: The solid curves show the free energy $F(r_{h},T)$ for AdS5 as
function of the black hole radius $r_{h}$ for three different temperatures
$T$: $T=T_{\rm min}$ (red upper curve), $T=T_{c}$ (blue middle curve), and
$T=1.2T_{c}$ (black lower curve). See text for the definition of $T_{\rm min}$
and $T_{c}$. The dashed line shows the equilibrium (on-shell) free energy
$F(r_{h},T_{0}(r_{h}))$ which traces out the extrema in the family of free
energy curves. The free energy is shown in units of $L/G$ where $G$ is the
gravitational constant.
As is evident from Fig. 7, the Hawking-Page transition for the conformal,
large-$N_{c}$ super-Yang-Mills theory is a first-order phase transition. This
is mirrored in the behavior of the pure non-supersymmetric SU($N_{c}$) gauge
theory, which also exhibits a first-order deconfinement transition for
$N_{c}\geq 3$ Panero (2009). Gravity dual models that resemble pure
SU($N_{c}$) gauge theory more quantitatively can be constructed by adding a
dilaton field to 5-dimensional Einstein gravity, which describes the running
of the gauge coupling constant. In general, dilaton gravity duals of confining
gauge theories exhibit a first-order deconfinement phase transition similar to
that of the conformal super-Yang-Mills theory Gürsoy _et al._ (2009a).
On the other hand, the deconfinement transition in QCD is known to be a smooth
crossover Bazavov _et al._ (2019); Borsanyi _et al._ (2020). The thermal
properties of dual dilaton gravity models including fundamental matter
(quarks) that mimic the running coupling and chiral properties of QCD have
been studied Mandal and Morita (2011), and models that can change from a
first-order phase transition to a crossover transition have been constructed
Attems _et al._ (2018).
The dynamics of the bulk transition depends on the rate at which the
temperature drops. If the cooling rate is slow compared with the microscopic
times scales, the transition will proceed at or near $T_{c}$ through a mixed
phase via bubble formation Aharony _et al._ (2006); Bigazzi _et al._ (2020);
Janik _et al._ (2021). In the case of rapid cooling, the transition occurs
via a Gregory-Laflamme instability Gregory and Laflamme (1993) from a
supercooled phase at $T_{\rm min}$ Hubeny and Rangamani (2002); Mandal and
Morita (2011); Buchel and Lehner (2015); Dias _et al._ (2016); Yaffe (2018).
The second scenario prevails in the limit $N_{c}\to\infty$, but for
holographic models of QCD with $N_{c}=3$ it is likely that the first scenario
is realized under the conditions of a heavy ion collision. For a smooth
crossover certainly the first scenario is realized.
## VIII Hadron emission as analogue of Hawking radiation
Decades of work by many theorists were needed to reach the present level of
understanding of the black hole information puzzle. Understanding the
mechanisms ruling the generation of entanglement in hadronization is most
probably a problem of comparable difficulty. Therefore, it is tempting to
profit from the insights of the black hole community by constructing a model
which treats hadronization in analogy to Hawking radiation. In this analogy
the hadrons correspond to the Hawking radiation outside of the black hole
horizon as hadrons cannot exist within the QGP, so its surface presents a
horizon for them. Photon pair creation at the black hole horizon corresponds
to hadronic particle-hole production at the QGP surface where the ingoing hole
state gets absorbed but transfers its entanglement with the outgoing hadron to
the interior of the (shrinking) QGP fireball in analogy to island formation in
black hole decay Almheiri _et al._ (2021), see the sketch in Fig.8.
Figure 8: Sketch of the transition from a highly entangled QGP fireball to a
highly entangled hadron gas based on the fact that the time-reversal
invariance of QCD forbids creation of von Neumann entropy.
The main idea motivating this analogy is that black hole decay leads to a Page
curve for the entropy of Hawking radiation outside of the horizon which is
what we also expect for the hadronic state outside of the QGP. Complete
entanglement among the hadrons is only reached at the end of hadronization.
There exist various phenomenological observations on the QCD side which could
fit into our model. As explained in the Introduction the uniform temperature
of all observed hadron yields, which is well described by the thermal hadron
resonance gas model, is difficult to understand. A possible line of argument
was suggested by us in Ref. Müller and Schäfer (2017) based on the
phenomenologically very successful quark recombination model Fries _et al._
(2008). In this model quarks and gluons from the fireball coalesce at its
surface to form hadrons. In doing so their energies $E_{i}$ add up and their
probability densities multiply, generating for any Fock-state of the hadron
$h$ with energy $E_{h}$ the common factor Müller _et al._ (2005)
$\prod_{i}e^{-E_{i}/T_{\rm ch}}=e^{-(\sum_{i}E_{i})/T_{\rm
ch}}=e^{-E_{h}/T_{\rm ch}},$ (14)
where the subscript “ch” indicates that the temperature parameter is derived
from the chemical composition of the hadron gas.
An ad hoc feature of this model, dictated by phenomenology, is that the
temperature of the QGP fireball and that of all produced hadrons has the same
value $T_{\rm ch}$. This feature is not easily understood without
entanglement, because different hadrons scatter differently and thus are not
expected to decouple at the same time from the expanding fireball. This could
be naturally explained if a highly entangled QGP state and a highly entangled
hadronic state are basically indistinguishable as we argued above based on the
ideas of van Raamsdonk Van Raamsdonk (2020).
## IX A holographic picture of hadronization
As long as information about the state of a time reversal invariant quantum
system is not lost by any kind of measurement and the associated (partial)
collapse of the complete many-body wave function, it must be possible “to run
the movie backwards”. The highly complex initial state $\Phi_{\mathrm{i}}$ of
a relativistic heavy ion collision has nearly zero entropy.666The ground state
of a colliding nucleus is unique, and any interaction of the nucleus with the
accelerator structure is completely negligible on the scale of nuclear
excitations. Finally, although the two nuclei can be Coulomb excited on their
approach to each other, the excitation is coherent and does not change the
fact that the nuclear quantum state is pure. In the parton (quark-gluon)
basis, this state is characterized by a density matrix with two blocks
describing the two nuclei approaching each other. This entangled initial state
evolves by a unitary transformation into another highly entangled final state
$\Phi_{\mathrm{f}}$ characterized by a full density matrix in the parton
basis. Eventually, this final many-parton quantum state is projected onto
hadron states by experimental measurements, which identify the asymptotic
eigenstates of the many-parton system. This happens at a time of order
$O(10^{-9}~{}{\rm s})$, much longer than the duration of the nuclear reaction
which is of order $O(10^{-23}~{}{\rm s})$. Since the detector acts as a heat
bath, this leads to decoherence and thus to entropy production.
For a realistic holographic description of a relativistic heavy ion collision
that includes hadronization it is crucial to describe entanglement at each
time. This is not possible in numerical solutions of the classical Einstein
equations, which depend only on the classical, local energy momentum tensor on
the AdS edge.777We remind the reader that use the term “AdS edge” to avoid
confusion of the asymptotic region of AdS space with the boundary of the QCD
fireball. It is thus unclear whether an approximately valid AdS model that
keeps track of entanglement during a transition of the Hawking-Page type
exists. We argue in this Section that it does.
Our argument rests crucially on the fact that for a many-particle quantum
state, such as the final state of a heavy ion collision, the effects of
entanglement are only relevant if the complete wave function is considered,
but are negligible for observables involving only a few hadrons. This is a
consequence of the ”monogamy of entanglement” which is well established in
quantum information theory.
Lots of text
Figure 9: Illustration of the different stages of a high energy heavy ion
collison. A) After the QCD fireball is formed hadrons are emitted from its
surface in analogy to Hawking radiation. B) This leads to volume growth of the
whole QCD system on the edge. In parallel the fireball cools, e.g. the dual
black hole sinks deeper into the AdS throat. However, the 3-dimensional volume
of the QGP fireball and thus the 3-dimensional volume of the AdS BH remains
roughly constant, see Fig. 4. C) When the temperature of the fireball, which
is identical to that of the AdS black hole, reaches $T_{c}$ the Hawking-Page-
like transition occurs corresponding to complete hadronization of the
remaining fireball on the AdS edge. The picture illustrates the moment of the
transition. D) Due to the monogamy of entanglement any pair of hadrons in the
state after the hadronization transition can only share a very small fraction
of entanglement, on average proportional to $O(1/N_{h})$ with $N_{h}$ being
the number of hadrons. Figure 10: E) As the Hawking-Page-like transition for
QCD at small baryon number density is a cross-over the difference between the
situation just before (at $T=T_{c}+\epsilon$, see Fig. 9C) and after (at
$T=T_{c}-\epsilon$, see Fig. 9D) is small. Therefore, E), which is strictly
valid only above $T_{c}$, is expected to provide a good approximation. Note
that because hadrons on the AdS edge are maximally entangled with states on
the horizon, they are uncorrelated with all other edge hadrons such that all
of them form a thermal ensemble with temperature $T_{c}$. This is just one of
the crucial elements of ETH. Note also, that the difference in this respect to
Fig. 9D is only of order $O(1/N_{h})$ and thus negligible. For more
discussions, please see the main text.
There exists a fundamental constraint, closely related to the no-cloning
theorem Wootters and Zurek (1982), which states that quantum entanglement
cannot be freely shared among many objects Wootters (1998); Coffman _et al._
(2000); Osborne and Verstraete (2006). This has been analyzed in detail for
systems of qubits for which quantitatively precise statements can be made and
formal proofs are possible. One can define a quantity $\tau(\rho_{AB})$,
called ”tangle”, which quantifies entanglement between the elements of bi-
partitions of multi-particle quantum states $A$, $B$, described by a density
matrix $\rho_{AB}$. $\tau$ can have values between 0 (no entanglement) and 1
(complete entanglement). For a quantum state of $n$ subsystems $A_{1}$,
$A_{2}$, …$A_{n}$ the following constraint holds Osborne and Verstraete
(2006):
$\sum_{k=2}^{n}\tau(\rho_{A_{1},A_{k}})\leq\tau(\rho_{A_{1},(A_{2}A_{3}...A_{n})})\leq
1.$ (15)
We identify the $A_{i}$ in our case with the $N_{h}$ individual hadron states
and assume that a bound of the type (15) exists that limits the average
entanglement between any two hadrons to a value of order $O(1/N_{h})$. We will
use this argument only in a very generic manner, arguing that for, e.g. a
heavy ion collision at LHC, in which thousands of hadrons are produced, the
effects of entanglement are negligible as long as only few-hadron observables
are measured.
As we have argued in Section VI heavy ion phenomenology provides compelling
evidence for the existence of two hadronization mechanisms: the emission of
individual hadrons from the QGP fireball surface ($\sim 20$%) and the
instantaneous hadronization of the remaining fireball at $T_{c}$ ($\sim 80$%).
Both mechanisms are characterized by the same temperature $T_{c}$. We
described specific models in the AdS/CFT context in Section VII and VIII. In
this Section we combine these models to an overall scenario.
On one hand, the presence of two mechanisms with quite different time
dependence complicates our endeavour compared with the usual black hole
information puzzle. On the other hand, we can make use of detailed
phenomenological knowledge based on a huge amount of high-precision data
amassed by heavy ion experiments. We hope that this advantage outweighs the
disadvantages.
There exist many concepts in the literature which could be adapted to our
situation. For example, Maldacena and Susskind suggested in Maldacena and
Susskind (2013) (see Fig. 13) a geometric, holographic interpretation of the
entanglement of Hawking radiation in terms of ER bridges for a Minkowski space
black hole, which is somewhat similar to ours. The difference between their
picture and ours is that we do not study a horizon in the edge but in the bulk
of AdS space and that we treat hadrons as entangled with the AdS horizon dual
to the QGP fireball rather than photons as entangled with a black hole
horizon.
Figures 9 and 10 present conceptual illustrations of our ideas. We will
outline in Section X some steps that could be taken to make these more
quantitative.
As shown in Fig. 4, the transverse size of the QGP fireball stays nearly the
same as function of time, as the internal cooling caused by the hydrodynamic
expansion is nearly compensated by evaporation from the surface. The surface
temperature of the QGP fireball stays fixed at $T_{c}$ during this period
while the volume of the hadron resonance gas (HRG) outside of the QGP fireball
grows continuously. When the QGP fireball temperature reaches $T_{c}$
throughout, the remaining volume hadronizes quickly. In the AdS dual
description, the spatially bounded region on the AdS edge corresponding to the
QGP fireball extends into the AdS bulk until it reaches an also spatially
bounded BH horizon. It appears plausible that this region in AdS space, which
is schematically depicted as a cylinder in Fig. 9 A), should be modeled as the
holographic dual of a BCFT Takayanagi (2011). The emitted hadrons are
entangled with states on the BH horizon via ER bridges, as illustrated in Fig.
13 of Ref. Maldacena and Susskind (2013), such that no entropy is produced.
As discussed in Section I, hadronization in heavy ion collisions is usually
modeled as “chemical freeze out”, which is hypothesized to explain the
observation that all hadron yields agree with the predictions of the thermal
model although hadronic interactions continue in the expanding hadron gas. As
we argue that the success of the thermal model is a consequence of quantum
coherence, e. g. in the form of ETH, we do not need any such additional
mechanism. However, even our mechanism requires a rapid transition from the
hydrodynamic QGP phase to the hadronic phase. In fact, microscopic simulations
indicate a rapid growth of the specific viscosity at temperatures below 150
MeV signalling a rapid breakdown of hydrodynamics when the expanding fireball
cools below $T_{c}$ Yang and Fries (2022).
Holographically, this transition corresponds to the transition between Figs. 9
B and D, where Fig.9 C is a schematic representation of an intermediate stage.
We do not yet have a reliable quantitative description of these intermediate
stages, neither in QCD nor in holography, but one future goal stated in
Section X is to develop such a description in the holographic model of the
transition.
With respect to phenomenology, however, we can argue that the mapping to a
purely hadronic late stage should be a smooth one, i. e. a cross-over rather
than a first order phase transition, in line with the fact that the
deconfinement/confinement phase transition of QCD is a cross-over.
We conclude this Section with the following remark: If a holographic dual of
QCD exists, it must be possible to develop a quantitative description of the
hadronization of a quark-gluon plasma. At present we do not know whether this
is the case. However, even if if it does not or turns out to be unachievable
in practice, much could be learned about the quantum physics of the
confinement transition by investigation of a holographic model that
incorporates salient features of the quark-hadron transition in QCD. In the
next Section we will discuss some steps that can be taken in this direction.
## X (Some) Open Questions
As already emphasized our reasoning is speculative and must be consolidated or
refuted by detailed investigations. Because the problem of decoherence and
thermalization of many-particle quantum states is a very generic one, there
exists a large and rapidly expanding literature on it, where the ETH plays a
prominent role. This leads to our first set of open questions:
* •
Does QCD exhibit ETH behavior?
We interpret the success of the thermal model as heuristic indication that it
does show ETH behavior, but it would be desirable to affirm this with rigorous
methods, such that the success of the thermal model becomes a prediction
rather than merely an observation. To make contact with the experimental data,
it is not only necessary to confirm ETH properties in principle, but to also
establish the time scale at which ETH behavior becomes manifest (see Dymarsky
(2019)).
In fact, this question is much discussed in quantum information theory and
answering it could become an early success of quantum simulators (see, e. g.,
Ref. Schuckert and Knap (2020) and references therein). There are two crucial
advantages of quantum computing in this context. First, for the arguments
given in Section III the extensive nature of the microcanonical entropy leads
to a strong exponential suppression of all off-diagonal matrix elements except
for very small systems, which are easiest to simulate. Second, the simulation
of realistic hadrons is not required to test the validity of ETH as long as
the Hamiltonian is well enough modeled. Also, showing ETH behavior for the
gauge group SU(2), which is easier to simulate, should be sufficient.
Another promising approach is lattice gauge theory. Just as RMT behavior was
established for the QCD Dirac operator by simulations on small lattices with
classical computers Berbenni-Bitsch _et al._ (1998); Verbaarschot and Wettig
(2000) the same may be possible for ETH.
* •
Is ETH necessary or only sufficient to explain success of the thermal model?
If it is necessary the relevant question becomes: How long does it take until
few particle observables look in good approximation thermal?
Recently a number of observations suggest that ETH is less universal than
often assumed and may not even be necessary for a system to thermalize Harrow
and Huang (2022). In Ref. Majidy _et al._ (2022) it was shown that non-
Abelian charges reduce entropy-production rates and may enhance finite-size
deviations from eigenstate thermalization. The occurrence of many-body scars
may also negate ergodic behavior in certain systems Turner _et al._ (2018)
and slow down thermalization Michailidis _et al._ (2020).
In general, one should be aware that ETH postulates very restrictive
constraints for matrix elements of generic operators between energy
eigenstates. It could well be that less restrictive requirements already imply
the emergence of thermal behavior to such an extent that the differences are
unobservable in realistic experiments. For example, it would be interesting to
understand whether monogamy of entanglement in many-particle systems Wootters
and Zurek (1982); Wootters (1998); Coffman _et al._ (2000); Osborne and
Verstraete (2006) is already sufficient to mimic thermalization as long as
only small subsystems of the complete, entangled quantum states are observed.
Obviously, the search for alternatives to, and generalizations of, the ETH
paradigm is a vast and rapidly developing area of research.
A second set of questions revolves around the search for useful holographic
models of the quark-hadron transition.
* •
Do recent advances in numerical solutions of gravity in AdS space make it
possible to better understand the dynamics of a Hawking-Page type transition?
Over the years much effort has been invested to better understand the
phenomenological implications of the original Hawking-Page transition without
leading to definitive conclusions Hubeny and Rangamani (2002); Mandal and
Morita (2011); Buchel and Lehner (2015); Dias _et al._ (2016); Yaffe (2018);
Janik _et al._ (2021). In the large-$N_{c}$ limit, the HP transition will
proceed by spinodal decomposition of a deeply supercooled phase unless the
cooling process is very slow. One interesting question is whether the system
re-equilibrates after the transition is complete Yaffe (2018); another is
whether quantum gravity corrections at finite ’t Hooft coupling Gubser _et
al._ (1998) help to smoothen out the HP transition. In a very slowly cooling
scenario, the transition will proceed via bubble nucleation. It is an
interesting question which of these scenarios is applicable to pure
SUN($N_{c}$) gauge theories at moderate $N_{c}$ where the equilibrium
deconfinement transition is known to be of first order.
Progress in the numerical treatment of matrix models Hanada and Watanabe
(2022); Pateloudis _et al._ (2022); Buividovich (2022); Buividovich _et al._
(2019) as well as AdS/CFT initial value problems Waeber and Yaffe (2022) may
make it possible to explore the real-time dynamics of Hawking-Page
transitions. However, none of these efforts addresses the question how the
entanglement structure of the quasi-thermal QGP is reflected in the
entanglement structure of the produced hadrons. In the holographic dual this
raises the challenge of computing the geometric form of an ER bridge between
the asymptotic (“edge”) regions of a holographic thermofield double state for
general initial conditions.
* •
Would holographic models in which the transition is a smooth crossover as in
QCD allow for an approximate identification of the HP transition with the
hadronization transition of a quark-gluon plasma?
The possible application of the HP transition in nonconformal dilaton-gravity
models Gürsoy _et al._ (2009a, b); Megias _et al._ (2011) as a holographic
model of the dynamical deconfinement transition in QCD was explored in Ref.
Gürsoy _et al._ (2013). More recently, the dynamical nature of the transition
including domain formation Attems _et al._ (2020) and domain wall propagation
Ecker _et al._ (2021) have been investigated. Even shock wave collisions were
studied in a holographic model that parametrizes the evolution of the quark-
hadron transition from a smooth crossover to a first-order phase transition
with increasing baryon chemical potential Attems (2021).
* •
Can hadron emission from the surface of a quark-gluon plasma be modeled as
splitting quenches?
There exist low-dimensional toy models for the splitting of one domain of
conformal field theory into two Shimaji _et al._ (2019); Caputa _et al._
(2019) or the formation of an Einstein-Rosen (ER) bridge between two horizons
Anderson _et al._ (2020). Can these models be generalized to multiple black
hole splittings, correponding to the production of multiple hadrons and to
higher dimensions? Is AdS/BCFT the adequate framework to describe
hadronization? If it is, does it imply, in the spirit of Ref. Van Raamsdonk
(2020), that the holographic description of the quark-gluon plasma and an
interacting hadron gas are quite similar? In fact this is exactly what is
observed on the field theoretical side of the duality. The properties of a
hadron resonance gas and the QGP are indistinguishable close to hadronization
(see, e. g., Bellwied _et al._ (2021).
Finally, one would like to know whether there are experimental signatures of
the entanglement structure among hadrons emitted in the decay of quark-gluon
plasma.
* •
How can the validity of the analogy between hadronization and black hole decay
proposed in Section VII be tested?
As discussed in Section IX the monogamy property of entanglement Osborne and
Verstraete (2006) implies that the states of any pair of hadrons among the $N$
emitted hadrons is only entangled at the $O(1/N)$ level. If one would know
what the most accessible entanglement signature is, one could test this
conclusion as a function of the QGP fireball size. One possibility is to look
for quantum correlations among the spins of hadrons Gong _et al._ (2022).
## XI Conclusions
The application of concepts from the AdS/CFT duality to describe the early
stage, spanning about $1-2$ fm/$c$, of high energy heavy ion collisions as
they are studied experimentally at LHC and RHIC has been remarkably successful
and brought many important insights. This suggests that one should try to
extend the dual holographic description to later times including the
hadronization stage. As we explained, any such effort has to face the full
complexity of the black hole information problem in the dual desription.
The time evolution of both the AdS black hole and QCD is unitary. No
information gets lost and no entropy is produced. Consequently, thermal
ensemble in the strict sense cannot be produced. However, what can be produced
is a system for which all realistic measurements produce results that are
indistinguishable from strictly thermal ones. Recent progress on black hole
evaporation has substantiated this conclusion. For QCD the corresponding field
theoretical calculation, which would have to treat quantum entanglement during
each step of the collision appears impractical. If, however, the AdS- and QCD-
based descriptions of a heavy ion collision were holographically dual up to
calculable corrections, a QCD-based calculation would be unnecessary, and one
could simply adapt the insights from black holes to heavy ion collisions.
Here we focused on the question whether and how the AdS dictionary could be
extended to times beyond the hadronization of the quark-gluon plasma and how
the phenomenological properties of heavy ion collisions up to and beyond the
Page time could be represented holographically. In doing so, we have not
reached a conclusive answer, but we also did not encounter obvious roadblocks,
and we formulated topics for future work that could help answer some of the
many remaining questions.
Acknowledgments: We thank M. Kaminski, L. Yaffe, and S. Waeber for many
valuable discussions and comments on a draft of this manuscript. We also thank
T. Hartman for insightful discussions on Euclidean wormholes. B. M.
acknowledges support from the U.S. Department of Energy Office of Science
(Grant DE-FG02-05ER41367) and from Yale University during a sabbatical stay in
Spring 2022.
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# Nuclear effects in coherent photoproduction of heavy quarkonia ††thanks:
Presented at “Diffraction and Low-$x$ 2022”, Corigliano Calabro (Italy),
September 24-30, 2022.
J. Nemchik B.Z. Kopeliovich Czech Technical University in Prague, FNSPE,
Břehová 7, 11519 Prague, Czech Republic
Institute of Experimental Physics SAS, Watsonova 47, 04001 Košice, Slovakia
Departamento de Física, Universidad Técnica Federico Santa María,
Avenida España 1680, Valparaíso, Chile
###### Abstract
Coherent photoproduction of heavy quarkonia on nuclear targets is studied
within the QCD color dipole formalism including several main phenomena: i) The
correlation between impact parameter of a collision $\vec{b}$ and dipole
orientation $\vec{r}$; ii) The higher-twist nuclear shadowing related to the
$\bar{Q}Q$ Fock state of the photon; iii) The leading-twist gluon shadowing
corresponding to higher Fock components of the photon containing gluons; iv)
Reduced effects of quantum coherence in a popular Balitsky-Kovchegov equation
compared to calculations, which are frequently presented in the literature.
Our calculations of differential cross sections are in good agreement with
recent ALICE data on charmonium production in ultra-peripheral nuclear
collisions. We present also predictions for coherent photoproduction of other
quarkonium states ($\psi^{\,\prime}$(2S), $\Upsilon$(1S) and
$\Upsilon^{\,\prime}$(2S)) that can be verified by future measurements at the
LHC.
14.40.Pq,13.60.Le,13.60.-r
## 1 Significance of $\vec{b}-\vec{r}$ correlation
The dipole-nucleon electroproduction amplitude within the color dipole
formalism has the following factorized form [1],
$\displaystyle\\!\\!\\!\\!\mathcal{A}^{N}(x,\vec{q}\,)=2\\!\int\\!d^{2}b\,e^{i\vec{q}\cdot\vec{b}}\int\\!d^{2}r\int_{0}^{1}\\!\\!d\alpha\Psi_{V}^{*}(\vec{r},\alpha)\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}\,)\Psi_{\gamma^{\ast}}(\vec{r},\alpha,Q^{2})\,,$
(1)
where $\vec{q}$ is the transverse component of the momentum transfer, $\alpha$
is the fractional light-front (LF) momentum carried by a heavy quark or
antiquark of the $\bar{Q}Q$ Fock component of the photon with the transverse
separation $\vec{r}$.
The dipole-proton amplitude in Eq. (1),
$\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}\,)$, depends also on the
impact parameter of collision $\vec{b}$. The LF distribution functions
$\Psi_{\gamma^{\ast}}(\vec{r},\alpha,Q^{2})$ and $\Psi_{V}(\vec{r},\alpha)$
correspond to transitions $\gamma^{\ast}\to\bar{Q}Q$ and $\bar{Q}Q\to V$,
respectively.
Higher Fock components containing gluons contribute by default to the dipole-
proton amplitude. Considering nuclear targets, these components must be taken
into account separately due to different coherence effects in gluon radiation.
The essential feature of
$\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}\,)$ is the
$\vec{b}-\vec{r}$ correlation [1],
$\displaystyle\mathrm{Im}\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}\,)=\frac{\sigma_{0}}{8\pi\mathcal{B}(x)}\,\Biggl{\\{}\exp\left[-\,\frac{\bigl{[}\vec{b}+\vec{r}(1-\alpha)\bigr{]}^{2}}{2\mathcal{B}(x)}\right]+$
$\displaystyle\exp\left[-\,\frac{(\vec{b}-\vec{r}\alpha)^{2}}{2\mathcal{B}(x)}\right]-\,2\,\exp\Biggl{[}-\,\frac{r^{2}}{R_{0}^{2}(x)}-\,\frac{\bigl{[}\,\vec{b}+(1/2-\alpha)\vec{r}\,\bigr{]}^{2}}{2\mathcal{B}(x)}\Biggr{]}\Biggr{\\}}\,,$
(2)
where the interaction vanishes if $\vec{r}\bot\vec{b}$ and reaches maximal
strength if $\vec{r}\parallel\vec{b}$.
From the known amplitude (1) one can calculate the differential cross section
$\displaystyle\frac{d\sigma^{\gamma N\to
VN}(x,t=-q^{2})}{dt}=\frac{1}{16\,\pi}\,\Bigl{|}\mathcal{A}^{N}(x,\vec{q}\,)\Bigr{|}^{2}\,,$
(3)
where $x=(M_{V}^{2}+Q^{2})/(W^{2}+Q^{2}-m_{N}^{2})$ with the quarkonium mass
$M_{V}$.
The real part of the $\gamma^{*}N\to VN$ amplitude and the skewness correction
have been incorporated as described in Ref. [2].
Figure 1: Demonstration of the importance of $\vec{b}-\vec{r}$ correlation in
photoproduction of the $\psi^{\,\prime}(2S)$-to-$J\\!/\\!\psi(1S)$ ratio of
differential cross sections (3) at $W=200\,\,\mbox{GeV}$.
As an example, Fig. 1 nicely demonstrates a manifestation of the
$\vec{b}-\vec{r}$ correlation in $\psi^{\,\prime}(2S)$-to-$J\\!/\\!\psi(1S)$
ratio of differential cross sections [1]. Model predictions (solid line) are
significantly different from results based on the standard assumption
$\vec{b}\parallel\vec{r}$, frequently used in the literature. However,
treating nuclear targets, the effect of the $\vec{b}-\vec{r}$ correlation is
diluted [3].
## 2 Higher-twist nuclear shadowing
The lowest Fock component of the projectile photon $|\bar{Q}Q\rangle$ has a
small transverse dipole size $\propto 1/m_{Q}$, where $m_{Q}$ denotes the
heavy quark mass. The corresponding shadowing correction is small as well
since is $\propto 1/m_{Q}^{2}$, and so it can be treated as a higher twist
effect.
For the amplitude of coherent quarkonium electroproduction on nuclear targets,
$\gamma^{*}A\to VA$, one can employ the above expression (1) for
$\mathcal{A}^{N}(x,\vec{q}\,)$, but replacing the dipole-nucleon by dipole-
nucleus amplitude,
$\displaystyle\mathcal{A}^{A}(x,\vec{q}\,)\\!=\\!2\\!\int\\!\\!d^{2}b_{A}\,e^{i\vec{q}\cdot\vec{b}_{A}}\\!\\!\\!\int\\!\\!d^{2}r\\!\int_{0}^{1}\\!\\!\\!d\alpha\Psi_{V}^{*}(\vec{r},\alpha)\mathcal{A}^{A}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}_{A})\Psi_{\gamma^{\ast}}(\vec{r},\alpha,Q^{2}),$
(4)
where $b_{A}$ is the nuclear impact parameter.
In ultra-peripheral collisions (UPC) at the LHC, the photon virtuality
$Q^{2}\sim 0$ and the photon energy in the target rest frame is sufficiently
high, so the coherence length exceeds substantially the nuclear radius
$R_{A}$,
$\displaystyle
l_{c}^{\bar{Q}Q}=1/q_{L}=\frac{W^{2}+Q^{2}-m_{N}^{2}}{m_{N}\,(M_{V}^{2}+Q^{2})}\Biggl{|}_{Q^{2}\sim
0}\approx\frac{W^{2}}{m_{N}M_{V}^{2}}\gg R_{A}\,.$
Then fluctuations of the dipole size are frozen due to Lorentz time dilation
and one can rely on the eikonal form for the dipole-nucleus partial amplitude
at impact parameter $\vec{b}_{A}$,
$\displaystyle\mathrm{Im}\mathcal{A}^{A}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}_{A})\biggl{|}_{l_{c}^{\bar{Q}Q}\gg
R_{A}}\\!\\!\\!\\!\\!\\!\\!\\!\\!=\\!1\\!-\\!\Biggl{[}1\\!-\\!\frac{1}{A}\,\\!\\!\int\\!\\!d^{2}b\,\mathrm{Im}\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}\,)T_{A}(\vec{b}_{A}+\vec{b}\,)\Biggr{]}^{A}\\!\\!.$
(5)
Here $T_{A}(\vec{b}_{A})=\int_{-\infty}^{\infty}dz\,\rho_{A}(\vec{b}_{A},z)$
is the nuclear thickness functions normalized as $\int
d^{2}b_{A}\,T_{A}(\vec{b}_{A})=A$, and $\rho_{A}(\vec{b}_{A},z)$ is the
nuclear density.
The corresponding expression for the differential cross section in the limit
$l_{c}^{\bar{Q}Q}\gg R_{A}$ is analogous to that for proton, Eq. (3), and
reads
$\displaystyle\frac{d\sigma^{\gamma^{\ast}A\to
VA}(x,t=-q^{2}\,)}{dt}\Biggl{|}_{l_{c}^{\bar{Q}Q}\gg
R_{A}}=\frac{1}{16\,\pi}\,\Bigl{|}\mathcal{A}^{A}(x,\vec{q}\,)\Bigr{|}^{2}\,.$
(6)
## 3 Leading-twist gluon shadowing
The gluon shadowing (GS) corrections are related to the higher Fock components
of the photon containing, besides the $\bar{Q}Q$ pair, additional gluons,
$|\bar{Q}Q\,g\rangle$, $|\bar{Q}Q\,2g\rangle$, … $|\bar{Q}Q\,ng\rangle$. In a
$\gamma^{*}p$ collision these components are included in the $\bar{Q}Q$-dipole
interaction with the proton. In an electro-production on a nucleus, such
multi-gluon fluctuations contribute to the amplitude
$\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b})$ in the eikonal formula
(5) for the dipole-nucleus amplitude.
At small photon energies we expect the Bethe-Heitler regime of radiation, when
each of multiple interactions produce independent gluon radiation.
However, the pattern of multiple interactions changes in the regime of long
$l_{c}^{\bar{Q}Qg}\gg d$, where $d\approx 2\,\mbox{fm}$ is the mean separation
between bound nucleons. The gluon radiation length reads
$\displaystyle
l_{c}^{\bar{Q}Qg}=\frac{2k\alpha_{g}(1-\alpha_{g})}{k_{T}^{2}+(1-\alpha_{g})m_{g}^{2}+\alpha_{g}M_{\bar{Q}Q}^{2}},$
(7)
where $k$ is the photon energy in the target rest frame, $\alpha_{g}$ is the
LF fraction of the photon momentum carried by the gluon, $M_{\bar{Q}Q}$ is the
effective mass of the ${\bar{Q}Q}$ pair and $m_{g}\approx 0.7\,\mbox{GeV}$ is
the effective gluon mass fixed by data on gluon radiation [4, 5]. Such a
rather large $m_{g}$ leads to a strong inequality $l_{c}^{\bar{Q}Qg}\ll
l_{c}^{\bar{Q}Q}$, namely $l_{c}^{\bar{Q}Qg}=l_{c}^{\bar{Q}Q}/f_{g}$, where a
large factor $f_{g}\approx 10$ has been obtained in [6].
At long $l_{c}^{\bar{Q}Qg}\gg d$ the Landau-Pomeranchuk effect is at work when
radiation does not resolve multiple interactions acting as one accumulated
kick. This leads to a reduction of intensity of gluon radiation compared to
the Bethe-Heitler regime. This is why it is called the GS correction.
Thus the gluon shadowing is a part of Gribov corrections corresponding to
higher multi-gluon Fock components of the photon and requiring eikonalization
of these components. Differently from $\bar{Q}Q$ fluctuations, a $\bar{Q}Qg$
component does not reach the ”frozen” size regime due to the divergent
$d\alpha_{g}/\alpha_{g}$ behavior. The corresponding variation of the
$\bar{Q}Q-g$ dipole size was taken into account adopting the Green function
technique [7, 2].
The $Q\bar{Q}g$ Fock state is characterized by two scales:
i) One scale determines the small $\bar{Q}Q$ separation, which is $\approx
1/m_{Q}$ and represents a higher twist effect. At large $m_{Q}$ it can be
treated as a point-like color-octet system; ii) The second scale determines a
much larger $\bar{Q}Q$-$g$ transverse size.
Thus the $\bar{Q}Q-g$ system is strongly asymmetric and controls GS, which is
the leading twist effect since is hardly dependent (only logarithmically) on
the $m_{Q}$. It can be treated with high precision as glue-glue dipole [4]
with the transverse size $\approx 1/m_{g}$. The gluon shadowing factor $R_{G}$
has been calculated as function of $b_{A}$ and rapidity $y$ adopting the Green
function formalism. The corresponding values of $R_{G}$ can be found in Fig.1
of Ref. [2].
## 4 Comparison with data
Model calculations of differential cross sections $d\sigma^{\gamma Pb\to
J\\!/\\!\psi(1S)Pb}/dt$ and $d\sigma^{\gamma Pb\to\psi^{\,\prime}(2S)Pb}/dt$
including effects of $\vec{b}-\vec{r}$ correlation, quark and gluon shadowing
are presented in Fig. 2 [2]. One can see a good agreement of our predictions
with ALICE data [8]. Predictions for other heavy quarkonium states
$\Upsilon(1S)$ and $\Upsilon^{\,\prime}(2S)$ can be found in [2].
Figure 2: Predictions for $d\sigma^{\gamma Pb\to VPb}/dt$ in comparison with
ALICE data [8] at $W\approx 125\,\,\mbox{GeV}$.
## 5 Coherence length for multi-gluon components
The virtual photon with energy $k$ develops a $\bar{Q}Q$ fluctuation for a
lifetime
$\displaystyle
l_{c}^{\bar{Q}Q}=\frac{2k}{Q^{2}+M_{Q\bar{Q}}^{2}}=\frac{1}{xm_{N}}P_{q}=l_{c}^{max}P_{q}\,$
(8)
where the $\bar{Q}Q$-effective mass squared
$M_{\bar{Q}Q}^{2}=(m_{Q}^{2}+k_{T}^{2})/\alpha(1-\alpha)$, so that the factor
$P_{q}=1/(1+M_{\bar{Q}Q}^{2}/Q^{2})$. Exact calculations in [6] led to mean
values $\langle P_{q}\rangle_{T,L}\approx 0.36(0.75)$ at $x\sim 10^{-4}\div
10^{-3}$. The inequality $\langle P_{q}\rangle_{L}>\langle P_{q}\rangle_{T}$
means that L photons develop lighter fluctuations than T ones.
The higher Fock component $\bar{Q}Qg$ has the coherence length
$\displaystyle
l_{c}^{\bar{Q}Qg}=\frac{2k}{Q^{2}+M_{\bar{Q}Qg}^{2}}=\frac{1}{xm_{N}}P_{g}=l_{c}^{max}P_{g}\,,$
(9)
where the $\bar{Q}Qg$-effective mass squared
$M_{\bar{Q}Qg}^{2}=\frac{M_{\bar{Q}Q}^{2}+k_{T}^{2}}{1-\alpha_{g}}+\frac{m_{g}^{2}+k_{T}^{2}}{\alpha_{g}}\approx
M_{\bar{Q}Q}^{2}(1+\frac{\gamma}{\alpha_{g}})$ with
$\gamma=2m_{g}/M_{\bar{Q}Q}^{2}$, giving thus the factor
$P_{g}=\alpha_{g}/(\alpha_{g}+\gamma)$. Averaging $P_{g}$ over the gluon
radiation spectrum $d\alpha_{g}/\alpha_{g}$ and fixing the $\bar{Q}Q$ \- $g$
transverse separation at the mean value $1/m_{g}$, we obtain $\langle
P_{g}\rangle/\langle P_{q}\rangle=0.12$.
For the $|\bar{Q}Q2g\rangle$ Fock state the effective mass squared
$M_{\bar{Q}Q2g}^{2}\approx
M_{\bar{Q}Q}^{2}(1+\frac{\gamma}{\alpha_{g1}}+\frac{\gamma}{\alpha_{g2}})$ and
the factor
$P_{2g}=\alpha_{g1}\alpha_{g2}/(\alpha_{g1}\alpha_{g2}+\gamma\alpha_{g1}+\gamma\alpha_{g2})$.
Performing the averaging process over radiation spectra
$d\alpha_{g1}/\alpha_{g1}$ and $d\alpha_{g2}/\alpha_{g2}$ we get $\langle
P_{2g}\rangle/\langle P_{q}\rangle=0.035$. Straightforward generalization for
higher multi-gluon photon components leads to strong inequalities,
$M_{\bar{Q}Q}^{2}\ll M_{\bar{Q}Qg}^{2}\ll\cdots\ll M_{\bar{Q}Qng}^{2}$ and
$l_{c}^{\bar{Q}Q}\gg l_{c}^{\bar{Q}Qg}\gg\cdots\gg l_{c}^{\bar{Q}Qng}$ with
corresponding mean values of $l_{c}$ at the LHC energy as presented in Tab. 1.
| $\langle P_{ng}\rangle/\langle P_{q}\rangle$ | $\langle l_{c}\rangle$ [fm]
---|---|---
$\bar{Q}Q$ | ——- | 120.0
$\bar{Q}Qg$ | 0.11940 | 14.2
$\bar{Q}Q2g$ | 0.03560 | 4.2
$\bar{Q}Q3g$ | 0.01630 | 1.9
$\bar{Q}Q4g$ | 0.00952 | 1.1
Table 1: Fractions of the coherence length for $\bar{Q}Q$ Fock state related
to higher photon components containing different number of gluons.
In heavy quarkonium production in UPC at the LHC there are two dominant
sources of shadowing; the higher twist quark and leading twist gluon shadowing
related to $\bar{Q}Q$ and $\bar{Q}Qg$ component of the photon, respectively.
However, the quark shadowing is suppressed due to large $m_{Q}$. Higher Fock
states, $|\bar{Q}Qng\rangle$ with $n\geq 2$, have rather small or negligible
contributions to shadowing.
Balitsky-Kovchegov (BK) equation [9, 10] in combination with the eikonal
expression (5) assumes that transverse sizes of all photon components are
”frozen” during propagation through the medium, $l_{c}^{\bar{Q}Q}$,
$l_{c}^{\bar{Q}Qg}$,…, $l_{c}^{\bar{Q}Qng}\gg R_{A}$. This leads to
exaggeration of shadowing effects.
Figure 3: Relative impact of reduced coherence effects in the BK equation for
$\gamma Pb\to J\\!/\\!\psi Pb$ in terms of the energy-dependent factor
$d_{V}(W)=|\sigma_{coh}^{GF}-\sigma_{coh}^{eik}|/\sigma_{coh}^{eik}$.
Fig. 3 [11] represents a relative comparison of the model predictions for
coherent $t$-integrated cross section based on a solution of BK equation
combined with the Green function formalism, $\sigma_{coh}^{GF}$, and with
eikonal expression (5), $\sigma_{coh}^{eik}$. One can see that even at the LHC
collision energy ($W=125\,\,\mbox{GeV}$), the frequently used traditional
”eikonal” calculations cause an overestimation of shadowing effects by about
20$\%$. The factor $d_{V}$ rather slowly decreases with c.m. energy $W$ and
one needs quite large $W\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}500\,\,\mbox{GeV}$ in order to use the
”frozen” eikonal approximation with a reasonable accuracy. So one can conclude
that the BK equation cannot be applied to nuclear targets.
Acknowledgments: This work was supported in part by ANID-Chile PIA/APOYO
AFB180002. The work of J.N. was partially supported by Grant No. LTT18002 of
the Ministry of Education, Youth and Sports of the Czech Republic, by the
project of the European Regional Development Fund No.
CZ.02.1.01/0.0/0.0/16_019/0000778 and by the Slovak Funding Agency, Grant No.
2/0020/22.
## References
* [1] B.Z. Kopeliovich, M. Krelina, J. Nemchik, Phys. Rev. D103, 094027 (2021).
* [2] B.Z. Kopeliovich, M. Krelina, J. Nemchik, I.K. Potashnikova, Phys. Rev. D105, 054023 (2022).
* [3] B. Z. Kopeliovich, H. J. Pirner, A. H. Rezaeian, I. Schmidt, Phys. Rev. D77, 034011 (2008).
* [4] B.Z. Kopeliovich, A. Schäfer, A.V. Tarasov; Phys. Rev. D62, 054022 (2000).
* [5] B.Z. Kopeliovich, I.K. Potashnikova, B. Povh, I. Schmidt, Phys. Rev. D76, 094020 (2007).
* [6] B.Z. Kopeliovich, J. Raufeisen, A.V. Tarasov, Phys. Rev. C62, 035204 (2000).
* [7] Y. Ivanov, B. Kopeliovich, A. Tarasov, J. Hüfner, Phys. Rev. C66, 024903 (2002).
* [8] S. Acharya et al. [ALICE], Phys. Lett. B817, 136280 (2021).
* [9] I. Balitsky, Nucl. Phys. B463, 99 (1996).
* [10] Y.V. Kovchegov, Phys. Rev. D60, 034008 (1999).
* [11] B.Z. Kopeliovich and J. Nemchik; “Quantum coherence in the Balitsky-Kovchegov equation,” (to be published).
|
[a]Nicholas Sale
# Persistent homology as a probe for center vortices and deconfinement in
SU(2) lattice gauge theory
Biagio Lucini Jeffrey Giansiracusa
###### Abstract
Topological Data Analysis (TDA) is a field that leverages tools and ideas from
algebraic topology to provide robust methods for analysing geometric and
topological aspects of data. One of the principal tools of TDA, persistent
homology, produces a quantitative description of how the connectivity and
structure of data changes when viewed over a sequence of scales. We propose
that this presents a means to directly probe topological objects in gauge
theories. We present recent work on using persistent homology to detect center
vortices in SU(2) lattice gauge theory configurations in a gauge-invariant
manner. We introduce the basics of persistence, describe our construction, and
demonstrate that the result is sensitive to vortices. Moreover we discuss how,
with simple machine learning, one can use the resulting persistence to
quantitatively analyse the deconfinement transition via finite-size scaling,
providing evidence on the role of vortices in relation to confinement in Yang-
Mills theories.
## 1 Introduction
Center vortices are topological defects that are observed in lattice quantum
chromodynamics (QCD) simulations [2]. They provide a potential explanation for
the mechanism of confinement in QCD and the deconfinement phase transition
[17, 4, 19], but existing methods to study them in configurations rely on
gauge fixing and projection [9] which suffers from the Gribov copies problem
[10, 16]. In this project we leverage persistent homology [6], a tool from
topological data analysis (TDA), to study center vortices in a gauge invariant
manner. Rather than working with lattice QCD, we consider the $\mathrm{SU}(2)$
lattice gauge theory as a toy model since this also exhibits center vortices
and a deconfinement phase transition.
## 2 Lattice Model and Center Vortices
The 4D $\mathrm{SU}(2)$ lattice gauge theory is specified by
$\mathrm{SU}(2)$-valued variables $U_{\mu}(x)$, taking the form of a $2\times
2$ complex matrix, located on each link $(x,\mu)$ of an $N_{t}\times
N_{s}^{3}$ lattice $\Lambda$ with periodic boundary conditions, where
$\mu\in\\{0,1,2,3\\}$ describes the direction in which the link emanates from
the lattice site $x\in\Lambda$. Gauge invariant observables are obtained as
traces of products of the link variables along closed paths $C$, also known as
Wilson loops $W(C)$. The simplest non-trivial example is the Wilson loop
around a $1\times 1$ plaquette $(x,\mu,\nu)$ of the lattice:
$W_{\mu,\nu}(x)=\frac{1}{2}\,tr\Big{[}U_{\mu}(x)\,U_{\nu}(x+\hat{\mu})\,U_{\mu}^{\dagger}(x+\hat{\nu})\,U_{\nu}^{\dagger}(x)\Big{]}.$
We use this to define the Wilson action given a configuration
$\mathbf{U}=\\{U_{\mu}(x)\\}_{(x,\mu)}$ as
$S(\mathbf{U})=-\frac{\beta}{4}\sum_{x,\mu<\nu}W_{\mu,\nu}(x)$ (1)
where $\beta=4/g^{2}$ and $g$ is the gauge coupling parameter. This in turn
allows us to define the vacuum expectation value of any given observable
$A(\mathbf{U})$ as
$\langle A\rangle=\frac{\int
d\mathbf{U}\,A(\mathbf{U})\,e^{-S(\mathbf{U})}}{\int
d\mathbf{U}\,e^{-S(\mathbf{U})}}$ (2)
where $d\mathbf{U}=\prod_{x,\mu}dU_{\mu}(x)$ is a product of Haar measures
over $\mathrm{SU}(2)$ for each link variable. In practice we estimate
expectations using Monte Carlo methods, where Eq. (2) becomes a simple mean of
the observed values.
We introduce center vortices following [17]. Fix a time slice at time $t$.
Given two closed oriented curves $C$ and $C^{\prime}$ in that 3-dimensional
slice with linking number $m$, a loop operator $B(C^{\prime},t)$ can be
defined that has the following commutation algebra with the Wilson loop
$W(C,t)$:
$W({C},t)B({C^{\prime}},t)-(-1)^{m}B({C^{\prime}},t)W({C},t)=0.$ (3)
The operator $B(C^{\prime},t)$ is called the ’t Hooft loop which, when acting
on a gauge configuration, creates a magnetic flux with the resulting
observable effect of multiplication of all Wilson loops around curves ${C}$
with linking number 1 with ${C^{\prime}}$ by $-1$. The ’t Hooft loop is
therefore said to be a vortex creation operator. Since the center of the group
$Z(\mathrm{SU}(2))=\\{I,-I\\}\cong\mathbb{Z}_{2}$ plays a role (as exposed by
the factor $(-1)^{m}$), the vortices are called center vortices. Allowing the
curve $C^{\prime}$ to vary continuously over time slices, we see that a vortex
traces out a surface in 4-space, closed by the periodic boundary conditions.
The above describes a ’thin’ vortex. In practice center vortices have some
finite thickness, so that only larger Wilson loops may fully link with them
and obtain the full center charge. Loops that partially link may still obtain
a partial charge, being multiplied by some matrix lying between $I$ and $-I$
in $\mathrm{SU}(2)$.
To explicitly insert a thin vortex into the system to study we will make use
of the trick of imposing twisted boundary conditions [18]. The idea is that we
negate the contribution to the action of the co-closed collection of
plaquettes
$T=\\{((0,0,y,z),0,1)\,\,|\,\,0\leq y,z<N_{s}\\}$
corresponding to a surface wrapping round the latter two spatial dimensions of
the lattice. The action with twisted boundary conditions becomes
$S_{T}(\mathbf{U})=-\frac{\beta}{4}\Bigg{[}\sum_{\begin{subarray}{c}x,\mu<\nu\\\
(x,\mu,\nu)\not\in
T\end{subarray}}W_{x,\mu,\nu}-\sum_{\begin{subarray}{c}x,\mu<\nu\\\
(x,\mu,\nu)\in T\end{subarray}}W_{x,\mu,\nu}\Bigg{]}$ (4)
which we refer to as the twisted action. This modification of the action
allows the lattice to support an odd number of center vortices wrapping in the
$yz$ plane, which is not allowed by the usual periodic boundary conditions of
the Wilson action. We can think of this twisted action as explicitly inserting
a thin vortex into the system on the surface defined by $T$, so that the
system is forced to generate a (thick) vortex to cancel it out. We shall
denote expectations calculated with respect to this twisted action by $\langle
A\rangle_{\text{twist}}$, where $A$ is a generic observable.
## 3 Persistent Homology
Persistent homology takes a nested sequence of topological spaces and produces
a topological summary called a barcode or persistence diagram [6] (see [3, 7,
13, 8] for useful references). In this case we assign a filtered cubical
complex $F_{\boldsymbol{U}}:\mathbb{R}\rightarrow\text{CubicalComplex}$ (such
that $s\leq t$ implies $F(s)\subseteq F(t)$) to each configuration
$\boldsymbol{U}=\\{U_{\sigma}(x)\\}$ and compute the persistent homology of
this filtered cubical complex. Our construction is based on Wilson loops and
therefore yields gauge-invariant persistence diagrams.
The idea is to explicitly construct a cubical model of vortex surfaces, under
the assumption that vortices are thin. Vortex sheets live on the dual lattice
$\Lambda^{*}$. We therefore consider a decomposition of the spacetime manifold
into a cubical complex $Y$ given by $\Lambda^{*}$ in which there is a vertex
for each dual lattice site, an edge for each link in the dual lattice, a
2-cube for each dual plaquette, etc. Note that there is a bijection between
the 2-cubes of this cubical complex (i.e. dual plaquettes) with the plaquettes
of the original lattice $\Lambda$, pairing a plaquette with the dual plaquette
that intersects it at a single point. We will construct a filtration of $Y$ by
letting each 2-cube enter at a filtration index given by the value of the
Wilson loop around the plaquette it is paired with in the bijection.
To define the filtered complex we give a filtration index $f(c)\in\mathbb{R}$
for each cube $c$ in $Y$ specifying when it appears. Then
$F_{\boldsymbol{U}}(s)$ is the subcomplex of $Y$ consisting of all cubes $c$
for which $f(c)\leq s$. That is,
$F_{\boldsymbol{U}}(s)=f^{-1}(-\infty,s].$
Since we are attempting to model vortex surfaces, we will initially specify
when the 2-cubes are to enter the filtered complex and then introduce the
cubes of other dimensions based on these.
Our construction of the function $f$ is the following:
1. 1.
For each 2-cube $c$, i.e. dual plaquette, we set $f(c)$ equal to the value of
the Wilson loop around the plaquette paired with it by the bijection.
2. 2.
Since a 2-cube is not allowed to be included before its constituent 1-cubes
and 0-cubes in a cubical complex, setting $f(c)$ for these to be the smallest
value of $f$ of any of the 2-cubes they are incident to.
3. 3.
For the 3-cubes and 4-cubes we follow a clique-like rule, setting $f(c)$ for
these to be the largest value of $f$ of any of the 2-cubes contained in their
boundary.
Thus for $s<-1$, $F_{\boldsymbol{U}}(s)$ is the empty complex and for $s\geq
1$, $F_{\boldsymbol{U}}(s)$ is the filled in tiling of spacetime, homeomorphic
to a 4-torus due to the periodic boundary conditions. Going between these
values, the first cubes to enter $F_{\boldsymbol{U}}$ are surfaces made up of
plaquettes in bijection with Wilson loops that are close to $-1$. The idea
therefore is that thin vortex surfaces will enter the filtered complex early.
Moreover, since small Wilson loops like those considered here still pick up a
partial charge from thick vortices, surfaces representing those thick vortices
ought to enter the filtered complex earlier than they otherwise would have. We
expect to detect these closed surfaces in persistent $H_{2}$. We may also see
other topological features such as the presence of handles or holes in
$H_{1}$, as well as the transient low-persistence points in persistent $H_{0}$
and $H_{1}$ that arise as the surface forms near the start of the filtration.
## 4 Detecting Vortices with Twisted Boundary Conditions
We first test if the persistent homology can distinguish between
configurations generated using the Wilson action and configurations generated
using the twisted action. That is, if it detects an inserted vortex.
For $N_{s}\in\\{12,16,20\\}$, fixing $N_{t}=4$, we generate $200$
configurations using the Wilson action (1) and $200$ configurations using the
twisted action (4) for each $\beta\in\\{1.5,1.6,\dots 2.9\\}$. Configurations
are generated using the HiRep software [5] with 1 heatbath step and 4
overelaxation steps for each Monte Carlo step and a sample taken every 100
Monte Carlo steps.
Figure 1 shows example persistence diagrams obtained using the two different
actions and in the two phases of the model.
Figure 1: Sample persistence diagrams of individual configurations obtained
using the following actions and values of $\beta$: (a) Wilson, $\beta=1.5$ (b)
twisted, $\beta=1.5$ (c) Wilson, $\beta=2.9$ (d) twisted, $\beta=2.9$. The
arrow in (d) indicates the point $(b,\infty)\in PH_{2}$ with the smallest
birth index $b$. Note the distance between it and the others.
In the deconfined phase, one of the infinite death points in $H_{2}$ is born
much earlier for the twisted action. This represents a surface which wraps the
periodic boundary conditions of the lattice entering our filtration early:
i.e. the inserted vortex. We therefore define the following observable based
on the persistence diagram of a configuration
$m_{2}=\min\big{\\{}\,b\,\,\big{|}\,\,(b,\infty)\in PH_{2}\,\big{\\}}.$
The expected value of $m_{2}$ for different lattice sizes with the Wilson
action and twisted action are shown in Figure 2(a). Note that there is no
difference between the expectations estimated using the different actions well
into the confined phase, but in the deconfined phase the curves separate. As
the lattice size increases, the point at which the curves diverge approaches
the critical $\beta$ of the phase transition from below. These observations
motivate measuring the difference between the expected values using different
actions
$O_{m_{2}}=\langle m_{2}\rangle-\langle m_{2}\rangle_{\text{twist}}$
as a phase indicator which will be zero in the confined phase and non-zero in
the deconfined phase, similar to the definition of an order parameter but
without the requirement to detect any symmetry breaking. A finite-size scaling
analysis of this quantity yields the curve collapse in Figure 2(b), computed
numerically using the Nelder-Mead method following [1]. The resulting
estimates of $\beta_{c}$ and $\nu$,
$\displaystyle\beta_{c}$ $\displaystyle=2.291\pm 0.019$ $\displaystyle\nu$
$\displaystyle=0.614\pm 0.079,$
are in agreement with our reference estimate $\beta_{c}=2.2986(6)$ from [12]
and $\nu=0.629971(4)$ from [11]. Error estimates are obtained by performing
$2000$ bootstraps.
(a)
(b)
Figure 2: (a) The expected value of the observable $m_{2}$ as a function of
$\beta$ plotted for different values of $N_{s}$ and with the Wilson and
twisted actions. (b) The curve collapse of $O_{m_{2}}$ using $\beta_{c}=2.291$
and $\nu=0.614$. Error bars are not shown for clarity but are comparable to
those in (a).
## 5 Detecting the Deconfinement Transition Without Twisted Boundary
Conditions
Using a machine learning framework inspired by that in [14], we investigate if
it is possible to extract the critical $\beta$ and critical exponent $\nu$ of
the deconfinement transition for $N_{t}=4$ using configurations sampled using
the Wilson action alone.
For lattices of size $4\times N_{s}^{3}$ with $N_{s}\in\\{12,16,20,24\\}$, we
train a $k$-nearest neighbours classifier ($k=30$) on the concatenated
$PH_{0}$, $PH_{1}$, $PH_{2}$ and $PH_{3}$ persistence images of $200$
configurations sampled at each $\beta$ in the confined and deconfined regions
given in Table 1. The classifier is then used to produce a predicted
classification $O_{k\mathrm{NN}}$ for $200$ configurations sampled for each
value of $\beta$ in the critical region. The resulting curve is shown in
Figure 3.
Region | $\beta$
---|---
Confined | 2.2 , 2.21, 2.22, 2.23, 2.24
Deconfined | 2.36, 2.37, 2.38, 2.39, 2.4
Critical | 2.25, 2.26, 2.27, 2.275, 2.28, 2.285, 2.29, 2.295, 2.298, 2.299, 2.3, 2.301, 2.302, 2.305, 2.31, 2.315, 2.32, 2.325, 2.33, 2.34, 2.35
Table 1: Values of $\beta$ sampled at for the $N_{t}=4$ phase transition.
Figure 3: Plot showing our phase indicator $\langle O_{k\mathrm{NN}}\rangle$
as a function of $\beta$ for $N_{t}=4$. The points show the measured
expectations and the curve is the output of histogram reweighting these
measurements.
Assuming a known value of $\nu=0.629971$, we can do a finite-size scaling
analysis by extracting the pseudo-critical point for each $N_{s}$ via the
implicit formula $\langle O_{k\mathrm{NN}}\rangle(\beta_{c}(N_{s}))=0.5$ then
fitting these to the straight line ansatz
$\beta_{c}(N_{s})-\beta_{c}(\infty)\propto N_{s}^{-1/\nu}$ to extract
$\beta_{c}=\beta_{c}(\infty)$. The resulting fit is shown in Figure 4(a). The
intercept yields $\beta_{c}=2.2989\pm 0.0009$, supporting our reference
estimate of $\beta_{c}=2.2986(6)$ from [12].
(a)
(b)
Figure 4: (a) Estimating $\beta_{c}$ using a linear fit, assuming known $\nu$.
The pseudo-critical values of $\beta$, obtained as the points where the curves
in Figure 3 cross $0.5$, are fitted to a straight line against
$N_{s}^{-1/\nu}$ with $\nu=0.629971$. Error bars are estimated by
bootstrapping. (b) The curve collapse of $\chi_{k\mathrm{NN}}$ using
$\beta_{c}=2.2988$ and $\nu=0.634$.
Alternatively, we can try to estimate both $\beta_{c}$ and $\nu$
simultaneously via a curve collapse of the variance curves
$\chi_{k\mathrm{NN}}=\langle O_{k\mathrm{NN}}^{2}\rangle-\langle
O_{k\mathrm{NN}}\rangle^{2}$ using a numerical procedure like that in [1]. The
result of using the Nelder-Mead method is shown in Figure 4(b). The obtained
estimates of $\beta_{c}$ and $\nu$
$\displaystyle\beta_{c}$ $\displaystyle=2.2988\pm 0.0007$ $\displaystyle\nu$
$\displaystyle=0.634\pm 0.014$
are consistent with previous estimates.
## 6 Conclusion
We designed a methodology to use persistent homology to detect center vortices
and tested the efficacy of this by using it to distinguish configurations
generated using a twisted action from configurations generated using the usual
Wilson action. We also performed a quantitative analysis of the deconfinement
transition using two different phase indicators derived from the persistent
homology. We argue that, since the methodology summarises center vortices but
is also sensitive to the phase transition, then center vortices must play some
role in the phase transition. For a stronger argument we would need to
consider the sensitivity of our methodology to the objects involved in other
pictures of confinement, e.g., monopoles.
## Acknowledgments
Numerical simulations have been performed on the Swansea SUNBIRD system. This
system is part of the Supercomputing Wales project, which is part-funded by
the European Regional Development Fund (ERDF) via Welsh Government.
Configurations of the $\mathrm{SU}(2)$ lattice gauge theory were sampled using
the HiRep software [5]. Persistent homology calculations were performed using
giotto-tda [20]. Histogram reweighting calculations were performed using
pymbar [15]. NS has been supported by a Swansea University Research Excellence
Scholarship (SURES). JG was supported by EPSRC grant EP/R018472/1. BL received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme under grant agreement No
813942. The work of BL was further supported in part by the UKRI Science and
Technology Facilities Council (STFC) Consolidated Grant ST/T000813/1, by the
Royal Society Wolfson Research Merit Award WM170010 and by the Leverhulme
Foundation Research Fellowship RF-2020-461\9.
## References
* [1] S. M. Bhattacharjee and F. Seno. A measure of data collapse for scaling. Journal of Physics A, 34:6375–6380, 2001.
* [2] James Biddle, Waseem Kamleh, and Derek Leinweber. Static quark potential from centre vortices in the presence of dynamical fermions, 2022, arXiv:2206.00844.
* [3] Gunnar Carlsson. Persistent homology and applied homotopy theory, 2020, arXiv:2004.00738.
* [4] John M. Cornwall. Quark confinement and vortices in massive gauge-invariant qcd. Nuclear Physics B, 157(3):392–412, 1979.
* [5] Luigi Del Debbio, Agostino Patella, and Claudio Pica. Higher representations on the lattice: Numerical simulations, su(2) with adjoint fermions. Phys. Rev. D, 81:094503, 2010.
* [6] H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete & Computational Geometry, 28:511–533, 2002.
* [7] Herbert Edelsbrunner and John Harer. Persistent homology—a survey. Discrete & Computational Geometry - DCG, 453, 01 2008.
* [8] Robert Ghrist. Barcodes: The persistent topology of data. BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY, 45, 02 2008.
* [9] Rudolf Golubich and Manfried Faber. A possible resolution to troubles of su(2) center vortex detection in smooth lattice configurations. Universe, 7(5), 2021.
* [10] V.N. Gribov. Quantization of non-abelian gauge theories. Nuclear Physics B, 139(1):1–19, 1978.
* [11] Filip Kos, David Poland, David Simmons-Duffin, and Alessandro Vichi. Precision islands in the ising and $o(n)$ models. Journal of High Energy Physics, 2016.
* [12] Biagio Lucini, Michael Teper, and Urs Wenger. The high temperature phase transition in su(n) gauge theories. Journal of High Energy Physics, 2004, 07 2003.
* [13] Nina Otter, Mason Porter, Ulrike Tillmann, Peter Grindrod, and Heather Harrington. A roadmap for the computation of persistent homology. EPJ Data Science, 6, 2015.
* [14] Nicholas Sale, Jeffrey Giansiracusa, and Biagio Lucini. Quantitative analysis of phase transitions in two-dimensional $xy$ models using persistent homology. Phys. Rev. E, 105:024121, 2022.
* [15] Michael R. Shirts and John D. Chodera. Statistically optimal analysis of samples from multiple equilibrium states. The Journal of chemical physics, 129 12:124105, 2008.
* [16] John D. Stack and William W. Tucker. The gribov ambiguity for maximal abelian and center gauges in su(2) lattice gauge theory. Nuclear Physics B - Proceedings Supplements, 94(1):529–531, 2001\. Proceedings of the XVIIIth International Symposium on Lattice Field Theory.
* [17] G. ’t Hooft. On the phase transition towards permanent quark confinement. Nuclear Physics B, 138(1):1–25, 1978.
* [18] G. ’t Hooft. A property of electric and magnetic flux in non-abelian gauge theories. Nuclear Physics B, 153:141–160, 1979.
* [19] Gerard ’t Hooft. A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories. Nucl. Phys. B, 153:141–160, 1979.
* [20] Guillaume Tauzin, Umberto Lupo, Lewis Tunstall, Julian Burella Pérez, Matteo Caorsi, Anibal Medina-Mardones, Alberto Dassatti, and Kathryn Hess. giotto-tda: A topological data analysis toolkit for machine learning and data exploration, 2020, arXiv:2004.02551.
|
# Differentiable User Models
Alex Hämäläinen Department of Computer Science
Aalto University Mustafa Mert Çelikok Department of Computer Science
Aalto University Samuel Kaski Department of Computer Science
Aalto University Department of Computer Science
University of Manchester
###### Abstract
Probabilistic user modeling is essential for building machine learning systems
in the ubiquitous cases with humans in the loop. However, modern advanced user
models, often designed as cognitive behavior simulators, are incompatible with
modern machine learning pipelines and computationally prohibitive for most
practical applications. We address this problem by introducing widely-
applicable differentiable surrogates for bypassing this computational
bottleneck; the surrogates enable computationally efficient inference with
modern cognitive models. We show experimentally that modeling capabilities
comparable to the only available solution, existing likelihood-free inference
methods, are achievable with a computational cost suitable for online
applications. Finally, we demonstrate how AI-assistants can now use cognitive
models for online interaction in a menu-search task, which has so far required
hours of computation during interaction.
## 1 Introduction
User modeling constructs informative representations of individual users to
enable computational systems to customize and adapt their behavior for them
[Li and Zhao, 2020]. It has been extensively studied over the years, also
recently in recommender systems [Yu et al., 2019, Yuan et al., 2020], human-
in-the-loop machine learning [Daee et al., 2018] and AI-assistants [Horvitz et
al., 2013, Dafoe et al., 2021]. Machine learning is needed in user modeling to
infer user-specific information based on observed user behavior. Depending on
the application, the inferred information can be the end result or, for
instance, used to parameterize a user simulator to form predictions of user
behaviors to guide the behavior of the system.
Traditionally, salient use cases of user modeling, such as recommendation
engines, have utilized user-specific preference profiles based on users’
history. However, these approaches are not sufficiently powerful in more
complex interactive applications, where the user plans and interacts
strategically, for instance in human-AI collaboration and human-in-the-loop
decision-making. On the other hand, while recent ML research has shown
significant success in learning accurate neural models directly from data,
this is an infeasible approach in user modeling in general, as typical user-
driven applications are data starved. In contrast, recent approaches, e.g.,
Kangasrääsiö et al. [2019], Moon et al. [2022], have utilized advanced
general-purpose behavioral models, based on cognitive science, in a Bayesian
setting and received encouraging results with limited data. The probabilistic
treatment of the problem enables taking uncertainty of the inferences into
account — which fundamentally allows the system to balance between exploration
and exploitation in interaction with the user.
A prominent body of these advanced cognitive models is based on computational
rationality [Lewis et al., 2014, Gershman et al., 2015], which posits that
seemingly irrational behaviors of humans are rational under their cognitive
bounds. It follows that human behaviors can be accurately modeled as a result
of RL-based optimization given that the underlying decision-theoretic
framework and the optimization procedure are specified such that they
accurately capture the appropriate bounds governing human cognition, such as
the limits of computational capacity. A concrete example of such a model,
which we consider in our experiments, is the model of rational menu search
[Chen et al., 2015]. This cognitive model describes human search behavior in
terms of eye movements when searching for a target item in a computer dropdown
menu, while encoding the limitations of human cognition and perception when
processing visual information.
Despite their many benefits, these advanced cognitive models are currently not
used beyond small-scale practical applications due to two important factors:
(1) they are expressed as non-differentiable simulators and hence incompatible
with modern machine learning frameworks and (2) they are computationally
infeasible to be used directly in realistic applications.
In this paper, we address these limitations: we enable computationally
efficient probabilistic user modeling suitable for real-time applications —
even with advanced cognitive models that lack a closed-form likelihood. We do
this by combining the best of gradient-based and Bayesian learning: we show
how one can develop differentiable user models which are sample-efficient by
leveraging prior knowledge from non-differentiable cognitive models and can
quantify uncertainty in their estimates. As a result, the surrogates become
widely applicable with online computational cost independent of the complexity
of the original models. The contributions of this work are:
* •
We introduce a way of enabling computationally efficient inference with
cognitive user models by building generalizable differentiable surrogates for
them through meta-learning.
* •
We demonstrate a flexible way of leveraging any existing user data during
surrogate training to address possible model misspecification in cognitive
models, especially in the case of action noise.
* •
With neural processes as example surrogate models, we demonstrate comparable
user modeling accuracy to current methods with a computational time suitable
for online applications.
Our work removes a key computational bottleneck currently hindering
incorporation of users into probabilistic programming models. Probabilistic
user modeling based on cognitive models can now be applied widely without
extensive computational budgets.
## 2 Differentiable user models
This work considers computationally efficient probabilistic user modeling for
interactive settings, between a user from a user population and a
computational system. User modeling is brought in to guide adaptation of the
behavior of such a system for individual users; user modeling is needed for
(i) inferring user-specific information from observed user behaviors and (ii)
then using the information for user behavior prediction. The following
subsections detail the specifics of current approaches and their limitations,
together with our approach for addressing these limitations.
### 2.1 Probabilistic user modeling with cognitive models
Following the intuition presented by Kangasrääsiö et al. [2019], we formulate
the probabilistic user modeling setting as follows: a population of users is
engaged with a distribution of decision-making tasks denoted by
$p(\theta_{T})$. Each modeling scenario involves a user $\theta_{U}\sim
p(\theta_{U})$ and is fully described by the respective user and task specific
parameters $\theta=\\{\theta_{T},\theta_{U}\\}$. The users are assumed to
generate their policies $\pi$ through an implicit process
$\mathcal{P}_{\theta}$ which they execute to generate pairs of states and
actions $(\mathbf{s},\mathbf{a})$. The system has access to a cognitive model
$p(\pi\mid\theta)$ approximating the true process and a corresponding prior
$p(\theta)$. An important task corresponding to this user modeling setting is
the inference problem of approximating the posterior
$p(\theta\mid(\mathbf{s},\mathbf{a}))\propto
p((\mathbf{s},\mathbf{a})\mid\pi)p(\pi\mid\theta)p(\theta).$ (1)
Computing the posterior $p(\theta\mid(\mathbf{s},\mathbf{a}))$ and then using
the likelihood model $p(\pi\mid\theta)$ for computing the corresponding
posterior predictive distribution $p(\pi\mid(\mathbf{s},\mathbf{a}))=\int
p(\pi\mid\theta)p(\theta\mid(\mathbf{s},\mathbf{a}))d\theta$ would be the
Bayesian choices for achieving the objectives (i) and (ii).
For cognitive models, the likelihood $p(\pi\mid\theta)$, required for solving
the Bayesian inference task for the posterior, is typically not evaluable in
closed-form due to the simulator-type nature of these models. So far, this
issue has been circumvented by utilizing exclusively likelihood-free inference
(LFI) methods, such as approximate Bayesian computation (ABC) [Sisson et al.,
2018, Sunnåker et al., 2013] and Bayesian optimization for likelihood-free
inference (BOLFI) [Gutmann et al., 2016], as proposed by Kangasrääsiö et al.
[2019] and Moon et al. [2022]. The basic idea of LFI is to replace the
computationally expensive simulator $p(\pi\mid\theta)$ with an approximation
that is separately learned on the observed data [Gutmann et al., 2016], in
this case the $(\mathbf{s},\mathbf{a})$ from each user. For user modeling,
this approach has two problems: this process requires numerous computationally
expensive simulations with the cognitive model and the data in typical user
modeling applications often is too scarce for learning a new model
independently for each user.
Moon et al. [2022] proposed circumventing the computational complexity by
learning a generalizable policy-modulation network as a surrogate for the
original model, i.e. $p(\pi\mid\theta)$, and obtained significant speed-ups
for inference. However, as noted by the authors, their approach is still
prohibited by the computational cost of LFI needed for approximating the
posterior, and is not feasible for real-time inference. Similarly, the
simulation costs of cognitive models $p(\pi\mid\theta)$ are often too
expensive to enable estimating the posterior predictive in real-time
applications, even if the posterior was readily available. Furthermore, even
though LFI methods are developing fast, practical interactive settings may
require hierarchical approaches for generalizing across the user models, which
has been traditionally difficult with LFI [Turner and Van Zandt, 2014]. An
additional problem with LFI-based modeling is the sensitivity to model
misspecification, which is very likely in user models.
### 2.2 Amortization for cognitive models
In this work, we seek to address the limitations of current approaches and to
enable efficient computation for both the likelihood and posterior models so
that the posterior predictive distribution is practical to approximate. Our
approach is to amortize posterior predictive inference through surrogate
modeling. Training generalizable surrogates offline enables using them during
online interaction without extensive computation.
While this approach would solve the issue of online computational complexity,
the offline complexity of simulating sufficient amounts of training data for
them will still be an issue due to the vast diversity of different behaviors
the cognitive models are able to express. In particular, even if training a
generalizable surrogate for a cognitive simulator would be achievable, as done
in the work of Moon et al. [2022], training a surrogate directly for
approximating the posterior can be ultimately be computationally intractable
if constructing the training data requires numerous repeated evaluations with
LFI (for the reference, Kangasrääsiö et al. [2019] reported that even a single
LFI result would require at least 700 CPU hours with the menu search model).
Furthermore, as we will later discuss in Section 2.5, data-efficiency in
training the surrogates is also otherwise a desirable factor as it helps
combating model misspecification in cognitive models.
### 2.3 Casting simulator-based modeling as meta-learning
In order to make the surrogate training more sample-efficient, we approach
amortization task from meta-learning perspective. Here, the key insight is
that both the likelihood and posterior models can be learned jointly with an
appropriate policy approximation task, without ever needing to approximate the
true posterior $p(\theta\mid(\mathbf{s},\mathbf{a}))$, if one is satisfied
with using a latent representation $z\in\mathcal{Z}$ to capture user-specific
information. Following this intuition, we generalize the likelihood and
posterior models to mappings $h$ and $g$:
###### Definition 2.1 (Amortization for cognitive models).
Let $\mathcal{S}$ and $\mathcal{A}$ denote the state and action spaces
corresponding to the user model and
$\mathcal{O}=\bigcup_{n}(\mathcal{S}\times\mathcal{A})^{n}$ be a collection of
$m$ observations over behavior generated by an individual user $\theta_{U}$ in
a task $\theta_{T}$. Amortization for cognitive models corresponds learning
the following functions such that they are evaluable during online
interaction:
1. 1.
Inference of user and task representations, done by the mapping
$h:\leavevmode\nobreak\ \mathcal{O}\rightarrow P(\mathcal{Z})$, where
$P(\mathcal{Z})$ denotes a probability distribution over a joint user and task
representation space $\mathcal{Z}$, which aims to capture the properties
governing user behavior.
2. 2.
User behavior prediction, done by the mapping $g:\leavevmode\nobreak\
\mathcal{S}\times\mathcal{Z}\rightarrow P(\mathcal{A})$, where
$P(\mathcal{A})$ is a probability distribution over user action space.
In line with the Definition 2.1, we amortize the computation for the posterior
predictive distribution over a cognitive model through learning generalizable
surrogates for the mappings $h$ and $g$. Intuitively, we are here building on
the conceptual similarity between Bayesian methods and meta-learning
(previously discussed, e.g., by Grant et al. [2018] and Garnelo et al.
[2018b]), and consider the mapping $h$, i.e., computing the posterior over the
user representation as equivalent to task-specific adaptation during meta-
testing and the mapping $g$, i.e., computing the likelihood as analogous to
prediction.
To formalize the idea, let $s\in\mathcal{S}$, $a\in\mathcal{A}$ and
$(\mathbf{s},\mathbf{a})=\\{(s_{1},a_{1}),\dots,(s_{n},a_{n})\\}\in\mathcal{O}$.
Our goal is to learn the mappings $h$ and $g$, with optimizable parameters
$\\{\psi,\phi\\}$, to approximate the posterior
$p_{\psi}(z\mid(\mathbf{s},\mathbf{a}))$ and the likelihood $p_{\phi}(a\mid
s,z)$ with respect to a latent representation $z\in\mathcal{Z}$. The
corresponding posterior predictive model can here be written as
$p_{\\{\psi,\phi\\}}(a\mid s,(\mathbf{s},\mathbf{a}))=\int p_{\phi}(a\mid
s,z)p_{\psi}(z\mid(\mathbf{s},\mathbf{a}))dz$. The surrogates should jointly
minimize the following objective for policy approximation, while generalizing
over the ground-truth user and task population ($\theta\sim p(\theta)$ and
$\pi\sim p(\pi\mid\theta)$):
$\min_{\phi,\psi}\mathbb{E}_{\theta\sim
p(\theta),s\in\mathcal{S}}\bigg{[}\delta\big{[}\pi(a\mid
s),p_{\\{\psi,\phi\\}}(a\mid s,(\mathbf{s},\mathbf{a}))\big{]}\bigg{]},$ (2)
where $\delta$ is a dissimilarity function (e.g., KL-divergence) and the
observations $(\mathbf{s},\mathbf{a})$ are assumed to have been generated by
executing $\pi$ in the underlying environment. In Section 3, we demonstrate
how a solution to this problem can be approximated with neural processes.
Consistently with numerous current meta-learning approaches (e.g., Finn et al.
[2017], Garnelo et al. [2018b]), we propose a modeling workflow with separate
offline (meta-training) and online (meta-testing) phases described below. We
additionally expand on mitigating the effects of possible model
misspecification in cognitive models.
### 2.4 Meta-training and meta-testing
Algorithm 1 specifies the proposed meta-training procedure, to enable
generalization of the surrogates $h$ and $g$ over the population of interest
$p(\theta)$. The procedure needs to be complemented with an appropriate meta-
learning loss for approximating a solution to Eqn. 2 in terms of
$\\{\psi,\phi\\}$, depending on the implementations of $h$ and $g$. In Section
3, we exemplify this with neural processes.
The corresponding meta-testing, i.e., task-specific adaptation phase is
straightforward: mappings $h$ and $g$ can be utilized for inferring user
representations $z\sim p_{\psi}(z\mid(\mathbf{s},\mathbf{a}))$ w.r.t. observed
$(\mathbf{s},\mathbf{a})$ and for predicting user behaviors $a\sim
p_{\phi}(a\mid s,z)$ on states of interest $s\in\mathcal{S}$.
Algorithm 1 Meta-training cognitive model surrogates
Require: A distribution over users: $p(\theta_{U})$
Require: A distribution over tasks: $p(\theta_{T})$
Require: A cognitive model: $p(\pi\mid\theta)$
Initialize $h$ and $g$ with $\\{\psi,\phi\\}$
repeat
Sample $\theta=\\{\theta_{U},\theta_{T}\\}$, $\theta_{U}\sim p(\theta_{U})$,
$\theta_{T}\sim p(\theta_{T})$
Generate $\pi\sim p(\pi\mid\theta)$
Generate $n$ trajectories $(\mathbf{s},\mathbf{a})$ by executing $\pi$
Optimize $\\{\psi,\phi\\}$ with respect to $(\mathbf{s},\mathbf{a})$ with an
appropriate training loss
until done
Note that consistently with Garnelo et al. [2018b], the proposed meta-learning
workflow deliberately differs from many other popular meta-learning
approaches, such as model-agnostic meta-learning (MAML) [Finn et al., 2017]
and Reptile [Nichol and Schulman, 2018], by fully excluding the gradient-based
optimization loop during task-specific adaptation phase. Instead, the
adaptation phase is here reduced to a forward pass through $h$. Not only is
this computationally faster, enabling online computation, the probabilistic
nature of our approach can also enable interactive systems to balance between
exploration-exploitation trade-offs. As we demonstrate in our experiments,
these benefits additionally translate into improved modeling accuracy.
### 2.5 Model misspecification in cognitive models.
Model misspecification is a relevant issue in behavioral user modeling. While
typical LFI-approaches are highly sensitive to misspecification, this can be
mitigated with our approach by combining observed user data with simulated
data and meta-training the surrogates again, when new observations become
available. We demonstrate in Section 4.2 that this approach enables balancing
between modeling accuracy and data requirements — especially in practical
interactive user modeling applications which only have limited collections of
user behavior datasets available.
## 3 User modeling with neural processes
We use neural processes (NP) [Garnelo et al., 2018b] as an example solution
for implementing and learning the mappings $h$ and $g$ of Definition 2.1.
First, we briefly cover the relevant background on NPs and then explain in
detail how they can be adapted for user modeling.
### 3.1 Background on neural processes
Neural processes [Garnelo et al., 2018b] are a family of neural latent-
variable models combining properties of neural networks and Gaussian processes
(GP). Specifically, they are differentiable solutions for representing
uncertainty over functions that may be utilized for few-shot approximation.
For our purposes, NPs are particularly fitting as they match Definition 2.1
and that the meta-learning objective (Eqn. 2) can be readily computed for
them.
NPs model a set of functions $\\{f_{d}\\}_{d}$ where each $f_{d}:X\rightarrow
Y$ is assumed to be drawn from an underlying stochastic process $f_{d}\sim F$.
NP approximates the underlying process $F$ with a neural network $g$. As each
function $f_{d}$ drawn from the process $F$ represents an individual
instantiation of the process, a latent variable $z$ is introduced for
capturing the instance-dependent variation in $F$ as $f_{d}(x)=g(x,z)$. NPs
consist of an encoder, an aggregator and a conditional decoder. The encoder is
a neural network for constructing representations $r_{i}=h_{\phi}((x,y)_{i})$
at given observations $(x,y)_{i}$. The aggregator, $\alpha$, constructs
permutation-invariant summaries of the encoded representations as
$r=\alpha(\\{r_{i}\\})=\frac{1}{n}\sum_{i=1}^{n}r_{i}$. The summaries are
further utilized to parametrize a (multivariate Gaussian) latent distribution
$z\sim\mathcal{N}(\mu(r),I\sigma(r))$. The conditional decoder,
$g_{\psi}(x_{T},z)$, is a neural network that is conditioned on samples from
the latent distribution to estimate $f_{d}(x_{T})=y$ at locations $x_{T}$.
NP meta-training procedure samples individual instantiations $f_{d}\sim F$ of
the stochastic process $F$. Here, each function $f_{d}$ is evaluated at a
varying number of inputs to produce a dataset of tuples $(x,y)_{i}^{d}$. Each
dataset is then divided into separate context $(x_{1:m},y_{1:m})$ and target
$(x_{m+1:n},y_{m+1:n})$ sets. Intuitively, here the context set represents the
fully observed function evaluations while the target $x_{m+1:n}$ represents
the locations at which the model aims to approximate $y_{m+1:n}$. The context
and target sets are input to the encoder and the conditional decoder
respectively, and the model parameters $\\{\phi,\psi\\}$ are optimized with
respect to the NP-variant of Evidence lower-bound (ELBO). For further
information about NPs and their training, see Garnelo et al. [2018b]. Finally,
note that the low complexity of NPs ($\mathcal{O}(n+m)$) makes them suitable
for real-time scenarios.
### 3.2 Adapting neural processes for user modeling
Neural processes can be adapted as concrete implementations for the required
mappings $h$ and $g$ and for approximating a solution for Equation 2 within
the proposed meta-training procedure (Algorithm 1). First, we recognize that
Equation 2 is essentially a function approximation problem to which NPs can be
applied — the true behavior-generative process $p(\pi\mid\theta)$ can
essentially be treated as a stochastic process $\mathcal{P}$ where each
instantiation $\pi\sim\mathcal{P}$ represents a policy. The NP latent variable
$z$ is utilized for capturing user/task representations and the mappings $h$
and $g$ can be implemented with the NP encoder $p_{\phi}(\pi\mid z)$ and
conditional decoder $p_{\psi}(z\mid(\mathbf{s},\mathbf{a}))$. The meta-
training procedure is adapted as follows: the sampled behaviors
$(\mathbf{s},\mathbf{a})$ are split into context and target sets and the
parameters $\\{\psi,\phi\\}$ can be optimized according to NP-ELBO.
In addition to the vanilla NPs, we consider also attentive neural processes
(ANP) Kim et al. [2019], conditional neural processes (CNP) [Garnelo et al.,
2018a] and attentive conditional neural processes (ACNP). ANPs are essentially
NPs with the difference of including attention in the encoder architecture.
The attention acts as a local latent variable, allowing ANPs to capture both
global and local information affecting user behaviors. CNPs (and ACNPs)
implement the latent encoding $h$ as a deterministic mapping, thus lacking an
important ability of sampling on $\mathcal{Z}$.
## 4 Experiments
We conduct three experiments where we compare our approach against other ways
one could conceivably try to solve the problem — although this has not been
previously done. The first is a demonstration in a benchmark gridworld
environment. The second is a menu search task where a cognitive user model,
justified and validated by earlier cognitive science studies, allows us to
study real-user performance with simulations. The third experiment is a
reasonably realistic menu search assistant scenario.
#### Comparison methods and baselines.
We assess the modeling capabilities of the proposed solution in terms of its
ability to predict the actions of individual agents, as a function of the
number of previous observations of their behavior in the modeling task of
Equation 2. This metric directly evaluates the posterior predictive but also
indirectly the quality of the posteriors over user representations
$z\in\mathcal{Z}$. Unless otherwise specified, the experiments aim to simulate
realistic user modeling applications by limiting the training data to
observations from $\sim 1000$ simulated users.
Table 1: Modeling accuracy as a function of the number of observed full episodes in the menu-search setting of Section 4.2. Episodes | ANP | Reptile | MAML | Transformer | Oracle | Population avg.
---|---|---|---|---|---|---
$0$ | $\mathbf{0.937\pm 0.011}$ | $0.829\pm 0.054$ | $0.774\pm 0.033$ | $0.920\pm 0.035$ | $\mathbf{0.970\pm 0.002}$ | $0.921\pm 0.012$
$1$ | $\mathbf{0.953\pm 0.011}$ | $0.921\pm 0.021$ | $0.916\pm 0.026$ | $0.922\pm 0.021$ | $\dots$ | $\dots$
$2$ | $\mathbf{0.954\pm 0.011}$ | $0.930\pm 0.020$ | $0.928\pm 0.025$ | $0.931\pm 0.017$ | $\dots$ | $\dots$
$5$ | $\mathbf{0.955\pm 0.010}$ | $0.944\pm 0.017$ | $0.943\pm 0.021$ | $0.926\pm 0.012$ | $\dots$ | $\dots$
$9$ | $\mathbf{0.955\pm 0.010}$ | $0.954\pm 0.016$ | $0.952\pm 0.014$ | $0.928\pm 0.009$ | $\dots$ | $\dots$
We compare our approach against two baselines and three alternative surrogate
architectures. The alternative surrogates are transformers trained with MAML
and Reptile, and a standard transformer. MAML and Reptile act as alternative
representative meta-learning approaches to the policy approximation task over
user population, while the transformer intends to provide a reference point
for the performance of sequential models which are frequently used in
alternative user modeling domains, such as sequential recommendation. None of
the alternative surrogate architectures are fully consistent with the proposed
meta-learning procedure and are applied to the policy approximation task on
simulated data directly instead. We also include comparisons between several
alternative NP architectures. Details are included in the Supplement.
Figure 1: Gridworld results. Left: Modeling accuracy as a function of the
number of observed full episodes. The best NP-based model (here ANP) achieves
comparable results to the upper bound given by an oracle; all NP-based models
are clearly better than alternatives. The figure illustrates the gradual
improvement of the predictions of NPs as more episodes are perceived. The BO
results are averaged over the number of context trajectories due to the small
sample size. Right: Illustration of ANP uncertainty update on policy
predictions. The predictions (gray arrows) align towards the implicitly
inferred possible goal states (green rectangles). In the upper figure, the
predictions are conditioned on one observed trajectory (orange arrows). In the
lower figure, we observe that the system implicitly infers the location of the
positive reward, within the accuracy of two squares, after perceiving the
second trajectory. The dark green and red squares are the true positive and
negative reward states.
The two baselines are a Bayesian Optimization (BO) model and a population
average predictor. Furthermore, we provide results from an oracle, acting as
an upper bound for the performance of any solution, including LFI. Both the BO
baseline and the oracle utilize the cognitive model $p(\pi\mid\theta)$
directly for prediction — the oracle parametrizes the cognitive model with the
true user parameters while BO utilizes MAP estimates. The population average
predictor directly approximates the population level action distribution
$p(a\mid s)$ for each state without any user-specific conditioning.
Even though it would be an important baseline, we were not able to produce any
representative results with LFI due to its immense computational complexity.
For reference, Kangasrääsiö et al. [2019] compared several LFI-methods,
including BO, on exactly the same Menu Search model used in our experiments.
They gave 700h of CPU-time for each method to run only one individual
inference task and noted that it is likely that none of the methods converged.
Obtaining conclusive accuracy results with LFI in our experiment setting is
practically intractable as at least hundreds of individual inference results
would be needed. Here, we consider the BO-baseline as an approximate lower-
bound for LFI performance. Although converging faster than LFI, BO is still
computationally very heavy, due to the expensive simulation costs, and
feasible only in our first experiment.
### 4.1 Experiment 1: Gridworld environment
#### Setting.
The first experiment scenario is based on a simple $10\times 10$ gridworld
environment. In this setting, we consider modeling Monte Carlo Tree Search
(MCTS) [Browne et al., 2012] agents with unknown reward functions and MCTS
parameters. This benchmark scenario evaluates the modeling system’s ability to
approximate the uncertainty over user policies. In this experiment, we assume
that a generative user model and a parameter prior are available, capturing
the true generative process of the population.
The gridworld environment is defined as a partially observable Markov decision
process (POMDP) with deterministic transition dynamics. The action space
consists of four possible actions that correspond to the agent relocating from
its current state to adjacent states. Each gridworld scenario always contains
two reward states - one with a positive reward and one with a negative reward.
The agents gain no rewards or penalties other than from the given states. The
full setting details are given in the Supplement.
#### Modeling task.
The modeling task is to predict the subsequent actions of agents sampled from
the population. Each scenario assigns the modeling system with observed
trajectories from a varying number of previous episodes and a partial
trajectory from the current episode generated by the agent. The task is to
predict the remaining actions of the trajectory in the current episode. All
information about the agent, excluding the observed trajectories, is hidden
during both training and evaluation (except for the oracle).
#### Results.
The NP-family models are mostly able to outperform all the baselines (Fig. 1)
with the ANP converging close to the performance of the oracle. It is likely
that the BO-baseline has not properly converged, although given clearly the
largest amount of computation time, and it may not act as a reliable
approximation for LFI performance. Finally, the transformer and MAML are
unable to generalize to the task, likely due to the too limited amount of
training data.
Comparisons between the NP-family models suggest that, in terms of NP
architecture, the most impactful factor contributing to the modeling
performance is attention, i.e., local latent variables, as ANP and ACNP
outperform their non-attentive counterparts. Consistent with the probabilistic
treatment of $h$ (Def. 2.1), stochasticity of the global latent variables $z$
(ANP and NP) also seems to improve the results. Because ANP was clearly the
best of the NP methods, and hence remaining NP-models would not affect
conclusions, we omit the NP, CNP and ACNP models for the following
experiments, to save computation.
### 4.2 Experiment 2: Menu search environment
#### Setting.
Our second experiment is based on the Menu Search model of Kangasrääsiö et al.
[2019], a modified version of Chen et al. [2015]. The Menu Search model is a
cognitive model describing human search behavior in terms of eye movements
(saccades) when searching for a target item in a computer dropdown menu.
Motivated by computational rationality [Gershman et al., 2015], the model
simulates user behavior as a result of optimizing the search behavior with RL
given their cognitive constraints of the user. The details are given in the
Supplement.
Table 2: Modeling accuracies for different numbers of observed full episodes with the ANP-based system when trained with data partially from a misspecified user model and partially from the true population. Here, the percentages denote the share of the training data generated with the misspecified user model. Episodes | ANP 0% | ANP 25% | ANP 50% | ANP 75% | ANP 100%
---|---|---|---|---|---
$0$ | $\mathbf{0.937\pm 0.011}$ | $0.920\pm 0.015$ | $0.895\pm 0.016$ | $0.888\pm 0.013$ | $0.852\pm 0.017$
$1$ | $\mathbf{0.953\pm 0.011}$ | $0.923\pm 0.012$ | $0.899\pm 0.014$ | $0.891\pm 0.013$ | $0.854\pm 0.016$
$2$ | $\mathbf{0.954\pm 0.011}$ | $0.925\pm 0.012$ | $0.901\pm 0.014$ | $0.892\pm 0.012$ | $0.857\pm 0.016$
$5$ | $\mathbf{0.955\pm 0.010}$ | $0.926\pm 0.012$ | $0.902\pm 0.014$ | $0.894\pm 0.012$ | $0.862\pm 0.016$
$9$ | $\mathbf{0.955\pm 0.010}$ | $0.926\pm 0.012$ | $0.902\pm 0.014$ | $0.895\pm 0.012$ | $0.865\pm 0.016$
#### Modeling task.
In this experiment, we apply our method for modeling agents whose search
behavior is specified by the Menu Search model. As in the first experiment, we
train the model parameters on data simulated with the given cognitive model.
For each simulation, we sample a new menu, together with its element-wise
information about the target word, as specified by Kangasrääsiö et al. [2019].
#### Modeling accuracy.
Table 1 summarizes the obtained modeling accuracies. We notice that after one
observed trajectory, the ANP-based model achieves results comparable to the
oracle upper bound. Unlike in the previous experiment, most of the users
seemed to converge to a relatively narrow and finite set of search strategies,
simplifying the difficulty of the modeling problem. As a result, MAML and the
transformer achieve clearly higher relative accuracy than in the previous
experiment, despite the limited training data.
#### Model misspecification.
We study the effects of model misspecification in cognitive models by
repeating the modeling task with a noisy model. This model represents an
otherwise accurate Menu Search model, but roughly $35\%$ of the saccades are
modeled randomly into incorrect locations instead of following the policy of
the correct model (full implementation details in the Supplement). We repeat
the meta-training with different percentages of data obtained from the true
user population. We explore both our solution’s robustness against the model
with action noise and its ability to adapt to the true generative process.
The results are gathered in Table 2. First, we observe that our approach can
remain robust against user model noise: even when trained solely on data
coming from the noisy model (ANP 100%), the modeling accuracy remains
reasonably good and surpasses the accuracy of the noisy model ($\approx
65\%$). Secondly, it can be seen how our solution adapts to the ground-truth
generative process when the proportion of the ground-truth data increases. We
repeated the scenario by meta-training the ANP only on data from the true user
population. We found that the noisy model improved the results when the number
of observed real users was under $200$ (i.e., here percentage $<20\%$), after
which it had a hindering effect on the predictions. However, we expect that
the utility of misspecified models can be significantly higher in more complex
modeling problems where more data is required to generalize to the problem.
### 4.3 Experiment 3: Menu search assistant
#### Setting.
In our third experiment, we aim to demonstrate the practical utility of the
proposed approach for interactive systems by extending the Menu Search
environment into a reasonably realistic AI-assistant scenario. First, we scale
the environment to consider two levels of hierarchy: the menu consists of a
main menu whose elements act as links to sub-menus; we use the menus of the
previous experiment.
Secondly, we introduce an AI assistant equipped with the proposed user
modeling system. The task of the assistant is to utilize the modeling system
to infer the target elements of the users based on observed search behaviors
in the current menu, and to propose sub-menus for the users. Intuitively, a
successful assistant should guide the users to menus that are likely to
contain the true target for them, to shorten their search time. The assistant
is allowed to provide any guidance only after the user is independently
searched through at least one sub-menu. We implement the assistant as a simple
rule-based agent that conditions its actions on the simulated user behaviors
$a\sim p_{\phi}(a\mid s,z)$, $z\sim p_{\psi}(z\mid(\mathbf{s},\mathbf{a}))$.
Further details on the experiment setting are in the Supplement.
Table 3: User search times and modeling/simulation times per assistant action with different assistant systems in Section 4.3. Assistant type | Search time (s) | Time saved (%) | Modeling time per action (ms)
---|---|---|---
No assistance | $4.774\pm 0.235$ | $-$ | $-$
MAML | $4.089\pm 0.645$ | $14.3$ | $1174.922\pm 43.760$
Reptile | $3.973\pm 0.519$ | $16.8$ | $1053.342\pm 37.988$
Transformer | $2.918\pm 0.191$ | $38.8$ | $\mathbf{1.140\pm 0.442}$
ANP | $\mathbf{2.590\pm 0.226}$ | $\mathbf{45.7}$ | $8.460\pm 6.495$
Full knowledge | $\mathbf{2.577\pm 0.162}$ | $\mathbf{46.0}$ | $-$
#### Results.
Table 3 compares the performance of the resulting assistant against a non-
assisted user, a MAML-based, a Reptile-based, and a transformer-based
solutions. The MAML and Reptile-based solutions require gradient-computation
during test-time leading to modeling times greatly higher than the response
time between human actions ($\approx 300$ms) in this experiment. This prevents
online user model updates, hence hindering the effectiveness of the
assistance. We also include results with an assistant that has full knowledge
of the users’ target elements to provide an upper-bound. We notice that the
ANP-guided assistant can significantly reduce the user’s search time and
almost reaches the upper-bound performance of the assistant that has perfect
knowledge. The observed results are encouraging regarding the ability of our
solution to guide the behaviors of real-time interactive systems.
## 5 Related work
Our work connects to a larger body of research considering user modeling in
interactive AI. For instance, Carroll et al. [2019] and Strouse et al. [2021]
share the idea that efficient interaction with humans requires the AI to have
an accurate model of the human. In contrast to many this line of works, our
work concentrates on using models based on cognitive and behavioral sciences
as priors, instead of ML-experts hand-crafting the models from scratch or
learning them from large collections of user data. Using such models has been
impractical up to now, and this the problem we now solve.
Inverse reinforcement learning (IRL) [Ng et al., 2000] considers a related
problem to ours, aiming to recover agents’ reward functions based on observed
behaviors. Although it has been previously utilized also in user-centric
problems [Chandramohan et al., 2011], our perspective is more general as we
consider inference over arbitrary user parameters (instead of only rewards)
and over varying policy-generative algorithms/processes. This allows our
approach to be utilized for inference with a wide range cognitive models,
where user behaviors are not necessarily optimal and are governed by human
biases. Imitation learning (IL) [Hussein et al., 2017], on the other hand,
considers learning models to imitate human (expert) behaviors on a given task.
The crucial difference to our setting is that, unlike with IL, we do not
necessarily seek to solve the task the human is solving, but to
probabilistically model humans and their behaviors.
Using transfer and meta-learning in RL problems has been previously widely
studied. For instance, Yao et al. [2018] used HiP-MDPs [Doshi-Velez and
Konidaris, 2016] for modeling differences in environment dynamics and to
further parametrize a policy. Similarly, Galashov et al. [2019] propose a
probabilistic framework for sequential decision-making that they instantiate
with NPs for meta-learning. In contrast to this line of works, the novelty of
our work is not about a generalizable solution to distributions of RL tasks,
but rather about a generalizable method for making modeling with cognitive
models practical. This is an important distinction because cognitive models
are not necessarily compatible with the RL formalism — even when they are,
they are based on computational rationality, and specifically tailored to
account for cognitive limitations. Adapting these limitations to existing
frameworks, such as HiP-MDPs, is not trivial and necessarily requires manual
effort.
Our work also connects to a line of research studying inference for decision
making agents in the context of probabilistic programming. However, most of
the approaches make restricting assumptions either regarding the behavior
generative processes of the users or the inference objectives and could be
applied only for very limited types of problems. For instance, Zhi-Xuan et al.
[2020] consider online inference of boundedly-rational agents but their
approach can be applied only in discrete and deterministic environments to
capture only agent goals. Furthermore, their solution assumes that the agents
start planning their policy from scratch during interaction — in practical
interactive settings, humans might already have a partial or complete plans at
the beginning of interaction. On the other hand, many other works, such as by
Seaman et al. [2018], assume that the likelihood for the generative process
$p(\pi\mid\theta)$ can be evaluated for MCMC, which is often an unrealistic
assumption with advanced cognitive models.
Many computational approaches motivated by cognitive science share parallels
with our objectives. For instance, computational rationality [Lewis et al.,
2014, Gershman et al., 2015] and Theory of Mind (ToM) [Premack and Woodruff,
1978] have motivated numerous computational approaches such as Bayesian ToM
[Baker et al., 2011], Machine Theory of Mind [Rabinowitz et al., 2018] and the
Menu Search model [Chen et al., 2015] for modeling human behaviors.
Furthermore, Peltola et al. [2019] utilize ToM for modeling users with their
own models of the interactive system in bandit settings. Among others, these
models are prime candidates our method can be applied to.
## 6 Discussion
In this work, we have addressed the so-far unaddressed problem of enabling
probabilistic user modeling with complex cognitive models in real-time
applications. We introduced a meta-learning approach for training widely
applicable differentiable surrogates for approximating posterior predictive
estimation with cognitive models. We studied neural process models as example
implementations for the surrogates and demonstrated comparable modeling
performance to likelihood-free inference with computational cost suitable for
online applications. We also showed that the proposed solution allows AI-
assistants to utilize cognitive user models computationally feasibly, for
instance in a previously studied menu-search task. In a larger scale, the
solution not only removes a computational bottleneck currently hindering
incorporation of users into probabilistic programming models, but also enables
real-time user modeling in various applications where they currently are not
possible within usually available computational budgets.
We also demonstrated how the effects of model misspecification in cognitive
models can be mitigated in the surrogates by incorporating observed user data
in the training. Importantly, we observed that our approach provided
robustness against action noise while adapting to the true population as more
behavior data became available. Based on these observations, we conclude that
the proposed solution can be particularly useful in application domains where
user data are limited or behavioral user models can be slightly misspecified,
although future studies are still required in settings where the
misspecification is caused by more systematic biases.
It is crucial to note that amortization for probabilistic user modeling with
cognitive models, as detailed in Section 2, has not been previously widely
studied. Apart from LFI, which is computationally intractable for our
problems, we are not aware of any solutions which could act as either relevant
baselines or alternatives to the proposed approach. Specifically, all the
experimented alternative surrogate implementations, such as MAML and
transformers, are not fully consistent with the probabilistic nature of the
problem, limiting their applicability in practice. We further note that also
neural processes feature several compromises in comparison to a fully Bayesian
setting with cognitive models: although supporting posterior predictive
estimation, they cannot be directly adapted for Bayesian inference in an
explicit, predefined parameter space and they do not necessarily follow all
constraints coming from the known structure of the behavioral model due to
amortization. Future research should adapt and develop alternative surrogate
solutions to address these drawbacks.
Interesting avenues for future research include utilizing the surrogates in
full probabilistic programming pipelines, although we hypothesize that this
should already be possible within certain limits with our approach. Other
attractive extensions could consider alternative surrogate architectures to
handle, for instance, non-stationarity in cognitive models and settings with
multiple data modalities. Regarding ethical considerations, user modeling has
always been a double-edged tool and can potentially be abused to serve other
interests than those of users — this should be taken carefully into account in
all of its applications. As a generic tool to mitigate some of these issues,
we recommend combining user systems with privacy preservation with
differential privacy.
###### Acknowledgements.
We would like to thank Sebastiaan De Peuter, Pierre-Alexandre Murena and
Sammie Katt for their valuable advice and feedback. This work was supported by
the Academy of Finland (Flagship programme: Finnish Center for Artificial
Intelligence FCAI and decision 345604) Humane-AI-NET and ELISE Networks of
Excellence Centres (EU Horizon: 2020 grant agreements 952026 and 951847), and
UKRI Turing AI World-Leading Researcher Fellowship (EP/W002973/1). We also
acknowledge the computational resources provided by the Aalto Science-IT
Project from Computer Science IT.
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## Appendix A Experiment 1 details
#### Setting.
The first experiment scenario considers modeling Monte Carlo Tree Search
(MCTS) [Browne et al., 2012] agents in a simple $10\times 10$ gridworld
environment. The gridworld environment is defined as a partially observable
Markov decision process (POMDP) with deterministic transition dynamics. The
state and action spaces are shared across the all agents, including the
transition function; the reward and observation functions are individual for
each agent.
The state space $\mathcal{S}=\\{1,\dots,10\\}^{2}$ captures all possible agent
locations, i.e., grid states. The action space is defined as
$\mathcal{A}=\\{\text{up},\text{down},\text{left},\text{right}\\}$, the
transition function corresponds to transitioning to adjacent grid states in
relation to the agent’s current location according to the selected action.
Actions attempting to relocate agents out of the grid do not cause state
transitions.
The agents are fully described by their generative process $\pi\sim
p(\pi\mid\theta)$ (MCTS) and parameters $\theta\sim p(\theta)$. Their
respective distributions are described in Table 4. In addition to the reward
function, determining one positive and one negative reward state, the prior
determines the agent memory, observation function, and MCTS planning-tree
depth. The agent memory is defined as a binary parameter determining if the
agents utilize the planning tree from previous time steps for subsequent
planning instead of starting from scratch. The agents are divided into two
subspecies depending on their observation function. Specifically, the complete
gridworld is fully observable for half of the agent population, while the
other half cannot observe reward states that are located beyond planning-tree
depth. In practice, such agents avoid exploring the environment if the
positive reward is not directly observed.
Table 4: Uniform prior on user model parameters for the user population in gridworld experiment. User Parameter | Distribution
---|---
Reward States ($x,y$) | $\mathcal{U}\\{1,\dots,10\\}$
Memory | $\mathcal{U}\\{0,1\\}$
Observation function | $\mathcal{U}\\{0,1\\}$
Tree Depth | $\mathcal{U}\\{5,\dots,10\\}$
Here the task is to model individual users sampled from the population
$\theta\sim p(\theta)$. The users generate $n\sim\mathcal{U}\\{1,\dots,8\\}$
trajectories of length $10$, each collected from one episode in the
environment. A new initial user state, $x,y\sim\mathcal{U}\\{1,\dots,10\\}$,
is sampled at each episode. The trajectories are then partitioned into context
and target data for NP training by randomly selecting and then truncating one
target trajectory at length $l\sim\mathcal{U}\\{1,\dots 9\\}$. The beginning
of the truncated trajectory, together with the other trajectories, forms the
context dataset while the remaining half is held-out as modeling target. Based
on the given context data, we evaluate prediction accuracy on the held-out
target datasests over tasks. Unlike in the rest of our experiments, each
method is here trained with data from $10000$ users.
#### Implementation.
All NP models are implemented on the basis of the `NeuralProcesses.jl`
library, a Julia variant of the neural process library of Dubois et al.,
(2020). The MCTS agents are implemented by utilizing the `POMDPs.jl` (Egorov
et al., 2017) library. All NP models are meta-trained on a single GPU (NVIDIA
Quadro P2200). The code used to produce all the results in this paper can be
found at https://github.com/hamalajaa/DifferentiableUserModels.
The base-architecture is shared among all NP models (summarized in Table 5).
We implement the MAML as first-order MAML [Finn et al., 2017] to reduce test-
time computation. The Reptile, MAML and transformer details are included in
Tables 6 and 7 (Remark: here, the ’fifth’ action corresponds to ’staying
still’ when the positive reward is found.)
Table 5: Base-architecture shared by all NP models in experiment 1. Encoder | | Decoder |
---|---|---|---
Number of layers | $6$ | Number of layers | $6$
Activations | Leaky ReLU | Activations | Leaky ReLU
Hidden dimensions | $128$ | Hidden dimensions | $128$
Latent distribution | Gaussian | Output distribution | Categorical
Input dimensions | $2$ | Output dimensions | $5$
Table 6: The architecture shared by Reptile, MAML, and transformer in experiment 1. Architecture |
---|---
Layers | Transformer + $6$ MLP
Activations | ReLU (+ softmax)
Hidden dimensions | $128$
Input dimensions | $2$
Output dimensions | $5$
Table 7: Reptile, MAML, and transformer training and evaluation details in experiment 1. Training | Reptile | MAML | Transformer
---|---|---|---
Optimizer and learning rate | - | - | Adam, $5\cdot 10^{-4}$
Meta optimizer | Adam, $5\cdot 10^{-3}$ | Gradient descent, $5\cdot 10^{-3}$ | -
Batch optimizer | Gradient descent, $5\cdot 10^{-3}$ | Gradient descent, $5\cdot 10^{-3}$ | -
Loss | Cross entropy | Cross entropy | Cross entropy
Evaluation | | |
Optimizer and learning rate | Gradient descent, $0.01$ | Gradient descent, $0.01$ | -
N of gradient steps | $32$ | $32$ | -
## Appendix B Experiment 2 details
#### Setting.
The second experiment is based on the Menu Search model of Kangasrääsiö et al.
[2019]. The Menu Search model is a cognitive model describing human search
behavior in terms of eye movements (saccades) when searching for a target item
in a computer dropdown menu. The model simulates user behaviors as a result of
optimization with RL given their cognitive constraints. In this experiment, we
implement the users as deep Q-learning agents to reduce data generation costs.
Formally, the environment is specified as a POMDP where states contain
information about (1) the user’s current knowledge about the menu elements,
(2) the current gaze focus, and (3) whether or not the user has closed the
menu (i.e., quit). We consider a menu of eight elements, where each element is
described with its semantic relevance and length given the user’s target
element. At each step, the user can either fixate on a menu element or quit
the scenario. Fixating on a menu element has a chance to reveal the given
element while also having a chance to reveal the adjacent elements via
peripheral vision. If user action results in revealing the target element, the
element is automatically selected, and a significant positive reward is given.
The target word is not present in $10\%$ of the generated menus. If the user
recognizes that the target element is not present and quits the menu, a large
reward is emitted. For each action, the user is otherwise given a small
negative reward based on the duration of the action specified by cognitive
parameters presented in Table 8 as priors. When entering a menu for the first
time, there is a chance, $p_{rec}$, that the user recalls the menu, completely
revealing the entire menu layout.
Table 8: Distributions for user cognitive properties used in the second experiment. User Parameter | | Distribution
---|---|---
Menu recall probability | $p_{rec}$ | $\text{{Beta}}(3.0,1.35)$
Eye fixation duration | $f_{dur}$ | $\mathcal{N}(3.0,1.0)$
Target item selection delay | $d_{sel}$ | $\mathcal{N}(0.3,0.3)$
Similarly as in the first experiment, each user completes
$n\sim\mathcal{U}\\{1,\dots,8\\}$ search tasks with independently sampled
menus and target elements constructing $n$ trajectories. Similarly, as in the
first experiment, we truncate one of the trajectories to form global and local
contexts and the prediction target for ANP training. The ANP, Reptile, MAML,
and transformer architectures follow the design used in the first experiment
(except for the input and output dimensions).
#### Implementation.
The implementation details for ANP, Reptile, MAML, and transformer
architectures follow the design used in the first experiment (except for the
input and output dimensions).
## Appendix C Experiment 3 details
The third experiment extends the menu search environment into a relatively
realistic AI-assistant scenario. First, the menu environments are scaled to
consider two levels of hierarchy: each full menu consists of a main menu whose
elements correspond to labels that (1) act as links to sub-menus and (2)
summarize the contents of these menus. Secondly, we introduce an AI assistant
equipped with the proposed user modeling system. The task of the assistant is
to utilize the modeling system to propose sub-menus for the users.
Intuitively, a successful assistant should guide the users to menus that are
likely to contain the true target to shorten their search time.
#### Environment.
The hierarchical menu search environment introduces an $8\times 8$ two-level
menu setting. Importantly, the environment behaves otherwise similarly to the
original non-hierarchical version, with the exception of introducing a main
menu that allows a user to navigate between multiple menus. In addition, we
introduce a simple mapping between user observations (semantic relevancies and
lengths w.r.t. the target element) and assistant observations (logical
groups). Specifically, each scenario introduces a set of $8$ logical groups
$\mathcal{S}_{AI}=\\{1,\dots,8\\}$ and $4$ semantic relevance groups
$\mathcal{S}_{user}=\\{target(1),high(2),medium(3),low(4)\\}$ and an
independently generated bidirectional mapping between $\mathcal{S}_{AI}$ and
$\mathcal{S}_{user}$. The mapping initializes an ordered set of relevancies as
$r=\\{4,4,4,3,3,2,3,3\\}$ and assigns a relevance for each logical group with
a randomized circular shift on $r$. The intuition of the mapping is simply to
mask the semantic information regarding the target element (via randomization)
while allowing a soft prior heuristic for the assistant by conserving semantic
similarity between similar logical groups. We similarly mask the item lengths
via randomization.
After the mapping between the observation spaces $\mathcal{S}_{AI}$ and
$\mathcal{S}_{user}$ is constructed, we sample two logical groups for each
sub-menu (such that each group occurs exactly twice in the full menu) and
determine a semantic label for the menus summarizing the relevancies of their
respective logical groups. The target element is then assigned randomly into
one of the sub-menus that includes a logical group with $high$ relevance. The
contents for each sub-menu are otherwise determined by mapping the semantic
labels of their logical groups into individual items according to the original
menu search model specifications. The main menu similarly follows the original
specifications — however, we utilize the semantic labels of the corresponding
sub-menus as the relevancies for the main menu elements. At the main menu
level, we also replace the item length information with a binary variable
denoting if the user has already opened the corresponding sub-menu. Finally,
the transition dynamics between the main menu and sub-menus are defined as
follows: selecting an element at the main menu -level transitions the user to
the corresponding sub-menu, while quitting a sub-menu transitions the
environment state back to the main menu. Otherwise, all the transition and
reward dynamics follow the original environment specifications.
#### Assistant.
The hierarchical menu search setting involves a simple search assistant guided
by a pre-trained ANP-based user model (user model implementation and training
details are described below). In each scenario, the assistant is initially
inactive and only activates if the user fails to find the target element from
the first sub-menu it explores. When activated, the assistant may suggest and
highlight an individual main menu element when the user is at the main menu
level. A highlighted main menu element is assumed to attract the attention of
the user at its next action and the user’s gaze is guided towards the
highlighted element. Simultaneously, we assume that the user features some
degree of trust towards the assistant’s suggestion and the semantic relevance
score of the highlighted element is increased by one level. In practice, this
allows the user also to reject poor suggestions.
We implement the assistant as a simple rule-based agent that continuously
updates the user model as new user actions are observed. We assume that the
assistant can track users’ gaze locations but that it does not have access to
the semantic relevancies of the items. Instead, the assistant updates its
estimate on the currently observed (and unobserved) menu elements in terms of
the observation space $\mathcal{S}_{AI}$ specified above. When activated, the
assistant simulates one user action at fully observed main menu -level
conditioned on the observed user search behavior: $a\sim p_{\phi}(a\mid s,z)$,
$z\sim p_{\psi}(z\mid(\mathbf{s},\mathbf{a}))$. The main menu element
corresponding to the estimated most likely user action is then selected as the
assistant’s suggestion.
#### Implementation and training details.
The ANP-based user model is meta-trained on a single GPU. Each user generates
$1$ trajectory which is split at length $l\sim\mathcal{U}\\{2,\dots,10\\}$
into context and target trajectories for ANP training. The base architectures
of the ANP, Reptile, MAML and transformer models are identical to the previous
experiment. The online prediction times are run on a laptop CPU (Intel Core
i7-7700HQ).
|
[a,b]Phuong Nguyen
# Performance Optimization of Baryon-block Construction in the Stochastic LapH
Method
Ben Hörz
###### Abstract
Implementations of measurement kernels in high-level Lattice QCD frameworks
enable rapid prototyping, but can leave hardware capabilities significantly
underutilized. This is an acceptable tradeoff if the time spent in unoptimized
routines is generally small. The computational cost of modern spectroscopy
projects however can be comparable to or even exceed the cost of generating
gauge configurations and computing solutions of the Dirac equation. One such
key kernel in the stochastic LapH method is the computation of baryon blocks;
we discuss several implementation strategies and achieve a 7.2x speedup over
the current implementation on a system with Intel® Xeon® Platinum 8358
processors, formerly Ice Lake.
## 1 Introduction
With the advent of modern spectroscopy methods for multi-hadron systems [1, 2,
3], ever more complicated physical systems are coming into reach. While a
variety of two-meson systems including with several coupled channels have been
investigated (see [4] for a plenary review at this conference), and more
recently even studies of three-meson systems have started appearing [5, 6, 7,
8, 9, 10, 11, 12] (also subject of a topical review this year [13]), the
situation for systems with baryons is comparably less advanced. Hadron
interactions involving baryons are however among the fundamental building
blocks for an understanding of nuclear physics rooted in QCD (for a review,
see for instance [14]). Examples include two-nucleon interactions, which –
when determined at sufficiently light pion mass – can serve as a validation
system to establish the reliability of these kinds of lattice QCD calculations
[15, 16, 17, 18]; three-nucleon interactions, which are difficult to access
phenomenologically and hence present a great opportunity for lattice QCD to
provide useful data; a variety of meson-baryon systems, for instance nucleon-
pion scattering in the isospin $I=3/2$ channel featuring the $\Delta(1232)$
resonance [19, 20, 21].
Even though modern spectroscopy methods present no conceptual difficulties
generalizing to systems involving baryons, there are a few challenges in
practice. Baryonic systems typically suffer from a worse signal-to-noise ratio
than purely mesonic systems, requiring larger amounts of statistics to obtain
meaningful results. In addition to necessitating more correlation-function
samples, every individual sample tends to be more computationally expensive
compared to the mesonic sector due to the increased number of quark fields. At
the correlator-construction level, algorithms eliminating redundant
computations have been devised to alleviate the proliferation of Wick
contractions [22, 23, 24, 5].
This work is concerned with improving the efficiency of the computation of
baryon functions in the stochastic LapH method [2]. In the stochastic LapH
framework, baryon blocks are rank-three tensors in dilution indices, carrying
additional labels identifying the baryon operator (flavor, spin, hadron
momentum) as well as the three noises used for the stochastic estimate of the
quark propagators in the LapH subspace. Correlators are then computed through
tensor contractions over dilution indices of those baryon functions as
governed by Wick’s theorem. This beneficial property of the stochastic LapH
method – the evaluation of complicated multi-hadron correlation functions is
reduced to tensor contractions involving blocks representing the constituent
hadrons – enables the re-use of baryon blocks for a wide variety of physical
systems. The optimizations presented in this work are hence immediately
applicable to a breadth of calculations involving baryons.
This contribution is organized as follows: section 2 defines the baryon-block
kernel and discusses its computational characteristics, section 3 contrasts
several implementation strategies, and benchmark results are presented in
section 4.
## 2 Baryon blocks in the stochastic LapH method
The stochastic LapH method [2] is a stochastic variant of distillation [1]
which avoids the $V^{2}$ scaling with the spatial simulation volume $V$ by
stochastically estimating the quark propagator projected into the LapH (or
distillation) subspace. A useful quantity for the computation of multi-hadron
correlation functions involving baryons is the (single-site) baryon function
defined per time slice of the simulation volume,
$\displaystyle
B_{d_{1}d_{2}d_{3}}^{(\vec{p},\Lambda,\mu,\eta)}=c^{(\vec{p},\Lambda,\mu)}_{\alpha\beta\gamma}\sum_{\vec{x}}\mathrm{e}^{-\mathrm{i}\,\vec{p}\vec{x}}\epsilon_{abc}q^{(\eta_{1},d_{1})}_{\alpha
a\vec{x}}q^{\prime(\eta_{2},d_{2})}_{\beta
b\vec{x}}q^{\prime\prime(\eta_{3},d_{3})}_{\gamma c\vec{x}},$ (1)
with color indices $a$, $b$, $c$, and the summation runs over the sites of a
three-dimensional time slice of the lattice. The relevant combinations of spin
indices $\alpha$, $\beta$, $\gamma$ are selected according to the group-
theoretical projection coefficients
$c^{(\vec{p},\Lambda,\mu)}_{\alpha\beta\gamma}$ for a given hadron momentum
$\vec{p}$, irrep $\Lambda$ and irrep row $\mu$ [25]. The quark fields $q$,
$q^{\prime}$, $q^{\prime\prime}$, which have been projected into the LapH
subspace, are obtained by repeatedly solving the Dirac equation with diluted
stochastic sources identified by a noise label $\eta$ and dilution index
$d_{1}=1,\dots,N_{\mathrm{dil}}$. Crucially, those solutions of the Dirac
equation for a given noise need only be computed once, and can then be cheaply
stored on disk and used for various multi-hadron projects by reconstructing
the smeared quark fields from their coefficients $Q$ in the basis of
eigenvectors of the three-dimensional gauge-covariant Laplacian $\phi^{(l)}$,
$l=1,\dots,N_{\mathrm{ev}}$, which is used to define the LapH subspace,
$\displaystyle q^{(\eta,d)}_{\alpha
a\vec{x}}=\sum_{l=1}^{N_{\mathrm{ev}}}Q^{(\eta,d)}_{\alpha
l}\phi^{(l)}_{a\vec{x}},$ (2)
and similarly for $q^{\prime}$ and $q^{\prime\prime}$, which differ only in
their coefficients $Q$, but with the same eigenvectors $\phi$.
Both the group-theoretical projection involving the spin indices and the
bookkeeping of noise indices in (1) are handled by the calling application,
leaving
$\displaystyle
B_{d_{1}d_{2}d_{3}}^{(\vec{p})}=\sum_{\vec{x}}\mathrm{e}^{-\mathrm{i}\,\vec{p}\vec{x}}\epsilon_{abc}Q^{(1)}_{d_{1}l_{1}}Q^{(2)}_{d_{2}l_{2}}Q^{(3)}_{d_{3}l_{3}}\phi^{(l_{1})}_{a\vec{x}}\phi^{(l_{2})}_{b\vec{x}}\phi^{(l_{3})}_{c\vec{x}},$
(3)
where summation over the eigenvector indices $l_{1}$, $l_{2}$ and $l_{3}$ is
implied, as the computational kernel, which is called many times for different
noise and spin combinations, i.e. different quark field coefficients, but with
the same momentum set and Laplacian eigenvectors.
Depending on the sizes of $N_{\mathrm{ev}}$ and $N_{\mathrm{dil}}$, as well as
how many times the kernel (3) is called per time slice, one of the following
two different approaches is preferable.
For moderate values of $N_{\mathrm{ev}}$, (3) can be efficiently computed
using a two-step procedure: During the setup stage, the mode-triplets
$\displaystyle
T_{l_{1}l_{2}l_{3}}^{\vec{p}}=\sum_{\vec{x}}\mathrm{e}^{-\mathrm{i}\,\vec{p}\vec{x}}\epsilon_{abc}\phi^{(l_{1})}_{a\vec{x}}\phi^{(l_{2})}_{b\vec{x}}\phi^{(l_{3})}_{c\vec{x}},$
(4)
which are spin-, noise- and flavor-blind, are computed and kept in memory. All
lattice-sized objects in (3) have then been consumed, and the baryon function
for a given set of quark-field coefficients can be computed by tensor-
contracting them onto the precomputed mode-triplet. Those tensor contractions
can be performed with high performance, so the majority of the runtime tends
to be associated with the initial setup phase, which needs to be amortized
over many kernel invocations. The major drawback of this mode-triplet approach
is the need to keep one $N_{\mathrm{ev}}^{3}$-sized object per momentum in
memory111Based on the symmetries of (4), only ${N_{\mathrm{ev}}\choose 3}$
elements of a mode-triplet are independent. Exploting that symmetry with a
sparse storage scheme however complicates the subsequent tensor contractions
of quark-field coefficients onto the mode-triplet..
For large number of eigenvectors $N_{\mathrm{ev}}$, a more economical approach
is to first reconstruct the quark fields from the coefficients as per (2), and
subsequently perform the reduction (3) over sets of lattice-sized objects. The
quark-field reconstruction can be efficiently implemented using matrix-matrix
multiplication and reduces the complexity of subsequent lattice-sized
reductions to $N_{\mathrm{dil}}^{3}$ (rather than $N_{\mathrm{ev}}^{3}$ for
the mode-triplet approach), which however must be performed for every kernel
invocation.
In view of the requirements of baryon calculations in large volumes – such as
the E250 ensemble [26] generated by the CLS effort [27, 28], where employing
the mode-triplet approach is not feasible222First results in the mesonic
sector presented in [29] used the analogous mode-doublet approach for meson
construction, which is still affordable due to its slightly weaker
$N_{\mathrm{ev}}^{2}$ scaling. For the $N_{\mathrm{ev}}=1536$ employed in that
work, the mode triplet on the other hand occupies $1.8\,\mathrm{TB}$ of memory
already for the moderate number of momenta $N_{\mathrm{mom}}=33$, clearly
making this approach impractical. – the goal of this work is to provide an
efficient implementation of (3), which utilizes the great compute capabilities
of modern hardware by exploiting the $N_{\mathrm{dil}}^{3}$ compute complexity
with only linear-in-$N_{\mathrm{dil}}$ memory traffic.
## 3 Implementation details
Algorithm 1 Cache blocking algorithm with pseudo code
1:$q_{1},q_{2},q_{3},phase$
2:$baryon$
3:
4:! $Block{D_{i}}\leftarrow N_{D_{i}}\,/\,Bsize{D_{i}}$ ($i=1,2,3$)
5:! $BlockX\leftarrow N_{X}\,/\,Bsize{X}$
6:function BaryonConstruct($q_{1},q_{2},q_{3},phase$)
7: for $Block{D_{1}}$ do in parallel
8: for $Block{D_{2}}$ do in parallel
9: tmpBuf $\leftarrow 0.$
10: for each $Block{X}$ do
11: for $d_{1}\leftarrow 1$ to $Bsize{D_{1}}$ do
12: for $d_{2}\leftarrow 1$ to $Bsize{D_{2}}$ do
13: for $x\leftarrow 1$ to $Bsize{X}$ do
14: $diq(d_{1},d_{2},:,x)\leftarrow q_{1}(\tilde{d_{1}},:,x)\times
q_{2}(\tilde{d_{2}},:,x)$
15: end for
16: end for
17: end for
18: for each $Block{D_{3}}$ do
19: for $d_{3}\leftarrow 1$ to $Bsize{D_{3}}$ do
20: for each $diq_{i}$ in $diq$ do
21: for $x\leftarrow 1$ to $Bsize{X}$ do
22: $singlet(d_{1},d_{2},d_{3},x)\leftarrow diq_{i}\times
q_{3}(\tilde{d_{3}},:,x)$space
23: end for
24: end for
25: end for
26: tmpBuf(:) $\leftarrow$ tmpBuf(:) \+ $singlet\times phase(:)$
$\triangleright$ Intel® MKL JIT GEMM
27: end for
28: end for
29: $baryon\leftarrow$ tmpBuf
30: end for parallel
31: end for parallel
32: return $baryon$
33:end function
Cache Blocking in $N_{D_{1}}$, $N_{D_{2}}$, $N_{D_{3}}$, $N_{X}$
The quark-field reconstruction can be performed efficiently using matrix-
matrix multiplication, for which highly optimized implementations are
available for all hardware architectures. Hence, in the following section, we
focus on optimizing the baryon-block calculation given the reconstructed quark
fields $q_{1}$, $q_{2}$, $q_{3}$.
The optimized algorithm is shown in Algorithm 1. Typically, the number of
requested hadron momenta $N_{\mathrm{mom}}$ is much smaller than the number of
allowed momenta (e.g. $33\ll 64^{3}$); therefore using a fast Fourier
transform is not beneficial. Hence, the phase factor
$e^{-\mathrm{i}\vec{p}\vec{x}}$ for the momentum projection in (3) can be
precomputed and re-used for several kernel invocations.
Cache blocking techniques are employed in conjunction with an appropriate data
layout to optimize data locality. Blocking is implemented both in the spatial
indices $x$ and the three dilution indices $d_{1},d_{2},d_{3}$. The blocking
in $x$ allows the kernel to exploit the available inherent input reuse. For
example, each block of $q_{1}$ input can be kept in the cache and re-used for
different $diq$ calculations with different $q_{2}$ since the input size is
small enough to stay in the cache (ll. 8-14). The blocking in
$d_{1},d_{2},d_{3}$ enables the kernel to keep the intermediate data (diq,
singlet, tmpBuf) in cache and use it for subsequent calculations (ll. 16-22).
A suitable data memory layout is (in row-major convention)
$N_{\mathrm{dil}}\times N_{\mathrm{BlockX}}\times N_{\mathrm{color}}\times
N_{\mathrm{BsizeX}}$ for the input $q_{1},q_{2},q_{3}$, ensuring that the data
is accessed contiguously in $x$ for each each color component. Furthermore,
this data layout stores the BlockX-sized chunks for the three colors of
$q_{1},q_{2},q_{3}$ adjacently, enhancing spatial locality in the calculations
of diq and singlet.
The small matrix-matrix multiplication in l. 23 to compute
$\textit{tmpBuf(:)}+=singlet\times phase(:)$, utilizes the Intel® Math Kernel
Library (Intel® MKL) with just-in-time (JIT) code generation for small
matrices with
$m=\mathrm{BsizeD}_{1}\times\mathrm{BsizeD}_{2}\times\mathrm{BsizeD}_{3}$,
$n=N_{\mathrm{mom}}$ and $k=\mathrm{BsizeX}$. These matrices – $singlet$ and
tmpBuf – should be sufficiently small to remain in the cache. General-purpose
GEMM implementations are typically optimized targeting larger matrix sizes.
Thus, for the small matrix-matrix multiplication required here, the Intel®
Math Kernel Library (Intel® MKL) with JIT compilation is used to generate
target microarchitecture code for the kernel which is optimized for small-
sized complex-valued matrix multiplication problems. As the matrix sizes are
fixed, the kernel can be produced once and then called many times, amortizing
the JIT compilation overhead.
Parallelization for multiple threads is achieved by distributing the work in
the loops over dilution-index blocks, (ll. 4-5) . In practice, the loops over
$\mathrm{BlockD}_{1}$ and $\mathrm{BlockD}_{2}$ are collapsed into a joint
iteration space and parallelized for multiple threads with OpenMP. In this
approach, there are no data dependencies since computations of each thread are
fully independent. Furthermore, as each block works on an identical amount of
data, the load is expected to be well-balanced between threads. Lastly,
parallelization is implemented at an outer level, encompassing plenty of work
per loop trip to amortize the OpenMP runtime overhead for instance for thread
scheduling.
## 4 Performance results
Figure 1: Performance of the previous and optimized kernel normalized to the
single-thread performance of the previous implementation. The optimized kernel
outperforms the baseline by up to 6.8x for the test with fixed frequency
(left) and by up to 7.2x for the test when turbo boost is enabled (right).
The implementation of Algorithm 1 is evaluated on a test system with two
Intel® Xeon® Platinum 8358 processors @ 2.60 GHz for a moderately large
problem size of $L=64$ with $N_{\mathrm{dil}}=64$ dilution indices per quark
field and number of requested momenta $n_{\mathrm{mom}}=33$. Figure 1 shows
the performance of the optimized kernel compared to the previous
implementation using 1 to 64 cores.
The block sizes are tunable parameters which generally depend on the target
architecture and should be tuned experimentally. For our test node, the block
sizes yielding the best performance are $\mathrm{BsizeX}=32$,
$\mathrm{BsizeD}_{1}=4$, $\mathrm{BsizeD}_{2}=8$, $\mathrm{BsizeD}_{3}=16$.
This can be understood as aligning the memory footprint of each single
intermediate object with the available cache hierarchy. For instance, the
$diq$ array stores $\mathrm{BsizeD}_{1}\times\mathrm{BsizeD}_{2}\times
N_{\mathrm{colors}}\times\mathrm{BsizeX}=3072$ complex double-precision
values, occupying 48 kB of memory, thus fitting perfectly into the L1 cache.
Similarly, the $singlet$ has a size of 256 kB which fits into the L2 cache.
When run on a single core, the optimized kernel is 1.8x and 2.5x faster than
the baseline in tests with and without turbo boost, respectively. With in-
depth profiling, the superior performance of the optimized implementation can
be traced back to better use of the memory system, as expected. The optimized
kernel operates at 1.6x and 3x higher arithmetic intensity in Read and Write,
respectively. In addition, the memory access patterns are also significantly
improved such that fewer cycles are spent on load and store operations in the
optimized kernel in comparison to the baseline (42% reduction in loads and 70%
reduction in stores). The optimized kernel is indicated to be L1-bound instead
of DRAM-bound. As a result, with a performance of 37.7 DP GFlops/s, the kernel
reaches 54% of the theoretical peak performance.
Figure 2: Strong-scaling behavior of the optimized kernel. Left: With fixed
frequency, the kernel achieves almost perfect parallel efficiency for a 62.5x
speedup with 64 threads. Right: With turbo enabled, the kernel scales well up
to eight cores. Beyond eight cores, the variable frequency gets throttled and
the decrease in parallel efficiency matches the decrease in frequency,
implying that the optimized kernel is compute-bound.
The improvement of the optimized kernel becomes particularly apparent in
multithreaded runs, where it outperforms the baseline by 1.8x to 6.8x at fixed
frequency and 2.5x to 7.2x for tests with turbo boost on (Figure 1). This
boost in performance can be understood as being due to improved temporal
locality. As long as threads progress at a comparable rate, adjacent threads
working on a different BlockD2 but sharing the same BlockD1 access the same
input data, which may be served from the cache hierarchy.
Figure 2 shows the strong-scaling behavior of the optimized kernel at fixed
clock frequency, achieving almost perfect parallel efficiency at a speedup of
62.5x when using 64 cores. In addition, the parallel efficiency is above 0.98
for all tests, implying that the kernel scales almost perfectly within a node.
For production runs with turbo boost enabled the clock frequency can increase
up to the boost frequency (3.6 GHz on our test system). While performance in
absolute terms is slightly better than at fixed frequency, the strong-scaling
parallel efficiency deteriorates, showing a 39.6x speedup with 64 cores
(Figure 2). The loss in parallel efficiency is due to frequency throttling.
While for one thread the average frequency is $3.285\,\mathrm{GHz}$, it drops
to $2.219\,\mathrm{GHz}$ when running with 32 threads. The frequency ratio
$0.67$ matches the parallel efficiency, indicating that the optimized kernel
is compute-bound and indeed limited by the frequency throttling.
Figure 3: Scaling performance of the optimized kernel with respect to the
problem size. The kernel scales as expected to within a few percent both with
the spatial volume and the number of dilution indices.
The scalability of the optimized kernel with respect to the problem size, of
importance in view of ever-increasing simulation volumes, is shown in Figure
3. As is evident from (3), the computational cost scales as
$\mathcal{O}(L^{3})$ and $\mathcal{O}(N_{\mathrm{dil}}^{3})$ with the spatial
volume and number of dilution indices, respectively. The optimized kernel
scales as expected to within a few percent as a function of the problem size
both with the spatial volume as well as the number of dilution indices.
## 5 Summary
We have presented an optimized implementation of the kernel computing baryon
blocks in the stochastic LapH method, achieving an up to 7.2x speedup over the
previos implementation. Exploiting the high arithmetic intensity of (3) by
blocking in dilution and spatial indices, and performing the momentum
projection with a JIT-compiled microkernel provided by the Intel® Math Kernel
Library (Intel® MKL), we achieve good single-core performance on a test system
with Intel® Xeon® Platinum 8358 processors. Parallelizing over blocks of
dilution indices using multithreading, we also observe good scalability all
the way to the maximum number of cores per socket333Depending on memory
requirements, this parallelization can straightforwardly be supplemented by a
standard domain decomposition over MPI ranks in order to enable scaling beyond
one socket or node at the expense of an $N_{\mathrm{dil}}^{3}$-sized MPI
reduction at the end of kernel execution.
This optimized implementation has been upstreamed into the chroma_laph
measurement suite and is ready for use in production runs on large lattice
volumes.
## Acknowledgments
We acknowledge useful conversations with Colin Morningstar as well as our
Intel colleagues Christoph Bauinger and Carsten Uphoff. Intel and the Intel
logo are trademarks of Intel Corporation or its subsidiaries.
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|
# A Reciprocity Formula on Multicurves
Juhan Kim Department of Mathematics, Seoul National University, 1 Gwanak-ro,
Gwanak-gu, Seoul 08826<EMAIL_ADDRESS>
###### Abstract.
Given a specific collection of curves on an oriented surface with punctures,
we associate a power series by counting its intersections with multicurves.
This paper presents a reciprocity formula on the power series when multicurves
with no component contractible to a puncture are concerned, as a
generalization of the reciprocity presented in [1].
## 1\. Introduction
The following reciprocity formula Theorem 1.1 is originally introduced in [1].
It is presented as a topological interpretation of log Calabi-Yau property of
a relative character variety of a surface. Let there be a closed disk. Attach
some “handles” to the disk in an orientable way to obtain a ribbon graph $R$,
just as in Figure 1. Define _multicurve_ on $R$ to be a finite disjoint union
of simple closed non-contractible curves on $R$. We say a multicurve is
_essential_ when it has no component isotopic to a boundary component of $R$.
For $r\in\mathbb{Z}_{\geq 0}$, Define $c(r)$ to be the number of essential
multicurves (up to isotopy) that passes through the handles $r$ times in
total. More details can be found in [1].
###### Theorem 1.1.
The series $Z(t)=\sum\limits_{r=0}^{\infty}c(r)t^{r}$is a rational function,
and
$Z(t^{-1})=Z\left(t\right).$
Figure 1. Examples of a multicurve and a counting curve on a ribbon graph
The goal of this paper is to provide a new proof of this formula, furthermore
generalizing it to a multivariate version on arbitrary surfaces with boundary
or punctured surfaces. Let $S_{g,n}$ be an oriented surface of genus $g$ with
$n\geq 1$ distinct punctures. First we define multicurve in the identical way.
###### Definition 1.2 (multicurve).
A _multicurve_ on $S_{g,n}$ is a finite disjoint union of simple closed non-
contractible curves on $S_{g,n}$. A multicurve is _essential_ when it has no
component contractible to a puncture. We allow a trivial multicurve which is
an empty set. Denote by $M$ the set of isotopy classes of multicurves, by $E$
those of essential multicurves.
In the case of ribbon graph, we have counted the number of curves passing
through handles. Since this amounts to counting intersections of a multicurve
and another collection of curves that cut across each handle(Figure 1), we
generalize this to define the following notion of counting curve.
###### Definition 1.3 (counting curve).
Consider a finite disjoint union of simple curves on $S_{g,n}$ that connect
between the punctures. When such disjoint union meets every nontrivial
multicurve, we say that union is a _counting curve_.
###### Example 1.4.
In Figure 2, $a\cup b\cup c$ is a counting curve on $S_{1,2}$. Its components
are simple non-contractible curves, and it meets every nontrivial multicurve.
$a\cup b\cup d$ is not a counting curve. Even though its component are simple
and non-contractible, it does not meet $m$, a nontrivial multicurve.
Figure 2. Example and non-example of counting curve on $S_{1,2}$
###### Definition 1.5.
Given a counting curve $C$ with $m$ components $C_{1},\ldots,C_{m}$, we define
a counting function $p:M\rightarrow\mathbb{Z}^{m}_{\geq 0}$ as
$p(x)_{i}=\min\limits_{c\in x}\\#(c\cap C_{i})\ (x\in M)$. Let
$G_{\alpha}=\\#\left\\{x\in M\ |\ p(x)=\alpha\right\\}$,
$F_{\alpha}=\\#\left\\{x\in E\ |\ p(x)=\alpha\right\\}$. Now define the
following power series.
$g_{C}(x)=\sum\limits_{\alpha\in\mathbb{Z}^{m}_{\geq
0}}G_{\alpha}x^{\alpha}=\sum\limits_{\alpha\in\mathbb{Z}^{m}_{\geq
0}}G_{\alpha}x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\ldots x_{m}^{\alpha_{m}}$
$f_{C}(x)=\sum\limits_{\alpha\in\mathbb{Z}^{m}_{\geq
0}}F_{\alpha}x^{\alpha}=\sum\limits_{\alpha\in\mathbb{Z}^{m}_{\geq
0}}F_{\alpha}x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\ldots x_{m}^{\alpha_{m}}$
Then we are ready to state the main theorem. The fact that $g_{C}$ and $f_{C}$
are well-defined (that is, $G_{\alpha}$ and $F_{\alpha}$ are always finite)
will be evident in the proof of Theorem 1.6.
###### Theorem 1.6.
$g_{C}$ and $f_{C}$ are rational functions, and
$f_{C}(x^{-1})=f_{C}(x)$
where $f_{C}(x^{-1})=f_{C}(x_{1}^{-1},x_{2}^{-1},\ldots,x_{m}^{-1})$.
###### Example 1.7.
A counting curve $a\cup b$ on $S_{1,1}$ is described in Figure 3. One easily
checks that there is a bijection $\phi:\mathbb{Z}^{2}/\left\\{\pm
1\right\\}\to E$ such that
$p\left((s,t)\right)=\left(\left|s\right|,\left|t\right|\right)$, as suggested
in Figure 3.
Figure 3. Some multicurves on $S_{1,1}$
From this we calculate $f_{a\cup b}$ as follows:
$f(x,y)=1+\frac{x}{1-x}+\frac{y}{1-y}+\frac{2xy}{(1-x)(1-y)}=\frac{1+xy}{(1-x)(1-y)}.$
Then
$f\left(x^{-1},y^{-1}\right)=\frac{1+x^{-1}y^{-1}}{\left(1-x^{-1}\right)\left(1-y^{-1}\right)}=\frac{1+xy}{(1-x)(1-y)}=f(x,y).$
The theorem follows from the observation that multicurves can be viewed as
integer points of a rational cone. In this case Stanley’s reciprocity theorem
on rational cones([2]) is applicable. The following theorem, which can be
found in [3, p. 179, p. 188], is one of its reformulations.
###### Theorem 1.8 (Stanley).
For $r\leq m$, let $\Phi$ be an $r\times m$ integer matrix, $K$ be the set of
its nonnegative integer solutions and $K^{\circ}$ its positive integer
solutions. Then $k(z)=\sum\limits_{\alpha\in K}z^{\alpha}$,
$k^{\circ}(z)=\sum\limits_{\alpha\in K^{\circ}}z^{\alpha}$ are rational
functions. Factors of their denominators are of the form
$\left(1-z^{\beta}\right)$ where $\beta$ is a nonnegative integer solution of
$\Phi$. Furthermore, if $k^{\circ}\neq 0$, $k$ and $k^{\circ}$ satisfy
$k(z^{-1})=(-1)^{m-\mathrm{rank}(\Phi)}k^{\circ}(z).$
###### Remark 1.9.
Equivalently, we could define everything on $\Sigma_{g,n}$, the compact
oriented surface of genus $g$ with $n$ boundaries, instead of $S_{g,n}$. In
this case counting curve is defined to be a disjoint union of curves
connecting between boundary curves just as in the case of ribbon graph. The
number of curves counted will not be affected. In this point of view Theorem
1.1 is a special case of Theorem 1.6, where the counting curves are defined to
be those curves cutting across the handles.
## 2\. Proof
###### Lemma 2.1.
Suppose a counting curve $C$ is given. By adding additional curve components
to $C$, we can obtain a new counting curve $C^{\prime}$ such that $S_{g,n}$ is
divided by open regions bounded by three (not necessarily distinct) curve
components of $C^{\prime}$ and those regions are homeomorphic to an open disk.
Proof. Each open region bounded by $C$ is homeomorphic to an open disk and
does not include any puncture. If not, it would allow non-contractible curves
not intersecting $C$. Add diagonal curves to those polygon-shaped regions to
divide them into smaller regions bounded by three curve components, and we
have $C^{\prime}.$ It is then obvious that $C^{\prime}$ is a counting curve.
$\square$
From now on, _edge_ refers to a component of $C^{\prime}$ and _triangle_
refers to a region bounded by $C^{\prime}$. It is possible that two of the
three edges of a triangle coincide; see the triangle bounded by $c$ and $d$ in
Figure 4. Since an edge bounds two triangles and a triangle is bounded by
three edges (if counted properly), we have $2N$ triangles and $3N$ edges for
some $N$. Then $2N$ triangles, $3N$ edges, $n$ points filling in the punctures
consist a cell structure on a surface of genus $g$ and we have $2N-3N+n=2-2g$,
hence $N\equiv n\ (mod\ 2)$.
Now observe how a multicurve appears in each triangles. When we consider a
multicurve intersecting $C^{\prime}$ at minimum in its isotopy class, their
should be no bigons bounded by part of an edge and part of the multicurve.
Thus such multicurve restricted to a triangle is a disjoint union of segments
connecting between edges of the triangle. Its isotopy type is characterized by
a triple of nonnegative integers representing the number of segments
connecting each pair of edges. Since we have $2N$ triangles the configuration
on the whole surface is represented by a $6N$-tuple of nonnegative integers.
There must be same number of segments connected from each side of an edge so
the tuple must satisfy linear relations corresponding to the edges. There are
$3N$ edges and the tuple must be a nonnegative integer solution of some
$3N\times 6N$ integer matrix $\Phi$. Conversely, if we have a nonnegative
linear solution of such $\Phi$, we can glue those curves in triangles to
obtain a multicurve. The argument above is suggests the following proposition.
###### Proposition 2.2.
Given a counting curve $C^{\prime}$ satisfying the condition of Lemma 2.1 with
$2N$ triangles and $3N$ edges, there is a one-to-one correspondence between
isotopy classes of multicurves and nonnegative integer solutions of some
$3N\times 6N$ integer matrix $\Phi$.
###### Example 2.3.
In Figure 4, $C=a\cup b\cup c$ is a counting curve and can be refined to
$C^{\prime}=a\cup b\cup c\cup d\cup e\cup f$ which satisfies the condition of
Lemma 2.1. In the view of Proposition 2.2, multicurves on $S_{1,2}$
corresponds(up to isotopy) to nonnegative integer solutions of the following
$\Phi$.
Figure 4. Edges and triangles on $S_{1,2}$
$\Phi=\bordermatrix{&1&2&3&4&5&6&7&8&9&10&11&12\cr
a&1&1&0&0&0&0&0&-1&-1&0&0&0\cr b&1&0&1&0&0&0&-1&-1&0&0&0&0\cr
c&0&0&0&0&0&0&0&0&0&0&1&-1\cr d&0&0&0&0&1&1&0&0&0&0&-1&-1\cr
e&0&1&1&-1&0&-1&0&0&0&0&0&0\cr f&0&0&0&1&1&0&-1&0&-1&0&0&0\cr}$
Proof of Proposition 2.2. We only need to show that different solutions
correspond to non-isotopic multicurves. Denote the edges by
$C^{\prime}_{1},C^{\prime}_{2},\ldots C^{\prime}_{3N}$. Fix their
orientations. For an oriented connected curve component $T$ that meets
$C^{\prime}$ transversely finitely many times, we designate a cyclic word by
moving along $T$ and concatenating free symbols $X_{i}$ or $X_{i}^{-1}$
(depending on the orientation) whenever it crosses $C_{i}$. Note that any
isotopy of $T$ can be viewed as consecutively inserting or deleting
$X_{i}X_{i}^{-1}$ or $X_{i}^{-1}X_{i}$’s(by making or removing bigons). For
example, in Figure 5 the cyclic word for the left figure is
$\left[X_{1}X_{5}X_{4}X_{4}^{-1}X_{6}\right]$ and after making a bigon(right
figure) it becomes
$\left[X_{1}X_{5}X_{5}^{-1}X_{5}X_{4}X_{4}^{-1}X_{6}\right]$. Hence the cyclic
word, when viewed as a conjugacy class in the free group with generators
$\left\\{X_{1},X_{2},\ldots X_{3N}\right\\}$, is well-defined up to isotopy.
In the conjugacy class there is a cyclically reduced word which is unique up
to cyclic shift([4, Theorem 1.3]). This means $T$ is isotopic to a unique
configuration without any bigon, hence no more than one solution of $\Phi$ can
correspond to $T$. This applies to every multicurve by considering for each
connected component.$\square$
Figure 5. A curve configuration modified by an isotopy
From now on, fix $C$, $C^{\prime}$, $\Phi$ as defined in
$Proposition~{}\ref{prop1}$.
###### Lemma 2.4.
$\mathrm{rank}(\Phi)=3N$.
Proof. Let $v_{1},v_{2}\cdots,v_{3N}$ be row vectors of $\Phi$. Suppose
$\sum\limits_{i=1}^{3N}a_{i}v_{i}=0$. Choose a triangle. Pick columns
corresponding to segments connecting between edges of the triangle. If the
triangle has three distinct edges, say $v_{1},v_{2},v_{3}$. Then $\Phi$ looks
as follows after reordering columns so that the picked columns come first, up
to sign of rows.
$\left(\begin{array}[]{cccccc}1&1&0&&\hbox{\multirowsetup$\ast$}&\\\
1&0&1&&&\\\ 0&1&1&&&\\\ 0&0&0&&&\\\ &\vdots&&&&\\\ 0&0&0&&&\end{array}\right)$
From this, we have $a_{1}=a_{2}=a_{3}=0$. Meanwhile, if two of edges coincide
so that the chosen triangle is bounded by two distinct edges, say
$v_{1},v_{2}$, then $\Phi$ looks as follows up to reordering of rows and
columns and sign of rows.
$\left(\begin{array}[]{cccccc}1&1&0&&\hbox{\multirowsetup$\ast$}&\\\
1&-1&0&&&\\\ 0&0&0&&&\\\ &\vdots&&&&\\\ 0&0&0&&&\end{array}\right)$
Thus $a_{1}=a_{2}=0$. Since every edge bounds some triangle, we have $a_{i}=0$
for all $i$. We conclude that $v_{1},v_{2},\cdots,v_{3N}$ are independent and
$\mathrm{rank}(\Phi)=3N$. $\square$
Next consider curves that are contractible to some puncture. On $S_{g,n}$ we
have $n$ curves up to isotopy, say $d_{1},d_{2},\ldots,d_{n}$. Denote their
corresponding $6N$-tuples by $\delta_{1},\delta_{2},\ldots,\delta_{n}$.
###### Lemma 2.5.
$\sum\limits_{i=1}^{n}\delta_{i}=\left(1,1,\ldots,1\right)$.
Proof. Consider a segment connecting between two (not necessarily distinct)
edges of a triangle. It corresponds to an angle between those two edges. By
collecting all segments we obtain a multicurve whose corresponding tuple is
$\left(1,1,\ldots,1\right)$. This multicurve passes along each angle exactly
once, so this is $\bigcup\limits_{i=1}^{n}d_{i}$ and the corresponding tuple
is $\sum\limits_{i=1}^{n}\delta_{i}$. $\square$
Proof of Theorem 1.6.
Note that $\left(1,1,\ldots,1\right)$ is the unique minimal positive integer
solution of $\Phi$. By applying Theorem 1.8 we have
$k\left(z^{-1}\right)=(-1)^{6N-\mathrm{rank}(\Phi)}k^{\circ}(z)=(-1)^{6N-\mathrm{rank}(\Phi)}z^{\left(1,1,\ldots,1\right)}k(z)$
where $k(z)$ and $k^{\circ}(z)$ are defined as in Theorem 1.8. Proposition 2.2
tells us that $g_{C^{\prime}}(x)=k(z)$ under the substitution
$z_{(i,j)}=x_{i}^{\frac{1}{2}}x_{j}^{\frac{1}{2}}$, where $x_{i},x_{j}$ are
variables corresponding to edges $C_{i},C_{j}$ and $z_{(i,j)}$ is the variable
corresponding to the segment connecting those edges. Define $\alpha_{i}$ by
$x^{\alpha_{i}}=z^{\delta_{i}}$. Since every multicurve is a disjoint union of
an essential multicurve and (possibly multiple copies of) $d_{i}$’s, we have
$f_{C^{\prime}}(x)\prod\limits_{i=1}^{n}\sum\limits_{k=0}^{\infty}x^{k\alpha_{i}}=g_{C^{\prime}}(x)$,
or equivalently
$f_{C^{\prime}}(x)=g_{C^{\prime}}(x)\prod\limits_{i=1}^{n}\left(1-x^{\alpha_{i}}\right)$.
With Lemma 2.4 and Lemma 2.5, we have
$\displaystyle f_{C^{\prime}}\left(x^{-1}\right)$
$\displaystyle=g_{C^{\prime}}\left(x^{-1}\right)\prod\limits_{i=1}^{n}\left(1-x^{-\alpha_{i}}\right)$
$\displaystyle=(-1)^{6N-\mathrm{rank}(\Phi)}z^{\left(1,1,\ldots,1\right)}g_{C^{\prime}}(x)\prod\limits_{i=1}^{n}\left(1-x^{-\alpha_{i}}\right)$
$\displaystyle=(-1)^{N}\left(\prod\limits_{i=1}^{n}x^{\alpha_{i}}\right)g_{C^{\prime}}(x)\prod\limits_{i=1}^{n}\left(1-x^{-\alpha_{i}}\right)$
$\displaystyle=(-1)^{n}g_{C^{\prime}}(x)\prod\limits_{i=1}^{n}\left(x^{\alpha_{i}}-1\right)$
$\displaystyle=g_{C^{\prime}}(x)\prod\limits_{i=1}^{n}\left(1-x^{\alpha_{i}}\right)$
$\displaystyle=f_{C^{\prime}}(x).$
By substituting $1$ to variables corresponding to components of $C^{\prime}$
that are not components of $C$, we obtain $g_{C}$, $f_{C}$ from
$g_{C^{\prime}}$, $f_{C^{\prime}}$, respectively. This is justified by the
following argument. The denominators of $g_{C^{\prime}}$ are of the form
$\left(1-x^{\beta}\right)$ by Theorem 1.8. Since every nontrivial multicurve
intersects $C$, at least one variable corresponding to a component of $C$ has
a positive exponent in $x^{\beta}$. In other words, $\left(1-x^{\beta}\right)$
does not vanish by the substitution. Hence $g_{C}$ is well-defined. Since
$f_{C^{\prime}}(x)=g_{C^{\prime}}(x)\prod\limits_{i=1}^{n}\left(1-x^{\alpha_{i}}\right)$,
we can say the same thing about $f_{C}$. By the substitution on the equation
$f_{C^{\prime}}\left(x^{-1}\right)=f_{C^{\prime}}(x)$, we conclude that
$f_{C}\left(x^{-1}\right)=f_{C}(x)$. $\square$
###### Acknowledgment.
The author would like to emphasize his sincerest gratitude towards his advisor
Junho Peter Whang for introducing one of his valuable works [1] and suggesting
the interesting challenge of generalizing it. His numerous counsels during the
writing of this paper must be also appreciated.
## 3\. References
##
* [1] Junho Peter Whang. Global geometry on moduli of local systems for surfaces with boundary. Compos. Math., 156(8):1517–1559, 2020.
* [2] Richard P. Stanley. Combinatorial reciprocity theorems. Advances in Math., 14:194–253, 1974.
* [3] Richard P. Stanley. Linear Diophantine equations and local cohomology. Invent. Math., 68(2):175–193, 1982.
* [4] Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinatorial group theory. Dover Publications, Inc., Mineola, NY, second edition, 2004. Presentations of groups in terms of generators and relations.
|
# Repeat Voting:
Two-Vote May Lead More People To Vote††thanks: The idea of repeat voting
evolved following discussions at the Annual Conference of the Federmann Center
for the Study of Rationality in February 2017\. The author thanks the Center’s
members, in particular Maya Bar-Hillel, Orit Kedar, and Motty Perry, for their
suggestions.
Sergiu Hart The Hebrew University of Jerusalem (Federmann Center for the Study
of Rationality, Department of Economics, and Institute of Mathematics). _E-
mail_<EMAIL_ADDRESS>_Web site_ : http://www.ma.huji.ac.il/hart
(October 17, 2017)
###### Abstract
A _repeat voting_ procedure is proposed, whereby voting is carried out in two
identical rounds. Every voter can vote in each round, the results of the first
round are made public before the second round, and the final result is
determined by adding up all the votes in both rounds. It is argued that this
simple modification of election procedures may well increase voter
participation and result in more accurate and representative outcomes.
Suppose that it is two weeks after the Brexit vote, and there is a new vote on
the same issue—what will the result be? Given the way the original vote went,
will people change their minds and vote differently? Will the original results
cause people who had not voted to cast their vote in this second round? Will
the final result be different?111Before the Brexit vote, a petition that
called for a second vote in case of low participation and a narrow winning
margin was launched; it got 22 signatures before the vote, and more than 2
million signatures in the two days after the result was announced.
(Interestingly, the initiator was a “leave” supporter who believed that
“leave” would lose.) See, e.g., http://www.bbc.com/news/uk-politics-eu-
referendum-36629324 (There are no clear answers to any of these questions, but
one can easily provide arguments either way.) Now carry out a similar thought
experiment regarding the latest presidential election in the U.S., or whatever
your latest favorite, or unsettling, election is …
Democratic elections are beset by many problems. One issue is low voter
turnout, which at times is only one-half of the eligible voters or even less.
Another issue is excessive reliance on polls: polls affect voters, despite
repeatedly turning out to being quite far from accurate. This also relates to
the low-turnout issue: “I will not waste my time voting, as my candidate is in
any case sure to win” (or “… sure to lose”). Polls may also lead people not to
cast their vote for their preferred candidate, if, for example, they do not
want him or her to win by too large a majority, or if they want to voice a
certain “protest” through their vote—only to find out that in the end their
candidate did not win at all. Yet another issue concerns unexpected events
that occur extremely close to election time, too late to be able to be
addressed by the candidates, such as a terrorist attack, the publication of
false information, bad weather, and so on. What is common to many of these
situations is that people might want to change their vote, or their non-
participation in the election, once they see the actual results and how these
came about.
To address these and other issues, I propose the use of the following REPEAT
VOTING procedure.
1. A.
Voting is carried out in two rounds.
2. B.
Every eligible voter is entitled (and encouraged) to vote in each of the two
rounds.
3. C.
All the votes of the two rounds are added up, and the final election result is
obtained by applying the current election rules222Be they plurality, special
majority, electoral college, and so on. to these two-round totals.
4. D.
The results of the first round are officially counted and published; the
second round takes place, say, two weeks after the first round, but no less
than one week after the official publication of the first round’s results.
What are the _advantages_ of repeat voting?
1. 1.
_Polls._ The first round becomes a de facto giant opinion poll; however,
because the votes of the first round count, it is a much more truthful poll
(in contrast to the usual pre-election polls, where giving untruthful
answers—whether intentionally or not—carries no cost333Someone once quipped
that Israelis tell the truth in polls, but lie when they cast their vote.).
The combination of the large sample size and incentivized truthfulness makes
the results of the first round a significantly more accurate predictor of the
electorate views. It is thus crucial for the votes of the first round to count
no less than the votes of the second round, which explains why we are adding
up the votes of the two rounds, rather than having only the second round
determine the outcome.
2. 2.
_Participation._ Voters who do have a preference that is not however strong
enough to make them vote in the first round may well be led to vote in the
second round because of the results of the first round. Thus, participation in
at least one round of the election is expected to increase. It is better that
people vote even in one round than not at all.444Voters who have strong or
extreme positions will most probably vote in both rounds; their relative
weight in the final result will decrease when enough people are motivated to
vote in the second round (which may well happen if such extreme positions get
higher shares of the vote in the first round). One indirect advantage is that
people who vote may feel closer to the elected officials, and to the
democratic system in general.
3. 3.
_Representative results._ The final results may be more representative,
because the second round makes it possible for the voters as well as for the
candidates to “correct” any problems of the first round. This includes the
effects of wrong predictions by the polls, as well as any special
circumstances and events that occurred close to election time (see the second
paragraph of the paper; it is unlikely that such unexpected events will happen
both times). All this, again, can only increase the robustness of the results:
they become more trustworthy and more accepted.
4. 4.
_New reference point._ The results of the first round become a new reference
point, which may well affect a person’s choice in the second round:
_imagining_ a new situation and _being_ in a new situation are not the same
thing.555Robert J. Aumann, awarded the Nobel Prize in Economics in 2005, tells
the following story (S. Hart, “An Interview with Robert Aumann,”
_Macroeconomic Dynamics_ 9, 2005, page 711; reprinted in: Paul A. Samuelson
and William A. Barnett, editors, _Inside the Economist’s Mind: Conversations
with Eminent Economists_ , Blackwell Publishing 2006). In 1956 he had two
offers: one from Bell Labs in New York, and another from the Hebrew University
of Jerusalem. It took him a long time to make up his mind, and he chose Bell
Labs. He phoned them and told them that he accepted their offer. Once he put
down the phone, he immediately started imagining the next few years at Bell
Labs, and reached the conclusion that he had made the wrong choice. A day
later he phoned Bell Labs and asked them if he could change his mind—which
they graciously agreed to. How come a leading game theorist couldn’t
understand all this before he made his initial decision? Aumann’s answer is
that until he found himself in the new situation of someone going to Bell
Labs, he could not really grasp what it meant!
5. 5.
_Strategic voting._ People seem to be more strategic in their voting than is
usually believed (again, see the examples in the second paragraph above), but
under current procedures they base their strategic decisions on possibly
inaccurate polls. Repeat voting provides a much more solid basis. In close
elections it is conceivable that the voting of the second round may be less
strategic (and the other way around when there is a large winning margin in
the first round).666An interesting related instance concerns the minimal
threshold for a party to be represented in a parliament. Many potential
entrants try to convince voters that they have support that is higher than the
threshold and so voting for them would not be a “waste” of one’s vote. In many
cases, however, it turns out that these parties do not pass the threshold;
once this is seen in the first round, there will be many fewer such wasted
votes in the second round.
What are the possible _disadvantages_ of repeat voting?
1. 1.
_Costs._ A second round adds costs (however, in future voting that may be
conducted online, these costs would become much smaller). The additional
electoral campaign between the two rounds also increases the costs (but one
should remember that two rounds are already used in various elections, albeit
not two identical rounds as proposed here). One way to save costs is to carry
out the second round only when the results of the first round are close (for
instance, when the winning margin is below a certain threshold that is
specified in advance).777Suggested by Motty Perry and Steve Brams.
2. 2.
_Participation._ There may be fewer voters in the first round (“I will have a
chance to vote in the second round”).
3. 3.
_Bandwagon effect._ Voters with strong or extreme positions, who are much more
likely to vote in the first round, may have a big effect on the results of the
first round, which may then have a bandwagon effect on the whole election.
One can think of other ways to overcome the issues pointed out above. For
example, one can repeat the vote three times, with the winner having to win at
least two rounds (this applies only to two-outcome elections, however, not to
multi-candidate and parliamentary elections, and is inherently more
complicated).888This procedure was also suggested by Shachar Kariv. Another
possibility is to make voting mandatory (as in certain countries); while this
may resolve the participation issue, it does not resolve the significant
“polls issue” discussed in advantage #1 above. Yet another is to have the
votes in the two rounds of repeat voting weighted differently (for instance,
depending on the total number of votes in each round999For example, averaging
the percentages of votes that each candidate received in the two rounds (which
amounts to giving weights to the two rounds that are inversely proportional to
the total number of votes in each) may perhaps increase participation in that
round where there are fewer voters (probably the first round).); at this
point, however, it seems best to leave it as simple and straightforward as
possible.
In summary: REPEAT VOTING is a simple modification of election procedures that
is capable of increasing voter participation and yielding more accurate and
representative results. Everyone deserves a second chance, as the saying goes.
Shouldn’t this include voters and candidates?
|
# Common Knowledge of Abstract Groups
Merlin Humml, Lutz Schröder
###### Abstract
Epistemic logics typically talk about knowledge of individual agents or groups
of explicitly listed agents. Often, however, one wishes to express knowledge
of groups of agents specified by a given property, as in ‘it is common
knowledge among economists’. We introduce such a logic of common knowledge,
which we term _ abstract-group epistemic logic (AGEL)_. That is, AGEL features
a common knowledge operator for groups of agents given by concepts in a
separate agent logic that we keep generic, with one possible agent logic being
$\mathcal{ALC}$. We show that AGEL is ExpTime-complete, with the lower bound
established by reduction from standard group epistemic logic, and the upper
bound by a satisfiability-preserving embedding into the full $\mu$-calculus.
Further main results include a finite model property (not enjoyed by the full
$\mu$-calculus) and a complete axiomatization.
## Introduction
Epistemic (modal) logic is concerned with the individual and collective
knowledge of agents. One of the most important modalities for collective
knowledge is _common knowledge_ : A fact $\phi$ is common knowledge in a given
group of agents if everyone in the group knows $\phi$, and everyone knows that
everyone knows $\phi$, etc. In the present work, our focus of attention is on
the involved notion of group of agents. The most basic variant of the common
knowledge operator, typically written $C$, refers to _all_ agents in a
predetermined finite set $Ag$ that forms a parameter of the logic as a whole
(Fagin et al. 1995). In a more fine-grained variant, $C$ can be annotated with
an explicitly given subset of the set of agents: For $A\subseteq Ag$,
$C_{A}\phi$ says that $\phi$ is common knowledge among the agents in $A$. For
instance, if $\mathsf{Alice}$ and $\mathsf{Bob}$ are legitimate participants
in a communication protocol and $\phi$ is a fact concerning a shared key, then
$\phi$ would ideally be common knowledge of $\mathsf{Alice}$ and
$\mathsf{Bob}$ but not of a malicious third party $\mathsf{Charlie}$ – i.e.
$C_{\\{\mathsf{Alice},\mathsf{Bob}\\}}\phi$ would hold but
$C_{\\{\mathsf{Alice},\mathsf{Bob},\mathsf{Charlie}\\}}\phi$ would not.
Listing agents in a group explicitly is appropriate in well-controlled
settings such as the above, where the participants in the epistemic situation
are fixed and previously known. In other application contexts, however, this
may not always be the case, in particular in statements found in real-world
argumentation. Consider, for example, the sentence ‘Doctors agree that smoking
is bad for your health.’ We take this sentence (maybe debatably) as making a
statement about common knowledge of all doctors. Encoding this claim as a
formula of the form $C_{A}\phi$ where $A$ is a finite set explicitly
enumerating all doctors is clearly neither feasible nor even semantically
desirable, as the statement is presumably meant to hold without regard to
exactly how many, and which, doctors are practising in the world at the
moment. Rather, one would want $A$ to be given by the defining property of
_being a doctor_.
In the present paper, we introduce an epistemic logic that allows precisely
this: _AGEL_ features a common knowledge operator for groups of agents
described by concepts in a dedicated _agent logic_. We keep the technical
development generic in the choice of agent logic, subject to some technical
requirements on the agent logic that are satisfied, for instance, by the
description logic $\mathcal{ALC}$; so we can describe groups of agents such as
‘doctors and pharmacists’ or ‘parents of teenagers’. We note that we treat the
agent logic as rigid, i.e. there is no uncertainty about membership in the
groups it describes. In other words, group descriptions are _de re_ rather
than _de dicto_. We further illustrate the logic on a variant of the muddy
children puzzle where the number of participants is unspecified and
potentially large. Our main results on AGEL are ExpTime-completeness of the
satisfiability problem; a bounded (specifically, doubly exponential) model
property; and a complete axiomatization. Technically, we establish the lower
complexity bound by a satisfiability-preserving translation of standard group
epistemic logic (Fagin et al. 1995) into AGEL, and the upper bound by a
satisfiability-preserving translation of AGEL into the full $\mu$-calculus
(i.e. $\mathcal{ALC}$ with inverse roles and fixpoint operators (Vardi 1998)).
Use of the full $\mu$-calculus avoids the exponential blow-up that would be
incurred by a more naive reverse translation of AGEL into group epistemic
logic. However, the full $\mu$-calculus does not have the finite model
property (Vardi 1998; Streett 1982). Instead, we show the bounded model
property and completeness by means of a filtered model construction that uses
ideas from the finite model construction for propositional dynamic logic
(Fischer and Ladner 1979; Blackburn, de Rijke, and Venema 2001), in particular
transitive closure of (small) canonical pseudo-models.
#### Related Work
There is a line of research on indexing knowledge modalities with _names_ that
designate groups of agents (Grove and Halpern 1993) (Fagin et al. 1995,
Chapter 6). We refer to such groups as _named groups_ ; they are similar to
the atoms of our agent logic but are _non-rigid_ , i.e. their interpretation
depends on the current world, in an approach that is focused on the analysis
of knowledge about the identity of agents. Although common knowledge of such
name-defined groups has been mentioned early on (Grove and Halpern 1993),
results have largely focused on operators of the type ‘every agent / some
agent with name $n$ knows’ (recall that generally, ‘everyone knows $\phi$’
differs from $\phi$ being common knowledge in that it need not imply that
everyone knows that everyone knows $\phi$, etc.). Recently, Bílková,
Christoff, and Roy (2021) have shown completeness and the finite model
property (but no complexity bound) for common knowledge of name-defined
groups. (A different form of common knowledge for non-rigid groups has been
considered earlier, without considering axiomatization (Moses and Tuttle
1988).)
Extending the descriptive means for groups of agents has been considered
already by Grove and Halpern (1993), who give an axiomatization (but no
complexity bound) of a logic that features ‘everyone knows’ and ‘someone
knows’ operators for propositional combinations of names. Additional
expressive means are provided by subsequent first-order extensions of the
logic that allow quantifying over agent names (Grove 1995; Naumov and Tao
2019) in the style of term modal logic (Fitting, Thalmann, and Voronkov 2001).
Quantifying over agents provides a straightforward way of encoding, e.g., the
‘everybody knows operator’; for instance, ‘every doctor knows $\phi$’ would be
expressed as $\forall x.\,\mathsf{doctor}(x)\rightarrow K_{x}\phi$ if $K_{x}$
denotes the usual single-agent knowledge modality ‘$x$ knows’. Common
knowledge, on the other hand, involves transitive closure and as such is not
first-order expressible, hence not easily accommodated in such frameworks.
Also, decidability results in a first-order setting will, of course, require
additional restrictions (e.g. it has recently been shown that the two-variable
fragment of term-modal logic is decidable, with complexity between NExpTime
and 2ExpSpace (Padmanabha and Ramanujam 2019)).
The above logics and ours employ ternary transition relations relating agents
to pairs of worlds. Beyond epistemic logic, ternary relations appear, e.g., in
the logic of Pierce algebras (de Rijke 1993), in arrow logic (Venema 1996),
and in Routley-Meyer-style semantics of _relevance logic_ (Mares 2020). While
such modalities are formally similar to _everyone-knows_ modalities for
abstract groups, the frame conditions imposed on models and, consequently, the
attached meta-theory are quite different from ours.
## Abstract-Group Epistemic Logic
We proceed to introduce the syntax and semantics of _ abstract-group epistemic
logic (AGEL)_.
##### Syntax
We parametrize the logic over the choice of an _agent logic_
${\mathcal{L}_{\mathsf{Ag}}}$ that serves to specify groups of agents. We will
discuss assumptions on the semantics of ${\mathcal{L}_{\mathsf{Ag}}}$ later in
this section. Syntactically, we require only that
${\mathcal{L}_{\mathsf{Ag}}}$ has a formula syntax where formulae are
expressions in some grammar, in particular giving rise to a standard notion of
_subformula_ , and includes propositional atoms from a set
$\mathsf{At}_{\mathsf{Ag}}$, referred to as _agent atoms_ , as well as the
full set of Boolean connectives. Properties of ${\mathcal{L}_{\mathsf{Ag}}}$
needed in our main results will be named explicitly in the respective
theorems. One choice for ${\mathcal{L}_{\mathsf{Ag}}}$ that satisfies all
requisite properties is the standard description logic $\mathcal{ALC}$ (Baader
et al. 2003).
We further assume a set $\mathsf{At}_{\mathsf{W}}$ of _world atoms_. The set
of _(world) formulae_ $\phi,\chi,\dots$ of AGEL is then defined by the grammar
$\phi,\chi:=\bot\mid p\mid\neg\phi\mid\phi\land\chi\mid
C_{\psi}\phi\qquad(p\in\mathsf{At}_{\mathsf{W}},\psi\in{\mathcal{L}_{\mathsf{Ag}}}).$
That is, we include propositional atoms and Boolean connectives; further
Boolean connectives $\lor$, $\rightarrow$, $\leftrightarrow$ are defined as
usual. The key feature is then the common knowledge operator $C_{\psi}$ for
groups of agents defined by an agent formula $\psi$; a formula $C_{\psi}\phi$
is read ‘$\phi$ is common knowledge among agents satisfying $\psi$’, using the
term _common knowledge_ in the sense recalled in the introduction.
###### Example 1.
We may encode the statement ‘parents of teenagers know that education is
pointless’ as the AGEL formula
$C_{\exists\mathsf{hasChild}.\,\mathsf{Teenager}}\,\mathsf{ep}$, using
$\mathcal{ALC}$ as the agent logic and understanding the world atom
$\mathsf{ep}$ as ‘education is pointless’. The even more frustrating fact that
parents know that their offspring know this as well would be captured by the
formula
$C_{\exists\mathsf{hasChild}.\,\mathsf{Teenager}}\,C_{\mathsf{Teenager}}\,\mathsf{ep}$.
##### Semantics
We assume that the agent logic comes with a notion of _agent model_ , and that
every agent model $\mathcal{A}$ is equipped with an underlying set $Ag$ of
_agents_ and a satisfaction relation $\models_{\mathcal{A}}\>\subseteq
Ag\times{\mathcal{L}_{\mathsf{Ag}}}$; we write $\mathcal{A},a\models\psi$ for
$(a,\psi)\in\;\models_{\mathcal{A}}$, and
$\llbracket\phi\rrbracket_{\mathcal{A}}=\\{a\in
Ag\mid\mathcal{A},a\models\psi\\}$. We require that
_${\mathcal{L}_{\mathsf{Ag}}}$ conservatively extends classical propositional
logic_. By this we mean more specifically that ${\mathcal{L}_{\mathsf{Ag}}}$
does not impose restrictions on valuations of agent atoms, i.e. given a set
$Ag$ and a valuation
$V_{\mathsf{Ag}}\colon\mathsf{At}_{\mathsf{Ag}}\to\mathcal{P}(Ag)$, there
always exists an agent model $\mathcal{A}$ with underlying set $Ag$ such that
$\mathcal{A},a\models q$ iff $a\in V_{\mathsf{Ag}}(q)$, for $a\in Ag$ and
$q\in\mathsf{At}_{\mathsf{Ag}}$.
Then, an _( AGEL) model_ $\mathcal{M}=(X,\mathcal{A},V_{\mathsf{W}},\sim)$
consists of a set $X$ of _worlds_ , an agent model $\mathcal{A}$, a _world
valuation_ $V_{\mathsf{W}}\colon\mathsf{At}_{\mathsf{W}}\to\mathcal{P}(X)$
interpreting the world atoms, and a family $\sim$ of _indistinguishability
relations_ $\sim_{a}\;\subseteq X\times X$ indexed over agents $a\in Ag$. We
require that each $\sim_{a}$ is an equivalence relation (in keeping with the
usual view that epistemic indistinguishability relations should be equivalence
relations); see however Remark 2. For a set $A\subseteq Ag$ of agents, we
write $\sim_{A}=(\bigcup_{a\in A}\sim_{a})^{*}$ 111This should not be confused
with similar notation used to denote the intersection of the $\sim_{a}$ in
work on epistemic logic with quantification over agents (Naumov and Tao 2019)
where $(-)^{*}$ denotes reflexive-transitive closure (note that $\sim_{A}$ is
symmetric, hence an equivalence). We define satisfaction
${\mathcal{M},x\models\psi}$ (_x satisfies $\psi$_) of a formula $\psi$ at a
world $x$ recursively by
$\displaystyle{\mathcal{M},x\not\models\bot}\qquad{\mathcal{M},x\models
p}\text{ iff }x\in V_{\mathsf{W}}(p)$
$\displaystyle{\mathcal{M},x\models\neg\phi}\text{ iff
}{\mathcal{M},x\not\models\phi}$
$\displaystyle{\mathcal{M},x\models\phi\land\chi}\text{ iff
}{\mathcal{M},x\models\phi}\text{ and }{\mathcal{M},x\models\chi}$
$\displaystyle{\mathcal{M},x\models C_{\psi}\phi}\text{ iff whenever
}x\sim_{{\llbracket\psi\rrbracket_{\mathcal{A}}}}y,\text{ then
}{\mathcal{M},y\models\phi};$
that is, $C_{\psi}$ is the standard box modality for
$\sim_{{\llbracket\psi\rrbracket_{\mathcal{A}}}}$. When
${\mathcal{M},x\models\phi}$, then we also say that $(\mathcal{M},x)$ is a
_model of $\phi$_ (and we will use this phrasing in general, also for other
logics). The formula $\phi$ is _satisfiable_ if there is a model of $\phi$,
and _valid_ (notation: $\models\phi$) if $\mathcal{M},x\models\phi$ for all
$\mathcal{M},x$. We write ${\llbracket\phi\rrbracket_{\mathcal{M}}}=\\{x\in
X\mid{\mathcal{M},x\models\phi}\\}$.
We record a fixpoint characterization of $C_{\psi}$: [end]lemma The set
${\llbracket C_{\psi}\phi\rrbracket_{\mathcal{M}}}$ is the greatest fixed
point of the function $F\colon\mathcal{P}(X)\to\mathcal{P}(X)$ given by $F(U)$
being the set of worlds $x$ such that ${\mathcal{M},x\models\phi}$ and
whenever ${\mathcal{A},a\models\psi}$ and $x\sim_{a}y$, then $y\in U$. By the
Knaster-Tarski fixpoint theorem, we can equivalently show that ${\llbracket
C_{\psi}\phi\rrbracket_{\mathcal{M}}}$ is the greatest postfixed point of $F$.
First, it is clear that ${\llbracket C_{\psi}\phi\rrbracket_{\mathcal{M}}}$ is
a postfixed point of $F$: Since $C_{\psi}$ is the box operator for the
transitive and reflexive closure of the $\psi$-successor relation, every world
$x$ in ${\llbracket C_{\psi}\phi\rrbracket_{\mathcal{M}}}$ satisfies $\phi$,
and all $\psi$-successors of $x$ satisfy $C_{\psi}\phi$ again. Second, if
$K\subseteq X$ is another postfixed point of $F$, i.e. $K\subseteq F(K)$, then
every world in $K$ satisfies $\phi$, and $K$ is closed under
$\psi$-successors, so by induction on the formation of $\sim_{\psi}$ as a
reflexive-transitive closure, every world in $K$ is in ${\llbracket
C_{\psi}\phi\rrbracket_{\mathcal{M}}}$.
###### Remark 2.
Since the modality $C_{\psi}$ takes reflexive-transitive closures, it is in
fact immaterial whether the indistinguishability relations $\sim_{a}$ are
individually reflexive and transitive; we impose the corresponding requirement
mainly to ease notation and discussion.
[end, text proof=Details for “autorefthm:prAtEnd]remark In logical settings
with non-rigid (i.e. world-dependent) agent models (e.g. (Grove and Halpern
1993; Grove 1995; Naumov and Tao 2019)), one can accommodate uncertainty about
the identity and properties of agents, and moreover specify agent groups by
their knowledge, as in ‘people who know $\phi$ also know $\psi$’. As indicated
in the introduction, AGEL, which assumes the agent logic to be rigid, trades
this mode of expression for a computationally and axiomatically tractable
treatment of common knowledge. One may still envisage extending AGEL with an
operator $I$, with $I\,\phi$ read as ‘knows about the truth or falsity of
$\phi$’. That is, an agent satisfies $I\,\phi$ if in every world, she knows
either $\phi$ or $\neg\phi$. For instance, if $\phi$ represents the publicly
contested fact whether or not transfer negotiations are under way concerning a
football player $P$, then $I\,\phi$ would hold for the relevant officials of
the involved clubs and maybe for $P$. However, harnessing such an operator
technically is likely to be challenging. To name one potential obstacle, the
first-order translation of $I\,\phi$ would presumably need to involve three
variables (i.e. unlike the translation of most modal logics would not end up
in the two-variable fragment): one for the agent $x$, and two for worlds that
are indistinguishable for $x$, and then required to agree on $\phi$ (see the
appendix). Using notation for models employed in the definition of the
semantics directly in first-order syntax, we translate satisfaction of the
formula $I\,\phi$ by an agent $a$ roughly into
$\forall x,y.\,x\sim_{a}y\to(\bar{\phi}(x)\leftrightarrow\bar{\phi}(y))$
where $\bar{\phi}(-)$ represents a first-order translation of $\phi$. This
formula needs three variables $a,x,y$. (This observation is purely heuristic;
we do not formally claim that use of three variables cannot be avoided, nor
would a two-variable translation directly imply decidability of the extension
of AGEL with $I$, as models are subject to constraints that fail to be first-
order definable.)
###### Remark 3.
Since AGEL is effectively a fixpoint logic, it is expected that compactness
fails. Indeed, given atomic agent concepts $A,B$ and a world atom $p$, the set
consisting of the formula $\neg C_{A\lor B}\>p$ and all formulae of the form
$C_{D_{1}}C_{D_{2}}\dots C_{D_{n}}p$, for $n\geq 0$ and
$D_{1},\dots,D_{n}\in\\{A,B\\}$, is unsatisfiable but all its finite subsets
are satisfiable. It follows that there is no finitary proof system for AGEL
that is _strongly_ complete, i.e. makes all unsatisfiable sets of formulae
inconsistent. We later give a proof system that is _weakly_ complete, i.e.
derives all valid formulae.
## Complexity
We show next that the satisfiability problem of AGEL is ExpTime-complete.
### Lower Bound:
Reduction from Group Epistemic Logic
We prove ExpTime-hardness by a satisfiability-preserving encoding of standard
group epistemic logic (GEL) (with common knowledge), which is known to be
ExpTime-hard (Fagin et al. 1995). (To facilitate the subsequent discussion, we
reduce from a slightly more expressive logic than strictly necessary for the
hardness proof.) We briefly recall the syntax and semantics of GEL: The logic
is parametrized over a finite set $Ag$ of agents and a set
$\mathsf{At}_{\mathsf{W}}$ of (world) atoms. The set of _formulae_
$\phi,\psi,\dots$ of GEL is then given by the grammar
$\phi,\psi::=\bot\mid p\mid\neg\phi\mid\phi\land\psi\mid C_{G}\phi$
where $p\in\mathsf{At}_{\mathsf{W}}$ and $\emptyset\neq G\subseteq Ag$, with
$C_{G}\phi$ read ‘$\phi$ is common knowledge among the agents in $G$’.
(Knowledge operators $K_{a}$ for individual agents $a$ are included as common
knowledge operators $C_{\\{a\\}}$.) As indicated in the introduction, the
difference with AGEL is that in GEL, groups $G$ of agents need to be given as
enumerated finite subsets of a known fixed set of named agents. Models
$\mathcal{M}=(X,V_{\mathsf{W}},\sim)$ consist of a set $X$ of worlds, a
(world) valuation
$V_{\mathsf{W}}\colon\mathsf{At}_{\mathsf{W}}\to\mathcal{P}(X)$, and a family
$\sim$ of indistinguishability relations $\sim_{a}\;\subseteq X\times X$,
indexed over agents $a\in Ag$ and again required to be equivalence relations.
For $G\subseteq Ag$, we write $\sim_{G}:=(\bigcup_{a\in G}\sim_{a})^{*}$.
Then, satisfaction ${\mathcal{M},x\models\phi}$ of a formula $\phi$ at a world
$x$ is defined recursively by the expected clauses for atoms and propositional
connectives, and
${\mathcal{M},x\models C_{G}\phi}\text{ iff whenever }x\sim_{G}y,\text{ then
}{\mathcal{M},y\models\phi}.$
The encoding $q$ of GEL into AGEL is given as follows. We introduce a fresh
agent atom $p_{a}$ for each $a\in Ag$. For a GEL formula $\phi$, $q(\phi)$ is
then defined recursively by
$q(C_{G}\phi):=C_{\bigvee_{a\in G}p_{a}}q(\phi)$
and commutation with all other constructs (i.e. $q(\neg\phi)=\neg q(\phi)$
etc.). Using the running assumption that ${\mathcal{L}_{\mathsf{Ag}}}$
conservatively extends classical propositional logic, we obtain
[end]theorem[Lower complexity bound] The satisfiability problem for AGEL is
ExpTime-hard.
###### Proof sketch.
We need to show that $q$ is indeed satisfiability-preserving. A model
$(\mathcal{M},x)$ of a GEL formula $\phi$ over the set $Ag$ of agents, with
$\mathcal{M}=(X,V_{\mathsf{W}},\sim)$, is transformed into a model
$(\mathcal{M}^{\prime},x)$ of the AGEL formula $q(\phi)$, with
$\mathcal{M}^{\prime}=(X,\mathcal{A},V_{\mathsf{W}},\sim^{\prime})$, by taking
$Ag$ to be the underlying set of $\mathcal{A}$, and
$V_{\mathsf{Ag}}(p_{a})=\\{a\\}$ for $a\in Ag$; this uses the running
assumption that ${\mathcal{L}_{\mathsf{Ag}}}$ conservatively extends classical
propositional logic.
Conversely, a model $(\mathcal{M},x)$ of $q(\phi)$, with
$\mathcal{M}=(X,\mathcal{A},V_{\mathsf{W}},\sim)$, is transformed into a model
$(\mathcal{M}^{\prime},x)$ of $\phi$, with
$\mathcal{M}^{\prime}=(X,V_{\mathsf{W}},\sim^{\prime})$, by taking
$\sim^{\prime}_{a}\;=\;\sim_{V_{\mathsf{Ag}}(p_{a})}$ (using notation like in
the formal semantics of AGEL). ∎
By reduction from GEL satisfiability. We show that the translation $q$ given
in the main body of the paper is satisfiability-preserving, i.e. that $\phi$
is satisfiable iff $q(\phi)$ is satisfiable.
_‘If’ :_ Given an AGEL-model
$\mathcal{M}=(X,(Ag,V_{\mathsf{Ag}}),V_{\mathsf{W}},$ ${\sim})$, we construct
a GEL-model $\mathcal{M}^{\prime}=(X,Ag,V_{\mathsf{W}},\sim^{\prime})$ by
taking
$\displaystyle\sim^{\prime}_{a}\;=\;\sim_{{\llbracket
p_{a}\rrbracket_{\mathcal{A}}}}.$
We show by induction on $\phi$ that for all $x\in X$, ${\mathcal{M},x\models
q(\phi)}$ iff ${\mathcal{M}^{\prime},x\models\phi}$. The cases for world atoms
and Boolean operators are trivial; we do the case for $C_{G}\phi$. By
definition, $q(C_{G}\phi)=C_{\bigvee_{a\in G}p_{a}}q(\phi)$. Now
$C_{\bigvee_{a\in G}p_{a}}$ is a standard modal box on the transitive-
reflexive closure of the union of the relations $\sim_{\llbracket
p_{a}\rrbracket_{\mathcal{A}}}$ over all $a\in G$, and by construction of
$\sim^{\prime}$, $C_{G}$ is a standard modal box over the same relation. We
are done by induction.
_‘Only if’ :_ Given a GEL model $\mathcal{M}=(X,V_{\mathsf{W}},\sim)$ over the
set $Ag$ of agents, we construct an AGEL model
$\mathcal{M}^{\prime}=(X,\mathcal{A},V_{\mathsf{W}},\sim^{\prime})$ by taking
$Ag$ to be the underlying set of $\mathcal{A}$, and
$V_{\mathsf{Ag}}(p_{a})=\\{a\\}$ for $a\in Ag$; this uses the running
assumption that ${\mathcal{L}_{\mathsf{Ag}}}$ conservatively extends classical
propositional logic, so an agent model $\mathcal{A}$ with the prescribed
valuation of agent atoms is guaranteed to exist. The truth conditions in the
respective semantics of GEL and AGEL are then literally the same.
###### Remark 4.
Depending on additional restrictions on the agent logic, it will sometimes be
possible to give also a fairly straightforward satisfiability-preserving
translation in the reverse direction, from AGEL to GEL. For instance, if the
agent logic is just classical propositional logic, then we can proceed as
follows: Let $\phi$ be an AGEL formula, and let
$A\subseteq\mathsf{At}_{\mathsf{Ag}}$ be the set of agent atoms mentioned in
$\phi$. Let $Ag$ be the (finite) set of truth valuations $\kappa\colon A\to 2$
in the set $2=\\{\bot,\top\\}$ of Boolean truth values, and write
$\kappa(\psi)$ for the truth value of a propositional formula $\psi$ over $A$
under $\kappa$; then a translation of $\phi$ into a satisfiability-equivalent
GEL formula $s(\phi)$ is defined recursively by
$s(C_{\psi}\chi)=C_{\\{\kappa\in Ag\mid\kappa(\psi)=\top\\}}s(\chi)$
and commutation with all other constructs. However, even in this basic case,
such a translation will be of limited use as it has exponential blowup (the
set ${\\{\kappa\in Ag\mid\kappa(\psi)=\top\\}}$ can be exponentially large).
For more expressive agent logics, e.g. whenever the agent logic extends
$\mathcal{ALC}$, one has (series of) formulae that are satisfiable only over
exponentially large agent models: Given agent formulae $\psi_{n}$ that are of
polynomial size in $n$ but satisfiable only over agent models of exponential
size in $n$ (in $\mathcal{ALC}$, such $\psi_{n}$ exist), the AGEL formulae
$p\land C_{\psi_{n}}\neg p$ are satisfiable only over models whose agent model
components are of exponential size in $n$. Indeed, from a purely computational
point of view (and for suitably restricted agent logics), one may see AGEL as
a way of dealing with exponentially many agents without incurring doubly
exponential computational cost. We realize this by an encoding into a
different target logic, discussed after the next example.
###### Example 5 (Lots of muddy children).
The classical _muddy children_ puzzle with $k$ many children can be seen as
$k$ many agents $A_{i}$ communicating according to a fixed protocol to gain
common knowledge of a length-$k$ bitstring where each agent $A_{i}$ can see
all bits except the one at index $i$ (Pavlovic 2021, Section 4.3).
Specifically, it is commonly known initially that at least one bit is set, and
the protocol then proceeds in rounds in which the agents announce whether they
have learned their missing bit. The full modelling of the puzzle thus requires
a dynamic epistemic logic with common knowledge and public announcements
(Baltag, Moss, and Solecki 1998; Lutz 2006). Here, we concentrate on
modelling, in an extended setting, how knowledge is gained in individual
rounds of the protocol, which does not require public announcements; we
generalize the textbook treatment by Huth and Ryan (2004).
We consider a variant of the puzzle that can essentially be seen as a product
of $n$ copies of the original puzzle. We then have an $n\times k$-matrix of
bits, and each agent has an _invisibility type_ consisting of one
_invisibility index_ per row determining the bit she cannot see in that row
(in the original puzzle, there is only one row, and the invisibility type is
the identity of the agent). We require that every bit of the matrix is seen by
at least one agent. We do not otherwise restrict which invisibility types are
realized; also, a given invisibility type may be realized by more than one
agent. Note that there are exponentially many (viz, $k^{n}$) invisibility
types; the point of these considerations being that the number of
(distinguishable) agents is a) not fixed, and b) potentially large. We
introduce propositional atoms $p_{(j,i)}$ for $1\leq j\leq n$ and $1\leq i\leq
k$ indicating whether the bit at position $(j,i$) in the matrix is _set_
($1$). We use agent atoms $h_{j,i}$ to describe agents who cannot see the
value of the bit at position $(j,i)$ of the matrix, in an agent logic that
extends propositional logic with a propositional background theory, in this
case hardwiring the above description of the scenario (every agent sees all
bits except one per row, and every bit is seen by some agent). The common
knowledge resulting from the visibility conditions is hence
$\textstyle C_{\top}\big{(}\bigwedge_{ij}(p_{j,i}\rightarrow C_{\neg
h_{j,i}}p_{j,i})\land(\neg p_{j,i}\rightarrow C_{\neg h_{j,i}}\neg
p_{j,i})\big{)}.$
We write $\alpha_{j}^{\leq x}$ for the (purely propositional) formula stating
that at most $x$ bits are set in row $j$; that is, $\alpha_{j}^{\leq x}$ is
the disjunction of all conjunctions of the form $\bigwedge_{i\in
H}p_{i,j}\land\bigwedge_{i\in\\{1,\dots,k\\}\setminus H}\neg p_{j,i}$ where
$H\subseteq\\{1,\dots,k\\}$ and $|H|\leq x$. (This formula is of exponential
size in the number $k$ of columns, but note that this happens already in the
original muddy children puzzle, i.e. in the case $n=1$.) The initial knowledge
available to the agents before the first round is that at least one bit is set
in each row:
$\textstyle C_{\top}(\bigwedge_{j}\neg\alpha_{j}^{\leq 0}).$
In each communication round the agents then choose one row, and communicate
whether or not they know the value of the bit they cannot see in that row.
Assuming all agents do not know the value of their respective bit, this
establishes common knowledge about everyone’s uncertainty:
$\textstyle C_{\top}(\bigwedge_{ij}\neg C_{h_{j,i}}p_{j,i}\land\neg
C_{h_{j,i}}\neg p_{j,i}).$
Of course the order of rounds is irrelevant here, and the state of the
protocol can hence simply be represented by a tuple $(x_{i},\dots,x_{n})$
where each $x_{j}$ counts how many communication rounds have taken place for
row $j$ (counting communication of the initial knowledge that at least one bit
is set in each row). The key invariant of the protocol is that if these
counters reach $(x_{1},\dots,x_{n})$ without anyone having learned new bits,
then this results in the accumulated common knowledge
$\textstyle C_{\top}(\bigwedge_{j}\neg\alpha_{j}^{\leq x_{j}}).$
This clearly holds in the beginning of the game due to the initial knowledge.
Then in state $(x_{1},\dots,x_{n})$, after querying row $j$, the common
knowledge increases according to the inference
$\Gamma,C_{\top}(\neg\alpha_{j}^{\leq x_{j}}),\textstyle
C_{\top}(\bigwedge_{i}\neg C_{h_{j,i}}p_{j,i}\land\neg C_{h_{j,i}}\neg
p_{j,i})\\\ \vDash C_{\top}(\neg\alpha_{j}^{\leq x_{j}+1}).$
where $\Gamma$ represents the visibility axioms and $\vDash$ denotes local
(i.e. per-world) consequence.. The formal proof is similar to the textbook
proof for the original puzzle (Huth and Ryan 2004), and sketched as follows:
* •
Assume $\alpha_{j}^{\leq x_{j}+1}$.
* •
From the accumulated knowledge of the previous rounds, the agents already know
that more than $x_{j}$ bits are set in row $j$. From the assumption, they can
conclude that exactly $x_{j}+1$ bits are set.
* •
Given that nobody knew whether their respective missing bit in row $j$ is set,
the agents can conclude that more than $x_{j}+1$ bits are set in row $j$,
contradicting the assumption. (Otherwise at least one agent whose bit is set
would see only $x_{j}$ many set bits and could hence have deduced that her
missing bit is set.)
Similar reasoning is used to conclude missing bits once enough communication
rounds have been performed.
### Upper Bound: Encoding into the Full $\mu$-Calculus
We establish the ExpTime upper bound on satisfiability checking by a
satisfiability-preserving translation of AGEL into the $\mu$-calculus with
converse, also known as the _full $\mu$-calculus_, whose satisfiability
problem is in ExpTime (Vardi 1998). We emphasize that the full $\mu$-calculus
does not have the finite model property (Vardi 1998; Streett 1982); we
therefore establish a bounded model property separately in the next section.
In the translation, we use fixpoints to take transitive-reflexive closures,
and inverse roles to close under symmetry. The main idea is then to view the
family $\sim$ of per-agent indistinguishability relations $\sim_{a}$ featuring
in the definition of AGEL models as a ternary relation on a single domain, and
to encode this ternary relation as a binary relation between worlds and
(agent, world)-pairs.
In fact, the single-variable fragment of the full $\mu$-calculus suffices for
the translation; we briefly recall its syntax and semantics. The syntax is
parametrized over sets $\mathsf{AP}$ and $\mathsf{Prog}$ of _atomic
propositions_ and _atomic programs_ , respectively. A _program_ is either an
atomic program or a _converse program_ $\alpha^{-}$ of
$\alpha\in\mathsf{Prog}$. Moreover, we fix a single _fixpoint variable_ $z$.
Then, the set of _formulae_ $\phi,\psi,\dots$ is given by the grammar
$\phi,\psi::=\bot\mid p\mid
z\mid\neg\phi\mid\phi\land\psi\mid[\alpha]\phi\mid\nu z.\,\phi$
where $p\in\mathsf{AP}$ and $\alpha$ is a program. The box operator $[\alpha]$
is read ‘for all $\alpha$-successors’. The $\nu z$ operator takes greatest
fixpoints, and binds $z$; that is, an occurrence of $z$ is _free_ if it lies
outside the scope of any $\nu z$. Application of $\nu z$ is restricted to
formulae $\phi$ in which every free occurrence of $z$ is _positive_ , i.e.
lies under an even number of negations $\neg$. Further propositional
connectives $\top$, $\lor$, $\rightarrow$, $\leftrightarrow$ are defined as
usual. Moreover, we define diamond operators as
$\langle\alpha\rangle\phi:=\neg[\alpha]\neg\phi$. (Also, one can define least
fixpoints $\mu z.\,\phi$.)
The semantics is defined over _models_ $\mathcal{M}=(X,R,V)$ that consist of a
domain $X$, an assignment $R$ of a transition relation $R(\alpha)\subseteq
X\times X$ to every atomic program $\alpha\in\mathsf{Prog}$; and a valuation
$V\colon\mathsf{AP}\to\mathcal{P}(X)$ of the atomic propositions. The
interpretation of converse programs is defined by
$R(\alpha^{-})=\\{(e,d)\mid(d,e)\in R(\alpha)\\}$. The semantics of a formula
$\phi$ is then given as a function
$\llbracket\phi\rrbracket_{\mathcal{M}}\colon\mathcal{P}(X)\to\mathcal{P}(X)$
whose argument serves as the extension of $z$, inductively defined by the
expected clauses for Boolean operators, $\llbracket
p\rrbracket_{\mathcal{M}}=V(p)$, and
$\displaystyle\llbracket z\rrbracket_{\mathcal{M}}(U)$ $\displaystyle=U$
$\displaystyle\llbracket[\alpha]\phi\rrbracket_{\mathcal{M}}(U)$
$\displaystyle=\\{w\in X\mid\forall v\in X.\,(w,v)\in R(\alpha)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\rightarrow
v\in\llbracket\phi\rrbracket_{\mathcal{M}}(U)\\}$ $\displaystyle\llbracket\nu
z.\,\phi\rrbracket_{\mathcal{M}}(U)$ $\displaystyle=\bigcup\\{Z\subseteq X\mid
Z\subseteq\llbracket\phi\rrbracket_{\mathcal{M}}(Z)\\}.$
One shows by induction that when $\nu z.\,\phi$ is well-formed, then
$\llbracket\phi\rrbracket_{\mathcal{M}}$ is monotone w.r.t. set inclusion, so
$\llbracket\nu z.\,\phi\rrbracket_{\mathcal{M}}(U)$ as defined above is, by
the Knaster-Tarski fixpoint theorem, the greatest fixpoint of
$\llbracket\phi\rrbracket_{\mathcal{M}}$.
In detail, the translation is then defined as follows. We assume for the
translation that the agent logic is given as a fragment of the full
$\mu$-calculus. For clarity, we write $u$ for the syntactic embedding; we
assume that each $u(\psi)$ is _closed_ , i.e. does not contain a free
occurrence of $z$. As indicated previously, a typical choice would be
$\mathcal{ALC}$ (modulo the usual slight syntactic shifts), but for purposes
of the complexity analysis, the agent logic could in fact be the full
$\mu$-calculus itself (whereas the axiomatization introduced in the next
section needs assumptions on the agent logic that are not satisfied by the
full $\mu$-calculus, such as the finite model property).
We use three fresh atomic programs $\pi_{1},\pi_{2},\mathsf{edge}$. The
intention is that $\mathsf{edge}$ relates worlds to (agent, world)-pairs, out
of which $\pi_{1}$ extracts the agent component, and $\pi_{2}$ the world
component. Speaking more precisely, we do not insist that $\pi_{1},\pi_{2}$
are functional, so when $(x,r)\in R(\mathsf{edge})$ in a model, then $r$ may
in fact represent several (agent, world) pairs, namely all pairs $(a,y)$ such
that $(r,a)\in R(\pi_{1})$ and $(r,y)\in R(\pi_{2})$ (and indeed it may happen
that $r$ represents no pair at all). As per Remark 2, we do not need to worry
at this point about the fact that the induced indistinguishability relations
are not forced to be equivalence relations. As an intermediate step in the
translation, we introduce forward and backward binary ‘next-step’ modalities
by abbreviation:
$\psi\leadsto\phi:=[\mathsf{edge}](\langle\pi_{1}\rangle\psi\rightarrow[\pi_{2}]\phi),$
is understood as ‘all worlds that are reached by a single forward
indistinguishability step for an agent satisfying $\psi$ satisfy $\phi$’, and
$\phi\mathrel{\reflectbox{$\leadsto$}}\psi:=[\pi_{2}^{-}](\langle\pi_{1}\rangle\psi\rightarrow[\mathsf{edge}^{-}]\phi)$
describes the opposite direction, ‘all worlds that are reached by a single
backward indistinguishability step for an agent satisfying $\psi$ satisfy
$\phi$’. Using these two abbreviations to model a step along the
symmetrization of the indistinguishability relation, we can encode the common
knowledge modality, modelling transitive-reflexive closure via a greatest
fixpoint as indicated above: We define the translation $t$ of a AGEL formula
$\phi$ into a formula $t(\phi)$ in the full $\mu$-calculus recursively by
$t(C_{\psi}\phi)=\nu z.\,t(\phi)\allowbreak\land(u(\psi)\leadsto
z)\allowbreak\land(z\mathrel{\reflectbox{$\leadsto$}}u(\psi))$
and commutation with all other constructs. (Since no recursive calls of $t$
are made on $\psi$, its duplication does not lead to exponential blowup;
recall that by the current assumption, the translation $u$ of agent formulae
does essentially nothing.) We thus obtain the following result.
[end]theorem[Upper complexity bound] If the agent logic is a fragment of the
full $\mu$-calculus, then the satisfiability problem of AGEL is in ExpTime. We
show that the translation $t$ is really satisfiability-preserving, i.e. that
an AGEL formula $\phi$ is satisfiable iff $t(\phi)$ is satisfiable in the full
$\mu$-calculus.
_‘Only if’ :_ Given a AGEL model
$\mathcal{M}=(X,\mathcal{A},V_{\mathsf{W}},\sim)$ we construct a
$\mu$-calculus model $\mathcal{M}^{\prime}=(X^{\prime}=Ag\uplus
X\uplus(Ag\times X),R,V)$ where we interpret the syntactic material (atomic
programs and atomic propositions) mentioned in agent formulae in $\phi$ over
$Ag\subseteq X^{\prime}$ like in the given agent model $\mathcal{A}$,
exploiting that the agent logic is a fragment of the full $\mu$-calculus. We
assume w.l.o.g. that the sets $X$, $Ag$, and $Ag\times X$ are already
disjoint. We define the valuation $V$ as extending the valuation induced by
$\mathcal{A}$ to interpret world atoms via $V_{\mathsf{W}}$, and moreover put
$\displaystyle R(\mathsf{edge})$ $\displaystyle=\\{(x,(a,y))\mid
x\sim_{a}y\text{ in }\mathcal{M}\\}$ $\displaystyle R(\pi_{1})$
$\displaystyle=\\{((a,x),a)\mid(a,x)\in Ag\times X\\}$ $\displaystyle
R(\pi_{2})$ $\displaystyle=\\{((a,x),x)\mid(a,x)\in Ag\times X\\}.$
This model translation encodes the ternary relation into a heterogeneous world
set where edges are encoded as pairs of agents and target states. The agent
and the target state are then extracted from a pair with $\pi_{1}$, $\pi_{2}$
serving as projections.
We show by induction on $\phi$ that $\mathcal{M},x\models\phi$ iff
$\mathcal{M}^{\prime},x\models t(\phi)$, for $x\in X$. Again, the cases for
world atoms and Boolean operators are trivial; we do the case for the common
knowledge operator $C_{\psi}$.
We first note that due to symmetry of $\sim$, we have that for $x,y,a\in
X^{\prime}$,
$(x,(a,y))\in R(\mathsf{edge}),$
(and then $((a,y),a)\in R(\pi_{1})$, $((a,y),y)\in R(\pi_{2})$) exactly when
$((a,x),y)\in R(\mathsf{edge}^{-})$
(and then $((a,x),a)\in R(\pi_{1})$, $(x,(a,x))\in R(\pi_{2}^{-})$), namely,
both hold iff $x\sim_{a}y$, equivalently $y\sim_{a}x$, in $\mathcal{M}$. It is
hence clear that $\psi\leadsto\chi$ and
$\chi\mathrel{\reflectbox{$\leadsto$}}\psi$ are equivalent over the model
$\mathcal{M}^{\prime}$. Over this model, we can thus equivalently modify the
translation to
$t(C_{\psi}\phi)=\nu z.\,\phi\land(\psi\leadsto z);$
also, the above implies that $\psi\leadsto(-)$ acts as a box modality on
$\psi$-successors ($\psi\leadsto\chi$ holds at a world iff $\chi$ holds at all
$\psi$-successors). By Lemma Semantics, the interpretation of $C_{\psi}$ in
$\mathcal{M}$ is a greatest fixpoint involving a box operator for the same
relation $\bigcup_{a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}}\sim_{a}$.
More formally, since $\mathcal{M}$ embeds into $\mathcal{M}^{\prime}$, and the
associated inclusion map is a $p$-morphism (Blackburn, de Rijke, and Venema
2001), i.e. a functional bisimulation, w.r.t. these relations, it follows by
the well-known fact that the (forward) $\mu$-calculus is bisimulation-
invariant that satisfaction of $C_{\psi}\phi$ is preserved.
_‘If’ :_ Given a $\mu$-calculus-model $\mathcal{M}=(X,R,V)$ we construct an
AGEL model $\mathcal{M}^{\prime}=(X,\mathcal{A},V_{\mathsf{W}},\sim)$. We use
the original set $X$ of worlds also as the underlying set $Ag$ of the agent
model $\mathcal{A}$, whose structure we then obtain by just suitably
restricting $\mathcal{M}$. Similarly, we take $V_{\mathsf{W}}$ to be the
restriction of $V$ to world atoms, i.e. we interpret world atoms in
$\mathcal{M}^{\prime}$ like in $\mathcal{M}$. We define the
indistinguishability relation $\sim_{a}$ for $a\in Ag=X$ as the symmetric
closure of the relation
$\\{(x,y)\mid\exists(x,x^{\prime})\in R(\mathsf{edge}).\\\ (x^{\prime},a)\in
R(\pi_{1}),(x^{\prime},y)\in R(\pi_{2})\\},$
exploiting that by Remark 2, we do not actually need to care whether
$\sim_{a}$ is reflexive or transitive. We do enforce symmetry to ensure more
straightforward agreement of the semantics on both sides.
It is then clear that for each $a\in Ag$ and each agent formula $\psi$,
${\mathcal{A},a\models\psi}$ iff
$a\in{\llbracket\psi\rrbracket_{\mathcal{M}}}(U)$, noting that the latter does
not depend on $U$ since agent formulae are assumed to be closed.
Now note that by the above construction of $\sim$, $\psi\leadsto(-)$ and
$(-)\mathrel{\reflectbox{$\leadsto$}}\psi$ jointly act as a box modality on
the $\psi$-successors according to $\sim$, where the symmetric closure is
emulated by considering both $\psi\leadsto(-)$ and
$(-)\mathrel{\reflectbox{$\leadsto$}}\psi$. Replacing the semantic clause for
$C_{\psi}$ by the fixpoint characterization as per Lemma Semantics, we thus
obtain that the truth conditions for $C_{\psi}$ is the same as for its
translation along $t$. As the clauses for Boolean operators agree as well, and
the valuation of world atoms in $\mathcal{M}^{\prime}$ is just inherited from
$\mathcal{M}$, we obtain that for each $x\in X$ and each world formula $\chi$,
$\mathcal{M}^{\prime},x\models\chi$ iff $x\in\llbracket
t(\chi)\rrbracket_{\mathcal{M}}(U)$ (noting again that the latter set does not
depend on $U$ because $t(\chi)$ is closed), so $\mathcal{M}^{\prime}$ serves
as a witness for satisfiability of any formula $\chi$ such that $t(\chi)$ is
satisfied within $\mathcal{M}$.
###### Proof sketch.
We need to show that the translation $t$ is really satisfiability-equivalent.
From a model $\mathcal{M},x_{0}$ of an AGEL formula $\phi$, with
$\mathcal{M}=(X,\mathcal{A},V_{\mathsf{W}},\sim)$, we construct a model
$(\mathcal{M}^{\prime},x_{0})$ of $t(\phi)$, with
$\mathcal{M}^{\prime}=(X^{\prime},R,V)$, as follows. Following the intention
of the translation $t(\phi)$ as indicated above, we take $X^{\prime}$ to be
the union of the set $X$ of worlds, the set $Ag$ of agents (the underlying set
of $\mathcal{A}$), and the set $Ag\times X$ of (agent, world) pairs, assuming
w.l.o.g. that these sets are disjoint. We interpret the syntactic material
(atomic programs and atomic propositions) mentioned in agent formulae in
$\phi$ over $Ag\subseteq X^{\prime}$ like in the given agent model
$\mathcal{A}$, exploiting that the agent logic is a fragment of the full
$\mu$-calculus. Similarly, we interpret world atoms on $X$ in the same way as
in $\mathcal{M}$. Finally, we interpret $\pi_{1},\pi_{2}$ as the expected
projections $R(\pi_{1})=\\{((a,x),a)\mid(a,x)\in Ag\times X\\}$,
$R(\pi_{2})=\\{((a,x),x)\mid(a,x)\in Ag\times X\\}$, and $\mathsf{edge}$ as
$R(\mathsf{edge})=\\{(x,(a,y))\mid x\sim_{a}y\text{ in }\mathcal{M}\\}.$
It is not hard to see that $(\mathcal{M}^{\prime},x)$ is really a model of
$\phi$.
In the converse direction, given a model $(\mathcal{M},x_{0})$ of the
$\mu$-calculus formula $t(\phi)$, with $\mathcal{M}=(X,R,V)$, we construct a
model $(\mathcal{M}^{\prime},x_{0})$ of $\phi$, with
$\mathcal{M}^{\prime}=(X,\mathcal{A},V_{\mathsf{W}},\sim)$, as follows. We use
the original set $X$ of worlds also as the underlying set $Ag$ of the agent
model $\mathcal{A}$, whose structure we then obtain by just suitably
restricting $\mathcal{M}$. Similarly, we take $V_{\mathsf{W}}$ to be the
restriction of $V$ to world atoms, i.e. we interpret world atoms in
$\mathcal{M}^{\prime}$ like in $\mathcal{M}$. Finally, as indicated above, we
define the indistinguishability relation $\sim_{a}$ for $a\in Ag=X$ as the
symmetric closure of the relation
$\\{(x,y)\mid\exists(x,e)\in R(\mathsf{edge}).\\\ (e,a)\in R(\pi_{1}),(e,y)\in
R(\pi_{2})\\}.$
Again, it is not hard to check that $(\mathcal{M},x)$ is really a model of
$\phi$. ∎
In combination with Theorem Lower Bound: Reduction from Group Epistemic Logic,
we thus have
###### Corollary 6 (Complexity of AGEL).
If the agent logic is a fragment of the full $\mu$-calculus, then the
satisfiability problem of AGEL is ExpTime-complete.
###### Remark 7.
Indeed the above encoding implies that one can raise the expressiveness of the
agent logic to extensions of the full $\mu$-calculus that remain decidable in
ExpTime. One candidate is the _full hybrid $\mu$-calculus_ (Sattler and Vardi
2001), which extends the full $\mu$-calculus with _nominals_ , i.e.
propositional atoms denoting single objects. This opens the possibility of
combining explicitly named agents in the standard sense with abstract groups
of agents, as in the formula
$C_{\mathsf{John}\vee\exists\,\mathsf{hasFriend}.\,\\!\mathsf{John}}\,\mathsf{Pub\\_on\\_Wednesdays},$
which says that John and his friends know that their regular pub night is on
Wednesdays.
## Completeness and Bounded Models
We axiomatize AGEL in Hilbert style using the following system $C5$ of axioms
and rules:
$\displaystyle(T)$ $\displaystyle C_{\psi}\phi\rightarrow\phi$
$\displaystyle(\bot)$ $\displaystyle\phi\rightarrow C_{\bot}\phi$
$\displaystyle(K)$ $\displaystyle
C_{\psi}(\phi\rightarrow\gamma)\rightarrow(C_{\psi}\phi\rightarrow
C_{\psi}\gamma)$ $\displaystyle(4)$ $\displaystyle C_{\psi}\phi\rightarrow
C_{\psi}C_{\psi}\phi$ $\displaystyle(5)$ $\displaystyle\neg
C_{\psi}\phi\rightarrow C_{\psi}\neg C_{\psi}\phi$
$\displaystyle(\mathit{Ind})$ $\displaystyle
C_{\psi\lor\chi}(\phi\rightarrow(C_{\psi}\phi\land
C_{\chi}\phi))\rightarrow(\phi\rightarrow C_{\psi\lor\chi}\phi)$
$\displaystyle(\mathit{Nec})$
$\displaystyle\frac{\phi}{C_{\psi}\phi}\qquad(\mathit{AM})\quad\frac{\gamma\rightarrow\psi}{C_{\psi}\phi\rightarrow
C_{\gamma}\phi}$
Recall that by our running assumptions, the agent logic
${\mathcal{L}_{\mathsf{Ag}}}$ is closed under all propositional connectives.
We write $C5\vdash\phi$ if an AGEL formula $\phi$ is derivable in this system;
$\phi$ is _consistent_ if $C5\not\vdash\neg\phi$. For a finite set $\Gamma$ of
formulae, we write $\widehat{\Gamma}$ for the conjunction of all formulae in
$\Gamma$, and we say that $\Gamma$ is _consistent_ if $\widehat{\Gamma}$ is
consistent. The system includes the usual $S5$ axioms for common knowledge,
reflecting normality (axiom $(K)$), reflexivity (axiom $(T)$), transitivity
(axiom $(4)$), and Euclideanity (axiom $(5)$), as well as the usual
necessitation rule $(\mathit{Nec})$. As usual, the ($K$) axiom implies
commutation of $C_{\psi}$ with conjunction, and hence, together with the
necessitation rule ($\mathit{Nec}$), monotonicity and replacement of
equivalents for $C_{\psi}$. Specific properties of AGEL are reflected in the
axiom $(\bot)$, which, together with $(T)$, says that $C_{\bot}\phi$ holds
almost vacuously, in the sense that it does not claim any agent to know
anything but requires $\phi$ to be true in the current world; in the rule
$(\mathit{AM})$, which says that $C_{\psi}\phi$ is antimonotone in $\psi$, as
requiring fewer agents to know $\phi$ is a weaker claim; and, centrally, in
the induction axiom $(\mathit{Ind})$, which captures the fact that the
indistinguishability relation $\sim_{A\cup B}$ for a union $A\cup B$ of two
sets $A,B$ of agents is the reflexive-transitive closure of the union of the
indistinguishability relations $\sim_{A}$, $\sim_{B}$ for the original sets.
The rule $(\mathit{AM})$ implies replacement of equivalents in the index of
$C_{\psi}$ (from equivalence of $\psi$ and $\psi^{\prime}$, derive equivalence
of $C_{\psi}\phi$ and $C_{\psi^{\prime}}\phi$). Via $(\mathit{AM})$, the
system depends on reasoning in the agent logic, which we will assume to be
completely axiomatized.
We note first that the induction axiom generalizes to multiple disjuncts:
[end]lemma Every formula of the form
$C_{\bigvee_{i=1}^{n}\psi_{i}}(\phi\rightarrow(\textstyle\bigwedge_{i=1}^{n}C_{\psi_{i}}\phi))\rightarrow(\phi\rightarrow
C_{\bigvee_{i=1}^{n}\psi_{i}}\phi)$
(for $n\geq 0$) is derivable in $C5$. Induction on $n$. We keep use of
monotonicity, replacement of equivalents, and propositional reasoning
implicit. The case $n=0$ is by ($\mathit{Nec}$) and $(\bot)$, and the case
$n=1$ by ($T$). For the inductive step $n\to n+1$, where $n\geq 1$, we note
first that the inductive hypothesis implies derivability of the formula
$C_{\bigvee_{i=1}^{n+1}\psi_{i}}C_{\bigvee_{i=1}^{n}\psi_{i}}(\phi\rightarrow(\textstyle\bigwedge_{i=1}^{n}C_{\psi_{i}}\phi))\rightarrow\\\
C_{\bigvee_{i=1}^{n+1}\psi_{i}}(\phi\rightarrow
C_{\bigvee_{i=1}^{n}\psi_{i}}\phi)$
using ($\mathit{Nec}$) (with $C_{\bigvee_{i=1}^{n+1}\psi_{i}}$) and ($K$).
Using ($\mathit{AM}$),and ($4$), we then derive
$C_{\bigvee_{i=1}^{n+1}\psi_{i}}(\phi\rightarrow(\textstyle\bigwedge_{i=1}^{n}C_{\psi_{i}}\phi))\rightarrow
C_{\bigvee_{i=1}^{n+1}\psi_{i}}(\phi\rightarrow
C_{\bigvee_{i=1}^{n}\psi_{i}}\phi)$
and thus, conjoining both sides of the implication with
$C_{\bigvee_{i=1}^{n+1}\psi_{i}}(\phi\to C_{\psi_{n+1}}\phi)$ and using
commutation of $C_{\bigvee_{i=1}^{n+1}\psi_{i}}$ with conjunction,
$C_{\bigvee_{i=1}^{n+1}\psi_{i}}(\phi\rightarrow(\textstyle\bigwedge_{i=1}^{n+1}C_{\psi_{i}}\phi))\rightarrow\\\
C_{\bigvee_{i=1}^{n+1}\psi_{i}}(\phi\rightarrow(C_{\bigvee_{i=1}^{n}\psi_{i}}\phi\land
C_{\psi_{n+1}}\phi)).$
The inductive claim for $n+1$ then follows by chaining this implication with
an instance of ($\mathit{Ind}$). We explicitly record soundness of the system:
[end]theorem[Soundness] If $C5\vdash\phi$ then $\models\phi$. Soundness of the
axioms $(T)$, $(4)$, $(5)$, $(K)$ and the rule $(\mathit{Nec})$ is standard
for modal logics interpreted over equivalence relations. Soundness of $(\bot)$
and the rule $(\mathit{AM})$ is clear. Soundness of the induction axiom
$C_{\psi\lor\chi}(\phi\rightarrow(C_{\psi}\phi\land
C_{\chi}\phi))\rightarrow(\phi\rightarrow C_{\psi\lor\chi}\phi)$ is seen as
follows. Let $x$ a world in an AGEL-model $\mathcal{M}$ that satisfies
$C_{\psi\lor\chi}(\phi\rightarrow(C_{\psi}\phi\land C_{\chi}\phi))$ and
$\phi$; we have to show that ${\mathcal{M},x\models C_{\psi\lor\chi}\phi}$. We
need to show that ${\mathcal{M},y\models\phi}$ for all worlds $y$ such that
$x\sim_{{\llbracket\psi\lor\chi\rrbracket_{\mathcal{A}}}}y$. Let $y$ be such a
world. Then there exists a finite chain
$x=x_{0}\sim_{a_{1}}x_{1}\sim_{a_{2}}\dots\sim_{a_{m-1}}x_{m-1}\sim_{a_{m}}x_{m}=y$
such that
$a_{1},\dots,a_{m}\in{\llbracket\psi\lor\chi\rrbracket_{\mathcal{A}}}$. By
straightforward induction on $i$, every state $x_{i}$ in this chain satisfies
$C_{\psi}\phi$, $C_{\chi}\phi$, and $\phi$. In particular, $x_{m}$ satisfies
$\phi$, as required. We show completeness via a finite canonical model
construction that is related to the standard treatment of propositional
dynamic logic (PDL) in that it needs to close canonical models under
transitivity (Fischer and Ladner 1979; Blackburn, de Rijke, and Venema 2001).
This construction requires some restrictions on the agent logic:
###### Definition 8.
We say that the agent logic ${\mathcal{L}_{\mathsf{Ag}}}$ has the _filtered
model property_ if, for each finite set $\Sigma_{\mathsf{Ag}}$ of agent
formulae that is closed under subformulae, there is an agent model
$\mathcal{A}(\Sigma_{\mathsf{Ag}})$, with underlying set denoted by
$Ag(\Sigma_{\mathsf{Ag}})$, such that on the one hand any two distinct agents
in $Ag(\Sigma_{\mathsf{Ag}})$ are distinguished by a formula in
$\Sigma_{\mathsf{Ag}}$ (in particular $Ag(\Sigma_{\mathsf{Ag}})$ is finite,
namely at most exponential in $|\Sigma_{\mathsf{Ag}}|$), and on the other hand
for every satisfiable subset $\Gamma\subseteq\Sigma_{\mathsf{Ag}}$, there is
an agent in $Ag(\Sigma_{\mathsf{Ag}})$ that satisfies all formulae in
$\Gamma$. Since the agent logic is closed under propositional connectives, we
then have, for each agent $a\in Ag(\Sigma_{\mathsf{Ag}})$, a _characteristic
agent formula_ $\widehat{a}$ (a propositional combination of
$\Sigma_{\mathsf{Ag}}$-formulae) such that
${\llbracket\widehat{a}\rrbracket_{\mathcal{A}(\Sigma_{\mathsf{Ag}})}}=\\{a\\}$.
We put
$\mathsf{Clo_{\mathsf{Ag}}}(\Sigma_{\mathsf{Ag}})=\Sigma_{\mathsf{Ag}}\cup\\{\widehat{}a\mid
a\in Ag(\Sigma_{\mathsf{Ag}})\\}.$
###### Example 9.
As indicated in the introduction, $\mathcal{ALC}$ has the filtered model
property (and is completely axiomatized), and in fact one can go beyond
$\mathcal{ALC}$ to some degree. In particular, the extension of
$\mathcal{ALC}$ with _nominals_ , $\mathcal{ALCO}$, still has the filtered
model property. (See also comments in Remark 7.)
We _fix from now on a consistent AGEL formula $\rho_{0}$_. We base our
canonical model construction on a suitable notion of closure.
###### Definition 10 (Normalized negation).
We let $\leavevmode\hbox{\smash{$\dot{\neg}$}}\phi=\chi$ if $\phi$ has the
form $\phi=\neg\chi$, and
$\leavevmode\hbox{\smash{$\dot{\neg}$}}\phi=\neg\phi$ otherwise.
###### Definition 11 (Closure).
Let $\Sigma_{\mathsf{Ag}}$ be the closure of the set of agent formulae
occurring in $\rho_{0}$ under taking subformulae. Then the _closure_ $\Sigma$
of $\rho_{0}$ is the least set of world formulae containing $\rho_{0}$ that is
closed under (world) subformulae and normalized negation, and moreover
satisfies
$\text{if }C_{\chi}\phi\in\Sigma\text{ and
}\psi\in\mathsf{Clo_{\mathsf{Ag}}}(\Sigma_{\mathsf{Ag}})\text{, then
}C_{\psi}\phi\in\Sigma$
(with $\mathsf{Clo_{\mathsf{Ag}}}(\Sigma_{\mathsf{Ag}})$ as per Definition 8).
We next construct a weak form of model of $\rho_{0}$ that assigns
indistinguishability relations to sets of agents without regard to their
definition via reflexive-transitive closure; this will be rectified in a
subsequent step.
###### Definition 12 (Pseudo-model).
An AGEL _pseudo-model_
$\mathcal{M}^{p}=(X,\mathcal{A},V_{\mathsf{W}},\sim^{p})$ consists of a set
$X$ of worlds, an agent model $\mathcal{A}$ with underlying set $Ag$ of
agents, a valuation
$V_{\mathsf{W}}\colon\mathsf{At}_{\mathsf{W}}\to\mathcal{P}(X)$ of the world
atoms, and an equivalence relation $\sim^{p}_{A}$ on $X$ for each subset
$A\subset Ag$. The _semantics_ of AGEL over pseudo-models is defined like over
models, except that the interpretation of $C_{\psi}$ uses the relation
$\sim^{p}_{{\llbracket\psi\rrbracket_{\mathcal{A}}}}$ in place of
$\sim_{{\llbracket\psi\rrbracket_{\mathcal{A}}}}$.
We construct a _canonical pseudo-model_
$\mathcal{C}^{p}_{\Sigma}=(X_{\Sigma},\allowbreak\mathcal{A}(\Sigma_{\mathsf{Ag}}),\allowbreak
V_{\mathsf{W}},\sim^{p})$ by taking $X_{\Sigma}$ to consist of the maximal
consistent subsets of $\Sigma$; $\mathcal{A}(\Sigma_{\mathsf{Ag}})$ as per
Definition 8; $V_{\mathsf{W}}(p)=\\{\Gamma\in X_{\Sigma}\mid p\in\Gamma\\}$;
and
$\Gamma\sim^{p}_{\llbracket\psi\rrbracket_{\mathcal{A}}}\Delta\leftrightarrow\widehat{\Gamma}\land
P_{\psi}\widehat{\Delta}\text{ consistent}$
where by $P_{\psi}$ we denote the dual of $C_{\psi}$, i.e. $P_{\psi}\phi:=\neg
C_{\psi}\neg\psi$. Well-definedness is guaranteed by the properties of
$\mathcal{A}(\Sigma_{\mathsf{Ag}})$; indeed we shall often write
$\sim^{p}_{\psi}$ in place of
$\sim^{p}_{\llbracket\psi\rrbracket_{\mathcal{A}}}$ for readability, similarly
for $\sim$. We note that $\Sigma$ inherits exponential size from
$\mathsf{Clo_{\mathsf{Ag}}}(\Sigma_{\mathsf{Ag}})$, so $X_{\Sigma}$ is of
doubly exponential size in the size of $\rho_{0}$.
[end]lemma The relations $\sim^{p}_{\psi}$ are reflexive and symmetric.
_Symmetry:_ Let $\widehat{\Gamma}\land P_{\psi}\widehat{\Delta}$ be
consistent. Then $\widehat{\Delta}\land P_{\psi}\widehat{\Gamma}$ is
consistent; for assume otherwise. Then $\vdash\widehat{\Delta}\rightarrow
C_{\psi}\neg\widehat{\Gamma}$, and hence $\widehat{\Gamma}\land
P_{\psi}C_{\psi}\neg\widehat{\Gamma}$ is consistent. By the contraposition of
axiom $(5)$, we obtain that $\widehat{\Gamma}\land
C_{\psi}\neg\widehat{\Gamma}$ is consistent, which contradicts $(T)$.
_Reflexivity:_ To show that $\Gamma\sim^{p}_{\psi}\Gamma$, suppose otherwise.
Then $\vdash\widehat{\Gamma}\rightarrow C_{\psi}\neg\widehat{\Gamma}$ and via
$(T)$, $\vdash\widehat{\Gamma}\rightarrow\neg\widehat{\Gamma}$, contradicting
consistency of $\Gamma$.
The point of using the canonical pseudo-model is that it allows for a
straightforward proof of the usual existence lemma: [end]lemma[Existence
Lemma] Let $\Gamma\in X_{\Sigma}$ be a world in the canonical pseudo-model
$\mathcal{C}^{p}_{\Sigma}$ such that $\neg C_{\psi}\phi\in\Gamma$. Then there
exists $\Delta\in X_{\Sigma}$ such that $\Gamma\sim^{p}_{\psi}\Delta$ and
$\neg\phi\in\Delta$ Enumerate the formulae
$\sigma_{1},\dots,\sigma_{m}\in\Sigma$. We define sets $\Delta_{n}$
iteratively to make $\widehat{\Gamma}\land P_{\psi}\widehat{\Delta_{n}}$
consistent:
$0$:
$\Delta_{0}=\\{\neg\phi\\}$. Since $\neg C_{\psi}\phi\in\Gamma$, we trivially
have that $\widehat{\Gamma}\land P_{\psi}\neg\phi$ is consistent.
$n\to n+1$:
As $\widehat{\Gamma}\land P_{\psi}\widehat{\Delta_{n}}$ is consistent, so is
$\widehat{\Gamma}\land
P_{\psi}(\widehat{\Delta_{n}}\land(\sigma_{n+1}\lor\leavevmode\hbox{\smash{$\dot{\neg}$}}\sigma_{n+1}))$.
Since $P_{\psi}$ derivably commutes with disjunction, it follows that
$\widehat{\Gamma}\land
P_{\psi}((\widehat{\Delta_{n}}\land\sigma_{n+1})\lor(\widehat{\Delta_{n}}\land\leavevmode\hbox{\smash{$\dot{\neg}$}}\sigma_{n+1}))$
is consistent, i.e. at least one of $\widehat{\Gamma}\land
P_{\psi}(\widehat{\Delta_{n}\cup\\{\sigma_{n+1}\\}})$ or
$\widehat{\Gamma}\land
P_{\psi}(\widehat{\Delta_{n}\cup\\{\leavevmode\hbox{\smash{$\dot{\neg}$}}\sigma_{n+1}\\}})$
is consistent, and correspondingly $\Delta_{n}\cup\\{\sigma_{n+1}\\}$ or
$\Delta_{n}\cup\\{\leavevmode\hbox{\smash{$\dot{\neg}$}}\sigma_{n+1}\\}$ can
then be picked as $\Delta_{n+1}$.
Then, $\Delta_{m}$ serves as the desired $\Delta$. Leveraging characteristic
agent formulae, we can derive a proper AGEL-model, the _canonical model_
$\mathcal{C}_{\Sigma}$, from the canonical pseudo-model
$\mathcal{C}^{p}_{\Sigma}$ by taking
$\sim_{a}\;=\;\sim^{p}_{\widehat{a}},$
where we exploit that by Remark 2, we do not need to care whether $\sim_{a}$
is transitive. The key point is then that the existence lemma survives this
transition thanks to the following fact, which hinges on the induction axiom
$(\mathit{Ind})$: [end]lemma For all formulae
$\psi\in\mathsf{Clo_{\mathsf{Ag}}}(\Sigma_{\mathsf{Ag}})$,
$\sim^{p}_{\psi}\subseteq\sim_{{\llbracket\psi\rrbracket_{\mathcal{A}}}}$ Let
$\Gamma,\Delta\in X_{\Sigma}$, and let
$\psi\in\mathsf{Clo_{\mathsf{Ag}}}(\Sigma_{\mathsf{Ag}})$ be an agent formula
such that $\Gamma\sim^{p}_{\psi}\Delta$. We have to show that there exist a
finite sequence of worlds $C_{0},\dots,C_{n}$ and
$a_{0},\dots,a_{n-1}\in{\llbracket\psi\rrbracket_{\mathcal{A}}}$ such that
$\Gamma=C_{0}\sim_{a_{0}}C_{1},\dots,C_{n-1}\sim_{a_{n-1}}C_{n}=\Delta$. Let
$D$ be the set of all worlds reachable from $\Gamma$ by such a sequence; we
thus have to show that $\Delta\in D$. Define $\delta:=\bigvee_{\Theta\in
D}\widehat{\Theta}$. Note that
$\delta\land\bigvee_{a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}}P_{\widehat{a}}\neg\delta$
is inconsistent, for suppose otherwise. Then (by derivable commutation of $P$
operators with disjunction), $\delta\land P_{\widehat{a}}\widehat{F}$ would be
consistent for some $a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}$ and some
$F_{a}\notin D$. Hence, also $\widehat{C}\land P_{\widehat{a}}\widehat{F_{a}}$
is consistent for at least one $C\in D$. But then
$C\sim^{p}_{\widehat{a}}F_{a}$, i.e. $C\sim_{a}F_{a}$, implying that $F_{a}$
is reachable from $\Gamma$, in contradiction to $F_{a}\notin D$. So,
$\vdash\delta\rightarrow\bigwedge_{a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}}C_{\widehat{a}}\delta$.
By $(\mathit{Nec})$, we obtain $\vdash
C_{\bigvee_{a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}}\widehat{a}}(\delta\rightarrow\bigwedge_{a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}}C_{\widehat{a}}\delta)$,
whence $\vdash\delta\rightarrow
C_{\bigvee_{a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}}\widehat{a}}\delta$
by Lemma Completeness and Bounded Models. From Lemma 12, $\Gamma\in D$ and
hence $\vdash\widehat{\Gamma}\rightarrow
C_{\bigvee_{a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}}\widehat{a}}\delta$.
By the assumptions on $\mathcal{A}(\Sigma_{\mathsf{Ag}})$,
$\vdash\psi\rightarrow\bigvee_{a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}}\widehat{a}$.
Using $(\mathit{AM})$, we obtain $\vdash\widehat{\Gamma}\rightarrow
C_{\psi}\delta$. As $\Gamma\sim^{p}_{\psi}\Delta$, $\widehat{\Gamma}\land
P_{\psi}\widehat{\Delta}$ is consistent, and hence
$P_{\psi}(\widehat{\Delta}\land\delta)$ is consistent by standard reasoning
with the $(K)$ axiom. This means that $\widehat{\Delta}\land\delta$ is
consistent, whence $\widehat{\Theta}\land\widehat{\Delta}$ is consistent for
at least one disjunct $\widehat{\Theta}$ of $\delta$ (with $\Theta\in D$). As
both $\Theta$ and $\Delta$ are maximal consistent subsets of $\Sigma$, we
obtain $\Theta=\Delta$ and hence $\Delta\in D$. It is then straightforward to
establish the expected truth lemma, making use of the fact that
$\mathsf{Clo_{\mathsf{Ag}}}(\Sigma_{\mathsf{Ag}})$ contains all requisite
characteristic agent-formulae: [end]lemma[Truth Lemma] Let $\Gamma$ be a world
in the canonical model $\mathcal{C}_{\Sigma}$. Then for all $\phi\in\Sigma$,
$\phi\in\Gamma\leftrightarrow{\mathcal{C}_{\Sigma},\Gamma\models\phi}.$
Induction on $\phi$. The propositional cases are trivial. In the case for
$\phi=C_{\psi}\chi$, we need to show that $C_{\psi}\chi\in\Gamma$ iff
${\mathcal{C}_{\Sigma},\Gamma\models C_{\psi}\chi}$.
_‘Only if’:_ Suppose that $C_{\psi}\chi\in\Gamma$. We need to show that
${\mathcal{C}_{\Sigma},\Gamma\models C_{\psi}\chi}$, i.e. that all worlds
reachable from $\Gamma$ in finitely many $\psi$-steps satisfy $\chi$. By the
inductive hypothesis, this means that all these worlds contain $\chi$, which
follows by $(T)$ once we show that all these worlds contain $C_{\psi}\chi$; to
this end, it suffices so show that the formula $C_{\psi}\chi$ is inherited
from any world (say, $\Gamma$) to all its $\psi$-successors. Assume the
contrary. Then there exists a world $\Delta$ such that $\widehat{\Gamma}\land
P_{\widehat{a}}\widehat{\Delta}$ is consistent for some
$a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}$ and $\neg
C_{\psi}\chi\in\Delta$. In particular, $\widehat{\Gamma}\land
P_{\widehat{a}}{\neg C_{\psi}\chi}$ is consistent, i.e.
$\widehat{\Gamma}\land\neg C_{\widehat{a}}C_{\psi}\chi$ is consistent. By the
assumptions on $\mathcal{A}(\Sigma_{\mathsf{Ag}})$, and because
$a\in{\llbracket\psi\rrbracket_{\mathcal{A}}}$, we have
$\vdash\widehat{}a\rightarrow\psi$. Using $(\mathit{AM})$, we obtain $\vdash
C_{\psi}C_{\psi}\chi\rightarrow C_{\widehat{a}}C_{\psi}\chi$, and hence
$\vdash\neg C_{\widehat{a}}C_{\psi}\chi\rightarrow\neg C_{\psi}C_{\psi}\chi$.
Thus, $\widehat{\Gamma}\land\neg C_{\psi}C_{\psi}\chi$ is consistent; by (the
contraposition of) $(4)$, it follows that $\widehat{\Gamma}\land\neg
C_{\psi}\chi$ is consistent, in contradiction to $C_{\psi}\chi\in\Gamma$.
_‘If’:_ By contraposition. Suppose that $C_{\psi}\chi\notin\Gamma$, so $\neg
C_{\psi}\chi\in\Gamma$. We need to show that
$\mathcal{C}_{\Sigma},\Gamma\not\models C_{\psi}\chi$. By Lemma 12, we have
$\Gamma\sim^{p}_{\psi}\Delta$ for some $\Delta$ such that $\neg\chi\in\Delta$,
i.e. $\chi\notin\Delta$. By Lemma Completeness and Bounded Models,
$\Gamma\sim_{{\llbracket\psi\rrbracket_{\mathcal{A}}}}\Delta$, and by the
inductive hypothesis, $\mathcal{C}_{\Sigma},\Delta\not\models\chi$, showing
the claim.
Completeness and the bounded model property are then immediate:
[end]corollary[Completeness over finite models] Suppose that
${\mathcal{L}_{\mathsf{Ag}}}$ has the filtered model property and is
completely axiomatized. Then $C5$, together with the axiomatization of
${\mathcal{L}_{\mathsf{Ag}}}$, is weakly complete, i.e. every valid AGEL
formula is derivable. Moreover, AGEL has the _bounded model property_ : if a
formula $\rho_{0}$ is satisfiable, then $\rho_{0}$ has a finite model with at
most doubly exponentially many worlds in the size of $\rho_{0}$. By
contraposition: Let $\rho_{0}$ a formula such that $C5\not\vdash\rho_{0}$,
then $\neg\rho_{0}$ is consistent with $C5$. Let $\Sigma$ be the closure of
$\rho_{0}$ as in Definition 11. Then, there is a world $\Gamma\in X_{\Sigma}$
in the canonical model $\mathcal{C}_{\Sigma}$ such that
$\neg\rho_{0}\in\Gamma$ (since in the finite ordered set of consistent subsets
of $\Sigma$, every element is below a maximal one). By Lemma Completeness and
Bounded Models, ${\mathcal{C}_{\Sigma},\Gamma\models\neg\rho_{0}}$.
## Conclusions
We have introduced _ abstract-group epistemic logic (AGEL)_, a logic for
reasoning about the common knowledge of groups of agents that are described
abstractly via defining properties. We have established ExpTime-completeness,
a bounded model property, and (necessarily weak) completeness of a natural
axiomatization. The ExpTime upper bound holds in spite of the fact that the
expected encoding into standard group epistemic logic (with a common knowledge
operator for enumerated groups of agents) incurs exponential blowup, and
relies instead on a satisfiability-preserving translation into the
$\mu$-calculus with converse. Key directions for future research concern in
particular extensions of the logic by a distributed knowledge operator; by
dynamic epistemic modalities such as public announcements; by expressive means
for describing groups of agents via their individual knowledge; and by
allowing non-rigid agent names and agent atoms to capture knowledge about
agents.
## Acknowledgements
This work was supported by DFG (German Research Foundation) as part of the
Research Training Group 2475 “Cybercrime and Forensic Computing” (grant number
393541319/GRK2475/1-2019) and by DFG project “RAND: Reconstructing Arguments
from Newsworthy Debates” (grant number 377333057). The authors also wish to
thank the anonymous reviewers for their suggestions and feedback.
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## Appendix A Details and Full Proofs
We give details and full proofs where these have been omitted in the main body
of the paper; for the convenience of the reader, we restate the relevant
claims. When discussing AGEL models, we use the following additional
terminology: When $x\sim_{a}y$ and ${\mathcal{A},a\models\psi}$, we call $y$ a
_$\psi$ -successor of $x$_.
*[AGEL]: abstract-group epistemic logic
*[ AGEL]: abstract-group epistemic logic
*[GEL]: group epistemic logic
*[PDL]: propositional dynamic logic
|
# Evaluating Unsupervised Text Classification:
Zero-shot and Similarity-based Approaches
Tim Schopf 0000-0003-3849-0394 Technical University of Munich, Department of
Computer ScienceGarchingGermany<EMAIL_ADDRESS>, Daniel Braun
0000-0001-8120-3368 University of Twente, Department of High-tech Business and
EntrepreneurshipEnschedeNetherlands<EMAIL_ADDRESS>and Florian Matthes
0000-0002-6667-5452 Technical University of Munich, Department of Computer
ScienceGarchingGermany<EMAIL_ADDRESS>
###### Abstract.
Text classification of unseen classes is a challenging Natural Language
Processing task and is mainly attempted using two different types of
approaches. Similarity-based approaches attempt to classify instances based on
similarities between text document representations and class description
representations. Zero-shot text classification approaches aim to generalize
knowledge gained from a training task by assigning appropriate labels of
unknown classes to text documents. Although existing studies have already
investigated individual approaches to these categories, the experiments in
literature do not provide a consistent comparison. This paper addresses this
gap by conducting a systematic evaluation of different similarity-based and
zero-shot approaches for text classification of unseen classes. Different
state-of-the-art approaches are benchmarked on four text classification
datasets, including a new dataset from the medical domain. Additionally, novel
SimCSE (Gao et al., 2021) and SBERT-based (Reimers and Gurevych, 2019)
baselines are proposed, as other baselines used in existing work yield weak
classification results and are easily outperformed. Finally, the novel
similarity-based Lbl2TransformerVec approach is presented, which outperforms
previous state-of-the-art approaches in unsupervised text classification. Our
experiments show that similarity-based approaches significantly outperform
zero-shot approaches in most cases. Additionally, using SimCSE or SBERT
embeddings instead of simpler text representations increases similarity-based
classification results even further.
Natural Language Processing, Unsupervised Text Classification, Zero-shot Text
Classification
††conference: 6th International Conference on Natural Language Processing and
Information Retrieval; December 16-18, 2022; Bangkok, Thailand††copyright:
none††ccs: Computing methodologies Natural language processing††ccs: Computing
methodologies Artificial intelligence††ccs: Computing methodologies Machine
learning††ccs: Computing methodologies Unsupervised learning††ccs: Information
systems Clustering and classification††ccs: Computing methodologies Neural
networks
NLP
Natural Language Processing
KE
keyword enrichment
PLM
Pretrained Language Model
ZSL
zero-shot learning
0SHOT-TC
zero-shot text classification
ESA
Explicit Semantic Analysis
LSA
Latent Semantic Analysis
LSI
Latent Semantic Indexing
SVD
Singular Value Decomposition
## 1\. Introduction
Unsupervised text classification approaches aim to perform categorization
without using annotated data during training and therefore offer the potential
to reduce annotation costs. Despite this possibility, unsupervised text
classification approaches have attracted significantly less attention in
contrast to supervised text classification approaches. As a result, extensive
work is already being done to structure and evaluate the field of text
classification with a focus on supervised approaches (Mirończuk and
Protasiewicz, 2018; Kadhim, 2019; Kowsari et al., 2019; Minaee et al., 2021)
while little research has been conducted on evaluating unsupervised text
classification approaches. This study bridges this gap by assessing the two
most popular categories of unsupervised text classification approaches.
Generally, unsupervised text classification approaches aim to map text to
labels based on their textual description, without using annotated training
data. To accomplish this, there exist mainly two categories of approaches. The
first category can be summarized under similarity-based approaches. Thereby,
the approaches generate semantic embeddings of both the texts and the label
descriptions, before attempting to match the texts to the labels using
similarity measures such as cosine similarity (Song and Roth, 2014; Sappadla
et al., 2016; Haj-Yahia et al., 2019; Schopf et al., 2021). The second
category uses zero-shot learning (ZSL) to classify texts of unseen classes.
ZSL uses labeled training instances belonging to seen classes to learn a
classifier that can predict testing instances belonging to different, unseen
classes (Wang et al., 2019). Although ZSL techniques employ annotated data for
training, they do not use labels to provide information about the target
classes and can use their knowledge of the previously seen classes to classify
instances of unseen classes. Since pretrained zero-shot text classification
(0SHOT-TC) models do not require training or fine-tuning on labeled data from
the target classes, we classify them as an unsupervised text classification
strategy. The highly successful deep learning performances of recent years
have also stimulated research initiatives for 0SHOT-TC (Pushp and Srivastava,
2017; Rios and Kavuluru, 2018; Zhang et al., 2019; Yin et al., 2019; Liu et
al., 2021). We argue, that one of the main differences between ZSL and
similarity-based approaches is, that ZSL approaches use annotated data for
seen classes to predict texts of unseen classes, whereas pure similarity-based
approaches do not require seen classes at all.
We summarize the contributions of our work as follows::
* •
We evaluate the similarity-based and zero-shot learning categories for
unsupervised text classification of topics. Thereby, we conduct experiments
with representative approaches of each category on four different benchmark
datasets, including a new text classification dataset from the medical domain.
* •
We propose simple but strong baselines for unsupervised text classification
based on SimCSE (Gao et al., 2021) and SBERT (Reimers and Gurevych, 2019)
embedding similarities. Previous work has mostly been evaluated against
different weak baselines such as Word2Vec (Mikolov et al., 2013) similarities
which are easy to outperform and tend to overestimate the performance of new
unsupervised text classification approaches.
* •
Since transformer-based text representations have been widely established as
state-of-the-art for semantic text similarity in recent years, we further
adapt Lbl2Vec (Schopf et al., 2021, 2023), one of the most recent and well-
performing similarity-based methods for unsupervised text classification, to
be used with transformer-based language models111Code available:
https://github.com/sebischair/Lbl2Vec.
## 2\. Related Work
Chang et al. (2008) investigated unsupervised text classification under the
umbrella name ”Dataless Classification” in one of their earliest works. They
used Explicit Semantic Analysis (ESA) (Gabrilovich and Markovitch, 2007) to
embed the text and label descriptions in a common semantic space and picked
the label with the highest matching score for classification. Semantic
embeddings are vector representations of texts that capture their semantic
meaning and can be used as input for a variety of different Natural Language
Processing (NLP) downstream tasks (Braun et al., 2021; Schneider et al., 2022;
Schopf et al., 2022a, b). Dataless classification is based on the idea that
semantic representations of labels are equally relevant as learning semantic
text representations and was subsequently further examined in (Song and Roth,
2014; Chen et al., 2015; Li et al., 2016; Song et al., 2016).
With the progress of text embeddings, the term ”Dataless Classification”
became less prevalent and was rather represented by the broad category of
similarity-based approaches for unsupervised classification. Within this
category, Sappadla et al. (2016) embedded text documents and textual label
descriptions with Word2Vec and used cosine similarity between text and label
embeddings to predict instances of unseen classes. Haj-Yahia et al. (2019)
proposed to enrich label descriptions with expert keywords and subsequently
conduct unsupervised classification based on Latent Semantic Analysis (LSA)
(Deerwester et al., 1990) similarities. Stammbach and Ash (2021) introduced
DocSCAN, which produces semantic representations of text documents and uses
Semantic Clustering by Adopting Nearest-Neighbors for unsupervised text
classification. Schopf et al. (2021) used Doc2Vec (Le and Mikolov, 2014) to
jointly embed word, document, and label vectors for subsequent similarity-
based unsupervised text classification.
Similarly, Nam et al. (2016) jointly embedded document, label, and word
representations with Doc2Vec. However, they learned a ranking function for
multi-label classification and attempted to predict instances of unseen
classes in a zero-shot setting for classification. Zhang et al. (2019)
integrated four types of semantic knowledge (word embeddings, class
descriptions, class hierarchy, and a general knowledge graph) in a two-phase
framework for 0SHOT-TC. Yin et al. (2019) proposed to treat 0SHOT-TC as a
textual entailment problem, while Ye et al. (2020) tackled 0SHOT-TC with a
semi-supervised self-training approach.
## 3\. Text Classification Approaches
### 3.1. Baselines
We compare the findings of current state-of-the-art unsupervised text
classification approaches to some basic baselines to evaluate their
performance.
LSA: Singular Value Decomposition (SVD) is used on term-document matrices to
learn a set of concepts (or topics) related to the documents and terms
(Deerwester et al., 1990). For each dataset, we apply LSA to learn $n=$ number
of classes concepts. Afterward, the text documents are classified according to
the highest cosine similarity of resulting LSA vectors of documents and label
keywords. A similar approach was used by Haj-Yahia et al. (2019) for
unsupervised text classification.
Word2Vec: This produces semantic vector representations of words based on
surrounding context words (Mikolov et al., 2013). A Skip-gram model with a
vector size of 300 and a surrounding window of 5 is trained for each dataset.
The average of word embeddings is then used to represent the text documents
and label keywords. The text documents are predicted according to the highest
cosine similarity of the resulting Word2Vec representations of documents and
label keywords for classification. Similar approaches were used by Yin et al.
(2019) and Ye et al. (2020) as baseline for 0SHOT-TC.
SimCSE: This is a contrastive learning framework that produces sentence
embeddings which acieve state-of-the-art results in semantic similarity tasks
(Gao et al., 2021). Algorithm 1 is first used to separate the text documents
into paragraphs because SimCSE models have a maximum input sequence length.
Then, the average of SimCSE paragraph embeddings as text document
representations and the average of SimCSE label keyword embeddings as class
representations are employed. Finally, the text documents are classified
according to the highest cosine similarity of the resulting SimCSE
representations of document and label keywords.
SBERT: This is a modification of BERT (Devlin et al., 2019) that uses siamese
and triplet network structures to derive semantically meaningful sentence
embeddings (Reimers and Gurevych, 2019). We use the same classification
approach as described in the paragraph above, except that we now use SBERT
embeddings instead of SimCSE embeddings.
Algorithm 1 Split text document into paragraphs
$d=$ text document
$m_{k}=$ max input sequence length of transformer-model $k$
len($x$) = numer of words in text $x$
procedure split-document($d,m_{k}$)
$\textrm{sentences}_{d}\leftarrow$ sentence_tokenize($d$)
$\textrm{paragraphs}_{d}\leftarrow\emptyset$
$p\leftarrow\emptyset$
for $s$ in sentences do
if len($p$) \+ len($s$) ¡ $\frac{m_{k}}{2}$ then
$p\leftarrow p$ \+ $s$
else
$\textrm{paragraphs}_{d}\leftarrow\textrm{paragraphs}_{d}$ \+ $p$
$p\leftarrow\emptyset$ return $\textrm{paragraphs}_{d}$
### 3.2. Similarity-based Text Classification
As previously stated, numerous similarity-based approaches for unsupervised
text classification exist. However, the recently introduced Lbl2Vec approach
(Schopf et al., 2021) is focused on in this study. We chose Lbl2Vec to
represent the similarity-based classification category since preliminary
experiments confirmed improved performance compared with other similarity-
based approaches. Lbl2Vec works by jointly embedding word, document, and label
representations. First, word and documented representations are learned with
Doc2Vec. Then, the average of label keyword representations for each class is
used to find a set of most similar candidate document representations via
cosine similarity. The average of candidate document representations, in turn,
generates the label vector for each class. For classification, eventually, the
documents are assigned to the class where the cosine similarity of the label
vector and the document vector is the highest.
We adapt the Lbl2Vec approach, using transformer-based text representations
instead of Doc2Vec to create jointly embedded word, document, and label
representations. Since transformer-based text representations currently
achieve state-of-the-art results in text-similarity tasks, we investigate the
effect of the different resulting text representations on this similarity-
based text classification strategy. In this paper, we use SimCSE (Gao et al.,
2021) and SBERT (Reimers and Gurevych, 2019) transformer-models to create text
representations. We use the average paragraph embeddings per document as
document representations. The paragraphs of documents are obtained by applying
Algorithm 1. To find candidate documents for label vectors, the transformer-
models create individual embeddings for each label keyword. Then, cosine
similarity is used to find the documents that are most similar to the average
of the label keyword embeddings for each class. After obtaining the candidate
documents this way, the label vectors as an average of the candidate document
representations for each class are computed. For classification, the documents
are assigned to the class where the cosine similarity between the label vector
and the document vector is the highest. In the following, the Lbl2Vec approach
adapted with transformer-based text representations is referred to as
Lbl2TransformerVec.
### 3.3. Zero-shot Text Classification
0SHOT-TC is still relatively less researched, but nevertheless yields some
promising approaches. Using pretrained 0SHOT-TC models can be considered an
unsupervised text classification strategy, since no label information of
target classes are required for training or fine-tuning. Although newer
approaches such as the one of Liu et al. (2021) exist, preliminary experiments
confirmed that the zero-shot entailment approach (Yin et al., 2019) still
produces state-of-the-art 0SHOT-TC results in predicting instances of unseen
classes. As the name already implies, the zero-shot entailment approach deals
with 0SHOT-TC as a textual entailment problem. The underlying idea is similar
to that of similarity-based text classification approaches. Conventional
0SHOT-TC classifiers fail to understand the actual problem since the label
names are usually converted into simple indices. Therefore, these classifiers
can hardly generalize from seen to unseen classes. Considering 0SHOT-TC as an
entailment problem, however, provides the classifier with a textual label
description and therefore enables it to understand the meaning of labels.
Similarly, TARS (Halder et al., 2020) also uses the textual label description
to classify text in a zero-shot setting. However, TARS approaches the task as
a binary classification problem, where a text and a textual label description
is given to the model, which makes a prediction about whether that label is
true or not. The TARS authors state that this approach significantly
outperforms GPT-2 (Radford et al., 2019) in 0SHOT-TC.
Since the zero-shot entailment approach currently produces state-of-the-art
results in predicting instances of unseen classes and TARS also promises
encouraging results, we select both approaches to represent the ZSL category
for unsupervised text classification.
## 4\. Datasets
Our evaluation is based on four text classification datasets from different
domains. As we use the semantic meaning of class descriptions for unsupervised
text classification, we infer label keywords from each class name that serves
the purpose of textual class descriptions. Thereby, the inference step simply
consists of using the class names provided by the official documentation of
the datasets as label keywords. In a few cases, we additionally substituted
the class names with synonymous or semantically similar keywords, if we
considered this to be a more appropriate description of a certain class.
### 4.1. 20Newsgroups
The 20Newsgroups222qwone.com/ jason/20Newsgroups dataset is a common text
classification benchmark dataset. It was introduced by Lang (1995) and
comprised approximately 20,000 newsgroup posts, equally distributed across 20
different newsgroups classes. Appendix A.1 summarizes the classes and inferred
label keywords.
### 4.2. AG’s Corpus
The original AG’s
Corpus333groups.di.unipi.it/$\scriptstyle\sim$gulli/AG_corpus_of_news_articles
dataset is a collection of over 1 million news articles on different topics.
The Zhang et al. (2015) version is used in this study, which was constructed
by choosing the 4 largest classes from the original corpus. Each class
contains 30,000 training samples and 1,900 testing samples. In total, the
dataset consists of 127,600 samples. Appendix A.2 summarizes the classes and
inferred label keywords.
### 4.3. Yahoo! Answers
The Yahoo! Answers dataset was constructed by Zhang et al. (2015) and contains
10 different topic classes. Each class contains 140,000 training samples and
6,000 testing samples. In total, the dataset consists of 1,460,000 samples.
Appendix A.3 summarizes the classes and inferred label keywords.
### 4.4. Medical Abstracts
We obtained the raw Medical Abstracts dataset through
Kaggle444https://www.kaggle.com/datasets/chaitanyakck/medical-text. The
original corpus contains 28.880 medical abstracts describing 5 different
classes of patient conditions, with only about half of the dataset being
annotated. Furthermore, the original annotations consist of numerical labels
only. A medical text classification dataset from this corpus by using only the
labeled medical abstracts was created, adding descriptive labels to the
respective classes, and splitting the data into a training set and a test set.
Table 1 shows a summary of the processed Medical Abstracts dataset.
Class Name | #training | #test | $\sum$
---|---|---|---
Neoplasms | 2530 | 633 | 3163
| Digestive system
---
diseases
1195 | 299 | 1494
| Nervous system
---
diseases
1540 | 385 | 1925
| Cardiovascular
---
diseases
2441 | 610 | 3051
| General pathological
---
conditions
3844 | 961 | 4805
$\sum$ | 11550 | 2888 | 14438
Table 1. Class distributions within the Medical Abstracts dataset.
The inferred label keywords for each class are summarized in Appendix A.4. We
make this corpus available under the Creative Commons CC BY-SA 3.0
license555https://creativecommons.org/licenses/by-sa/3.0 at
https://github.com/sebischair/Medical-Abstracts-TC-Corpus.
## 5\. Experimental Design
For evaluation of different unsupervised text classification approaches, we
use the datasets described in Section 4. Since we don’t use label information
to train the classifiers, we concatenate the training and test sets for each
dataset and use the respective entire concatenated datasets for training and
testing. After checking the Yahoo! Answers dataset for consistency, we observe
that some answers we try to classify are empty or contain simple yes/no
statements. Therefore, answers that are empty or consist of only one word are
removed. We use the label keywords described in Appendix A for all text
classification approaches to create class representations. Additionally, for
the baselines and similarity-based approaches, we use the average of the
respective label keyword embeddings as class representations. In contrast, for
the zero-shot approaches, the respective label keywords of the 20Newsgroups,
AG’s Corpus, and Yahoo! Answers classes are concatenated with ”and” and then
used as hypotheses/label descriptions. For the Medical Abstracts dataset just
the class names are used as hypotheses/label descriptions.
We use the approaches described in Section 3.1 as baselines for unsupervised
text classification. For our SimCSE experiments, we use the sup-simcse-
roberta-large666princeton-nlp/sup-simcse-roberta-large model. To create
embeddings for the SBERT baseline approach, we use two different pretrained
SBERT models. We choose the general purpose models all-mpnet-
base-v2777sentence-transformers/all-mpnet-base-v2 and all-
MiniLM-L6-v2888sentence-transformers/all-MiniLM-L6-v2, trained on more than
one billion training pairs and expected to perform well on sentence similarity
tasks. The all-mpnet-base-v2 model is larger than the all-MiniLM-L6-v2 model
and guarantees slightly better quality sentence embeddings. The smaller all-
MiniLM-L6-v2 model, on the other hand, guarantees a five times faster encoding
time while still providing sentence embeddings of high quality.
For evaluation of similarity-based text classification, we apply the
approaches described in Section 3.2. Similar to the SimCSE and SBERT baseline
approaches, we generate text embeddings for the Lbl2TransformerVec approach
using the sup-simcse-roberta-large, all-mpnet-base-v2, and all-MiniLM-L6-v2
models.
For evaluation of 0SHOT-TC, we use the zero-shot approaches described in
Section 3.3. We conduct experiments with three different pretrained zero-shot
entailment models: a DeBERTa (He et al., 2020) model
999MoritzLaurer/DeBERTa-v3-base-mnli-fever-docnli-ling-2c trained on the
MultiNLI (Williams et al., 2018), Fever-NLI (Thorne et al., 2018), LingNLI
(Parrish et al., 2021), and DocNLI (Yin et al., 2021) datasets, a large BART
(Lewis et al., 2020) model101010facebook/bart-large-mnli trained on the
MultiNLI dataset, and a smaller DistilBERT (Sanh et al., 2019)
model111111typeform/distilbert-base-uncased-mnli trained on the MultiNLI
dataset. For TARS experiments, we use the BERT-based pretrained tars-
base-v8121212https://flair.informatik.hu-berlin.de/resources/models/tars-base/
model. Since tars-base-v8 pretraining is partly done on AG’s Corpus, we don’t
conduct TARS experiments on this dataset.
| | 20Newsgroups | AG’s Corpus | Yahoo! Answers | Medical Corpus
---|---|---|---|---|---
Baselines | LSA | 17.89 | 41.17 | 15.82 | 31.61
Word2Vec | 12.87 | 28.22 | 12.55 | 25.00
SimCSE | 42.84 | 80.10 | 49.90 | 34.94
| SBERT
---
(all-MiniLM-L6-v2)
57.89 | 68.57 | 43.77 | 46.53
| SBERT
---
(all-mpnet-base-v2)
59.75 | 70.84 | 51.25 | 46.34
Similarity-based TC | Lbl2Vec | 65.71 | 74.63 | 44.26 | 43.03
| Lbl2TransformerVec
---
(SimCSE)
58.79 | 83.79 | 53.32 | 39.60
| Lbl2TransformerVec
---
(all-MiniLM-L6-v2)
63.01 | 80.88 | 52.87 | 54.57
| Lbl2TransformerVec
---
(all-mpnet-base-v2)
64.69 | 80.05 | 55.84 | 56.46
0SHOT-TC | TARS | 17.65 | - | 34.60 | 10.92
| Zero-shot Entailment
---
(DistilBERT)
16.27 | 59.48 | 31.81 | 25.74
| Zero-shot Entailment
---
(BART-large)
38.54 | 68.24 | 40.21 | 56.86
| Zero-shot Entailment
---
(DeBERTa)
47.19 | 72.57 | 43.09 | 57.28
Table 2. $\text{F}_{1}\text{-scores}$ (micro) of examined text classification
approaches on different datasets. The best results on the respective dataset
are displayed in bold. Since we use micro-averaging to calculate our
classification metrics, we realize equal $\text{F}_{1}$, Precision, and Recall
scores respectively.
### 5.1. Hypotheses
We had four main hypotheses prior to conducting the experiments.
1. (1)
0SHOT-TC models yield better text classification results than similarity-based
approaches:
The 0SHOT-TC models investigated in this paper use a cross-encoder
architecture which allows them to compare the input text and the textual label
description simultaneously, while performing self-attention over both. In
contrast, the similarity-based approaches encode the input text and label
description separately. For semantic text similarity tasks, cross-encoders
have proven to perform better than calculating cosine similarities for
separately encoded texts. Hence we expect a similar outcome for unsupervised
text classification.
2. (2)
Using larger Pretrained Language Models results in better classification
performances:
Although this may seem obvious, we nevertheless want to examine whether the
outcomes of using larger PLMs justify their drawbacks during training and
inference.
3. (3)
Classification results of PLM-based approaches are highly domain dependent:
We assume that, PLM-based approaches lose some of their classification
performance when dealing with very domain-specific corpora, since this
specific domain may be underrepresented in the training data. Therefore, we
anticipate that for certain domains, approaches like Lbl2Vec that trains
unsupervised models on the classification data from scratch might perform
comparably better.
4. (4)
With increasing length of text documents, the performance of SimCSE and SBERT-
based approaches decreases:
SimCSE and SBERT representations are most effective if the texts are embedded
as a whole and no truncation strategy is used. Since we compute the document
representations as the average of their respective paragraph embeddings, we
assume that the quality of SimCSE and SBERT document embeddings decreases with
increasing text length, resulting in worse classification performance
accordingly.
## 6\. Evaluation
Table 2 shows the performances of unsupervised text classification approaches
for each dataset, measured in $\text{F}_{1}\text{-scores}$. We can observe
that none of the baselines achieves the highest $\text{F}_{1}\text{-score}$ on
any dataset based on these data. This indicates that the use of advanced
unsupervised text classification approaches usually yields better results than
simple baseline approaches. However, we observe that the LSA and Word2Vec
approaches generally yield the worst results and are easy to outperform. In
contrast, the SimCSE and SBERT baselines produce strong
$\text{F}_{1}\text{-scores}$ that even some of the advanced approaches could
not surpass in certain cases. Furthermore, the SimCSE and SBERT baseline
approaches may produce better results than the Lbl2Vec similarity-based
approach on three datasets. We nevertheless can deduce that the use of
advanced similarity-based approaches generally produces better unsupervised
text classification results than the use of simple baseline approaches.
Specifically, the Lbl2TransformerVec approaches using SBERT embeddings appear
to be promising, as they consistently perform well across all datasets and
outperform the baseline results. In contrast, the 0SHOT-TC approaches perform
consistently weak and in the majority of cases did not even manage to
outperform the baseline results. However, the DeBERTa zero-shot entailment
model could classify the domain-specific medical abstracts surprisingly well
and achieved the best $\text{F}_{1}\text{-score}$ of all classifiers on this
dataset. Nevertheless, considering that all 0SHOT-TC models yielded
disappointing results in all remaining experiments and also failed to
outperform the baselines, our first hypothesis can be rejected.
Concerning our second hypothesis, the results are less obvious. On the one
hand, the large DeBERTa zero-shot entailment model always significantly
outperforms the smaller BART-large and DistilBERT zero-shot entailment models.
Additionally, the BERT-based TARS model performs slightly better than the
smaller DistilBERT zero-shot entailment model, except in case of the domain-
specific Medical Abstracts dataset. Conversely, all-mpnet-base-v2 and all-
MiniLM-L6-v2-based approaches tend to produce unsupervised classification
results that are fairly close to each other. Although these results are quite
similar and sometimes even approaches based on the smaller all-MiniLM-L6-v2
model perform better, we nevertheless see that approaches based on the larger
all-mpnet-base-v2 produce slightly better results in most cases. Therefore, we
find sufficient support for our second hypothesis in the case of similarity-
based unsupervised text classification approaches, with even stronger support
in case of 0SHOT-TC.
Figure 1. $\text{F}_{1}\text{-scores}$ of classification models for the
individual classes of all four benchmark datasets.
Figure 1 shows a more detailed view of the classification results by
visualizing the $\text{F}_{1}\text{-scores}$ of classification models for the
individual classes of all datasets. Here we observe that the overall
performance of classifiers is class-dependent. While all classifiers generally
yield good results for some classes (e.g. the sports classes of the
20Newsgroups and AG’s Corpus datasets), all classifiers performed considerably
worse for other classes (e.g. ”talk.religion.misc [20Newsgroups]” or
”Education & Reference [Yahoo! Answers”]). When we compare the performance of
the Lbl2Vec model, which was trained from scratch, to that of PLM-based
approaches, we discover that all approaches produce similar results for many
classes. In some classes, however, Lbl2Vec clearly outperforms
$\text{F}_{1}\text{-scores}$ of all other PLM-based approaches (e.g. in the
”comp.sys.mac.hardware”, ”misc.forsale”, or ”alt.atheism” classes of the
20Newsgroups dataset). Unfortunately, this fact can’t be generalized from
individual classes to the entire domains. For example, Lbl2Vec scores
relatively well in ”comp.sys.mac.hardware (20Newsgroups)” and ”comp.windows.x
(20Newsgroups)” classes, but performs significantly worse than PLM-based
models in ”comp.os.ms-windows.misc (20Newsgroups)”, despite all classes
belonging to the same domain. We conclude that although a model trained from
scratch can yield better results than PLM-based approaches in some cases, as
demonstrated by the Lbl2Vec results on the 20Newsgroups dataset, we do not
find sufficient support for our third hypothesis.
Model | Kendall’s $\bm{\tau}$ | p-value
---|---|---
SimCSE | -0.16 | 0.16
| SBERT
---
(all-MiniLM-L6-v2)
0.07 | 0.52
| SBERT
---
(all-mpnet-base-v2)
0.04 | 0.73
| Lbl2TransformerVec
---
(SimCSE)
-0.08 | 0.46
| Lbl2TransformerVec
---
(all-MiniLM-L6-v2)
-0.03 | 0.82
| Lbl2TransformerVec
---
(all-mpnet-base-v2)
0.03 | 0.80
Table 3. Results of the correlation analysis to measure the relationship
between ${X=}$ average number of document words of each class in all four
benchmark datasets and ${Y=}$ $\text{F}_{1}\text{-scores}$ of each class in
all four benchmark datasets.
To test our fourth hypothesis, we perform a correlation analysis measuring
monotonic relationships between the $\text{F}_{1}\text{-scores}$ of the
transformer-based classification approaches per class and the average number
of document words per class. We choose Kendall’s $\tau$ as correlation
coefficient, because of its robustness against outliers and the small dataset.
Further, we determine a significance level of 0.05. Table 3 shows the results
of this correlation analysis. We can observe that all correlation coefficients
are close to zero. Therefore, we can’t identify a correlation trend. Moreover,
all p-values exceed our defined significance level of 0.05 by far, indicating
our test results are statistically insignificant. As a result, we find no
support for our fourth hypothesis and reject it.
## 7\. Limitations
One significant limitation of this evaluation is that only unsupervised text
classification results for the topic aspect are considered. This means that we
consider classification results based on topics that describe what a text
document is about only. However, text classification can be seen in a broader
context where aspects such as emotion or situation are predicted as well (Yin
et al., 2019). We only focus on unsupervised similarity-based approaches and
0SHOT-TC approaches that can classify the entire datasets without requiring
training or fine-tuning on parts of the datasets. Self-training approaches
which address the problem as a semi-supervised task or ZSL approaches that use
parts of the datasets for training or fine-tuning, may lead to different
results. Although we try to generalize from the datasets and approaches
examined in the experiments, our evaluation is limited to those datasets and
approaches nonetheless.
## 8\. Conclusion
The evaluation of unsupervised text classification approaches in Section 6,
has shown that similarity-based approaches generally outperform 0SHOT-TC
approaches in a variety of different domains. 0SHOT-TC approaches tend to
produce relatively bad results and are therefore hardly eligible for
unsupervised text classification problems. In comparison, similarity-based
approaches appear to predict instances of unknown classes more accurately. The
characteristics of text embeddings enable representations of similar topics or
classes to be located close to each other in embedding space. This implies
that text representation approaches which are able to cluster topics in
embedding space coherently also perform well in unsupervised text
classification. This characteristic is also evident in our work. DensMap
(Narayan et al., 2020) visualizations of document representations in embedding
space used for classification in this work are shown in Appendix A.5 in Figure
2. To improve similarity-based text classification results even further, we
can use additional, different, or more descriptive label keywords than the
ones we used for evaluation (Haj-Yahia et al., 2019; Schopf et al., 2021).
We showed that using larger PLMs yield better results for 0SHOT-TC, but this
is not always the case for similarity-based approaches. Therefore,
unsupervised text classification using smaller PLMs can be conducted in order
to benefit from faster inference without necessarily sacrificing much
performance in terms of $\text{F}_{1}\text{-score}$.
Our evaluation shows that simple approaches such as LSA or Word2Vec are easy
to outperform and therefore are not recommended to be used as baselines for
text classification of unseen classes. However, our proposed SimCSE and SBERT
baseline approaches generate strong unsupervised text classification results,
outperforming even some more advanced classifiers. Therefore, we propose to
use SimCSE and SBERT baselines for evaluating unsupervised text classification
approaches and 0SHOT-TC performance on unseen classes in future work.
Lbl2TransformerVec, our proposed similarity-based text classification approach
yields best $\text{F}_{1}\text{-scores}$ for almost all datasets. This is
largely due to the great text-similarity characteristics of SimCSE and SBERT
representations. Therefore, we believe that future unsupervised text
classification work will benefit considerably from enhanced text embedding
representations.
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## Appendix A Appendix
### A.1. 20Newsgroups Class Summary
Class Name | Label Keywords
---|---
alt.atheism | atheism
comp.graphics | computer, graphics
comp.os.ms-windows.misc | | computer, os,
---
microsoft, windows
comp.sys.ibm.pc.hardware | | computer, system,
---
ibm, pc, hardware
comp.sys.mac.hardware | | computer, system,
---
mac, hardware
comp.windows.x | computer, windows
misc.forsale | forsale
rec.autos | cars
rec.motorcycles | motorcycles
rec.sport.baseball | sport, baseball
rec.sport.hockey | sport, hockey
sci.crypt | encryption
sci.electronics | electronics
sci.med | medical
sci.space | space
soc.religion.christian | religion, christianity
talk.politics.guns | politics, guns
talk.politics.mideast | politics, arab
talk.politics.misc | politics
talk.religion.misc | religion
Table 4. 20Newsgroups class names and inferred label keywords.
### A.2. AG’s Corpus Class Summary
Class Name | Label Keywords
---|---
World | government
Sports | sports
Business | business
Science/Technology | science, technology
Table 5. AG’s Corpus class names and inferred label keywords.
### A.3. Yahoo! Answers Class Summary
Class Name | Label Keywords
---|---
Society & Culture | society, culture
Science & Mathematics | science, mathematics
Health | health
Education & Reference | education, reference
Computers & Internet | computers, internet
Sports | sports
Business & Finance | business, finance
Entertainment & Music | entertainment, music
Family & Relationships | family, relationships
Politics & Government | politics, government
Table 6. Yahoo! Answers class names and inferred label keywords.
### A.4. Medical Abstracts Class Summary
Class Name | Label Keywords
---|---
Neoplasms | neoplasms
| Digestive system
---
diseases
| intestine, system,
---
diseases
| Nervous system
---
diseases
| nervous, system,
---
diseases
| Cardiovascular
---
diseases
| cardiovascular,
---
diseases
| General pathological
---
conditions
| general, pathological,
---
conditions
Table 7. Medical Abstracts class names and inferred label keywords.
### A.5. DensMAP Dataset Visualizations
Figure 2. DensMAP visualizations of the document representations for each
dataset described in Section 4. The document representations were created by
applying the average paragraph embedding strategy described in Section 3.1
using SBERT (all-mpnet-base-v2).
*[ZSL]: zero-shot learning
*[0SHOT-TC]: zero-shot text classification
*[ESA]: Explicit Semantic Analysis
*[NLP]: Natural Language Processing
*[LSA]: Latent Semantic Analysis
*[SVD]: Singular Value Decomposition
*[PLMs]: Pretrained Language Model
*[PLM]: Pretrained Language Model
|
| Shear jamming and fragility in fractal suspensions under confinement
---|---
| Sarika C. K.,a Sayantan Majumdar,a,∗ and A. K. Sood b
| a Soft Condensed Matter Group, Raman Research Institute, Bengaluru 560080,
India.
| b Department of Physics, Indian Institute of Science, Bengaluru 560012,
India.
| (Dated August 28, 2024)
| Under applied stress, the viscosity of many dense particulate suspensions
increases drastically, a response known as discontinuous shear-thickening
(DST). In some cases, the applied stress can even transform the suspension
into a solid-like shear jammed state. Although shear jamming (SJ) has been
probed for dense suspensions with particles having well-defined shapes, such a
phenomenon for fractal objects has not been explored. Here, using rheology and
in situ optical imaging, we study the flow behaviour of ultra-dilute fractal
suspensions of multi-walled carbon nanotubes (MWCNT) under confinement. We
show a direct transition from flowing to SJ state without a precursory DST in
fractal suspensions at an onset volume fraction, $\phi\sim$ 0.5%,
significantly lower than that of conventional dense suspensions ($\phi\sim$
55%). The ultra-low concentration enables us to demonstrate the fragility and
associated contact dynamics of the SJ state, which remain experimentally
unexplored in suspensions. Furthermore, using a generalized Wyart-Cates model,
we propose a generic phase diagram for fractal suspensions that captures the
possibility of SJ without prior DST over a wide range of shear stress and
volume fractions.
††footnotetext: ∗<EMAIL_ADDRESS>
##
## 1 Introduction
Dense particulate suspensions formed by dispersing inorganic or polymeric
particles in Newtonian fluids show an array of striking non-Newtonian flow
properties at high volume fractions. Among these, shear-induced increase in
viscosity, commonly known as shear thickening, has attracted significant
recent interest from experimental as well as theoretical points of view.
During shear thickening, the increase in viscosity under increasing applied
stress can be mild or abrupt, giving rise to continuous shear thickening (CST)
or discontinuous shear thickening (DST) respectively 1, 2, 3, 4, 5. Many of
the dense suspensions showing DST also exhibit shear jamming (SJ) at high
enough applied stress ($\tau$) and volume fraction ($\phi$) where the system
develops finite yield stress like a solid. Recent studies point out that
beyond a critical onset stress for contact formation, system-spanning contact
networks similar to those observed for dry granular materials can explain the
phenomenon of SJ in dense suspensions 5, 6, 7, 8. Importantly, the SJ state is
reversible: once the applied stress is removed, the system goes back to the
initial fluid-like state in a short span of time 9. Such reversible tuning of
viscosity implies many potential applications, particularly in the field of
designing smart shock-absorbing materials 10, 11.
In general, SJ can be observed only over a limited range of volume fractions:
$\phi_{\mu}<\phi<\phi_{0}$, where $\phi_{\mu}$ and $\phi_{0}$ are the critical
jamming packing fractions with and without friction respectively. For
suspensions in which the friction and other stress-induced short-range
interactions between the particles are negligible, $\phi_{\mu}$ remains close
to $\phi_{0}$ and the range of $\phi$ over which shear jamming is observed
diminishes. With greater friction, $\phi_{\mu}$ reduces and moves away from
$\phi_{0}$ enabling SJ in suspensions over a considerable range of
concentrations. Recent numerical studies 12, 13 show that introducing
constraints on the sliding and rolling motion of the particles by increasing
the inter-particle friction and adhesion can lower $\phi_{\mu}$. Non-spherical
particle shapes can also constrain the rolling motion and consequently reduce
$\phi_{\mu}$ 14, 15. Furthermore, studies show that tuning hydrogen bonding
and solvent molecular weight enhances the particle contact interaction
eliciting shear jamming in conventional suspensions 15, 16, 17, 18. In
colloidal suspensions, it is demonstrated that the surface roughness of
particles enhances DST due to the interlocking of surface asperities 19, 20. A
recent study shows that the critical volume fractions for both CST and DST can
be lowered by tuning the surface chemistry and surface roughness of rigid
particles in suspensions 21. The key aspect emerging from these observations
is the governing role of the contact network structure and the frictional
contact strength between particles in the shear jamming of suspensions. Hence,
a fractal suspension of stiff particles is a model system to probe shear
jamming due to the strong frictional interaction between the particles.
Confinement can facilitate the formation of the characteristic system-spanning
network for shear jamming. With significantly high effective surface roughness
and occupation volume, fractal suspensions are expected to have much lower
$\phi_{\mu}$ compared to the non-fractal suspensions mentioned above.
In this paper, we study the rheological behaviour of very dilute suspensions
of multiwalled carbon nanotubes (MWCNT) dispersed in a Newtonian solvent,
N-methyl-2-pyrrolidone (NMP), which form fractal aggregates (will be referred
as flocs) due to cohesive interparticle interactions. Previously, the colossal
increase in viscosity in this system by almost four orders of magnitude was
thought to be due to DST 22. The present study establishes that this
divergence in viscosity has all the signatures of shear jamming transition.
For dense suspensions of frictional, non-fractal compact grains, as $\phi$
increases from a very low value, the system progressively passes through
Newtonian/shear thinning (flowing), CST, DST, SJ and isotropically jammed (IJ)
phases 18, 23. Equivalently, for dry granular systems the sequence of phases
is: unjammed, fragile, SJ and IJ 24. Interestingly, we observe a direct
transition from flowing to SJ state with increasing $\phi$, completely
bypassing any intermediate CST/DST phases. Moreover, the ultra-low
concentration range ($\phi\geq 0.5\%$) for observing the SJ in the system
allows us to experimentally probe, for the first time, the fragility and the
associated structural reorganization in shear jammed suspensions.
## 2 Results and Discussion
Fig. 1: a) Flow curve of MWCNT suspension ($\phi$ = 2.5%), plotted as
viscosity versus applied shear stress in log-log scale. The duration of each
measuring point of shear stress is 100 s. Three different states, CJ-state,
flowing-state and SJ-state are marked as I, II and III respectively, in
different colors. b) Variation of yield stress (circle) and shear jamming
stress (square) with respect to $\phi$, along with power law fit (solid
lines).
We measure the flow curve, viscosity ($\eta$) versus shear stress ($\tau$), of
flocculated fractal suspensions of MWCNT under controlled shear stress using
parallel plate geometry of the rheometer. The detailed fractal structure
analysis of MWCNT flocs is described in the Supplementary Information (S.I.)
and in Fig. S1 where we obtain 2-D fractal dimension of the dried clusters
$\sim$ 1.5 - 1.6 indicating that the flocs have a much larger effective volume
compared to the actual volume of the constituent MWCNTs. We consider a range
of volume fractions varying from 0.26 to 5.35% to measure the flow curves. We
show a typical flow curve for $\phi$ = 2.5% in Fig. 1a. The complete set of
flow curves is shown in Fig. S2. The flow curve exhibits three distinct
rheological signatures over the shear stress range of 0.01 - 10 Pa. The
suspension is initially in a jammed (unyielded) state (Region I) due to
cohesive interaction between the flocs until the yield stress ($\tau_{y}$) is
reached. We term this region as the cohesive jammed state (CJ state). Beyond
$\tau_{y}$, the suspension begins to flow and shear thins upon a further
increase in $\tau$ (Region II). The flowing-state continues until the
viscosity shoots up instantaneously at higher shear stress, $\tau^{*}$ (Region
III). The flow curves plotted as shear stress versus shear rate in Fig. S3a
and b show that in both the regions I and III, the shear rate drops down to
zero, reaching the resolution limit of the rheometer and fluctuates around
zero with positive and negative values. This indicates that regions I and III
are solid-like jammed states as the average shear rate remains negligible in
these regions. Typically, for systems showing DST and SJ, the stress
dependence of viscosity in the shear-thickening regime is given by a power-law
behaviour: $\eta\sim\tau^{\delta}$, with $\delta=1$ marking the onset of DST.
At higher stress values, $\delta$ becomes larger than 1 indicating unstable
flows in the system due to shear-induced local jammed regions. In this regime,
when the flow curve is represented by shear rate ($\dot{\gamma}$) versus
stress ($\tau$), the so-called ‘S-shaped’ curves 7 are obtained showing a
negative slope ($\frac{d\,\dot{\gamma}}{d\,\tau}<0$) just beyond the onset
stress for shear thickening. At higher stress values the system becomes solid-
like with a vanishing $\dot{\gamma}$. As $\dot{\gamma}\rightarrow 0$ at a
finite $\tau$, the viscosity diverges at this point. Hence, region III of the
flow curve manifests clear signatures of SJ transition for $\tau\geq\tau^{*}$.
For solid-like CJ and SJ states, $\eta\rightarrow\infty$. However, the
observed finite values of viscosity for the CJ and SJ states (Fig. 1a) is an
artifact originating from the instrumental limitation in measuring very small
shear rates. Thus, rotational rheometry is not sufficient to characterize the
CJ and SJ states and one requires oscillatory rheology for such
quantification. Oscillatory rheology measurements reveal that both the CJ and
SJ states are viscoelastic solid-like in nature (storage modulus $G^{\prime}$
much higher than the loss modulus $G^{\prime\prime}$), with the magnitude of
$G^{\prime}$ for the SJ state much higher compared to that for the CJ state
22. Nonetheless, stress-controlled rotational rheometry reliably quantifies
the flowing state, as well as, the yielding and SJ transitions that remain the
main focus of the present study.
Fig. 2: Stress reversal in CJ state (a, b) and SJ state (c, d). Applied shear
stress versus time (upper panels) and corresponding viscosity profiles (lower
panels) are plotted. e-f) Optical images of SJ state at the time of reversal,
300 s (e) and at 300.6 s (f) showing the contact breaking of MWCNT flocs upon
stress reversal. Arrow indicates the direction of applied shear and red
circles aid to see contact breaking. Scale bar: 200 $\mu$m.
We observe shear jamming for low volume fractions $\phi\geq 0.5\%$.
Remarkably, we obtain a direct transition to SJ state without any DST regime
for the entire range of applied stress and particle volume fractions. To our
knowledge, such behaviour has never been observed for suspensions. We will be
looking into the details of this atypical feature in the following
discussions. In Fig. 1b, we show the variation of threshold stress for the
onset of yielding ($\tau_{y}$) and shear jamming ($\tau^{*}$) as functions of
volume fraction. The variation of yield stress with volume fraction exhibits a
power law scaling, $\tau_{y}\sim\phi^{2.8}$ similar to other flocculated
suspensions 25. Similarly, the onset stress for SJ, $\tau^{*}\sim\phi^{2}$.
For all the concentrations, we find that the SJ states cannot withstand
significant applied stresses. The SJ state typically fails beyond $\sigma\sim$
15 Pa as also reported earlier 22. Such failure at relatively low stress
values indicates that the rigidity of the MWCNT flocs is much lower compared
to the rigid grains (e.g. corn starch, silica particles).
Fig. 3: Optical images taken at the time of stress reversal and after 0.3 s
for CJ state (a, b) and SJ state (c, d). Scale bar: 1 mm. Arrows indicate the
direction of applied shear stress and the red circles aid to see contact
breaking upon the stress reversal. e and f) Temporal evolution of the
structure of CJ state (e) and SJ state (f), plotted as the variation of light
intensity vector at a fixed line (dashed line) on the images.
One of the main characteristic features that differentiates a shear jammed
state from an isotropically jammed state is the fragility under applied
perturbations 26, 27. To investigate the fragility and associated structural
changes in MWCNT suspensions, we carried out stress reversal experiments
coupled with in situ optical imaging. The latter is possible in the present
case because shear jamming in MWCNT suspensions starts at much lower volume
fractions due to the fractal nature of flocs as compared to the high volume
fraction ($\sim$ 55%) requirement for spherical colloids. This enables us to
track the contact dynamics during the SJ transition using in situ optical
imaging. We first apply a constant shear stress of 0.02 Pa ($<\tau_{y}$) for
300 s to the suspension at rest ($\phi$ = 0.77%). The suspension stays
undisturbed in the CJ state. After 300 s, the direction of the stress is
reversed as shown in Fig. 2a. We observe that the viscosity remains unchanged
during the reversal (Fig. 2b) as the suspension remains jammed, confirming the
non-fragile, isotropic nature of the CJ state. To investigate the fragility of
the SJ state, a constant shear stress of 3 Pa ($>\tau^{*}$) is applied to a
freshly loaded suspension ($\phi$ = 0.77%) as shown in Fig. 2c. Initially, the
suspension flows for $\sim$ 90 s and enters an SJ state (marked as SJ1) which
can support the applied shear stress. The structural transformations
associated with this rheological response, captured using in situ optical
imaging, are shown in Fig. S4. It shows diffuse MWCNT network structure in the
quiescent state, vorticity aligned rolling log-like flocs and dispersed broken
flocs at the flowing-state 28, 29 and interconnected dense floc network
structure at the SJ state 22. After 300 s, the stress direction is reversed
which results in an immediate melting of the SJ1 state characterized by a drop
in viscosity by several orders of magnitude (Fig. 2d). This flowing-state
continues for $\sim$ 20 s and then it enters a re-entrant SJ state (marked as
SJ2) corresponding to the new direction of shear. Optical images captured
before the reversal of stress direction ($t$ = 300 s, SJ1 state) and just
after the reversal ($t$ = 300.6 s) are shown in Fig. 2e and 2f, respectively.
The images clearly show contact breaking between the fractal flocs within a
fraction of a second as the SJ state is melted upon the stress reversal. In
contrast, no change in the contact between the flocs is observed in the case
of CJ state before and after the stress reversal. Such anisotropy in the SJ
state indicates the presence of frictional contacts in the system that can
statically counter the applied stress in one direction but becomes unstable on
stress reversal 30, 26. Although the correlation between fragility and contact
breaking has been studied for SJ in granular matter 6, direct experimental
observation of such correlation remained unexplored as yet in dense
suspensions. In the flowing-state just before shear jamming, the effective
volume fraction of the flocs, $\phi_{eff}\sim 36.3\%$ remains much higher than
the actual volume fraction ($\phi=0.77\%$) (Fig. S4c). We have estimated
$\phi_{eff}$ using optical imaging as described in the S.I.. Such high
effective volume ensures that the system can show SJ even at a very low volume
fraction of MWCNT. Estimation of the 3-D fractal dimension ($d_{f}$) of the
flocs in the flowing-state, where the individual flocs can be identified,
yields a value $d_{f}\sim$ 2.1 (S.I.).
Fig. 4: Temporal evolution of probability distribution functions of pixel
intensity at the stress reversal (a) and across the re-entrant shear jamming
(inset). Color bar represents the time. Inset axis labels are the same as that
of the main graph. b) Computable Information Density (CID) variation with time
at the stress reversal in CJ state and SJ state. Vertical dash lines indicate
the time of reversal (300 s) and that of the re-entrance of SJ state (320 s).
Fig. 5: a) The phase diagram for fractal suspensions in $\tau$ \- $\phi$
parameter space. Stacked columns represent the experimental data and solid
lines denote the phase boundaries extracted from the model for $\alpha=10$.
Different phases marked are cohesively jammed (unyielded), shear
thinning/flowing and shear jammed states. The narrow red region indicates the
DST phase. b) The flow curve of MWCNT suspension ($\phi$ = 2.5%) fitted with
the combined constitutive model of Herschel-Bulkley and Wyart-Cates. Symbols
show the experimental data and the solid line denotes the constitutive model
fit. c-e) Narrowing of the DST phase (shaded area) in the phase diagram with
respect to growth rate, $\alpha$ = 1, 3 and 10 respectively. Boundaries of DST
(red) and shear jamming (green) are shown.
The isotropic nature of the CJ state and the fragile, anisotropic nature of
the SJ state are further confirmed by a large field of view, transmission mode
imaging (Fig. 3). There is no structural change in the CJ state upon stress
reversal (Fig. 3a and 3b) whereas the floc structure in the SJ state
immediately starts to break upon stress reversal (Fig. 3c and 3d). Such
distinct behaviour of CJ and SJ states can be captured by space-time plots
(Fig. 3e and 3f) depicting the time evolution of intensity along the dotted
lines marked in Figs. 3a - 3d. We find that the intensity distribution does
not show any time evolution for the CJ state, but for the SJ state a clear
change is observed during the transition from the SJ1 to the SJ2 state upon
stress reversal (Fig. 3f).
High field of view imaging and sampling the dynamics of a large number of
flocs enable us to probe the statistics of the stress-induced structural
evolution of the SJ state. In Fig. 4a, we show the time evolution of the
probability distribution of pixel intensity $P(I)$ recorded in the
transmission mode during the stress reversal. Before reversal, $P(I)$ shows a
bimodal nature with two distinct peaks at low (the dark region where MWCNT
flocs are present) and high (void space) intensities. Immediately after the
reversal ($t>$ 300.1), the two peaks start to merge giving rise to a single
peak at an intermediate intensity closer to the low intensity peak. In the
transmission mode imaging, the intensity at each pixel of the image is a
direct measure of transmitted light intensity through the sample across the
gap between the shearing plates: $I=I_{0}\leavevmode\nobreak\ e^{-\lambda d}$,
where $I_{0}$ is the incident light intensity, $\lambda$ is the absorption
coefficient and $d$ is the thickness of MWCNT flocs across the shear gap.
Under the assumption that the value of $\lambda$ remains constant for the
flocs, the increase in the intensity at any pixel indicates a reduction in the
thickness of MWCNT flocs across the gap between shear plates at the
corresponding position. Hence, the merging of two intensity bands reflects the
dissociation of fractal flocs and redistribution of disjointed flocs in the
lateral direction. We observe an exact opposite trend during the recurrence of
shear jamming as shown in the inset of Fig. 4a: a single peak of $P(I)$ in the
flowing-state splits into two distinct peaks in the re-entrant SJ state,
signifying the building up of system spanning contacts.
To further quantify the complex structural transition associated with SJ, we
measure the computable information density (CID) that can capture such
transitions in a wide range of equilibrium and non-equilibrium systems 31. We
show the variation of CID obtained from the ratio of the compressed
($L^{\prime}$) to the uncompressed ($L$) image size as a function of time
during the stress reversal in Fig. 4b. We observe that in the SJ state CID
remains almost constant. Interestingly, CID shows a sharp drop after the
stress reversal as the system starts to flow. As the re-entrant SJ state sets
in, CID increases again and reaches almost the same value as in the initial SJ
state. The strong correlation between the floc structure and SJ is remarkable
and novel. We conjecture that the lower value of CID in the flowing-state is
due to the uniform distribution of broken flocs as compared to the SJ state
which shows larger and distinct fractal structures. No such time-variation is
observed under stress reversal for the CJ state as seen in Fig. 4b.
Now we model the rheological behaviour of our system which is strikingly
different from the conventional dense suspensions. We require a constitutive
relation that incorporates a cohesive jammed (CJ) state for $\tau<\tau_{y}$,
shear thinning for $\tau_{y}<\tau<\tau^{*}$ and shear jamming for
$\tau>\tau^{*}$. For our system, yielding from the CJ state to shear thinning
state is well described by Herschel-Bulkley (HB) equation,
$\tau_{{}_{\text{HB}}}=\tau_{y}+k\dot{\gamma}^{n}$ ($k$: consistency index;
$n$: power-law exponent). Since the stress reversal experiments show the
existence of frictional contacts in our system, we consider the Wyart-Cates
(WC) model 7 which is widely used for capturing DST and SJ in suspensions of
frictional particles. In WC model the stress dependence of jamming
concentration is given by
$\phi_{J}(\tau)=\phi_{\mu}f(\tau)+\phi_{0}\left(1-f(\tau)\right)$. Here, the
function, $f(\tau)\in[0,1]$, indicates the fraction of frictional contacts in
the system. At low shear stress, when almost all the contacts are lubricated,
$f(\tau)\sim 0$. At higher applied stress, the lubrication layers between the
particles are breached to form frictional contacts. When a sufficient number
of the contacts are transformed into frictional ones at very high stress,
$f(\tau)$ approaches 1. For MWCNT suspensions, the values of $\phi_{0}$ and
$\phi_{\mu}$ are 5.4% and 0.51%, respectively. Such stress-induced increase of
$f(\tau)$ is empirically modeled by an exponential function 30, 32. The
strength of frictional contact at a microscopic scale also affects $f(\tau)$.
The sharp rise to the SJ state in our system indicates a rapid increase in the
sufficient number of contacts between the flocs. We describe this behaviour
with a stretched exponential function 33, 34,
$f(\tau)=1-e^{-(\frac{\tau-\tau_{y}}{\tau^{*}-\tau_{y}})^{\alpha}}$ for
$\tau>\tau_{y}$ and $f(\tau)=0$ for $\tau\leq\tau_{y}$, where the exponent
$\alpha$ determines the growth rate of $f$ with respect to $\tau$ (see Fig.
S5). It ensures that all the contacts are lubricated till the yielding and the
frictional contacts start to form at higher stress values. For non-fractal
frictional particles, $\alpha\sim 1$. With $\alpha>>1$, the rapid increase of
$f(\tau)$ makes $\phi_{J}$ to drop towards $\phi_{\mu}$ giving rise to an
abrupt divergence of viscosity. We find that a constitutive model combining
Herschel-Bulkley and Wyart-Cates relations for viscosity 35,
$\eta(\tau,\phi)=\frac{k^{1/n}\leavevmode\nobreak\
\tau_{y}}{(\tau-\tau_{y})^{1/n}}+\frac{k^{1/n}}{(\tau-\tau_{y})^{1/n-1}}+\eta_{0}\left(1-\frac{\phi}{\phi_{J}}\right)^{-\beta}$
(1)
where $\eta_{0}$ is the solvent viscosity, captures the flow curve very well
in the limit of $\alpha>>1$, as shown in Fig. 5b. Unlike earlier models, the
onset stress ($\tau^{*}$) to overcome the lubrication barrier between
particles is not a constant for our system but increases with $\phi$ as shown
in Fig. 1b. This indicates that the stress-induced frictional contacts are
possible only when the applied stress is high enough to overcome the yield
stress arising from the cohesive interparticle interactions. The values of
parameters $\eta_{0}$, $n$ and $k$ used for the modeling of experimental data
is given in the Supplementary Information.
Based on the combined constitutive model, we map out a phase diagram for
fractal suspensions in $\tau-\phi$ parameter space as shown in Fig. 5a. It
demarcates different phases, viz. unyielded (CJ state), shear thinning
(flowing-state), DST and shear jamming. DST occurs when there is a
discontinuous jump in the shear stress at a given shear rate, i.e.,
$\frac{d\,log(\tau)}{d\,log(\dot{\gamma})}=\infty$ and jamming occurs when
$\dot{\gamma}=0$ at a higher stress value. Yielding and shear jamming
boundaries are estimated from Eqn. 1. Interestingly, the growth rate of
$f(\tau)$ with shear stress controls the DST and SJ boundaries: the DST regime
shrinks as the growth rate of $f(\tau)$ increases. Narrowing of the DST phase
in the phase diagram is demonstrated in Fig. 5c-e, for growth rate, $\alpha=$
1, 3 and 10. With high values of $\alpha$ ($>>$1), the DST regime almost
disappears as the DST and SJ boundaries overlap. Consequently, the system can
directly transit from flowing to SJ state, bypassing the narrow DST zone. DST
is a prominent phase for suspensions of compact particles with well-defined
geometries. In contrast, we experimentally observe a direct transition from
flowing to SJ phase in MWCNT fractal suspensions at all volume fractions,
$\phi_{\mu}\leq\phi\leq\phi_{0}$. The absence of DST stems from the rapid
growth of a sufficient number of interlocking flocs forming a network
structure that can easily span the system due to the confinement. Another
notable point from the phase diagram is the rise of the shear jamming boundary
with $\phi$ showing an opposite trend compared to the conventional, non-
fractal systems showing DST and SJ. This is a direct consequence of $\phi$
dependent $\tau_{y}$ due to cohesive interactions in the system. The power-law
exponent in the WC model (Eqn. 1) is taken as $\beta$ = 2, as in the case of
conventional systems. We note that the SJ phase boundary is not very sensitive
to the value of $\beta$. Although the experimental data show a direct
transition from a flowing (yellow region) to SJ state (green region) as
indicated in Fig. 5a, we use a moderate value ($\alpha$ = 10) to indicate a
narrow DST region. This establishes the generality of the model capturing all
the possible phases. However, if we use very high $\alpha$ values
($\alpha\geq$ 50), the DST regime disappears completely.
Fig. 6: a) Scaling function $F(x)=(\eta-\psi(\phi))(\phi_{0}-\phi)^{2}$ vs.
scaling variable $x=\frac{f(\tau)C(\phi)}{(\phi_{0}-\phi)}$ for
0.5%$\leq\phi\leq$ 5.35%. Colors indicate different volume fractions as
depicted in panel (b). b) Variation of $C(\phi)$ with volume fraction.
A very recent study shows that by recasting the WC model, the viscosity of
shear thickening suspensions can be collapsed onto a universal curve 36. The
underlying scaling function $F(x)$ which is related to the anisotropy of the
system, is obtained from the relation,
$(\eta-\psi(\phi))(\phi_{0}-\phi)^{2}=F(x)$ with the scaling variable,
$x=\frac{f(\tau)C(\phi)}{(\phi_{0}-\phi)}$. Here, $\eta-\psi(\phi)$ is the
reduced viscosity where a volume fraction dependent part of the viscosity,
$\psi(\phi)$, ensures the data collapse below the shear-thickening/jamming
onset. Variation of $F(x)$ vs $x$ for a range of $\phi$ values is shown in
Fig. 6a, where we obtain a good scaling of the data. The scaling function
$F(x)$ diverges at $x=x_{c}=(\phi_{0}-\phi_{\mu})^{-1}$. In our case,
$x_{c}\approx 20.4$. We find from Fig. 6b that $C(\phi)$ decreases linearly
with increasing $\phi$ values and eventually vanishes at the isotropic jamming
point, $\phi_{0}$. This linear decrease in $C(\phi)$ is similar to that
observed for the shear-thickening suspensions of silica particles 36. As
mentioned in ref. 36, the function $C(\phi)$ is related to the volume fraction
dependent anisotropy of the SJ state.
## 3 Conclusion
We have presented stress-induced divergence of viscosity in fractal MWCNT
suspensions showing all the signatures of shear jamming at very low
concentrations ($\phi_{\mu}\sim$ 0.5%). By combining stress reversal
experiments along with in situ optical imaging, we bring out the fragility of
the SJ state and associated contact dynamics and structural reorganization.
Such correlations have remained unexplored so far in the experimental studies
of shear jamming in dense suspensions. Interestingly, from in situ optical
imaging, we observe that the effective area coverage of fractal clusters is
significantly reduced in the SJ state as compared to the initial CJ state (see
Fig. 3). This indicates that at stress, $\tau>\tau_{y}$, the flow induces
densification of the flocs making them stiffer which facilitates the contact
network formation required to achieve the SJ state. For dense suspensions of
frictional non-fractal particles, when the phase diagram is plotted in the
parameter plane of stress ($\tau$) and particle volume fraction ($\phi$) 18,
the DST regime narrows down as $\phi\rightarrow\phi_{0}$. However, in our
case, we observe a direct transition from flowing to SJ with no CST/DST regime
for all volume fractions ($\phi$: 0.5 % - 5.4 %) up to the isotropic jamming
point ($\phi_{0}$ = 5.4 %), as shown in Fig. 5a. This signifies that the SJ
phenomenon reported here is quite different from that of compact granular
systems. The very high effective friction related to the stress-induced
interlocking of fractal clusters is responsible for such striking stress
response.
Importantly, we present a phase diagram for fractal suspensions based on the
generalized Wyart-Cates model showing a cohesive jammed state, shear thinned
state and shear jammed state. The generality of the phase diagram sets a
framework for the rheology of fractal suspensions. The sharp increase in the
number of frictional contacts over a narrow range of stress is the key reason
to observe a shear jammed state from the flowing state without going through
the DST phase as commonly observed in dense suspensions of spherical or
anisotropic particles. Controlling fractal dimension in these systems by
tuning the inter-particle interactions and observing the resulting flow
behaviour can shed light on the sharp rise in stress-induced contact formation
or contact proliferation. Exploring the stability properties of SJ state in
fractal suspensions using a similar approach used for dry granular materials
37 can be an interesting future direction to explore. We hope that our study
will motivate further experimental and theoretical studies of SJ in
suspensions of fractal particles.
## Acknowledgements
AKS thanks the Department of Science and Technology, India for the support
under Year of Science Professorship and the Nanomission Council. SM thanks
Science and Engineering Research Board, India for the support through
Ramanujan Fellowship.
## Materials and Methods
For the preparation of suspensions, MWCNTs having a diameter of 30-50 nm and
length of a few microns purchased from Sun NanoTech are mixed with an organic
solvent, N-methyl-2-pyrrolidone (NMP), from Sigma Aldrich. Suspensions of
different volume fractions, $\phi=0.26-5.35\%$, are prepared by a continuous
process of mechanical stirring (10 min) and ultra-sonication (10 min) followed
by mechanical stirring again (10 min). This preparation protocol is
implemented to ensure a homogeneous distribution of flocs in the medium. The
volume fraction of MWCNT suspension is calculated from the weight fraction
using density of NMP (1.03 g/cm3) and the true density of MWCNT (1.7 - 2.1
g/cm3).
Rheological measurements are carried out in a stress-controlled rheometer (MCR
102, Anton Paar) at a fixed temperature of 25 oC. Flow curves are obtained
using a sandblasted cone-plate measuring system (diameter: 50 mm, cone-angle:
2o) and a steel bottom plate. Opto-rheological measurements are done on
suspensions ($\phi$ = 0.77$\%$) using a glass parallel plate measuring system
(diameter: 43 mm) along with a glass bottom plate at a fixed shear gap of 60
$\mu$m. Images are captured using a color CCD camera (Luminera, 640 X 480
pixels) fitted with a microscope objective lens. A white light source is used
for illuminating the imaging area. Images are collected at a rate of 50 fps in
transmission mode (Leica lens 5x) and in reflection mode (Mitutoyo lens 5x
with condenser). Optical images of drop cast samples are taken using Olympus
bright-field microscope.
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Supplementary Information
## Fractal structure of MWCNT flocs
For the fractal structure analysis, MWCNT suspension ($\phi$ = 0.03$\%$) is
drop cast onto a glass substrate and dried. Optical microscopic images of the
drop cast samples show the fractal nature of MWCNT flocs. The 2D fractal
dimension ($d_{f}$) is estimated by two methods: Area-Perimeter analysis
($log(A)\propto d_{f}\leavevmode\nobreak\ log(P)$) and box-counting method
($d_{f}=-\lim_{r\to 0}\frac{log(n)}{log(r)}$ where $n$ is the number of filled
boxes and $r$ is the box size). The Area-Perimeter analysis using the image of
discrete flocs (Fig. S1a) yields the value $d_{f}=1.64$ (see Fig. S1b). For
the box-counting method, images of interconnected flocs are collected in which
the fractal arrangement of nanotubes is clear. A typical image and the fractal
dimension estimation ($d_{f}=1.53$) are shown in Fig. S1c and d respectively.
As we increase the concentration for drop casting, such analysis becomes
difficult due to the dense packing of the dried clusters. So, we have used a
very dilute sample ($\phi$ = 0.03$\%$) for such analysis.
Fig. S1: a) Optical image of discrete MWCNT fractal flocs (scale bar: 50
$\mu$m). b) Log(A) versus log(P) of the individual flocs and the linear fit.
c) Optical image of the interconnected fractal flocs (Scale bar: 20 $\mu$m).
d) Log(n) versus log(r) of the interconnected flocs and the linear fit.
The concentration of the drop cast sample is low compared to that of the
sample used for rheology measurements. At those higher concentrations, it
becomes difficult to image the fractal arrangement of nanotubes due to
clustering. To address this issue, we have directly estimated the 3D fractal
dimension of the suspension used for the rheology measurement from the
effective volume fraction: $\phi_{eff}\approx\phi(R/a)^{3-d_{f}}$, where $R$
is the radius of gyration and $a$ is the size of MWCNT 1. Using the image of
dispersed broken flocs in the flowing state (Fig. S4c), the estimated value of
3D fractal dimension is, $d_{f}\approx 2.1$ ($\phi_{eff}=36.3\%$,
$\phi=0.77\%$, $R=100\mu$m and $a=1\mu$m). Here, we have considered only the
flowing state of the sample just beyond yielding, where the isolated clusters
are observed that enables us to estimate the value of $R$. In the CJ and SJ
states, the determination of $R$ is difficult using optical imaging due to the
connectivity of the clusters. For the estimation of effective volume fraction,
the area coverage of the dispersed broken flocs (Fig. S4c) is measured using
ImageJ software. Assuming a uniform distribution of flocs across the shear
gap, $\phi_{eff}$ is estimated.
## Flow Curves of MWCNT suspensions
Flow curves of MWCNT suspensions of different volume fractions ranging from
0.26$\%$ to 5.35$\%$ are plotted as viscosity versus applied shear stress in
Fig. S2. Shear jamming transition is observed for $\phi\geq 0.5\%$. From the
variation of viscosity with shear stress, $\eta\sim\tau^{\delta}$, the onset
of DST is characterized with, $\delta=1$. Here, the divergence of viscosity at
finite shear stress and the corresponding drop in shear rate,
$\dot{\gamma}\sim 0$, imply direct manifestation of shear jamming without
prior DST.
Fig. S2: a - l) Flow curves of MWCNT suspensions of different volume
fractions, 0.26, 0.51, 1.02, 1.52, 2.01, 2.51, 2.99, 3.47, 3.95, 4.42, 4.89,
5.35$\%$ respectively, plotted as viscosity versus shear stress in log-log
scale.
Since MWCNT suspensions belong to the category of frictional suspensions which
exhibit DST or SJ or both, we use S-shaped flow curve representation 2 of the
rheological data in Fig. S3a. In the S-shaped flow curve, SJ is indicated by
arch-shapes with a vanishing $\dot{\gamma}$ at higher stress. A magnified
portion of the flow curve ($\phi$ = 2.5%) is shown in Fig. S3b. It clearly
shows that the shear rate drops down to the resolution limit of the rheometer
and fluctuates around zero in regions I and III. Since the average shear rate
is negligible in these regions, they are solid-like jammed states; namely,
cohesive jammed state and shear jammed state.
Fig. S3: a) S-shaped flow curve of MWCNT suspension ($\phi$ = 2.5%), plotted
as shear stress versus shear rate in log-log scale. The dashed line denotes
the shear rate at the resolution limit of the rheometer. b) Magnified portion
of the flow curve showing the fluctuation of shear rate around zero (vertical
dashed line) in both CJ and SJ states.
## Stress reversal images
In situ optical images of MWCNT suspension ($\phi=0.77\%$) collected in
transmission mode during the stress reversal with a wide-angle lens (Leica,
5x) are shown in Fig. S4. It shows the structural transformations associated
with the initial flow of suspensions when shear stress of 3 Pa is applied. The
structural deformation pathways leading to the shear jamming can be clearly
seen in the images.
Fig. S4: Optical images taken after the application of shear stress (3 Pa). It
shows diffuse MWCNT network structure at the quiescent state (a), rolling log-
like flocs (b) and dispersed broken flocs (c) at the flowing-state and
interconnected dense floc network structure at the SJ state (d). Scale bar: 1
mm.
## Parameters for the modeling
For modeling the rheology data of MWCNT suspensions using the combined
constitutive model of Herschel-Bulkley and Wyart-Cates, we used parameter
values, $n=0.2$, $k=0.1$ and $\eta_{0}=1.65$ mPa s. The critical volume
fraction of jamming without friction, $\phi_{0}$ = 5.4%, is the volume
fraction above which the system remains in the CJ state under the application
of shear stress. The critical volume fraction of jamming with friction,
$\phi_{\mu}$ = 0.51%, is the volume fraction above which the system exhibits
shear jamming.
## $f(\tau)$ growth at different rates
Growth of fraction of frictional contacts with shear stress,
$f(\tau)=1-e^{-(\frac{\tau-\tau_{y}}{\tau^{*}-\tau_{y}})^{\alpha}}$ for
$\tau>\tau_{y}$ and $f(\tau)=0$ for $\tau\leq\tau_{y}$, for different values
of growth rates, $\alpha$ = 1, 3 and 10, is shown in Fig. S5.
Fig. S5: Growth of $f(\tau)$ with shear stress for different growth rates,
$\alpha$ = 1, 3 and 10.
## Notes and references
* 1 D. B. Genovese, Advances in Colloid and Interface Science, 2012, 171–172, 1–16.
* 2 M. Wyart and M. Cates, Physical review letters, 2014, 112, 098302.
|
# On Controller Reduction in Linear Quadratic Gaussian Control with
Performance Bounds††thanks: The work of Zhaolin Ren and Na Li is supported by
NSF CNS 2003111, NSF AI institute 2112085, and ONR YIP N00014-19-1-2217. The
work of Yang Zheng is supported by NSF ECCS-2154650. The work of Maryam Fazel
was supported by NSF TRIPODS II 2023166, CCF 1839291, CCF 2007036, and CCF
2212261. Emails<EMAIL_ADDRESS>(Zhaolin Ren<EMAIL_ADDRESS>(Yang Zheng<EMAIL_ADDRESS>(Maryam Fazel); and<EMAIL_ADDRESS>(Na Li).
Zhaolin Ren School of Engineering and Applied Science, Harvard University,
USA. Yang Zheng ECE Department, University of California San Diego, La
Jolla, USA Maryam Fazel ECE Department, University of Washington, Seattle,
USA. Na Li School of Engineering and Applied Science, Harvard University,
USA.
###### Abstract
The problem of controller reduction has a rich history in control theory. Yet,
many questions remain open. In particular, there exist very few results on the
order reduction of general non-observer based controllers and the subsequent
quantification of the closed-loop performance. Recent developments in model-
free policy optimization for Linear Quadratic Gaussian (LQG) control have
highlighted the importance of this question. In this paper, we first propose a
new set of sufficient conditions ensuring that a perturbed controller remains
internally stabilizing. Based on this result, we illustrate how to perform
order reduction of general non-observer based controllers using balanced
truncation and modal truncation. We also provide explicit bounds on the LQG
performance of the reduced-order controller. Furthermore, for single-input-
single-output (SISO) systems, we introduce a new controller reduction
technique by truncating unstable modes. We illustrate our theoretical results
with numerical simulations. Our results will serve as valuable tools to design
direct policy search algorithms for control problems with partial
observations.
## 1 Introduction
In many control applications, low-order controllers are often preferred over
high-order controllers, because they are simpler to maintain, more
interpretable, and computationally less demanding [1]. Thus, given a high-
order controller, one often would like to approximate it using a lower-order
controller that still stabilizes the plant whilst performing similarly on
relevant closed-loop performance metrics, such as the Linear Quadratic
Gaussian (LQG) cost. This problem is known as _controller reduction_.
Traditional approaches to controller reduction in LQG control have typically
centered on reducing the order of observer-based controllers and providing
error bounds between the performance of the truncated controller and that of
the original controller [1, 2].
However, the problem of order-reduction for general non-observer based
controllers has been less studied, especially in the context of LQG control.
Recent progress in model-free policy optimization for linear control has
highlighted the importance of order-reduction for general controllers [3]. In
particular, a natural problem in model-free policy optimization is to learn an
optimal policy iteratively using policy gradient methods [4]. It has recently
been shown that the optimization landscape of LQG control may contain saddle
points in state-space dynamic controllers [5]. While vanilla policy gradient
ensures the convergence to stationary points under mild assumptions, these
stationary points may be saddle points that are sub-optimal. As shown very
recently in [3], when a saddle point corresponds to a non-minimal controller,
it is possible to escape the saddle point by finding a lower-order controller
and adding a suitable random perturbation during policy gradient. It is thus
natural to consider order-reduction for general non-observer based
controllers, such that we find a lower-order controller with approximately
equivalent or lower LQG cost. Moreover, policy gradient for LQG control may
also meet unstable controllers111A dynamic controller that has unstable modes
itself but internally stabilizes the plant., but the results on order-
reduction for unstable controllers are far less complete [1, 6]. This
motivates the main questions in this paper:
1. 1.
Can we perform controller reduction on general, possibly unstable, LQG
controllers, such that the reduced-order controller remains internally
stabilizing?
2. 2.
Can we provide explicit error bounds on the LQG performance of the reduced-
order controller compared to the original controller?
These questions are not only relevant for the reasons relating to policy
optimization of LQG control [5], but are interesting in their own right for
the model and controller reduction literature [1, 7].
Our contributions. In this paper, we provide positive answers to both
questions. We first identify a novel set of sufficient conditions that ensure
the stability of a perturbed controller (Theorem 1), and then derive a new
bound on the LQG cost of a perturbed controller under the assumption that the
truncated component is stable and appropriately small (Theorem 2). For general
multiple-input and multiple-output (MIMO) systems, building on Theorem 1 and
Theorem 2, we then show (in Section 4 and Section 5.1 respectively) how
balanced truncation and modal truncation may be applied to general (non
observer-based, possibly unstable) LQG controllers to yield lower-order
controllers with bounded LQG performance gap (compared to that of the original
controller). Furthermore, for single-input and single-output (SISO) systems,
we discuss in Section 5.2 how internal stability may be preserved even when
the reduced-order controller has fewer unstable poles than the original
controller. This opens the path of controller reduction via truncating
unstable poles, a novel controller reduction technique which we illustrate
both theoretically and empirically. Finally, in Section 6, we characterize the
connection between the existence of a “small” Jordan block with near pole-zero
cancellation in SISO systems. This allows us to connect modal reduction with
order reduction for minimal systems that are close to being non-minimal.
### 1.1 Related work
Our work follows a long line of work in the controller reduction literature. A
classical result is that when a controller ${\mathbf{K}}$ and its reduced-
order counterpart ${\mathbf{K}}_{r}$ have the same number of unstable poles
and no poles on the imaginary axis, assuming some other conditions involving
the truncated component ${\mathbf{\Delta}}={\mathbf{K}}-{\mathbf{K}}_{r}$ hold
(see Lemma 3), the reduced-order controller ${\mathbf{K}}_{r}$ will stabilize
the original system [1, 8]. Thus, one common approach to controller reduction
is to truncate the stable part of a controller, whilst keeping its unstable
part intact. Popular methods to perform truncation of the stable part of a
controller include modal truncation [9, 10], balanced truncation [11], and
Hankel norm approximation [12]. When the difference of the stable portion and
its truncated portion satisfies a (frequency-weighted) error bound (see Lemma
3; cf. [1, Section II.A]), it guarantees that the truncated controller remains
internally stabilizing. However, there appear to be no existing results
providing an error bound on the LQG cost of the truncated controller from such
a procedure for general (possibly non observer-based) controllers. In
contrast, for observer-based controllers, there has been a significant line of
work based on coprime factorizations [13], which not only yields reduced-order
controllers that are internally stabilizing, but also guarantees LQG
performance bounds for the resulting truncated controllers (cf. [6, 1]).
However, a key limitation is that these method only work for observer-based
controllers.
A closely related but distinct research direction to controller reduction is
the topic of open-loop model reduction [7, 14, 15, 16]. While techniques in
model reduction often overlap with those in controller reduction (e.g.
balanced truncation [17], based on the theory of balanced realization in
[18]), modal truncation, and Hankel norm approximation [19]), tight
approximation bounds for the open-loop controller do not translate, a priori,
to tight performance bounds for the closed-loop system. For instance, while
there has been work studying model reduction for general unstable systems [20,
21, 22], it is unclear if such techniques can produce a stabilizing lower-
order approximation of an unstable (but stabilizing) controller.
Another important related topic is policy optimization for linear control
problems. There has been significant recent work studying policy optimization
for linear quadratic (LQ) control problems, for linear-quadratic-regulator
(LQR) [23], $\mathcal{H}_{2}$ linear control with $\mathcal{H}_{\infty}$
guarantees [24, 25], as well as LQG problems [5, 3]. In particular, as we
explained earlier, the considerations outlined in [3] on escaping saddle
points of the LQG problem was an important motivation for our work, where
controller reduction is required. See [4] for a recent review.
### 1.2 Paper outline
The rest of this paper is structured as follows. We present the problem
statement in Section 2. In Section 3, we first introduce Theorem 1, which
provides sufficient conditions such that a perturbed controller
${\mathbf{K}}_{r}$ of ${\mathbf{K}}$ is internally stabilizing. We further
derive an upper bound on the LQG cost $J({\mathbf{K}}_{r})$ (Theorem 2). In
Section 4, we study balanced truncation on the stable part of a controller. In
Section 5, we discuss modal truncation, where Section 5.1 studies modal
truncation on the stable part of a controller, and Section 5.2 discusses
controller reduction via truncation of unstable poles for SISO systems.
Section 6 presents the connection between near pole-zero cancellation and
small Jordan block. We provide numerical experiments to illustrate our
theoretical results in Section 7. We conclude the paper in Section 8.
### 1.3 Notation
We denote the set of real-rational proper stable transfer functions as
$\mathcal{RH}_{\infty}$ (i.e., all the poles are on the open left-half complex
plane). For simplicity, we omit the dimension of transfer matrices. The state-
space realization of a transfer function $\mathbf{G}(s)=C(sI-A)^{-1}B+D$ is
denoted as
$\mathbf{G}(s)=\left[\begin{array}[]{c|c}A&B\\\ \hline\cr
C&D\end{array}\right].$
We define the $\mathcal{L}_{\infty}$ norm for a transfer function
${\mathbf{G}}(s)$ as
$\left\lVert{\mathbf{G}}\right\rVert_{\mathcal{L}_{\infty}}\coloneqq\sup_{w\in\mathbb{R}}\sigma_{\max}({\mathbf{G}}(jw)),$
where $\sigma_{\max}(\cdot)$ denotes the maximum singular value. When
${\mathbf{G}}\in\mathcal{RH}_{\infty}$, its $\mathcal{H}_{\infty}$ norm is the
same as its $\mathcal{L}_{\infty}$ norm [26, Chapter 4.3]. We define the
$\mathcal{L}_{2}$ norm for ${\mathbf{G}}(s)$ as
$\left\lVert{\mathbf{G}}\right\rVert_{\mathcal{L}_{2}}\coloneqq\sqrt{\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathbf{Tr}}({\mathbf{G}}(-jw)^{{\mathsf{T}}}{\mathbf{G}}(jw)dw}.$
When ${\mathbf{G}}$ is stable and strictly proper, its $\mathcal{H}_{2}$ norm
is the same as its $\mathcal{L}_{2}$ norm [26, Chapter 4.3]. We denote the set
of transfer functions in $\mathcal{R}\mathcal{H}_{\infty}$ with finite
$\mathcal{L}_{2}$ norm as $\mathcal{R}\mathcal{H}_{2}$. Throughout our proofs
later, we often utilize submultiplicative-like inequalities to bound the
$\mathcal{H}_{\infty}$ or $\mathcal{H}_{2}$ norm of products of transfer
functions. For completeness, we summarize a list of such inequalities in
Appendix B.
## 2 Preliminaries and Problem Statement
### 2.1 General non-observer based controllers
Consider a strictly proper linear time-invariant (LTI) plant222For simplicity,
we assume that there is a common $B$ in front of both the $u(t)$ and $w(t)$
terms.
$\displaystyle\dot{x}(t)$ $\displaystyle=Ax(t)+Bu(t)+Bw(t),$ (1)
$\displaystyle y(t)$ $\displaystyle=Cx+v(t),$
where $x(t)\in\mathbb{R}^{n},u(t)\in\mathbb{R}^{m},y(t)\in\mathbb{R}^{p}$ are
the state vector, control action, and measurement vector at time $t$,
respectively; $w(t)\in\mathbb{R}^{m}$ and $v(t)\in\mathbb{R}^{p}$ are external
disturbances on the state and measurement vectors at time $t$, respectively.
One basic yet fundamental control problem is to design a feedback controller
(or policy) to stabilize the plant 1. A standard approach for this problem is
to use an observer-based controller of the form
$\displaystyle\dot{\xi}(t)$ $\displaystyle=A\xi(t)+Bu(t)+L(y(t)-C\xi(t))$ (2)
$\displaystyle u(t)$ $\displaystyle=-K\xi(t),$
where $\xi(t)\in\mathbb{R}^{n}$ is an estimated state,
$L\in\mathbb{R}^{n\times p}$ is an observer gain, and $K\in\mathbb{R}^{m\times
n}$ is a feedback gain. The observer and feedback gains are chosen such that
$A-LC$ and $A-BK$ are stable, and this guarantees the closed-loop internal
stability when applying the controller 2 to the plant 1 [26, Chapter 3.5].
Note that the order of this observer-based controller must be the same with
the system plant (i.e., the controller state $\xi(t)$ and the system state
$x(t)$ have the same dimension). Order reduction for controllers in the form
of (2) is discussed in [6, 1, 2].
In this paper, we consider a general non-observer based dynamic controller of
the form
$\displaystyle\dot{\xi}(t)$
$\displaystyle=A_{{\mathsf{K}}}\xi(t)+B_{{\mathsf{K}}}y(t),$ (3)
$\displaystyle u(t)$ $\displaystyle=C_{{\mathsf{K}}}\xi(t),$
where $\xi(t)\in\mathbb{R}^{q}$ is the internal state of the controller, and
$A_{{\mathsf{K}}},B_{{\mathsf{K}}},C_{{\mathsf{K}}}$ are matrices of proper
dimensions that specify the dynamics of the controller. The dimension $q$ of
the internal control variable $\xi$ is called the order of the dynamical
controller 3. The controller in 3 is more suitable for model-free policy
optimization as it does not explicitly depend on the system dynamics [5, 3].
It is clear that the observer-based controller 2 is a special case of 3 by
taking $q=n$, and $A_{\mathsf{K}}=A-BK-
LC,B_{{\mathsf{K}}}=L,C_{{\mathsf{K}}}=-K$. By combining 3 with 1, the closed-
loop system is internally stable if and only if the closed-loop matrix
$A_{\mathrm{cl}}:=\begin{bmatrix}A&BC_{{\mathsf{K}}}\\\
B_{{\mathsf{K}}}C&A_{{\mathsf{K}}}\end{bmatrix}$ (4)
is stable [26, Lemma 5.2].
### 2.2 Problem statement
Given an internally stabilizing controller
$(A_{{\mathsf{K}}},B_{\mathsf{K}},C_{\mathsf{K}})$ satisfying (4), the
controller reduction problem [1] is to find a new controller
$(\hat{A}_{{\mathsf{K}}},\hat{B}_{\mathsf{K}},\hat{C}_{\mathsf{K}})$ of lower
order $\hat{q}<q$ such that it still internally stabilizes the plant and does
not significantly affect the closed-loop performance. In particular, we
consider a normalized LQG control performance [27, 28], defined as
$J=\lim_{T\to\infty}\mathbb{E}\left[\frac{1}{T}\int_{0}^{T}x(t)^{{\mathsf{T}}}C^{{\mathsf{T}}}C{x}(t)+u(t)^{{\mathsf{T}}}u(t)dt\right],$
(5)
We make the following two assumptions.
###### Assumption 1.
The plant (1) is minimal, i.e., $(A,B)$ is controllable and $(C,A)$ is
observable.
###### Assumption 2.
In plant (1), the signals $w(t)\in\mathbb{R}^{m}$ and $v(t)\in\mathbb{R}^{p}$
are zero mean Gaussian white noise, each with a spectrum equal to the
identity.
Assumption 1 is standard and guarantees the existence of internally
stabilizing controllers333The existence of internally stabilizing controllers
only requires stabilizability and detectablity.. If the plant is not minimal,
we can always perform a lower-order minimal realization before designing a
controller. Assumption 2 was used to define the normalized LQG control problem
in [27, 28]. As we shall see next, this assumption simplifies the expression
of LQG cost 5 in the frequency domain. For the controller reduction problem,
it may not be easy to work directly with the internal stability condition 4 in
the state-space domain due to non-uniqueness of state-space realizations. It
is more convenient to consider equivalent conditions in the frequency domain.
In particular, the controller 3 can be represented as a transfer function
$\mathbf{K}:=C_{{\mathsf{K}}}(sI-A_{\mathsf{K}})^{-1}B_{\mathsf{K}}$. Let us
define a new performance signal $\tilde{y}=Cx$. Some simple manipulations show
that the closed-loop transfer function from $(w,v)$ to $(\tilde{y},u)$ is
$\displaystyle\begin{bmatrix}\mathbf{\tilde{y}}\\\
\mathbf{u}\end{bmatrix}=\begin{bmatrix}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\mathbf{K}\\\
\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\end{bmatrix}\begin{bmatrix}\mathbf{w}\\\
\mathbf{v}\end{bmatrix},$ (6)
where $\mathbf{G}(s)=C(sI-A)^{-1}B$. Then, we have the following condition for
internal stability.
###### Lemma 1 ([26, Lemma 5.3]).
The controller $\mathbf{K}$ in (3) internally stabilizes the plant (1) if and
only if the closed-loop transfer function from $(w,v)$ to $(\tilde{y},u)$ is
stable444The standard result in [26, Lemma 5.3] uses a slightly different set
of closed-loop transfer functions. Simple manipulations via
$(I-\mathbf{G}\mathbf{K})^{-1}=I+(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\mathbf{K}$
can show the equivalence., i.e.,
$\begin{bmatrix}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\mathbf{K}\\\
\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\end{bmatrix}\in\mathcal{RH}_{\infty}$
Furthermore, it is not difficult to see that the normalized LQG control cost
(5) can be expressed conveniently in the frequency domain. For completeness,
we provide a proof of Lemma 2 in Section C.1.
###### Lemma 2.
Under Assumption 2, given an internally stabilizing controller $\mathbf{K}$ in
(3), the normalized LQG control cost 5 can be expressed as follows
$J(\mathbf{K})=\left\|\begin{bmatrix}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\mathbf{K}\\\
\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\end{bmatrix}\right\|_{\mathcal{H}_{2}}^{2}.$
## 3 Robust stability and LQG performance
Our main goal is to properly perturb the controller $\mathbf{K}$ to get a
lower-order controller $\mathbf{K}_{r}$ such that the closed-loop performance
remains similar. In this section, we present two technical results that
underpin our controller reduction results in Sections 4 and 5: 1) a new robust
stability result (Theorem 1), and 2) an upper bound on the LQG performance for
the new controller $\mathbf{K}_{r}$ (Theorem 2).
### 3.1 A novel sufficient condition for internal stability
Classical results on controller reduction often focus on the case when the
truncated controller $\mathbf{K}_{r}$ has the same number of unstable poles as
the original controller $\mathbf{K}$ (cf. [6, 1]). Under such a condition,
sufficient conditions are available using the $\mathcal{L}_{\infty}$ norm of
terms involving the difference ${\mathbf{K}}_{r}-{\mathbf{K}}$ such that
${\mathbf{K}}_{r}$ can still stabilize ${\mathbf{G}}$ if ${\mathbf{K}}$
stabilizes ${\mathbf{G}}$. In particular, a widely-used condition is as
follows.
###### Lemma 3 ([1, Section II.A]).
Let $\mathbf{G}$ be the transfer function of an LTI plant 1, the controller
$\mathbf{K}$ 3 internally stabilize the plant, and $\mathbf{K}_{r}$ be another
dynamic controller. Denote
${\mathbf{\Delta}}\coloneqq{\mathbf{K}}_{r}-{\mathbf{K}}$. If
1. 1.
$\mathbf{K}$ and $\mathbf{K}_{r}$ have the same number of poles in $Re(s)>0$,
and no poles on the imaginary axis,
2. 2.
either
$\left\lVert{\mathbf{\Delta}}\mathbf{G}(I-\mathbf{K}\mathbf{G})^{-1}\right\rVert_{\mathcal{L}_{\infty}}\\!<\\!1$
or
$\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}\\!}\\!<\\!1$,
then $\mathbf{K}_{r}$ also internally stabilizes the plant $\mathbf{G}$.
This classical result underpins many controller reduction techniques in the
literature; see [1] for a review. Lemma 3 can be proved by combining Nyquist
stability criterion [8, Section IV] with a classical result [26, Theorem 5.7].
For completeness, we provide a proof in Section C.2. If the controller
$\mathbf{K}$ is stable in the first place, then the first condition in Lemma 3
can be naturally satisfied by using any stable controller $\mathbf{K}_{r}$.
When the controller $\mathbf{K}$ is unstable (i.e., $A_{\mathsf{K}}$ in 3 is
unstable), we might always need to preserve the unstable part in $\mathbf{K}$
in order to use Lemma 3. However, it is unclear if it is necessary for a
lower-order truncated controller ${\mathbf{K}}_{r}$ to have the same number of
unstable poles as ${\mathbf{K}}$ in order to maintain closed-loop stability.
In this section, we provide a novel set of sufficient conditions in Theorem 1
ensuring that ${\mathbf{K}}_{r}$ is still stabilizing, which makes no explicit
assumptions on whether ${\mathbf{K}}_{r}$ and ${\mathbf{K}}$ have the same
number of unstable poles. This technical result may be of independent
interest.
###### Theorem 1.
Let $\mathbf{G}$ be the transfer function of an LTI plant 1, the controller
$\mathbf{K}$ 3 internally stabilize the plant, and let $\mathbf{K}_{r}$ denote
another controller. Denote
${\mathbf{\Delta}}\coloneqq{\mathbf{K}}_{r}-{\mathbf{K}}$. If
${\mathbf{\Delta}}(I-{\mathbf{G}}{\mathbf{K}})^{-1}$ is stable and
$\displaystyle{\max\left\\{\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}},\left\lVert{\mathbf{\Delta}}(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{\infty}}\right\\}<1,}$
(7)
then, ${\mathbf{K}}_{r}$ also internally stabilizes ${\mathbf{G}}$.
Unlike the proof of Lemma 3 that is based on Nyquist stability [8, Section
IV], Theorem 1 can be proved directly from Lemma 1. If we can show
$\begin{bmatrix}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}&(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}_{r}\\\
\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}&\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\end{bmatrix}\in\mathcal{RH}_{\infty},$
(8)
Lemma 1 confirms that ${\mathbf{K}}_{r}$ internally stabilizes ${\mathbf{G}}$.
Since $\mathbf{K}$ internally stabilizes $\mathbf{G}$, we know that
$\begin{bmatrix}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\mathbf{K}\\\
\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\end{bmatrix}\in\mathcal{RH}_{\infty}.$
(9)
Motivated by [29, Appendix C], the key idea in our proof is to relate the
transfer functions in (8) with those in 9, and then to show that each of the
four subblocks in (8) is stable.
###### Proof.
For notational simplicity, denote
$\mathbf{Y}\coloneqq(I-\mathbf{G}\mathbf{K})^{-1},\;\mathbf{X}\coloneqq(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}.$
(10)
By this definition, we naturally have $\mathbf{X}=\mathbf{Y}\mathbf{G}$, and
$\mathbf{G}=\mathbf{Y}^{-1}\mathbf{X}.$ Since $\mathbf{K}$ internally
stabilizes $\mathbf{G}$, we know that $\mathbf{Y}$ and $\mathbf{X}$ are both
stable. Observe that
${\mathbf{Y}}=(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}{\mathbf{K}}+(I-{\mathbf{G}}{\mathbf{K}})^{-1}(I-{\mathbf{G}}{\mathbf{K}})=(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}{\mathbf{K}}+I,$
(11)
which implies
$\mathbf{X}{\mathbf{K}}=(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}{\mathbf{K}}$
is stable. By our definition in 10, the condition 7 is equivalent to
$\left\lVert{\mathbf{X}}{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}<1$,
$\left\lVert{\mathbf{\Delta}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}}<1$
and ${\mathbf{\Delta}}{\mathbf{Y}}$ is stable. According to the small-gain
theorem [26, Theorem 9.1], we know that both
$(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}$ and
$(I-{\mathbf{\Delta}}{\mathbf{X}})^{-1}$ exist and they are stable.
We now proceed to show stability of the four subblocks in (8). A helpful
calculation is that
$\displaystyle(I-\mathbf{G}\mathbf{K}_{r})^{-1}=(I-\mathbf{G}(\mathbf{K}+\mathbf{\Delta}))^{-1}$
$\displaystyle=(I-\mathbf{Y}^{-1}\mathbf{X}(\mathbf{K}+\mathbf{\Delta}))^{-1}$
$\displaystyle=(\mathbf{Y}-\mathbf{X}(\mathbf{K}+\mathbf{\Delta}))^{-1}\mathbf{Y}$
$\displaystyle=(I-\mathbf{X}\mathbf{\Delta})^{-1}\mathbf{Y},$ (12)
where we have applied that fact that $\mathbf{Y}=\mathbf{X}\mathbf{K}+I$ from
11.
First, noting that $\mathbf{Y}\coloneqq(I-{\mathbf{G}}{\mathbf{K}})^{-1}$ and
using 12, we have
$\displaystyle(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}=(I-\mathbf{X}\mathbf{\Delta})^{-1}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G},$
(13)
which is stable since $(I-\mathbf{X}\mathbf{\Delta})^{-1}$ and
$(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}$ are both stable.
Next observe that
$\displaystyle(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}_{r}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{(i)}}}{{\mathstrut{=}}}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}+(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}{\mathbf{X}}\mathbf{\Delta}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{(ii)}}}{{\mathstrut{=}}}(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}(I-{\mathbf{G}}{\mathbf{K}})^{-1}\mathbf{G}\mathbf{K}+(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}{\mathbf{X}}\mathbf{\Delta},$
(14)
where we have applied 12 to derive (3.1) and (14). Since
$(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}$,
$(I-{\mathbf{G}}{\mathbf{K}})^{-1}\mathbf{G}\mathbf{K}$, and
${\mathbf{X}}\mathbf{\Delta}$ are all stable, we know that
$(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}_{r}$ is stable.
Next, we consider the term $\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}$.
We have
$\displaystyle\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}=$
$\displaystyle\
{\mathbf{K}}\left(I-\mathbf{X}\mathbf{\Delta}\right)^{-1}{\mathbf{Y}}+{\mathbf{\Delta}}\left(I-\mathbf{X}\mathbf{\Delta}\right)^{-1}{\mathbf{Y}}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{(iii)}}}{{\mathstrut{=}}}$
$\displaystyle\
{\mathbf{K}}(I+\mathbf{X}{\mathbf{\Delta}}(I-\mathbf{X}{\mathbf{\Delta}})^{-1}){\mathbf{Y}}+{\mathbf{\Delta}}\left(I-\mathbf{X}\mathbf{\Delta}\right)^{-1}{\mathbf{Y}}$
$\displaystyle=$ $\displaystyle\
{\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}+{\mathbf{K}}(\mathbf{X}\mathbf{\Delta})(I-\mathbf{X}\mathbf{\Delta})^{-1}{\mathbf{Y}}+\mathbf{\Delta}(I-\mathbf{X}\mathbf{\Delta})^{-1}{\mathbf{Y}}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{(iv)}}}{{\mathstrut{=}}}$
$\displaystyle\
{\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}+{\mathbf{K}}{\mathbf{X}}(I-{\mathbf{\Delta}}\mathbf{X})^{-1}\mathbf{\Delta}{\mathbf{Y}}+(I-{\mathbf{\Delta}}\mathbf{X})^{-1}{\mathbf{\Delta}}{\mathbf{Y}}.$
(15)
To derive (3.1), we used the fact that
$(I-\mathbf{X}{\mathbf{\Delta}})^{-1}=(I-\mathbf{X}{\mathbf{\Delta}}+\mathbf{X}{\mathbf{\Delta}})(I-\mathbf{X}{\mathbf{\Delta}})^{-1}=I+\mathbf{X}{\mathbf{\Delta}}(I-\mathbf{X}{\mathbf{\Delta}})^{-1}.$
To derive (15), we used the push through identity
${\mathbf{\Delta}}(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}=(I-{\mathbf{\Delta}}{\mathbf{X}})^{-1}{\mathbf{\Delta}}$.
It is now clear $\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}$ is stable,
since ${\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}$,
${\mathbf{K}}{\mathbf{X}}$, $(I-{\mathbf{\Delta}}\mathbf{X})^{-1}$, and
$\mathbf{\Delta}{\mathbf{Y}}$ are all stable.
Finally, to study the stability of
${\mathbf{K}}_{r}(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}{\mathbf{G}}$, we reuse
the calculations from ${\mathbf{K}}_{r}(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}$
in 15, leading to
$\displaystyle\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}{\mathbf{G}}=$
$\displaystyle{\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}{\mathbf{G}}+(I-{\mathbf{\Delta}}\mathbf{X})^{-1}{\mathbf{\Delta}}{\mathbf{Y}}{\mathbf{G}}+{\mathbf{K}}{\mathbf{X}}(I-{\mathbf{\Delta}}\mathbf{X})^{-1}\mathbf{\Delta}{\mathbf{Y}}{\mathbf{G}},$
$\displaystyle=$
$\displaystyle{\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}{\mathbf{G}}+(I-{\mathbf{\Delta}}\mathbf{X})^{-1}{\mathbf{\Delta}}\mathbf{X}+{\mathbf{K}}{\mathbf{X}}(I-{\mathbf{\Delta}}\mathbf{X})^{-1}\mathbf{\Delta}\mathbf{X},$
which is stable, since
${\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}{\mathbf{G}}$,
${\mathbf{K}}{\mathbf{X}}$, $(I-{\mathbf{\Delta}}\mathbf{X})^{-1}$, and
$\mathbf{\Delta}\mathbf{X}$ are all stable. We have thus shown the stability
of all four subblocks in (8). This completes the proof. ∎
###### Remark 1 (Lemma 3 versus Theorem 1).
Both Lemma 3 and Theorem 1 provide a set of sufficient conditions for a
reduced-order controller to internally stabilize the original plant. We
comment here on some similarities and differences between the two sets of
sufficient conditions. First, condition 2 in Lemma 3 is stated in terms of the
$\mathcal{L}_{\infty}$ norm, while 7 in Theorem 1 requires the
$\mathcal{H}_{\infty}$ norm, thus condition 2 in Lemma 3 is in fact a weaker
requirement than 7555Note that an unstable transfer function may have finite
$\mathcal{L}_{\infty}$ norm and only stable transfer functions can have finite
$\mathcal{H}_{\infty}$ norm; for the definition of the $\mathcal{L}_{\infty}$
and $\mathcal{H}_{\infty}$ norm, the reader can refer to the notation in
Section 1.2.. In addition, Theorem 1 requires that
${\mathbf{\Delta}}(I-{\mathbf{G}}{\mathbf{K}})^{-1}$ should be stable, whilst
Lemma 3 makes no such requirement. However, a key advantage of Theorem 1 over
Lemma 3 is that it does not require ${\mathbf{K}}$ and its reduced-order
counterpart ${\mathbf{K}}_{r}$ to have the same number of poles in $Re(s)>0$
or to have no poles on the imaginary axis. This suggests that effective
controller reduction may not necessitate preserving all unstable poles of
${\mathbf{K}}$ — this intuition will turn out to be useful for Theorem 3
(Section 5.2), which shows that controller reduction via unstable modal
truncation is in fact possible.
### 3.2 A new bound on the perturbed LQG cost
Theorem 1 presents sufficient conditions to guarantee the closed-loop
stability using the reduced-order controller ${\mathbf{K}}_{r}$. In many
situations (such as policy optimization for LQG in [5, 3]), we also need to
understand the closed-loop performance under this new controller
${\mathbf{K}}_{r}$. Our next technical result show that if the error
${\mathbf{\Delta}}\coloneqq{\mathbf{K}}_{r}-{\mathbf{K}}$ is stable, the
change of the LQG cost 5 can also be bounded. The proof builds on the analysis
techniques in Theorem 1.
###### Theorem 2.
Let $\mathbf{G}$ be the transfer function of an LTI plant 1, the controller
$\mathbf{K}$ 3 internally stabilize the plant, and $\mathbf{K}_{r}$ denote
another controller. Denote
${\mathbf{\Delta}}\coloneqq{\mathbf{K}}_{r}-{\mathbf{K}}$. If
$\displaystyle\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}<\frac{1}{\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{\infty}}},$
(16)
then the controller ${\mathbf{K}}_{r}$ internally stabilizes the plant 1, and
the resulting LQG cost 5 satisfies
$J({\mathbf{K}}_{r})\leq\frac{1}{\left(1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}\right)^{2}}(J(\mathbf{K})+S_{1}+S_{2}),$
(17)
where with the notation $\mathbf{Y}:=(I-\mathbf{G}{\mathbf{K}})^{-1}$,
$\mathbf{X}:=(I-\mathbf{G}{\mathbf{K}})^{-1}\mathbf{G}$ and
$\mathbf{\Delta}:=\mathbf{K}_{r}-\mathbf{K}$, we have
$\displaystyle
S_{1}:=2\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}(\left\lVert\mathbf{X}\mathbf{K}\right\rVert_{\mathcal{H}_{2}})+2\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}\\!(\\!\left\lVert{\mathbf{K}}\mathbf{Y}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}\\!+\\!\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\\!)\\!(1\\!+\\!\left\lVert{\mathbf{K}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}}),$
(18) $\displaystyle
S_{2}:=\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}^{2}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}^{2}\\!+\\!\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}^{2}(\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}^{2}+\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}^{2})(1\\!+\\!\left\lVert{\mathbf{K}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}})^{2}.$
###### Proof.
We first show via Theorem 1 that ${\mathbf{K}}_{r}$ internally stabilizes
${\mathbf{G}}$. If 16 holds, it follows that
$\displaystyle\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}$
$\displaystyle\leq\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}<1.$
Similarly, it can be verified that
$\left\lVert{\mathbf{\Delta}}(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{\infty}}<1.$
We also have that ${\mathbf{\Delta}}(I-{\mathbf{G}}{\mathbf{K}})^{-1}$ is
stable since ${\mathbf{\Delta}}$ and $(I-{\mathbf{G}}{\mathbf{K}})^{-1}$ are
both stable. Therefore, the conditions in Theorem 1 are all satisfied, which
implies then that ${\mathbf{K}}_{r}$ internally stabilizes ${\mathbf{G}}$.
We then proceed to upper bound the LQG cost $J(\mathbf{K}_{r})$ for the new
reduced-order controller $\mathbf{K}_{r}$. From Lemma 2, we know that
$J(\mathbf{K}_{r})\\!=\\!\left\lVert(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}{\mathbf{K}}_{r}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert{\mathbf{K}}_{r}(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert{\mathbf{K}}_{r}(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}\right\rVert_{\mathcal{H}_{2}}^{2}.$
Our strategy is to bound each term above using the norms of the corresponding
terms when the original controller $\mathbf{K}$ is applied666This process will
frequently use some standard norm inequalities listed in Appendix B..
We first observe that by 13, the following inequality holds
$\displaystyle\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle=\left\lVert(I-\mathbf{X}\mathbf{\Delta})^{-1}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq\left\lVert(I-\mathbf{X}\mathbf{\Delta})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq\frac{1}{1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}}\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}},$
(19)
where we have applied standard norm inequalities in Lemma 7 and Lemma 9. Next,
observe that
$\displaystyle\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}_{r}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle=\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}(\mathbf{K}+\mathbf{\Delta})\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}\right\rVert_{\mathcal{H}_{2}}+\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq\frac{1}{1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}}\left(\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\mathbf{K}\right\rVert_{\mathcal{H}_{2}}\right)$
$\displaystyle\qquad\qquad+\frac{\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}}{1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}}\left(\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}\right),$
(20)
where we used 19 and similar norm inequalities to derive the last inequality.
We then consider the term $\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}$.
From 15 in the proof of Theorem 1, we have
$\displaystyle\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}={\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}+{\mathbf{K}}{\mathbf{X}}(I-{\mathbf{\Delta}}\mathbf{X})^{-1}\mathbf{\Delta}(I-\mathbf{G}\mathbf{K})^{-1}+(I-{\mathbf{\Delta}}\mathbf{X})^{-1}{\mathbf{\Delta}}(I-\mathbf{G}{\mathbf{K}})^{-1},$
leading to the following upper bound
$\displaystyle\left\lVert\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\right\rVert_{\mathcal{H}_{2}}\leq\left\lVert{\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}\right\rVert_{\mathcal{H}_{2}}+\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}\frac{\left\lVert(I-\mathbf{G}{\mathbf{K}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\left(1+\left\lVert{\mathbf{K}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}}\right)}{1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}}.$
(21)
Similarly, we can derive the following upper bound
$\displaystyle\left\lVert\mathbf{K}_{r}(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{2}}\leq\left\lVert{\mathbf{K}}(I\\!-\\!\mathbf{G}{\mathbf{K}})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{2}}+\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}\frac{\left\lVert(I\\!-\\!\mathbf{G}{\mathbf{K}})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{\infty}}\left(1\\!+\\!\left\lVert{\mathbf{K}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}}\right)}{1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}}.$
(22)
Combining the bounds in 19, 20, 21 and 22, we arrive at the desired bound as
follows
$\displaystyle
J(\mathbf{K}_{r})\leq\left(\frac{1}{1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}}\right)^{2}(J(\mathbf{K})+S_{1}+S_{2}),$
where $S_{1}$ and $S_{2}$ are defined in 18 (recalling the notation
$\mathbf{Y}=(I-\mathbf{G}{\mathbf{K}})^{-1},\mathbf{X}=(I-\mathbf{G}{\mathbf{K}})^{-1}{\mathbf{G}}$
in 10), and
$J(\mathbf{K})\\!=\\!\left\lVert(I\\!-\\!\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert(I\\!-\\!\mathbf{G}\mathbf{K})^{-1}\mathbf{G}{\mathbf{K}}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert{\mathbf{K}}(I\\!-\\!\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert{\mathbf{K}}(I\\!-\\!\mathbf{G}\mathbf{K})^{-1}\right\rVert_{\mathcal{H}_{2}}^{2}.$
This finishes the proof. ∎
Theorem 2 shows that as long as the truncation error is bounded as in 16, the
reduced-order controller $\mathbf{K}_{r}$ still internally stabilizes the
plant. Note that the bound in 16 can be verified since $\mathbf{K}$ and
$\mathbf{G}$ are known by the designer. Furthermore, the upper bound (17)
implies that
$\left|\frac{J(\mathbf{K}_{r})-J(\mathbf{K})}{J(\mathbf{K})}\right|\leq\mathcal{O}(\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}).$
When the truncation error is small (measured by
$\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}$), the change
of LQG cost is also small. Similar bounds like 17 seem to be less studied for
general non-observer based controllers in the literature [1]. Most existing
bounds assume an observed-based controller in 2 (cf. [6, 1, 2]), and most of
the techniques therein rely on coprime factorization [13]. Theorem 1 and
Theorem 2 work for any perturbed controller $\mathbf{K}_{r}$ satisfying the
assumptions therein. In the next two sections, we show how to use balanced and
modal truncation strategies to derive suitable reduced-order controllers
$\mathbf{K}_{r}$.
## 4 Controller reduction via balanced truncation
In this section, we discuss controller reduction strategies using balanced
truncation and apply Theorem 1 and Theorem 2 to derive stability and
performance guarantees.
### 4.1 Balanced truncation
We begin with a result reviewing the following well-known fact about balanced
truncation: for asymptotically stable transfer functions, under appropriate
assumptions, a reduced-order transfer function resulting from balanced
truncation is also asymptotically stable.
###### Lemma 4 ([17, Theorem 3.2], [16, Theorem 7.9]).
Let $\mathbf{P}$ be an asymptotically stable transfer function (i.e. all poles
are in the open left-half plane) with a balanced minimal state-space
realization
$\displaystyle\mathbf{P}=\left[\begin{array}[]{c c |c}A_{11}&A_{12}&B_{1}\\\
A_{21}&A_{22}&B_{2}\\\ \hline\cr C_{1}&C_{2}&D\end{array}\right],\mbox{ s.t.
}$ $\displaystyle\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{bmatrix}\\!\begin{bmatrix}\Sigma_{1}&0\\\
0&\Sigma_{2}\end{bmatrix}\\!+\\!\begin{bmatrix}\Sigma_{1}&0\\\
0&\Sigma_{2}\end{bmatrix}\\!\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{bmatrix}^{{\mathsf{T}}}\\!=\\!-\begin{bmatrix}B_{1}\\\
B_{2}\end{bmatrix}\\!\begin{bmatrix}B_{1}\\\
B_{2}\end{bmatrix}^{{\mathsf{T}}}$
$\displaystyle\begin{bmatrix}\Sigma_{1}&0\\\
0&\Sigma_{2}\end{bmatrix}\\!\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{bmatrix}\\!+\\!\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{bmatrix}^{{\mathsf{T}}}\\!\begin{bmatrix}\Sigma_{1}&0\\\
0&\Sigma_{2}\end{bmatrix}\\!=\\!-\begin{bmatrix}C_{1}^{{\mathsf{T}}}\\\
C_{2}^{{\mathsf{T}}}\end{bmatrix}\\!\begin{bmatrix}C_{1}^{{\mathsf{T}}}\\\
C_{2}^{{\mathsf{T}}}\end{bmatrix}^{{\mathsf{T}}}$
where $\Sigma_{i}\succ 0$ is a positive-definite diagonal matrix for each
$i\in\\{1,2\\}$. If $\Sigma_{1}$ and $\Sigma_{2}$ share no eigenvalues in
common, then the sub-blocks $A_{11}$ and $A_{22}$ are both asymptotically
stable. Furthermore, we have the following error bound
$\|\mathbf{P}-\mathbf{P}_{r}\|_{\mathcal{H}_{\infty}}\leq
2\mathrm{trace}(\Sigma_{2}),\;\text{where}\;\mathbf{P}_{r}\\!=\\!\left[\begin{array}[]{c
|c}A_{11}&B_{1}\\\ \hline\cr C_{1}&D\end{array}\right].$ (23)
Note that Lemma 4 assumes that the transfer function $\mathbf{P}$ has a
balanced state-space realization. It is well-known from classical control that
any stable transfer function has a balanced minimal realization. For
completeness, we state (and prove) this result in Proposition 1 in Appendix A.
In addition, in Algorithm 1, we state the standard balanced truncation
procedure for a minimal realization of a stable controller $\mathbf{K}$.
Algorithm 1 Balanced truncation for stable systems
0: 1) A stable system ${\mathbf{K}}$ with minimal state-space realization
${\mathbf{K}}=\left[\begin{array}[]{c|c}A&B\\\ \hline\cr
C&D\end{array}\right]$, 2) post-truncation order parameter $r$
1: Form the controllability and observability Gramians $W_{c}$ and $W_{o}$ via
solving
$\displaystyle{A}W_{c}+W_{c}{A}^{{\mathsf{T}}}+{B}{B}^{{\mathsf{T}}}=0,\qquad{A}^{{\mathsf{T}}}W_{o}+W_{o}{A}+{C}^{{\mathsf{T}}}{C}=0.$
2: Compute $\Sigma={\rm
diag}\left(\left\\{\sqrt{\lambda_{i}(W_{c}W_{o})}\right\\}_{i=1}^{n}\right)$.
3: Factorize $W_{c}=QQ^{{\mathsf{T}}}$, and compute an orthonormal $U$ such
that $Q^{{\mathsf{T}}}W_{o}Q=U\Sigma^{2}U^{{\mathsf{T}}}$.
4: Let $T=\Sigma^{1/2}U^{{\mathsf{T}}}Q^{-1}$, and compute a balanced minimal
realization ${\mathbf{K}}=\left[\begin{array}[]{c|c}\tilde{A}&\tilde{B}\\\
\hline\cr\\\\[-10.0pt] \tilde{C}&\tilde{D}\end{array}\right],$ where
$\tilde{A}=TAT^{-1},\tilde{B}=TB,\tilde{C}=CT^{-1},\tilde{D}=D.$
5: Partition $\tilde{A},\tilde{B},\tilde{C},\Sigma$ as
$\tilde{A}=\begin{bmatrix}\tilde{A}_{11}&\tilde{A}_{12}\\\
\tilde{A}_{21}&\tilde{A}_{22}\end{bmatrix},\tilde{B}=\begin{bmatrix}\tilde{B}_{1}\\\
\tilde{B}_{2}\end{bmatrix},\tilde{C}=\begin{bmatrix}\tilde{C}_{1}&\tilde{C}_{2}\end{bmatrix},\Sigma=\begin{bmatrix}\Sigma_{1}&0\\\
0&\Sigma_{2}\end{bmatrix},$ where
$\tilde{A}_{11},\Sigma_{1}\in\mathbb{R}^{r\times
r},\tilde{B}_{1}\in\mathbb{R}^{r\times p},\tilde{C}_{1}\in\mathbb{R}^{m\times
r}$.
6: return the reduced order controller
${\mathbf{K}}_{r}\coloneqq\left[\begin{array}[]{c|c}\tilde{A}_{11}&\tilde{B}_{1}\\\
\hline\cr\\\\[-10.0pt] \tilde{C}_{1}&\tilde{D}\end{array}\right].$
### 4.2 Controller reduction
In general, the dynamical controller ${\mathbf{K}}$ is not stable itself,
i.e., $A_{\mathsf{K}}$ in 3 has unstable eigenvalues. The standard balanced
truncation procedure cannot be applied to unstable systems directly. Our
strategy is to divide the controller ${\mathbf{K}}$ into a stable part and
unstable part
${\mathbf{K}}={\mathbf{K}}_{<}+{\mathbf{K}}_{\geq},$ (24)
where ${\mathbf{K}}_{<}$ of order $n_{1}$ contains all stable poles (i.e.,
those on the open left-half plane) and ${\mathbf{K}}_{\geq}$ of order $n_{2}$
contains the remaining poles (i.e., those on the closed right-half plane), and
$n_{1}+n_{2}=n$. In this section, we assume the controller contains at least
one stable pole ($n_{1}\geq 1$).
The separation 24 is always possible by computing the Jordan normal form of
$A_{\mathsf{K}}$ such that
$A_{\mathsf{K}}=Q_{\mathsf{K}}\hat{A}_{\mathsf{K}}Q_{\mathsf{K}}^{-1},\qquad\text{with}\;\;\hat{A}_{\mathsf{K}}=\begin{bmatrix}\hat{A}_{{\mathsf{K}},<}&0\\\
0&\hat{A}_{{\mathsf{K}},\geq}\end{bmatrix},$ (25)
where $Q_{\mathsf{K}}\in\mathbb{R}^{n\times n}$ is an invertible coordinate
transformation777Since any real-valued matrix can be expressed in a Jordan
canonical form, such a transformation $Q_{\mathsf{K}}$ always exists., the
eigenvalues of $\hat{A}_{{\mathsf{K}},<}\in\mathbb{R}^{n_{1}\times n_{1}}$ are
in the open left-half plane, and the eigenvalues of
$\hat{A}_{{\mathsf{K}},\geq}\in\mathbb{R}^{n_{2}\times n_{2}}$ are in the
closed right-half plane, and $n_{1}+n_{2}=n$. Therefore, the stable and
unstable parts in 24 can be expressed as
${\mathbf{K}}_{<}=\left[\begin{array}[]{c|c}\hat{A}_{{\mathsf{K}},<}&\hat{B}_{{\mathsf{K}},<}\\\
\hline\cr\hat{C}_{{\mathsf{K}},<}&0\end{array}\right],\qquad{\mathbf{K}}_{\geq}=\left[\begin{array}[]{c|c}\hat{A}_{{\mathsf{K}},\geq}&\hat{B}_{{\mathsf{K}},\geq}\\\
\hline\cr\hat{C}_{{\mathsf{K}},\geq}&0\end{array}\right],$ (26)
where $\hat{C}_{\mathsf{K}}\coloneqq C_{\mathsf{K}}Q_{\mathsf{K}}$ and
$\hat{B}_{\mathsf{K}}\coloneqq Q_{\mathsf{K}}^{-1}B_{\mathsf{K}}$ are
partitioned into
$\hat{C}_{\mathsf{K}}=\begin{bmatrix}\hat{C}_{{\mathsf{K}},<}&\hat{C}_{{\mathsf{K}},\geq}\end{bmatrix},\qquad\hat{B}_{\mathsf{K}}=\begin{bmatrix}\hat{B}_{{\mathsf{K}},<}\\\
\hat{B}_{{\mathsf{K}},\geq}\end{bmatrix}$
with $\hat{C}_{{\mathsf{K}},<}\in\mathbb{R}^{m\times
n_{1}},\hat{C}_{{\mathsf{K}},\geq}\in\mathbb{R}^{m\times(n-n_{1})}$, and
$\hat{B}_{{\mathsf{K}},<}\in\mathbb{R}^{n_{1}\times
p},\hat{B}_{{\mathsf{K}},\geq}\in\mathbb{R}^{(n-n_{1})\times p}$.
We can then perform a balanced truncation on the stable part
${\mathbf{K}}_{<}$ and get a reduced-order controller
${\mathbf{K}}_{<,{r}}=\left[\begin{array}[]{c|c}\tilde{A}_{<,11}&\tilde{B}_{<,1}\\\
\hline\cr\\\\[-12.0pt] \tilde{C}_{<,1}&0\end{array}\right],$
where the order is $n_{r}<n_{1}$. The final reduced-order controller becomes
${\mathbf{K}}_{r}={\mathbf{K}}_{<,r}+{\mathbf{K}}_{\geq}=\left[\begin{array}[]{cc|c}\tilde{A}_{<,11}&0&\tilde{B}_{<,1}\\\
0&\hat{A}_{{\mathsf{K}},\geq}&\hat{B}_{{\mathsf{K}},\geq}\\\
\hline\cr\\\\[-12.0pt]
\tilde{C}_{<,1}&\hat{C}_{{\mathsf{K}},\geq}&0\end{array}\right]$ (27)
which has order $r:=n_{r}+n_{2}<n$. This process is summarized in Algorithm 2.
Algorithm 2 Balanced truncation for an unstable controller with stable part
0: 1) A controller ${\mathbf{K}}$ with a minimal order-$n$ state-space
realization
${\mathbf{K}}=\left[\begin{array}[]{c|c}A_{\mathsf{K}}&B_{\mathsf{K}}\\\
\hline\cr C_{\mathsf{K}}&0\end{array}\right]$, 2) the post-truncation order
$r\geq n_{2}$ with $n_{2}$ defined in 24.
1: Compute the Jordan normal form of $A_{\mathsf{K}}$ in 25.
2: Separate the controller ${\mathbf{K}}$ into
${\mathbf{K}}={\mathbf{K}}_{<}+{\mathbf{K}}_{\geq}$ as 26.
3: Perform balanced truncation on the stable and minimal system
${\mathbf{K}}_{<}$ with post-truncation order parameter $n_{r}<n_{1}$ to
obtain an order-$n_{r}$ system ${\mathbf{K}}_{<,r}$.
4: return the reduced-order controller
${\mathbf{K}}_{r}={\mathbf{K}}_{r,<}+{\mathbf{K}}_{\geq}$ (of order
$r=n_{r}+n_{2}<n)$.
Based on Theorems 1 and 2, under appropriate conditions, the resulting
controller ${\mathbf{K}}_{r}$ from Algorithm 2 remains a stabilizing
controller and has a similar LQG cost compared to the original controller
${\mathbf{K}}$.
###### Corollary 1.
Consider a minimal $n$-th order controller $\mathbf{K}$ which stabilizes the
plant ${\mathbf{G}}$. Suppose we obtain an $r$-th order controller
${\mathbf{K}}_{r}$ via Algorithm 2, where $r<n$, such that
${\mathbf{K}}_{r}={\mathbf{K}}_{<,r}+{\mathbf{K}}_{\geq}$, where
${\mathbf{K}}_{<,r}$ is a lower-order balanced truncation of
${\mathbf{K}}_{<}$. Suppose that $\Sigma_{<,1}$ and $\Sigma_{<,2}$ in the
balanced truncation of ${\mathbf{K}}_{<}$ share no eigenvalues. If
$\displaystyle\sigma_{n_{r}+1}+\cdots+\sigma_{n_{1}}<\frac{1}{2\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{\infty}}},$
(28)
where $\sigma_{n_{r}+1},\dots,\sigma_{n_{1}}$ are the diagonal elements of
$\Sigma_{<,2}$ in the balanced truncation of ${\mathbf{K}}_{<}$, then the
reduced-order controller ${\mathbf{K}}_{r}$ internally stabilizes the plant 1,
and the resulting LQG cost 5 satisfies
$J({\mathbf{K}}_{r})\leq\frac{1}{\left(1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}\right)^{2}}(J(\mathbf{K})+S_{1}+S_{2}),$
where with the notation $\mathbf{Y}:=(I-\mathbf{G}{\mathbf{K}})^{-1}$,
$\mathbf{X}:=(I-\mathbf{G}{\mathbf{K}})^{-1}\mathbf{G}$ and
$\mathbf{\Delta}:=\mathbf{K}_{r}-\mathbf{K}$,
$\displaystyle
S_{1}:=2\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}(\left\lVert\mathbf{X}\mathbf{K}\right\rVert_{\mathcal{H}_{2}})+2\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}(\left\lVert{\mathbf{K}}\mathbf{Y}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}+\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}})(1+\left\lVert{\mathbf{K}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}}),$
$\displaystyle
S_{2}:=\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}^{2}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}^{2}(\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}^{2}+\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}^{2})(1+\left\lVert{\mathbf{K}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}})^{2}.$
The proof is based on the analysis in Theorem 2. The only difference is that
Corollary 1 truncates the singular values in the stable part
${\mathbf{K}}_{r}$ and imposes the condition 28, which is the same 16 in
Theorem 2 when applying 23 in Lemma 4. Note that Corollary 1 is reduced to
Theorem 2 when the controller $\mathbf{K}$ is stable itself.
## 5 Controller reduction via modal truncation
In this section, we proceed to discuss controller reduction by modal
truncation, which may apply to the truncation of either stable or unstable
component(s) in a controller. In particular, we first apply modal truncation
on the stable part of a controller in Section 5.1, and then discuss the
performance of modal truncation on possibly unstable component(s) for SISO
systems in Section 5.2.
### 5.1 Modal truncation on stable component(s)
Algorithm 3 Modal truncation
0: 1) A controller ${\mathbf{K}}$ with a minimal order-$n$ state-space
realization
${\mathbf{K}}=\left[\begin{array}[]{c|c}A_{\mathsf{K}}&B_{\mathsf{K}}\\\
\hline\cr C_{\mathsf{K}}&0\end{array}\right]$; 2) the order reduction
parameter $r_{\mathrm{red}}$ (a positive integer less than $k$)
1: Convert $A_{\mathsf{K}}$ into the standard Jordan normal form (29).
2: Decompose ${\mathbf{K}}$ as ${\mathbf{K}}(s)=\sum_{i=1}^{k}C_{i}(sI-
A_{i})^{-1}B_{i}$ that is consistent with the Jordan block 29.
3: For each $i$, compute the index $d_{i}$ in (30).
4: Let $o_{i}$ be the ranking of $i$ according to $\\{d_{i}\\}_{i=1}^{k}$,
such that $o_{i}=j$ if $d_{i}$ is the $j$-th smallest value.
5: Set ${\mathbf{\Delta}}=\sum_{i\in[k],o_{i}\leq r_{\mathrm{red}}}C_{i}(sI-
A_{i})^{-1}B_{i}$.
6: return the reduced order controller
${\mathbf{K}}_{r}:={\mathbf{K}}-{\mathbf{\Delta}}$.
The basic idea of modal truncation begins with writing the controller
${\mathbf{K}}$ 3 into ${\mathbf{K}}(s)=\sum_{i=1}^{k}C_{i}(sI-
A_{i})^{-1}B_{i}$, where $A_{i}$ contains a mode corresponding to an
eigenvalue of $\lambda_{i}$ in ${\mathbf{K}}$. This is always possible by
considering its standard Jordan form
$A_{\mathsf{K}}=\begin{bmatrix}A_{1}&0&\dots&0\\\ 0&A_{2}&\dots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\dots&A_{k}\end{bmatrix},$ (29)
where each $A_{i}$ is a Jordan block of order $n_{i}$, and
$\sum_{i=1}^{k}n_{i}=n.$ Let $\lambda_{i}$ denote the eigenvalue associated
with each Jordan block $A_{i}$. We then directly remove some modes that are
less significant according the criterion defined below
$d_{i}=\begin{cases}\left\lVert C_{i}(sI-
A_{i})^{-1}B_{i}\right\rVert_{\mathcal{H}_{\infty}}&\text{if}\;\lambda_{i}<0\\\
\left\lVert
C_{i}A_{i}^{-1}B_{i}\right\rVert_{2}&\text{if}\;\lambda_{i}>0.\end{cases}$
(30)
The detailed steps are listed in Algorithm 3.
As a counterpart to balanced truncation, we can derive upper bounds the LQG
cost change when performing modal truncation on the stable part of a
controller ${\mathbf{K}}$.
###### Corollary 2.
Consider a minimal $n$-th order controller $\mathbf{K}$ which stabilizes the
plant ${\mathbf{G}}$. Consider the decomposition
${\mathbf{K}}={\mathbf{K}}_{<}+{\mathbf{K}}_{\geq}$ in 24, and suppose we
obtain a lower-order approximation ${\mathbf{K}}_{r,<}$ of ${\mathbf{K}}_{<}$
using the modal truncation algorithm in Algorithm 3. Let
${\mathbf{K}}_{r}:={\mathbf{K}}_{r,<}+{\mathbf{K}}_{\geq}$. Denote
$\mathbf{\Delta}={\mathbf{K}}_{<}-{\mathbf{K}}_{r,<}$. Suppose that
$\displaystyle\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}<\frac{1}{\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{\infty}}}.$
(31)
Then, we have that ${\mathbf{K}}_{r}$ internally stabilizes ${\mathbf{G}}$,
and the resulting LQG cost 5 satisfies
$J({\mathbf{K}}_{r})\leq\frac{1}{\left(1-\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}\right)^{2}}(J(\mathbf{K})+S_{1}+S_{2}),$
where with the notation $\mathbf{Y}:=(I-\mathbf{G}{\mathbf{K}})^{-1}$,
$\mathbf{X}:=(I-\mathbf{G}{\mathbf{K}})^{-1}\mathbf{G}$, we have
$\displaystyle
S_{1}:=2\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}(\left\lVert\mathbf{X}\mathbf{K}\right\rVert_{\mathcal{H}_{2}})+\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}(\left\lVert{\mathbf{K}}\mathbf{Y}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}+\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}})\left(1+\left\lVert{\mathbf{K}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}}\right),$
$\displaystyle
S_{2}:=\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{\infty}}^{2}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}^{2}(\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}^{2}+\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}^{2})\left(1+\left\lVert{\mathbf{K}}{\mathbf{X}}\right\rVert_{\mathcal{H}_{\infty}}\right)^{2}.$
###### Proof.
Since 31 holds, the condition in 7 holds, and we can thus apply Theorem 1 to
conclude that ${\mathbf{K}}_{r}$ internally stabilizes ${\mathbf{G}}$. Then,
the same calculations in Theorem 2 can be used to show the upper bound on
$J({\mathbf{K}}_{r})$. ∎
### 5.2 Modal truncation on unstable component(s)
Next, we introduce the following result, which studies the LQG cost change
when truncating unstable mode(s) of a controller ${\mathbf{K}}$, for single-
input single-output (SISO) systems.
###### Theorem 3 (Order reduction of unstable SISO controllers).
Consider a minimal $n$-th order controller $\mathbf{K}$ which stabilizes the
plant ${\mathbf{G}}$. Suppose both ${\mathbf{G}}(s)$ and ${\mathbf{K}}(s)$ are
univariate rational polynomial functions (SISO systems). Let an $r$-order
controller ${\mathbf{K}}_{r}$ computed via Algorithm 3, where $r<n$, where we
denote $\mathbf{\Delta}={\mathbf{K}}-{\mathbf{K}}_{r}$ as in Algorithm 3 (note
that ${\mathbf{\Delta}}$ may be unstable). Suppose ${\mathbf{\Delta}}$ has no
unstable mode at 0, and
$\displaystyle(1-(1-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}{\mathbf{\Delta}})^{-1}\in\mathcal{RH}_{\infty},$
(32)
Then, ${\mathbf{K}}_{r}$ internally stabilizes ${\mathbf{G}}$, and
$J({\mathbf{K}}_{r})\leq\left\lVert(1-{\mathbf{X}}{\mathbf{\Delta}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}^{2}(J(\mathbf{K})+S_{1}+S_{2}),$
(33)
where with the notation $\mathbf{Y}:=(I-\mathbf{G}{\mathbf{K}})^{-1}$,
$\mathbf{X}:=(I-\mathbf{G}{\mathbf{K}})^{-1}\mathbf{G}$, we have
$\displaystyle
S_{1}\\!:=\\!2\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{L}_{\infty}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{X}\mathbf{K}\right\rVert_{\mathcal{H}_{2}}+2\\!\left\lVert{\mathbf{K}}\mathbf{Y}\right\rVert_{\mathcal{H}_{2}}\\!\left(\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{L}_{2}}\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}\right)$
$\displaystyle\quad\quad+2\left\lVert{\mathbf{K}}\mathbf{Y}\right\rVert_{\mathcal{H}_{2}}\\!\left(\\!\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{L}_{\infty}}\\!\left(\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{2}}\\!+\\!\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}\\!\\!+\\!\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\\!\right)\\!\right)$
$\displaystyle
S_{2}\\!:=\\!\left\lVert\mathbf{\Delta}\right\rVert_{\mathcal{L}_{\infty}}^{2}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}^{2}+\left(\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}\left(\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{2}}+\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{2}}\right)\right)^{2}+\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}^{2}\left(\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}+\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\right)^{2}.$
###### Proof.
Without loss of generality, we suppose ${\mathbf{\Delta}}(s)=C_{k}(sI-
A_{k})^{-1}B_{k}$ and assume that $\lambda_{k}>0$. From the proof of Theorem
1, to show that ${\mathbf{K}}_{r}$ internally stabilize ${\mathbf{G}}$, it
suffices for us to show that
$\displaystyle\begin{bmatrix}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}&(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}_{r}\\\
\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}&\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\end{bmatrix}\in\mathcal{RH}_{\infty}.$
From 13 in the proof of Theorem 1, since ${\mathbf{K}}$ stabilizes
${\mathbf{G}}$, the stability of $(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}$
is ensured when $(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}$ is stable.
Meanwhile, from 14 in the proof of Theorem 1, assuming ${\mathbf{K}}$
stabilizes ${\mathbf{G}}$, the stability of
$(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}{\mathbf{K}}_{r}$ is achieved when
$(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}$ and ${\mathbf{X}}{\mathbf{\Delta}}$
are stable.
Next, to show that $\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}$ is
stable, from 15 in the proof of Theorem 1, assuming ${\mathbf{K}}$ stabilizes
${\mathbf{G}}$, it suffices for us to show that
$(I-{\mathbf{\Delta}}{\mathbf{X}})^{-1}$ and ${\mathbf{\Delta}}{\mathbf{Y}}$
are stable. Observe that in the SISO case,
$(I-{\mathbf{X}}{\mathbf{\Delta}})^{-1}=(I-{\mathbf{\Delta}}{\mathbf{X}})^{-1}$.
Note that in the SISO case, since
$(1-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}_{r}=\mathbf{K}_{r}(1-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}$,
we do not have to separately show the stability of
$\mathbf{K}_{r}(1-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}$.
Thus, collecting the conditions above, given that ${\mathbf{K}}$ internally
stabilizes ${\mathbf{G}}$, to show that ${\mathbf{K}}_{r}$ internally
stabilizes ${\mathbf{G}}$, recalling that
${\mathbf{X}}:=(1-\mathbf{G}\mathbf{K})^{-1}{\mathbf{G}}$ and
${\mathbf{Y}}:=(1-\mathbf{G}\mathbf{K})^{-1}$, it suffices for us to show that
$\displaystyle(1-(1-\mathbf{G}\mathbf{K})^{-1}{\mathbf{G}}{\mathbf{\Delta}})^{-1}\in\mathcal{RH}_{\infty},\quad(1-\mathbf{G}\mathbf{K})^{-1}{\mathbf{G}}{\mathbf{\Delta}}\in\mathcal{RH}_{\infty},\quad{\mathbf{\Delta}}(1-{\mathbf{G}}{\mathbf{K}})^{-1}\in\mathcal{RH}_{\infty}.$
Our assumption in 32 assumes that
$(1-(1-\mathbf{G}\mathbf{K})^{-1}{\mathbf{G}}{\mathbf{\Delta}})^{-1}\in\mathcal{RH}_{\infty}$.
We proceed to show now that (with the aid of the assumption that
$(1-(1-\mathbf{G}\mathbf{K})^{-1}{\mathbf{G}}{\mathbf{\Delta}})^{-1}\in\mathcal{RH}_{\infty}$)
$\displaystyle(1-\mathbf{G}\mathbf{K})^{-1}{\mathbf{G}}{\mathbf{\Delta}}\in\mathcal{RH}_{\infty},\quad{\mathbf{\Delta}}(1-{\mathbf{G}}{\mathbf{K}})^{-1}\in\mathcal{RH}_{\infty}.$
We first show that
$(1-\mathbf{G}\mathbf{K})^{-1}{\mathbf{G}}{\mathbf{\Delta}}\in\mathcal{RH}_{\infty}$.
Since $(1-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}$ is stable, the only
possible pole for
$(1-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}{\mathbf{\Delta}}$ is at
$s=\lambda_{k}$, where $\lambda_{k}$ is the eigenvalue associated with
$A_{k}$. Thus, it suffices to show that
$\left\lvert(1-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}{\mathbf{\Delta}}(\lambda_{k})\right\rvert<\infty$.
Observe that for any $i\in[k]$, the transfer function $C_{i}(sI-
A_{i})^{-1}B_{i}$ can be written as a rational function in the form
$\displaystyle C_{i}(sI-
A_{i})^{-1}B_{i}=\frac{\alpha_{i}(s)}{(s-\lambda_{i})^{r_{i}}},$
where $r_{i}=n_{i}$ (this is due to the minimality of the state-space
realization $(A_{{\mathsf{K}}},B_{{\mathsf{K}}},C_{{\mathsf{K}}},0)$), and
$\alpha_{i}(s)$ is coprime to $(s-\lambda_{i})$. Then, we can express
${\mathbf{K}}(s)$ as
$\displaystyle{\mathbf{K}}(s)=\sum_{i=1}^{k}C_{i}(sI-
A_{i})^{-1}B_{i}=\sum_{i=1}^{k}\frac{\alpha_{i}(s)}{(s-\lambda_{i})^{n_{i}}}=\frac{\alpha(s)}{\prod_{i=1}^{k}(s-\lambda_{i})^{n_{i}}},$
where $\alpha(s)=\sum_{i=1}^{k}\left(\alpha_{i}(s)\prod_{j\neq
i}(s-\lambda_{j})^{n_{j}}\right)$. Since this
$(A_{{\mathsf{K}}},B_{{\mathsf{K}}},C_{{\mathsf{K}}},0)$ state-space
realization of ${\mathbf{K}}$ is minimal, $\alpha(s)$ is coprime to
$(s-\lambda_{i})$ for each $i\in[k]$.
Note that it is possible that $\lambda_{i}=\lambda_{k}$ for $i\neq k$. Hence,
by extracting the factors of $(s-\lambda_{k})$, we may decompose
$\prod_{i=1}^{k-1}(s-\lambda_{i})^{n_{i}}=(s-\lambda_{k})^{n_{k}^{\prime}}\prod_{i=1,\lambda_{i}\neq\lambda_{k}}^{k-1}(s-\lambda_{i})^{n_{i}},$
where $n_{k}^{\prime}=\sum_{i=1,\lambda_{i}=\lambda_{k}}^{k-1}n_{i}$. Then, we
have
$\displaystyle\frac{{\mathbf{G}}(s){\mathbf{\Delta}}(s)}{1-{\mathbf{G}}(s){\mathbf{K}}(s)}$
$\displaystyle=\frac{{\mathbf{G}}(s)\frac{\alpha_{k}(s)}{(s-\lambda_{k})^{n_{k}}}}{1-{\mathbf{G}}(s)\frac{\alpha(s)}{\prod_{i=1}^{k}(s-\lambda_{i})^{n_{i}}}}$
$\displaystyle=\frac{\alpha_{k}(s)(s-\lambda_{k})^{n_{k}^{\prime}}}{\frac{(s-\lambda_{k})^{n_{k}+n_{k}^{\prime}}}{{\mathbf{G}}(s)}+\frac{\alpha(s)}{\prod_{i=1,\lambda_{i}\neq\lambda_{k}}^{k-1}(s-\lambda_{i})^{n_{i}}}}.$
(34)
Now, since ${\mathbf{K}}$ internally stabilizes ${\mathbf{G}}$, there is no
unstable pole-zero cancellation between ${\mathbf{G}}$ and ${\mathbf{K}}$ [26,
Theorem 5.7]. Thus, if we express ${\mathbf{G}}(s)$ in rational polynomial
form as ${\mathbf{G}}(s)=\frac{g_{n}(s)}{g_{d}(s)}$, then $g_{n}(s)$ is
coprime to $(s-\lambda_{k})$ (recall that $\lambda_{k}$ is in the open right-
half plane, by our assumption at the start of the proof). Thus, the polynomial
$\frac{(s-\lambda_{k})^{n_{k}+n_{k}^{\prime}}}{{\mathbf{G}}(s)}$
evaluated at $s=\lambda_{k}$ is zero. Hence, continuing from 34, we have
$\displaystyle\frac{{\mathbf{G}}(\lambda_{k}){\mathbf{\Delta}}(\lambda_{k})}{1-{\mathbf{G}}(\lambda_{k}){\mathbf{K}}(\lambda_{k})}$
$\displaystyle=\frac{\alpha_{k}(\lambda_{k})(\lambda_{k}-\lambda_{k})^{n_{k}^{\prime}}}{\frac{(\lambda_{k}-\lambda_{k})^{n_{k}+n_{k}^{\prime}}}{{\mathbf{G}}(\lambda_{k})}+\frac{\alpha(\lambda_{k})}{\prod_{i=1,\lambda_{i}\neq\lambda_{k}}^{k-1}(\lambda_{k}-\lambda_{i})^{n_{i}}}}$
$\displaystyle=\frac{\alpha_{k}(\lambda_{k})(\lambda_{k}-\lambda_{k})^{n_{k}^{\prime}}}{\frac{\alpha(\lambda_{k})}{\prod_{i=1,\lambda_{i}\neq\lambda_{k}}^{k-1}(\lambda_{k}-\lambda_{i})^{n_{i}}}},$
which is finite, since $\alpha_{k}(s)$ and $\alpha(s)$ are both coprime to
$(s-\lambda_{k})$ and hence their pointwise evaluation at $s=\lambda_{k}$ is
finite. Thus, we conclude that
$\frac{{\mathbf{G}}{\mathbf{\Delta}}}{1-{\mathbf{G}}{\mathbf{K}}}$ does not
have a pole at $\lambda_{k}$. This implies that it has a finite
$\mathcal{H}_{\infty}$ norm.
Next we proceed to show that
${\mathbf{\Delta}}(1-{\mathbf{G}}{\mathbf{K}})^{-1}\in\mathcal{RH}_{\infty}$.
The argument for this is similar to the one above. Recall that we can write
${\mathbf{G}}(s)=\frac{g_{n}(s)}{g_{d}(s)},$
where $g_{n}(s)$ is coprime to $(s-\lambda_{k})$ and $g_{n}(s)$ and $g_{d}(s)$
are coprime. In particular, we can also decompose $g_{d}(s)$ as
$(s-\lambda_{k})^{r_{g}}g_{d}^{\prime}(s)$, where $r_{g}$ is a nonnegative
integer. Then,
$\displaystyle\frac{{\mathbf{\Delta}}(s)}{1-{\mathbf{G}}(s){\mathbf{K}}(s)}=\frac{\frac{\alpha_{k}(s)}{(s-\lambda_{k})^{n_{k}}}}{1-\frac{g_{n}(s)}{g_{d}(s)}\frac{\alpha(s)}{\prod_{i=1}^{k}(s-\lambda_{i})^{n_{i}}}}=\frac{\alpha_{k}(s)(s-\lambda_{k})^{n_{k}^{\prime}+r_{g}}}{(s-\lambda_{k})^{n_{k}+n_{k}^{\prime}+r_{g}}+\frac{g_{n}(s)}{g_{d}^{\prime}(s)}\frac{\alpha(s)}{\prod_{i=1,\lambda_{i}\neq\lambda_{k}}^{k-1}(s-\lambda_{i})^{n_{i}}}}$
From the above expression, since $g_{n}(s),g_{d}^{\prime}(s)$ and $\alpha(s)$
are all coprime to $(s-\lambda_{k})$, it is not hard to see that
$\frac{{\mathbf{\Delta}}(\lambda_{k})}{1-{\mathbf{G}}(\lambda_{k}){\mathbf{K}}(\lambda_{k})}$
is finite (and must in fact be zero if $n_{k}^{\prime}+r_{g}>0$). Recall that
since $\frac{1}{1-{\mathbf{G}}{\mathbf{K}}}$ is stable (as ${\mathbf{K}}$
internally stabilizes ${\mathbf{G}}$), the only possible pole for
$\frac{{\mathbf{\Delta}}}{1-{\mathbf{G}}{\mathbf{K}}}$ is at a pole of
${\mathbf{\Delta}}$, which is at $s=\lambda_{k}$. Since we have just excluded
this possibility, it follows that
$\frac{{\mathbf{\Delta}}}{1-{\mathbf{G}}{\mathbf{K}}}$ is stable.
The proof for the bound on the change in LQG performance 33 is similar to that
in Theorem 2. We provide the details in Section C.3. ∎
###### Remark 2.
We note a limitation of our result, namely the requirement that the truncated
Jordan blocks have non-zero eigenvalues. Hence, the procedure does not work
when we wish to truncate a Jordan block corresponding to an zero eigenvalue.
In addition, due to the relative simplicity of defining zeros for SISO
systems, we chose to limit our attention to SISO systems. Extending to general
MIMO remains future work.
## 6 Connecting near pole-zero cancellation to small Jordan block
An intuitive way of defining “near non-minimality” for a transfer function
${\mathbf{G}}(s)$ is the existence of a pair of pole $p_{i}$ and zero $q_{i}$
which are “close” to each other. Assuming that $p_{i}$ is a simple pole, when
this happens, we then conjecture that the coefficient corresponding to the
term $\frac{1}{s-p_{i}}$ in the partial fraction decomposition of
${\mathbf{G}}(s)$ is small, i.e. the Jordan block corresponding to the pole
$p_{i}$ is small.
Below, we formalize this idea in the case $p_{i}$ is a simple pole888We
believe a similar result holds for the general case when $p_{i}$ is a repeated
pole, and leave the precise characterization to future work..
###### Lemma 5.
Consider a minimal transfer SISO transfer function ${\mathbf{G}}(s)$. Suppose
$p_{i}$ is a simple pole of $G(s)$. Suppose we have the factorization
$\displaystyle{\mathbf{G}}(s)=\frac{s-q_{i}}{s-p_{i}}\frac{n(s)}{d(s)},$
where $n(s)$ is coprime to $d(s)$ and $s-p_{i}$, and $d(s)$ is also coprime to
$s-p_{i}$. By Bezout’s identity, we know there exists $a(s),b(s)$ such that
$a(s)d(s)+b(s)(s-p_{i})=1$, where $a(s)$ is a constant, and $b(s)$ has degree
less than $d(s)$.
Observe that we can write $n(s)a(s)=f(s)(s-p_{i})+r$ for some complex
polynomial $f(s)$ and remainder term $r\in\mathbb{C}$. Then, we can write
$G(s)$ as the partial fraction sum
${\mathbf{G}}(s)=\frac{(p_{i}-q_{i})r}{s-p_{i}}+\frac{e(s)}{d(s)},$
where $e(s)=n(s)(s-q_{i})b(s)+n(s)a(s)d(s)+(p_{i}-q_{i})f(s)d(s)$.
As a consequence, we note that if $p_{i}-q_{i}$ is small (relative to the
remainder term $r$), then the Jordan block corresponding to the $p_{i}$ pole
also has a small coefficient.
###### Proof.
Since $s-p_{i}$ and $d(s)$ are coprime, there exists polynomials $a(s)$ and
$b(s)$ where $a(s)d(s)+b(s)(s-p_{i})=1$, and
$\operatorname{deg}(a(s))=0,\operatorname{deg}(b(s))<\operatorname{deg}(d(s))$.
We can then write
$\displaystyle\frac{n(s)(s-q_{i})}{d(s)(s-p_{i})}$
$\displaystyle=\frac{n(s)(s-q_{i})(a(s)d(s)+b(s)(s-p_{i}))}{d(s)(s-p_{i})}$
$\displaystyle=\frac{n(s)(s-q_{i})b(s)}{d(s)}+\frac{n(s)(s-q_{i})a(s)}{s-p_{i}}$
$\displaystyle=\frac{n(s)(s-q_{i})b(s)}{d(s)}+n(s)a(s)+\frac{(p_{i}-q_{i})n(s)a(s)}{s-p_{i}}.$
Observe that we can write $n(s)a(s)=f(s)(s-p_{i})+r$, where $r\in\mathbb{C}$.
Thus, continuing from the derivations, we have
$\displaystyle\frac{n(s)(s-q_{i})}{d(s)(s-p_{i})}=\frac{n(s)(s-q_{i})b(s)+n(s)a(s)d(s)+(p_{i}-q_{i})f(s)d(s)}{d(s)}+\frac{(p_{i}-q_{i})r}{s-p_{i}}.$
∎
We now consider the reverse direction, where we go from the existence of a
small Jordan block corresponding to a pole to a near pole-zero cancellation at
that pole.
We first recall Rouché’s theorem, a standard result from complex analysis [30]
which will be useful for us.
###### Theorem 4 (Rouché’s theorem).
Suppose $f$ and $g$ are holomorphic on an open set $U$ containing a circle $C$
and its interior. Suppose $\left\lvert f(z)\right\rvert>\left\lvert
g(z)\right\rvert$ for all $z$ in $C$.999Technically, we only need that
$\left\lvert f(z)+tg(z)\right\rvert>0$ for all $t\in[0,1]$ and all $z\in C$.
Then, $f$ and $f+g$ have the same number of zeros (including multiplicity)
inside the circle $C$.
###### Lemma 6.
Consider a minimal SISO transfer function ${\mathbf{G}}(s)$. Suppose $p$ is a
(possibly repeated) pole of $G$, with order $n_{p}$. By applying partial
fraction factorization, ${\mathbf{G}}(s)$ can be (uniquely) decomposed
additively as
$\displaystyle{\mathbf{G}}(s)=\sum_{j=1}^{n_{p}}\frac{\alpha_{j}}{(s-p)^{j}}+\frac{u(s)}{d(s)},$
where $u(s)$ is coprime to $d(s)$, and $d(s)$ is coprime to $(s-p)$. Then,
$\min_{p^{\prime}\neq p:(s-p)^{j}u(s)=0}\left\lvert
p-p^{\prime}\right\rvert>0$, and for any $\epsilon>0$ such that
$\epsilon<\min_{p^{\prime}\neq p:(s-p)^{j}u(s)=0}\left\lvert
p-p^{\prime}\right\rvert,$
there exists $\delta(\epsilon,G)>0$ such that if
$\max_{j}\left\lvert\alpha_{j}\right\rvert<\delta(\epsilon,G)$, then
$\displaystyle{\mathbf{G}}(s)=\frac{\prod_{j=1}^{n_{p}}(x-p+\epsilon_{j})n(s)}{(x-p)^{n_{p}}d(s)}$
such that $\left\lvert\epsilon_{j}\right\rvert<\epsilon$ for some polynomial
$n(s)$ where $n(s)$ is coprime to both $(x-p)$ and $d(s)$.
###### Proof.
Since
${\mathbf{G}}(s)=\sum_{j=1}^{n_{p}}\frac{\alpha_{j}}{(s-p)^{j}}+\frac{u(s)}{d(s)},$
it follows that
${\mathbf{G}}(s)=\frac{\left(\sum_{j=1}^{n_{p}}\alpha_{j}(s-p)^{n_{p}-j}\right)d(s)+u(s)(s-p)^{n_{p}}}{(s-p)^{n_{p}}d(s)}.$
Let
$e(s)\coloneqq\left(\sum_{j=1}^{n_{p}}\alpha_{j}(s-p)^{n_{p}-j}\right)d(s)$,
and let $f(s)\coloneqq u(s)(s-p)^{n_{p}}$. Clearly, $f(s)$ has a repeated zero
with multiplicity $n_{p}$ at $s=p$. Consider any $\epsilon>0$ such that
$\epsilon<\min_{p^{\prime}\neq p:(s-p)^{j}u(s)=0}\left\lvert
p-p^{\prime}\right\rvert$; we note that $\min_{p^{\prime}\neq
p:(s-p)^{j}u(s)=0}\left\lvert p-p^{\prime}\right\rvert$ exists and is
positive101010As an aside, we note it may be infinite if $u(s)$ is a constant
or if $u(s)$ only has a root at $s=p$).. Then, it follows that $\left\lvert
f(s)\right\rvert>0$ for any $s\in\partial
B(p,\epsilon)\coloneqq\\{s\in\mathbb{C}:\left\lvert
p-s\right\rvert=\epsilon$}; moreover, by definition of $\epsilon$, the only
zeros of $f(s)$ within the closed disc
$B(p,\epsilon)\coloneqq\\{s\in\mathbb{C}:\left\lvert
s-p\right\rvert\leq\epsilon\\}$ are at $s=p$. By continuity, there exists some
$\delta(\epsilon,G)>0$ such that when
$\max_{j}\left\lvert\alpha_{j}\right\rvert<\delta(\epsilon,G)$, it follows
that $\left\lvert e(s)\right\rvert<\left\lvert f(s)\right\rvert$ for any
$s\in\partial B(p,\epsilon)$. In the event that holds, since $e(s)$ and $f(s)$
are holomorphic on $\mathbb{C}$, it follows from Rouché’s theorem that
$e(s)+f(s)$ has the same number of zeros in the closed disc $B(p,\epsilon)$ as
$f(s)$. Since $f(s)$ has precisely $n_{p}$ zeros within $B(p,\epsilon)$, our
desired result follows.
∎
## 7 Numerical Examples
### 7.1 Comparing balanced truncation to modal truncation
We here compare the performance of balanced truncation versus modal
truncation. Consider the following plant and controller pair
$\mathbf{G}=\left[\begin{array}[]{c|c}\begin{bmatrix}-6.00&-13.84&-11.95\\\
1&0&0\\\ 0&1&0\end{bmatrix}&\begin{bmatrix}1\\\ 0\\\ 0\end{bmatrix}\\\
\hline\cr\begin{bmatrix}-1.74&-7.63&-8.37\end{bmatrix}&0\end{array}\right],\quad{\mathbf{K}}\\!=\\!\left[\begin{array}[]{c|c}\begin{bmatrix}-2.1541&-0.0104&0\\\
-0.0104&-2.1731&0\\\ 0&0&0.2\end{bmatrix}&\begin{bmatrix}0\\\ 1.2815\\\
0.5\end{bmatrix}\\\
\hline\cr\begin{bmatrix}-0.8097&-1.2368&0.5\end{bmatrix}&0\end{array}\right]$
It is easy to check numerically that ${\mathbf{K}}$ internally stabilize
$\mathbf{G}$. Indeed, the closed-loop poles (i.e., eigenvalues of
$A_{\mathrm{cl}}$ in 4) are (-0.38,-2.53, -2.15), each with multiplicity 2.
Note that this controller $\mathbf{K}$ has an unstable mode $\lambda=0.2$,
thus the standard balanced truncation in Lemma 4 is inapplicable directly. As
discussed in Section 4.2, we separate the controller as
${\mathbf{K}}={\mathbf{K}}_{<}+{\mathbf{K}}_{\geq}$, and perform reduction on
the stable part ${\mathbf{K}}_{<}$ (i.e. Algorithms 2 and 3).
###### Remark 3 (System generation).
To illustrate Theorem 2 and our results in Sections 4 and 5, we need to
consider a plant $\mathbf{G}$ and a (possibly unstable) controller
$\mathbf{K}$ that is internally stabilizing. To generate such instances, we
first generate a random stable and minimal system, ${\mathbf{K}}_{<}$ and
another unstable part ${\mathbf{K}}_{\geq}$. We define the augmented system
${\mathbf{K}}={\mathbf{K}}_{<}+{\mathbf{K}}_{\geq}$, and compute a stabilizing
controller for ${\mathbf{K}}$, which we call ${\mathbf{G}}$. Finally, we treat
${\mathbf{G}}$ as the system plant and ${\mathbf{K}}$ as the controller. It is
thus clear that ${\mathbf{K}}$ internally stabilizes ${\mathbf{G}}$ (duality
between plant and controller).
After performing balanced truncation and modal truncation, we get two reduced-
order controllers (of order 2) ${\mathbf{K}}_{r,\mathrm{bt}}$ and
${\mathbf{K}}_{r,\mathrm{mt}}$ respectively. Both
${\mathbf{K}}_{r,\mathrm{bt}}$ and ${\mathbf{K}}_{r,\mathrm{mt}}$ satisfy the
bound in 16, thus internally stabilizes the plant, guaranteed by Theorem 2.
Indeed, under Assumption 2, the LQG cost of the two truncated controllers are
listed in Table 1. In this example, the LQG cost of the balanced truncation is
very close to the original performance, while the modal truncated controller
has a slightly higher LQG cost. Note that the quantity
$\left\lVert{\mathbf{K}}-{\mathbf{K}}_{r}\right\rVert_{\mathcal{H}_{\infty}}$
is significantly smaller in the case of balanced truncation. Theorem 2 states
that the upper bound on $J({\mathbf{K}}_{r})-J({\mathbf{K}})$ is tighter when
$\left\lVert{\mathbf{K}}-{\mathbf{K}}_{r}\right\rVert_{\mathcal{H}_{\infty}}$
is smaller; since
$\left\lVert{\mathbf{K}}-{\mathbf{K}}_{r,\mathrm{bt}}\right\rVert_{\mathcal{H}_{\infty}}<\left\lVert{\mathbf{K}}-{\mathbf{K}}_{r,\mathrm{mt}}\right\rVert_{\mathcal{H}_{\infty}}$,
this explains why $J({\mathbf{K}}_{r,\mathrm{bt}})$ is lower than
$J({\mathbf{K}}_{r,\mathrm{mt}})$ in this case. We provide more extensive
comparisons in Appendix D.
Table 1: Comparison of bal. & mod. trunc. | Original | Bal. Trunc. | Mod. Trunc
---|---|---|---
LQG cost | 8.0552 | 8.0552 | 8.9928
$\left\lVert{\mathbf{K}}-{\mathbf{K}}_{r}\right\rVert_{\mathcal{H}_{\infty}}$ | — | $2.6572\cdot 10^{-6}$ | 0.0580
Figure 1: Relationship between LQG cost gap
$J({\mathbf{K}}_{r})-J({\mathbf{K}})$ and the $\mathcal{H}_{\infty}$ norm of
the truncated component (under modal reduction approach).
Scaling effect of
$\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}$. We next
study how the performance gap behaves as a function of the size of the
truncated component. For this analysis, we (randomly) generate five stable and
minimal SISO systems of order 4, ${\mathbf{K}}_{r}$, and augment the system by
adding a stable mode ${\mathbf{\Delta}}$, where
$\displaystyle{\mathbf{K}}_{r}\\!=\\!\left[\begin{array}[]{c|c}\begin{bmatrix}1.5&-1&-0.21\\\
3.00&-0.43&-1.00\\\ 2.00&-0.07&-5.00\end{bmatrix}&\begin{bmatrix}0.18\\\
0.97\\\ 1.2\end{bmatrix}\\\ \hline\cr\\\\[-8.0pt]
\begin{bmatrix}1.00&2.00&3.00\end{bmatrix}&0\end{array}\right],\
{\mathbf{\Delta}}\\!=\\!\left[\begin{array}[]{c|c}-1&\sqrt{\epsilon}\\\
\hline\cr\sqrt{\epsilon}&0\end{array}\right].$
We denote the augmented system as
${\mathbf{K}}:={\mathbf{K}}_{r}+{\mathbf{\Delta}}$. We then generate a
stabilizing controller, ${\mathbf{G}}$, which stabilizes ${\mathbf{K}}$.
Viewing ${\mathbf{G}}$ as the system plant, we then compare the LQG cost of
${\mathbf{K}}$ and ${\mathbf{K}}_{r}$ on the system ${\mathbf{G}}$, as the
$\mathcal{H}_{\infty}$ norm of the truncated component, i.e.
$\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}$, varies (to
be precise, we plot 30 ${\mathbf{\Delta}}$’s, each corresponding to a
different $\epsilon$, where we let $\epsilon$ range (equally spaced) between
0.0001 and 0.05). As we can see in Figure 1, for the five set of controllers,
there is close to a linear relationship (with different slopes) between the
LQG cost gap ratio
$\frac{J({\mathbf{K}}_{r})-J({\mathbf{K}})}{J({\mathbf{K}})}$ and the
$\mathcal{H}_{\infty}$ norm of the truncated component, i.e.
$\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}$; this is
consistent with the upper bound on $J({\mathbf{K}}_{r})$ in Theorem 3.
### 7.2 Truncating unstable mode(s)
We here consider an example to illustrate Theorem 3. Consider the following
plant ${\mathbf{G}}$
$\displaystyle{\mathbf{G}}=\left[\begin{array}[]{c|c}\begin{bmatrix}-5.86&-9.50&0.56\\\
1&0&0\\\ 0&1&0\end{bmatrix}&\begin{bmatrix}1\\\ 0\\\ 0\end{bmatrix}\\\
\hline\cr\\\\[-10.0pt]
\begin{bmatrix}-7.18&-25.61&-8.41\end{bmatrix}&0\end{array}\right].$
It can be verified that the controller
${\mathbf{K}}=\left[\begin{array}[]{c|c}A_{\mathsf{K}}&B_{\mathsf{K}}\\\
\hline\cr C_{\mathsf{K}}&0\end{array}\right],$ with
$A_{\mathsf{K}}\\!=\\!\begin{bmatrix}1.37&0&0\\\ 0&-0.37&0\\\
0&0&0.34\end{bmatrix},\ B_{\mathsf{K}}\\!=\\!\begin{bmatrix}0.19\\\ 0.04\\\
0.04\end{bmatrix},\ C_{\mathsf{K}}\\!=\\!\begin{bmatrix}3.79\\\ 4.14\\\
-1.57\end{bmatrix}^{{\mathsf{T}}}.$
internally stabilizes ${\mathbf{G}}$. Applying Algorithm 3, we obtain an order
2 controller ${\mathbf{K}}_{r}$, which removes the last (unstable) mode of
$A_{\mathsf{K}}$, leading to
${\mathbf{K}}_{r}=\left[\begin{array}[]{c|c}A_{{\mathsf{K}},r}&B_{{\mathsf{K}},r}\\\
\hline\cr C_{{\mathsf{K}},r}&0\end{array}\right],$ with
$\quad A_{{\mathsf{K}},r}\\!=\\!\begin{bmatrix}1.37&0\\\
0&-0.37\end{bmatrix},\;B_{{\mathsf{K}},r}\\!=\\!\begin{bmatrix}0.19\\\
0.04\end{bmatrix},\;C_{{\mathsf{K}},r}\\!=\\!\begin{bmatrix}3.79\\\
4.14\end{bmatrix}^{{\mathsf{T}}},$
The truncated component ${\mathbf{\Delta}}$ is unstable and takes the form
$\displaystyle{\mathbf{\Delta}}=\left[\begin{array}[]{c|c}0.34&0.04\\\
\hline\cr-1.57&0\end{array}\right],$
which satisfies the bound 32. Thus, Theorem 3 guarantees that the reduced-
order controller ${\mathbf{K}}_{r}$ still internally stabilizes the plant.
Indeed, numerical computation shows that the LQG cost of the original
controller, $J({\mathbf{K}})$, is 343.2, while the LQG cost of the truncated
controller, $J({\mathbf{K}}_{r})$, is 58.2. In this case , modal truncation
not only yields a stabilizing lower-order controller, but also a cost of lower
LQG cost.111111For this instance, the theoretical upper bound posited in
Theorem 3 is significantly larger than the original cost.
## 8 Conclusion
We have presented on controller reduction for general non observer-based
controllers using balanced truncation and modal truncation. For SISO systems,
we demonstrate how LQG control may be performed even when there are no stable
components in the controller. We hope that our work will be useful not only
for policy optimization in LQG control but also for the controller reduction
community. Two interesting future directions are 1) extending truncation of
unstable modes to MIMO systems and 2) applying the results to escape saddle
points in the LQG policy optimization [3].
## Appendix A Background on Balanced realization
The following result, which states that any stable transfer function has a
balanced realization, is well-known from classical control theory.
###### Proposition 1 (cf. [31, Section 26.2]).
Consider a stable transfer function ${\mathbf{G}}$ with a minimal state-space
realization
${\mathbf{G}}=\left[\begin{array}[]{c|c}A&B\\\ \hline\cr
C&D\end{array}\right],$
i.e. the transfer function can be implemented in state-space as the linear
system
$\displaystyle\dot{x}=Ax+Bu,$ $\displaystyle y=Cx+Du,$
for some $A\in\mathbb{R}^{n\times n},B\in\mathbb{R}^{n\times
p},C\in\mathbb{R}^{m\times n}$ and $D\in\mathbb{R}^{m\times p}$, where
$A\in\mathbb{R}^{n\times n}$ is a stable matrix, and $(A,B)$ is controllable
and $(A,C)$ is observable. Recall that the corresponding controllability and
observability gramians $W_{c}$ and $W_{o}$ are defined as the positive
definite121212The positive definite nature of the solutions follows from the
definition of minimality. solutions to the Lyapunov equations
$\displaystyle AW_{c}+W_{c}A^{{\mathsf{T}}}+BB^{{\mathsf{T}}}=0,$ (35a)
$\displaystyle A^{{\mathsf{T}}}W_{o}+W_{o}A+C^{{\mathsf{T}}}C=0$ (35b)
respectively. Then, there exists a balanced state-space realization of
${\mathbf{G}}$ in the form
${\mathbf{G}}=\left[\begin{array}[]{c|c}\tilde{A}&\tilde{B}\\\
\hline\cr\\\\[-10.0pt] \tilde{C}&\tilde{D}\end{array}\right],$
where
$\tilde{A}=TAT^{-1},\ \tilde{B}=TB,\ \tilde{C}=CT^{-1},\ \ \tilde{D}=D$
for some invertible $T\in\mathbb{R}^{n\times n}$, such that the corresponding
controllability and observability gramians
$\tilde{W}_{c}=TW_{c}T^{{{\mathsf{T}}}}$ and
$\tilde{W}_{o}=T^{-{{\mathsf{T}}}}W_{o}T^{-1}$ satisfy the relation
$\tilde{W}_{c}=\tilde{W}_{o}=\Lambda$
for some positive diagonal matrix $\Lambda\in\mathbb{R}^{n\times n}$.
###### Proof.
First, for any invertible transformation $T\in\mathbb{R}^{n\times n}$, we note
that if we define the coordinate $\tilde{x}=Tx$, then the transfer function
${\mathbf{G}}$ can be implemented in state-space as the linear system
$\displaystyle\tilde{x}=Tx=TAT^{-1}\tilde{x}+TBu,$ $\displaystyle
y=CT^{-1}\tilde{x}+Du.$
Henceforth, for notational convenience we denote
$\tilde{A}=TAT^{-1},\tilde{B}=TB,\tilde{C}=CT^{-1},\tilde{D}=D$. The
corresponding controllability and observability gramians $\tilde{W}_{c}$ and
$\tilde{W}_{o}$ are the solutions to the Lyapunov equations
$\displaystyle
TAT^{-1}\tilde{W}_{c}+\tilde{W}_{c}(TAT^{-1})^{{\mathsf{T}}}+TB(TB)^{{\mathsf{T}}}=0,$
$\displaystyle(TAT^{-1})^{{\mathsf{T}}}\tilde{W}_{o}+\tilde{W}_{o}TAT^{-1}+(CT^{-1})^{{\mathsf{T}}}CT^{-1}=0$
By multiplying 35a on the left by $T$ and on the right by $T^{{\mathsf{T}}}$,
and similarly multiplying 35b on the left by $T^{-1}$ and on the right by
$T^{-{{\mathsf{T}}}}$, it follows that
$\tilde{W}_{c}=TW_{c}T^{{\mathsf{T}}},\qquad\tilde{W}_{o}=T^{-{{\mathsf{T}}}}W_{o}T^{-1},$
where both $\tilde{W}_{c}$ and $\tilde{W}_{o}$ are symmetric, and also
positive definite since $W_{c}$ and $W_{o}$ are positive definite. Our goal is
to find a $T$ such that $\tilde{W}_{c}=\tilde{W}_{o}=\Lambda$ for some
positive definite diagonal matrix $\Lambda$, i.e. $\tilde{W}$ and
$\tilde{W}_{o}$ are equal to each other and are a diagonal matrix with
positive entries along the diagonal. Moreover, we will see that $\Lambda^{2}$
will share the same eigenvalues as the matrix product $W_{c}W_{o}$.
Suppose such a $T$ exists. That implies then that
$TW_{c}W_{o}T^{-1}=(TW_{c}T^{{\mathsf{T}}})(T^{-{{\mathsf{T}}}}W_{o}T^{-1})=\Lambda\Lambda=\Lambda^{2}.$
Now, note that $W_{c}$ can be written in the form $W_{c}=QQ^{{\mathsf{T}}}$
for some invertible $Q$, for instance by taking its Cholesky decomposition.
Then, we have
$\Lambda^{2}=TW_{c}W_{o}T^{-1}=TQQ^{{\mathsf{T}}}W_{o}T^{-1}=(TQ)Q^{{\mathsf{T}}}W_{o}Q(TQ)^{-1}.$
This implies that $Q^{{\mathsf{T}}}W_{o}Q$ is similar to $\Lambda^{2}$, and
hence there exists an orthonormal matrix $U$ such that
$Q^{{\mathsf{T}}}W_{o}Q=U\Lambda^{2}U^{{\mathsf{T}}}$. Thus, we have
$\Lambda^{2}=(TQ)Q^{{\mathsf{T}}}W_{o}Q(TQ)^{-1}=(TQ)U\Lambda^{2}U^{{\mathsf{T}}}(TQ)^{-1}=TQU\Lambda^{-1/2}\Lambda^{2}\Lambda^{1/2}U^{{\mathsf{T}}}Q^{-1}T^{-1}.$
Set $TQU\Lambda^{-1/2}=I$, such that $T=\Lambda^{1/2}U^{{\mathsf{T}}}Q^{-1}$.
Then,
$\displaystyle
TW_{c}T^{{\mathsf{T}}}=(\Lambda^{1/2}U^{{\mathsf{T}}}Q^{-1})QQ^{{\mathsf{T}}}(\Lambda^{1/2}U^{{\mathsf{T}}}Q^{-1})^{{\mathsf{T}}}=\Lambda^{1/2}U^{{\mathsf{T}}}U\Lambda^{1/2}=\Lambda,$
$\displaystyle
T^{-{{\mathsf{T}}}}W_{o}T^{-1}=(\Lambda^{1/2}U^{{\mathsf{T}}}Q^{-1})^{-{{\mathsf{T}}}}W_{o}(\Lambda^{1/2}U^{{\mathsf{T}}}Q^{-1})^{-1}=\Lambda^{-1/2}UQ^{{\mathsf{T}}}W_{o}QU^{{\mathsf{T}}}\Lambda^{1/2}=\Lambda^{-1/2}\Lambda^{2}\Lambda^{1/2}=\Lambda.$
Hence, the desired result follows by applying the coordinate transformation
$\tilde{x}=Tx,$ where $T=\Lambda^{1/2}U^{{\mathsf{T}}}Q^{-1}$. Note that we
have not explicitly specified what the positive definite diagonal matrix
$\Lambda$ should be. To this end, we note that since $TW_{c}W_{o}T^{-1}$ is a
similarity transformation of $W_{c}W_{o}$, $\Lambda^{2}=TW_{c}W_{o}T^{-1}$
must share the same eigenvalues as $W_{c}W_{o}$. Hence one possible choice for
$\Lambda$ is choosing $\Lambda\coloneqq{\rm
diag}\left(\\{\lambda_{i}(W_{c}W_{o})\\}_{i=1}^{n}\right)$ for each $i\in[n]$.
This completes our proof.
∎
## Appendix B Useful norm identities and inequalities
For the reader’s reference, we collect several useful well-known norm
identities and inequalities that recur frequently in our proofs. The reader
may refer to [26, Chapter 4] for more details.
We begin with the following norm triangular inequalities
$\displaystyle\left\lVert{\mathbf{G}}_{1}+{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{2}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{L}_{2}}+\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{2}},\quad\quad\forall{\mathbf{G}}_{1},{\mathbf{G}}_{2},$
$\displaystyle\left\lVert{\mathbf{G}}_{1}+{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{\infty}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{L}_{\infty}}+\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{\infty}},\quad\forall{\mathbf{G}}_{1},{\mathbf{G}}_{2}.$
As a corollary,
$\displaystyle\left\lVert{\mathbf{G}}_{1}+{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{2}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{2}}+\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{2}},\quad\quad\forall{\mathbf{G}}_{1},{\mathbf{G}}_{2}\in\mathcal{R}\mathcal{H}_{2},$
$\displaystyle\left\lVert{\mathbf{G}}_{1}+{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{\infty}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}+\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{\infty}},\quad\forall{\mathbf{G}}_{1},{\mathbf{G}}_{2}\in\mathcal{R}\mathcal{H}_{\infty}$
The following result, bounding the $\mathcal{H}_{\infty}$ norm of the products
of two transfer functions using a sub-multiplicative property, is used
throughout our proofs.
###### Lemma 7.
Let ${\mathbf{G}}_{1}\in\mathcal{R}\mathcal{H}_{\infty}$. Suppose that
${\mathbf{G}}_{2}$ has finite $\mathcal{L}_{\infty}$ norm, and
${\mathbf{G}}_{1}$ and ${\mathbf{G}}_{2}$ have compatible dimensions. If
${\mathbf{G}}_{1}{\mathbf{G}}_{2}$ is stable, then
$\displaystyle\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{\infty}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{\infty}}.$
As a special case, when ${\mathbf{G}}_{2}\in\mathcal{RH}_{\infty}$ as well, we
have
$\displaystyle\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{\infty}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{\infty}}.$
###### Proof.
Suppose ${\mathbf{G}}_{1}{\mathbf{G}}_{2}$ is stable. Then, by definition, its
$\mathcal{H}_{\infty}$ norm is the same as its $\mathcal{L}_{\infty}$ norm.
Hence,
$\displaystyle\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{\infty}}=\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{\infty}}$
$\displaystyle=\sup_{w\in\mathbb{R}}\sigma_{\max}({\mathbf{G}}_{1}(jw){\mathbf{G}}_{2}(jw))$
$\displaystyle\leq\sup_{w\in\mathbb{R}}\sigma_{\max}({\mathbf{G}}_{1}(jw))\sup_{w\in\mathbb{R}}\sigma_{\max}({\mathbf{G}}_{2}(jw))$
$\displaystyle\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{\infty}}.$
The special case follows from the definition of the $\mathcal{H}_{\infty}$
norm. ∎
The next result, which bounds the $\mathcal{H}_{2}$ norm of the products of
two transfer functions, is also quite useful.
###### Lemma 8.
Let ${\mathbf{G}}_{1}\in\mathcal{R}\mathcal{H}_{\infty}$. Suppose that
${\mathbf{G}}_{2}$ has finite $\mathcal{L}_{2}$ norm, and ${\mathbf{G}}_{1}$
and ${\mathbf{G}}_{2}$ have compatible dimensions. If
${\mathbf{G}}_{1}{\mathbf{G}}_{2}$ is stable, then
$\displaystyle\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{2}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{2}}.$
(36)
In addition, when ${\mathbf{G}}_{1}\in\mathcal{R}\mathcal{H}_{2}$, and suppose
that ${\mathbf{G}}_{2}$ has finite $\mathcal{L}_{\infty}$ norm, where
${\mathbf{G}}_{1}$ and ${\mathbf{G}}_{2}$ have compatible dimensions. If
${\mathbf{G}}_{1}{\mathbf{G}}_{2}$ is stable, then
$\displaystyle\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{2}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{2}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{\infty}}.$
(37)
As a corollary, when ${\mathbf{G}}_{1},{\mathbf{G}}_{2}$ are both in
$\mathcal{R}\mathcal{H}_{2}$, we have
$\displaystyle\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{2}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{2}},\quad\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{2}}\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{2}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{\infty}}.$
###### Proof.
Suppose ${\mathbf{G}}_{1}{\mathbf{G}}_{2}$ is stable. Then, by definition, its
$\mathcal{H}_{2}$ norm is the same as its $\mathcal{L}_{2}$ norm. Suppose that
${\mathbf{G}}_{1}\in\mathcal{R}\mathcal{H}_{\infty}$ and that
${\mathbf{G}}_{2}$ has finite $\mathcal{L}_{2}$ norm. Then,
$\displaystyle\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{2}}^{2}=\left\lVert{\mathbf{G}}_{1}{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{2}}^{2}$
$\displaystyle=\int_{0}^{\infty}{\mathbf{Tr}}\left(({\mathbf{G}}_{1}(jw){\mathbf{G}}_{2}(jw))^{*}({\mathbf{G}}_{1}(jw){\mathbf{G}}_{2}(jw))\right)dw$
$\displaystyle=\int_{0}^{\infty}{\mathbf{Tr}}\left({\mathbf{G}}_{2}(jw)^{*}{\mathbf{G}}_{1}(jw)^{*}{\mathbf{G}}_{1}(jw){\mathbf{G}}_{2}(jw)\right)dw$
$\displaystyle\leq\int_{0}^{\infty}\sigma_{\max}({\mathbf{G}}_{1}^{*}(jw){\mathbf{G}}_{1}(jw)){\mathbf{Tr}}\left({\mathbf{G}}_{2}(jw)^{*}{\mathbf{G}}_{2}(jw)\right)dw$
$\displaystyle\leq\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{L}_{\infty}}^{2}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{2}}^{2}=\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}^{2}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{2}}^{2}$
Using analogous calculations we may derive the other inequalities in our
result as well. Our corollary follows from the definition of the
$\mathcal{H}_{\infty}$ and $\mathcal{H}_{2}$ norm. ∎
We end with the following inequality which follows from the small-gain theorem
[26, Theorem 9.1].
###### Lemma 9.
Suppose ${\mathbf{G}}_{1}{\mathbf{G}}_{2}\in\mathcal{R}\mathcal{H}_{\infty}$
and
$\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{\infty}}<1$.
It follows that
$\displaystyle\left\lVert(I-{\mathbf{G}}_{1}{\mathbf{G}}_{2})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\leq\frac{1}{1-\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{L}_{\infty}}}.$
As a corollary, if ${\mathbf{G}}_{2}\in\mathcal{R}\mathcal{H}_{\infty}$ as
well, then
$\displaystyle\left\lVert(I-{\mathbf{G}}_{1}{\mathbf{G}}_{2})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\leq\frac{1}{1-\left\lVert{\mathbf{G}}_{1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert{\mathbf{G}}_{2}\right\rVert_{\mathcal{H}_{\infty}}}.$
## Appendix C Technical proofs
### C.1 Proof of Lemma 2
###### Proof of Lemma 2.
It follows from the definition of the LQG cost in 5 that we have
$J({\mathbf{K}})=\lim_{t\to\infty}\mathbb{E}\left[z(t)^{\top}z(t)\right],\quad
z(t)\coloneqq\begin{bmatrix}\tilde{y}(t)\\\ u(t)\end{bmatrix}.$
Denote $\tilde{w}(t)\coloneqq\begin{bmatrix}w(t)&v(t)\end{bmatrix}^{\top}.$
Denote
$\displaystyle\mathbf{T}\coloneqq\begin{bmatrix}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\mathbf{K}\\\
\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}&\mathbf{K}(I-\mathbf{G}\mathbf{K})^{-1}\end{bmatrix}.$
Then, from 6, we have that $\mathbf{z}=\mathbf{T}\mathbf{\tilde{w}}$.
Equivalently, there exists some (proper) state-space realization
$(A_{\mathbf{T}},B_{\mathbf{T}},C_{\mathbf{T}},0)$ of $\mathbf{T}$, such that
we have the dynamical representation
$\displaystyle\tilde{\xi}(t)=A_{\mathbf{T}}\tilde{\xi}(t)+B_{\mathbf{T}}\tilde{w}(t)$
$\displaystyle z(t)=C_{\mathbf{T}}\tilde{\xi}(t).$
This implies in particular that
$\displaystyle
z(t)=C_{\mathbf{T}}\int_{\tau=0}^{t}e^{A_{\mathbf{T}}(t-\tau)}B_{\mathbf{T}}\tilde{w}(\tau)d\tau.$
Thus, assuming that $w$ and $v$ are white noise terms (so $\tilde{w}$ is also
a white noise term), we have
$\displaystyle J({\mathbf{K}})$
$\displaystyle=\lim_{t\to\infty}\mathbb{E}\left[z(t)^{\top}z(t)\right]$
$\displaystyle=\lim_{t\to\infty}\mathbb{E}\left[\left(C_{\mathbf{T}}\int_{\tau=0}^{t}e^{A_{\mathbf{T}}(t-\tau)}B_{\mathbf{T}}\tilde{w}(\tau)d\tau\right)^{\top}\left(C_{\mathbf{T}}\int_{\tau=0}^{t}e^{A_{\mathbf{T}}(t-\tau)}B_{\mathbf{T}}\tilde{w}(\tau)d\tau\right)\right]$
$\displaystyle=\mathbb{E}\left[{\mathbf{Tr}}\left(\int_{0}^{\infty}\left(C_{\mathbf{T}}e^{A_{\mathbf{T}}t}B_{\mathbf{T}}\right)^{\top}\left(C_{\mathbf{T}}e^{A_{\mathbf{T}}t}B_{\mathbf{T}}\right)dt\right)\right]$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathbf{Tr}}\left(\mathbf{T}^{*}(j\omega)\mathbf{T}(j\omega)\right)d\omega$
$\displaystyle=\left\lVert\mathbf{T}\right\rVert_{\mathcal{H}_{2}}^{2}$
where the second-to-last equality holds by Parseval’s theorem, and the final
equality follows from definition of the $\mathcal{H}_{2}$ norm. This completes
our proof. ∎
### C.2 Proof of Lemma 3
Although Lemma 3 is widely used, we cannot find a proof easily in the
literature. We provide here a proof of Lemma 3, which relies on the MIMO
version of the Nyquist stability theorem [26, Theorem 5.8] and the discussion
in [8, Section IV].
Throughout our proof below, given a transfer function $\mathbf{F}$, we use
$n_{z}(\mathbf{F})$ to denote the number of zeros of $\mathbf{F}$ in the open
right-half complex plane, and $n_{p}(\mathbf{F})$ to denote the number of
poles of $\mathbf{F}$ in the open right-half complex plane.
###### Proof of Lemma 3.
Recall that ${\mathbf{\Delta}}:={\mathbf{K}}-{\mathbf{K}}_{r}$. Without loss
of generality, it suffices for us to show the result in the case that
$\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}<1$;
the other case, when
$\left\lVert{\mathbf{\Delta}}{\mathbf{G}}(I-{\mathbf{K}}{\mathbf{G}})^{-1}\right\rVert_{\mathcal{L}_{\infty}}<1$,
follows by symmetry of ${\mathbf{K}}$ and ${\mathbf{G}}$ in the closed-loop
transfer matrix.
By [26, Theorem 5.7], to prove that ${\mathbf{K}}_{r}$ internally stabilizes
${\mathbf{G}}$, it suffices to show two items.
1. 1.
$n_{p}({\mathbf{G}})+n_{p}({\mathbf{K}}_{r})=n_{p}({\mathbf{G}}{\mathbf{K}}_{r})$,
i.e., no unstable pole-zero cancellation between ${\mathbf{G}}$ and
${\mathbf{K}}_{r}$.
2. 2.
$(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}$ is stable, i.e.
$\det((I-{\mathbf{G}}{\mathbf{K}})^{-1})$ has all its poles in the open left-
half complex plane.
We will first show that
$W(\det(I-{\mathbf{G}}{\mathbf{K}}_{r}))=W(\det(I-{\mathbf{G}}{\mathbf{K}}))$,
where $W(\cdot)$ denotes the winding number (around the origin) of a
trajectory in $\mathbb{C}$. By homotopy invariance of the winding number, if
$\det((I-{\mathbf{G}}{\mathbf{K}}+\epsilon{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r}))(s))\neq
0$ for all $s\in D$ and all $0\leq\epsilon\leq 1$, where $D$ denotes the
Nyquist contour in the right-half-plane131313This is a contour comprising a
path travelling up the $iw$ axis, from $0-i\infty$ to $0+i\infty$, and another
semicircular arc, with radius $r\to\infty$, that travels clockwise from
$0+i\infty$ to $0-i\infty$., it follows that
$W(\det(I-{\mathbf{G}}{\mathbf{K}}_{r}))=W(\det(I-{\mathbf{G}}{\mathbf{K}}))$.
We now prove that $\forall 0\leq\epsilon\leq 1$, we have
$\det((I-{\mathbf{G}}{\mathbf{K}}+\epsilon{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r}))(s))\neq
0$ for all $s\in D$.
* •
For the semicircular arc at infinity of $D$, it is not hard to see that
$\det\left((I-{\mathbf{G}}{\mathbf{K}}+\epsilon{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r}))\right)\not\equiv
0$ for any $0\leq\epsilon\leq 1$. This is because the determinant of any
transfer function is a complex polynomial and thus has finitely many roots.
Therefore it must be non-vanishing as we approach infinity.
* •
We here show that
$\det\left((I-{\mathbf{G}}{\mathbf{K}}+\epsilon{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r}))(i\omega)\right)\neq
0$ for any $\omega\in\mathbb{R}$. This is equivalent to showing
$\underline{\sigma}(I-{\mathbf{G}}{\mathbf{K}}+\epsilon{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r}))>0,$
where for any transfer function $\mathbf{F}$, we denote
$\underline{\sigma}(\mathbf{F}):=\min_{\omega\in\mathbb{R}}\,\sqrt{\lambda_{\min}(\mathbf{F}(-i\omega)^{{\mathsf{T}}}\mathbf{F}(i\omega))},$
i.e., the minimum singular value of $\mathbf{F}(s)$ over the imaginary axis.
Since
$\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r})\right\rVert_{\mathcal{L}_{\infty}}<1$,
it follows that for any $0\leq\epsilon\leq 1$,
$\displaystyle\underline{\sigma}(I-{\mathbf{G}}{\mathbf{K}}+\epsilon{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r}))$
$\displaystyle=\underline{\sigma}((I-{\mathbf{G}}{\mathbf{K}})(I+\epsilon(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r})))$
$\displaystyle\stackrel{{\scriptstyle\textnormal{(i)}}}{{\mathstrut{\geq}}}\underline{\sigma}((I-{\mathbf{G}}{\mathbf{K}}))(1-\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r})\right\rVert_{\mathcal{L}_{\infty}})$
$\displaystyle\stackrel{{\scriptstyle\textnormal{(ii)}}}{{\mathstrut{>}}}0.$
To derive (• ‣ C.2), we used the fact that
$\sigma_{\min}(AB)\geq\sigma_{\min}(A)\sigma_{\min}(B)$ and
$\sigma_{\min}(I+A)\geq 1-\sigma_{\max}(A)$ for compatible matrices. To derive
(• ‣ C.2), we used the fact that
$1-\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r})\right\rVert_{\mathcal{L}_{\infty}}>0,$
and the fact that ${\mathbf{K}}$ internally stabilizes ${\mathbf{G}}$ (which
implies that $\det(I-{\mathbf{G}}{\mathbf{K}})$ has all its zeros in the open
left-half plane), indicating that
$\underline{\sigma}((I-{\mathbf{G}}{\mathbf{K}}))>0$ over the imaginary axis.
We thus conclude that
$\displaystyle
W(\det(I-{\mathbf{G}}{\mathbf{K}}))=W(\det(I-{\mathbf{G}}{\mathbf{K}}_{r}))$
(38)
By the multivariate Nyquist criterion, we know that for any square transfer
function $\mathbf{F}$,
$W(\det(I-\mathbf{F}))=n_{z}(I-\mathbf{F})-n_{p}(\mathbf{F}).$
Since ${\mathbf{K}}$ internally stabilizes ${\mathbf{G}}$, we know that
$n_{z}(I-{\mathbf{G}}{\mathbf{K}})=0$. Thus, we have
$W(\det(I-{\mathbf{G}}{\mathbf{K}}))=-n_{p}({\mathbf{G}}{\mathbf{K}}).$
Combining this with 38, we then have that
$\displaystyle
W(\det(I-{\mathbf{G}}{\mathbf{K}}_{r}))=-n_{p}({\mathbf{G}}{\mathbf{K}}).$
(39)
Since $n_{z}(I-{\mathbf{G}}{\mathbf{K}}_{r})\geq 0$, we also know that
$\displaystyle
W(\det(I-{\mathbf{G}}{\mathbf{K}}_{r}))=n_{z}(I-{\mathbf{G}}{\mathbf{K}}_{r})-n_{p}({\mathbf{G}}{\mathbf{K}}_{r})\geq-
n_{p}({\mathbf{G}}{\mathbf{K}}_{r})$ (40)
Combining 39 and 40 leads to
$\displaystyle n_{p}({\mathbf{G}}{\mathbf{K}})\leq
n_{p}({\mathbf{G}}{\mathbf{K}}_{r}).$ (41)
Since ${\mathbf{K}}$ internally stabilizes ${\mathbf{G}}$,
$n_{p}({\mathbf{G}}{\mathbf{K}})=n_{p}({\mathbf{G}})+n_{p}({\mathbf{K}})$.
Meanwhile, by definition we have
$n_{p}({\mathbf{G}}{\mathbf{K}}_{r})\leq
n_{p}({\mathbf{G}})+n_{p}({\mathbf{K}}_{r})=n_{p}({\mathbf{G}})+n_{p}({\mathbf{K}})=n_{p}({\mathbf{G}}{\mathbf{K}}),$
where we used the assumption that
$n_{p}({\mathbf{K}})=n_{p}({\mathbf{K}}_{r})$. Continuing from 41, we then
conclude that
$n_{p}({\mathbf{G}}{\mathbf{K}}_{r})=n_{p}({\mathbf{G}}{\mathbf{K}})=n_{p}({\mathbf{G}})+n_{p}({\mathbf{K}})=n_{p}({\mathbf{G}})+n_{p}({\mathbf{K}}_{r}).$
Thus, there is no unstable pole-zero cancellation between ${\mathbf{G}}$ and
${\mathbf{K}}_{r}$.
Meanwhile, we also have that
$\displaystyle n_{z}(I-{\mathbf{G}}{\mathbf{K}}_{r})$
$\displaystyle=W(\det(I-{\mathbf{G}}{\mathbf{K}}_{r}))+n_{p}({\mathbf{G}}{\mathbf{K}}_{r})$
$\displaystyle=W(\det(I-{\mathbf{G}}{\mathbf{K}}))+n_{p}({\mathbf{G}}{\mathbf{K}})=0.$
So, $(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}$ has no pole in the closed RHP.
Since we already showed earlier that for any $0\leq\epsilon\leq 1,$
$\det(I-{\mathbf{G}}{\mathbf{K}}+\epsilon{\mathbf{G}}({\mathbf{K}}-{\mathbf{K}}_{r}))\neq
0,$
since $(I-{\mathbf{G}}{\mathbf{K}})^{-1}$ has no pole along the imaginary
axis, it follows that $(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}$ has no pole on
the imaginary axis as well. Thus, $(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}$ is
stable. This concludes our proof. ∎
### C.3 Proof of the LQG bound in 33
We here prove the results on the change in LQG performance in 33. We have
proved that the conditions in Theorem 1 are all satisfied and thus
${\mathbf{K}}_{r}$ internally stabilizes ${\mathbf{G}}$.
From Lemma 2, we know that
$J(\mathbf{K}_{r})\\!=\\!\left\lVert(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}{\mathbf{K}}_{r}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert{\mathbf{K}}_{r}(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert{\mathbf{K}}_{r}(I\\!-\\!\mathbf{G}\mathbf{K}_{r})^{-1}\right\rVert_{\mathcal{H}_{2}}^{2}.$
We proceed to upper bound the $\mathcal{H}_{2}$ norm of each sub-block in
terms of norms of the sub-blocks corresponding to ${\mathbf{K}}$.
Observe that by 13, we have that
$\displaystyle\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle=\left\lVert(I-\mathbf{X}\mathbf{\Delta})^{-1}(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq\left\lVert(1-{\mathbf{X}}{\mathbf{\Delta}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\left(\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}\right).$
(42)
Next, observe that
$\displaystyle\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}_{r}\right\rVert_{\mathcal{H}_{2}}=\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}(\mathbf{K}+\mathbf{\Delta})\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq\,$
$\displaystyle\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{K}\right\rVert_{\mathcal{H}_{2}}+\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\mathbf{\Delta}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq\,$
$\displaystyle\left\lVert(1-{\mathbf{X}}{\mathbf{\Delta}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\left(\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\mathbf{K}\right\rVert_{\mathcal{H}_{2}}\right)+\left\lVert(1-{\mathbf{X}}{\mathbf{\Delta}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{\Delta}\right\rVert_{L_{\infty}}\left(\left\lVert(I-\mathbf{G}\mathbf{K})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}\right),$
where we used 19 and an analogous calculation to 13 (but keeping the
$\mathbf{K}$ on the right) to derive the last inequality. We also used 37 in
Appendix B to bound the $\mathcal{H}_{2}$ norm of the term
$\left\lVert(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}{\mathbf{G}}{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{2}}\leq\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\left\lVert(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{2}}.$
We next consider the term $\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}$.
From 15 in the proof of Theorem 1, we have
$\displaystyle\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}={\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}+{\mathbf{K}}{\mathbf{X}}(I-{\mathbf{\Delta}}\mathbf{X})^{-1}\mathbf{\Delta}(I-\mathbf{G}\mathbf{K})^{-1}+(I-{\mathbf{\Delta}}\mathbf{X})^{-1}{\mathbf{\Delta}}(I-\mathbf{G}{\mathbf{K}})^{-1}.$
Thus, we have that
$\displaystyle\
\left\lVert\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq$ $\displaystyle\
\left\lVert{\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}\right\rVert_{\mathcal{H}_{2}}+\left\lVert(1-\mathbf{X}{\mathbf{\Delta}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\left(\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{2}}+\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{2}}\right)$
where the inequality uses 36 in Appendix B. From 15, we know that
$\displaystyle\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}{\mathbf{G}}={\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}{\mathbf{G}}+{\mathbf{K}}{\mathbf{X}}(I-{\mathbf{\Delta}}\mathbf{X})^{-1}\mathbf{\Delta}(I-\mathbf{G}\mathbf{K})^{-1}{\mathbf{G}}+(I-{\mathbf{\Delta}}\mathbf{X})^{-1}{\mathbf{\Delta}}(I-\mathbf{G}{\mathbf{K}})^{-1}{\mathbf{G}}.$
By a similar analysis to the preceding calculation for
$\left\lVert{\mathbf{K}}_{r}(I-{\mathbf{G}}{\mathbf{K}}_{r})^{-1}\right\rVert_{\mathcal{H}_{2}}$,
we have
$\displaystyle\left\lVert\mathbf{K}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{2}}$
$\displaystyle\leq$
$\displaystyle\left\lVert{\mathbf{K}}(I-\mathbf{G}{\mathbf{K}})^{-1}\right\rVert_{\mathcal{H}_{2}}+\left\lVert(1-\mathbf{X}{\mathbf{\Delta}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert(I-{\mathbf{G}}{\mathbf{K}})^{-1}{\mathbf{G}}\right\rVert_{\mathcal{H}_{2}}\left(\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}+\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\right),$
where the inequality uses 36 in Appendix B. Combining the results above, we
have that
$\displaystyle J(\mathbf{K}_{r})$
$\displaystyle=\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}{\mathbf{K}}_{r}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert{\mathbf{K}}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\mathbf{G}\right\rVert_{\mathcal{H}_{2}}^{2}+\left\lVert{\mathbf{K}}_{r}(I-\mathbf{G}\mathbf{K}_{r})^{-1}\right\rVert_{\mathcal{H}_{2}}^{2}$
$\displaystyle\leq\left\lVert(1-{\mathbf{X}}{\mathbf{\Delta}})^{-1}\right\rVert_{\mathcal{H}_{\infty}}^{2}(J(\mathbf{K})+S_{1}+S_{2}).$
where (recalling the notation that
$\mathbf{Y}=(I-\mathbf{G}{\mathbf{K}})^{-1},\mathbf{X}=(I-{\mathbf{G}}{\mathbf{K}})^{-1}$),
$\displaystyle
S_{1}=2\\!\left(\left\lVert\mathbf{\Delta}\right\rVert_{L_{\infty}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{X}\mathbf{K}\right\rVert_{\mathcal{H}_{2}}\\!+\\!\left\lVert{\mathbf{K}}\mathbf{Y}\right\rVert_{\mathcal{H}_{2}}\\!\left(\left\lVert\mathbf{\Delta}\right\rVert_{L_{2}}\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}\\!\\!+\\!\left\lVert\mathbf{\Delta}\right\rVert_{L_{\infty}}\\!\left(\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{2}}\\!+\\!\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}\\!\\!+\\!\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}\\!\right)\\!\right)\\!\right)\\!,$
$\displaystyle
S_{2}=\left\lVert\mathbf{\Delta}\right\rVert_{L_{\infty}}^{2}\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}^{2}+\left(\left\lVert\mathbf{Y}\right\rVert_{\mathcal{H}_{\infty}}\left(\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{2}}+\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{2}}\right)\right)^{2}+\left\lVert\mathbf{X}\right\rVert_{\mathcal{H}_{2}}^{2}\left(\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\left\lVert{\mathbf{K}}\mathbf{X}\right\rVert_{\mathcal{H}_{\infty}}+\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{L}_{\infty}}\right)^{2}.$
## Appendix D More numerical examples
(a) Box plot of LQG cost ratio $\frac{J(K_{r})}{J(K)}$
(b) Boxplot of $\left\lVert\Delta\right\rVert_{\mathcal{H}_{\infty}}$
Figure 2: Boxplot of LQG cost ratio $\frac{J(K_{r})}{J(K)}$ and size of
truncated component $\left\lVert\Delta\right\rVert_{\mathcal{H}_{\infty}}$ for
balanced truncation and modal truncation approaches, across 30 trials. The
lower and upper blue lines indicate the 25th percentile (Q1) and 75th
percentile (Q3) respectively. The lower and upper (shorter) black bars
indicate the minimum and maximum non-outlier values, where an outlier is any
value that lies below Q1 - 1.5 IQR or Q3 + 1.5 IQR, where IQR refers to the
interquartile range, i.e. Q3 - Q1. Note that the relative to the boxplots for
modal truncation, the boxplots for balanced truncation are significantly
narrower and concentrates about 1.
Comparing balanced truncation to modal reduction. We provide here more
comparison between the performance of balanced truncation and that of modal
truncation. To do so, we generate a random stable and minimal SISO system of
order $3$ which we denote as ${\mathbf{K}}_{<}$, and augment it with an
unstable mode ${\mathbf{K}}_{\geq}$, which has the form
${\mathbf{K}}_{\geq}=\left[\begin{array}[]{c|c}0.2&0.5\\\ \hline\cr
0.5&0\end{array}\right].$
We define the augmented SISO system ${\mathbf{K}}$ as by setting
${\mathbf{K}}={\mathbf{K}}_{<}+{\mathbf{K}}_{\geq}$; note its order is 4. We
then compute a stabilizing controller for ${\mathbf{K}}$, which we call
${\mathbf{G}}$. Treating ${\mathbf{G}}$ as the system plant (using the plant-
controller duality in SISO systems), we then perform balanced truncation and
modal reduction (each reducing the system order by 1) on the stable part of
${\mathbf{K}}$ as illustrated in Algorithm 2 and Algorithm 3, yielding the
reduced order controllers (of order 3) ${\mathbf{K}}_{r,\mathrm{bt}}$ and
${\mathbf{K}}_{r,\mathrm{mt}}$ respectively. As the box plot in Figure 2(a)
illustrates, balanced truncation tends to yield reduced-order controllers that
essentially has the same LQG cost as the original controller, while modal
truncation has a larger spread. For a reason why this might be true, consider
the boxplot in Figure 2(b), which also indicates that
$\left\lVert{\mathbf{\Delta}}_{r,\mathrm{mt}}\right\rVert_{\mathcal{H}_{\infty}}$
has significantly more variability than
$\left\lVert{\mathbf{\Delta}}_{r,\mathrm{bt}}\right\rVert_{\mathcal{H}_{\infty}}$.
This suggests a positive relationship between the size of the truncated
component $\left\lVert{\mathbf{\Delta}}\right\rVert_{\mathcal{H}_{\infty}}$,
and the deviation of the truncated controller cost from the original
controller’s cost (in either direction). For the case where
$J({\mathbf{K}}_{r})>J({\mathbf{K}})$, this is consistent with our result in
Theorem 2, which states that upper bound on
$J({\mathbf{K}}_{r})-J({\mathbf{K}})$ is tighter when
$\left\lVert{\mathbf{K}}-{\mathbf{K}}_{r}\right\rVert_{\mathcal{H}_{\infty}}$
is smaller.
For the practitioner, this suggests that it may be worthwhile to consider both
modal and balanced truncation approaches, since for some systems modal
truncation may yield relatively large improvements in LQG cost, while for some
balanced truncation may be a better choice.
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# Unified Container Shipping Industry Data From 1966: Freight Rate, Shipping
Quantity, Newbuilding, Secondhand, and Scrap Price††thanks: Declarations of
interest: none
Takuma Matsuda Suguru Otani Faculty of Commerce, Takushoku University. Email:
<EMAIL_ADDRESS>of Economics, Rice University. Email:
<EMAIL_ADDRESS>
( First version: November 29, 2022
Current version: )
###### Abstract
We construct a new unified panel dataset that combines route-year-level
freight rates with shipping quantities for the six major routes and industry-
year-level newbuilding, secondhand, and scrap prices from 1966 (the beginning
of the industry) to 2009. We offer detailed instructions on how to merge
various datasets and validate the data’s consistency by industry experts and
former executives who have historical knowledge and experience. Using this
dataset, we provide a quantitative and descriptive analysis of the industry
dynamics known as the container crisis. Finally, we identify structural breaks
for each variable to demonstrate the impact of the shipping cartels’ collapse.
Keywords: container shipping industry; exemption agreement; container crisis;
shipping cartel; container freight rate
“When we think about technology that changes the world, we think about
glamorous things like the Internet. But if you try to figure out what happened
to world trade, there is a strong case to be made that it was the container …”
– Krugman, Paul.111Citigroup Foundation Special Lecture, Festschrift paper in
honor of Alan V. Deardorff, University of Michigan IPC working paper 91, 2009
“[W]ithout [the container], the tremendous expansion of world trade in the
last forty years—the fastest growth in any major economic activity ever
recorded—could not possibly have taken place.”
– Drucker, Peter F.222Innovation and Entrepreneurship, Butterworth-Heinemann,
2007 (p.28)
## 1 Introduction
### 1.1 Research background, motivation and purpose
Container shipping is a crucial component of global trade that has
revolutionized the world. According to IHS Markit and Descartes Datamyne,
container shipping accounts for 45.4% of amount-based imports to the U.S.,
21.3% of amount-based exports from the U.S., and 10.12% of quantity-based
world trade are shipped in 2021. Additionally, the container shipping industry
offers a fascinating opportunity to explore industry dynamics. Since the
industry began global shipping operations in 1966, the market’s initial state
can be determined with a significant entry and exit of firms. Furthermore,
many countries have competition laws that exempt international cartels and
consortia, which are called “shipping conferences” and “shipping
alliances”333Transport Policy Council (2007) of the Ministry of Land,
Infrastructure and Transportation in Japan defines a shipping conference as an
agreement on freight rates and other business matters to restrain competition
among container shipping lines along the same route. A consortium is a
technical agreement between several shipping lines to form a group and operate
a joint service for the diversification of services and cost reduction in the
liner service. Consortiums include alliances that are widely used. in the
shipping industry.444There is an important difference between shipping
conferences and alliances: conferences are organized by specific routes and
directions, while alliances are formed globally. There may exist one
conference covering trade on the Transatlantic route and another covering
trade between Northern Europe and the U.S. Gulf ports. In addition, firms do
not always participate in conferences on all routes and directions they serve.
Thus, conferences are heterogeneous in their structure and membership. Despite
its significance, there is a lack of a panel dataset for the route-year-level
freight rate and shipping quantity in the container shipping industry,
particularly for the years 1966 to 1990, which limits quantitative research
during this period555As other related studies focus on the container freight
rate panel data, Luo et al. (2009) use shipping demand and freight rate data
between 1980 and 2007. As shipping demand, they used the world container
throughput reported in the Drewry Annual Container Market Review and Forecast.
The container freight rate is calculated as the weighted average of the
Transpacific, Europe-Far East, and Transatlantic trades from the same data
source. Due to data limitations, they calculated the missing period
(1980–1993) from the General Freight Index in the Shipping Statistics Yearbook
2007, using a simple statistical equation between the container freight rate
and the general freight index from 1994 to 2008.. This study provides a
unified dataset based on published books and publicly available data sources
to the extent possible, combining route-year-level data with new industry-
level data regarding shipbuilding, secondhand, and scrap prices.666The data
sources are listed in Appendix B.
To construct the freight index, we adopted the tractable imputation approach,
which involves linking multiple data sources that overlap information for some
years. This approach is simpler than formal but complicated processes or
estimation approaches that assume the AR(1) process under stationary
assumptions (Jeon 2022). We also validated our approach by presenting
interview-based evidence from ex-executive officers of shipping companies.
This confirms that our Twenty-foot Equivalent Unit (TEU)-based price data
provide a reasonable benchmark measure, even though container freight rates
were not determined by container units in the 1970s.
Using our new dataset, we have conducted an analysis of the historical
shipping price reductions in the 1980s, known as the “container crisis”
(Broeze 2002), by implementing the unknown multiple structural breaks test
(Bai and Perron 1998, 2003). It is anecdotally known that the crisis was
triggered by two events: (1) the withdrawal of Sea-Land, which was the biggest
cartel member from shipping cartels in 1980, and (2) the enactment of the
Shipping Act of 1984. The first event changed the market share of shipping
cartels, whereas the second event neutralized the shipping cartels. We also
applied the test to industry-year-level newbuilding, secondhand, and scrap
prices to determine the relationship between shipping markets and shipbuilding
markets.
Based on the test, we found that the container crisis occurred on all six
routes by 1980, which was triggered by the withdrawal of Sea-Land, although
non-U.S. routes reacted earlier than U.S. routes. Additionally, we found that
industry-year-level newbuilding, secondhand, and scrap prices displayed a
relatively stable trend during the container crisis, unlike shipping prices.
We interpret these differences as naturally capturing the distinctions between
shipping markets with shipping conferences and shipbuilding markets without
cartel groups. Thus, the container crisis was a specific event in the shipping
market.
### 1.2 Literature review
This study contributes to two strands of the literature: the historical
connection to the recent growing empirical research on the shipping industry
and the effect of an explicit cartel on the price in the shipping industry.
First, this study provides the necessary data to connect the history of the
container shipping industry from its beginning to its development after 2000,
a topic that has gained attention in the industrial organization literature
(Aguirregabiria et al. 2021). The most relevant paper in this regard is Jeon
(2022), in which she examined the relationship between demand uncertainty and
firm-market-year-level investment decisions using container shipping demand
and freight rate data for the years 1997 to 2014. To facilitate her learning-
based model, she needed to obtain the initial prices and demand in the
container industry. However, due to data limitations, she had to resort to
imputation and truncation approaches for missing shipping demand data from
1966: Q2 to 1996: Q4.777On cartel issues in a similar industry, the most
relevant paper is by Asker (2010). He studied how the presence of a cartel
affects market conduct following its dissolution and how the dissolution might
be affected by the obligations imposed on firms that seek leniency in the
tanker shipping market between 2001 and 2002. Kalouptsidi (2014), Brancaccio
et al. (2020), and Greenwood and Hanson (2015) investigated the bulk shipping
industry after 2000. Bai and Li (2021) studied the tanker market during
2017-2020 and investigated how the imbalance between the demand for and the
supply of shipping services determines congestion. Although these industries
are closely related to the container shipping industry, each of the industries
is characterized differently because the market structure and competition are
different. In addition, Kalouptsidi (2017) and Barwick et al. (2019) studied
the shipbuilding industry, which is an upstream industry for the container and
bulk shipping industries. These papers rely on the use of Clarksons Research
database and focus on the period after 2000. In the literature on
international trade, Bernhofen et al. (2016) use country-level panel data
regarding containerization adoption for the period 1962–1990. They focused on
the effect of containerization adoption on the trade quantity; therefore, they
refer to the Containerization International Yearbook only for obtaining the
first presence of container technology in each country. Rua (2014) added port-
level data to their study and investigated the diffusion of initial adoptions
of containerized transportation. Coşar and Demir (2018) exploited rich Turkish
export data and examined modal choice between containerization and breakbulk
shipping. Our study overcomes this limitation and complements the findings of
previous studies.
Second, this study examines the effect of explicit shipping cartels on
shipping prices.888The ongoing study of one of the authors of this paper
investigates the effect by constructing a structural model and focuses on the
shipping conferences as a single explicit cartel entity, and departs from the
traditional focus on tacit collusion in the literature, e.g., Porter (1983),
Bresnahan (1987), Miller and Weinberg (2017), and Byrne and De Roos (2019).
The institutional background is similar to Igami (2015)’s findings. He studied
the impact of market power on international coffee prices and evaluated the
impact of a cartel treaty on coffee prices and its global welfare consequences
under counterfactual competition regimes, that is, collusion versus
Cournot–Nash in a single homogenous good market. Shipping cartels have been
recognized in survey papers, such as Levenstein and Suslow (2006) and Asker
and Nocke (2021). Some studies, such as Morton (1997) and Podolny and Morton
(1999), examine specific shipping cartels in the U.K. in the 1800s. The most
relevant studies are Wilson and Casavant (1991), Pirrong (1992), and Clyde and
Reitzes (1998). Wilson and Casavant (1991) provided evidence of regime change
by the Shipping Act of 1984 using data on quarterly freight rates and shipping
quantities of five selected commodities only on the Transpacific route.
Pirrong (1992) tested the model prediction of the core theory surveyed in
Sjostrom (2013) using data from two specific trade routes between 1983 and
1985. Clyde and Reitzes (1998) studied the relationship between market power
and the market share of shipping conferences after the act. However, they did
not exploit cross-sectional variations, especially between non-U.S. and U.S.
routes. Our study is the first empirical research to detect the effect of
shipping cartels on the six main routes after global containerization in the
1970s, and it overcomes data limitations to complement the findings of
previous studies.
The remainder of this paper is organized as follows. Section 2 summarizes the
data and institutional background of the container shipping industry. Section
3 presents interview-based evidence from an ex-chairperson and an ex-vice
chairperson to demonstrate the consistency of our recovered data. Section 4
implements structural break tests to detect price reductions in market
variables. Finally, Section 6 presents our conclusions, and Appendix B
presents the details of data construction.
## 2 Data and Institutional Details
We provide details of the data source in Section 2.1. Next, we provide
graphical interpretations in Section 2.2 and summary statistics for the
variables in Section 2.3. We also introduce institutional details. For
clarification, in this industry, a market refers to a non-directional location
pair. For instance, if a firm operates a container ship in the eastbound of
the transpacific, it should also operate the ship in the westbound. A route
refers to a directional round-trip between two locations. For example, both
the westbound and eastbound of the transpacific. An industry refers to the
entire market consisting of all six routes.
### 2.1 Data source
We constructed route-year-level and industry-year-level data for the container
shipping industry.999Based on the firm-level data, each route is divided into
conference and non-conference routes. For example, the Transatlantic eastbound
conference route is a single route and the Transatlantic conference market is
a single market. A conference market is a market where all conference firms
conducted collusive behavior under the shipping conferences before the
Shipping Act, but have competed since the act, whereas a non-conference market
is a market where all non-conference firms have competed throughout the whole
period. In the interview with ex-practitioners in shipping companies, they
told that container freight rates for nonmember carriers are 20% to 30% lower
than conference firms on the same routes. However, as far as we know, there
was no data available on the trends in freight rates offered by the non-
conference carriers. In addition, although we checked The Japan Maritime
Daily, the newspaper of the maritime industry, we could not find any articles
that continuously reported on the level of freight rates offered by the non-
conference carriers. First, we collected route-year-level data on container
freight rates and shipping quantities. Collecting data, particularly before
1994, was not trivial because no single data source covers the period from
1966 to 1993. Thus, we needed to carefully combine data from multiple sources
carefully.101010For reference, we merge “Issues of Our Ocean Shipping”
(“Wagakuni no Gaikou Kaiun Ni Tsuite,” in Japanese), Global Container Markets
Drewry Shipping Consultants, Review of Maritime Transport, Containerization
International 1973, World Sea Trade Service, Container transportation cost and
profitability 1980/2000, The Container Crisis 1982, World Container Data 1985,
and World’s sea trades. Then, we converted merged data into TEU-based data. In
Appendix B, we provide the data source and a detailed guide with some
assumptions on data construction for each container freight rate and shipping
quantity on the six major routes: mainhaul and backhaul (separately) on the
Transpacific (Asia and North America), Transatlantic (North America and
Europe), and Asia-Europe routes. Finally, we used price and quantity data from
the six routes between 1966 and 2009. The price was adjusted according to the
CPI in the U.S. in 1995. To check the accuracy and validity of our data, in
Section 3, we provide interview-based evidence on the consistency of our
recovered data with the historical experience of industry experts, Akimitsu
Ashida and Hiroyuki Sato.
Second, we utilized industry-year-level data for newbuilding, secondhand, and
scrap prices.111111For reference, we merged Review (1971-1998) and Lloyd’s
Shipping Economist (1983-1990). Then, we converted merged data into TEU-based
data. Specifically, we used newbuilding and scrap prices per TEU between 1966
and 1998 and the secondhand price per TEU between 1968 and 1998. These prices
were adjusted to the CPI in the U.S. in 1995.
It is worth noting that recent data on newbuilding, secondhand, and scrap
prices after 1998 and shipping prices and quantities after 2009 are available
for purchase from companies such as Clarksons and IHS Markit. While our data
can be merged with proprietary data, this study does not include such data in
order to provide public access to our data.
### 2.2 Graphical interpretation
#### 2.2.1 Route-year-level shipping price and quantity data
Figure 1 illustrates the nonstationary trends in container freight rates and
shipping quantity between 1966 and 1990. The container freight rate decreased
with fluctuations, and suddenly dropped significantly, with the transition of
freight rates on the Asia-Europe eastbound and westbound routes being more
unstable than those on Transpacific and Transatlantic routes in the 1970s and
1980s. This was mainly due to the reopening of the Suez Canal in 1976, which
increased the supply of container shipping services (Saito et al. 2022).
Additionally, freight rates for the Asia to Europe trade were higher than for
other trade routes, possibly due to the strong influence of shipping
conferences on Asia to Europe trade in the 1970s and early 1980s. The figure
shows a sharp decline in freight rates in the second half of the 1980s,
possibly because of the significant impact of the conferences’ loss of power
in the early 1980s.121212In the interview with an ex-executive of a Japanese
shipping company, he pointed out the difference between conferences.
Conferences related to Asia-Europe had a strict membership screening process,
and the number of voyages by member companies was clearly defined (closed
conference). In addition, some members in the conference pooled their freight
and then redistributed them. In contrast, the conferences for Transpacific
routes were free to join or leave (open conference), and freight pooling was
explicitly prohibited under the Shipping Act of 1916 in the U.S.
The shipping quantity on all routes increased monotonically between 1973 and
1990 due to the increase in containerized cargo and the increase in the size
of ships.131313When Sea-land began container transport services in 1966, 226
35-foot containers were deployed. In the 1970s, with the development of
international maritime container transport, container vessels began to
increase in size, with vessels of approximately 2000 TEU becoming the mainstay
of liner services. Owing to increased competition following the Shipping Act
of 1984, shipping companies wanted to introduce even larger vessels. In 1988,
American President Lines (APL) opened a 39.1-meter wide, 4300 TEU vessel, the
first container vessel that could not pass the Panama Canal. Further, in the
1990s, 6000TEU-class vessels come into the container shipping market. In
response to the rapid increase in cargo exported from Asian countries,
especially China, to Europe and the U.S., the number of vessels deployed on
the Trans-Pacific East and Asia-Europe East routes has increased exponentially
since 2000. Jeon (2022) has examined these characteristics in detail since
2000.
The enactment of the Shipping Act of 1984 in the U.S. sharply divided the
conference market regime into two periods, before and after 1984, as
illustrated by the trend of the container freight rate in Figure 1. The
Shipping Act of 1984 in the U.S. made it much easier to form shipping
conferences. As a result, after 1984, the Transpacific and Transatlantic
routes saw significant price reductions due to market competition.141414This
is consistent with an interview article to ex-executives of Japanese shipping
companies (Ishida and Sato 2006) that were in charge of container trades in
the 1980s. The change in the market regime played a significant role in
shaping the container crisis.
(a) Price
(b) Quantity
Figure 1: Trends in route-year-level shipping prices and quantities.
Note: Container freight rates before 1992 refer to conference prices. The
container freight rates after 1993 are unified prices based on conference and
non-conference prices, and the difference between them is known to vanish
because of the Shipping Act of 1984. The latter are standard data often used
in the literature, such as Jeon (2022). Prices were adjusted to the CPI in the
U.S. in 1995.
#### 2.2.2 Industry-year-level newbuilding, secondhand, and scrap prices data
Figure 2: Trends in industry-year-level newbuilding, secondhand, and scrap
prices.
The data span 33 years (1966-1998) for the entire container shipping industry.
The data cover newbuilding and scrap prices between 1966 and 1998 and
secondhand prices between 1968 and 1998. All prices were adjusted according to
the CPI in the U.S. in 1995.
Figure 2 illustrates the transitions of newbuilding, secondhand, and scrap
prices. First, we observe four peaks in the newbuilding price in 1974, 1980,
1987, and 1991, indicating that the newbuilding price peaks at similar times
to the shipping price. Surprisingly, the shipping price, after adjusting CPI,
did not decrease significantly, which means that the trend of the newbuilding
price did not decrease unlike the shipping price. This is a new finding
because it is commonly known that vessels themselves were expensive in the
1970s and the shipbuilding price per TEU decreased in the 1990s due to the
gradual increase in the size of ships. We find that the increasing CPI
corresponds with a seemingly decreasing shipbuilding price in the nominal
sense.
Second, the secondhand price sharply decreased between 1974 and 1979. However,
after 1980, it remained relatively stable compared to shipping and newbuilding
prices. In the 1970s, when full container vessels started operating on liner
routes, the number of vessels was insignificant, leading to a small number of
vessels traded in the secondhand market, which resulted in more fluctuation in
secondhand prices. However, since the 1980s, the number of vessels traded in
the secondhand market has increased, leading to more stable secondhand freight
rates and a more significant correlation with new building prices.
Third, the scrap price remained stable, without significant fluctuations
relative to other price variables. Moreover, the scrap price level was smaller
than that of the newbuilding and secondhand prices because the scrap price is
linked to the input price, such as the steel price. Thus, the trend in scrap
price reflects the dynamics of the input price.
### 2.3 Summary statistics
#### 2.3.1 Route-year-level shipping price and quantity data
Table 1: Summary statistics.
| N | mean | sd | min | max
---|---|---|---|---|---
Price ($ per TEU): $P_{rt}$ | 240 | 2105.35 | 1250.84 | 561.86 | 6654.79
Quantity (1 mil TEU): $Q_{rt}$ | 240 | 2.34 | 2.75 | 0.00 | 17.70
(a) Route-year-level variables (1966-2008)
| N | mean | sd | min | max
---|---|---|---|---|---
Newbuilding price ($ per TEU): $P_{t}^{new}$ | 33 | 20633.82 | 5555.70 | 13186.28 | 31966.62
Secondhand price ($ per TEU): $P_{t}^{second}$ | 31 | 18371.82 | 14502.46 | 7020.96 | 46792.94
Scrap price ($ per TEU): $P_{t}^{scrap}$ | 33 | 609.97 | 196.54 | 334.25 | 1098.38
(b) Industry-year-level variables: $(P_{t}^{new},P_{t}^{scrap})$ (1966-1998)
and $P_{t}^{second}$ 1968-1998)
Note: Panel (a) of the six routes covers a data span of 44 years (1966-2009).
The Transatlantic routes opened in 1966, Transpacific routes opened in 1967,
and Asia-Europe routes opened in 1971. The shipping prices before 1992 in
Panel(a) refer to conference prices, while the shipping prices after 1993 are
unified prices based on conference and non-conference prices. The difference
between them vanished because of the Shipping Act of 1984. The latter are
standard data that are often used in the literature such as Jeon (2022). Panel
(b) of the container shipping industry covers a data span of 33 years
(1966-1998). The data cover newbuilding and scrap prices between 1966 and
1998, and secondhand prices between 1968 and 1998. All prices were adjusted
according to the CPI in the U.S. in 1995.
Panel (a) in Table 1 presents the summary statistics of the route-year-level
variables for the shipping price $P_{rt}$ and quantity $Q_{rt}$. The data
cover the Transatlantic routes that opened in 1966, the Transpacific routes
that opened in 1967, and the Asia-Europe routes that opened in 1971.151515For
Table 1, the shipping prices before 1992 refer to conference prices, while the
shipping prices after 1993 are unified prices based on conference and non-
conference prices, while the difference between them vanished in the
Containerization International Yearbook. Thus, we observe the full history of
the conference market. On average, the shipping price was $2105.35 per TEU and
the shipping quantity was 2.34 million TEUs.
#### 2.3.2 Industry-year-level newbuilding, secondhand, and scrap prices data
Panel (b) in Table 1 shows the summary statistics of industry-year-level
variables for the newbuilding price $P_{t}^{new}$, scrap price $P_{t}^{scrap}$
between 1966 and 1998, and secondhand price $P_{t}^{second}$ between 1968 and
1998. The average newbuilding price per TEU was $ 20633.82. The average
secondhand price per TEU was $18371.82, which was less than the newbuilding
price after 1971. The average scrap price per TEU was $609.97.
### 2.4 Institutional details
Here, we introduce the institutional background of the shipping conference,
the early history of the container shipping industry, and two key events
triggering the change in competition regime.
Table 2 summarizes the historical events of the liner shipping industry before
and after global containerization.
Table 2: The historical background of the liner shipping industry.
Timing / Period | Event | Related Study
---|---|---
1875 | the first liner shipping conference (The U.K.-Calcutta) established | Morton (1997)
| | Podolny and Morton (1999)
1916 | The Shipping Act of 1916 prohibited deferred rebate |
1918 | The end of WW1 | Deltas et al. (1999)
1936 | U.S. Maritime Commission established |
1945 | The end of WW2 |
1956 | The world’s first container ship sailed |
1961 | The 1961 Amendment enacted |
| Federal Maritime Commission (FMC) established |
| Technical Committee on Cargo Containers established (ISO/TC104) |
1964 | ISO adopted seven sizes of containers as ISO standards |
1966 | The first Full-container (FC) ship sailed the Transatlantic route |
1967 | The first FC ship sailed the Transpacific route |
1971 | The first FC ship sailed the Asia-Europe route |
1973 | Oil shock |
1974 | The UNCTAD Code of Conduct for Liner Conferences adopted | Fox (1992, 1995)
1980 | Withdrawal of Sea-Land from shipping conferences | Sjostrom (1989)
1983\. April | The UNCTAD Code of Conduct for Liner Conferences accomplished |
1984\. June | The Shipping Act of 1984 in the U.S. enacted, conference breakdown | Clyde and Reitzes (1995, 1998)
| | Wilson and Casavant (1991)
| | Pirrong (1992)
1985.Sep | Yen appreciation due to the Plaza Accord |
1991 | Strategic alliance boom started |
1998\. Oct | The Ocean Shipping Reform Act (OSRA) of 1998, effective in 1999 | Reitzes and Sheran (2002)
| | Fusillo (2006, 2013)
* a
Note: This table is based on unified survey papers such as Sjostrom (2004,
2013) and Martin (2012) and specific papers (Clyde and Reitzes 1995, 1998).
Industry and legislative changes after 1990 are beyond the scope of this
study. For reference, see Reitzes and Sheran (2002), which states that “OSRA
alters the role of the Federal Maritime Commission as a cartel enforcer. Under
the Shipping Act of 1984, all carriers, both conference carriers, and
independent carriers, had to file their tariffs with the FMC. The FMC then
policed these rates, issuing fines to carriers that engaged in secret
discounting, known as “rebating.” Under the OSRA, carriers are obliged to make
their freight rates publicly available, but the FMC’s enforcement obligations
are eliminated. OSRA’s elimination of tariff-filing requirements and rate
enforcement by the Federal Maritime Commission raises the cost of monitoring
their members’ pricing activities. (page 56).”
#### 2.4.1 Shipping conferences
The world’s first shipping conference, the “Calcutta Conference,” was formed
in 1875 on the route between England and Calcutta to establish a unified
freight rate.161616In this subsection, we refer to The Actual State of
Competition in Oceangoing Shipping and Problems with Competition Policy
(“Gaikoukaiun no Kyousoujittai To Kyousouseisakujou no Mondaiten ni Tsuite,”
in Japanese) reported by Study Group on Government Regulation and Competition
Policy (2006) of the Japan Fair Trade Commission, Branch and Stopford (2013),
and Sjostrom (2013). In particular, Sjostrom (1989) discusses the economic
insights of the shipping conferences from an historical viewpoint. Morton
(1997) provide detailed historical evidence on British shipping industry
before 1900. In 1879, the Chinese Alliance, the ancestor of the Far East
Freight Conference, was formed on routes between Asia and
Europe.171717Sjostrom (2004) conducted a survey on shipping conferences, and
pointed out that systems similar to shipping conferences had existed in
Atlantic shipping and British coastal shipping before the Calcutta Conference.
The shipping conferences appeared 15 years before the enactment of the Sherman
Antitrust Act in 1890, one of the first competition laws in the modern world,
so cartels themselves were not illegal at that time. Table 2 summarizes the
history of the liner shipping industry before 2000, with related studies and
key legislative changes.
##### Internal mechanism to conference firms.
The internal mechanism of conference firms is mainly aimed at market
stabilization by controlling entry via excess capacity (Fusillo 2003),
predatory pricing (Morton 1997, Podolny and Morton 1999), price discrimination
(Fox 1992, Clyde and Reitzes 1998), and loyalty contracts (Marin and Sicotte
2003), among other things. To achieve this, shipping conferences agreed on
various matters, in addition to freight rates. The content of the agreements
covered: (1) alternatives to suppress freight rate competition among member
shipping companies, (2) alternatives to prevent shippers from moving to non-
conference shipping companies, and (3) alternatives to directly exclude non-
conference vessels.
To avoid freight rate competition, rate agreements and vessel allocation
agreements were concluded among the member shipping companies. Rate agreements
are signed by members to agree on rates for each product and to update the
rates jointly. Vessel allocation agreements adjust the amount of tonnage to be
allocated, the number of voyages, ports of call, operation schedules, and
cargo to be loaded. These features were modeled as price and quantity-fixing
cartels.181818For example, Clyde and Reitzes (1998) model the market with the
shipping conference as a collusion equilibrium.
##### External mechanism for non-conference firms.
The conference tariff rates, or freight rates, determined by these agreements
had been publicly noticed, and no entry restrictions imposed non-conference
container shipping market. The market was, however, subject to competition
from non-conference vessels and new entrants. Consequently, shipping
conferences introduced the Dual Rate System, the Fidelity Rebate System, and
the Differed Rebate System to ensure effectiveness of freight rate agreements
and prevent shippers from flowing to non-conference shipping companies that
offer lower rates than the conferences’ rates.191919In the Dual Rate System, a
shipper and a shipping conference conclude an exclusive patronage
contract/loyalty agreement and provide transportation service for the specific
route. The contract rate is lower than the spot rate on the condition that the
shipper uses only conference member carriers’ service within a specific
contract period. The Fidelity Rebate refunds a portion of the freight if the
shipper uses only the conference carriers within a specified period (4-6
months). The Differed Rebate System is also an incentive to use conference
carriers. Under the system, if a shipper had used only members’ services for a
specified refundable period (4-6 months), and if it does not use any non-
conference members’ service for the deferment period following the refundable
period, a certain amount of money is refunded upon the shipper’s request. The
refund amount was usually around 10% of the freight. Fox (1992) uses the U.S.
port pair-level data in 1977 to examine the effect of the dual rate contract
and consumer loyalty. These entry deterrents promoted the stability of freight
rates and liner services.202020For instance, Marin and Sicotte (2003) examine
the economic effects of exclusive contracts of ocean shipping cartels during
the 1950s between firms and the ultimate consumers of their product. They
record that, “During the congressional investigations of shipping conferences
in the late 1950s and early 1960s documents obtained from an ocean carrier
contained an admission that ‘the entire contract system is a fighting measure
to get rid of outside competition’ ” (p.198).
Conferences also utilized “fighting ships,” which are vessels that temporarily
put into service at a similar schedule as non-conference shipping companies
and at lower rates.212121See Marshall and Marx (2014) (page 148) and
Harrington Jr et al. (2018) for reference to put the strategy of shipping
conferences in general cartel literature. Harrington Jr et al. (2018) classify
the general response of cartels to the expansion of noncartel supply into four
strategies: takeover, starvation, coercion, and bribery. The fighting ships
are classified into coercion strategy. Fighting ships were used to force non-
conference shipping companies to leave routes, and all members shared the
associated operational losses.
Since the 1960s, the nature of shipping alliances have changed. Technological
innovation centered on containerization, and pro-competitive amendments to
shipping laws in the United States, particularly, have had an impact on the
functioning of shipping alliances and market competition in the shipping
market.
#### 2.4.2 The inception of the container shipping industry
The history of container shipping began with Malcolm P. McLean, the founder of
the U.S. Land Transportation Company Sea-Land Service.222222Bernhofen et al.
(2016) and Rua (2014) explain the detailed history of the container shipping
industry from the viewpoint of the global trade and country-level development.
Levinson (2016) provides an overview of the industry with anecdotal and
qualitative evidence. The world’s first container ship sailed from the Port of
Newark, New Jersey, to the Port of Houston, Texas in 1956. The first
international container ship was employed by Sea-Land Service for the
Transatlantic Route in 1966, and for the Transpacific route in 1967. However,
for the Asia-Europe route, the first international semi-container ship,
Cornelia Maersk, was employed in 1967. The first international full-container
ship, Kamakura-maru, was delayed in 1971.
Global containerization induced several market changes between 1966 and 1990,
including transforming the cost structure, lowering barriers to entry,
stimulating the rise of non-conference shipping companies in developing
countries, shifting from “closed” conferences to “open” conferences, and
forming a consortium. Appendix A provides further details on each of these
changes.
#### 2.4.3 Withdrawal of Sea-Land from shipping conferences in 1980
In the late 1970s, the increase in competition for Transpacific routes led to
pressure on shipping rates. To address this, Sea-Land introduced eight SL-7
high-speed container vessels with a speed of 33 knots. However, the high
operation costs of these vessels negatively impacted the company’s
profitability. Consequently, Sea-Land withdrew from shipping conferences to
avoid tonnage-based cargo allocation imposed by the conferences and secure
profitability. Former executives of a Japanese shipping company stated that
Sea-Land had to increase loaded cargo but could not achieve its goals without
leaving the conference that set freight rates and that it felt it had no
choice but to become a non-alliance carrier and establish its own freight rate
to retain customers (Ishida and Sato 2006).
After Sea-Land’s withdrawal from shipping conferences, shippers with contracts
with conference carriers hesitated paying the penalty to switch to Sea-Land
under the conference’s Dual Rate System. Sea-Land focused on import shippers
and intermodal cargoes not covered by the double-freight rate system. The
company also paid the freight cost of returning inland containers to the port
instead of passing them on to shippers, decreasing freight rates.
#### 2.4.4 The Shipping Act of 1984 in the U.S.
In June 1984, the United States enacted the new Shipping Act as part of the
Reagan administration’s deregulation policy to allow member carriers to make
individual agreements with shippers on freight rates and services, and to
unbind shippers from shipping conferences to enable them to make more
appropriate choices in shipping companies. This drastically changed the
competition regime.232323Wilson and Casavant (1991) provide anecdotal evidence
and a case study of the effect of the Shipping Act of 1984.
The Shipping Act of 1984 included the right to Independent Action (IA),
allowing member firms to define freight rates or services that deviate from
the conference tariff rates and guaranteeing of members carriers in the
shipping conferences to set their own rates or services.
It required conferences to allow the right to form service contracts (SCs),
which are contracts in which the shipper commits in advance to load a specific
quantity or more of cargo to the shipping company during a specific period.
Under this contract, the shipping company reserves the space necessary to
carry cargo and applies discounted freight rates242424The minimum number of
containers promised by the shipper to the shipping company is called the MQC
(Minimum Quantity Commitment)..
The act explicitly prohibited the Dual Rate System and the shipping conference
lost its binding power over shippers, and individual shipping companies
frequently exercised their right to independent action, which encouraged
competition and led to a significant decline in freight rates on U.S.-related
routes.252525Japan Maritime Center (2008) pointed out that the Dual Rate
System was the most effective way in which shipping companies kept their
shippers when the conference system was functioning.
##### Stabilization agreement.
Containerization and the Shipping Act of 1984, weakened the price-binding
power of shipping conferences by increasing contracts based on IAs and SCs.
Shipping companies thus set their freight rates, leading to intense price
competition and sharp drops in Transpacific freight rates. To stabilize
shipping routes, conference shipping companies formed a “stabilization
agreement” inviting most shipping companies, including non-conference ones.
Agreements shared information on supply and demand trends and agreed
guidelines for rate restoration and surcharges, but binding power on rates.
The Transpacific Stabilization Agreement (TSA) was formed in 1989 by 13
shipping companies to stabilize shipping routes from Asia to North
America.262626TSA was dissolved in 2018.
## 3 Interviews
In this section, we provide interview-based evidence on the consistency of our
recovered data on container freight rates with the historical experience of
industry experts, Akimitsu Ashida and Hiroyuki Sato.272727We obtained
additional interview-based evidence. Mikio Tasaka, who belonged to the Nittsu
Group and worked in the U.S. in the 1980s, stated that there was no
discrepancy in the development of freight rates. In addition, Carolyn Almquist
experienced pricing and conference division in the container shipping business
at American President Lines, stated that provide a reasonable basis for
analysis. Professor Yutaka Yamamoto (University of Nagasaki), who had 20 years
of experience in the container shipping business at American President Lines,
stated that the trend was generally reasonable. We also received responses
from Kwon Oh In who worked for 40 years at Korea Maritime Transport Company, a
major Korean container shipping company. He said that our graphs seemed
convincing as a general trend of the container shipping industry during that
period.
### 3.1 Akimitsu Ashida, an ex-chairperson of Mitsui O.S.K Lines
Concerning the increase in freight rates in Transatlantic and Asia-Europe
routes in the 1980s, Mr. Akimitsu Ashida, former chairman of Mitsui O.S.K.
Lines (MOL), answered about the situation on liner routes. He served as the
company’s European Division Manager from 1985-86 and responded to our e-mail
inquiry on February 28, 2022. We asked him about his thoughts and related
memory on the figures and tables.
Regarding Shipping Quantity in Asia-Europe routes
Ashida: Unlike Transpacific routes (where U.S. Customs publishes data),
European routes did not have a system whereby customs authorities published
statistics on container transport volumes. Therefore, the FEFC did not have to
compile and notify its members of actual container transport volumes.
Consequently, it is not easy to find the data even today.
Regarding Freight Rates in Asia-Europe routes
Ashida: Freight rates were rising regularly in nominal terms. One of the
reasons was that shipping conferences had relatively strong power on Asia-
Europe routes. In addition, surcharges such as the Bunker Adjustment Factor
(BAF) and Currency Adjustment Factor (CAF) were collected without fail. Under
these circumstances, the Plaza Accord of 1985 raised the yen-dollar exchange
rate from 240 yen to approximately 140 yen. This caused dollar-based freight
rates to rise sharply, especially for cargoes originating in Japan. Non-
Japanese carriers earned even higher profits because their yen-based costs
were relatively small.
The Reason Why the Shipping Conferences Were Still Functioning on the Asia-
Europe Routes
Ashida: One reason is that there were no powerful non-conference member
shipping lines. At that time, Japanese shippers avoided non-conference members
when exporting cargo from Japan. The exports accounted for 50 % of all cargo
shipped from Asia. Therefore, the export cargo was mainly assigned to
conference member firms. This trend was disrupted by the introduction of large
newly built vessels by non-conference members from 1990 onward.
I recall that about 35 % of cargo on the Asia-Europe routes was handled
through forwarders, unlike the Transpacific routes. Furthermore, the Asia-
Europe routes were less active in discount negotiations than the Transpacific
routes because the inland transport distances were shorter and lower ocean
freight rates would reduce the profit for the company.
### 3.2 Hiroyuki Sato, an ex-vice president of Mitsui O.S.K Lines
We interviewed Mr. Hiroyuki Sato, former vice president of MOL, about the
shipping conferences in the Transpacific routes and the situation in liner
routes in the 1970s and the 1980s. He had been in charge of sales for Asia-
Europe service routes from 1969 and Transpacific service routes from 1974. We
conducted onsite interview with Mr. Sato on November 17, 2021 and our e-mail
inquiries were responded by him on December 15, 2021 and February 28, 2022. We
asked him about his thoughts on the figures and tables used in this study.
Regarding Freight Rate
Sato: In the 1970s, containerization was progressing on all routes, but
freight rates were determined based on weight or measured in the same way as
conventional vessels, instead of box rates (rates per container). There was
also an arrangement called “minimum revenue per container.” In addition, a few
percent of discounts were applied to long-term contract shippers using a
double freight contract with the shipper.
In the 1980s, there was a shift to box rates; the enactment of the Shipping
Act of 1984 was a significant reason for this shift. The period after the Act
was when space charters were no longer viable, leading to the formation of
alliances in the 1990s.
## 4 Results of Structural Breaks Test
We test multiple unknown structural breaks for each route using Bai and Perron
(1998, 2003), to confirm anecdotal evidence of container crises between the
1970s and the 1980s, documented in Broeze (2002), and to find when the
container crisis started and how long it lasted. Methodologically, we follow
Bai and Perron (2003) to address the problem of estimating the break dates and
the number of breaks and present an efficient algorithm.282828The most
relevant paper is Fan and Yin (2016), which applies Bai and Perron (2003)’s
method to the semi-annual data on the newbuilding price index, the time
charter rate index, and the second-hand price index for each ship size (i.e.,
Feeder, Feedermax, Handy, Sub-Panamax, and Panamax) between October 1996 and
July 2013. They focus on the unknown structural breaks in the relationship
between the three abovementioned global-level indices, whereas we are
interested in the unknown structural breaks in the route-level container
freight rate corresponding with competition regime changes.
We consider a multiple linear regression with $k$ breaks ($k+1$ regimes) as
follows:
$\displaystyle y_{t}=z_{t}^{\prime}\delta_{j}+u_{t}\quad
t=T_{j-1}+1,\ldots,T_{j}$ (1)
for $j=1,\ldots,k+1$. In this model, $y_{t}$ is the targeted price at time
$t$, $z_{t}(q\times 1)$ is a vector of $q$-dimensional covariates,
$\delta_{j}(j=1,\ldots,k+1)$ is the corresponding vector of coefficients, and
$u_{t}$ is the disturbance at time $t$. We allow serial correlation in the
errors and variances of the residuals across segments. The indices
$\left(T_{1},\ldots,T_{k}\right)$, that is., the breakpoints, are explicitly
treated as unknown (we use the convention that $T_{0}=0$ and $T_{k+1}=T$ ).
The purpose of this section is to estimate the unknown regression coefficients
together with the breakpoints and the number, therefore we specify $z_{t}=1$
as the simplest case. Bai and Perron (2003) suggested that the trimming
parameter $\varepsilon=h/T=0.10$ holds where $h$ is the minimum distance
between each break. Following Bai and Perron (2003), we assume a minimum
length equal to 10% of the sample size, given that we allow for a maximum of
four breaks. Our data specify $T=40$ (i.e., yearly data between 1966-2007),
thus we impose $h=4$ for the route-year-level analysis of the shipping price.
For consistency, we also impose $h=4$ for the industry-year-level analysis of
newbuilding, secondhand, and scrap prices.
### 4.1 Route-year-level shipping price data
| BIC: $m$ = 0 | $m$ = 1 | $m$ = 2 | $m$ = 3 | $m$ = 4 | $m$ = 5 | $m$ = 6 | $m$ = 7 | $m$ = 8
---|---|---|---|---|---|---|---|---|---
Transatlantic westbound | 647.86 | 555.88 | 538.08 | 538.10 | 538.89 | 538.95 | 545.65 | 552.91 | 560.26
Transatlantic eastbound | 685.71 | 620.56 | 596.29 | 582.23 | 581.03 | 584.06 | 587.28 | 592.78 | 598.48
Transpacific westbound | 679.59 | 609.32 | 592.63 | 568.14 | 568.96 | 573.59 | 579.50 | 586.58 | 593.87
Transpacific eastbound | 714.70 | 640.64 | 599.18 | 592.70 | 592.22 | 595.82 | 600.01 | 606.82 | 614.56
Asia to Europe | 684.17 | 620.42 | 593.91 | 567.82 | 571.46 | 576.96 | 584.01 | 591.20 | 627.68
Europe to Asia | 631.38 | 581.33 | 553.76 | 539.17 | 545.08 | 551.75 | 558.62 | 566.59 | 588.79
Figure 3: The estimated breakpoints and 95 % confidence intervals with BIC.
* a
Note: Each segment shows the estimated breakpoints and the 95% confidence
intervals for $\hat{T}_{i}$ for $i=1,\cdots,4$. Following Bai and Perron
(2003), the number of breakpoints is selected by minimizing BIC, as shown in
the bottom table. The estimated parameters $\hat{\delta}_{j}$ and standard
errors for all $j=1,\cdots,k+1$ for all possible numbers of breakpoints are
omitted. We use the sample between 1968 and 2008 for the Transatlantic and
Transpacific routes and the sample between 1971 and 2008 for the Asia-Europe
routes. We used the strucchange R-package developed by Zeileis et al. (2002).
Figure 3 illustrates the estimated breakpoints $\hat{T}_{j}$ and 95%
confidence intervals for route-year-level analysis of shipping freight rates.
The bottom table shows BIC for all possible number of breaks under $h=4$,
where the minimizer of the BIC is the optimal break number $\hat{k}$. First,
we find that the 95% confidence intervals of the structural break cover the
period between 1979 and 1980 on the U.S. routes (left and center panels),
whereas the intervals cover the period between 1976 and 1978 on the non-U.S.
routes (right panels). This indicates that the container crisis occurred on
all six routes by 1980, triggered by the withdrawal of Sea-Land, although
non-U.S. routes reacted earlier than U.S. routes. The reason for the response
in non-U.S. routes may be related to the reopening of the Suez Canal in 1976,
which increased the supply of container shipping services (Saito et al. 2022).
Second, we find remarkable differences between U.S. routes and non-U.S. routes
between 1984 and 1990. The 95% confidence intervals of the structural break
cover some period between 1984-1990 on the U.S. routes, whereas there were no
breaks on the non-US routes in the period. This implies that the enactment of
the Shipping Act of 1984 generated some structural breaks in the treatment
group — the U.S. routes. Surprisingly, we also find that the container crisis
lasted heterogeneously, even along the U.S. routes. For example, Transpacific
routes seemed to recover in 1986, whereas Transatlantic routes remained sour
until 1990. This result may support the fact that the supply-demand
relationship on the Transpacific and Transatlantic routes changed differently
in the late 1980s. The substantial change in exchange rates after the Plaza
Accord in 1985 led to a recovery in the demand for transportation on the
Transpacific route. However, no such situation was observed between Europe and
the U.S.
### 4.2 Industry-year-level newbuilding, secondhand, and scrap prices data
| BIC ($m$ = 0) | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
Newbuilding | 668.72 | 668.35 | 665.16 | 661.39 | 663.70 | 667.99 | 674.65 | 694.26
Secondhand | 687.91 | 610.21 | 609.25 | 606.64 | 612.79 | 619.02 | 625.65 | -
Scrap | 448.16 | 446.64 | 440.37 | 443.81 | 440.36 | 446.38 | 452.99 | 470.14
Figure 4: The estimated breakpoints and 95 % confidence intervals with BIC.
* a
Note: Each segment shows the estimated breakpoints and the 95% confidence
intervals for $\hat{T}_{i}$ for $i=1,\cdots,4$ for each route. Following Bai
and Perron (2003), the number of breakpoints is selected by minimizing BIC, as
shown in the bottom table. The estimated parameters $\hat{\delta}_{j}$ and the
standard errors for all $j=1,\cdots,k+1$ for all possible numbers of break
points are omitted. We use samples from 1968 to 1998 and the strucchange
R-package developed by Zeileis et al. (2002).
Figure 4 illustrates the estimated breakpoints $\hat{T}_{j}$ and 95%
confidence intervals for industry-level analysis of newbuilding, secondhand,
and scrap prices. First, we find that 95% confidence intervals of the
structural break cover the period between 1974 and 1976 for newbuilding and
secondhand prices. These results correspond with the shipping price trend in
the Asia-Europe market. Second, downward structural breaks in prices were not
detected between 1984 and 1990. These results imply that, unlike route-level
shipping prices, the container crisis did not significantly decrease industry-
level prices in the shipbuilding market. We interpret these differences as
naturally capturing the differences between shipping markets with shipping
conferences and those without cartel groups.
## 5 Practical Implications, Discussion, and Future Research
### 5.1 Practical implications
Our study makes an important contribution to the development of consistent
data on the history of the container shipping industry and policy discussions
for practitioners. The data contribution enables practitioners to obtain
empirical knowledge on the main container transport markets, as well as to
develop a methodology to construct container market trends on other routes,
where information such as freight rates and vessel volumes deployed is only
partially available. For example, container transport market trends on routes
between the Indian subcontinent, Africa and Europe, South East Asia and the
U.S. are important information for manufacturing companies to establish their
supply chains and respond to specific policies, but the available information
is fragmented. Thus, our data contribution sheds light on the calculation of
partially known logistics costs for shippers.
### 5.2 Discussion
We summarize the potential concerns for the data and structural break tests.
First, our imputation approach is ad hoc, which potentially makes the data
sensitive to other imputation approaches or assumptions. However, we believe
that our approach is the best alternative among feasible approaches because
other data-driven imputation approaches require a large sample size, several
overlapping years, and the no-structural-break assumption of the data, which
our data does not satisfy (discussed in Section 2.4 and Appendix B). Second,
we may face a small sample problem for the structural breaks tests because we
use route-year-level data between 1968 and 2008. If route-quarter-level data
for a longer period can be consistently constructed, the results of the
structural breaks may change.
Regarding the possibility of combining our data with data from 2009 and
beyond, we have a couple of reasons for not doing so. First, our main data
source, Review of Maritime Transport, only provides route-level price data up
until 2009. Second, there are some challenges with merging our data with
Drewry’s data for a fee since their price data is based on a different
observation unit than ours. Drewry’s data includes monthly port-to-port level
price data, while our data is route-year-level. Third, the industry
experienced significant changes around 2009, such as the Global Financial
Crisis and the Repeal of the EU’s competition law exemption for shipping
conferences in October 2008. These industry changes not only make it
inappropriate to simply merge data from 2009 and beyond with our existing
data, but they can also contaminate the results of structural break tests.
These concerns are beyond the scope of this study and are left for future
research.
### 5.3 Future research
Our data are useful for understanding the related historical policy debates in
the container shipping industry. In ongoing research, one of the authors merge
our data with the firm-market-level shipbuilding data between 1966 and 1990 ;
then construct a dynamic structural model of firms’ entry, exit, and
shipbuilding investment decisions in homogenous good markets under collusive
and perfect competition regimes, which are exogenously determined by the
enactment of the Shipping Act of 1984. Understanding the effect of shipping
conferences based on the structural model makes the specific features of the
inner allocation mechanism of shipping conferences contributable to the
general literature on competition law and industrial policy which have been
investigated in the shipbuilding and shipping industry (Kalouptsidi 2017,
Barwick et al. 2019). Thus, our data sheds light on the historical questions
of future research in the container shipping industry.
Our data are also useful for understanding the current policy debates in the
container shipping industry. For example, we merge our data with post-1990
firm-market-level shipbuilding data to evaluate the effects of EU competition
law exemptions (by Regulation (EEC) No 4056/86) for liner shipping firms,
which terminated in 2008, on merger waves after 1990. Jeon (2022) uses data
from 2006 to 2014 and conducts counterfactuals with respect to competition and
industry consolidation by simulating the industry under a multi-plant
monopolist and a merger of the top two firms. Kalouptsidi (2017) and Barwick
et al. (2019) investigated the Asian shipbuilding market in the 2000s to study
China’s expansion. Thus, our data fill the gap between prior studies focusing
on the 2000s and the 2010s and future research examining the 1980s and the
1990s.
Our data provides a historical benchmark of shipping prices for updated issues
in the container shipping industry. First, our data enables us to make
quantitative comparisons between price increase during Covid period and
historical prices that were under active shipping conferences. For instance,
during the Covid period, the highest peak of shipping prices per TEU from
Shanghai to Los Angeles was recorded in February 2022, adjusted according to
the CPI in the U.S. in 1995, and amounted to $4462. This level is comparable
to the highest peak of cartel prices observed in the Transpacific eastbound
route, as illustrated in Figure 1. This will offer valuable insights into how
prices have evolved over time, as well as how significant the shipping price
increase during Covid was. Second, our 1966–2009 data provides a helpful
benchmark for predicting the impacts of significant industry changes that are
expected to occur, such as the updates to the EU competition law exemption for
consortium in 2024 and the breaking of the 2M alliance between Mediterranean
Shipping Company (MSC) and Maersk in 2025. By using our data, we can gain a
better understanding of the potential effects of these changes on shipping
prices.
## 6 Conclusion
This study provides a new unified panel dataset of route-year-level freight
rate and shipping quantities for the six major routes and industry-year-level
newbuilding, secondhand, and scrap prices from 1966 (beginning of the
industry) to 2009. The data provide a fundamental basis for understanding the
container shipping industry. The data and structural break tests provide
historical insights into the nonstationary dynamics of these price variables,
known as the container crisis. We find that the container crisis was a
specific event in the shipping market. Our data shed light on the industry
dynamics of shipping cartels. We leave a detailed analysis based on these data
for future work.
Acknowledgement
We benefited from anonymous referees and participants at the International
Association of Maritime Economists 2022. We thank Akimitsu Ashida and Hiroyuki
Sato for sharing industry knowledge and expertise as ex-chairperson in the
1980s. And we thank Mikio Tasaka, Yasuhiro Fujita, Jong-khil Han and Yutaka
Yamomoto for professional comments. This study was supported by JSPS KAKENHI
Grant Numbers 20K22129 and 22K13501.
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Container freight rate | Year | Source | Measured Units | Note
---|---|---|---|---
| 1965,1970,1975,1979 | Issues of Our Ocean Shipping | dollars per 100 tons-mile | Liner sector, three main cargo
| 1973-1976 | Current status of marine transportation | converted dollars per TEU | Revenue and quantity on only Transpacific and Asia-Europe routes
| 1976-1994 | Global Container Markets Drewry Shipping Consultants | dollars by FEU | Missing 1976-1989 in Asia-Europe
| 1994-2009 | Review of Maritime Transport | dollars by TEU | All market-level info is available
Liner freight rate | | | |
| 1965-2009 | Review of Maritime Transport | index rate based on 1995(=100) | Global liner freight rate
Quantity | | | |
| 1966-1972 | Containerization International 1973 | TEU | Aggregate carrying capacity for each route
| 1970,1974,1978,1980 | The Container Crisis 1982 | 1 million ton | Aggregate quantity for each route
| 1975,1978,1981,1984,1987,1990 | Container transportation cost and profitability 1980/2000 | 1000TEU | Aggregate quantity for each route
| 1985-1997 | World Sea Trade Service | 1000TEU | All market-level info is available
| 1994-2009 | Review of Maritime Transport | 1000TEU | All market-level info is available
| (1970,1975,1977,1978,1979) | *Issues of Our Ocean Shipping | D/W tons | Aggregate quantity for each route
| (1973) | *Containerization International 1975 | 1000TEU | Aggregate quantity for each route
| (1978,1981) | *The Container market to 1990 | TEU | Aggregate carrying capacity for each route and container type
| (1983) | *World Container Data 1985 | 1000TEU | Aggregate quantity for each route
* a
Note: The data sources with ($*$) are used for reference to check the
consistency of the trend.
Table 3: Overview of data sources.
Figure 5: The observed and imputed variable cells for market-level data.
* a
Note: In the yellow X cells, the route-level variables are recorded in data.
In the green Y cells, the route-level variables that cannot identify the
eastbound and westbound routes are recorded, so that we assign the values
based on the fixed ratio of the eastbound to the westbound in the closest
period. In the orange Z cells, although route-level variables are not
recorded, ship-level variables with targeted routes are available, so that we
convert firm-level information into route-level variables based on the fixed
ratio in the closest period. The red cells are missing values. Thus, we impute
the values via interpolation based on the observed points in each route and
global liner freight rate.
## Appendix A Details of global containerization (Not for publication)
##### Transforming the cost structure.
In the 1960s, the volume of cargo movement on ocean liners and port handling
costs increased, putting pressure on the management of shipping companies.
However, in the U.S., the cargo was loaded into standardized containers for
transport, which not only eliminated the need for onboard cargo handling
equipment but also enabled cargo handling in the rain. In addition, cargo
handling in container shipping has saved a significant amount of work done by
workers. Furthermore, they significantly reduce the time and cost required for
cargo handling at ports. However, containerization has made the industry more
capital-intensive.
##### Lowering barriers to entry.
The development of containerization has dramatically lowered barriers to entry
for new shipping companies. Prior to the development of containerization,
large and small cargoes had to be loaded and unloaded individually, which
required skilled stacking techniques and expertise to prevent cargo collapse
during the voyage and a significant amount of time and workforce for cargo
handling and management. As containerization has progressed, cargo sizes have
been standardized, and many stacking techniques and expertise have become
unnecessary, significantly reducing the time and labor required for cargo
handling.
##### Rise of (non-conference) shipping companies in developing countries.
Several national and local governments competed to build container terminals,
which resulted in the loss of technological and equipment advantages of
existing shipping companies. Thus, barriers to entry for new shipping
companies that did not have the technology or bases for loading and unloading
dropped significantly.292929Further, containerization has made it possible to
quickly reload cargo onto freight trains and trailers, which has led to the
rapid development of international intermodal transportation. In the past,
shipping companies were contracted to transport goods only by sea between
ports and were not involved in inland transportation. The relationship between
inland and vessel transportation is investigated by Bernhofen et al. (2016)
and Levinson (2016) explained it anecdotally. In parallel with the
development of containerization, developing and socialist countries began to
develop the ocean shipping business as a core industry, and state-owned
shipping companies in these countries began to enter the market as non-
conference members.
##### Shifting from “closed” conferences to “open” conferences.
Furthermore, conferences came to be strongly perceived as an impediment to the
entry of the shipping industries of developing countries into the world trade
market, mainly in the UNCTAD arena. In response to this trend, the United
Nations Convention on a Code of Conduct for Linear Shipping Conferences was
concluded in 1974. This convention allows shipping companies from trading
parties to participate in shipping conferences. Moreover, when a conference
determines the ratio of cargo transportation by shipping companies, the ratio
of origin-destination shipping companies to third-country shipping companies
should be 4:4:2. In the 1980s, conferences related to European routes began to
ask non-conference companies to join conferences to prevent their market share
from declining.303030Theoretically, in the 1970s, the potential entrants in
the conference markets consisted of non-containerized liner shipping companies
in the shipping conferences. In the 1980s, the potential entrants added non-
conference firms. However, the number of non-conference firms joining shipping
conferences is marginal.
##### Forming a consortium.
As containerization progressed, container routes required considerable
investment in ship construction and container terminal ownership. Therefore,
multiple companies began to form consortiums to reduce the scale of investment
while maintaining a certain level of service.313131For example, these
consortia include the TRIO Group, formed by European shipping lines, and the
Ace Group, consisting of Japanese, Korean, French, and other shipping lines.
Specifically, consortiums are engaged in business co-operation such as space
chartering, joint use of container terminals, and coordination of operation
schedules.323232 Major container shipping companies are forming global
alliances such as THE Alliance and the Ocean Alliance. They are utilizing a
consortium framework. Both “conference” and “consortium” are defined as
concerted actions by competition laws in various countries, which then require
exemptions from competition laws. However, member companies at a conference
agree on freight rates, and consortium members do not have any consent.
Individual shipping companies in the consortium determine freight rates and
sales activities. Thus, the consortium is not treated as a single firm.
## Appendix B Data construction (Not for publication)
The container shipping industry is a fascinating laboratory for investigating
industry dynamics. Because the industry started global shipping in 1966, we
can determine the initial state of the market’s dynamic structure. There is
also substantial firm entry and exit in the industry, which makes it ideal for
studying industry dynamics in the global markets. Finally, the markets observe
the dynamics of shipping alliances, mergers, and consolidations.
Despite its significance, a panel dataset regarding the container freight rate
and shipping quantity on the three major trade routes (front-haul and back-
haul separately) between 1966 and 2009 was not available.333333Similar to
related papers focusing on the container freight rate panel data, Luo et al.
(2009) use shipping demand and freight rate data between 1980 and 2007. As for
shipping demand, they used the world container throughput reported in the
Drewry Annual Container Market Review and Forecast. The container freight rate
is calculated as the weighted average of Transpacific, Europe-Far East and
Transatlantic trades from the same data source. Because of the data
limitation, they calculated the missing period (1980–1993) from the General
Freight Index in the Shipping Statistics Yearbook 2007, using a simple
statistical equation between the container freight rate and the general
freight index from 1994 to 2008. Although, unified instructions and guidance
to construct the dataset are provided by published books and available data,
the construction and merging processes involve conversion and imputation
errors from multiple datasets. Table 3 presents the data sources. Instead of
manually constructing the freight index from the commodity-level freight rate
via formal but complicated processes, we adopted at tractable imputation
approach, which links multiple data sources that overlap information for a few
years. The corresponding code is also provided to ensure that an interested
researcher can replicate the construction.
Collecting data on container freight rates and shipping quantity, particularly
before 1994, is not trivial because there is no single data source. In the
subsequent subsections, we provide detailed data construction for each
container freight rate and shipping quantity.
### B.1 Container freight rate
To construct the data regarding the container freight rate, we refer to the
data sources presented in Table 3 and the liner freight rate before global
containerization.
#### B.1.1 Shipping conference and Liner freight rate
The liner shipping industry has traditionally formed shipping conferences
including explicit cartels. Until 1984, shipping conferences played an
important role in the container-shipping market. For example, in February
1967, a container shipping firm, Matson, started an operation on the Atlantic
route under the freight rate contracted with one of conferences, the TPFC.
Thus, the container freight rate was determined by liner shipping conferences.
This unique feature allows us to use the liner freight rate to approximate the
container freight rate. Figure 6 compares the liner freight rate with the
container freight rate recorded since 1995. In particular, “Issues of Our
Ocean Shipping” recorded that in 1978, 76.0% of U.S. liner transports, 66.4%
of European liner transports, and 68.8% of Asian and Australian liner
transports were containerized. Therefore, more than two-thirds of the liner
freight rate was determined by the container freight rate in the 1970s and the
1990s.
Figure 6: The trend of the liner freight rate.
* a
Note: Source: Review of Maritime Transport, published annually by UNCTAD. The
measure of the price index is not explicitly mentioned in the Review of
Maritime Transport.
#### B.1.2 Container freight rate between 1966 and 1975
Data on freight rates between 1966 (i.e., the beginning of the industry) and
1975 were missing from a single data source. Thus, we need to infer and impute
data from multiple sources and information in subsequent years. Fortunately,
we can use the institutional knowledge in Section B.1.1, the liner freight
rate, and the published data that overlap the data in subsequent years in
multiple data sources.
The “Issues of Our Ocean Shipping” (“Wagakuni no Gaikou Kaiun Ni Tsuite”, in
Japanese) records the freight rate of liner shipping on three major routes in
1965, 1970, 1975, and 1979. The freight rates for the three types of cargo
(electric appliances, clothes, and ceramic products) are available. The
freight rates were measured in dollars for 100 tons per mile. Shipping miles
between specific ports are listed.343434For the Transpacific and Asia-Europe
routes, the shipping miles between specific ports, Japan-Hamburg and Japan-San
Francisco routes are mentioned. However, only for the transatlantic route, the
specific ports were not mentioned. Thus, we take the average of the shipping
miles for Hamburg-San Francisco (i.e., west coast) and Hamburg-Halifax routes
(i.e., east coast). Using this information, we can recover route-level
container freight rates for 1965, 1970, and 1975. Finally, we assumed that the
proportions of liner and container freight rates for each route were fixed
before 1975. Under this assumption, we recovered and interpolated the
eastbound and westbound container freights for each route based on the fixed
proportion of the liner and container freight rates and that of subsequent
years, which overlaps container freight rates in multiple data
sources.353535The overlapped information in 1979 is key for merging multiple
data before and after 1975. However, the freight rate swung irregularly and
non-proportionally in both data sources. For making the smooth transition of
the freight rate in the merged dataset, we calculated the conversion rate
based on the freight rate of 1975 from the Issues of our ocean shipping and
that of 1976 from Global container markets year.
#### B.1.3 Container freight rate between 1976 and 1994
The data on the freight rate between 1976 and 1994 were recorded in the Global
Container Markets Drewry Shipping Consultants. The recording of the freight
rate started in 1976 for the transatlantic and Transpacific routes and in 1990
for the Europe-Asia route. Thus, data on the freight rate of the Europe-Asia
route between 1976 and 1989 are missing.363636Global Container Markets Drewry
Shipping Consultants stated that, “the Europe-Far East trade is the least well
documented of the axial routes (page 109)”.
For the imputation of the missing data, we assumed that the proportion of
liner and container freight rates for eastbound and westbound Asia-Europe
routes were fixed between 1976 and 1990. This assumption implies that there is
no time-varying difference between the eastbound and westbound Asia-Europe
routes. Under this assumption, we recovered the container freight rates of the
eastbound and westbound Asia-Europe routes.
#### B.1.4 Container freight rate after 1995
The freight rate data were recorded in the Review of Maritime Transport which
refers to the Containerization International Yearbook. The recording of
freight rates began in June 1994.373737Jeon (2022) stated: “The first dataset
on firm-level investment and capital is therefore supplemented with the
historical price and quantity data compiled from the Review of Maritime
Transport published by the United Nations that goes back to 1997.” (p. 9).
However, we confirm that the data in 1994 are available in Containerization
International Yearbook. The Containerization International Yearbook mentioned
that “the information is derived mainly from confidential reports provided by
some of the main carriers and other public sources.” Review of Maritime
Transport is often used because of the consistency of data sources and public
availability. For example, Jeon (2022) used quarterly freight data.
The above steps generate time-series data of the container freight rate for
each route between 1966 and 2009. Figure 7 shows the trend of the container
freight rate (CPI-adjusted to 1995) with the liner freight rate depicted in
Figure 6. In particular, the recovered and imputed data points in the 1960s
and the 1970s quantitatively capture anecdotal and institutional evidence.
First, for the transatlantic and Transpacific routes, a peak in 1976 and a
sharp declining trend in 1979 were observed. Second, the container freight
rate on each route corresponds to fluctuations in the liner freight rate.
Figure 7: The trend of the container freight rate.
### B.2 Shipping quantity
To construct data on container trade volume, we refer to the five data sources
shown in Table 3.
#### B.2.1 Shipping quantity between 1966 and 1975
Official data on shipping quantities between 1966 and 1969 were not available.
However, we could keep track of the development of the container vessels
during this period. First, data regarding vessel capacity were recorded in
Containerization International 1973.383838“Theory and Practice of Container
Shipping” (“Container Yusou no Riron to Jissai”, in Japanese) contains
information about the launch of container ships owned by the United States’
four main firms, Moore-McCormack, U.S. Lines, SeaLand, and CML (American
Export Isbrandtsen). This helps us to identify the exact launch year of full-
container ships by the largest players, that is, United States’ container
shipping companies. We use the data source only to check institutional
evidence. This information identifies the carrier composition of the
Transatlantic, Transpacific, and Asia-Europe routes between 1966 and 1972.
Second, the Review of Maritime Transport (1971) summarizes the relationship
between annual global container carrying capacity and container capacity in
1969 and 1970 on the transatlantic route. We assume that container capacities
were fully exploited during this period. This assumption is not restrictive,
because the demand for container shipping was considerably high during this
period. Under this assumption, the shipping quantity is recovered by
calculating the total shipping quantity from the container capacity and
invariant conversion rate.
Third, the data on shipping quantity in 1970 and 1974 were recorded in The
Container Crisis 1982, and the data for 1973 were recorded in Containerization
International 1975. However, data for 1971 and 1972 were missing. The data
contains regional levels of container shipping, that is, container traffic in
Asia, North America, and Europe. One alternative is the imputation approach,
which employs external data that provide information about missing
data.393939Jeon (2022) stated that: “We set the start date for firms”
information as the second quarter of 1966, which is the date of the first
international container voyage. Then, we employed quarterly data on the value
of trade by origin-destination pair from the IMF Direction of Trade Statistics
database to impute the missing data on demand states from 1966: Q2-1996: Q4.”
(p. 25), and “To translate the value of trade to the quantity of container
trade, the demand state for the 1997–-2014 period was regressed on the de-
trended value of trade. Then, the demand states for periods with missing data
are constructed as predicted values from the regression. For the 1997–-2014
period, actual demand states are used.” (footnote 37). We assign the recorded
trade volume to each route in proportion based on subsequent
years.404040Alternatively, we could assign the aggregate container shipping
quantity to westbound and eastbound routes based on the data on the value of
trade by origin-destination pair from the IMF Direction of Trade Statistics
database. We did not take the full-imputation approach because the value of
trade includes goods transported by the bulk shipping service. For missing
years, we interpolated the trade volume using the observed data points.
#### B.2.2 Shipping quantity between 1976 and 1994
Data on shipping quantity between 1976 and 1994 were recorded in World Sea
Trade Service, Container transportation cost and profitability 1980/2000, The
Container Crisis 1982, and World Container Data 1985. World Sea Trade Service
started recording data in 1985, whereas the other data sources recorded the
data before 1985 but irregularly. Specifically, the quantity data for 1971,
1972, 1973, 1976, 1977, and 1979 were missing for all routes, and the quantity
data for 1970, 1974, 1980, 1982, and 1983 were recorded in the total shipping
quantity, summing the eastbound and westbound for each market.
#### B.2.3 Shipping quantity after 1995
Data on shipping quantity after 1995 were recorded in the Review of Maritime
Transport and World’s sea trades. The former refers to World Sea Trade Service
Review, various issues, 1996; Journal of Commerce, various issues, 1996; and
Containerization International, various issues, 1996. As in Section B.1.4,
shipping quantity data are easily constructed from a single data source, the
Review of Maritime Transport. Before merging the data before and after 1995,
we confirm that the discrepancy between the two data sources is marginal for
almost all market-year observations. However, only the quantities on the
transatlantic eastbound and westbound routes in 1995 were somewhat different.
The above steps generate time-series data of container shipping quantity
measured by 1000 TEU for each route between 1966 and 2009. Figure 8
illustrates the trend in the container shipping quantity. Shipping quantity on
all routes increased monotonically between 1973 and 2000. After 2000, this
trend increased exponentially, particularly on the Transpacific eastbound and
Europe-to-Asia routes.
Figure 8: The trend of the container shipping quantity. The trend before 1976
is based on ship-level carrying capacity information on each route.
### B.3 Newbuilding, Secondhand, and Scrap prices
We provide recommendations for constructing newbuilding, secondhand, and scrap
prices for container ships from 1967. Our data were collected from a series of
Review (1971-1998) published by Fearnley and Lloyd’s Shipping Economist
(1983-1990) published by Lloyd’s of London Press.
#### B.3.1 Newbuilding prices
We collected industry-year-level newbuilding prices per 18000 DWT bulk ships
from Review (1971-1998)). Using the overlapped year, we converted prices into
consistent prices under the assumption that the conversion rate is invariant
across years. Then, we divided the price per 12000 dwt by 10 to convert it to
the price per 1200 TEU and then converted it into the price per TEU. Finally,
we convert bulk prices into container prices based on Lloyd’s Shipping
Economist (1983-1990).
#### B.3.2 Secondhand prices
We collected industry-year-level secondhand prices per 16000 DWT CSD (liner
type) ships from Review (1971-1998). Using the overlapped year, we converted
prices into consistent prices under the assumption that the conversion rate is
invariant across years.
We assume a fixed depreciation rate ($=X$) and a fixed conversion rate ($=a$)
for 1981 and 1983 and observe the following:
$\displaystyle 1981:2.7+2X$ $\displaystyle=16.0a\quad\text{(18-year
depreciation)}$ $\displaystyle 1983:0.8+0X$
$\displaystyle=11a\quad\text{(20-year depreciation)}$
We solved the equations for $X$ and $a$. To obtain the price per
$1600dwt+15X$, we multiplied the above price by 12000/16000 and obtained the
price per 12,000dwt. We then divided the price by 10 to convert it to the
price per 1200TEU. We then converted it into the price per TEU of 5 years ship
by dividing it by 1200. Finally, we converted liner prices into container
prices based on Lloyd’s Shipping Economist (1983-1990).
#### B.3.3 Scrap prices
We collected industry-year-level scrap prices per LTD in the Far-East. Using
the overlapped year, we converted prices into consistent prices under the
assumption that the conversion rate is invariant across years. We then divided
each LTD by four to obtain the price per dwt, multiplied it by 10, and
converted it into the price per TEU.414141The rate is based on
https://nippon.zaidan.info/seikabutsu/2002/00264/contents/030.htm.
|
# The Adversary Bound Revisited: From Optimal Query Algorithms to Optimal
Control
Duyal Yolcu https://github.com/qudent
###### Abstract
This note complements the paper ”One-Way Ticket to Las Vegas and the Quantum
Adversary” [6]. I develop the ideas behind the adversary bound - universal
algorithm duality therein in a different form, using the same perspective as
[2, 3] in which query algorithms are defined as sequences of feasible reduced
density matrices rather than sequences of unitaries. This form may be faster
to understand for a general quantum information audience: It avoids defining
the ”unidirectional relative $\gamma_{2}$-bound” and relating it to query
algorithms explicitly. It is also more general because the lower bound (and
universal query algorithm) apply to a class of time-optimal control problems
rather than just query problems. That is in addition to advantages to be
discussed in [6], namely the more elementary algorithm and correctness proof
that avoids phase estimation and spectral analysis, allows for limited
treatment of noise, and improves the runtime by another
$\Theta(\log(1/\varepsilon))$ compared to [15, 5].
The approach - not new - starts with considering an optimal query problem for
state conversion - in which we are given an unknown oracle $\displaystyle
L_{a}$ for $\displaystyle a\in A$, and want to construct an algorithm that
transforms initial states $\displaystyle\ket{\xi_{a}}$ to acceptable target
states $\displaystyle\ket{\tau_{a}}$ by invoking that oracle - as an optimal
control problem, in which $\displaystyle a\in A$ is stored as a quantum ”input
state” in another Hilbert space $\displaystyle\mathcal{A}$, and we want to
transform $\displaystyle\ket{\xi}=\sum_{a\in A}\ket{a}\otimes\ket{\xi_{a}}$ to
$\displaystyle\ket{\tau}=\sum_{a\in A}\ket{a}\otimes\ket{\tau_{a}}$ by
invoking $\displaystyle L=\sum_{a\in A}\ket{a}\bra{a}\otimes L_{a}$ without
being allowed to access $\displaystyle\mathcal{A}$ directly. Then we track
feasible reduced density matrices on $\displaystyle\mathcal{A}$ (which
correspond to the transpose of the Gram matrices in the adversary bound
literature). This information is sufficient to track the system’s state by
standard facts on purifications and their unitary equivalence. This is close
to [2, 3].
The adversary bound is the inverse maximal speed we can achieve in reduced
density matrix space from any starting RDM, travelling in the desired
direction. We describe a new universal control algorithm that matches this
speed up to an error-dependent factor by slightly perturbing initial and
target state by that ideal starting RDM; by a linearity argument, an algorithm
going along a straight line between these perturbed states is feasible. We can
then bound the error in the final state if we apply that algorithm to the
original initial state, rather than the perturbed one.
Importantly, this approach doesn’t assume that $\displaystyle L$ is ”read-
only”, i.e. block-diagonal in $\displaystyle\mathcal{A}$, anymore - as long as
there is still an ”idle subspace” as defined in the note. We can therefore
apply it to problems in which the register $\displaystyle\mathcal{A}$ is to be
manipulated, rather than just read out. The argument also works with problems
in which $\displaystyle L$ is only subunitary, i.e. may correspond to noise
occuring and the algorithm ”giving up” in some instances.
To follow this text, one only needs basic knowledge in quantum physics
including reduced density operators, purifications and their local
equivalence. Therefore, I hope it has expository value for a general quantum
information audience that wants to understand adversary bound - universal
query algorithm dualities.
## Acknowledgments
I thank Alexander Belov (Aleksandrs Belovs) and Berare Göktürk for helpful
discussions while writing the draft.
###### Contents
1. 1 The problem - discrete time
2. 2 Quantum algorithms as sequences of feasible reduced density matrices on AB
3. 3 Adversary bound
4. 4 Matching the lower bound by a universal algorithm
5. 5 Further remarks
1. 5.1 From control to query algorithms
2. 5.2 Continuous time
3. 5.3 Other characterizations of quantum processes
6. 6 Conclusion and outlook
7. 7 Further references
## 1 The problem - discrete time
We consider a tripartite quantum system $\mathcal{ABC},$ assumed to be finite-
dimensional for convenience. We want to construct an algorithm that transforms
an initial state $\displaystyle\ket{\xi}\neq 0\in\mathcal{ABC}$ to a target
state $\displaystyle\ket{\tau}\in\mathcal{ABC}$ (or similar, e.g. allowing a
range of acceptable target states) as fast as possible. The catch is that the
algorithm is not allowed to arbitrarily act on $\displaystyle\mathcal{A}$ \-
instead, there is a fixed subunitary interaction operator $\displaystyle L$
(i.e. fulfilling
$\displaystyle\left\|L\ket{\varphi}\right\|\leq\left\|\ket{\varphi}\right\|$
for any $\ket{\varphi}$) that acts on $\displaystyle\mathcal{AB}$ (not
$\mathcal{C}$) once per timestep. On the other hand, we are allowed to apply
arbitrarily complex unitaries on $\displaystyle\mathcal{BC}$ at any time,
without cost. We also assume that $\displaystyle\mathcal{C}$ is ”as large as
the algorithm could need it to be”; in the proof of Proposition 1, we’ll see
that $\displaystyle\dim\mathcal{C}\geq\dim(\mathcal{AB})$ works for any
algorithm. We optimize over the number of timesteps (or, conversely, lower-
bound the number of timesteps necessary).
Finally, we fix a designated normalized state
$\displaystyle\ket{\mathrm{idle}}\in\mathcal{B}$. We assume that
$\displaystyle L$ acts trivially on $\displaystyle\ket{\mathrm{idle}}$, i.e.
$\displaystyle
L\left(\ket{\varphi}\ket{\mathrm{idle}}\right)=\ket{\varphi}\ket{\mathrm{idle}}$
for all $\displaystyle\ket{\varphi}\in\mathcal{A}.$ This is equivalent to
assuming that $\displaystyle L$ can be applied in a ”controlled” way, and the
algorithm can always choose to do nothing.
Allowing subunitary, rather than only unitary, $\displaystyle L$ allows a
limited discussion of noise: If the true transformation a physical system
undergoes involves a noisy quantum channel, one can choose $\displaystyle L$
to be one Kraus operator of that quantum channel and consider it the
”successful” Kraus operator. The other Kraus operators can be considered
”errors”, and one stops tracking the computation in case one of them is
applied. Then the norm of the state will decay over time - corresponding to
losing probability mass in case an error occurs - but one can still define
sets of acceptable, sub-normalized target states, and achieving them with that
$\displaystyle L$ is sufficient to solve the entire problem. However, it is
necessary for our universal algorithm that $\displaystyle L$ preserves the
norm when acting on $\displaystyle\ket{\mathrm{idle}}.$
As mentioned in the abstract, we can encode a query complexity problem by
choosing $\displaystyle L$ block-diagonal. For a more thorough introduction to
query problems, see [5].
## 2 Quantum algorithms as sequences of feasible reduced density matrices on
AB
We now define quantum algorithms in terms of the feasible intermediate reduced
density matrices on $\displaystyle\mathcal{AB}$. This section uses essentially
the same ideas as [2, 3] (aside from pointing out that they apply to a wider
class of $\displaystyle L$).
###### Definition 1.
In our discussion, a T-step quantum algorithm is a list of positive
semidefinite operators
$\displaystyle\left(\pi^{0},\pi^{1},\dotsc,\pi^{T-1}\right)$ representing
(non-normalized) reduced density matrices on $\displaystyle\mathcal{AB}$ such
that for all $\displaystyle j$,
$\mathrm{tr}_{\mathcal{B}}\pi^{j+1}=\mathrm{tr}_{\mathcal{B}}\left(L\pi^{j}L^{\dagger}\right).$
(1)
###### Proposition 1.
In the model of Section 1, it is possible to transform
$\displaystyle\ket{\xi}\mapsto\ket{\tau}$ in $\displaystyle T$ timesteps iff
such a list exists with
$\displaystyle\mathrm{tr}_{\mathcal{BC}}\ket{\xi}\bra{\xi}=\mathrm{tr}_{\mathcal{B}}\pi^{0}$
and
$\displaystyle\mathrm{tr}_{\mathcal{BC}}\ket{\tau}\bra{\tau}=\mathrm{tr}_{\mathcal{B}}\left(L\pi^{T-1}L^{\dagger}\right).$
###### Proof.
First suppose that such a transformation is possible and let
$\displaystyle\ket{\Phi^{j}}\in\mathcal{ABC}$ be the system’s state directly
before the $\displaystyle j+1$th application of $\displaystyle L$ (counting
from $\displaystyle 1$). Then set
$\displaystyle\pi^{j}:=\mathrm{tr}_{\mathcal{C}}\ket{\Phi^{j}}\bra{\Phi^{j}}.$
As reduced density operators of a nonnormalized state, these are positive
semidefinite. Directly after the $\displaystyle j+1$th application of
$\displaystyle L,$ the reduced density matrix on $\displaystyle\mathcal{A}$ is
$\displaystyle\mathrm{tr}_{\mathcal{B}}\left(L\pi^{j}L^{\dagger}\right)$ by
standard quantum physics. Similarly, directly before the $\displaystyle j+2$nd
application, the RDM is
$\displaystyle\mathrm{tr}_{\mathcal{B}}\left(\pi^{j+1}\right).$ But these
matrices must be equal because between the $\displaystyle j+1$th and the
$\displaystyle j+2$nd application, the quantum computer may only act on
$\displaystyle\mathcal{BC},$ and not on $\displaystyle\mathcal{A}.$
Conversely, suppose there is an algorithm as in Definition 1 and consider a
sequence of purifications of the $\displaystyle\pi^{j}$ on
$\displaystyle\mathcal{C},$ i.e. $\displaystyle\ket{\Phi^{j}}\in\mathcal{ABC}$
such that
$\displaystyle\mathrm{tr}_{\mathcal{C}}\ket{\Phi^{j}}\bra{\Phi^{j}}=\pi^{j}.$
By standard quantum physics again, these always exist if
$\displaystyle\dim\mathcal{AB}\leq\dim\mathcal{C}$ [16]. Then by Equations 1,
$\displaystyle(L\otimes I_{\mathcal{C}})\ket{\Phi^{j}}$ and
$\displaystyle\ket{\Phi^{j+1}}$ are purifications of the same reduced density
matrix on $\displaystyle\mathcal{A}.$ The same is true for
$\displaystyle\ket{\Phi^{0}}$ and $\displaystyle\ket{\xi},$ as well as
$\displaystyle(L\otimes I_{\mathcal{C}})\ket{\Phi^{T-1}}$ and
$\displaystyle\ket{\tau}.$ As all such purifications are related by local
unitaries (i.e. unitaries acting only on $\displaystyle\mathcal{BC}$) [16,
17], a valid quantum algorithm exists that starts with
$\displaystyle\ket{\xi}$ and applies these connecting unitaries between
applications of $\displaystyle L.$ ∎
As promised in the abstract, this argument essentially works by tracking the
reduced density matrix on $\displaystyle\mathcal{A}.$ This set of lists of
operators is also a convex set in a natural way, which gives rise to nice
properties - see [6] for details.
## 3 Adversary bound
Now consider a T-step quantum query algorithm transforming
$\displaystyle\ket{\xi}\mapsto\ket{\tau}$ and consider the sum of all
$\displaystyle\pi^{j}$,
$\overline{\pi}:=\sum_{j=0}^{T-1}\pi^{j}\in\mathbb{S}\mathcal{{}_{\mathcal{AB}}},$
(2)
where $\displaystyle\mathbb{S}_{\mathcal{AB}}$ denotes the set of positive
semidefinite operators on $\displaystyle\mathcal{AB}.$ Then
$\displaystyle\mathrm{tr_{\mathcal{B}}\left(L\overline{\pi}L^{\dagger}\right)=}\sum_{j=0}^{T-1}\mathrm{tr}_{\mathcal{B}}\left(L\pi^{j}L^{\dagger}\right)=\sum_{j=0}^{T-2}\mathrm{tr}_{\mathcal{B}}\pi^{j+1}+\mathrm{tr}_{\mathcal{BC}}\left(\ket{\tau}\bra{\tau}\right)$
$\displaystyle=\mathrm{tr_{\mathcal{B}}\overline{\pi}+tr}_{\mathcal{BC}}\left(\ket{\tau}\bra{\tau}-\ket{\xi}\bra{\xi}\right).$
(3)
Furthermore,
$\mathrm{tr}(\overline{\pi})\leq\ T\ \bra{\xi}\ket{\xi}:$ (4)
If we turn the sequence of $\displaystyle\pi^{j}$ into a quantum algorithm
involving a sequence of $\displaystyle\ket{\Phi^{j}}$ as in 2,
$\displaystyle\mathrm{tr}\left(\pi^{j}\right)=\left\|\ket{\Phi^{j}}\right\|^{2}\leq\left\|\ket{\xi}\right\|^{2}$
by subunitarity of $\displaystyle L$; Inequality 4 is the result of adding
these inequalities.
If a $\displaystyle T$-query algorithm exists, some
$\displaystyle\overline{\pi}$ must exist, which yields the following
bound:111The motivation for this term is clearer in other expositions, such as
Childs’s lecture notes [9].
###### Definition 2 (Adversary bound).
The adversary bound of a state conversion problem, denoted
$\displaystyle\mathrm{Adv}\left(\ket{\xi}\rightarrow\ket{\tau}\right)$ with
$\ket{\xi}\neq 0$, is the optimal value of the minimization problem (which we
call the primal problem)
$\displaystyle\mathrm{minimize}$ $\displaystyle\ \mathrm{tr}\
(\overline{\pi})/\bra{\xi}\ket{\xi}$ (5) $\displaystyle\mathrm{subject\ to}$
$\displaystyle\
\mathrm{tr}_{\mathcal{B}}\left(L\overline{\pi}L^{\dagger}-\overline{\pi}\right)=\mathrm{tr}_{\mathcal{BC}}\left(\ket{\tau}\bra{\tau}-\ket{\xi}\bra{\xi}\right),$
(6) $\displaystyle\ \overline{\pi}\in\mathbb{S}_{\mathcal{AB}}.$ (7)
By the discussion above,
$\displaystyle\mathrm{Adv}\left(\ket{\xi}\rightarrow\ket{\tau}\right)$ lower-
bounds the number of queries of quantum query algorithms solving the state
conversion problem exactly.
The inverse of this problem’s optimal solution is also the answer to the
question ”In the space of reduced density matrices on
$\displaystyle\mathcal{A},$ what is the maximum fraction of the desired change
$\displaystyle\mathrm{tr}_{\mathcal{BC}}\left(\ket{\tau}\bra{\tau}-\ket{\xi}\bra{\xi}\right)$
achievable, starting from any state with the correct normalization?”
Subsection 5.1 discusses a strengthening relevant for query problems, omitted
here to reduce technicality.
The following remark uses semidefinite programming duality; the result isn’t
necessary for the remainder of the discussion, and a reader unfamiliar with
the technique may take it on faith. The optimal value of the problem is lower-
bounded by the optimal value of the maximization problem (the dual problem)
$\displaystyle\mathrm{maximize}$ $\displaystyle\left(\bra{\tau}\Gamma\otimes
I_{\mathcal{BC}}\ket{\tau}-\bra{\xi}\Gamma\otimes
I_{\mathcal{BC}}\ket{\xi}\right)/\bra{\xi}\ket{\xi}$ (8)
$\displaystyle\mathrm{subject\ to}$ $\displaystyle\ L^{\dagger}(\Gamma\otimes
I_{\mathcal{B}})L-\Gamma\otimes I_{\mathcal{B}}\preceq I_{\mathcal{AB}},$ (9)
$\displaystyle\ \Gamma\in\mathbb{H}_{\mathcal{A}},$ (10)
where $\displaystyle\mathbb{H}_{\mathcal{A}}$ denotes the space of Hermitian
matrices on $\displaystyle\mathcal{A}$. This means that finding any feasible
$\displaystyle\Gamma$ for this problem corresponds to a proof that no
algorithm can be faster - which is more convenient for finding lower bounds on
the number of steps necessary for a conversion. We can see that Slater’s
strong duality condition is fulfilled by choosing $\displaystyle\Gamma=0$ in
the dual problem. This means that the best solution to Problem 8-10 results in
a value equal to the best solution of Problem 5-7.
## 4 Matching the lower bound by a universal algorithm
Now assume we have some feasible solution $\displaystyle\overline{\pi}$ of the
optimization problem in Definition 2, which doesn’t have to be optimal. Can we
”turn it around” and obtain an algorithm to transform
$\displaystyle\ket{\xi}\rightarrow\ket{\tau}$ in $\displaystyle\mathrm{tr}\
\overline{\pi}$ steps? The answer will turn out to be ”almost”.
###### Proposition 2.
Using the notations above, for any integer $\displaystyle T^{\prime}>0$, the
sequence of $\displaystyle\pi^{j}$
$\displaystyle\left(\pi^{j}\right)_{0\leq j<T^{\prime}}$
$\displaystyle=\left(\left(\frac{T^{\prime}-j}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\xi}\bra{\xi}+\frac{j}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\tau}\bra{\tau}\right)\otimes\ket{\mathrm{idle}}\bra{\mathrm{idle}}+\frac{\overline{\pi}}{T^{\prime}}\right)_{0\leq
j<T^{\prime}}$
constitutes a $\displaystyle T^{\prime}$-query quantum query algorithm solving
the state conversion problem
$\displaystyle\ket{\xi}\otimes\ket{0}+\frac{\ket{v}}{\sqrt{T^{\prime}}}\otimes\ket{1}\mapsto\ket{\tau}\otimes\ket{0}+\frac{\ket{v}}{\sqrt{T^{\prime}}}\otimes\ket{1},$
where $\displaystyle\ket{v}$ is a purification of
$\displaystyle\overline{\pi}$ on $\displaystyle\mathcal{C}$ (and we add a
qubit to the ancilla space).
Intuitively, this algorithm works by using a scaled down version of
$\overline{\pi}$ to go from initial to target density matrix, along a shifted
straight line in the space of reduced density matrices — by Equation 6,
$\overline{\pi}$ is able to induce exactly the change in the desired
direction.
###### Proof.
We first show that $\left(\pi^{j}\right)_{0\leq j<T^{\prime}}$ is a quantum
query algorithm in the sense of Definition 1. This is equivalent to the
conditions that
* •
Each $\pi^{j}\succeq 0$: This follows from the facts that density matrices are
positive semidefinite and closed under convex mixtures.
* •
Equation 1 holds for $j=0,\ldots,T-1$. Plugging in our $\pi^{j}$, we need to
show that
$\mathrm{tr}_{\mathcal{B}}\left(\left(\frac{T^{\prime}-j-1}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\xi}\bra{\xi}+\frac{j+1}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\tau}\bra{\tau}\right)\otimes\ket{\mathrm{idle}}\bra{\mathrm{idle}}+\frac{\overline{\pi}}{T^{\prime}}\right)\\\
=\mathrm{tr}_{\mathcal{B}}\left(L\left(\left(\frac{T^{\prime}-j}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\xi}\bra{\xi}+\frac{j}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\tau}\bra{\tau}\right)\otimes\ket{\mathrm{idle}}\bra{\mathrm{idle}}+\frac{\overline{\pi}}{T^{\prime}}\right)L^{\dagger}\right)$
for these $j$. We know that $L$ acts trivially on $\ket{\mathrm{idle}}$, so
the condition is equivalent to
$\left(\frac{T^{\prime}-j-1}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\xi}\bra{\xi}+\frac{j+1}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\tau}\bra{\tau}\right)+\mathrm{tr}_{\mathcal{B}}\left(\frac{\overline{\pi}}{T^{\prime}}\right)\\\
=\left(\frac{T^{\prime}-j}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\xi}\bra{\xi}+\frac{j}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\ket{\tau}\bra{\tau}\right)+\mathrm{tr}_{\mathcal{B}}\left(L\frac{\overline{\pi}}{T^{\prime}}L^{\dagger}\right).$
Rearranging the terms, we transform this condition into
$\frac{1}{T^{\prime}}\
\mathrm{tr}_{\mathcal{BC}}\left(\ket{\tau}\bra{\tau}-\ket{\xi}\bra{\xi}\right)=\frac{1}{T^{\prime}}\mathrm{tr}_{\mathcal{B}}\left(L\overline{\pi}L^{\dagger}-\overline{\pi}\right).$
Up to a factor, this is exactly the condition of Equation 6, and we are
guaranteed it is fulfilled for a feasible $\overline{\pi}$.
Similarly, we may show that
$\displaystyle\mathrm{tr}_{\mathcal{B}}\left(\pi^{0}\right)$
$\displaystyle=\mathrm{tr}_{\mathcal{BC}}\ket{\xi}\bra{\xi}+\frac{\mathrm{tr}_{\mathcal{B}}\left(\overline{\pi}\right)}{T^{\prime}},$
(11)
$\displaystyle\mathrm{tr}_{\mathcal{B}}\left(L\pi^{T^{\prime}-1}L^{\dagger}\right)$
$\displaystyle=\mathrm{tr}_{\mathcal{BC}}\ket{\tau}\bra{\tau}+\frac{\mathrm{tr}_{\mathcal{B}}\left(\overline{\pi}\right)}{T^{\prime}},$
(12)
which equals the reduced density matrices of the claimed initial and final
states. By Proposition 1, this implies that the claimed transformation is
indeed possible. ∎
As $\displaystyle T^{\prime}\rightarrow\infty$, initial and target states
converge to our desired $\displaystyle\ket{\xi},\ket{\tau}$ (apart from having
redefined the ancilla space. Let $E\colon\mathcal{ABC}\to\mathcal{ABC}$ be the
total effective evolution operator applied by the algorithm; this operator
must be subunitary. If we apply the algorithm to our true initial state
$\displaystyle\ket{\xi}\otimes\ket{0},$ and project to the
$\displaystyle\ket{0}$ subspace in the end (i.e. measures that qubit and
outputs ”failure” in case the result is $\displaystyle 1$, generally reducing
the norm of the state), the resulting state fulfills
$\displaystyle\left\|P_{0}E\left(\ket{\xi}\otimes\ket{0}\right)-\ket{\tau}\otimes\ket{0}\right\|/\left\|\ket{\xi}\right\|$
$\displaystyle=\left\|P_{0}E\left(\ket{\xi}\otimes\ket{0}+T^{\prime-1/2}\ket{\nu}\otimes\ket{1}\right)-P_{0}E\left(T^{\prime-1/2}\ket{\nu}\otimes\ket{1}\right)-\ket{\tau}\otimes\ket{0}\right\|/\left\|\ket{\xi}\right\|$
$\displaystyle=\left\|\ket{\tau}\otimes\ket{0}-P_{0}E\left(T^{\prime-1/2}\ket{\nu}\otimes\ket{1}\right)-\ket{\tau}\otimes\ket{0}\right\|/\left\|\ket{\xi}\right\|$
$\displaystyle=T^{\prime-1/2}\left\|-P_{0}E\left(\ket{\nu}\otimes\ket{1}\right)\right\|\leq
T^{\prime-1/2}\sqrt{\mathrm{tr}\overline{\pi}}/\left\|\ket{\xi}\right\|,$ (13)
where we used subunitarity of $E$ and $\displaystyle P_{0}$ in the last line.
In fact, a similar argument would show that the norm difference would be at
most twice that if we were not allowed to throw away part of the state in the
last step. For an optimal $\displaystyle\overline{\pi},$
$\displaystyle\sqrt{\mathrm{tr}\overline{\pi}}/\left\|\ket{\xi}\right\|=\sqrt{\mathrm{Adv}\left(\ket{\xi}\mapsto\ket{\tau}\right)}.$
For large $\displaystyle T^{\prime},$ each individual step puts most of its
amplitude into the $\displaystyle\ket{\mathrm{idle}}$ subspace. Remarkably,
one can show (proof omitted) that for the algorithm’s intermediate states
$\displaystyle\ket{\Phi^{j}},$ the quantity
$\displaystyle\sum_{j=0}^{T^{\prime}-1}\bra{\Phi^{j}}I-P_{\mathrm{idle}}\ket{\Phi^{j}}/\bra{\xi}\ket{\xi}\leq\mathrm{tr\overline{\pi}/\bra{\xi}\ket{\xi}}$
independent of $\displaystyle T^{\prime}.$ The analogue for query problems -
called ”Las Vegas complexity” - is defined and studied in [6].
In conclusion:
###### Theorem 1.
1. 1.
A control algorithm converting $\displaystyle\ket{\xi}$ to
$\displaystyle\ket{\tau}$ uses at least
$\displaystyle\mathrm{Adv}\left(\ket{\xi}\rightarrow\ket{\tau}\right)$ steps,
2. 2.
Conversely, for any acceptable error $\displaystyle\varepsilon$, we can find
an algorithm converting $\displaystyle\ket{\xi}\otimes\ket{0}$ to
$\displaystyle\ket{\tau^{\prime}}\otimes\ket{0}+\ket{\Delta}\otimes\ket{1}$,
with
$\displaystyle\left\|\ket{\tau^{\prime}}-\ket{\tau}\right\|/\left\|\ket{\xi}\right\|\leq\varepsilon,$
that takes
$T^{\prime}=\left\lceil\frac{\mathrm{Adv}\left(\ket{\xi}\rightarrow\ket{\tau}\right)}{\varepsilon^{2}}\right\rceil$
(14)
steps and fulfills
$\displaystyle\sum_{j=0}^{T-1}\bra{\Phi^{j}}I-P_{\mathrm{idle}}\ket{\Phi^{j}}/\bra{\xi}\ket{\xi}\leq\mathrm{Adv}\left(\ket{\xi}\rightarrow\ket{\tau}\right)$
on the intermediate states.
As remarked in the abstract, this algorithm corresponds to going along a
straight line with constant velocity in the space of reduced density
operators. As $\displaystyle\overline{\pi}$ is not ”used up” during this
transformation, we can interpret it as a ”catalyst” in the spirit of catalytic
states in LOCC transformations (see [20]).
## 5 Further remarks
### 5.1 From control to query algorithms
I briefly discuss how to modify this argument for quantum query complexity
problems in state conversion problems; I skipped this before for simplicity.
This note completely ignores function evaluation and output conditions - i.e.
the question of what final states allow calculating some function of the input
in a query problem. See e.g. [6], [5], [3] for a more thorough discussion of
query complexity problems.
Start directly after Equation 3. Let $\displaystyle P_{\mathcal{A}^{\prime}}$
be a projector onto a subspace
$\displaystyle\mathcal{A}^{\prime}\subseteq\mathcal{A}$ such that
$\displaystyle P_{\mathcal{A}^{\prime}}L=LP_{\mathcal{A}^{\prime}}.$ Choosing
$\displaystyle\mathcal{A^{\prime}=A}$ and $\displaystyle
P_{\mathcal{A}^{\prime}}=I$ will always work; when dealing with a query
problem and $\displaystyle L$ is block-diagonal in some basis
$\displaystyle\left\\{\ket{a}\right\\}_{a\in A}$ of
$\displaystyle\mathcal{A}$, we could choose
$\displaystyle\mathcal{A}^{\prime}:=\mathrm{span}\left\\{\ket{a}\right\\}$ as
well for any $\displaystyle a\in A.$ The argument that shows
$\displaystyle\mathrm{tr}(\overline{\pi})\leq\ T\bra{\xi}\ket{\xi}$
(Inequality 4) is in fact sufficient to show that
$\mathrm{tr}(P_{\mathcal{A}^{\prime}}\overline{\pi})\leq\ T\
\bra{\xi}P_{\mathcal{A}^{\prime}}\ket{\xi}$ (15)
for any such $\displaystyle\mathcal{A}^{\prime},$ because we can commute
$\displaystyle P_{\mathcal{A}^{\prime}}$ through the entire evolution.
So each suitable $\displaystyle\mathcal{A}^{\prime}$ yields a lower bound on
$\displaystyle T,$ and we can replace the optimization target
$\displaystyle\mathrm{tr}\overline{\pi}/\bra{\xi}\ket{\xi}$ given in
Definition 2 by
$\underset{\mathcal{A}^{\prime}\subseteq\mathcal{A}\colon
P_{\mathcal{A}^{\prime}}L=LP_{\mathcal{A}^{\prime}},P_{\mathcal{A}^{\prime}}\ket{\xi}\neq
0}{\mathrm{sup}}\
\left(\mathrm{tr}(P_{\mathcal{A}^{\prime}}\overline{\pi})/\bra{\xi}P_{\mathcal{A}^{\prime}}\ket{\xi}\right)$
(16)
and add the constraint
$\mathrm{tr}\left(P_{\mathcal{A^{\prime}}}\overline{\pi}\right)=0$ (17)
for any $\mathcal{A}^{\prime}$ such that
$P_{\mathcal{A}^{\prime}}\ket{\xi}=0.$
We can also fix a set of $\displaystyle\mathcal{D^{\prime}}$ that fit, and
consider the optimization problem that considers only these. For a block-
diagonal $\displaystyle L$ as above and $\left\|P_{a}\ket{\xi}\right\|=1$ for
all $a,$ this results in an optimization problem equivalent to the
unidirectional relative $\displaystyle\gamma_{2}$-bound of [6].
Conversely, suppose we have a optimal solution of that modified optimization
problem. Then we can insert any $\displaystyle P_{\mathcal{A}^{\prime}}$ with
$P_{\mathcal{A}^{\prime}}\ket{\xi}\neq 0$ into the derivation of Inequality
13. Using the fact that it commutes with all operators involved in that
derivation, we derive that
$\frac{\left\|P_{\mathcal{A}^{\prime}}P_{0}A\left(\ket{\xi}\otimes\ket{0}\right)-P_{\mathcal{A}^{\prime}}\ket{\tau}\otimes\ket{0}\right\|}{\left\|P_{\mathcal{A}^{\prime}}\ket{\xi}\right\|}\leq\sqrt{\frac{\mathrm{Adv}\left(\ket{\xi}\mapsto\ket{\tau}\right)}{T^{\prime}}}$
(18)
for each individual $\displaystyle P_{\mathcal{A}^{\prime}},$ rather than just
$\displaystyle P_{\mathcal{A}^{\prime}}=I.$ In , this allows us to consider
the error bound
$\displaystyle\left\|\ket{\tau^{\prime}}-\ket{\tau}\right\|/\left\|\ket{\xi}\right\|\leq\varepsilon$
with
$\displaystyle\left\|P_{\mathcal{A}^{\prime}}\left(\ket{\tau^{\prime}}-\ket{\tau}\right)\right\|/\left\|P_{\mathcal{A}^{\prime}}\ket{\xi}\right\|\leq\varepsilon$
for each individual ones. If we have considered a query problem as a control
problem as in the abstract, and want to ensure that the error in the state
conversion is small for all possible inputs, such a strengthening is
necessary.
### 5.2 Continuous time
This note discusses everything in discrete time; however, quantum physics as
we know it is continuous and described by differential equations. In a
physical system, the interaction between $\displaystyle\mathcal{A}$ and
$\displaystyle\mathcal{B}$ would be described by a Hamiltonian $\displaystyle
H;$ we may model a situation in which the wavefunction may decohere, and we
stop considering the decohered parts, by choosing a non-Hermitian
$\displaystyle H.$
One approach to bridging the gap is to choose $\displaystyle\epsilon>0$ and
consider a discrete-time query model with $\displaystyle
L_{\epsilon}:=e^{-iH\epsilon}.$ Then $\displaystyle T$ steps correspond to an
elapsed time $\displaystyle T\epsilon.$ Intuitively, the associated family of
lower bounds and algorithms should converge to a description of the
continuous-time situation as $\displaystyle\epsilon\mapsto 0^{+}.$ However, I
didn’t succeed in making all associated analysis statements rigorous.
### 5.3 Other characterizations of quantum processes
As mentioned, Section 2 is very similar to the semidefinite programming (SDP)
characterization of quantum algorithms by [2, 3]. Incidentally, an SDP
characterization of the success probability is also possible if the
transformations aren’t subunitaries, but arbitrary quantum channels between
mixed states (e.g. because they introduce errors). This can be done by an
application of the frameworks developed independently in [11, 10]. However,
the matrix size necessary here is exponential in $\displaystyle T.$
In continuous time, [14] discuss time-optimal control in a still more general
setting based on the Pontryagin maximum principle.
## 6 Conclusion and outlook
The main novelty in this note is the universal algorithm, which is simpler and
more general than the previous one based on phase detection [15, 5] and shaves
another factor of $\Theta\left(\log(1/\varepsilon)\right)$ off the runtime.
The way we obtained this algorithm, and proved its correctness, is also
unusual:
* •
Instead of specifying gates and families of states directly, we considered all
inputs at once in an associated control problem and feasible ways to
manipulate reduced density matrices involving these inputs,
* •
Instead of proving correctness starting with the correct initial state, and
showing that the final state is not too wrong after application of the
algorithm, we started with a slightly wrong initial state, and proved that the
final state will be correct when applied to that modified state.
These ideas may be useful to devise other quantum algorithms.
A query-efficient algorithm doesn’t necessarily translate into a gate-
efficient one in the usual model of quantum complexity, as the algorithm’s
unitaries may be hard to construct. For example, the query complexity of the
$\displaystyle k$-distinctness problem was characterized by Belovs in 2012 [4]
using the adversary method, but an algorithm matching this complexity (up to a
polylogarithmic factor) was only presented in 2022 by Jeffery and Zur [13]. So
it would be interesting to find conditions that $\displaystyle\overline{\pi}$
needs to fulfill so that the unitaries involved in the associated universal
algorithm are efficiently representable.
Though it is not obvious from the presentation, perhaps the closest relative
to the algorithm presented here is the adiabatic algorithm given by Brandeho
and Roland [8] in the continuous-time setting. In particular, they use the
idea of slowly moving from modified initial to modified target states as well,
and their algorithm has the same $\Theta(\log(1/\varepsilon))$ speedup
compared to [15]. The essential difference to the algorithm presented here is
that we avoid any error term for the intermediate steps of the computation —
while, for finite runtime, an adiabatic algorithm incurs a nonzero error
during the computation as well. So, conversely, it may be worth investigating
which Gram matrix evolutions more concrete adiabatic algorithms correspond to,
and attempting to optimize them based on the results.
## 7 Further references
The adversary method for quantum query algorithms has evolved over multiple
decades from the BBBV lower bound on Grover’s search problem [7]; after being
defined in [1], [19, 12, 15, 18] were some contributions along the way. The
previous most general, ”state of the art” discussion is [5]; a more
pedagogical one in [9].
## References
* [1] Andris Ambainis. Quantum lower bounds by quantum arguments. In Proceedings of the thirty-second annual ACM symposium on Theory of computing, pages 636–643, 2000.
* [2] H. Barnum, M. Saks, and M. Szegedy. Quantum query complexity and semi-definite programming. In 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings. IEEE Comput. Soc.
* [3] Howard Barnum. Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries. CoRR, abs/quant-ph/0703141, 2007.
* [4] Aleksandrs Belovs. Learning-Graph-Based Quantum Algorithm for k-Distinctness. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 207–216, 2012.
* [5] Aleksandrs Belovs. Variations on Quantum Adversary, 2015.
* [6] Aleksandrs Belovs and Duyal Yolcu. One-way ticket to las vegas and the quantum adversary. In 2023 Conference on Quantum Information Processing, 2023.
* [7] Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and Weaknesses of Quantum Computing. SIAM Journal on Computing, 26(5):1510–1523, 10 1997.
* [8] Mathieu Brandeho and Jérémie Roland. A universal adiabatic quantum query algorithm. https://arxiv.org/abs/1409.3558, 2015.
* [9] Andrew M Childs. Lecture notes on quantum algorithms. http://www.cs.umd.edu/~amchilds/qa/.
* [10] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. Theoretical framework for quantum networks. Physical Review A, 80(2), 2009.
* [11] Gus Gutoski and John Watrous. Toward a general theory of quantum games. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC ’07. ACM Press, 2007.
* [12] Peter Hoyer, Troy Lee, and Robert Spalek. Negative weights make adversaries stronger. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 526–535, 2007.
* [13] Stacey Jeffery and Sebastian Zur. Multidimensional Quantum Walks, with Application to k-Distinctness. arXiv, 2022.
* [14] Navin Khaneja, Roger Brockett, and Steffen J. Glaser. Time optimal control in spin systems. Physical Review A, 63(3), 2001.
* [15] Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert palek, and Mario Szegedy. Quantum Query Complexity of State Conversion. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science. IEEE, 10 2011.
* [16] Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information. American Association of Physics Teachers, 2002.
* [17] Maris Ozols. Unitary equivalence of purifications. https://marozols.wordpress.com/2012/05/09/unitary-equivalence-of-purifications/, 2012\.
* [18] Ben W Reichardt. Reflections for quantum query algorithms. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 560–569. SIAM, 2011.
* [19] Robert Spalek and Mario Szegedy. All quantum adversary methods are equivalent. arXiv preprint quant-ph/0409116, 2004.
* [20] Wikipedia contributors. Quantum catalyst — Wikipedia, the free encyclopedia. https://en.wikipedia.org/w/index.php?title=Quantum_catalyst&oldid=1087475350, 2022\. [Online].
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# Spin-liquid insulators can be Landau’s Fermi liquids
Michele Fabrizio International School for Advanced Studies (SISSA), Via
Bonomea 265, I-34136 Trieste, Italy
###### Abstract
The long search for insulating materials that possess low-energy
quasiparticles carrying electron’s quantum numbers except charge – inspired by
the neutral spin-1/2 excitations, the so-called spinons, exhibited by
Anderson’s resonating-valence-bond state – seems to have reached a turning
point after the discovery of several Mott insulators displaying same thermal
and magnetic properties as metals, including quantum oscillations in a
magnetic field. Here, we show that such anomalous behaviour is not
inconsistent with Landau’s Fermi liquid theory of quasiparticles at a
Luttinger surface. That is the manifold of zeros within the Brillouin zone of
the single-particle Green’s function at zero frequency, and which thus defines
the spinon Fermi surface conjectured by Anderson.
Common sense would suggest that Mott insulators and Landau’s Fermi liquids are
antinomic phases of matter that can turn one into the other only through a
Mott transition.
However, there is growing, intriguing evidence of quasiparticle-like
excitations in some Mott insulating materials. For instance, the Kondo
insulators SmB6 and YbB12 show quantum oscillations in a magnetic field,
finite specific heat, $C_{v}/T$, and thermal conductivity, $\kappa/T$,
coefficients for $T\to 0$ [1, 2, 3, 4, 5, 6], though $\kappa\sim T$ is still
debated in SmB6 [7, 8].
Evidence of finite $C_{v}/T$ and $\kappa/T$ for $T\to 0$ is also found in
candidate spin-liquid insulators: 1$T$-TaS2 [9, 10, 11], and, with some
caveats, in the organic salts EtMe3Sb[Pd(dmit)2]2 [12, 13, 14, 15, 16, 17] and
$\kappa$-(BEDT-TTF)2Cu2(CN)3 [18, 19]. Quantum oscillations in the
magnetothermal conductivity of the field induced spin-liquid state of
$\alpha$-RuCl3 have also been reported [20], even though their origin is
controversial [21].
All the above properties, at odds with the conventional view of insulators,
are commonly interpreted by the existence of neutral quasiparticles [22, 23,
24, 25, 26, 27, 28], not necessarily gapless [26], although alternative
explanations have been proposed [29, 30, 31]. Those quasiparticles are dubbed
spinons [32, 33] when they only carry the spin quantum number, which is the
case of systems whose low energy behaviour is determined by just a single
band, as we shall assume hereafter.
Despite the observed Fermi-liquid-like thermal and magnetic properties of
spinons, their emergence from spin-charge deconfinement [34] is at first sight
incompatible with Landau’s Fermi liquid theory [35, 36, 37]. This is obviously
the case of conventional Landau’s quasiparticles at a Fermi surface, the
location of poles of the single-particle Green’s function at zero frequency
and temperature, since these poles entail metallicity.
However, it has been recently shown [38] that Landau’s quasiparticles also
exist at a Luttinger surface, the manifold of zeros of the single-particle
Green’s function at zero frequency and temperature. These quasiparticles are
invisible in the single-particle spectrum, and are also incompressible [39],
thus perfectly allowed in insulators. Nonetheless, the insulating character
poses constraints to Landau’s Fermi liquid theory, most notably the vanishing
of Drude weight and of charge compressibility. Here, we show that these
constraints can be fulfilled. We conclude that a Landau Fermi liquid can well
be insulating, and analyse its physical properties with special emphasis on
the quantum oscillations in a magnetic field.
Uncovering Landau quasiparticles – In what follows, we consider a periodic
model with a single band of interacting electrons, and assume that neither
translational symmetry nor spin rotational one are broken.
The single-particle Green’s function is therefore diagonal in momentum
$\mathbf{k}$ and spin $\sigma=\uparrow,\downarrow$, and independent of the
latter. In Matsubara frequencies, ${\epsilon}=(2n+1)\pi T$, the Green’s
function satisfies Dyson’s equation
$\displaystyle G(i{\epsilon},\mathbf{k})$
$\displaystyle=\frac{\displaystyle\;1\;}{\displaystyle\;\;i{\epsilon}-{\epsilon}(\mathbf{k})-\Sigma(i{\epsilon},\mathbf{k})\;\;}\;,$
(1)
where ${\epsilon}(\mathbf{k})$ is the non-interacting energy dispersion in
momentum space measured with respect to the chemical potential, and
$\Sigma(i{\epsilon},\mathbf{k})$ the self-energy that, like
$G(i{\epsilon},\mathbf{k})$, has a real part even in ${\epsilon}$, while
$\displaystyle\text{Im}\,\Sigma(i{\epsilon},\mathbf{k})=-\text{Im}\,\Sigma(-i{\epsilon},\mathbf{k})\begin{cases}<0&{\epsilon}>0\,,\\\
>0&{\epsilon}<0\,.\end{cases}$ (2)
We define the real function
$\displaystyle Z({\epsilon},\mathbf{k})$
$\displaystyle=Z(-{\epsilon},\mathbf{k})=\left(1-\frac{\displaystyle\;\;\text{Im}\,\Sigma(i{\epsilon},\mathbf{k})\;\;}{\displaystyle\;{\epsilon}\;}\right)^{-1}\,,$
(3)
which, because of (2), varies in the interval $[0,1]$. Through
$Z({\epsilon},\mathbf{k})$ we can rewrite Eq. (1) as
$\displaystyle G(i{\epsilon},\mathbf{k})$
$\displaystyle=\frac{\displaystyle\;Z({\epsilon},\mathbf{k})\;}{\displaystyle\;\;i{\epsilon}-{\epsilon}_{*}({\epsilon},\mathbf{k})\;\;}\;,$
(4)
with real
$\displaystyle{\epsilon}_{*}({\epsilon},\mathbf{k})={\epsilon}_{*}(-{\epsilon},\mathbf{k})=Z({\epsilon},\mathbf{k})\,\Big{(}{\epsilon}(\mathbf{k})+\text{Re}\,\Sigma(i{\epsilon},\mathbf{k})\Big{)}\,.$
(5)
Landau’s Fermi liquid theory can be formally derived under the assumption that
${\epsilon}_{*}({\epsilon},\mathbf{k})$ and $Z({\epsilon},\mathbf{k})$ are
analytic, at least to leading order, in ${\epsilon}$ around ${\epsilon}=0$, as
well as in $\mathbf{k}$ close to the surface defined by
${\epsilon}_{*}(0,\mathbf{k})=0$ [38]. This assumption is equivalent to
assuming that $\Sigma(i{\epsilon},\mathbf{k})$ is analytic at any non-zero
${\epsilon}$, which includes conventional Fermi liquids as the special case of
$\Sigma(i{\epsilon},\mathbf{k})$ analytic also at ${\epsilon}=0$, but also
allows for poles of $\Sigma(i{\epsilon},\mathbf{k})$ for ${\epsilon}\to 0$.
The actual quasiparticles have energy dispersion
${\epsilon}_{*}(\mathbf{k})\equiv{\epsilon}_{*}(0,\mathbf{k})$ and residue
$Z(\mathbf{k})\equiv Z(0,\mathbf{k})$. The roots of
${\epsilon}_{*}(\mathbf{k})$ in momentum space define the quasiparticle Fermi
surface that, because of the definition (5), correspond
* •
either to the roots of
${\epsilon}(\mathbf{k})+\text{Re}\,\Sigma(0,\mathbf{k})$, the conventional
Fermi surface,
* •
or those of $Z(0,\mathbf{k})$, the so-called Luttinger surface [40].
Therefore, well-defined quasiparticles exist at Fermi as well at Luttinger
surfaces, and that despite the vanishing quasiparticle residue $Z(\mathbf{k})$
at the Luttinger surface implies the absence of quasiparticle peaks in the
physical electron density of states.
Fermi liquid properties – We recall that Landau’s Fermi liquid theory allows
calculating linear response functions at low temperature, low frequency and
long wavelength in terms of two unknown functions: the quasiparticle
dispersion ${\epsilon}_{*}(\mathbf{k})$ and the Landau parameters
$f_{\mathbf{k}\sigma,\mathbf{k^{\prime}}\sigma^{\prime}}$, where $\sigma$ and
$\sigma^{\prime}$ are the spins of the quasiparticles with momentum
$\mathbf{k}$ and $\mathbf{k^{\prime}}$, respectively. In reality, this huge
simplification just applies to densities of conserved quantities and their
currents defined through the continuity equation. Indeed, only in those cases
one can exploit the Ward-Takahashi identities and relate vertex to self-energy
corrections [36].
In a single-band periodic model, the conserved quantities are the electron
number $N=N_{\uparrow}+N_{\downarrow}$, the energy $E$, and the magnetisation
along a given axis, e.g., $M=N_{\uparrow}-N_{\downarrow}$. We denote by
$\chi_{\rho_{Q}}(\omega,\mathbf{q})$ and
$\chi^{\phantom{q}}_{J_{Q}}(\omega,\mathbf{q})$, the proper response
functions, respectively, of the density, $\rho_{Q}$, and current, $J_{Q}$,
operators associated to the conserved quantity $Q=N,E,M$, i.e., the response
functions irreducible with respect to cutting a Coulomb interaction line. The
thermodynamic susceptibilities are simply obtainable through
$\chi_{Q}=-\chi^{q}_{\rho_{Q}}$, where
$\chi^{q}_{\rho_{Q}}\equiv\chi_{\rho_{Q}}(\omega=0,\mathbf{q}\to{\boldsymbol{0}})$
is the so-called $q$-limit of the density response function. We recall that
the specific heat is actually defined through $C_{v}=\chi_{E}/T$.
In absence of impurities, the low-temperature conductivities have the standard
Drude-like expression $\sigma_{Q}(\omega)=i\,D_{Q}/(\omega+i0^{+})$, where the
Drude weights $D_{Q}$ coincide with the so-called $\omega$-limit of the
corresponding current response functions:
$D_{Q}=\chi^{\omega}_{J_{Q}}\equiv\chi^{\phantom{q}}_{J_{Q}}(\omega\to
0,\mathbf{q}={\boldsymbol{0}})$. Similarly to the specific heat, the thermal
conductivity is defined by $\sigma_{E}(\omega)/T$.
According to Landau’s Fermi-liquid theory [36, 37]
$\displaystyle\chi_{N/M}$
$\displaystyle=-2\\!\int\frac{\displaystyle\;d\mathbf{k}\;}{\displaystyle\;(2\pi)^{d}\;}\,\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k})\;}\,\Big{(}1-\text{A}_{S/A}(\mathbf{k})\Big{)}\,,$
(6) $\displaystyle D_{N/M}$
$\displaystyle=-\frac{\displaystyle\;2\;}{\displaystyle\;d\;}\\!\int\frac{\displaystyle\;d\mathbf{k}\;}{\displaystyle\;(2\pi)^{d}\;}\,\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k})\;}\;{\boldsymbol{v}}_{*}(\mathbf{k})\cdot{\boldsymbol{v}}_{S/A}(\mathbf{k})\,,$
where $d>1$ is the dimension (in $d=1$ Landau’s Fermi liquid theory is not
applicable [41]), $f(x)$ the Fermi distribution function,
${\boldsymbol{v}}_{*}(\mathbf{k})=\partial{\epsilon}_{*}(\mathbf{k})/\partial\mathbf{k}$
the quasiparticle group velocity, and
$\displaystyle\text{A}_{S/A}(\mathbf{k})$
$\displaystyle=-\int\frac{\displaystyle\;d\mathbf{k^{\prime}}\;}{\displaystyle\;(2\pi)^{d}\;}\,\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k^{\prime}})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k^{\prime}})\;}\;\text{A}^{S/A}_{\mathbf{k},\mathbf{k^{\prime}}}\,,$
(7) $\displaystyle\overline{{\boldsymbol{v}}}_{S/A}(\mathbf{k})$
$\displaystyle={\boldsymbol{v}}_{*}(\mathbf{k})+\int\frac{\displaystyle\;d\mathbf{k^{\prime}}\;}{\displaystyle\;(2\pi)^{d}\;}\,\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k^{\prime}})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k^{\prime}})\;}\;{\boldsymbol{v}}_{*}(\mathbf{k^{\prime}})\,f^{S/A}_{\mathbf{k},\mathbf{k^{\prime}}}\,.$
The parameters $\text{A}^{S/A}_{\mathbf{k},\mathbf{k^{\prime}}}$ correspond to
the $q$-limit of the quasiparticle scattering amplitudes in the spin-singlet
($S$) and spin-triplet ($A$) particle-hole channels, and are related to the
$f$-parameters, the $\omega$-limit counterparts,
$\displaystyle f_{\text{S}\,\mathbf{k},\mathbf{k^{\prime}}}$
$\displaystyle=f_{\mathbf{k}\uparrow,\mathbf{k^{\prime}}\uparrow}+f_{\mathbf{k}\uparrow,\mathbf{k^{\prime}}\downarrow}\,,$
(8) $\displaystyle f_{\text{A}\,\mathbf{k},\mathbf{k^{\prime}}}$
$\displaystyle=f_{\mathbf{k}\uparrow,\mathbf{k^{\prime}}\uparrow}-f_{\mathbf{k}\uparrow,\mathbf{k^{\prime}}\downarrow}\,,$
through the Bethe-Salpeter equation
$\displaystyle\text{A}^{S/A}_{\mathbf{k},\mathbf{k^{\prime}}}=f^{S/A}_{\mathbf{k},\mathbf{k^{\prime}}}+\int\frac{\displaystyle\;d\mathbf{p}\;}{\displaystyle\;(2\pi)^{d}\;}\,\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{p})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{p})\;}f^{S/A}_{\mathbf{k},\mathbf{p}}\,\text{A}^{S/A}_{\mathbf{p},\mathbf{k^{\prime}}}\,.$
(9)
Similarly, the specific heat $C_{v}$ and the Drude weight $K$ of the thermal
conductivity read
$\displaystyle C_{v}$
$\displaystyle=-\frac{\displaystyle\;2\;}{\displaystyle\;T\;}\,\int\frac{\displaystyle\;d\mathbf{k}\;}{\displaystyle\;(2\pi)^{d}\;}\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k})\;}\,{\epsilon}_{*}(\mathbf{k})^{2}$
(10)
$\displaystyle\quad-\frac{\displaystyle\;2\;}{\displaystyle\;T\;}\,\int\frac{\displaystyle\;d\mathbf{k}\,d\mathbf{k^{\prime}}\;}{\displaystyle\;(2\pi)^{2d}\;}\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k})\;}\,\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k}^{\prime})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k}^{\prime})\;}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad{\epsilon}_{*}(\mathbf{k})\,{\epsilon}_{*}(\mathbf{k^{\prime}})\;\text{A}^{S}_{\mathbf{k},\mathbf{k^{\prime}}}\,,$
$\displaystyle K$
$\displaystyle=-\frac{\displaystyle\;2\;}{\displaystyle\;dT\;}\,\int\frac{\displaystyle\;d\mathbf{k}\;}{\displaystyle\;(2\pi)^{d}\;}\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k})\;}\,{\epsilon}_{*}(\mathbf{k})^{2}\,\big{|}{\boldsymbol{v}}_{*}(\mathbf{k})\big{|}^{2}$
$\displaystyle\qquad+\frac{\displaystyle\;2\;}{\displaystyle\;dT\;}\,\int\frac{\displaystyle\;d\mathbf{k}\,d\mathbf{k^{\prime}}\;}{\displaystyle\;(2\pi)^{2d}\;}\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k})\;}\,\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k}^{\prime})\big{)}\;}{\displaystyle\;\partial{\epsilon}_{*}(\mathbf{k}^{\prime})\;}$
$\displaystyle\qquad\qquad\qquad\qquad{\epsilon}_{*}(\mathbf{k})\,{\epsilon}_{*}(\mathbf{k^{\prime}})\,{\boldsymbol{v}}_{*}(\mathbf{k})\cdot{\boldsymbol{v}}_{*}(\mathbf{k^{\prime}})\;f^{\text{S}}_{\mathbf{k},\mathbf{k^{\prime}}}\,.$
The first term on the right hand side of both equations is linear in
temperature $T$. Conversely, the second terms give a finite contribution at
low $T$ only upon expanding $\text{A}^{S}_{\mathbf{k},\mathbf{k^{\prime}}}$
and $f^{\text{S}}_{\mathbf{k},\mathbf{k^{\prime}}}$ in
${\epsilon}_{*}(\mathbf{k})$ and ${\epsilon}_{*}(\mathbf{k^{\prime}})$, as
well as including higher order corrections in the heat vertex as obtained
through the Ward-Takahashi identity. All those corrections yield at first
sight terms of order $T^{3}$. In reality, the expansion is not regular. For
instance, the corrections to the linear term of the specific heat are actually
of order $T^{d}$ [42, 43], with logarithmic corrections in $d=3$, $T^{3}\,\ln
1/T$. Nonetheless, at leading order in $T$ only the first terms contribute,
and thus
$\displaystyle C_{v}$
$\displaystyle\simeq\frac{\displaystyle\;2\pi^{2}\;}{\displaystyle\;3\;}\,T\,\rho_{*}\,,$
$\displaystyle K$ $\displaystyle\simeq
C_{v}\,\frac{\displaystyle\;v_{*}^{2}\;}{\displaystyle\;d\;}\,,$ (11)
where
$\displaystyle\rho_{*}\equiv\int\frac{\displaystyle\;d\mathbf{k}\;}{\displaystyle\;(2\pi)^{d}\;}\,\delta\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\,,$
(12)
is the quasiparticle density of states at the chemical potential, and
$\displaystyle
v_{*}^{2}\equiv\frac{\displaystyle\;1\;}{\displaystyle\;\rho_{*}\;}\,\int\frac{\displaystyle\;d\mathbf{k}\;}{\displaystyle\;(2\pi)^{d}\;}\,\delta\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\,\big{|}{\boldsymbol{v}}_{*}(\mathbf{k})\big{|}^{2}\,.$
(13)
Mott insulators with a Luttinger surface – Let us now consider a hypothetical
model that has only a Luttinger surface in the Brillouin zone, with finite
quasiparticle density of states at the chemical potential, $\rho_{*}\not=0$ in
Eq. (12). Since quasiparticles at the Luttinger surface are invisible in the
single-particle density of states and incompressible [39], the system
describes a non-symmetry breaking Mott insulator that may only occur at half-
filling in a single-band model.
In a Mott insulator with localised electrons, we expect that
$f_{\mathbf{k}\uparrow,\mathbf{k^{\prime}}\uparrow}\simeq 0$, which implies
$f^{S}_{\mathbf{k},\mathbf{k^{\prime}}}\simeq-f^{A}_{\mathbf{k},\mathbf{k^{\prime}}}$
and
$\text{A}^{S}_{\mathbf{k},\mathbf{k^{\prime}}}\simeq-\text{A}^{A}_{\mathbf{k},\mathbf{k^{\prime}}}$.
However, for the system to be a charge insulator, we need to impose that the
compressibility $\chi_{N}$ and charge Drude weight $D_{N}$ in Eq. (6) vanish,
which implies, through Eq. (7), that $\text{A}_{S}(\mathbf{k})=1$ plus a
correction that averages to zero on the Luttinger surface, as well as that the
flux of ${\boldsymbol{v}}_{S}(\mathbf{k})$ out of the Luttinger surface is
zero. In turn, since
$\text{A}_{A}(\mathbf{k})\simeq-\text{A}_{S}(\mathbf{k})=-1$ and
${\boldsymbol{v}}_{A}(\mathbf{k})\simeq
2{\boldsymbol{v}}_{*}(\mathbf{k})-{\boldsymbol{v}}_{S}(\mathbf{k})$, then,
through Eqs. (6) and (13), the spin susceptibility $\chi_{M}$ and Drude weight
$D_{M}$ become simply
$\displaystyle\chi_{M}$ $\displaystyle\simeq 4\rho_{*}\,,$ $\displaystyle
D_{M}$
$\displaystyle\simeq\frac{\displaystyle\;4\;}{\displaystyle\;d\;}\,\rho_{*}\,v_{*}^{2}\,.$
(14)
Comparing (14) with (11), we find that the Wilson ratio, which measures the
effective correlation strength, is
$\displaystyle
R_{\text{W}}=\frac{\displaystyle\;\pi^{2}T\;}{\displaystyle\;3C_{v}\;}\;\chi_{M}\simeq
2\,.$ (15)
Therefore, a Landau Fermi liquid characterised by a Luttinger surface without
Fermi pockets may indeed have charge properties of an insulator, while spin
and thermal ones of a metal, in that not dissimilar from a spin-liquid
insulator with gapless spinons.
We mention that conventional Fermi liquids often do not survive down to $T=0$,
since they may encounter an instability at $T_{c}>0$ towards a different phase
that, most of the times, breaks symmetries and opens gaps in the quasiparticle
spectrum. Well known examples are the superconducting and superfluidity
instabilities in normal metals and 3He, respectively. A Fermi liquid
description of such an instability is justified when quasiparticles have
already reached quantum degeneracy at $T_{c}$, which implies that $T_{c}$ must
be much smaller than the quasiparticle Fermi energy ${\epsilon}_{F}$.
Similarly, we cannot exclude that also quasiparticles at a Luttinger surface,
the gapless spinons, may become unstable at $T_{c}\ll{\epsilon}_{F}$ towards,
e.g., a magnetically ordered phase, and eventually acquire a gap. In this
case, which presumably corresponds to highly frustrated magnets, the above
Fermi liquid properties would still be observable for $T_{c}\ll
T\ll{\epsilon}_{F}$. On the contrary, if $T_{c}\sim{\epsilon}_{F}$, likely the
case of unfrustrated magnets, the quantum degenerate behaviour of
quasiparticles at the Luttinger surface cannot set in before the instability.
Quantum oscillations – The next relevant question to be addressed is whether
quasiparticles at a Luttinger surface contribute to quantum oscillations in a
magnetic field $B$. On one hand, the semiclassical approach to the de Haas-van
Alphen (dAvH) effect by Lifshitz and Kosevich [44], which just relies on the
existence of quasiparticles, would suggest a positive answer. However, the
vanishing Drude weight implies, through (6) and (7), that
$\displaystyle 0$ $\displaystyle=-\int
d\mathbf{k}\,\frac{\displaystyle\;\partial
f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\;}{\displaystyle\;\partial\mathbf{k}\;}\cdot{\boldsymbol{v}}_{S}(\mathbf{k})$
(16) $\displaystyle=\int
d\mathbf{k}\,f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\,{\boldsymbol{\nabla}}_{\mathbf{k}}\cdot{\boldsymbol{v}}_{S}(\mathbf{k})$
$\displaystyle=\int
d\mathbf{k}\,f\big{(}{\epsilon}_{*}(\mathbf{k})\big{)}\,\mathrm{Tr}\Big{(}\hat{m}_{c}(\mathbf{k})^{-1}\Big{)}\,,$
where $\hat{m}_{c}(\mathbf{k})$ is the cyclotron mass tensor as it emerges
from the Landau-Boltzmann transport equation. Considering, for simplicity, an
isotropic $\hat{m}_{c}(\mathbf{k})=m_{c}(\mathbf{k})\,\hat{I}$, it follows
that vanishing Drude weight is equivalent to vanishing $1/m_{c}(\mathbf{k})$,
or, equivalently, vanishing cyclotron frequency, once integrated over the
volume enclosed by the Luttinger surface. That hints at the absence of quantum
oscillations, in contrast to the previous observation.
To resolve this issue, we resort to Luttinger’s theory of the de Haas-van
Alphen effect in interacting electron systems [45]. Luttinger showed that the
leading oscillatory part of the free energy derives from
$\displaystyle\Delta F_{\text{osc}}$
$\displaystyle=-T\,\sum_{\epsilon}\,\text{e}^{i{\epsilon}0^{+}}\,\mathrm{Tr}\ln\Big{(}i{\epsilon}-\hat{H}_{0}-\hat{\Sigma}(i{\epsilon})\Big{)}\,,$
(17)
where $\hat{H}_{0}$ is the non-interacting Hamiltonian, which includes the
static and uniform magnetic field $B$, represented in a generic basis of
single particle wavefunctions. The self-energy matrix
$\hat{\Sigma}(i{\epsilon})$ in (17) must include any polynomial in $B$ but not
oscillatory terms in $1/B$ [45]. In matrix notations, we now define
$\displaystyle\hat{Z}({\epsilon})^{-1}$ $\displaystyle\equiv
1-\frac{\displaystyle\;\;\text{Im}\,\hat{\Sigma}({\epsilon})\>\;}{\displaystyle\;{\epsilon}\;}\;,$
(18)
which is a positive-definite matrix with eigenvalues $\geq 1$, and the
hermitian matrix
$\displaystyle\hat{H}_{*}({\epsilon})$
$\displaystyle=\sqrt{\,\hat{Z}({\epsilon})\;}\,\Big{(}\hat{H}_{0}+\text{Re}\,\hat{\Sigma}(i{\epsilon})\Big{)}\,\sqrt{\,\hat{Z}({\epsilon})\;}\,.$
(19)
With these definitions that generalise (3) and (5), the free energy component
(17) becomes
$\displaystyle\Delta F_{\text{osc}}$
$\displaystyle=-T\,\sum_{\epsilon}\,\text{e}^{i{\epsilon}0^{+}}\,\mathrm{Tr}\ln\Big{(}i{\epsilon}-\hat{H}_{*}({\epsilon})\Big{)}$
(20)
$\displaystyle\qquad+T\,\sum_{\epsilon}\,\text{e}^{i{\epsilon}0^{+}}\,\mathrm{Tr}\ln\hat{Z}({\epsilon})$
$\displaystyle\equiv\Delta F^{(1)}_{\text{osc}}+\Delta
F^{(2)}_{\text{osc}}\,.$
In conventional Fermi liquids, where $\hat{Z}(0)$ has no null eigenvalue, the
first term, $\Delta F^{(1)}_{\text{osc}}$, is the only that contributes and
yields the Lifshitz and Kosevich theory of the dHvA effect, as shown by
Luttinger [45]. Indeed, in the semiclassical limit, $\hat{H}_{*}({\epsilon})$
becomes the representation in the chosen basis of the operator
${\epsilon}_{*}\big{(}{\epsilon},\mathbf{K}(\mathbf{r})\big{)}$, Eq. (5) with
$\mathbf{k}$ replaced by
$\displaystyle\mathbf{K}(\mathbf{r})=-i\hbar\,\frac{\displaystyle\;\partial\;}{\displaystyle\;\partial\mathbf{r}\;}+\frac{\displaystyle\;e\;}{\displaystyle\;2c\;}\,{\boldsymbol{B}}\wedge\mathbf{r}\,,$
(21)
and thus
$\displaystyle\Delta F^{(1)}_{\text{osc}}$
$\displaystyle\simeq-T\sum_{\epsilon}\,\text{e}^{i{\epsilon}0^{+}}\;\mathrm{Tr}\ln\Big{(}i{\epsilon}-{\epsilon}_{*}\big{(}\mathbf{K}(\mathbf{r})\big{)}\Big{)}\,.$
(22)
After that, one can simply follow Lifshitz and Kosevich [44] and derive the
expression of the dHvA oscillations.
However, in the present case of a Luttinger surface, also $\Delta
F^{(2)}_{\text{osc}}$ in (20) may contribute since $\hat{Z}({\epsilon})$ has
zero eigenvalues at ${\epsilon}=0$. To assess their role, we note that
$\hat{Z}({\epsilon})$ in the semiclassical limit is the representation of the
operator $Z\big{(}{\epsilon},\mathbf{K}(\mathbf{r})\big{)}$, i.e., of
$Z({\epsilon},\mathbf{k})$ in Eq. (3) with
$\mathbf{k}\to\mathbf{K}(\mathbf{r})$. Moreover, the contribution of $\Delta
F^{(2)}_{\text{osc}}$ to quantum oscillations only derives from the region
around the zeros of $Z({\epsilon},\mathbf{k})$ [46], i.e., small ${\epsilon}$
and $\mathbf{k}$ close to the Luttinger surface. In that region, we can write,
without loss of generality and consistently with the analytic assumption, that
[47, *Alexei-RPP2019, 38]
$\displaystyle\Sigma(i{\epsilon},\mathbf{k})\underset{{\epsilon}\to
0}{\simeq}\frac{\displaystyle\;\Delta(\mathbf{k})^{2}\;}{\displaystyle\;\;i{\epsilon}-E(\mathbf{k})\;\;}\;,$
(23)
where $\mathbf{k}_{L}:\,E(\mathbf{k}_{L})=0$ defines the Luttinger surface
provided $\Delta(\mathbf{k}_{L})\not=0$, so that, for ${\epsilon}\simeq 0$ and
$\mathbf{k}\simeq\mathbf{k}_{L}$,
$\displaystyle Z({\epsilon},\mathbf{k})$
$\displaystyle=\frac{\displaystyle\;{\epsilon}^{2}+E(\mathbf{k})^{2}\;}{\displaystyle\;\;{\epsilon}^{2}+E(\mathbf{k})^{2}+\Delta(\mathbf{k})^{2}\;\;}$
(24)
$\displaystyle\simeq\frac{\displaystyle\;\;{\epsilon}^{2}+E(\mathbf{k})^{2}\;\;}{\displaystyle\;\;\Delta(\mathbf{k})^{2}\;\;}\;,$
$\displaystyle{\epsilon}_{*}({\epsilon},\mathbf{k})$
$\displaystyle=\frac{\displaystyle\;\;{\epsilon}(\mathbf{k})\,\big{(}{\epsilon}^{2}+E(\mathbf{k})^{2}\big{)}-E(\mathbf{k})\,\Delta(\mathbf{k})^{2}\;\;}{\displaystyle\;\;{\epsilon}^{2}+E(\mathbf{k})^{2}+\Delta(\mathbf{k})^{2}\;\;}$
$\displaystyle\simeq-E(\mathbf{k})\,,$
which, as anticipated, are analytic. Therefore,
$\displaystyle
Z\big{(}{\epsilon},\mathbf{K}(\mathbf{r})\big{)}\simeq{\epsilon}^{2}+{\epsilon}_{*}\big{(}\mathbf{K}(\mathbf{r})\big{)}^{2}$
(25)
$\displaystyle\qquad\qquad=\Big{(}i{\epsilon}-{\epsilon}_{*}\big{(}\mathbf{K}(\mathbf{r})\big{)}\Big{)}\,\Big{(}-i{\epsilon}-{\epsilon}_{*}\big{(}\mathbf{K}(\mathbf{r})\big{)}\Big{)}\,,$
and, correspondingly,
$\displaystyle\Delta F^{(2)}_{\text{osc}}$ $\displaystyle\simeq
T\sum_{\epsilon}\,\text{e}^{i{\epsilon}0^{+}}\;\bigg{[}\ln\Big{(}i{\epsilon}-{\epsilon}_{*}\big{(}\mathbf{K}(\mathbf{r})\big{)}\Big{)}$
(26)
$\displaystyle\qquad\qquad\qquad\quad+\ln\Big{(}-i{\epsilon}-{\epsilon}_{*}\big{(}\mathbf{K}(\mathbf{r})\big{)}\Big{)}\bigg{]}\,,$
so that, through (22) and (26), Eq. (20) becomes
$\displaystyle\Delta F_{\text{osc}}$ $\displaystyle\simeq
T\sum_{\epsilon}\,\text{e}^{i{\epsilon}0^{+}}\;\ln\Big{(}-i{\epsilon}-{\epsilon}_{*}\big{(}\mathbf{K}(\mathbf{r})\big{)}\Big{)}$
(27) $\displaystyle\simeq-\Delta F^{(1)}_{\text{osc}}\,,$
as can be readily verified following Lifshitz and Kosevich [44]. As a result,
quasiparticles at the Luttinger surface of a Mott insulator do yield dHvA
oscillations in the magnetisation $-\partial\Delta F_{\text{osc}}/\partial B$
alike conventional quasiparticles with dispersion
${\epsilon}_{*}(\mathbf{k})$, apart from a $\pi$-shift.
Concluding remarks – Few remarks are now in order. Conventional theories of
spin-liquids [49, 50, 51, 52, 53, 54, 55] predict that a spinon Fermi surface
is most likely associated to so-called $U(1)$ spin liquids, apart from few
known exceptions [56, 57, 58, 59]. In that $U(1)$-case, the specific heat
behaves at low temperature as $T^{2/3}$ and $T\ln 1/T$ in $d=2$ and $d=3$,
respectively [54, 60, 61], and, correspondingly, $\kappa/T$ diverges for $T\to
0$ [22]. These thermal properties, different from the observed ones, challenge
the spin-liquid interpretation. Finite $C_{v}/T$ and $\kappa/T$ for $T\to 0$
may be, for instance, attributed to magnetic impurities, assuming a gapped
spin liquid phase lacking a spinon Fermi surface [26]. However, this
explanation implies that also quantum oscillations are not due to spinons, and
thus that all intriguing thermal and magnetic properties observed in
experiments are unrelated to the purported spin liquid nature of the material,
which is a bit disappointing.
On the contrary, the Fermi liquid properties of a Mott insulator with a
Luttinger surface seem to account for all experimental evidences. Nonetheless,
the analyticity assumption on the self-energy underlying Landau’s Fermi liquid
theory is evidently incompatible with the above mentioned non-analytic
behaviour of $U(1)$ spin liquids with a spinon Fermi surface. Therefore,
either that analytic behaviour never occurs in physical models, or Mott
insulators with a Luttinger surface realise one of the above mentioned
exceptions [56, 57, 58, 59] of spin liquids with a spinon Fermi surface.
Indeed, an example of a spin liquid with $C_{v}\sim T$ is very well known: the
half-filled Hubbard model in one dimension. Even though interacting electrons
in $d=1$ behave as Luttinger liquids [62], their low-frequency, low-
temperature and long-wavelength properties are just alike conventional Fermi
liquids [63, 41, 62], including the specific heat that, as we mentioned, is
obtainable by the $q$-limit of the heat-heat response function. In particular,
the half-filled Hubbard model in $d=1$ is an insulator that has a Luttinger
surface at $k=\pm\pi/2$ as well as gapless spinons that yield a finite spin
susceptibility, a finite $C_{v}/T$, apart from corrections vanishing as powers
of $1/\ln T$, and a Wilson ratio $R_{W}=2$ for $T\to 0$ [64]. That is
precisely what our Fermi-liquid analysis predicts.
In conclusion, we have shown that non-symmetry breaking Mott insulators with a
Luttinger surface realise gapless spin liquids, where the spinons are actually
Landau’s quasiparticles at the Luttinger surface, which thus provides the
rigorous definition of Anderson’s spinon Fermi surface [32, 33]. These
quasiparticles contribute to thermal and magnetic properties, including
quantum oscillations, just like conventional quasiparticles do, despite the
system is a charge insulator.
The author is very grateful to Andrey Chubukov and Erio Tosatti for helpful
discussions and comments. This work was funded by the European Research
Council (ERC), under the European Union’s Horizon 2020 research and innovation
programme, Grant agreement No. 692670 ”FIRSTORM”.
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|
# Positivity-preserving and entropy-bounded discontinuous Galerkin method for
the chemically reacting, compressible Euler equations. Part II: The
multidimensional case
Eric J. Ching, Ryan F. Johnson, and Andrew D. Kercher Laboratories for
Computational Physics and Fluid Dynamics, U.S. Naval Research Laboratory, 4555
Overlook Ave SW, Washington, DC 20375
###### Abstract
In this second part of our two-part paper, we extend to multiple spatial
dimensions the one-dimensional, fully conservative, positivity-preserving, and
entropy-bounded discontinuous Galerkin scheme developed in the first part for
the chemically reacting Euler equations. Our primary objective is to enable
robust and accurate solutions to complex reacting-flow problems using the
high-order discontinuous Galerkin method without requiring extremely high
resolution. Variable thermodynamics and detailed chemistry are considered. Our
multidimensional framework can be regarded as a further generalization of
similar positivity-preserving and/or entropy-bounded discontinuous Galerkin
schemes in the literature. In particular, the proposed formulation is
compatible with curved elements of arbitrary shape, a variety of numerical
flux functions, general quadrature rules with positive weights, and mixtures
of thermally perfect gases. Preservation of pressure equilibrium between
adjacent elements, especially crucial in simulations of multicomponent flows,
is discussed. Complex detonation waves in two and three dimensions are
accurately computed using high-order polynomials. Enforcement of an entropy
bound, as opposed to solely the positivity property, is found to significantly
improve stability. Mass, total energy, and atomic elements are shown to be
discretely conserved.
###### keywords:
Discontinuous Galerkin method; Combustion; Detonation; Minimum entropy
principle; Positivity-preserving
††footnotetext:
DISTRIBUTION STATEMENT A. Approved for public release. Distribution is
unlimited.
## 1 Introduction
This paper is the second part of a series of two that introduces a fully
conservative, positivity-preserving, and entropy-bounded discontinuous
Galerkin (DG) method for simulating the multicomponent, chemically reacting,
compressible Euler equations. In Part I [1], we addressed the one-dimensional
case; here, we focus on the multidimensional case. The starting point of our
methodology is the fully conservative, high-order scheme previously developed
by Johnson and Kercher [2] that does not produce spurious pressure
oscillations in smooth regions of the flow or across material interfaces when
the temperature is continuous. The generation of such oscillations is a major,
well-known issue that inhibits fully conservative numerical schemes [3, 4, 5].
Nonconservative methods are a common alternative to circumvent this drawback.
The fully conservative formulation in [2] instead maintains both pressure
equilibrium and discrete conservation of mass and total energy through
consistent evaluation of both the complex thermodynamics and the resulting
semidiscrete form, as well as the proper choice of nodal basis. A series of
complex multicomponent reacting flows was computed, including a three-
dimensional reacting shear flow, which did not require additional
stabilization to calculate. Also simulated was a two-dimensional moving
detonation wave. With artificial viscosity to stabilize the flow-field
discontinuities, the correct cellular structure was predicted. However, a
linear polynomial approximation and a very fine mesh were required to maintain
robustness, illustrating the challenge of using high-order methods to achieve
stable and accurate solutions to multidimensional detonation-wave problems on
relatively coarse meshes. In light of this difficulty, we aim to develop a
positivity-preserving and entropy-bounded DG method that can robustly and
efficiently simulate complex reacting flows in multiple dimensions using high-
order polynomial approximations.
### 1.1 Summary of Part I
In Part I [1], we introduced the groundwork to construct such a formulation.
First, we established a minimum entropy principle satisfied by entropy
solutions to the multicomponent Euler equations with chemical source terms,
extending to the reacting case the proof by Gouasmi et al. [6] of a minimum
entropy principle in the nonreacting case. This principle, which states that
the spatial minimum of the specific thermodynamic entropy increases with time,
is a critical component of the theoretical basis for the formulation. In the
remainder of Part I [1], we introduced the mathematical framework in one
spatial dimension that ensures the discrete solution satisfies the following:
nonnegative species concentrations, positive density, positive pressure, and
specific thermodynamic entropy bounded from below. We discussed how to
maintain compatibility with the strategies devised in [2] to maintain pressure
equilibrium between adjacent elements. In addition, a two-point numerical
state function was derived, as part of an entropy-stable DG scheme based on
diagonal-norm summation-by-parts operators for treating stiff chemical
reactions in the reaction step of an operator splitting procedure. Applying
the coupled solver to canonical one-dimensional test cases, we demonstrated
that the formulation can achieve robust and accurate solutions on relatively
coarse meshes. Optimal high-order convergence in a smooth flow, namely
thermal-bubble advection, was obtained. Furthermore, we found that the
enforcement of only the positivity property (i.e., nonnegative concentrations
and positive density and pressure) can fail to suppress large-scale nonlinear
instabilities, but the added enforcement of an entropy bound significantly
improves robustness, to a greater degree than in the monocomponent,
calorically perfect case.
### 1.2 Contributions of Part II
In this paper, we extend the solver to two and three spatial dimensions. Our
main contributions are as follows:
* 1.
The mathematical framework in Part I [1] is enhanced to account for curved,
multidimensional elements of arbitrary shape. Our multidimensional extension
further generalizes current multidimensional positivity-preserving/entropy-
bounded DG schemes in the literature [7, 8, 9, 10] by relaxing restrictions on
the volume and surface quadrature rules, physical modeling, polynomial order
of the geometric approximation, and/or numerical flux function. We also
discuss how to deal with curved elements in the reaction step.
* 2.
We propose strategies to maintain compatibility with the procedures introduced
in [2] to preserve pressure equilibrium, which is less straightforward than in
one dimension.
* 3.
We compute complex, large-scale detonation-wave problems in two and three
dimensions with detailed chemistry. Enforcement of only the positivity
property often fails to provide adequate stabilization (even with artificial
viscosity), while enforcing an entropy bound enables robust calculations on
coarse meshes.
### 1.3 Additional background
Positivity-preserving DG schemes have emerged in recent years as a popular
numerical technique to simulate fluid flows in a robust manner. [1, Section 1]
briefly reviews the history of the development of positivity-preserving and
related entropy-bounded DG methods pioneered by Zhang and Shu [11, 7] for the
monocomponent, nonreacting Euler equations; here, we focus on the
multidimensional aspect. The key idea in constructing such positivity-
preserving/entropy-bounded schemes is to expand the element average of the
solution as a convex combination of first-order three-point systems and
pointwise values. Zhang and Shu [7] devised one such expansion on rectangular
meshes based on tensor products of Gauss-Legendre and Gauss-Lobatto rules,
which was later extended to straight-sided triangular elements [8] by
utilizing triangle-rectangle transformations [12, 13]. Lv and Ihme [9]
introduced an expansion compatible with curved elements of arbitrary shape.
The only restriction on the volume and surface quadrature rules is that the
weights be positive; the surface quadrature points need not be part of the set
of volume quadrature points, and the surface quadrature rules can be different
among the faces of a given element, which is crucial for prismatic elements
and $p$-adaptive calculations. However, the expansion in [9] relies on the
Lax-Friedrichs numerical flux, and the resulting time-step-size constraint
assumes a calorically perfect gas. Jiang and Liu [10] proposed an expansion
that is compatible with certain polygonal elements and does not rely on the
Lax-Friedrichs numerical flux, but the following assumptions are made: (a) for
a given face, the geometric Jacobian and surface normal are constant and (b)
the same surface quadrature rule is employed for each face. Said expansion can
be viewed as a generalization of that by Zhang et al. [8] for straight-sided
triangular elements. In this paper, we introduce an expansion that is
compatible with:
* 1.
curved elements of arbitrary shape
* 2.
any invariant-region-preserving numerical flux
* 3.
any combination of volume and surface quadrature rules (which can differ among
faces) with positive weights
Furthermore, no physics-based assumptions are placed on the resulting time-
step-size constraint.
### 1.4 Outline
The remainder of this paper is organized as follows. The governing equations
and basic DG discretization are reviewed in Sections 2 and 3, respectively.
The following section presents the positivity-preserving and entropy-bounded
multidimensional DG formulation for the transport step. Results for complex
two- and three-dimensional moving detonation-wave simulations are given in
Section 6. We close the paper with concluding remarks.
It is recommended that Part I [1] be read first since the formulation
developed here relies on many of the key ideas introduced in detail in the
first part. For conciseness, important concepts already discussed in Part I
are only briefly summarized in this paper.
## 2 Governing equations
The governing equations are the compressible, multicomponent, chemically
reacting Euler equations, written as
$\frac{\partial y}{\partial
t}+\nabla\cdot\mathcal{F}\left(y\right)-\mathcal{S}\left(y\right)=0$ (2.1)
where $t\in\mathbb{R}^{+}$ is time and
$y(x,t):\mathbb{R}^{d}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{m\times d}$
is the conservative state vector, given by
$y=\left(\rho v_{1},\ldots,\rho v_{d},\rho
e_{t},C_{1},\ldots,C_{n_{s}}\right)^{T}.$ (2.2)
$x=(x_{1},\ldots,x_{d})$ denotes the physical coordinates, with $d$ indicating
the number of spatial dimensions, $v=\left(v_{1},\ldots,v_{d}\right)$ is the
velocity vector, $e_{t}$ is the specific total energy, and $C_{i}$ is the
concentration of the $i$th species. $\rho$ is the density, computed as
$\rho=\sum_{i=1}^{n_{s}}\rho_{i}=\sum_{i=1}^{n_{s}}W_{i}C_{i},$
where $\rho_{i}$ is the partial density and $W_{i}$ is the molecular weight of
the $i$th species. $n_{s}$ is the total number of species, and $m=d+n_{s}+1$
is the total number of state variables. $Y_{i}=\rho_{i}/\rho$ and
$X_{i}=C_{i}/\sum_{i=1}^{n_{s}}C_{i}$ are the mass fraction and mole fraction,
respectively, of the $i$th species.
$\mathcal{F}(y):\mathbb{R}^{m}\rightarrow\mathbb{R}^{m\times d}$ in Equation
(2.1) is the convective flux, the $k$th spatial component of which is defined
as
$\mathcal{F}_{k}^{c}\left(y\right)=\left(\rho
v_{k}v_{1}+P\delta_{k1},\ldots,\rho v_{k}v_{d}+P\delta_{kd},v_{k}\left(\rho
e_{t}+P\right),v_{k}C_{1},\ldots,v_{k}C_{n_{s}}\right)^{T},$ (2.3)
where $P$ is the pressure, computed as
$P=R^{0}T\sum_{i=1}^{n_{s}}C_{i},$ (2.4)
with $T$ denoting the temperature and $R^{0}$ the universal gas constant. The
total energy is given by
$e_{t}=u+\frac{1}{2}\sum_{k=1}^{d}v_{k}v_{k},$
where $u$ is the mixture-averaged specific internal energy, calculated as
$u=\sum_{i=1}^{n_{s}}Y_{i}u_{i},$
with $u_{i}$ indicating the specific internal energy of the $i$th species. In
this work, we employ the thermally perfect gas model, with $u_{i}$
approximated by a polynomial function of temperature as
$u_{i}=\sum_{k=0}^{n_{p}+1}b_{ik}T^{k}.$ (2.5)
The specific heats at constant volume and constant pressure, $c_{v}$ and
$c_{p}$, the specific enthalpy, $h_{i}$, and the specific thermodynamic
entropy, $s_{i}$, of the $i$th species can be calculated by appropriately
differentiating/integrating Equation (2.5) and incorporating the integration
constants from the NASA curve fits [14, 15]. The mixture-averaged
thermodynamic entropy, $s$, which will be important in subsequent sections, is
given by
$s=\sum_{i=1}^{n_{s}}Y_{i}s_{i}.$
Finally, $\mathcal{S}(y):\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$ in Equation
(2.1) is the chemical source term, given by
$\mathcal{S}\left(y\right)=\left(0,\ldots,0,0,\omega_{1},\ldots,\omega_{n_{s}}\right)^{T},$
(2.6)
where $\omega_{i}$ is the production rate of the $i$th species. Additional
details on the thermodynamic relationships and chemical production rates can
be found in Part I [1].
## 3 Discontinuous Galerkin discretization
Let $\Omega\subset\mathbb{R}^{d}$ be the computational domain, which is
partitioned by $\mathcal{T}$, comprised of non-overlapping cells $\kappa$ with
boundaries $\partial\kappa$. Let $\mathcal{E}$ denote the set of interfaces
$\epsilon$, with
$\cup_{\epsilon\in\mathcal{E}}\epsilon=\cup_{\kappa\in\mathcal{T}}\partial\kappa$.
At interior interfaces, there exists $\kappa^{+},\kappa^{-}\in\mathcal{T}$
such that $\epsilon_{\mathcal{}}=\partial\kappa^{+}\cap\partial\kappa^{-}$.
$n^{+}$ and $n^{-}$ denote the outward facing normal of $\kappa^{+}$ and
$\kappa^{-}$, respectively, with $n^{+}=-n^{-}$. The discrete subspace
$V_{h}^{p}$ over $\mathcal{T}$ is defined as
$\displaystyle V_{h}^{p}$ $\displaystyle=$
$\displaystyle\left\\{\mathfrak{v}\in\left[L^{2}\left(\Omega\right)\right]^{m}\,\middle|\,\forall\kappa\in\mathcal{T},\left.\mathfrak{v}\right|_{\kappa}\in\left[\mathcal{P}_{p}(\kappa)\right]^{m}\right\\},$
(3.1)
where, for $d=1$, $\mathcal{P}_{p}(\kappa)$ is the space of polynomial
functions of degree less than or equal to $p$ in $\kappa$. For $d>1$, the
choice of polynomial space typically depends on the type of element [16].
The semi-discrete problem statement is as follows: find $y\in V_{h}^{p}$ such
that
$\displaystyle\sum_{\kappa\in\mathcal{T}}\left(\frac{\partial y}{\partial
t},\mathfrak{v}\right)_{\kappa}-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{F}\left(y\right),\nabla\mathfrak{v}\right)_{\kappa}+\sum_{\epsilon\in\mathcal{E}}\left(\mathcal{F}^{\dagger}\left(y^{+},y^{-},n\right),\left\llbracket\mathfrak{v}\right\rrbracket\right)_{\mathcal{E}}-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{S}\left(y\right),\mathfrak{v}\right)_{\kappa}=0\qquad\forall\>\mathfrak{v}\in
V_{h}^{p},$ (3.2)
where $\left(\cdot,\cdot\right)$ denotes the inner product,
$\mathcal{F}^{\dagger}\left(y^{+},y^{-},n\right)$ is the numerical flux, and
$\left\llbracket\cdot\right\rrbracket$ is the jump operator, defined such that
$\left\llbracket\mathfrak{v}\right\rrbracket=\mathfrak{v}^{+}-\mathfrak{v}^{-}$
at interior interfaces and
$\left\llbracket\mathfrak{v}\right\rrbracket=\mathfrak{v}^{+}$ at boundary
interfaces. Due to the stiff chemical source terms. we employ Strang splitting
[17] over a given interval $(t_{0},t_{0}+\Delta t]$ as
$\displaystyle\frac{\partial y}{\partial
t}+\nabla\cdot\mathcal{F}\left(y\right)=0$ $\displaystyle\textup{ in
}\Omega\times\left(t_{0},t_{0}+\nicefrac{{\Delta t}}{{2}}\right],$ (3.3)
$\displaystyle\frac{\partial y}{\partial t}-\mathcal{S}\left(y\right)=0$
$\displaystyle\textup{ in }\left(t_{0},t_{0}+\Delta t\right],$ (3.4)
$\displaystyle\frac{\partial y}{\partial
t}+\nabla\cdot\mathcal{F}\left(y\right)=0$ $\displaystyle\textup{ in
}\Omega\times\left(t_{0}+\nicefrac{{\Delta t}}{{2}},t_{0}+\Delta t\right],$
(3.5)
where Equations (3.3) and (3.5) are integrated in time with an explicit
scheme, while Equation (3.4) is solved using a fully implicit, temporal DG
discretization for ODEs.
The volume and surface terms in Equation 3.2 are evaluated with a quadrature-
free approach [18, 19]. A nodal basis is employed, such that the element-local
polynomial approximation of the solution is given by
$y_{\kappa}=\sum_{j=1}^{n_{b}}y_{\kappa}(x_{j})\phi_{j},$ (3.6)
where $n_{b}$ is the number of basis functions,
$\left\\{\phi_{1},\ldots,\phi_{n_{b}}\right\\}$ are the basis functions, and
$\left\\{x_{1},\ldots,x_{n_{b}}\right\\}$ are the node coordinates. The
nonlinear convective flux in the second and third integrals in Equation (3.2)
can be approximated as
$\mathcal{F_{\kappa}}\approx\sum_{k=1}^{n_{c}}\mathcal{F}\left(y_{\kappa}\left(x_{k}\right)\right)\varphi_{k},$
(3.7)
where $n_{c}\geq n_{b}$ and
$\left\\{\varphi_{1},\ldots,\varphi_{n_{c}}\right\\}$ is a set of polynomial
basis functions that may be different from those in Equation (3.6). As
discussed by Johnson and Kercher [2], pressure equilibrium within and between
elements is maintained under either of the following conditions:
* 1.
$n_{c}=n_{b}$ and the integration points are in the set of solution nodes
* 2.
If $n_{c}>n_{b}$ (i.e., over-integration), then the flux interpolation in
Equation (3.8) is replaced with
$\mathcal{F_{\kappa}}\approx\sum_{k=1}^{n_{c}}\mathcal{F}\left(\widetilde{y}_{\kappa}\left(x_{k}\right)\right)\varphi_{k},$
(3.8)
where $\widetilde{y}:\mathbb{R}^{m}\times\mathbb{R}\rightarrow\mathbb{R}^{m}$
is a modified state defined as
$\widetilde{y}\left(y,\widetilde{P}\right)=\left(\rho v_{1},\ldots,\rho
v_{d},\widetilde{\rho
u}\left(C_{1},\ldots,C_{n_{s}},\widetilde{P}\right)+\frac{1}{2}\sum_{k=1}^{d}\rho
v_{k}v_{k},C_{1},\ldots,C_{n_{s}}\right)^{T}.$ (3.9)
$\widetilde{\rho u}$ is a modified internal energy is evaluated from the
unmodified species concentrations and a polynomial approximation of the
pressure that interpolates onto the span of
$\left\\{\phi_{1},\ldots,\phi_{n_{b}}\right\\}$ as
$\widetilde{P}_{\kappa}=\sum_{j=1}^{n_{b}}P\left(y_{\kappa}\left(x_{j}\right)\right)\phi_{j}.$
Standard over-integration typically fails to preserve pressure equilibrium,
resulting in the generation of spurious pressure oscillations that can lead to
solver divergence. On the other hand, employing the modified flux
interpolation (3.8) in both the volume and surface flux integrals in Equation
(3.2) achieves pressure equilibrium both internally and between adjacent
elements (except in severely underresolved computations, in which appreciable
deviations from pressure equilibrium are inevitable).
As discussed in Part I [1], the linear-scaling limiter used to enforce the
positivity property and entropy boundedness does not completely eliminate
small-scale nonlinear instabilities, particularly in the vicinity of flow-
field discontinuities. Therefore, the artificial dissipation term [16]
$-\sum_{\kappa\in\mathcal{T}}\left(\mathcal{F}^{\mathrm{AV}}\left(y,\nabla
y\right),\nabla\mathfrak{v}\right)_{\kappa},$ (3.10)
where $\mathcal{F}^{\mathrm{AV}}(y,\nabla y)=\nu_{\mathrm{AV}}\nabla y$, is
added to the LHS of Equation (3.2). $\nu_{\mathrm{AV}}$ is the artificial
viscosity, computed as [2]
$\nu_{\mathrm{AV}}=\left(C_{\mathrm{AV}}+S_{\mathrm{AV}}\right)\left(\frac{h^{2}}{p+1}\left|\frac{\partial
T}{\partial y}\cdot\frac{\mathcal{R}\left(y,\nabla
y\right)}{T}\right|\right),$
where $S_{\mathrm{AV}}$ is a shock sensor based on intra-element variations
[20], $C_{\mathrm{AV}}$ is a user-defined coefficient, $h$ is the element
size, and $\mathcal{R}\left(y,\nabla y\right)$ is the strong form of the
residual (2.1). This artificial viscosity formulation was used to successfully
dampen nonphysical oscillations near flow-field discontinuities in a variety
of multicomponent-flow simulations [1, 2]. Other types of artificial viscosity
or limiters can be employed as well. Note that the integral (3.10) vanishes
for $\mathfrak{v}\in V_{h}^{0}$, which will be important in Section 4. For
further details on the basic DG discretization, boundary-condition
implementation, and conditions under which pressure oscillations are
generated, we refer the reader to [2].
## 4 Transport step: Entropy-bounded discontinuous Galerkin method in
multiple dimensions
In this section, we extend the one-dimensional positivity-preserving/entropy-
bounded DG scheme presented in Part I to multiple dimensions.
### 4.1 Preliminaries
We first review the geometric mapping from reference space to physical space,
as well as volumetric and surface quadrature rules. The minimum entropy
principle is then summarized. Finally, the split, multidimensional, first-
order three-point system, which is a building block for the general high-order
scheme, is discussed.
#### 4.1.1 Geometric mapping
Consider the mapping $x(\xi):\widehat{\kappa}\rightarrow\kappa$, with
$\widehat{\kappa}$ denoting the reference element, given by
$x(\xi)=\sum_{m=1}^{n_{g}}x_{\kappa,m}\Phi_{m}(\xi),$
where $\xi\in\mathbb{R}^{d}$ are the reference coordinates, $x_{\kappa,m}$ is
the physical coordinate of the $m$th node of $\kappa,$ $\Phi_{m}$ is the $m$th
basis function for the geometry interpolation, and $n_{g}$ is the number of
basis functions. The unit (or bi-unit) square and right triangle with unit (or
bi-unit) side length are common reference elements for quadrilateral and
triangular elements, respectively. The local solution approximation can then
be written as
$y_{\kappa}=\sum_{j=1}^{n_{b}}y_{\kappa}(x_{j})\phi(\xi),\quad
x=x(\xi)\in\kappa,\;\forall\xi\in\widehat{\kappa}.$
Let $\partial\kappa^{(f)}$ be the $f$th face of $\kappa.$ The geometric
mapping
$x\left(\zeta^{(f)}\right):\widehat{\epsilon}\rightarrow\partial\kappa^{(f)}$
for interfaces, where $\zeta^{(f)}\in\mathbb{R}^{d-1}$ are the reference
coordinates and $\widehat{\epsilon}$ is the reference face, is given by
$x\left(\zeta^{(f)}\right)=\sum_{m=1}^{n_{g,f}^{\partial}}x_{\kappa,m}^{(f)}\Phi_{m}^{(f)}\left(\zeta^{(f)}\right),$
where $x_{\kappa,m}^{(f)}$ is the physical coordinate of the $m$th node of
$\partial\kappa^{(f)}$, $\Phi_{m}^{(f)}$ is the $m$th basis function, and
$n_{g,f}^{\partial}$ is the number of basis functions. The reference face can
also be mapped to the reference element (i.e.,
$\xi\left(\zeta^{(f)}\right):\widehat{\epsilon}\rightarrow\widehat{\kappa}$).
#### 4.1.2 Quadrature rules
Consider a volumetric quadrature rule with points $\xi_{v}$ and positive
weights $w_{v}$ that satisfy
$\sum_{v=1}^{n_{q}}w_{v}=\left|\widehat{\kappa}\right|$, where
$\left|\widehat{\kappa}\right|$ is the volume of the reference element. The
volume integral over $\kappa$ of a generic function, $g(x)$, can be evaluated
as
$\int_{\kappa}g(x)dx=\int_{\widehat{\kappa}}g(x(\xi))\left|J_{\kappa}(\xi)\right|d\xi\approx\sum_{v=1}^{n_{q}}g\left(x(\xi_{v})\right)\left|J_{\kappa}(\xi_{v})\right|w_{v},$
where $J_{\kappa}$ is the geometric Jacobian and $\left|J_{\kappa}\right|$ is
its determinant. The integral evaluation is exact if $g(x)$ is a polynomial
and the quadrature rule is sufficiently accurate. Similarly, let $\zeta_{l}$
and $w_{l}^{\partial}$ be the points and positive weights of a surface
quadrature rule, with
$\sum_{l=1}^{n_{q}^{\partial}}w_{l}^{\partial}=\left|\widehat{\epsilon}\right|$,
where $\left|\widehat{\epsilon}\right|$ is the surface area of the reference
face. The surface integral over $\partial\kappa^{(f)}$ of a generic function
can be evaluated as
$\int_{\partial\kappa^{(f)}}g(x)ds=\int_{\widehat{\epsilon}}g\left(x\left(\zeta^{(f)}\right)\right)\left|J_{\partial\kappa}^{(k)}\left(\zeta^{(f)}\right)\right|d\zeta\approx\sum_{l=1}^{n_{q}^{\partial}}g\left(x\left(\zeta_{l}^{(f)}\right)\right)\left|J_{\partial\kappa}^{(f)}\left(\zeta_{l}^{(f)}\right)\right|w_{f,l}^{\partial}=\sum_{l=1}^{n_{q}^{\partial}}g\left(x\left(\zeta_{l}^{(f)}\right)\right)\nu_{f,l}^{\partial},$
where
$\nu_{f,l}^{\partial}=\left|J_{\partial\kappa}^{(f)}\left(\zeta_{l}^{(f)}\right)\right|w_{f,l}^{\partial}$
and $J_{\partial\kappa}^{(f)}$ is the surface Jacobian. The integral
evaluation is again exact if $g(x)$ is a polynomial and the quadrature rule is
sufficiently accurate. The closed surface integral over $\partial\kappa$ of a
generic function is evaluated as
$\int_{\partial\kappa}g(x)ds=\sum_{f=1}^{n_{f}}\int_{\partial\kappa^{(f)}}g(x)ds=\sum_{f=1}^{n_{f}}\int_{\widehat{\epsilon}}g\left(x\left(\zeta^{(f)}\right)\right)\left|J_{\partial\kappa}^{(f)}\left(\zeta^{(f)}\right)\right|d\zeta\approx\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}g\left(x\left(\zeta_{l}^{(f)}\right)\right)\nu_{f,l}^{\partial},$
where a different quadrature rule can be employed for each face. $n_{f}$, the
number of faces, is allowed to change among elements, but a slight abuse of
notation is done for simplicity.
Given that, as mentioned in Section 3, a quadrature-free approach [18, 19] is
employed in this work to evaluate the integrals in Equation (3.2), the reader
may question why quadrature rules are reviewed in such detail. As in Part I,
the first step in our analysis in Section 4.2 is to take $\mathfrak{v}\in
V_{h}^{0}$ in Equation (3.2) to obtain the scheme satisfied by the element
averages, which remains the same whether a quadrature-based or quadrature-free
approach is employed. Furthermore, in Section 4.2 we present our analysis in
terms of a quadrature-based approach for consistency with previous studies on
positivity-preserving and/or entropy-bounded DG methods.
#### 4.1.3 Minimum entropy principle
Let $U(y):\mathbb{R}^{m}\rightarrow\mathbb{R}$ be a given convex
(mathematical) entropy function and
$\mathcal{F}^{s}(y):\mathbb{R}^{m}\rightarrow\mathbb{R}^{d}$ the corresponding
spatial entropy flux. Satisfaction of the entropy inequality
$\frac{\partial U}{\partial t}+\nabla\cdot\mathcal{F}^{s}\leq 0,$ (4.1)
distinguishes physical solutions from nonphysical solutions when
discontinuities are present. Specifically, _entropy solutions_ are those that
satisfy (4.1) for all entropy/entropy-flux pairs. $U=-\rho s$ and
$\mathcal{F}^{s}=-\rho sv$ form a common, admissible entropy/entropy-flux pair
for the multicomponent Euler equations [6, 21].
The minimum entropy principle, which states that the spatial minimum of the
specific thermodynamic entropy is non-decreasing in time, follows from a
particular choice of entropy/entropy-flux pair,
$\left(U,\mathcal{F}^{s}\right)=\left(-\rho f_{0}(s),-\rho vf_{0}(s)\right),$
(4.2)
where $f_{0}(s)=\min\left\\{s-s_{0},0\right\\}$. Although $f_{0}(s)$ is not a
smooth function of $s$, it can be written as the limit of a sequence of smooth
functions, $f_{0}(s)=\underset{\epsilon\rightarrow 0}{\lim}f_{\epsilon}(s)$,
where $f_{\epsilon}(s)$ is given by [6]
$f_{\epsilon}(s)=\frac{1}{\epsilon}\int_{-\infty}^{\infty}f_{0}(s-\mathfrak{s})\frac{\exp\left(-\frac{\mathfrak{s}^{2}}{\epsilon^{2}}\right)}{\sqrt{\pi}}d\mathfrak{s},\quad\epsilon>0.$
According to the minimum entropy principle, we have, for $\left|x\right|\leq
R$,
$s(x,t)\geq s_{0}=\underset{\left|x\right|\leq R+v_{\max}t}{\text{Ess
inf}}s(x,0),$ (4.3)
where $R>0$ and $v_{\max}$ is the maximum speed in the domain at $t=0$. The
inequality (4.3) was first proved by Tadmor [22] for the monocomponent Euler
equations. It was recently generalized to the multicomponent, nonreacting
Euler equations by Gouasmi et al. [6] and then extended to the reacting case
in Part I [1].
#### 4.1.4 Multidimensional three-point system
Let $\mathcal{G}_{\sigma}$ be the following set:
$\mathcal{G_{\sigma}}=\left\\{y\mid C_{1}>0,\ldots,C_{n_{s}}>0,\rho
u^{*}>0,s\geq\sigma\right\\},$ (4.4)
where $\sigma\in\mathbb{R}$ and $u^{*}$ is the “shifted” internal energy [23],
$u^{*}=u-u_{0}=u-\sum_{i=1}^{n_{s}}Y_{i}b_{i0},$ (4.5)
such that $u^{*}>0$ if and only if $T>0$, provided
$c_{v,i}>0,\>i=1,\ldots,n_{s}$ [21]. Note that $y\in\mathcal{G}_{\sigma}$
implies $P(y)>0$. By the quasi-concavity of $s(y)$ (which follows from the
convexity of the entropy function $U=-\rho s$ [24]) and concavity of $\rho
u^{*}(y)$ [1] with respect to the state, for a given $\sigma$,
$\mathcal{G}_{\sigma}$ is a convex set. Note that $\mathcal{G}_{\sigma}$ is
similar to the corresponding set of admissible states in [25, 26], but with
the addition of the entropy constraint. Under the assumption that the exact
solution to the classical Riemann problem with initial data
$y\left(x,0\right)=\begin{cases}y_{1},&x<0\\\ y_{2},&x>0\end{cases}$
is an entropy solution that preserves positivity, $\mathcal{G}_{\sigma}$ is an
invariant set [27, 28, 1]. Specifically, $y_{1},y_{2}\in\mathcal{G}_{\sigma}$
implies that the average of the exact Riemann solution over a domain that
includes the Riemann fan is also in $\mathcal{G}_{\sigma}$.
Consider the one-dimensional three-point system,
$\displaystyle y_{\kappa}^{j+1}=y_{\kappa}^{j}-\frac{\Delta
t}{h}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j},y_{\kappa^{(1)}}^{j},-1\right)+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j},y_{\kappa^{(2)}}^{j},1\right)\right],$
(4.6)
which corresponds to a $p=0$, one-dimensional, element-local DG discretization
with forward Euler time stepping, where $j$ indexes the time step, $h$ is the
element size, and $\kappa^{(1)}$ and $\kappa^{(2)}$ are the elements to the
left and right of $\kappa$, respectively. If $\mathcal{F}^{\dagger}$ is an
_invariant-region-preserving_ numerical flux, then
$y_{\kappa}^{j},y_{\kappa_{L}}^{j},y_{\kappa_{R}}^{j}\in\mathcal{G}_{\sigma}$
implies $y_{\kappa}^{j+1}\in\mathcal{G}_{\sigma}$ under the condition [10]
$\frac{\Delta t\lambda}{h}\leq\frac{1}{2},$ (4.7)
where $\lambda$ is an upper bound on the maximum wave speed of the system. In
particular, $y_{\kappa}^{j+1}$ then satisfies [10, 29]
$s\left(y_{\kappa}^{j+1}\right)\geq\min\left\\{s\left(y_{\kappa_{L}}^{j}\right),s\left(y_{\kappa^{(1)}}^{j}\right),s\left(y_{\kappa^{(2)}}^{j}\right)\right\\}.$
(4.8)
Examples of invariant-region-preserving fluxes are the Godunov, Lax-
Friedrichs, HLL, and HLLC fluxes (see [10]).
Figure 4.1: Schematic of the multidimensional three-point system in Equation
(4.9).
A $p=0$, element-local DG discretization in multiple spatial dimensions in
conjunction with foward Euler time stepping gives rise to the following
multidimensional three-point system:
$\displaystyle y_{\kappa}^{j+1}=y_{\kappa}^{j}-\frac{\Delta
t}{L}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j},y_{\kappa^{(1)}}^{j},n\right)+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j},y_{\kappa^{\left(2\right)}}^{j},-n\right)\right],$
(4.9)
where a schematic of this multidimensional three-point system is given in
Figure 4.1. Our goal here is to apply the above results of the one-dimensional
three-point system to the multidimensional system. To analyze the classical
Riemann problems associated with the interfaces, we introduce a rotated
coordinate system with orthonormal basis
$\left\\{n,t_{1},\ldots,t_{d-1}\right\\}$. With this change of basis and
letting $x^{\parallel}$ denote the coordinate in the $n$-direction, the
projected (homogeneous) governing equations are given by [30]
$\frac{\partial y}{\partial t}+\frac{\partial}{\partial
x^{\parallel}}\left(n\cdot\mathcal{F}(y)\right)=0,$ (4.10)
with
$y=\left(\rho v^{\parallel},\rho v_{1}^{\perp}\ldots,\rho v_{d-1}^{\perp},\rho
e_{t},C_{1},\ldots,C_{n_{s}}\right)^{T}$
and
$n\cdot\mathcal{F}(y)=\left(\rho\left(v^{\parallel}\right)^{2}+P,\rho
v^{\parallel}v_{1}^{\perp},\ldots,\rho
v^{\parallel}v_{d-1}^{\perp},v^{\parallel}\left(\rho
e_{t}+P\right),v^{\parallel}C_{1},\ldots,v^{\parallel}C_{n_{s}}\right)^{T},$
where the velocity is expanded as
$\displaystyle v$ $\displaystyle=$
$\displaystyle\left(v^{\parallel},v_{1}^{\perp},\ldots,v_{d-1}^{\perp}\right)$
$\displaystyle=$ $\displaystyle\left(v\cdot n,v\cdot t_{1},\ldots,v\cdot
t_{d-1}\right).$
An equivalent system is
$\begin{cases}\frac{\partial\rho_{1}}{\partial t}+\frac{\partial}{\partial
x_{1}}\left(\rho_{1}v^{\parallel}\right)=0\\\ \quad\quad\quad\vdots\\\
\frac{\partial\rho_{n_{s}}}{\partial t}+\frac{\partial}{\partial
x_{1}}\left(\rho_{n_{s}}v^{\parallel}\right)=0\\\ \frac{\partial
v^{\parallel}}{\partial t}+v^{\parallel}\frac{\partial}{\partial
x_{1}}\left(v^{\parallel}\right)+\frac{1}{\rho}\frac{\partial
P(\rho,s)}{\partial x_{1}}=0\\\ \frac{\partial v_{1}^{\perp}}{\partial
t}+v^{\parallel}\frac{\partial}{\partial x_{1}}\left(v_{1}^{\perp}\right)=0\\\
\quad\quad\quad\vdots\\\ \frac{\partial v_{d-1}^{\perp}}{\partial
t}+v^{\parallel}\frac{\partial}{\partial
x_{1}}\left(v_{d-1}^{\perp}\right)=0\\\ \frac{\partial s}{\partial
t}+v^{\parallel}\frac{\partial}{\partial x_{1}}\left(s\right)=0\end{cases},$
(4.11)
where the last equation is the entropy transport equation (see [1, Section
4]). The Jacobian of (4.11) is
$\left(\begin{array}[]{cccccccc}v^{\parallel}&0&0&\rho_{1}&0&\ldots&0&0\\\
0&\ddots&0&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&v^{\parallel}&\rho_{n_{s}}&0&0\ldots&0&0\\\ \frac{1}{\rho}\frac{\partial
P}{\partial\rho}&\ldots&\frac{1}{\rho}\frac{\partial
P}{\partial\rho}&v^{\parallel}&0&0\ldots&0&\frac{1}{\rho}\frac{\partial
P}{\partial s}\\\ 0&\ldots&0&0&v^{\parallel}&0&\ldots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\\
0&\ldots&0&0&\ldots&0&v^{\parallel}&0\\\
0&\ldots&0&0&0&\ldots&0&v^{\parallel}\end{array}\right),$
which does not depend on
$v^{\perp}=\left(v_{1}^{\perp},\ldots,v_{d-1}^{\perp}\right)$. The exact
solution of the classical Riemann problem associated with (4.10) with initial
data
$y\left(x^{\parallel},0\right)=\begin{cases}y_{1},&x^{\parallel}<0\\\
y_{2},&x^{\parallel}>0\end{cases}$
can then be obtained in two steps [30, 31, Chapter 4.8]. The first step is to
solve the one-dimensional classical Riemann problem associated with the
equations
$\frac{\partial y^{\parallel}}{\partial
t}+\frac{\partial\mathcal{F}\left(y^{\parallel}\right)}{\partial
x^{\parallel}}=0,$ (4.12)
where
$y^{\parallel}=\left(\rho_{1},\ldots,\rho_{n_{s}},\rho v^{\parallel},\rho
e_{t}^{\parallel}\right)^{T}$
and
$\mathcal{F}\left(y^{\parallel}\right)=\left(\rho_{1}v^{\parallel},\ldots,\rho_{n_{s}}v^{\parallel},\rho\left(v^{\parallel}\right)^{2}+P,v^{\parallel}\left(e_{t}^{\parallel}+P\right)\right)^{T},$
with $e_{t}^{\parallel}=e_{t}-\left(v^{\parallel}\right)^{2}/2$. The second
step is to solve
$\frac{\partial}{\partial t}\left(\begin{array}[]{c}\rho v_{1}^{\perp}\\\
\vdots\\\ \rho v_{d-1}^{\perp}\end{array}\right)+\frac{\partial}{\partial
x^{\parallel}}\left(\begin{array}[]{c}\rho v^{\parallel}v_{1}^{\perp}\\\
\vdots\\\ \rho v^{\parallel}v_{d-1}^{\perp}\end{array}\right)=0.$
See [30] and [31, Chapter 4.8] for more information. Notice that
$\displaystyle\rho(y)=\rho\left(y^{\parallel}\right),\quad\rho u^{*}(y)=\rho
u^{*}\left(y^{\parallel}\right),\quad s(y)=s\left(y^{\parallel}\right),$
$\displaystyle C_{i}(y)=C_{i}\left(y^{\parallel}\right),i=1,\ldots,n_{s}.$
Consequently, since $\mathcal{G}_{\sigma}$ is an invariant set for (4.12), it
is also an invariant set for the projected equations (4.10). The invariant-
region-preserving numerical fluxes for the one-dimensional three-point system
can then be shown to be invariant-region-preserving for the multidimensional
three-point system (4.9). Specifically, as in [10] for the one-dimensional
case, the RHS of (4.9) can be rewritten as a convex combination of
$y_{\kappa}^{j}$ and exact-Riemann-solution averages. As such, if
$y_{\kappa}^{j}$, $y_{\kappa^{(1)}}^{j}$, and $y_{\kappa^{(2)}}^{j}$ are in
$\mathcal{G}_{\sigma}$, then $y_{\kappa}^{j+1}$ is also in
$\mathcal{G}_{\sigma}$ under the time-step-size constraint
$\frac{\Delta t\lambda}{L}\leq\frac{1}{2},$
where $\lambda$ is an upper bound on the maximum wave speed of the system.
Throughout this work, we employ the HLLC numerical flux [31].
The analysis of the multidimensional three-point system (4.9) is essential for
the construction of a positivity-preserving and entropy-bounded DG scheme for
$p>0$ on arbitrary, curved elements. Specifically, as we demonstrate in
Section 4.2, the element average of the solution (for $p>0$) at the $(j+1)$th
time step, $\overline{y}_{\kappa}^{j+1}$, can be decomposed into a convex
combination of both pointwise values of $y_{\kappa}^{j}(x)$ and three-point
systems involving pointwise values of $y_{\kappa}^{j}(x)$. Therefore, with the
aid of a simple limiting procedure to ensure that said pointwise values of
$y_{\kappa}^{j}(x)$ are in $\mathcal{G}_{\sigma}$,
$\overline{y}_{\kappa}^{j+1}$ will also be in $\mathcal{G}_{\sigma}$ under a
time-step-size constraint.
### 4.2 Entropy-bounded, high-order discontinuous Galerkin method in multiple
dimensions
We are now in a position to derive a time-step constraint that ensures that
the evolved element average is in $\mathcal{G}$. The average of $y_{\kappa}$
is given by
$\overline{y}_{\kappa}=\frac{1}{\left|\kappa\right|}\int_{\kappa}ydx,$ (4.13)
where $\left|\kappa\right|$ is the volume of $\kappa$. Let
$\partial\mathcal{D}_{\kappa}$ be the set of surface integration points used
to evaluate the surface integrals in Equation (3.2) (i.e., the numerical flux
terms), defined as
$\partial\mathcal{D}_{\kappa}=\bigcup_{f=1}^{n_{f}}\left\\{x\left(\zeta_{l}^{(f)}\right),l=1,\ldots,n_{q,f}^{\partial}\right\\}.$
As discussed by Lv and Ihme [9], given a sufficiently accurate quadrature
rule, the element average can be expanded as
$\displaystyle\overline{y}_{\kappa}$
$\displaystyle=\sum_{v=1}^{n_{q}}\frac{\left|J_{\kappa}(\xi_{v})\right|w_{v}}{|\kappa|}y_{\kappa}\left(\xi_{v}\right),$
$\displaystyle=\sum_{v=1}^{n_{q}}\theta_{v}y_{\kappa}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}y_{\kappa}\left(\xi\left(\zeta_{l}^{(f)}\right)\right).$
(4.14)
If
$\partial\mathcal{D}_{\kappa}\subseteq\left\\{x(\xi_{v}),v=1,\ldots,n_{q}\right\\}$,
i.e., this set of volume quadrature points contains the surface integration
points, then we can simply take
$\theta_{v}=\begin{cases}\frac{\left|J_{\kappa}(\xi_{v})\right|w_{v}}{|\kappa|}&x\left(\xi_{v}\right)\notin\partial\mathcal{D}_{\kappa}\\\
0&x\left(\xi_{v}\right)\in\partial\mathcal{D}_{\kappa}\end{cases}$
and
$\theta_{f,l}=\frac{\left|J_{\kappa}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)\right|w_{f,l}}{|\kappa|},$
where $w_{f,l}$ is the volume quadrature weight corresponding to the volume
quadrature point that satisfies $\xi_{v}=\xi\left(\zeta_{l}^{(f)}\right)$. If
$\partial\mathcal{D}_{\kappa}\nsubseteq\left\\{x(\xi_{v}),v=1,\ldots,n_{q}\right\\}$,
we can instead take
$\theta_{v}=\frac{\left|J_{\kappa}(\xi_{v})\right|w_{v}}{|\kappa|}-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}\psi_{v}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),$
where $\left\\{\psi_{1},\ldots,\psi_{n_{d}}\right\\}$ is a set of Lagrange
basis functions, with $n_{b}\leq n_{d}\leq n_{q}$, whose interpolation nodes
are located at a subset of the $n_{q}$ quadrature points, while $\psi_{v}=0$
for $v=n_{d}+1,\ldots,n_{q}$, such that
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}y_{\kappa}\left(\xi_{v}\right)$
$\displaystyle=\sum_{v=1}^{n_{q}}\left[\frac{\left|J_{\kappa}(\xi_{v})\right|w_{v}}{|\kappa|}-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}\psi_{v}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)\right]y_{\kappa}\left(\xi_{v}\right)$
$\displaystyle=\sum_{v=1}^{n_{q}}\frac{\left|J_{\kappa}(\xi_{v})\right|w_{v}}{|\kappa|}y_{\kappa}\left(\xi_{v}\right)-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}\sum_{v=1}^{n_{q}}y_{\kappa}\left(\xi_{v}\right)\psi_{v}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)$
$\displaystyle=\sum_{v=1}^{n_{q}}\frac{\left|J_{\kappa}(\xi_{v})\right|w_{v}}{|\kappa|}y_{\kappa}\left(\xi_{v}\right)-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}y_{\kappa}\left(\xi\left(\zeta_{l}^{(f)}\right)\right).$
$\theta_{f,l}$ will be related to a time-step-size constraint later in this
section. Note that
$\sum_{v=1}^{n_{q}}\theta_{v}+\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}=1.$The
positivity of the quadrature weights guarantees the existence of positive
values of $\theta_{f,l}$ that yield $\theta_{v}\geq 0$ [9]. Since
$\sum_{v}\theta_{v}+\sum_{f}\sum_{l}\theta_{f,l}=1$, the RHS of Equation
(4.14) is a convex combination of $y_{\kappa}$ evaluated at a set of points in
$\kappa$. Let $\kappa^{(f)}$ denote the $f$th neighbor of $\kappa$, and let
$\mathcal{D}_{\kappa}$ be the set of points at which the solution is evaluated
in Equation (4.14),
$\mathcal{D_{\kappa}}=\partial\mathcal{D}_{\kappa}\bigcup\left\\{x(\xi_{v}),v=1,\ldots,n_{q}\right\\}=\bigcup_{f=1}^{n_{f}}\left\\{x\left(\zeta_{l}^{(f)}\right),l=1,\ldots,n_{q,f}^{\partial}\right\\}\bigcup\left\\{x(\xi_{v}),v=1,\ldots,n_{q}\right\\}.$
Without loss of generality, we assume that the $n_{f}$th face is such that
$N=\max_{f}\left\\{n_{q,f}^{\partial}\right\\}=n_{q,n_{f}}^{\partial}$ and
define $\nu_{f,l}^{\partial}$ as
$\nu_{f,l}^{\partial}=\begin{cases}\left|J_{\partial\kappa}^{(f)}(\zeta_{l})\right|w_{f,l}^{\partial},&l=1,\ldots,n_{q,f}^{\partial}\\\
0,&l=n_{q,f}^{\partial}+1,\ldots,N\end{cases},$ (4.15)
such that
$\sum_{f=1}^{n_{f}}\sum_{l=1}^{N}\nu_{f,l}^{\partial}=\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\nu_{f,l}^{\partial}=\sum_{f=1}^{n_{f}}\left|\partial\kappa^{(f)}\right|=\left|\partial\kappa\right|,$
where $|\partial\kappa|$ is the surface area of $\kappa$ and
$\left|\partial\kappa^{(f)}\right|$ is the surface area of the $f$th face.
Standard flux interpolation, as in Equation (3.7), is assumed here. In Section
4.2.2, we will account for the modified flux interpolation in Equation (3.8).
Taking $\mathfrak{v}\in V_{h}^{0}$ in Equation (3.2), the scheme satisfied by
the element averages can then be expanded as
$\displaystyle\overline{y}_{\kappa}^{j+1}$ $\displaystyle=$
$\displaystyle\overline{y}_{\kappa}^{j}-\frac{\Delta
t}{\left|\kappa\right|}\sum_{f=1}^{n_{f}}\int_{\partial\kappa^{(f)}}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j},y_{\kappa^{(f)}}^{j},n\right)ds$
$\displaystyle=$
$\displaystyle\overline{y}_{\kappa}^{j}-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),y_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),n\left(\zeta_{l}^{(f)}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}y_{\kappa}^{j}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\left[\theta_{f,l}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),y_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),n\left(\zeta_{l}^{(f)}\right)\right)\right]$
$\displaystyle=$
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}y_{\kappa}^{j}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}-1}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}A_{f,l}+\sum_{l=1}^{N-1}\theta_{n_{f},l}B_{l}+\theta_{n_{f},N}C.$
(4.17)
Note that the volume quadrature rule used to expand
$\overline{y}_{\kappa}^{j}$ need not be explicitly used to evaluate any
integrals in Equation (3.2). Equation (4.2) and thus Equation (4.17) still
hold for the quadrature-free approach [18, 19] employed in this work since the
weights in Equation (4.15) can be taken to be the integrals of the basis
functions over the reference element, corresponding to a generalized Newton-
Cotes quadrature rule [32]. Notice that $\overline{y}_{\kappa}^{j+1}$ in
Equation (4.17) is expressed as a convex combination of $A_{f,l}$ , $B_{l}$,
$C$, and pointwise values $y_{\kappa}^{j}(x_{v})$. $A_{f,l}$, $B_{l}$, and $C$
in Equation (4.17) are given by
$\displaystyle A_{f,l}$ $\displaystyle=$ $\displaystyle
y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}}{\theta_{f,l}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),y_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),n\left(\zeta_{l}^{(f)}\right)\right)\right.$
(4.18)
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{(f)}\right)\right)\right],$
for $f=1,\ldots,n_{f}-1,\;l=1,\ldots,n_{q,f}^{\partial}$;
$\displaystyle B_{l}$ $\displaystyle=$ $\displaystyle
y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)$
(4.19) $\displaystyle-\frac{\Delta
t\nu_{n_{f},l}^{\partial}}{\theta_{n_{f},l}|\kappa|}\biggl{[}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa^{\left(n_{f}\right)}}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)$
$\displaystyle+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\biggr{]}$
$\displaystyle-\sum_{f=1}^{n_{f}-1}\frac{\Delta
t\nu_{f,l}^{\partial}}{\theta_{n_{f},l}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right],$
for $l=N-1$; and
$\displaystyle C$ $\displaystyle=$ $\displaystyle
y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)$
(4.20) $\displaystyle-\frac{\Delta
t\nu_{n_{f},N}^{\partial}}{\theta_{n_{f},N}|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa^{\left(n_{f}\right)}}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)$
$\displaystyle-\sum_{l=1}^{N-1}\frac{\Delta
t\nu_{n_{f},l}^{\partial}}{\theta_{n_{f},N}|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)$
$\displaystyle-\sum_{f=1}^{n_{f}-1}\frac{\Delta
t\nu_{f,N}^{\partial}}{\theta_{n_{f},N}|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(f\right)}\right)\right),n\left(\zeta_{N}^{\left(f\right)}\right)\right)$
$\displaystyle-\sum_{f=1}^{n_{f}-1}\sum_{l=1}^{N-1}\frac{\Delta
t\nu_{f,l}^{\partial}}{\theta_{n_{f},N}|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right).$
The above expansion relies on the conservation property of the numerical flux:
$\mathcal{F}^{\dagger}\left(y_{1},y_{2},n\right)=-\mathcal{F}^{\dagger}\left(y_{2},y_{1},-n\right).$
$A_{f,l}$ in Equation (4.18) takes the form of the three-point system (4.9).
Invoking the identity
$y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)=\sum_{f=1}^{n_{f}}\frac{\left|\partial\kappa^{(f)}\right|}{|\partial\kappa|}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),$
(4.21)
$B_{l}$ can be rewritten as
$\displaystyle B_{l}$ $\displaystyle=$
$\displaystyle\frac{\left|\partial\kappa^{(n_{f})}\right|}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t\nu_{n_{f},l}^{\partial}|\partial\kappa|}{\theta_{n_{f},l}|\kappa|\left|\partial\kappa^{(n_{f})}\right|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa^{\left(n_{f}\right)}}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right.$
(4.22)
$\displaystyle+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\biggr{]}\Biggr{\\}}$
$\displaystyle+\sum_{f=1}^{n_{f}-1}\frac{\left|\partial\kappa^{(f)}\right|}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}|\partial\kappa|}{\theta_{n_{f},l}|\kappa|\left|\partial\kappa^{(f)}\right|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right]\Biggr{\\}},$
for $l=1,\ldots,N-1$. The RHS of Equation (4.22) is a convex combination of
three-point systems and, if some $\nu_{f,l}^{\partial}$ are zero, the
pointwise value
$y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)$.
To analyze $C$, we invoke the following identities:
$y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)=\sum_{f=1}^{n_{f}}\sum_{l=1}^{N}\frac{\nu_{f,l}^{\partial}}{|\partial\kappa|}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)$
and
$\displaystyle\sum_{f=1}^{n_{f}}\sum_{l=1}^{N}\nu_{f,l}^{\partial}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\nu_{f,l}^{\partial}\mathcal{F}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)\right)\cdot
n\left(\zeta_{l}^{\left(f\right)}\right)$ $\displaystyle=$
$\displaystyle\int_{\partial\kappa}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)\right)\cdot
nds$ $\displaystyle=$
$\displaystyle\int_{\kappa}\nabla\cdot\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)\right)dx$
$\displaystyle=$ $\displaystyle 0,$
where the first line is due to the consistency property of the numerical flux
and the second line assumes sufficient accuracy of the surface quadrature
rule. $C$ in Equation (4.20) can then be rewritten as
$\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{\nu_{n_{f},N}^{\partial}}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t|\partial\kappa|}{\theta_{n_{f},N}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa^{\left(n_{f}\right)}}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)\right.$
$\displaystyle+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)\biggr{]}\Biggr{\\}}$
$\displaystyle+\sum_{l=1}^{N-1}\frac{\nu_{n_{f},l}^{\partial}}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t|\partial\kappa|}{\theta_{n_{f},N}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right]\Biggr{\\}}$
$\displaystyle+\sum_{f=1}^{n_{f}-1}\frac{\nu_{f,N}^{\partial}}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t|\partial\kappa|}{\theta_{n_{f},N}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(f\right)}\right)\right),n\left(\zeta_{N}^{\left(f\right)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{N}^{\left(f\right)}\right)\right)\right]\Biggr{\\}}$
$\displaystyle+\sum_{f=1}^{n_{f}-1}\sum_{l=1}^{N-1}\frac{\nu_{f,l}^{\partial}}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t|\partial\kappa|}{\theta_{n_{f},N}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)\right)\right],$
which is a convex combination of three-point systems (regardless of whether
some $\nu_{f,l}^{\partial}$ are zero). As such, $\overline{y}_{\kappa}^{j+1}$
is a convex combination of the following components:
* 1.
Pointwise values, $y_{\kappa}^{j}(x_{v})$, in $\kappa$
* 2.
Three-point systems involving pointwise values,
$y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)$ and
$y_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)$,
along $\partial\kappa$
* 3.
If some $\nu_{f,l}^{\partial}$ are zero, pointwise values,
$y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)$,
along $\partial\kappa$
The above results lead to the following theorem, where we use $y_{\kappa}^{-}$
to denote the exterior state along $\partial\kappa$.
###### Theorem 1.
If $y_{\kappa}^{j}(x)\in\mathcal{G}_{\sigma},\;\forall
x\in\mathcal{D_{\kappa}}$, and
$y_{\kappa}^{-,j}\in\mathcal{G}_{\sigma},\;\forall
x\in\partial\mathcal{D}_{\kappa}$, with
$\sigma\leq\min\left\\{\min\left\\{s\left(y_{\kappa}^{j}(x)\right)|x\in\mathcal{D_{\kappa}}\right\\},\min\left\\{s\left(y_{\kappa}^{-,j}(x)\right)|x\in\mathcal{\partial
D_{\kappa}}\right\\}\right\\},$ (4.23)
then $\overline{y}_{\kappa}^{j+1}$ in Equation (4.2) is also in
$\mathcal{G}_{\sigma}$ under the constraint
$\displaystyle\frac{\Delta t\lambda}{|\kappa|}$
$\displaystyle\leq\frac{1}{2}\min\left\\{L_{A},L_{B},L_{C}\right\\},$ (4.24)
$\displaystyle L_{A}$
$\displaystyle=\min\left\\{\left.\frac{\theta_{f,l}}{\nu_{f,l}^{\partial}}\right|f=1,\ldots,n_{f}-1,\;l=1,\ldots,n_{q,f}^{\partial}\right\\},$
$\displaystyle L_{B}$
$\displaystyle=\min\left\\{\left.\frac{\theta_{n_{f},l}}{\nu_{f,l}^{\partial}}\frac{\left|\partial\kappa^{(f)}\right|}{|\partial\kappa|}\right|,f=1,\ldots,n_{f},\;l=1,\ldots,\min\left\\{n_{q,f}^{\partial},N-1\right\\}\right\\},$
$\displaystyle L_{C}$
$\displaystyle=\frac{\theta_{n_{f},N}}{|\partial\kappa|},$
and the conditions
$\begin{cases}\theta_{v}\geq 0,&v=1,\ldots,n_{q}\\\
\theta_{f,l}>0,&f=1,\ldots,n_{f},\;l=1,\ldots,n_{q,f}^{\partial}.\end{cases}$
(4.25)
###### Proof.
The proof follows similar logic to that of the one-dimensional version in Part
I [1, Theorem 1]. The constraint $\Delta t\lambda/|\kappa|\leq L_{A}/2$
ensures that $A_{f,l}$ (Equation (4.18)) is in $\mathcal{G}_{\sigma}$, the
constraint $\Delta t\lambda/|\kappa|\leq L_{B}/2$ ensures that $B_{l}$
(Equation (4.22)) is in $\mathcal{G}_{\sigma}$, and the constraint $\Delta
t\lambda/|\kappa|\leq L_{C}/2$ ensures that $C$ (Equation (4.20)) is in
$\mathcal{G}_{\sigma}$. It follows from Equation (4.17) that
$\overline{y}_{\kappa}^{j+1}$ is in $\mathcal{G}_{\sigma}$. ∎
###### Remark 2.
A direct result of Theorem 1 is that
$s\left(\overline{y}_{\kappa}^{j+1}\right)\geq\min\left\\{\min\left\\{s\left(y_{\kappa}^{j}(x)\right)|x\in\mathcal{D_{\kappa}}\right\\},\min\left\\{s\left(y_{\kappa}^{-,j}(x)\right)|x\in\mathcal{\partial
D_{\kappa}}\right\\}\right\\}.$
###### Remark 3.
A simple linear-scaling limiter, described in Section 4.2.1, is directly
applied to enforce
$y_{\kappa}^{j+1}(x)\in\mathcal{G}_{s_{b,\kappa}^{j+1}},\>\forall
x\in\mathcal{D_{\kappa}}$, where $s_{b}$ is a lower bound on the specific
thermodynamic entropy. Motivated by the minimum entropy principle (4.3),
$s_{b}$ is computed in this work in an element-local manner as
$s_{b,\kappa}^{j+1}(y)=\min\left\\{s\left(y^{j}(x)\right)\left|x\in\bigcup_{f=1}^{n_{f}}\mathcal{D}_{\kappa^{(f)}}\bigcup\mathcal{D}_{\kappa}\right.\right\\},$
(4.26)
which is an approximation of the minimum entropy over $\kappa$ and the
neighboring elements. An alternative to the local entropy bound in Equation
(4.26) is the following global entropy bound:
$s_{b}(y)=\min\left\\{s\left(y(x)\right)|x\in\bigcup_{\kappa\in\mathcal{T}}\mathcal{D_{\kappa}}\right\\}.$
(4.27)
It was demonstrated that the local entropy bound (4.26) can more effectively
dampen nonlinear instabilities [1], particularly when the entropy varies
signifcantly throughout the domain. Additional information on these local and
global entropy bounds can be found in Part I [1]. This completes the
construction of a positivity-preserving, entropy-bounded, high-order DG
scheme.
###### Remark 4.
As discussed in Part I [1], in practice, we loosen some of the above
requirements. First, the maximum wave speed at a given point is simply
approximated as $\left|v\right|+c$, where $c$ is the speed of sound. Simple
algorithms to compute bounds on the wave speeds exist for the monocomponent
case [30, 33]; extending these to the multicomponent, thermally perfect case
may indeed be worthy of future investigation. A similar comment can be made
for most invariant-region-preserving numerical fluxes, which often require
wave-speed estimates. Second, we revise the definition of
$\mathcal{G}_{\sigma}$ as
$\mathcal{G}_{\sigma}=\left\\{y\mid\rho>0,\rho u^{*}>0,C_{1}\geq
0,\ldots,C_{n_{s}}\geq 0,\chi_{\sigma}\geq 0\right\\},$ (4.28)
where $\chi_{\sigma}=\rho s-\rho\sigma$ (introduced in [10]), which is
concave, and the species concentrations are permitted to be equal to zero,
given that requiring only positive concentrations would preclude the
calculation of practical problems. However, it is important to note that if
$C_{i}=0$ for some $i$, the specific thermodynamic entropy, $s$, becomes ill-
defined and the entropy functions $U=-\rho s$ and $U=-\rho f_{\epsilon}(s)$
lose convexity. Nevertheless, by making use of $0\log 0=0$ [21, Chapter 6],
$\rho s$ remains well-defined. We did not face any issues associated with
relaxing these two requirements in this study.
In the case that
$\partial\mathcal{D}_{\kappa}\nsubseteq\left\\{x(\xi_{v}),v=1,\ldots,n_{q}\right\\}$,
as described by Lv and Ihme [9], an optimization problem can be solved for
each element in a pre-processing step to maximize the RHS of (4.24). They also
introduced another simpler but more restrictive approach that reformulates the
time-step-size constraint as
$\frac{\Delta t\lambda}{\mathsf{L}}\leq\frac{1}{2}\mathsf{B},$
where $\mathsf{L}$ is a proposed length scale and $\mathsf{B}$ is tabulated in
[9] for common element shapes and quadrature rules. The constraint (4.24) can
be reformulated in a similar manner, utilizing the values of $\mathsf{B}$ in
[9]. Here, we simply prescribe the time-step size according to the linear
stability constraint with $\mathrm{CFL}=\mathcal{O}(0.1)$. Note that the
condition (4.24) is sufficient but not necessary for
$\overline{y}_{\kappa}^{j+1}$ to be in $\mathcal{G}_{\sigma}$. Therefore, as
recommended by Zhang [34], larger time-step sizes can be initially taken and,
if $\overline{y}_{\kappa}^{j+1}\notin\mathcal{G}_{\sigma}$, the time step can
be restarted with smaller $\Delta t$. The condition (4.24) excludes the
possibility of infinite loops.
If the faces are straight-sided (i.e., the surface Jacobian,
$J_{\partial\kappa}^{(f)}$, and the normal, $n$, are constant over any
$\partial\kappa^{(f)}$, for all $f$) and identical surface quadrature rules
are employed for each face, the time-step-size constraint (4.24) can be
simplified. Equation (4.17) can instead be written as
$\displaystyle\overline{y}_{\kappa}^{j+1}$ $\displaystyle=$
$\displaystyle\overline{y}_{\kappa}^{j}-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q}^{\partial}}\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),y_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)$
(4.29) $\displaystyle=$
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}y_{\kappa}^{j}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q}^{\partial}}\left[\theta_{f,l}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),y_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right]$
$\displaystyle=$
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}y_{\kappa}^{j}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}-1}\sum_{l=1}^{n_{q}^{\partial}}\theta_{f,l}A_{f,l}+\sum_{l=1}^{n_{q}^{\partial}}\theta_{n_{f},l}B_{l},$
where $\nu_{f,l}^{\partial}$ is now defined as
$\nu_{f,l}^{\partial}=\left|\partial\kappa^{(f)}\right|\widehat{w}_{l}^{\partial},$
with $\widehat{w}_{l}^{\partial}$ denoting normalized weights such that
$\sum_{l}\widehat{w}_{l}^{\partial}=1$. $A_{f,l}$ in Equation (4.29) is still
of the form (4.18) for $f=1,\ldots,n_{f}-1,\;l=1,\ldots,n_{q}^{\partial}$,
while $B_{l}$ is given by
$\displaystyle B_{l}$ $\displaystyle=$ $\displaystyle
y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)$
$\displaystyle-\frac{\Delta
t\nu_{n_{f},l}^{\partial}}{\theta_{n_{f},l}|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa^{\left(n_{f}\right)}}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)$
$\displaystyle-\sum_{f=1}^{n_{f}-1}\frac{\Delta
t\nu_{f,l}^{\partial}}{\theta_{n_{f},l}|\kappa|}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)$
for $l=1,\ldots,n_{q}^{\partial}$. With the identity
$\displaystyle\sum_{f=1}^{n_{f}}\left|\partial\kappa^{(f)}\right|\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{f=1}^{n_{f}}\left|\partial\kappa^{(f)}\right|\mathcal{F}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right)\cdot
n\left(\zeta_{l}^{\left(f\right)}\right)$ $\displaystyle=$
$\displaystyle\int_{\partial\kappa}\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right)\cdot
nds$ $\displaystyle=$
$\displaystyle\int_{\kappa}\nabla\cdot\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right)dx$
$\displaystyle=$ $\displaystyle 0,$
$B_{l}$ can be rewritten as
$\displaystyle B_{l}$ $\displaystyle=$
$\displaystyle\frac{\left|\partial\kappa^{(n_{f})}\right|}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t\widehat{w}_{l}^{\partial}|\partial\kappa|}{\theta_{n_{f},l}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa^{\left(n_{f}\right)}}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right.$
$\displaystyle+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\biggr{]}\Biggr{\\}}$
$\displaystyle+\sum_{f=1}^{n_{f}-1}\frac{\left|\partial\kappa^{(f)}\right|}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t\widehat{w}_{l}^{\partial}|\partial\kappa|}{\theta_{n_{f},l}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right]\Biggr{\\}}.$
$B_{l}$ is therefore a convex combination of three-point systems. The time-
step-size constraint can then be modified as
$\displaystyle\frac{\Delta t\lambda}{|\kappa|}$
$\displaystyle\leq\frac{1}{2}\min\left\\{L_{A},L_{B}\right\\},$ (4.30)
$\displaystyle L_{A}$
$\displaystyle=\min\left\\{\left.\frac{\theta_{f,l}}{\left|\partial\kappa^{(f)}\right|\widehat{w}_{l}^{\partial}}\right|f=1,\ldots,n_{f}-1,\;l=1,\ldots,n_{q}^{\partial}\right\\},$
$\displaystyle L_{B}$
$\displaystyle=\min\left\\{\left.\left.\frac{\theta_{n_{f},l}}{\left|\partial\kappa\right|\widehat{w}_{l}^{\partial}}\right|\right|,f=1,\ldots,n_{f},\;l=1,\ldots,n_{q}^{\partial}\right\\},$
which is similar to that in [10].
#### 4.2.1 Limiting procedure
The limiting procedure to enforce
$y_{\kappa}^{j+1}(x)\in\mathcal{G}_{s_{b}},\>\forall
x\in\mathcal{D_{\kappa}}$, is the same as in Part I. For completeness, we
summarize it here. The $j$+1 superscript and $\kappa$ subscript are dropped
for brevity. $\overline{y}_{\kappa}^{j}(x)$ is assumed to be in
$\mathcal{G}_{s_{b}}$. The limiting operator is of the same form as in [35],
[34], [10], [29], and related papers.
1. 1.
Positive density: if $\rho(x)>\epsilon,\>\forall x\in\mathcal{D}_{\kappa}$,
where $\epsilon>0$ is a small number (e.g., $\epsilon=10^{-10}$), then set
$C_{i}^{(1)}=C_{i}=\sum_{j=1}^{n_{b}}C_{i}(x_{j})\phi_{j},i=1,\ldots,n_{s}$;
otherwise, compute
$C_{i}^{(1)}=\overline{C}_{i}+\theta^{(1)}\left(C_{i}-\overline{C}_{i}\right),\quad\theta^{(1)}=\frac{\rho(\overline{y})-\epsilon}{\rho(\overline{y})-\underset{x\in\mathcal{D}}{\min}\rho(y(x))}.$
for $i=1,\ldots,n_{s}.$
2. 2.
Nonnegative concentrations: if $C_{i}^{(1)}(x)\geq 0,\>\forall
x\in\mathcal{D}_{\kappa}$, then set
$C_{i}^{(2)}=C_{i}^{(1)},i=1,\ldots,n_{s}$; otherwise, compute
$C_{i}^{(2)}=\overline{C}_{i}+\theta^{(2)}\left(C_{i}^{(1)}-\overline{C}_{i}\right),\quad\theta^{(2)}=\frac{\overline{C}_{i}}{\overline{C}_{i}-\underset{x\in\mathcal{D}}{\min}C_{i}^{(1)}(x)}.$
Let $y^{(2)}=\left(\rho v_{1},\ldots,\rho v_{d},\rho
e_{t},C_{1}^{(2)},\ldots,C_{n_{s}}^{(2)}\right)$.
3. 3.
Positive temperature: if $\rho u^{*}\left(y^{(2)}(x)\right)>\epsilon,\>\forall
x\in\mathcal{D}_{\kappa}$, then set $y^{(3)}=y^{(2)}$; otherwise, compute
$y^{(3)}=\overline{y}+\theta^{(3)}\left(y^{(2)}-\overline{y}\right),\quad\theta^{(3)}=\frac{\rho
u^{*}(\overline{y})-\epsilon}{\rho
u^{*}(\overline{y})-\underset{x\in\mathcal{D}}{\min}\rho u^{*}(y^{(2)}(x))}.$
The “positivity-preserving limiter” refers to the limiting procedure up to
this point.
4. 4.
Entropy constraint: if $\chi\left(y^{(3)}(x)\right)\geq 0,\>\forall
x\in\mathcal{D}_{\kappa}$, then set $y^{(4)}=y^{(3)}$; otherwise, compute
$y^{(4)}=\overline{y}+\theta^{(4)}\left(y^{(3)}-\overline{y}\right),\quad\theta^{(4)}=\frac{\chi(\overline{y})}{\chi(\overline{y})-\underset{x\in\mathcal{D}}{\min}\chi(y^{(3)}(x))}.$
The “entropy limiter” refers to the limiting procedure up to this point.
$y^{(4)}$ then replaces $y$ as the solution. This limiting procedure is
applied at the end of every RK stage. It is conservative and in general
preserves the formal order of accuracy for smooth solutions [7, 34, 24, 9,
10].
#### 4.2.2 Modified flux interpolation
We now discuss how to account for over-integration with the modified flux
interpolation in Equation (3.8). The scheme satisfied by the element averages
becomes
$\displaystyle\overline{y}_{\kappa}^{j+1}$ $\displaystyle=$
$\displaystyle\overline{y}_{\kappa}^{j}-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\widetilde{y}_{\kappa^{(k)}}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)$
(4.31) $\displaystyle=$
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}y_{\kappa}^{j}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\left[\theta_{f,l}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\widetilde{y}_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right]$
$\displaystyle=$
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}y_{\kappa}^{j}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}-1}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}\widetilde{A}_{f,l}+\sum_{l=1}^{N-1}\theta_{n_{f},l}\widetilde{B}_{l}+\theta_{n_{f},N}\widetilde{C}.$
Following the same steps as in Section 4.2, $\widetilde{A}_{f,l}$,
$\widetilde{B}_{l}$, and $\widetilde{C}$ can be written as
$\displaystyle\widetilde{A}_{f,l}$ $\displaystyle=$ $\displaystyle
y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}}{\theta_{f,l}|\kappa|}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),\widetilde{y}_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),n\left(\zeta_{l}^{(f)}\right)\right)\right.$
(4.32)
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{(f)}\right)\right)\right],$
$\displaystyle\widetilde{B}_{l}$ $\displaystyle=$
$\displaystyle\frac{\left|\partial\kappa^{(n_{f})}\right|}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t\nu_{n_{f},l}^{\partial}|\partial\kappa|}{\theta_{n_{f},l}|\kappa|\left|\partial\kappa^{(n_{f})}\right|}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa^{\left(n_{f}\right)}}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right.$
(4.33)
$\displaystyle+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\biggr{]}\Biggr{\\}}$
$\displaystyle+\sum_{f=1}^{n_{f}-1}\frac{\left|\partial\kappa^{(f)}\right|}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}|\partial\kappa|}{\theta_{n_{f},l}|\kappa|\left|\partial\kappa^{(f)}\right|}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right]\Biggr{\\}},$
and
$\displaystyle\widetilde{C}$ $\displaystyle=$
$\displaystyle\frac{\nu_{n_{f},N}^{\partial}}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t|\partial\kappa|}{\theta_{n_{f},N}|\kappa|}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa^{\left(n_{f}\right)}}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)\right.$
$\displaystyle+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)\biggr{]}\Biggr{\\}}$
$\displaystyle+\sum_{l=1}^{N-1}\frac{\nu_{n_{f},l}^{\partial}}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t|\partial\kappa|}{\theta_{n_{f},N}|\kappa|}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right)\right]\Biggr{\\}}$
$\displaystyle+\sum_{f=1}^{n_{f}-1}\frac{\nu_{f,N}^{\partial}}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t|\partial\kappa|}{\theta_{n_{f},N}|\kappa|}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(f\right)}\right)\right),n\left(\zeta_{N}^{\left(f\right)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{N}^{\left(f\right)}\right)\right)\right]\Biggr{\\}}$
$\displaystyle+\sum_{f=1}^{n_{f}-1}\sum_{l=1}^{N-1}\frac{\nu_{f,l}^{\partial}}{|\partial\kappa|}\Biggl{\\{}y_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right)-\frac{\Delta
t|\partial\kappa|}{\theta_{n_{f},N}|\kappa|}\left[\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),n\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{N}^{\left(n_{f}\right)}\right)\right),-n\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)\right)\right].$
Unfortunately, the corresponding three-point systems are not necessarily of
the type (4.9) since in general,
$y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)\neq\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)$.
The incompatibility is a result of expressing $\overline{y}_{\kappa}$ as a
convex combination of pointwise values of $y_{\kappa}(x)$ (as opposed to
$\widetilde{y}_{\kappa}(x)$). Since the element average of
$\widetilde{y}_{\kappa}$, denoted $\overline{\widetilde{y}}_{\kappa}$, is not
necessarily equal to $\overline{y}_{\kappa}$, $\overline{y}_{\kappa}$ cannot
be directly written as a convex combination of pointwise values of
$\widetilde{y}_{\kappa}(x)$. In the one-dimensional case, if the nodal set
includes the endpoints, then this issue is circumvented since
$y_{\kappa}^{j}(x_{L})=\widetilde{y}_{\kappa}^{j}(x_{L})$ and
$y_{\kappa}^{j}(x_{R})=\widetilde{y}_{\kappa}^{j}(x_{R})$. However, the
multidimensional case is more complicated. One simple approach, assuming that
the nodal set includes surface points that can be used for integration, is as
follows:
* 1.
Compute $y_{\kappa}^{j+1}$ using over-integration with the modified flux
interpolation (3.8).
* 2.
If $\overline{y}_{\kappa}^{j+1}\in\mathcal{G}_{s_{b}}$, then proceed to the
next time step. This will typically be true since in general,
$y_{\kappa}^{j}\left(x^{(f)}\left(\zeta_{l}\right)\right)\approx\widetilde{y}_{\kappa}^{j}\left(x^{(f)}\left(\zeta_{l}\right)\right)$
and the conditions laid out in Section 4.2 are not necessary for
$\overline{y}_{\kappa}^{j+1}$ to be in $\mathcal{G}_{s_{b}}$.
* 3.
In the rare case that $\overline{y}_{\kappa}^{j+1}\notin\mathcal{G}_{s_{b}}$,
recompute $y_{\kappa}^{j+1}$ with integration points that are in the nodal
set, which trivially maintains pressure equilibrium.
$\overline{y}_{\kappa}^{j+1}$ is then guaranteed to be in
$\mathcal{G}_{s_{b}}$ since $y_{\kappa}=\widetilde{y}_{\kappa}$ at the
solution nodes.
Note that this would only need to be done for the surface integrals; since the
second term in Equation (3.2) (i.e., the volumetric flux integral) does not
factor into the scheme satisfied by the element averages, over-integration
with the modified flux interpolation can be freely employed in said integral.
Furthermore, if $\overline{y}_{\kappa}^{j+1}$ satisfies the positivity
property but $s\left(\overline{y}_{\kappa}^{j+1}\right)<s_{b}$ (and therefore
$\overline{y}_{\kappa}^{j+1}\notin\mathcal{G}_{s_{b}}$), then it may still be
reasonable to proceed to the next time step, provided that
$s\left(y_{\kappa}^{j}(x)\right)\not\ll s_{b},\>\forall
x\in\mathcal{D_{\kappa}}$.
It should also be noted that using over-integration with the modified flux
interpolation (3.8), $\overline{y}_{\kappa}^{j+1}$ is guaranteed to at least
satisfy the positivity property under a different time-step-size constraint.
To show this, let $\mathcal{G}$ denote the set
$\mathcal{G}=\left\\{y\mid C_{1}\geq 0,\ldots,C_{n_{s}}\geq 0,\rho>0,\rho
u^{*}>0\right\\},$
and, as a representative example, rewrite Equation (4.32) as
$\displaystyle\widetilde{A}_{f,l}$ $\displaystyle=$
$\displaystyle\acute{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}}{\theta_{f,l}|\kappa|}\Delta\mathcal{F}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),$
where
$\displaystyle\acute{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)=$
$\displaystyle
y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}}{\theta_{f,l}|\kappa|}\left[\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),y_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),n\left(\zeta_{l}^{(f)}\right)\right)\right.$
$\displaystyle\left.+\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{(f)}\right)\right)\right]$
$\displaystyle\Delta\mathcal{F}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)=$
$\displaystyle\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),\widetilde{y}_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),n\left(\zeta_{l}^{(f)}\right)\right)$
$\displaystyle-\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),y_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right),n\left(\zeta_{l}^{(f)}\right)\right)$
$\displaystyle+\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{(f)}\right)\right)$
$\displaystyle-\mathcal{F}^{\dagger}\left(y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),y_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(n_{f}\right)}\right)\right),-n\left(\zeta_{l}^{(f)}\right)\right).$
$\acute{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{(f)}\right)\right)$ is
expressed as a three-point system of the type (4.9) and is therefore in
$\mathcal{G}$ under the constraint (4.24). Since the modified flux
interpolation (3.8) does not alter species concentrations or momentum, the
components of $\Delta\mathcal{F}$ corresponding to the concentrations are
equal to zero. Then, according to Lemma (5) in B, $\widetilde{A}_{f,l}$ is
also in $\mathcal{G}$ if
$\frac{\Delta
t}{\left|\kappa\right|}<\frac{\theta_{f,l}}{\nu_{f,l}^{\partial}\alpha^{*}\left(\acute{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\Delta\mathcal{F}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)\right)},$
(4.34)
where $\alpha^{*}$ is defined as in (B.1). In general, since
$y_{\kappa}^{j}\left(x^{(f)}\left(\zeta_{l}\right)\right)\approx\widetilde{y}_{\kappa}^{j}\left(x^{(f)}\left(\zeta_{l}\right)\right)$,
$\Delta\mathcal{F}$ and thus $\alpha^{*}$ are small, such that the constraint
(4.34) is not restrictive. The same analysis can be performed for
$\widetilde{B}_{l}$ and $C$ to yield similar constraints on $\Delta t$. In
practice, we do not find it necessary to explicitly account for these
additional constraints.
Though not pursued in this work, in A, we present an alternative approach
compatible with over-integration. Said approach utilizes an additional
auxiliary polynomial to guarantee
$\overline{y}_{\kappa}^{j+1}\in\mathcal{G}_{s_{b}}$ while employing the
modified flux interpolation (3.8).
## 5 Reaction step
In this section, we briefly discuss the reaction step, most of which is
independent of the number of spatial dimensions. Only the multidimensional
considerations are detailed here; the reader is referred to Part I [1] for
more information.
The element-local, semi-discrete form of Equation (3.4) is written as
$\displaystyle\int_{\kappa}\mathfrak{v}^{T}\frac{\partial y}{\partial
t}dx-\int_{\kappa}\mathfrak{v}^{T}\mathcal{S}(y)dx=0,\;\forall\mathfrak{v}\in
V_{h}^{p}.$ (5.1)
We approximate $\mathcal{S}\left(y\right)$ locally as a polynomial in
$V_{h}^{p}$ as
$\mathcal{S}_{\kappa}\approx\sum_{j=1}^{n_{b}}\mathcal{S}\left(y\left(x_{j}\right)\right)\phi_{j},$
giving the following spatially decoupled system of ODEs advanced at the
solution nodes from $t=t_{0}$ to $t=t_{f}$:
$\frac{d}{dt}y_{\kappa}\left(x_{j},t\right)-\mathcal{S}\left(y_{\kappa}\left(x_{j},t\right)\right)=0,\quad
j=1,\ldots,n_{b}.$
Suppose it can be guaranteed that
$\displaystyle y_{\kappa}\left(x_{j},t_{f}\right)$
$\displaystyle\in\mathcal{G}_{s\left(y_{\kappa}\left(x_{j},t_{0}\right)\right)},\quad
j=1,\ldots,n_{b}.$ (5.2)
With $s_{b}$ now given by
$s_{b}=\min_{j=1,\ldots,n_{b}}s\left(y_{\kappa}\left(x_{j},t_{0}\right)\right),$
$\overline{y}_{\kappa}(t_{f})$ is in $\mathcal{G}_{s_{b}}$ under the following
two conditions:
* 1.
The nodal set corresponds to the quadrature points of a rule with positive
weights (e.g., Gauss-Lobatto nodes).
* 2.
Said quadrature rule is sufficiently accurate to compute
$\overline{y}_{\kappa}$.
With $\overline{y}_{\kappa}(t_{f})\in\mathcal{G}_{s_{b}}$, the limiting
procedure in Section 4.2.1 can then be applied to enforce
$y_{\kappa}\left(x,t_{f}\right)\in\mathcal{G}_{s_{b}},\>\forall
x\in\mathcal{D}_{\kappa}$ (unless
$\mathcal{D}_{\kappa}=\left\\{x_{j},j=1,\ldots,n_{b}\right\\}$, in which case
the limiting procedure is superfluous). However, if the geometric Jacobian is
not constant, the quadrature rule may no longer be sufficiently accurate to
compute $\overline{y}_{\kappa}$. One remedy is to first expand $y_{\kappa}$ as
$y_{\kappa}=\sum_{i=1}^{n_{a}}y_{\kappa}(x_{i})\Psi_{i},$
where $\left\\{\Psi_{1},\ldots,\Psi_{n_{a}}\right\\}$ is a basis of
$V_{h}^{\widehat{p}}$, with $n_{a}>n_{b}$ and $\widehat{p}>p$, and then solve
$\displaystyle\int_{\kappa}\mathfrak{v}^{T}\frac{\partial y}{\partial
t}dx-\int_{\kappa}\mathfrak{v}^{T}\mathcal{S}(y)dx=0,\;\forall\mathfrak{v}\in
V_{h}^{\widehat{p}}$ (5.3)
for $y\in V_{h}^{\widehat{p}}$. Following the same procedure as above, the
resulting spatially decoupled system of ODEs is
$\frac{d}{dt}y_{\kappa}\left(x_{i},t\right)-\mathcal{S}\left(y_{\kappa}\left(x_{i},t\right)\right)=0,\quad
i=1,\ldots,n_{a}.$ (5.4)
Finally, $L^{2}$ projection is applied to project $y$ from
$V_{h}^{\widehat{p}}$ to $V_{h}^{p}$. Since $L^{2}$ projection is a
conservative operation, $\overline{y}_{\kappa}(t_{f})\in\mathcal{G}_{s_{b}}$,
where $s_{b}$ is now given by
$s_{b}=\min_{i=1,\ldots,n_{a}}s\left(y_{\kappa}\left(x_{i},t_{0}\right)\right).$
The limiter can then be employed to enforce
$y_{\kappa}\left(x,t_{f}\right)\in\mathcal{G}_{s_{b}},\>\forall
x\in\mathcal{D}_{\kappa}$, unless
$\mathcal{D}_{\kappa}=\left\\{x_{i},i=1,\ldots,n_{a}\right\\}$. Note that in
[36] and [37], slightly modified ODEs are similarly solved at a generic set of
quadrature points.
Each system of ODEs is solved using a DG discretization in time. Since the
remainder of the reaction step is identical to that in the one-dimensional
case, we refer the reader to Part I [1] for further details on the temporal
discretization, as well as techniques to ensure that (5.2) is satisfied.
## 6 Multidimensional results
First, we present solutions to a two-dimensional moving detonation on a series
of increasingly refined meshes over a range of polynomial orders. Next, we
present the solution to a three-dimensional, large-scale, moving detonation
wave in order to demonstrate the utility of the proposed formulation. The
SSPRK2 time integration scheme is used to temporally advance the solutions
presented here. All simulations are performed using a modified version of the
JENRE® Multiphysics Framework [38, 2] that incorporates the developments and
extensions described in this work.
### 6.1 Two-dimensional detonation wave
This test case is the two-dimensional extension of the one-dimensional
hydrogen-oxygen detonation wave diluted in Argon presented in Part I [1]. The
computational domain is
$\text{$\Omega$}=\left(0,0.45\right)\mathrm{m}\times\left(0,0.06\right)\mathrm{m}$,
with walls at the left, right, bottom, and top boundaries. The time step size
is prescribed according to the linear-stability constraint with
$\mathrm{CFL}=0.1$. The detonation wave moves from left to right. To perturb
the detonation, two high-temperature/high-pressure circular regions,
$\displaystyle\mathcal{C}_{1}$
$\displaystyle=\left\\{x\left|\sqrt{\left(x_{1}-0.021\right)^{2}+\left(x_{2}-0.015\right)^{2}}<0.0025\text{
m}\right.\right\\},$ $\displaystyle\mathcal{C}_{2}$
$\displaystyle=\left\\{x\left|\sqrt{\left(x_{1}-0.022\right)^{2}+\left(x_{2}-0.044\right)^{2}}<0.0025\text{
m}\right.\right\\},$
are placed to the right of the initial detonation wave. The initial conditions
are given by
$\begin{array}[]{cccc}\qquad\qquad\qquad\left(v_{1},v_{2}\right)&=&\left(0,0\right)\text{
m/s},\\\
X_{Ar}:X_{H_{2}O}:X_{OH}:X_{O_{2}}:X_{H_{2}}&=&\begin{cases}8:2:0.1:0:0\\\
7:0:0:1:2\end{cases}&\begin{array}[]{c}x_{1}<0.015\text{
m},x\in\mathcal{C}_{1},x\in\mathcal{C}_{2}\\\
\mathrm{otherwise}\end{array},\\\ \qquad\qquad\qquad\qquad
P&=&\begin{cases}{5.50}\mathrm{e}{5}&\text{ Pa}\\\ {6.67}\mathrm{e}{3}&\text{
Pa}\end{cases}&\begin{array}[]{c}x_{1}<0.015\text{
m},x\in\mathcal{C}_{1},x\in\mathcal{C}_{2}\\\
\mathrm{otherwise}\end{array},\\\ \qquad\qquad\qquad\qquad
T&=&\begin{cases}3500&\text{ K}\\\ 300\text{\hskip 10.00002pt\hskip
10.00002pt}&\text{ K}\end{cases}&\begin{array}[]{c}x_{1}<0.015\text{
m},x\in\mathcal{C}_{1},x\in\mathcal{C}_{2}\\\
\mathrm{otherwise}\end{array}.\end{array}$ (6.1)
No smoothing of the discontinuities in the initial conditions is performed.
This flow was also computed by Oran et al. [39], but with a slightly longer
domain. Their simulations revealed a complex cellular structure wherein each
cell is of size $0.055\text{ m}\times 0.03\text{ m}$. These results were
supported by the experiments conducted by Lefebvre et al. [40]. In addition,
Houim and Kuo [41] and Lv and Ihme [36] simulated this case with a domain of
height $0.03$ m. In all the aforementioned simulations (not including those
here), the solutions were initialized from a one-dimensional detonation.
In previous work, Johnson and Kercher [2] computed this flow with $p=1$ and
artificial viscosity for stabilization. The Westbrook mechanism [42] was
employed. Complex flow features, such as Kelvin-Helmholtz instabilities,
pressure waves, and triple points, were well-captured, and the correct
cellular structure was predicted. In particular, given that their domain
height was $0.06$ m, there were two cells in the vertical direction. However,
a very fine mesh, with spacing $h=9\times 10^{-5}$ m, was required. Increasing
$h$ or $p$ led to instabilities that could not be cured with the artificial
viscosity, resulting in solver divergence. Although increasing $p$ enhances
the resolution, the solution also generally becomes more susceptible to
spurious oscillations. Here, we aim to achieve robust solutions with $p\geq 1$
and relatively coarse meshes using the proposed entropy-bounded DG
formulation. Gmsh [43] is used to generate unstructured triangular meshes with
the following characteristic mesh sizes: $2h$, $4h$, $8h$, $16h$, $32h$, and
$64h$.
Figure 6.1 displays the distribution of OH mole fraction obtained from $p=2$
solutions at $t=200$ $\mu\mathrm{s}$ on a sequence of meshes. For $64h$,
although the solution is stable, the post-shock flow is extremely smeared due
to the exceedingly low resolution, and some spurious artifacts near the
leading shock are present. Interestingly, the detonation-front location is
nevertheless accurately predicted. For $32h$, despite evident smearing of the
post-shock flow, the overall flow topology can be discerned. With each
successive refinement of the mesh, the flow features behind the detonation
front become sharper. For $4h$ and $2h$, the flow topology is very well-
captured, including the triple points that connect the Mach stems and incident
shock, the transverse waves traveling in the vertical directions, and the
vortices that form behind the detonation front, demonstrating the ability of
the developed entropy-bounded DG formulation to achieve accurate solutions to
complex reacting flows. Furthermore, in all cases, the solution is stable
throughout the simulation, illustrating the robustness of the proposed
formulation.
Figure 6.1: OH mole-fraction field for a two-dimensional moving detonation
wave at $t=200$ $\mu\mathrm{s}$, computed with $p=2$ on a sequence of meshes,
where $h=9\times 10^{-5}$ m. The initial conditions are given in Equation
(6.1).
Figure 6.2 shows the maximum-pressure history, $P^{*}$, where
$P^{*,j+1}(x)=\max\left\\{P^{j+1}(x),P^{*,j}(x)\right\\}$, for the $p=2$
solutions at $t=200$ $\mu\mathrm{s}$. This quantity reveals the expected
diamond-like cellular structure, with two cells in the vertical direction. The
$64h$ lacks any clear cellular structure due to the excessive smearing.
However, detonation cells can be discerned in the $32h$ case, despite the
coarse resolution. From $32h$ to $8h$, the cellular structure begins to
dissipate towards the right of the domain, though it nevertheless remains
intact. At $4h$ and $2h$, the detonation cells remain sharp throughout.
Figure 6.2: Maximum-pressure history,$P^{*}$, where
$P^{*,j+1}(x)=\max\left\\{P^{j+1}(x),P^{*,j}(x)\right\\}$, for a two-
dimensional moving detonation wave at $t=200$ $\mu\mathrm{s}$ computed with
$p=2$ on a sequence of uniformly refined meshes, where $h=9\times 10^{-5}$ m.
The initial conditions are given in Equation (6.1).
It remains to be seen how much smearing of the solution at and behind the
detonation front is acceptable in a larger-scale configuration, although it is
encouraging that the front location and cellular structure are overall well-
predicted even with extremely coarse meshes. However, a thorough investigation
of this matter is outside the scope of this study. Figure 6.3 compares the
solutions for $p=1$, $p=2$, and $p=3$ on the $8h$ mesh. Some smearing of the
flow field is observed in the $p=1$ solution. Increasing $p$ results in
sharper predictions of the complex flow structures. Note that a rigorous
comparison among polynomial orders on the basis of accuracy vs. cost is left
for future work; our goal here is to demonstrate (as in Figures 6.2 and 6.3)
that the proposed methodology can robustly compute accurate solutions using
high-order polynomial approximations.
Figure 6.3: OH mole-fraction field for a two-dimensional moving detonation
wave at $t=200$ $\mu\mathrm{s}$ computed with the $8h$ mesh, where $h=9\times
10^{-5}$ m, using the following polynomial orders: $p=1$, $p=2$, and $p=3$.
The initial conditions are given in Equation (6.1).
It is worth noting that the entropy limiter, as opposed to solely the
positivity-preserving limiter, significantly improves the stability of these
calculations. Specifically, throughout each simulation, the entropy limiter
maintains a minimum temperature of around 300 K. Without it, however, the
temperature can dip significantly below 300 K, where the curve fits for
thermodynamic properties are no longer valid. Consequently, the nonlinear
solver in the reaction step can slow down significantly and even stall. As an
example, for $p=2$, $64h$, the simulation with the entropy limiter is over 14
times less expensive than a corresponding simulation with only the positivity-
preserving limiter. In other types of flows, these errors in temperature may
pollute the solution in different ways as well. This further highlights not
only the numerical difficulties of calculating multicomponent, reacting flows
with realistic thermodynamics, but also the advantages of using the entropy
limiter in simulations of such flows.
Figure 6.4 presents the percent error in mass, energy, and atom conservation
for $p=3$, $64h$ as a representative example, calculated every
$0.200\;\mu\mathrm{s}$ (for a total of 1000 samples). Specifically,
$\mathsf{N}_{O}$, $\mathsf{N}_{H}$, and $\mathsf{N}_{Ar}$ denote the total
numbers of oxygen, hydrogen, and argon atoms in the mixture. The error
profiles oscillate around $10^{-13}$% due to minor numerical-precision issues.
Overall, the errors remain close to machine precision, confirming that the
entropy-bounded DG formulation is conservative. Figure 6.4 also shows the
error in mass conservation (calculated every time step) for a solution
computed using a commonly employed clipping procedure in which negative
species concentrations are set to zero, instead of the limiting procedure
described in Section 4.2.1. Although artificial viscosity is still employed,
the error increases rapidly until the solver diverges.
Figure 6.4: Percent error in mass, energy, and atom conservation for $p=3$,
$64h$, where $h=9\times 10^{-5}$ m. The initial conditions for this two-
dimensional hydrogen detonation problem are given in Equation (6.1). Also
included is the error in mass conservation for a solution computed using a
commonly employed clipping procedure in which negative species concentrations
are set to zero instead of the limiting procedure described in Section 4.2.1.
Finally, we recompute the $p=2$, $64h$ case with curved elements of quadratic
geometric order. In particular, high-order geometric nodes are inserted at the
midpoints of the vertices of each element. At interior edges, the midpoint
nodes are then slightly perturbed from their initial positions. These
perturbations are performed only for $x>0.05\text{ m}$ to guarantee identical
initial conditions. We intentionally run this low-resolution case to ensure
extensive activation of the limiter and aggressively test the robustness of
the formulation when curved elements are employed. The distributions of OH
mole fraction are given in Figure 6.5 for both the linear and curved meshes.
Though slight mesh imprinting can be observed for the curved mesh, likely due
to the lower-quality elements, the two solutions are extremely similar and
remain stable throughout. This result confirms that the proposed formulation
is indeed compatible with nonlinear elements.
Figure 6.5: OH mole-fraction field for a two-dimensional moving detonation
wave at $t=200$ $\mu\mathrm{s}$ computed with $p=2$ and $64h$, where
$h=9\times 10^{-5}$ m, on linear and curved meshes. The curved mesh, which is
of quadratic order, is obtained by inserting high-order geometric nodes into
the linear mesh and then perturbing the inserted nodes. The initial conditions
are given in Equation (6.1).
### 6.2 Three-dimensional detonation wave
This test case is the three-dimensional extension of the two-dimensional
detonation wave computed in Section 6.1. The computational domain is
$\text{$\Omega$}=\left(0,0.3\right)\mathrm{m}\times\left(0,0.03\right)\mathrm{m}\times\left(0,0.03\right)\;\mathrm{m}$,
with walls at all boundaries. To reduce computational and memory demands, the
size of the domain in the $x_{1}$\- and $x_{2}$-directions is smaller than in
Section 6.1. The time-step size is prescribed according to the linear-
stability constraint with $\mathrm{CFL}=0.6$. The initial conditions are given
by
$\begin{array}[]{cccc}\qquad\qquad\qquad\left(v_{1},v_{2}\right)&=&\left(0,0\right)\text{
m/s},\\\
X_{Ar}:X_{H_{2}O}:X_{OH}:X_{O_{2}}:X_{H_{2}}&=&\begin{cases}7:0:0:1:2\\\
8:2:0.1:0:0\end{cases}&\begin{array}[]{c}x_{1}\geq 0.015\text{
m},x\in\mathcal{C}\\\ \mathrm{otherwise}\end{array},\\\
\qquad\qquad\qquad\qquad P&=&\begin{cases}{6.67}\mathrm{e}{3}&\text{ Pa}\\\
{5.50}\mathrm{e}{5}&\text{ Pa}\end{cases}&\begin{array}[]{c}x_{1}\geq
0.015\text{ m},x\in\mathcal{C}\\\ \mathrm{otherwise}\end{array},\\\
\qquad\qquad\qquad\qquad T&=&\begin{cases}300\text{\hskip 10.00002pt\hskip
10.00002pt}&\text{ K}\\\ 3500&\text{ K}\end{cases}&\begin{array}[]{c}x_{1}\geq
0.015\text{ m},x\in\mathcal{C}\\\ \mathrm{otherwise}\end{array},\end{array}$
(6.2)
where $\mathcal{C}$ represents a pocket of unburnt gas, similar to the two-
dimensional configuration in [36], defined as
$\mathcal{C}=\left\\{x\left|\sqrt{\left(x_{1}-0.014\right)^{2}+\left(x_{2}-0.015\right)^{2}+x_{3}^{2}}<0.005\text{
m}\right.\right\\}.$
No smoothing of the discontinuities in the initial conditions is performed.
The same type of diamond-like cellular structure is expected for this case,
with one cell in the $x_{2}$\- and $x_{3}$-directions [44, 45]. Our objective
with this demonstration test case is to achieve an accurate solution to this
complex, large-scale flow problem in a robust manner using high-order
polynomials. We leverage previous studies that detail the physics of this flow
configuration [44, 45]. Based on the results in Section 6.1, we select $p=2$
and a characteristic mesh spacing of $4.8h$, in order to balance accuracy with
computational and memory costs. These choices result in an unstructured mesh
containing approximately 15 million tetrahedral elements.
Figures 6.6 and 6.7 display the $X_{OH}$ and temperature distributions,
respectively, along the center $x_{3}=0.015\text{ mm}$ plane as the detonation
front traverses the domain. At early times, transverse waves, a large
induction zone, and a high-temperature, highly reactive region are present as
a result of the initialization. As the detonation is established, the flow
field behind the detonation front becomes characterized by unsteady,
multidimensional flow features, such as vortical structures, wave interactions
and reflections, and Kelvin-Helmholtz instabilities. In the vicinity of the
detonation front, transverse waves traveling in the vertical directions and
triple points can be observed. Although some small-scale numerical
instabilities are present, they do not impede the temporal advancement of the
solution or cause solver divergence.
Figure 6.6: $X_{OH}$ distributions along the center $x_{3}=0.015\text{ mm}$
plane for a three-dimensional moving detonation wave computed with $p=2$ and
$4.8h$, where $h=9\times 10^{-5}$ m. The initial conditions are given in
Equation (6.2). Figure 6.7: Temperature distributions along the center
$x_{3}=0.015\text{ mm}$ plane for a three-dimensional moving detonation wave
computed with $p=2$ and $4.8h$, where $h=9\times 10^{-5}$ m. The initial
conditions are given in Equation (6.2).
Figure 6.8 displays $X_{OH}$ and temperature distributions along various
$x_{1}x_{2}$-planes at $t=176\>\mu\mathrm{s}$, shortly before the detonation
front collides with the wall. The distributions along the corresponding
$x_{1}x_{3}$-planes are qualitatively very similar and therefore omitted for
brevity. In the post-shock region, $X_{OH}$ and temperature generally decrease
with distance from the shock front. Nevertheless, small pockets of low OH mole
fraction, surrounded by regions of higher OH mole fraction, can be observed.
These results highlight the multidimensional nature of this flow and further
illustrate that the complex flow topology is well-captured using the proposed
formulation.
(a) $X_{OH}$ (b) Temperature
Figure 6.8: $X_{OH}$ and temperature distributions along various
$x_{1}x_{2}$-planes at $t=176\>\mu\mathrm{s}$ for a three-dimensional moving
detonation wave computed with $p=2$ and $4.8h$, where $h=9\times 10^{-5}$ m.
The initial conditions are given in Equation (6.2).
Figure 6.9 presents several instantaneous pressure isosurfaces colored by the
quantity $x_{1}-\overline{x}_{1}$, where $\overline{x}_{1}$ is the mean
$x_{1}$-coordinate of the given isosurface, in order to illustrate the
structure of the detonation front, which is characterized by two orthogonal,
two-dimensional waves. Two vertical triple-point lines and two horizontal
triple-point lines move in-phase with each other in the horizontal and
vertical directions, respectively, indicating an in-phase rectangular mode
[45]. Each pair of triple-point lines propagates outwards, collides with the
boundary walls, and then reflects inwards. Subsequently, they collide with
each other and then reflect outwards, continuing in a periodic manner.
Figure 6.9: Instantaneous pressure isosurfaces colored by the quantity
$x_{1}-\overline{x}_{1}$, where $\overline{x}_{1}$ is the mean
$x_{1}$-coordinate of the given isosurface, at various times for a three-
dimensional moving detonation wave computed with $p=2$ and $4.8h$, where
$h=9\times 10^{-5}$ m. The “1” arrows indicate the vertical triple-point
lines, and the “2” arrows indicate the horizontal triple-point lines. The
initial conditions are given in Equation (6.2).
Distributions of maximum-pressure history along various $x_{1}x_{2}$-planes
and $x_{1}x_{3}$-planes at $t=176\>\mu\mathrm{s}$ are given in Figure 6.10.
Approximately two and a half cells are observed in the region
$x_{1}>0.15\text{ mm}$. High-pressure explosions occur when the triple-point
lines collide with each other and with the corners. “Slapping” waves result
from reflections of the triple-point lines, indicated with the white arrow in
Figure 6.10a (top) [45]. The $x_{1}x_{2}$-distributions are nearly identical
to the corresponding $x_{1}x_{3}$-distributions. Figure 6.11 shows isosurfaces
of maximum-pressure history for $x_{1}\in\left[0.236,0.292\right]$ m, which
illustrate, from a three-dimensional perspective, not only the temporal
evolution of triple-point-line intersections, but also the slapping waves
along the walls and the strong, high-pressure explosions. Overall symmetry in
the $x_{2}$\- and $x_{3}$-directions is observed. Figures 6.10 and 6.11
further confirm that the detonation front is characterized by an in-phase
rectangular mode.
(a) $x_{1}x_{2}$-planes (b) $x_{1}x_{3}$-planes
Figure 6.10: Distributions of maximum-pressure history along various
$x_{1}x_{2}$-planes and $x_{1}x_{3}$-planes at $t=176\>\mu\mathrm{s}$ for a
three-dimensional moving detonation wave computed with $p=2$ and $4.8h$, where
$h=9\times 10^{-5}$ m. The white arrow in Figure 6.10a (top) indicates a
“slapping” wave. The initial conditions are given in Equation (6.2). Figure
6.11: Isosurfaces of maximum-pressure history at $t=176\>\mu\mathrm{s}$ for
$x_{1}\in\left[0.236,0.292\right]$ m for a three-dimensional moving detonation
wave computed with $p=2$ and $4.8h$, where $h=9\times 10^{-5}$ m. The
isosurfaces are colored by $x_{1}$-position. Each of the “1” arrows indicates
a slapping wave, while the “2” arrow indicates the high-pressure explosion
resulting from triple-point-line collisions. The initial conditions are given
in Equation (6.2).
## 7 Concluding remarks
In this second part of our two-part paper, we introduced a positivity-
preserving, entropy-bounded, multidimensional DG methodology for the
chemically reacting, compressible Euler equations, extending the one-
dimensional version presented in Part I [1]. Compared to current
multidimensional positivity-preserving and/or entropy-bounded DG schemes in
the literature, restrictions on the quadrature rules, numerical flux function,
polynomial order of the geometric approximation, and physical modeling are
relaxed. In particular, the formulation is compatible with arbitrary, curved
elements, any invariant-region-preserving flux, and mixtures of thermally
perfect gases. A simple linear-scaling limiter enforces nonnegative species
concentrations, positive density, positive pressure, and bounded entropy.
Artificial viscosity aids in suppressing small-scale instabilities not
eliminated by the linear scaling. We discussed how to maintain compatibility
with the strategies introduced in [2] to maintain pressure equilibrium and
avoid generating spurious pressure oscillations.
The formulation was applied to complex moving detonation waves in two and
three dimensions. In the two-dimensional case, a variety of mesh sizes and
polynomial orders was considered. In [2], a linear polynomial approximation of
the solution and a very fine mesh were required to obtain a stable solution.
With the developed formulation, we achieved robust and accurate solutions
using high-order polynomials and a relatively coarse mesh. Increasing the
polynomial order resulted in sharper predictions of the rich flow topology.
Mass, total energy, and atomic elements were shown to be conserved. An
important finding is that the entropy limiter was crucial for the coarser two-
dimensional detonation problems; without it (i.e., only the positivity-
preserving limiter was applied), the nonlinear solver during the reaction step
can slow down substantially or even stall, due to the large undershoots in
temperature. In the three-dimensional detonation test case, we demonstrated
that our methodology can compute accurate and robust solutions to large-scale
reacting-flow problems. Future work will entail the simulation of larger-scale
detonation applications involving more complex geometries.
## Acknowledgments
This work is sponsored by the Office of Naval Research through the Naval
Research Laboratory 6.1 Computational Physics Task Area. Discussions with Dr.
Brian Taylor are gratefully acknowledged.
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## Appendix A Alternative approach with over-integration
Here, we present an alternative approach to guarantee
$\overline{y}_{\kappa}^{j+1}\in\mathcal{G}_{s_{b}}$, even when over-
integration with the modified flux interpolation (3.8) for preserving pressure
equilibrium is employed. The main idea is to construct a separate polynomial,
denoted $\check{y}_{\kappa}$, that satisfies the following:
$\displaystyle\overline{\check{y}}_{\kappa}$
$\displaystyle=\overline{y}_{\kappa},$
$\displaystyle\check{y}_{\kappa}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)$
$\displaystyle=\widetilde{y}_{\kappa}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\quad
f=1,\ldots,n_{f},\quad l=1,\ldots,n_{q,f}^{\partial},$
such that the scheme satisfied by the element averages becomes
$\displaystyle\overline{y}_{\kappa}^{j+1}$ $\displaystyle=$
$\displaystyle\overline{y}_{\kappa}^{j}-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(\widetilde{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\widetilde{y}_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)$
(A.1) $\displaystyle=$
$\displaystyle\overline{\check{y}}_{\kappa}^{j}-\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(\check{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\check{y}_{\kappa^{(f)}}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}\check{y}_{\kappa}^{j}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}}\sum_{l=1}^{n_{q,f}^{\partial}}\left[\theta_{f,l}\check{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right)-\frac{\Delta
t\nu_{f,l}^{\partial}}{|\kappa|}\mathcal{F}^{\dagger}\left(\check{y}_{\kappa}^{j}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),\check{y}_{\kappa^{(f)}}\left(\xi\left(\zeta_{l}^{\left(f\right)}\right)\right),n\left(\zeta_{l}^{\left(f\right)}\right)\right)\right]$
$\displaystyle=$
$\displaystyle\sum_{v=1}^{n_{q}}\theta_{v}\check{y}_{\kappa}^{j}\left(\xi_{v}\right)+\sum_{f=1}^{n_{f}-1}\sum_{l=1}^{n_{q,f}^{\partial}}\theta_{f,l}\check{A}_{f,l}+\sum_{l=1}^{N-1}\theta_{n_{f},l}\check{B}_{l}+\theta_{n_{f},N}\check{C},$
where the definitions of $\check{A}_{f,l}$, $\check{B}_{l}$, and $\check{C}$
can be deduced based on Section 4.2. Thus, $\overline{y}_{\kappa}^{j+1}$ can
be written as a convex combination of three-point systems and pointwise
values, and an analogous version of Theorem 1 holds. The degree of
$\check{y}_{\kappa}$ may be different from that of $y_{\kappa}$, provided that
the volume quadrature rule in Equation (A.1) is sufficiently accurate. First,
we propose a strategy to construct $\check{y}_{\kappa}$ on two-dimensional
quadrilateral elements. As an illustrative example, consider the Gauss-Lobatto
nodal set for $\check{y}_{\kappa}$ displayed in Figure A.1. Nodes 1 to 8
comprise the integration points used in the surface integrals in Equation
(A.1), whereas Node 9 serves as a degree of freedom to ensure
$\overline{\check{y}}_{\kappa}=\overline{y}_{\kappa}$. The coefficients of
$\check{y}_{\kappa}$ are specified as
$\displaystyle\check{y}_{\kappa}\left(x_{k}\right)$
$\displaystyle=\widetilde{y}_{\kappa}\left(x_{k}\right),k=1,\ldots,8,$
$\displaystyle\check{y}_{\kappa}\left(x_{9}\right)$
$\displaystyle=\frac{\overline{y}_{\kappa}|\kappa|-\sum_{v=1}^{n_{q}}\left|J_{\kappa}(\xi_{v})\right|w_{v}\sum_{k=1}^{8}\widetilde{y}_{\kappa}\left(x_{k}\right)\varphi_{k}\left(\xi_{v}\right)}{\sum_{v=1}^{n_{q}}\left|J_{\kappa}(\xi_{v})\right|w_{v}\varphi_{9}\left(\xi_{v}\right)}$
$\displaystyle=\widetilde{y}_{\kappa}\left(x_{9}\right)+\frac{\sum_{v=1}^{n_{q}}\left|J_{\kappa}(\xi_{v})\right|w_{v}\left[\sum_{i=1}^{n_{b}}y_{\kappa}(x_{i})\phi_{i}(\xi_{v})-\sum_{k=1}^{9}\widetilde{y}_{\kappa}\left(x_{k}\right)\varphi_{k}\left(\xi_{v}\right)\right]}{\sum_{v=1}^{n_{q}}\left|J_{\kappa}(\xi_{v})\right|w_{v}\varphi_{9}\left(\xi_{v}\right)}.$
In general, $\check{y}_{\kappa}\left(x_{9}\right)$ is not expected to differ
significantly from $\widetilde{y}_{\kappa}\left(x_{9}\right)$ or
$y_{\kappa}(x_{9})$; in fact, $\check{y}_{\kappa}\left(x_{9}\right)$ reduces
to $\widetilde{y}_{\kappa}\left(x_{9}\right)$ as the discrepancies between
$y_{\kappa}(x)$ and $\widetilde{y}_{\kappa}(x)$ vanish. If necessary, the
limiting procedure in Section 4.2.1 is applied to ensure that
$\check{y}_{\kappa}^{j}(x)\in\mathcal{G}_{s_{b}},\;\forall
x\in\mathcal{D_{\kappa}}$. Note that it is possible for the limiter to modify
the pressure at the nodes; nevertheless, as observed in the thermal-bubble
test case in Part I [1], the limiter typically will not destroy pressure
equilibrium or cause large-scale pressure oscillations in smooth regions of
the flow. This will likely remain true when limiting
$\check{y}_{\kappa}^{j}(x)$, in part because $y_{\kappa}^{j}(x)$ will already
have been limited.
Figure A.1: Illustrative nodal set for $\check{y}_{\kappa}$ on two-dimensional
quadrilateral elements.
Constructing $\check{y}_{\kappa}$ on two-dimensional triangular elements can
be done in a similar manner, provided that the nodal set for
$\check{y}_{\kappa}$ includes the surface integration points and the
corresponding surface quadrature rules have positive weights. However, typical
$p=2$ nodal sets, which have six nodes, do not include any interior nodes;
therefore, there are no degrees of freedom to ensure
$\overline{\check{y}}_{\kappa}=\overline{y}_{\kappa}$. Here, we propose a more
general strategy that makes use of the transformation between the reference
quadrilateral, $\widehat{\kappa}_{\mathrm{quad}}$, which is a bi-unit square,
and the reference triangle, $\widehat{\kappa}_{\mathrm{tri}}$, which is an
isosceles right triangle with side length of two, as shown in Figure A.2a [12,
13]. The bi-unit square can be mapped to the reference triangle as
$\displaystyle\xi_{1}$ $\displaystyle=\frac{(1+\eta_{1})(1-\eta_{2})}{2}-1,$
$\displaystyle\xi_{2}$ $\displaystyle=\eta_{2},$
where $\eta\in\mathbb{R}^{2}$ are the reference coordinates of
$\widehat{\kappa}_{\mathrm{quad}}$ and $\xi\in\mathbb{R}^{2}$ are the
reference coordinates of $\widehat{\kappa}_{\mathrm{tri}}$. The inverse
mapping is given by
$\displaystyle\eta_{1}$ $\displaystyle=2\frac{1+\xi_{1}}{1-\xi_{2}}-1,$
$\displaystyle\eta_{2}$ $\displaystyle=\xi_{2}.$
Consider the seven-node triangle in Figure A.2b, obtained by degeneration of
the Gauss-Lobatto nine-node quadrilateral [46]. Specifically, Nodes 3, 4, and
7 of $\widehat{\kappa}_{\mathrm{quad}}$ are coalesced into Node 3 of
$\widehat{\kappa}_{\mathrm{tri}}$, such that
$\displaystyle\check{y}_{\kappa}\left(\eta\right)$
$\displaystyle=\underset{k\neq
3,4,7}{\sum_{k=1}^{9}}\check{y}_{\kappa}\left(x_{k}\right)\varphi_{k}\left(\eta\right)+\check{y}_{\kappa}\left(x_{3}\right)\left[\varphi_{3}\left(\eta\right)+\varphi_{4}\left(\eta\right)+\varphi_{7}\left(\eta\right)\right],$
$\displaystyle=\sum_{k=1}^{7}\check{y}_{\kappa}\left(x_{k}\right)\check{\varphi}_{k}\left(\eta\right),$
where $\left\\{\varphi_{1},\ldots,\varphi_{9}\right\\}$ is ordered according
to the $\widehat{\kappa}_{\mathrm{quad}}$ node numbering and
$\left\\{\check{\varphi}_{1},\ldots,\check{\varphi}_{7}\right\\}$ is ordered
according to the $\widehat{\kappa}_{\mathrm{tri}}$ node numbering. Note that
$\check{\varphi}_{7}$ is equal to $\varphi_{3}+\varphi_{4}+\varphi_{7}$. Nodes
1 to 6 of $\widehat{\kappa}_{\mathrm{tri}}$ make up the Gauss-Lobatto points
used in the surface integrals. $\overline{\check{y}}_{\kappa}$ can be computed
using a sufficiently accurate quadrature rule for quadrilaterals as [13]
$\displaystyle\overline{\check{y}}_{\kappa}$
$\displaystyle=\frac{1}{|\kappa|}\int_{\kappa}\check{y}_{\kappa}(x)dx=\frac{1}{|\kappa|}\int_{\widehat{\kappa}_{\mathrm{tri}}}\check{y}_{\kappa}(\xi)\left|J_{\kappa}(\xi)\right|d\xi$
$\displaystyle=\frac{1}{|\kappa|}\int_{\widehat{\kappa}_{\mathrm{quad}}}\check{y}_{\kappa}(\eta)\left|J_{\kappa}(\xi(\eta))\right|\left|J_{\widehat{\kappa}}(\eta)\right|d\eta$
$\displaystyle=\frac{1}{|\kappa|}\sum_{v=1}^{n_{q}}\check{y}_{\kappa}(\eta_{v})\left|J_{\kappa}(\xi(\eta_{v}))\right|\left|J_{\widehat{\kappa}}(\eta_{v})\right|w_{v},$
where $\left|J_{\widehat{\kappa}}(\eta)\right|$ is the Jacobian determinant of
the mapping from the reference quadrilateral to the reference triangle, given
by
$\left|J_{\widehat{\kappa}}(\eta)\right|=\frac{1-\eta_{2}}{2},$
which is positive everywhere in $\widehat{\kappa}_{\mathrm{tri}}$ except along
the collapsed face, such that the analysis in Section (4.2) holds. Using the
$\widehat{\kappa}_{\mathrm{tri}}$ node numbering, the coefficients of
$\check{y}_{\kappa}$ are then prescribed as
$\displaystyle\check{y}_{\kappa}\left(x_{k}\right)$
$\displaystyle=\widetilde{y}_{\kappa}\left(x_{k}\right),k=1,\ldots,6$
$\displaystyle\check{y}_{\kappa}\left(x_{7}\right)$
$\displaystyle=\frac{\overline{y}_{\kappa}|\kappa|-\sum_{v=1}^{n_{q}}\left|J_{\kappa}(\xi(\eta_{v}))\right|\left|J_{\widehat{\kappa}}(\eta_{v})\right|w_{v}\sum_{k=1}^{6}\widetilde{y}_{\kappa}\left(x_{k}\right)\check{\varphi}_{k}\left(\eta_{v}\right)}{\sum_{v=1}^{n_{q}}\left|J_{\kappa}(\xi(\eta_{v}))\right|\left|J_{\widehat{\kappa}}(\eta_{v})\right|w_{v}\check{\varphi}_{7}\left(\eta_{v}\right)}$
$\displaystyle=\widetilde{y}_{\kappa}\left(x_{7}\right)+\frac{\sum_{v=1}^{n_{q}}\left|J_{\kappa}(\xi(\eta_{v}))\right|\left|J_{\widehat{\kappa}}(\eta_{v})\right|w_{v}\left[\sum_{i=1}^{n_{b}}y_{\kappa}(x_{i})\phi_{i}(\xi(\eta_{v}))-\sum_{k=1}^{7}\widetilde{y}_{\kappa}\left(x_{k}\right)\check{\varphi}_{k}\left(\eta_{v}\right)\right]}{\sum_{v=1}^{n_{q}}\left|J_{\kappa}(\xi(\eta_{v}))\right|\left|J_{\widehat{\kappa}}(\eta_{v})\right|w_{v}\check{\varphi}_{7}\left(\eta_{v}\right)}.$
This approach can be generalized to other orders and shapes as well.
(a) Mapping from reference quadrilateral, $\widehat{\kappa}_{\mathrm{quad}}$,
to reference triangle, $\widehat{\kappa}_{\mathrm{tri}}$. (b) Seven-node
triangle obtained via degeneration of the Gauss-Lobatto nine-node
quadrilateral.
Figure A.2: Transformation between reference quadrilateral and reference
triangle, as well as an illustrative nodal set for $\check{y}_{\kappa}$ on
triangular elements.
## Appendix B Supporting lemma
In the following, let $\Delta\mathcal{F}$ denote the quantity
$\Delta\mathcal{F}=\left(\Delta\mathcal{F}_{\rho v},\Delta\mathcal{F}_{\rho
e_{t}},\Delta\mathcal{F}_{C_{1}},\ldots,\Delta\mathcal{F}_{C_{n_{s}}}\right)^{T},$
and assume that $\Delta\mathcal{F}_{C_{i}}=0,i=1,\ldots,n_{s}$. The proof
below is similar to that in [34, Lemma 6].
###### Lemma 5.
Assume that $y=\left(\rho v,\rho e_{t},C_{1},\ldots,C_{n_{s}}\right)^{T}$ is
in $\mathcal{G}$. Then $\check{y}=y-\alpha^{-1}\Delta\mathcal{F}$, where
$\alpha>0$, is also in $\mathcal{G}$ under the following conditions:
$\alpha>\alpha^{*}\left(y,\Delta\mathcal{F}\right)=\max\left\\{\left.\alpha_{T}\right|_{\left(y,\Delta\mathcal{F}\right)},0\right\\},$
(B.1)
where
$\alpha_{T}=\begin{cases}\frac{-\mathsf{b}+\sqrt{\mathsf{b}^{2}-4\rho^{2}u\mathsf{g}}}{2\rho^{2}u},&\mathsf{b}^{2}-4\rho^{2}u\mathsf{g}\geq
0\\\ 0,&\mathrm{otherwise}\end{cases},$ (B.2)
$\mathsf{b}=-\rho\Delta\mathcal{F}_{\rho e_{t}}+\rho
v\cdot\Delta\mathcal{F}_{\rho v}$, and
$\mathsf{g}=-\frac{1}{2}\left|\Delta\mathcal{F}_{\rho v}\right|^{2}$.
###### Proof.
$\check{y}=y-\alpha^{-1}\Delta\mathcal{F}$ can be expanded as
$\begin{split}\check{y}=&\left(\check{\rho v},\check{\rho
e_{t}},\check{C_{1}},\ldots,\check{C}_{n_{s}}\right)^{T}\\\ =&\left(\rho
v-\alpha\Delta\mathcal{F}_{\rho v},\rho e_{t}-\alpha\Delta\mathcal{F}_{\rho
e_{t}},C_{1}-\alpha\Delta\mathcal{F}_{C_{1}},\ldots,C_{n_{s}}-\alpha\Delta\mathcal{F}_{C_{n_{s}}}\right)^{T}.\end{split}$
Since $\Delta\mathcal{F}_{C_{i}}=0,i=1,\ldots,n_{s}$, we have
$\check{C_{i}}=C_{i}\geq 0$. Density also remains unchanged.
Next, we focus on positivity of temperature. For a given $y=\left(\rho v,\rho
e_{t},C_{1},\ldots,C_{n_{s}}\right)^{T}$, let $Z(y)$ be defined as
$Z(y)=\rho^{2}u^{*}(y)=\rho(y)\rho e_{t}-\left|\rho
v\right|^{2}/2-\rho^{2}u_{0}(y).$ (B.3)
Note that if $Z(y)>0$, then $T(y)>0$. $Z\left(\check{y}\right)$ can be
expressed as
$\displaystyle Z\left(y-\alpha^{-1}\Delta\mathcal{F}\right)=$
$\displaystyle\sum_{i=1}^{n_{s}}W_{i}\left(C_{i}-\alpha^{-1}\Delta\mathcal{F}_{C_{i}}\right)\left(\rho
e_{t}-\alpha^{-1}\Delta\mathcal{F}_{\rho e_{t}}\right)$
$\displaystyle-\frac{1}{2}\left|\rho v-\alpha^{-1}\Delta\mathcal{F}_{\rho
v}\right|^{2}-\left[\sum_{i=1}^{n_{s}}W_{i}\left(C_{i}-\alpha^{-1}\Delta\mathcal{F}_{C_{i}}\right)\right]^{2}u_{0},$
which, after multiplying both sides by $\alpha^{2}$, can be rewritten as
$\displaystyle\alpha^{2}Z\left(y-\alpha^{-1}\Delta\mathcal{F}\right)=$
$\displaystyle\rho^{2}u\alpha^{2}-\mathsf{b}\alpha+\mathsf{g},$ (B.4)
where $\mathsf{b}=-\rho e_{t}\mathsf{M}-\rho\Delta\mathcal{F}_{\rho
e_{t}}+\rho v\cdot\Delta\mathcal{F}_{\rho v}+2\rho u_{0}\mathsf{M}$,
$\mathsf{g}=\mathsf{M}\Delta\mathcal{F}_{\rho
e_{t}}-\frac{1}{2}\left|\Delta\mathcal{F}_{\rho
v}\right|^{2}-u_{0}\mathsf{M}^{2}$, and
$\mathsf{M}=\sum_{i=1}^{n_{s}}W_{i}\Delta\mathcal{F}_{C_{i}}$. Since
$\mathsf{M}=0$, $\mathsf{b}$ and $\mathsf{g}$ reduce to
$\mathsf{b}=-\rho\Delta\mathcal{F}_{\rho e_{t}}+\rho
v\cdot\Delta\mathcal{F}_{\rho v}$ and
$\mathsf{g}=-\frac{1}{2}\left|\Delta\mathcal{F}_{\rho v}\right|^{2}$. Setting
the RHS of Equation (B.4) equal to zero yields a quadratic equation with
$\alpha$ as the unknown. Since $\rho^{2}u$ is positive, the quadratic equation
is convex. As such, if $\mathsf{b}^{2}-4\rho^{2}u\mathsf{g}<0$, then no real
roots exist, and $Z\left(y-\alpha^{-1}\Delta\mathcal{F}\right)>0$ for all
$\alpha\neq 0$; otherwise, at least one real root exists, in which case a
sufficient condition to ensure
$Z\left(y-\alpha^{-1}\Delta\mathcal{F}\right)>0$ is $\alpha>\alpha_{0}$, where
$\alpha_{0}$ is given by
$\alpha_{0}=\max\left\\{\frac{-\mathsf{b}+\sqrt{\mathsf{b}^{2}-4\rho^{2}u\mathsf{g}}}{2\rho^{2}u},0\right\\}.$
∎
|
# Transferability Estimation Based On Principal Gradient Expectation
Huiyan Qi1, Lechao Cheng2, Jingjing Chen1, Yue Yu1, Xue Song1, Zunlei Fengg3,
Yu-Gang Jiang1
1School of Computer Science & Shanghai Collaborative Innovation Center of
Intelligent Visual Computing,
Fudan University
2Zhejiang Lab
3Zhejiang University
###### Abstract
Transfer learning aims to improve the performance of target tasks by
transferring knowledge acquired in source tasks. The standard approach is pre-
training followed by fine-tuning or linear probing. Especially, selecting a
proper source domain for a specific target domain under pre-defined tasks is
crucial for improving efficiency and effectiveness. It is conventional to
solve this problem via estimating transferability. However, existing methods
can not reach a trade-off between performance and cost. To comprehensively
evaluate estimation methods, we summarize three properties: stability,
reliability and efficiency. Building upon them, we propose Principal Gradient
Expectation (PGE), a simple yet effective method for assessing
transferability. Specifically, we calculate the gradient over each weight unit
multiple times with a restart scheme, and then we compute the expectation of
all gradients. Finally, the transferability between the source and target is
estimated by computing the gap of normalized principal gradients. Extensive
experiments show that the proposed metric is superior to state-of-the-art
methods on all properties.
## 1 Introduction
Traditional machine learning work for predicting unseen testing instances with
the knowledge learned from training data [7, 53]. Since there is always a
distribution gap between training data and testing data, the inference
performance is lower than expected [58, 47]. However, as obtaining appropriate
training data is challenging, a surge of interest emerges in transferring
knowledge from the source task into the target task based on the pre-training
mechanism. In the realm of computer vision, the prevalent approaches are
linear probing [5] and fine-tuning. Linear probing adapts the model outputs by
learning a task-related head layer while fine-tuning adapts the parameters of
the entire model. Both linear probing and fine-tuning leverage models pre-
trained on large-scale datasets for downstream tasks.
Figure 1: The figure on the left denotes that the model’s parameters are
updated by first-order gradients in the optimization process. Intuitively, the
gap between the optimal points of the source and target domains can reflect
transferability. However, the point of the target is invisible. To solve this
problem, our PGE collects first-order gradients and computes the expectation
of all gradients as on the right.
Prominent models, including BERT [10] and GPT-3 [3] for natural language
processing, the ResNet [20] and vision transformer [12, 32, 31] variants pre-
trained on ImageNet [41] for vision tasks, and the CLIP [39, 40, 42] variants
for multimodal learning, have significantly outperformed previous approaches
in their fields of expertise. However, it is intuitive that a proper source
domain is more crucial than the scale of pre-training data. Take CIFAR100 [25]
for example, CIFAR10 [25] may be a better source than ImageNet [41].
Therefore, researchers are dedicated to designing transferability estimating
algorithms to determine the best source for the target domain efficiently.
To estimate transferability, researchers have proposed several transferability
metrics, including LEEP [35], LogMe [55], H-score [1], NCE [49], and GBC [36].
These metrics utilize the distribution in label and feature spaces of the
target domain. However, we found that sampling from the target domain with
certain approaches as previous works [35, 36] could change the data
distribution, which may lead to inferior results lacking in stability and
accuracy. Furthermore, the calculations of mentioned metrics demand the
distributions in feature spaces of pre-trained models. But the pre-training
process is time-consuming, which is harmful to efficiency.
As mentioned above, existing metrics can not reach a trade-off between
performance and cost, leading to low usability. To comprehensively evaluate
these estimation methods, we summarize three properties that a good
transferability estimation should possess for the first time. We consider that
a qualified measurement should be stable, reliable, and efficient. Stability
implies that the transferability metric value would not change violently with
a slight oscillation of data distribution. Reliability reflects the accuracy
of selecting a proper source domain. Efficiency requires concise calculations
and low time costs.
In this paper, we suggest a straightforward yet effective technique for
transferability estimation that satisfies the three properties, dubbed
Principal Gradient Expectation (PGE). Similarly to previous metric methods
[35, 55, 1, 49, 36], our approach applies to the scenario where both the
source and target tasks are a single task. Specifically, we consider the
optimization procedure to be an approximation between the initial and the
optimal state in the parameter space. Moreover, the optimal state gap between
the source and target domain indicates transferability. However, as shown in
Figure LABEL:idea, the optimal point of the target is invisible. To quantify
the gap, we collect first-order gradients of the model’s backbone multiple
times and compute the expectation of all gradients (on the right of Figure
LABEL:idea). The expectation is named principal gradient expectation, which
guarantees stability. Then, we design a scheme to compute the gap between
principal gradients. Specifically, we deflate the gap of the principal
gradient expectations between the source and target by Schwarz inequality [46]
to achieve a lower bound as the quantitative transfer gap. In the calculation
process of the proposed metric described above, the three properties are
guaranteed. As we do not utilize the distribution of the target domain and
collect gradients multiple times to compute the expectation of gradient, our
PGE could produce stable results. Considering that supervised pre-training is
more sensitive than unsupervised pre-training [57, 22], we obtain the first-
order gradient in an unsupervised mode. This operation facilitates the
promotion of accuracy in selecting a proper source domain. In addition, the
process of estimating transferability with principal gradient expectation does
not depend on pre-training parameters, which helpfully reduces the time cost.
In the experiments, we show that our proposed method outperforms state-of-the-
art methods in terms of stability, reliability and efficiency. To conclude,
the main contributions of this work can be summarized as follows:
* •
For the first time, we have summarized the properties that a qualified
transferability metric should satisfy and proposed corresponding evaluation
methods. The properties are stability, reliability and efficiency.
* •
We propose a novel approach (PGE) based on the principal gradient expectation
for assessing transferability. Moreover, our method satisfied the above
properties.
* •
We show through experiments that the proposed PGE is strongly related to the
transferred performance. Our approach is simple yet effective and shows its
superiority over other methods.
## 2 Related Work
Our research is closely aligned with the techniques of network adaptation and
transferability estimation. Consequently, we will now delve into the body of
literature surrounding these fields.
### 2.1 Network Adaptation
Network adaptation is prevalent in the computer vision field nowadays. This
technique involves adapting a pre-trained neural network to suit a new task in
situations where data is scarce. Linear probing (LP) and Fine-tuning (FT) are
typical methods for network adaptation. In the training process of LP, the
pre-trained parameters are frozen, while training the head layers associated
with the task [2, 37]. As for FT, the parameters of the entire network are
fine-tuned for downstream tasks to improve transfer accuracy [21]. Researchers
have also explored various methods to improve network adaptation. To alleviate
overfitting, [56]combined a lightweight “side” network with pre-trained
models. [26]integrated each layer of a model into the fine-tuned model. This
method can balance the performance of the source and target without increasing
the model size. [4]improved the ability of a model to extract features by
learning residual feature maps and significantly lessened memory usage.
Upstream bias mitigation(UBM) [23] has been found to benefit fine-tuning
language models. Sparse pruning [9]reduced the predictive error via allocating
pruning weights according to the zero-value weights in each layer.
[18]evaluated three pruning methods and found simple method may yield better
results. Meta-learning has been applied to network adaptation and training
dynamic networks [29]. Some studies [50, 43] have combined meta-learning with
reinforcement learning to improve the performance on new tasks. Recent work
[17] generalized given tasks to new tasks.
In addition, Neural Architecture Search(NAS) has also received significant
attention. Previous NAS works [59, 48, 30] are based on reinforcement learning
algorithms and their cost was high. ENAS [38] simplified the search process by
sharing parameters but it led to lower accuracy. Different NAS works [30, 11,
15] have attempted to reduce the search cost and improve accuracy in recent
years. [16] achieved good results by automatically adjusting the network
architecture to adapt to new tasks. More recently, [27, 28]utilized prompts
which consist of task-specific vectors and optimized prompts via gradients.
Different studies [24, 19, 54] applied prompts to multimodal tasks.
[45]trained vision transformers with prompts and generative knowledge. In this
work, we adopt LP and FT as the adaptation approaches to evaluate our
algorithm by measuring the correlation between the adaptation results and the
transferability computed with the principal gradient.
### 2.2 Transferability Estimation
To properly evaluate the transferability of pre-trained models, Negative
Conditional Entropy (NCE) [49] utilizes a metric derived from information
theory to measure the transferability and difficulty of classification tasks.
NCE assumes that the images in the source domain and the target domain are the
same, but their labels are distinct. They then calculate the transferability
score with the negative conditional entropy between the target and the source
labels. Contemporary work H-score [1] leverages statistics and information
theory to quantify the transferability of feature representations. However,
H-score can only be employed in classification tasks. While LEEP [35] focuses
on computing the joint probability over pseudo labels and the target labels to
yield the log expectation of the empirical predictor. But LEEP will obtain
different transferability scores when models with identical feature extractors
and distinct classification heads. LogMe [55] solve this issue by directly
estimating the maximum value of label evidence given features extracted by
pre-trained models. It models each target label as a linear model with
Gaussian noise, and then optimizes the prior distribution parameters to obtain
the average maximum (log) evidence of labels given the target instances’
embeddings. Recently, GBC [36] desires to measure the amount of overlap
between target classes in the feature space of the pre-trained model. While in
this work, the gradients of the model’s backbone are employed in our
estimation instead of distribution.
## 3 Method
### 3.1 Preliminaries
Considering a deep model $\boldsymbol{\mathit{M}}(w,h)$, we denote the feature
extractor as $w$ and the task-related layer as $h$. For $N$ alternative
sources $\\{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})\\}_{i=1}^{N}$,
$\mathcal{D}_{s(i)}$ and $\mathcal{T}_{s(i)}$ represent the source domain and
the source task, respectively. The target is denoted as
$(\mathcal{D}_{t},\mathcal{T}_{t})$, where $\mathcal{D}_{t}$ and
$\mathcal{T}_{t}$ separately indicate the target domain and the target task.
The goal of this work is to determine the appropriate source via quantify the
transferability between each source $(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})$
and the target $(\mathcal{D}_{t},\mathcal{T}_{t})$, as selecting a proper
source is critical to improving efficiency and effectiveness. Moreover, we use
$\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]$
to denote the transfer gap from an available source to the target.
After analyzing the previous methods [49, 1, 35, 55, 36], we recapitulate
three characteristics of a good transferability metric: stability,
reliability, and efficiency. Next, we provide a comprehensive explanation of
these propositions.
Proposition 1 (Stability). Let I denote a random sampling function (further
details about the sampling methods are described in Sec. 3.4).
$\emph{I}(\mathcal{D}_{t})$ represents a subset drawn according to
$\mathcal{D}_{t}$. The stability indicating the difference between
$\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\boldsymbol{\mathcal{D}_{t}},\mathcal{T}_{t})]$
and
$\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\boldsymbol{\emph{I}(\mathcal{D}_{t})},\mathcal{T}_{t})]$
is bounded, which can be described as:
$|\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\boldsymbol{\mathcal{D}_{t}},\mathcal{T}_{t})]-\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\boldsymbol{\emph{I}(\mathcal{D}_{t})},\mathcal{T}_{t})]|\leq\epsilon.$
(1)
Stability is essential since a finite dataset is a sub-sample in the manifold
space of the data distribution.
Proposition 2 (Reliability).
$\mathcal{A}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]$
is defined as the performance of the target domain transferred from an
available source. Moreover, a lower transfer gap indicates better transfer
performance. Thus,
$\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]<\mathcal{G}[(\mathcal{D}_{s(j)},\mathcal{T}_{s(j)});(\mathcal{D}_{t},\mathcal{T}_{t})]$
always enables
$\mathcal{A}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]>\mathcal{A}[(\mathcal{D}_{s(j)},\mathcal{T}_{s(j)});(\mathcal{D}_{t},\mathcal{T}_{t})]$.
Proposition 3 (Efficiency). The estimation of the transferability from source
to target is supposed to have low computation and complexity.
Previous studies such as [57, 22], have demonstrated that the transferability
is mainly related to the model’s backbone. However, existing methods employ
task-specific layers $h$ besides the model’s backbone to calculate the
transferability, which may harm reliability and lead to high costs. Our method
only utilizes the gradient of the backbone to estimate the transferability.
Furthermore, to reach a trade-off between computing efficiency and efficacy,
we use the first-order approximation of the loss.
### 3.2 Principal Gradient Expectation (PGE)
We consider the optimization procedure to be a distance approximation between
the initial and the optimal points in the parameter space. To balance
efficiency and effectiveness, we adopt the first-order gradient expectation.
Given a model $\boldsymbol{\mathit{M}}$ and a target
$(\mathcal{D}_{t},\mathcal{T}_{t})$, $\mathcal{L}$ denotes the loss function.
The model is initialized with random weights $\theta_{0}$ where
$\theta_{0}\sim\mathscr{N}(\mathbf{0},I)$. We first compute the gradient of
backbone parameters, denoted as $\nabla\mathcal{L}(\theta_{0})$. Instead of
updating model parameters $\theta$ with $\nabla\mathcal{L}(\theta_{0})$, we
then re-initialize the weights of the model and collect
$\nabla\mathcal{L}(\theta_{0})$ multiple times to compute the expectation of
gradients. We believe gradients reflect the inherent characteristics of the
source task (target task). Furthermore, re-initializing multiple times and
computing the expectation reduces the impact of abnormal gradients. This
expectation value is defined as Principal Gradient Expectation (PGE), which
can be formulated as follows:
$PGE=\mathbb{E}_{\theta_{0}}[\nabla\mathcal{L}(\theta_{0})].$ (2)
By making use of the definition of PGE, we compute the PGE for source
$(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})$ and target
$(\mathcal{D}_{t},\mathcal{T}_{t})$ with their own loss, respectively:
$PGE_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}=\mathbb{E}_{\theta_{0}}[\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})],$
(3)
$PGE_{(\mathcal{D}_{t},\mathcal{T}_{t})}=\mathbb{E}_{\theta_{0}}[\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})].$
(4)
### 3.3 Transferability Metric based on PGE
In our transfer setting, the restriction for the source and target tasks is
that the backbone of models must be the same. Moreover, we require the source
task $\mathcal{T}_{s}$ and target task $\mathcal{T}_{t}$ to be a single task
as in most existing works. And $\mathcal{T}_{t}$ is not necessary to be the
same as $\mathcal{T}_{s}$. So computing transferability with PGE is task-
irrelevant.
We consider a situation in which the model $\boldsymbol{\mathit{M}}$ only has
one single parameter $\theta$. The initialized value of $\theta$ is denoted as
$\theta_{0}$ and the optimal value is denoted $\theta^{*}$.
$\mathcal{L}(\theta_{0})$ is the initial loss value at $\theta_{0}$ and
$\mathcal{L}(\theta^{*})$ is the loss value at $\theta^{*}$.
$\mathcal{L}^{{}^{\prime}}(\theta_{0})$ represents the derivative at
$\theta_{0}$. According to Taylor’s formula, the loss value at $\theta^{*}$
could be expressed in the following form:
$\mathcal{L}(\theta^{*})=\mathcal{L}(\theta_{0})+\mathcal{L}^{{}^{\prime}}(\theta_{0})(\theta^{*}-\theta_{0})+R(\theta_{0}).$
(5)
$R(\theta_{0})$ is the remainder beyond the first-order approximation. Then we
rewrite the Eq. 5.
$(\theta^{*}-\theta_{0})+\frac{R(\theta_{0})}{\mathcal{L}^{{}^{\prime}}(\theta_{0})}=\frac{\mathcal{L}(\theta^{*})-\mathcal{L}(\theta_{0})}{\mathcal{L}^{{}^{\prime}}(\theta_{0})}.$
(6)
In Eq. 6, $(\theta^{*}-\theta_{0})$ can be regarded as the optimization
distance of $\theta$ from the initial state to the optimal state. However,
since the optimal parameter $\theta^{*}$ is always invisible in practice, we
require a more simplified definition for the gap. The term
$\frac{R(\theta_{0})}{\mathcal{L}^{{}^{\prime}}(\theta_{0})}$ contains high-
order derivatives, which implies the complexity of the optimization surface.
We propose a hypothesis that the optimization distance and the optimization
surface’s complexity can reflect the optimization difficulty of $\theta$, and
they are proportional to $1/\mathcal{L}^{{}^{\prime}}(\theta_{0})$. Therefore,
we define $1/\mathcal{L}^{{}^{\prime}}(\theta_{0})$ as a factor of the
optimization difficulty from $\theta_{0}$ to $\theta^{*}$ as follows:
$\mathcal{f}(\theta^{*},\theta_{0})=\frac{1}{\mathcal{L}^{{}^{\prime}}(\theta_{0})}.$
(7)
Next, we utilize the factors on both source
$(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})$ and target
$(\mathcal{D}_{t},\mathcal{T}_{t})$.
$\mathcal{f}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta^{*},\theta_{0})=\frac{1}{\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{{}^{\prime}}(\theta_{0})}.$
(8)
$\mathcal{f}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta^{*},\theta_{0})=\frac{1}{\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{{}^{\prime}}(\theta_{0})}.$
(9)
We define the gap between
$\theta_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{*}$ and
$\theta_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{*}$ as
$\mathcal{g}(\theta_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{*},\theta_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{*})$,
where $\theta_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{*}$ and
$\theta_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{*}$ represents the optimal model
parameters on the source and target, respectively. Since the value of the
factor could be positive or negative, the absolute value of the two factors’
subtraction is used to represent the gap between
$\mathcal{f}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta^{*},\theta_{0})$
and $\mathcal{f}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta^{*},\theta_{0})$.
$\displaystyle\mathcal{g}(\theta_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{*},\theta_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{*})$
$\displaystyle=\quad$
$\displaystyle\lvert\mathcal{f}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta^{*},\theta_{0})-\mathcal{f}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta^{*},\theta_{0})\rvert$
$\displaystyle=\quad$
$\displaystyle\lvert\frac{1}{\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{{}^{\prime}}(\theta_{0})}-\frac{1}{\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{{}^{\prime}}(\theta_{0})}\rvert$
$\displaystyle=\quad$
$\displaystyle\lvert\frac{\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{{}^{\prime}}(\theta_{0})-\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{{}^{\prime}}(\theta_{0})}{\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{{}^{\prime}}(\theta_{0})\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{{}^{\prime}}(\theta_{0})}\rvert$
$\displaystyle=\quad$
$\displaystyle\frac{\lvert\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{{}^{\prime}}(\theta_{0})-\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{{}^{\prime}}(\theta_{0})\rvert}{\lvert\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{{}^{\prime}}(\theta_{0})\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{{}^{\prime}}(\theta_{0})\rvert}.$
(10)
In practice, a model has numerous parameters, thus we expand the above
derivation to a high dimension version
$\mathcal{G}^{\prime}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]$.
$\displaystyle\mathcal{G}^{\prime}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]$
$\displaystyle=\quad$
$\displaystyle\parallel\mathcal{F}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta^{*},\theta_{0})-\mathcal{F}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta^{*},\theta_{0})\parallel_{2}$
$\displaystyle=\quad$
$\displaystyle\frac{\parallel\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})-\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\parallel_{2}}{\parallel\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})\enspace\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\parallel_{2}}.$
(11)
Similar with Eq. 7,
$\mathcal{F}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta^{*},\theta_{0})$
and $\mathcal{F}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta^{*},\theta_{0})$
are two vectors, measuring the optimization difficulty from $\theta_{0}$ to
$\theta_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}^{*}$ and
$\theta_{(\mathcal{D}_{t},\mathcal{T}_{t})}^{*}$, respectively. We then
deflate
$\frac{\parallel\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})-\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\parallel_{2}}{\parallel\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})\enspace\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\parallel_{2}}$
with Schwarz inequality [46].
$\frac{\parallel\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})-\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\parallel_{2}}{\parallel\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})\enspace\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\parallel_{2}}\quad\geq$
$\\\
\quad\quad\quad\quad\quad\quad\frac{\parallel\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})-\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\parallel_{2}}{\parallel\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})\parallel_{2}\enspace\parallel\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\parallel_{2}}.$
(12)
Considering calculating the gradient once might obtain abnormal gradient, we
employ PGE (Sec. 3.2) to calculate the transferability gap which is defined as
$\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]$.
$\displaystyle\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]=\frac{\parallel
PGE_{(\mathcal{D}_{t},\mathcal{T}_{t})}-PGE_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}\parallel_{2}}{\parallel
PGE_{(\mathcal{D}_{t},\mathcal{T}_{t})}\parallel_{2}\enspace\parallel
PGE_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}\parallel_{2}}$
$\displaystyle=\frac{\parallel\mathbb{E}_{\theta_{0}}\left[\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})\right]-\mathbb{E}_{\theta_{0}}\left[\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\right]\parallel_{2}}{\parallel\mathbb{E}_{\theta_{0}}\left[\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})\right]\parallel_{2}\enspace\parallel\mathbb{E}_{\theta_{0}}\left[\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\right]\parallel_{2}}.$
(13)
It should be noted that only the gradient of the model’s backbone $w$ is
applied when calculating
$\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]$.
Essentially, this metric measures the disparity between
$(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})$ and
$(\mathcal{D}_{t},\mathcal{T}_{t})$. Finally, we calculate the transferability
scores between each $(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})$ in
$\\{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})\\}_{i=1}^{N}$ and
$(\mathcal{D}_{t},\mathcal{T}_{t})$ with Eq. 3.3. The algorithm for acquiring
the transferability score with PGE is outlined in Algorithm 1.
Algorithm 1 Principal Gradient Expectation
A model $\boldsymbol{\mathit{M}}$ with random initialization $\theta_{0}$,
several different source
$\\{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})\\}_{i=1}^{N}$, and a target
$(\mathcal{D}_{t},\mathcal{T}_{t})$.
Ranking of the results of transfer of all sources to the target with
$\boldsymbol{\mathit{M}}$.
for $n=1\to N+1$ do //$N+1$ denotes N sources and a target.
for $i=1\to I$ do //$I$ denotes the number of iterations.
Input some instances into the model $\boldsymbol{\mathit{M}}$.
Computing $\nabla\mathcal{L}(\theta_{0})$
$\mathbb{E}_{\theta_{0}}[\nabla\mathcal{L}(\theta_{0})]\leftarrow((i-1)\ast\mathbb{E}_{\theta_{0}}[\nabla\mathcal{L}(\theta_{0})]+\nabla\mathcal{L}(\theta_{0}))/i$
//The gradients obtained this time are added to the previously collected
gradients and averaged.
end for
end for
for $i=1\to N$ do //$N$ denotes the number of sources
Calculate
$\frac{\parallel\mathbb{E}_{\theta_{0}}\left[\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})\right]-\mathbb{E}_{\theta_{0}}\left[\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\right]\parallel_{2}}{\parallel\mathbb{E}_{\theta_{0}}\left[\nabla\mathcal{L}_{(\mathcal{D}_{t},\mathcal{T}_{t})}(\theta_{0})\right]\parallel_{2}\enspace\parallel\mathbb{E}_{\theta_{0}}\left[\nabla\mathcal{L}_{(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)})}(\theta_{0})\right]\parallel_{2}}$
end for
Sort
return Ranking
### 3.4 Robust Evaluation with Multiple Subsampling
In this section, we introduce the standardized evaluation for the
transferability between different domains.
We define $\mathcal{A}$ as the performance on the target transferred from the
pre-trained model. If
$\mathcal{A}((\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t}))>\mathcal{A}((\mathcal{D}_{s(j)},\mathcal{T}_{s(j)});(\mathcal{D}_{t},\mathcal{T}_{t})))$,
we expect
$\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]<\mathcal{G}[(\mathcal{D}_{s(j)},\mathcal{T}_{s(j)});(\mathcal{D}_{t},\mathcal{T}_{t})]$.
The reliability of our method is validated by computing the correlations
between the ranking of
$\\{\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]\\}_{i=1}^{N}$
and the ranking of
$\\{\mathcal{A}((\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t}))\\}_{i=1}^{N}$.
Kendall’s $\tau$ [14] coefficient, a measure of rank correlation, quantifies
the similarity between two rankings. Kendall’s $\tau$ coefficient is defined
as:
${\displaystyle\tau={\frac{2}{n(n-1)}}\sum_{i<j}\operatorname{sgn}(x_{i}-x_{j})\operatorname{sgn}(y_{i}-y_{j})}.$
(14)
$x_{i}\in\mathcal{X}$ and $y_{i}\in\mathcal{Y}$ ($\mathcal{X}$ and
$\mathcal{Y}$ represent two rankings respectively). And n indicates the number
of items in the ranking. The $sgn(\cdot)$ function is a symbolic function.
Specifically, the range of $\tau$ is $\left[-1,1\right]$, a higher $\tau$
indicates a stronger correlation between
$\\{\mathcal{G}[(\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t})]\\}_{i=1}^{N}$
and
$\\{\mathcal{A}((\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t}))\\}_{i=1}^{N}$.
$\tau=0$ shows no correlation between them.
Following [35, 36], we generate multiple subsets of the target domain
$\mathcal{D}_{t}$ with two approaches: $\emph{I}_{1}$ randomly selects
$\eta_{1}\%$ categories in $\mathcal{D}_{t}$ and all samples in the selected
categories are used. $\emph{I}_{2}$ randomly selects $\eta_{2}\%$ samples from
each category in $\mathcal{D}_{t}$. Concretely, we construct one hundred
different $(\mathcal{D}_{t},\mathcal{T}_{t})$ from $\mathcal{D}_{t}$ with the
two approaches mentioned above and evaluate the performance of the pre-trained
model with the selected target test set. A curve is drawn in which the
horizontal axis represents the sizes of the subsets and the vertical axis
shows the performance of the different test sets. We use the area under the
curve as the measurement of the
$\mathcal{A}((\mathcal{D}_{s(i)},\mathcal{T}_{s(i)});(\mathcal{D}_{t},\mathcal{T}_{t}))$.
(a) The subsets are constructed by SI
(b) The subsets are constructed by SII
Figure 2: The stability comparison of different methods. $\uparrow$ denotes
that a higher value represents a better source, and $\downarrow$ denotes that
a lower value indicates a better source. We show the transferability scores’
variation of different measurement techniques along with increasing sampling
ratios of subsets. The top row of the figure illustrates that we build target
subsets with strategy SI on CIFAR100. We compare the results with five
existing transferability metrics. In each plot, the horizontal axis means the
sampling ratio of each subset while the vertical axis is the transferability
score. The bottom row of the figure indicates that we construct subsets with
strategy SII on CIFAR100.
## 4 Experiment
In this section, we scrutinize and affirm the proposed transferability metric
with diverse image data. Initially, we introduce the experimental settings,
followed by an evaluation of the proposed PGE from four key perspectives:
stability, reliability, efficiency, and generalizability.
### 4.1 Experimental Settings
Datasets. Our experiments are conducted on the following eight datasets:
CIFAR10 [25], CIFAR100 [25], STL10 [6], Mini-ImageNet [51], CUB [52], MNIST
[8], FGVC-Aircraft (Aircraft) [34], and PASCAL VOC 2012 [13]. Table 1 provides
a summary of the characteristics of these datasets, including resolution size
(Resolution), number of images (Img), and number of classes (Class).
Dataset | Resolution | Img | Class
---|---|---|---
CIFAR10 [25] | $32\times 32$ | 60k | 10
CIFAR100 [25] | $32\times 32$ | 60k | 100
STL10 [6] | $96\times 96$ | 13k | 10
Mini-ImageNet [51] | $84\times 84$ | 60k | 200
CUB [52] | $512\times 512$ | 11.7k | 200
MNIST [8] | $28\times 28$ | 70k | 10
Aircraft [34] | $512\times 512$ | 10k | 100
PASCAL VOC 2012 [13] | $320\times 480$ | 1.7k | 21
Table 1: The summary of datasets.
Transfer Methods. While previous studies have concentrated more on linear
probing, we examine both fine-tuning and linear probing.
(1) _Linear Probing_. This technique involves freezing the feature extractor
of the model and training a task-specific layer from scratch with the target
dataset.
(2) _Fine-Tuning_. Here, we replace the model’s task-specific layer with a new
one and fine-tune the entire model, including the feature extractor and the
task-specific layer, on the target dataset.
Evaluation. We construct the subsets with two distinct methods as follows:
SI: The first approach randomly samples 5% to 100% of the target categories
and uses all images within these categories.
SII: The second approach randomly selects a percentage between 10% to 100%
images within each category.
Notably, we do not sample all target datasets with either of the two sampling
approaches. And we generate subsets with SI when the number of samples in each
category is relatively small.
Implementation Details. We run 600 epochs of linear probing and fine-tuning
for each subset of the target domain (using SGD without Momentum and Cosine
annealing) [33]. And we adjust the learning rate with an initial one as
$0.1\ast batchsize/256$. Three classic backbones are adopted in the
experiments, i.e., Resnet18 [20], Resnet50 [20], and VGG16 [44]. Additional
experiments with more datasets and large-scale models are shown in Appendix.
### 4.2 Stability Comparison
In this section, we compare the proposed PGE with existing techniques,
including LEEP [35], LogMe [55], H-score [1], NCE [49], and GBC [36].
Moreover, we use CIFAR10, STL10, Mini-ImageNet, CUB, and MNIST as source
datasets and CIFAR100 as the target dataset.
Figure 2 shows the variation of transfer scores among different measurement
techniques when increasing the sampling ratio of each subset. It is
acknowledged that even with different subsets of a target domain, the
measurement results should be stable. In Figure 2(a), the subsets are
constructed by SI. We observed that as the sampling ratio arises, H-score [1]
and LogMe [55] increase while LEEP, NCE, and GBC decrease. By contrast, our
proposed PGE is superior to all compared methods for stability, as the
sampling ratio has no effect on its outputs. So we argue that the distribution
of the target domain is an important factor to compute the transferability for
the compared methods, as the scores vary with different distributions. Since
our calculation does not involve the distribution, it is more stable than
others. Moreover, the curves of LEEP [35], LogMe [55], and NCE [49] almost
overlap when randomly sampling categories over CIFAR100 as SI, which
demonstrate that it’s hard to discriminate which is the best source for the
target because they tend to yield similar transferability scores. In addition,
although H-score and GBC gain the same correct source (i.e., CIFAR 10) as our
PGE, PGE produces more stable and distinguishable results. And we can get the
correct result with only a portion of categories from the target dataset. In
Figure 2(b), the subsets are constructed by SII. A similar conclusion could be
drawn as in Figure 2(a) that PGE shows remarkable advantages over other
methods. We think these approaches may fail when the number of images is small
since the data distribution of a small target dataset usually cannot represent
the true distribution of real-world data. In comparison, PGE uses the
expectation of the principle gradient to estimate the transferability gap, as
the expectation can effectively reduce the impact of abnormal gradients.
Therefore, the proposed PGE can still obtain the correct result. In summary,
the proposed PGE yields more consistent and distinguishable results for both
strategies.
| CIFAR10 [25] | STL10 [6] | CIFAR100 [25] | CUB [52] | Aircraft [34] | Average Kendall’s $\tau$
---|---|---|---|---|---|---
| LP | FT | LP | FT | LP | FT | LP | FT | LP | FT
LEEP) [35] | 0.19 | 0 | 0 | 0 | 0 | -0.6 | 0.19 | 0.2 | -0.19 | -0.19 | -0.04
H-score) [1] | 1 | 0.79 | 1 | 1 | 1 | 0 | -0.19 | 0.19 | 0.19 | 0.19 | 0.52
NCE) [49] | 0.39 | 0.19 | -0.2 | -0.2 | 0 | -0.6 | -0.19 | -0.39 | -0.79 | -0.79 | -0.26
LogMe) [55] | 1 | 0.79 | 1 | 1 | 1 | 0 | -0.39 | -0.4 | -0.79 | -0.4 | 0.281
GBC) [36] | 1 | 0.79 | 0.4 | 0.4 | 1 | 0 | 0.39 | 0.39 | 0.19 | -0.19 | 0.44
PGE (Ours) | 1 | 0.79 | 1 | 1 | 1 | 0 | 0.39 | 0.39 | 0.79 | 0.39 | 0.68
| | | | | | | | | | |
Table 2: The reliability comparison of different methods. The value (higher is better) indicates the correlation between the ranking calculated by the transferability estimation technique and the ranking of transfer performances. Both the source and target tasks are image classification. And we describe the results for five target domains with two transfer methods (Linear Probing (LP) and Fine-Tuning (FT)). It could be found that the proposed method obtains the highest Kendall’s $\tau$) [14] coefficient in most experiments and the highest average Kendall’s $\tau$ [14] coefficient for the different domains. | PGE Gap $\downarrow$ ($10^{-2}$) | LP(%) | FT(%)
---|---|---|---
CIFAR100 | 0.284 | 50.57 | 66.69
STL10 | 0.773 | 39.86 | 65.39
Mini-ImageNet | 1.118 | 32.04 | 65.28
CUB | 1.125 | 26.20 | 60.67
MNIST | 4.842 | 17.32 | 62.66
| | $\tau$ : 1 | $\tau$ : 0.79
| | |
Table 3: The reliability of PGE scores. $\downarrow$ denotes that a lower gap which represents a better source. Transfer performances were obtained with two transfer methods, i.e., Linear Probing (LP) and Fine-Tuning (FT). | PGE Gap $\downarrow$ ($10^{-2}$) | MIoU(%)
---|---|---
CIFAR100 | 3.86 | 65.09
CIFAR10 | 4.04 | 65.03
Mini-ImageNet | 1.17 | 67.85
STL10 | 1.23 | 66.07
CUB | 1.18 | 65.53
| $\tau$ : 0.79
| |
Table 4: The generalizability of PGE with PASCAL VOC 2012 dataset.
$\downarrow$ denotes that a lower gap which represents a better source.
### 4.3 Reliability Comparison
In this section, we adopt the consistency between the estimated results and
the real transfer performances to compare the reliability of different
methods. As mentioned in Section 3, the transferability is supposed to
correlate well with the final performance of a model after fine-tuning/linear
probing on the target task. To validate the reliability of the proposed PGE,
we conduct experiments with both fine-tuning and linear probing on CIFAR10,
CIFAR100, STL10, CUB, and Aircraft. We choose five different sources for each
target. As shown in Table 2, the relevance between the transfer performances
and transferability is computed with Kendall’s $\tau$ [14] coefficient. In the
linear probing process, PGE identifies the optimal source for all five
targets. In the fine-tuning process, PGE determines the optimal source for
four targets. Moreover, we achieve the highest correlation for CIFAR10, CUB,
and Aircraft with both fine-tuning and linear probing. Overall, the proposed
PGE obtains the highest average correlation of $0.68$ among all existing
approaches.
The results in Table 2 reveal that H-score and LogMe can produce competitive
results on simple datasets (CIFAR10, STL10, and CIFAR100). However, most of
the existing approaches show poor performance on more challenging datasets
(CUB and Aircraft). We consider that existing methods are sensitive to the
resolution of the images while the proposed PGE is robust enough to alleviate
this issue.
Specifically, Table 3 presents the results of the estimated transferability
and the transfer performance on CIFAR10. When linear probing is adopted, the
transfer performances completely match ($\tau=1$) the ranking of the PGE
results. For fine-tuning, we also obtain Kendall’s $\tau=0.79$, illustrating
the transfer performances and PGE results are well-aligned.
Pre-trained task | Transfer Target | H-score [1] | LEEP [35] | NCE [49] | GBC [36] | LogMe [55] | PGE (ours)
---|---|---|---|---|---|---|---
Classification | Classification | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Classification | Regression | ✗ | ✗ | ✗ | ✗ | ✓ | ✓
Unsupervised | Classification | ✗ | ✗ | ✗ | ✗ | ✓ | ✓
Unsupervised | Regression | ✗ | ✗ | ✗ | ✗ | ✓ | ✓
| | | | | | |
Table 5: Generalizability of different techniques for different transfer
settings. “✓ / ✗” indicates whether the technique is applicable/unapplicable
for this transfer setting.
| PGE Gap $\downarrow$ (Sup.) | PGE Gap $\downarrow$ ( Uns. ($10^{-2}$)) | LP (%) | FT (%)
---|---|---|---|---
Mini-ImageNet | 4.35 | 5.00 | 48.21 | 61.83
CIFAR10 | 1.91 | 5.80 | 45.43 | 61.36
CIFAR100 | 2.56 | 7.79 | 45.34 | 59.47
CUB | 4.75 | 7.81 | 37.93 | 55.82
| | | |
Table 6: The ablation study with supervised (Sup.) and unsupervised (Uns.)
strategies of obtaining gradients. $\downarrow$ denotes that a lower gap which
represents a better source.
### 4.4 Efficiency Comparison
Regarding a new source, the existing methods need the pre-trained parameters
trained on the new domain to measure the transferability, which is time-
consuming. In contrast, our proposed method is more computationally efficient
for the following reasons. First, by employing the gradient of the first-order
optimization, our method does not require any pre-training process for the
source. Second, by analyzing the stability of the proposed method above, we
can find that the proposed PGE can significantly reduce the computational cost
by reducing the amount of data, which is not applicable to all other methods.
Table 7 compares the efficiency. We speed up the estimation process by up to
$7\times$ faster than existing works while obtaining more accurate
transferability.
Method | H-score | LEEP | NCE | GBC | LogMe | PGE (ours)
---|---|---|---|---|---|---
Time(s) | 327159 | 327135 | 327131 | 327212 | 327299 | 45488
| | | | | |
Table 7: The time burden of transferability estimation from CIFAR10 to
CIFAR100 with ResNet18.
### 4.5 Generalizability Comparison
We further extend our proposed PGE to compute transferability between source
and target tasks when they are different. The source domains are CIFAR10,
CIFAR100, STL10, Mini-ImageNet, and CUB while the target domain is PASCAL VOC
2012. The estimation process aims to find the most suitable source domain for
the segmentation task based on those classification tasks. The backbone here
is VGG16 and the epochs for pre-training is 100. The results are shown in
Table 4. The PGE values in Table 4 suggest that Mini-ImageNet could be a
suitable source, which is verified by the segmentation accuracy.
We list four common transfer settings, as shown in Table 5. It could be
concluded that all of the approaches can be easily adapted to classification
tasks. However, when the source task and target task are different, H-score,
LEEP, NCE and GBC can not estimate transferability. By contrast, our proposed
PGE and LogMe are more practical to adapt to this setting. For classification
and regression tasks, LogMe designs different modules while our proposed PGE
computes the transferability uniformly without pre-training.
### 4.6 Ablation Study
Recent researches [57, 22] has shown that unsupervised techniques acquire more
low-level and mid-level information. The information is more readily adaptable
to a new domain than supervised techniques. In supervised learning processes,
models tend to learn high-level semantics. In this subsection, we examine
transferability with principle gradient expectation which is obtained by
supervised and unsupervised techniques. The results are shown in Table 6. It
has been experimentally found that the supervised method is more susceptible
than the unsupervised method. A reasonable explanation for this is that
category information is embedded into the model to improve discrimination but
is harmful to calculating transferability.
## 5 Conclusion
Determining which source is the best for a particular target task is
challenging. Moreover, it is computationally costly to determine the source by
fine-tuning/linear probing all possible combinations of the sources and target
task. In this work, we summarize the properties that a good transferability
metric should possess. Building upon them, we propose a simple yet effective
transferability estimation approach termed PGE based on principal gradient
expectation. To properly evaluate the method’s validity, we applied two sub-
sampling techniques to the target domain. The experimental results on both
fine-tuning and linear probing demonstrate that PGE is superior to existing
metrics. Furthermore, since PGE computes the gradient of the backbone under
the unsupervised mode, it is more flexible and can be extended to different
tasks. In the future, it is worth exploring choosing a proper source task or
even a rational composition of source tasks for a given target task.
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# Dual adaptive MPC using an exact set-membership reformulation
Anilkumar Parsi Diyou Liu Andrea Iannelli Roy S. Smith Automatic Control
Laboratory, ETH Zurich, Zürich 8092 (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS> University of Stuttgart, Institute for Systems Theory
and Automatic Control, Stuttgart 70569 (e-mail<EMAIL_ADDRESS>stuttgart.de)
###### Abstract
Adaptive model predictive control (MPC) methods using set-membership
identification to reduce parameter uncertainty are considered in this work.
Strong duality is used to reformulate the set-membership equations exactly
within the MPC optimization. A predicted worst-case cost is then used to
enable performance-oriented exploration. The proposed approach guarantees
robust constraint satisfaction and recursive feasibility. It is shown that
method can be implemented using homothetic tube and flexible tube
parameterizations of state tubes, and a simulation study demonstrates
performance improvement over state-of-the-art controllers.
###### keywords:
Adaptive MPC, dual control, set-membership, uncertain linear systems
††thanks: This work was supported by the Swiss National Science Foundation
under Grant 200021_178890.
## 1 Introduction
In order to optimally control systems affected by model uncertainty, control
actions must have two features, (i) excite the system to obtain new
information (called exploration); (ii) use available information to minimize a
control cost (called exploitation). Achieving the optimal exploration-
exploitation trade-off is the dual control problem, which is a challenging
problem in control theory and has been studied for over six decades (Feldbaum
(1960-61); Mesbah (2018)). With recent advances in computing capabilities,
there is an increase in research on controllers which can learn model
uncertainties while performing control tasks (e.g., Klenske and Hennig (2016);
Zanon and Gros (2020)), raising the relevance of the dual control problem.
One popular class of controllers which has been studied in this regard is
model predictive control (MPC). The ability of MPC to ensure state and input
constraint satisfaction in the face of uncertainty has been viewed as an
important feature for learning-based control, because it can guarantee safety
of the system (Hewing et al. (2020)). Robust adaptive MPC is one such method,
where features from robust MPC methods (such as tube MPC (Kouvaritakis and
Cannon (2015))) are combined with online identification techniques (such as
set-membership (Milanese and Vicino (1991))) to ensure safe learning of the
model uncertainty. Using various combinations of these two features, multiple
robust adaptive MPC algorithms have been proposed in the recent past, e.g.,
Lorenzen et al. (2019); Bujarbaruah et al. (2020); Soloperto et al. (2020);
Köhler et al. (2021).
However, most existing robust adaptive MPC algorithms use passive exploration.
That is, the controller consists of two separately designed components: the
MPC optimizer which selects inputs based only on exploitation; and the online
identification algorithm which uses the collected measurements to update model
uncertainty. This approach can result in suboptimality, because the dual
control effect is unmodeled. One way to partially address this issue is to
enforce a perstistency of excitation condition on the control inputs (Lorenzen
et al. (2019)). However, optimal dual control trade-off would require
extensive tuning of these methods, and tuning methods have been proposed in
Soloperto et al. (2020). An alternative strategy is to use performance-based
exploration, where exploration is induced by modeling the expected performance
improvement from reduction in uncertainty. This approach was considered in
Parsi et al. (2022), where set-membership identification is used to reduce
model uncertainty, and the key novelty consisted of including the effect of
identification in the MPC optimizer. Nevertheless, this required
approximations which result in overestimating the capabilities of the
identification algorithm, and can lead to suboptimal exploratory actions.
In this work, we propose a novel way to exactly reformulate set-membership
identification inside MPC optimization problems. The proposed method has two
advantages. First, it results in an improved navigation of the dual control
trade-off compared to existing methods because the identification is exactly
modeled within the MPC controller. Second, the method can be used to generate
dual control algorithms starting from multiple existing formulations of
passive adaptive MPC. We highlight this feature by implementing dual control
using homothetic and flexible state tube parameterizations, and demonstrate
performance improvement using a simulation study.
### 1.1 Notation
The sets of real numbers and non-negative real numbers are denoted by
$\mathbb{R}$ and $\mathbb{R}_{\geq 0}$ respectively. The sequence of integers
from $n_{1}$ to $n_{2}$ is represented by $\mathbb{N}_{n_{1}}^{n_{2}}$. For a
vector $b$, $||b||_{\infty}$ represents its $\infty-$norm. The $i^{th}$ row of
a matrix $A$ is denoted by $[A]_{i}$, and $\mathbf{1}$ denotes a column vector
of appropriate length with all entries 1. The convex hull operator is
represented by $\text{co}\\{\cdot\\}$. The notation $a_{l|k}$ denotes the
value of $a$ at time $k{+}l$ computed at time $k$, $I_{n}$ denotes the
identity matrix with $n$ rows, and $\oplus$ denotes the Minkowski sum of two
sets.
## 2 Problem formulation
Consider an uncertain, linear system described by
$x_{t+1}=A(\theta)x_{t}+B(\theta)u_{t}+w_{t},$ (1)
where $x_{t}{\in}\mathbb{R}^{n}$ represents the state,
$u_{t}{\in}\mathbb{R}^{m}$ the control input and $w_{t}{\in}\mathbb{R}^{n}$
the additive disturbance at time $t$. The state space matrices have the
parametric description
$\displaystyle\begin{split}A(\theta)=A_{0}+\displaystyle\sum_{i=1}^{p}A_{i}[\theta]_{i},\quad
B(\theta)=B_{0}+\displaystyle\sum_{i=1}^{p}B_{i}[\theta]_{i},\end{split}$ (2)
where $\theta{\in}\mathbb{R}^{p}$ is a constant but unknown parameter, and
$\\{A_{i},B_{i}\\}_{i=0}^{p}$ are known matrices which model structured
uncertainty. The true value of $\theta=\theta^{*}$ is known to lie inside a
bounded set
$\Theta:=\\{\theta\in\mathbb{R}^{p}|H_{\theta}\theta\leq h_{\theta}\\},$ (3)
where $H_{\theta}\in\mathbb{R}^{n_{\theta}\times p}$ and
$h_{\theta}\in\mathbb{R}^{n_{\theta}}$. The disturbance $w_{t}$ always lies
within the bounded set
$\mathbb{W}:=\\{w\in\mathbb{R}^{n}|H_{w}w\leq h_{w}\\},$ (4)
where $H_{w}\in\mathbb{R}^{n_{w}\times n},h_{w}\in\mathbb{R}^{n_{w}}$. The
state of the system is perfectly measured at each time step. Given an initial
state $x_{0}$, the objective is to track a reference trajectory given by
$r_{t}\in\mathbb{R}^{n}$ for time steps t in $[0,T]$. This objective is
specified as a minimization of the worst-case deviation of the states and
inputs from the optimal reference, i.e.,
$J=\min_{u_{t}}\max_{w_{t}\in\mathbb{W}}\sum_{t=0}^{T}\left\lVert
Q(x_{t}{-}r_{t})\right\rVert_{\infty}{+}\left\lVert
R(u_{t}{-}u^{*}_{t})\right\rVert_{\infty},$ (5)
where $Q$ and $R$ are positive definite matrices, and
$\\{u^{*}_{t}\\}_{t=0}^{T}$ is a sequence of (possibly) unknown input
setpoints which track the reference trajectory while satisfying the dynamics
of the true system. The states and inputs of the system are constrained to lie
inside the compact set
$\mathbb{Z}=\left\\{(x,u)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\bigr{|}Fx+Gu\leq\mathbf{1}\right\\},$
(6)
where $F\in\mathbb{R}^{n_{c}\times n}$ and $G\in\mathbb{R}^{n_{c}\times m}$
are known matrices.
In the following sections, a dual control algorithm will be described to
perform the above task. In Section 3, an online identification scheme will be
used to reduce uncertainty in $\theta$, and a robust state tube will be
constructed to ensure constraint satisfaction. In Section 4, future set-
membership is formulated as a function of the MPC input variables, and the MPC
cost function is defined using a predicted state tube to enable performance-
based exploration. The dual adaptive MPC algorithm and its properties are
discussed in Section 5. The proposed method will be implemented using two
popular parameterizations of state tubes from literature, namely homothetic
tubes (HT) (Lorenzen et al. (2019)) and flexible tubes (FT) (Lu and Cannon
(2019)).
## 3 Parameter identification and robust state tubes
### 3.1 Set membership and parameter estimates
In order to reduce the uncertainty in $\theta$, set-membership identification
(Milanese and Vicino (1991)) is used to construct a sequence of sets
$\\{\Theta_{k}\\}_{k=0}^{T}$ satisfying
$\displaystyle\begin{split}\Theta_{k-1}&\cap\Delta_{k}\subseteq\Theta_{k}\subseteq\Theta_{k-1},\quad\Theta_{0}=\Theta,\\\
\Delta_{k}=\\{&\theta|x_{i+1}{-}A(\theta)x_{i}{-}B(\theta)u_{i}\in\mathbb{W},\>\forall
i\in\mathbb{N}_{k-\tau}^{k-1}\\}\\\ =\\{&\theta|-H_{w}D_{i}\theta\leq
h_{w}+H_{w}d_{i+1},\>\forall i\in\mathbb{N}_{k-\tau}^{k-1}\\},\end{split}$ (7)
where $\Delta_{k}$ is the set of all parameters $\theta$ that could have
generated the last $\tau$ measurements $\\{x_{i}\\}_{i=k-\tau+1}^{k}$, and
$\displaystyle\begin{split}D_{i}&:=D(x_{i},u_{i})=\begin{bmatrix}A_{1}x_{i}{+}B_{1}u_{i},&\ldots,&A_{p}x_{i}{+}B_{p}u_{i}\end{bmatrix},\\\
d_{i+1}&:=\>A_{0}x_{k}+B_{0}u_{k}-x_{i+1},\quad\forall
i\in\mathbb{N}_{k-\tau}^{k-1}.\end{split}$ (8)
The set $\Theta_{k}$ is described by a fixed number of hyperplanes for
computational efficiency, i.e.,
$\Theta_{k}:=\\{\theta|H_{\theta}\theta\leq h_{\theta_{k}}\\}.$ (9)
Here $h_{\theta_{k}}$ is initialized with the initial estimate of uncertainty
$h_{\theta}$, and is updated such that (7) is satisfied by solving a set of
linear programs, as proposed in Lorenzen et al. (2019). Whereas HT adaptive
MPC methods allow $H_{\theta}$ in (9) to be arbitrarily chosen, FT methods
further restrict $\Theta_{k}$ to polytopes with hyperplanes parallel to
coordinate axes (or hyperboxes). The numerical advantage is that, the vertices
of $\Theta_{k}$, denoted by $\\{\theta^{j}_{k}\\}_{j=1}^{q_{\theta}}$, can be
readily computed as a predefined linear combination of $h_{\theta_{k}}$ (Liu
(2022a)).
###### Lemma 1
(Lorenzen et al., 2019, Theorem 14(ii)) Under the set-membership scheme
(7)-(9), $\theta^{*}\in\Theta_{k}$ for all $k\geq 0.$
Additionally, an estimate $\bar{\theta}_{k}\in\Theta_{k}$ of the parameter
$\theta^{*}$ is computed using a least mean squares filter
$\displaystyle\begin{split}&\tilde{\theta}_{k+1}=\bar{\theta}_{k}{+}\mu
D(x_{k},u_{k})^{T}(x_{k+1}{-}A(\bar{\theta}_{k})x_{k}{-}B(\bar{\theta}_{k})u_{k}),\\\
&\bar{\theta}_{k+1}=\Pi_{\Theta_{k}}(\tilde{\theta}_{k+1}),\end{split}$ (10)
where $\Pi_{\Theta_{k}}$ represents a projection operator, and
$\bar{\theta}_{0}$ is an initial guess.
### 3.2 Control policies and tracking formulation
The control input is parameterized as
$u_{l|k}(x)=K({x}-r_{k+l})+v_{l|k},\quad\forall l\in\mathbb{N}_{0}^{N-1},$
(11)
where $v_{l|k}$ are online optimization variables, $K$ is a stabilizing
feedback gain and $N$ is the MPC prediction horizon.
###### Assumption 1
The parameter set $\Theta$ is such that there exists a feedback gain $K$ which
asymptotically stabilizes
$A_{\text{cl}}(\theta)=A(\theta)+B(\theta)K,\>\forall\theta\in\Theta$.
The input setpoints to be tracked are unknown due to the uncertainty in the
true parameter. Hence, the MPC problem estimates $u^{*}_{k+l}$ using
$\bar{\theta}_{k}$ and optimization variables $\bar{u}_{l|k}$ such that
$\displaystyle\begin{split}\begin{bmatrix}A(\bar{\theta}_{k})&B(\bar{\theta}_{k})\end{bmatrix}\begin{bmatrix}r_{k+l}\\\
\bar{u}_{l|k}\end{bmatrix}&=\begin{bmatrix}r_{k+l+1}\end{bmatrix},\quad\forall
l\in\mathbb{N}_{0}^{N-1},\\\
\begin{bmatrix}A(\bar{\theta}_{k})&B(\bar{\theta}_{k})\end{bmatrix}\begin{bmatrix}r_{k+N}\\\
\bar{u}_{N|k}\end{bmatrix}&=\begin{bmatrix}r_{k+N}\end{bmatrix}.\end{split}$
(12)
The setpoint for timestep $k+N$ is chosen to be an equilibrium point, in order
to define the terminal components of MPC (discussed in Section 3.4). Under
assumptions on the rank of the system matrices and the number of available
inputs (Parsi et al., 2022, Assumption 2), it can be guaranteed that there
always exists a feasible input sequence satisfying (12).
### 3.3 State tube construction
Using the control policy (11), a sequence of sets
$\\{\mathbb{X}_{l|k}\\}_{l=0}^{N}$ is constructed such that
$\mathbb{X}_{0|k}\ni x_{k}$ and for all
$x\in\mathbb{X}_{l|k},\theta\in\Theta_{k}$ and $w\in\mathbb{W}$,
$\displaystyle\mathbb{X}_{l+1|k}\ni A(\theta)x+B(\theta)u(x)+w,\quad
l\in\mathbb{N}_{0}^{N-1}.$ (13)
This sequence of sets is called the robust state tube (RST), as it contains
all feasible state trajectories within the MPC prediction horizon. The RST is
constructed by outer approximating the reachable states using parameterized
sets for computational efficiency.
#### 3.3.1 Homothetic tubes.
In an HT formulation, the RST is parameterized as
$\displaystyle\mathbb{X}^{H}_{l|k}$
$\displaystyle=\\{z_{l|k}\\}\oplus\alpha_{l|k}\mathbb{X}_{0},$ (14)
where $z_{l|k}\in\mathbb{R}^{n}$ and $\alpha_{l|k}\in\mathbb{R}_{\geq 0}$ are
translation and scaling variables and $\mathbb{X}_{0}$ is a predefined
polytope described in both hyperplane and vertex forms
$\displaystyle\mathbb{X}^{H}_{0}:=\\{x|H_{x}x\leq\mathbf{1}\\}=\text{co}\\{x^{1},x^{2},\ldots,x^{q}\\}.$
(15)
The following notation is introduced
$\displaystyle\begin{split}x_{l|k}^{j}&=z_{l|k}+\alpha_{l|k}x^{j},\quad
u_{l|k}^{j}{=}K(x_{l|k}^{j}-r_{k+l}){+}v_{l|k},\\\
D_{l|k}^{j}&=D(x_{l|k}^{j},u_{l|k}^{j}),\quad
d_{l|k}^{j}{=}A_{0}x_{l|k}^{j}{+}B_{0}u_{l|k}^{j}-z_{l+1|k},\\\
[\bar{f}]_{i}&=\underset{x\in\mathbb{X}_{0}}{\text{max}}[F+GK]_{i}x,\quad[\bar{w}]_{j}=\underset{w\in\mathbb{W}}{\text{max}}\>[H_{x}]_{j}w,\end{split}$
(16)
where $\bar{f}$ and $\bar{w}$ are computed offline.
###### Proposition 1
(Lorenzen et al., 2019, Proposition 9) Let the RST be parameterized according
to (14). Then, (6) and (13) are satisfied if,
$\forall\>j{\in}\mathbb{N}_{1}^{q}$ and $l{\in}\mathbb{N}_{0}^{N-1}$, there
exist $\Lambda_{l|k}^{H,j}\in\mathbb{R}^{n_{x}\times n_{\theta}}_{\geq 0}$
such that
$\displaystyle(F+GK)z_{l|k}+G(v_{l|k}-Kr_{k+l})+\alpha_{l|k}\bar{f}$
$\displaystyle\leq\mathbf{1},$ (17a) $\displaystyle-
H_{x}z_{0|k}-\alpha_{0|k}\mathbf{1}$ $\displaystyle\leq-H_{x}x_{k},$ (17b)
$\displaystyle\Lambda_{l|k}^{H,j}h_{\theta_{k}}+H_{x}d_{l|k}^{j}-\alpha_{l+1|k}\mathbf{1}$
$\displaystyle\leq-\bar{w},$ (17c) $\displaystyle H_{x}D_{l|k}^{j}$
$\displaystyle=\Lambda_{l|k}^{H,j}H_{\theta}.$ (17d)
The reformulation (17c)-(17d) is obtained by imposing (13) at each vertex of
$\mathbb{X}_{l|k}$, and using strong duality (Boyd and Vandenberghe (2004)) to
ensure (13) for all $\theta\in\Theta_{k}$.
#### 3.3.2 Flexible tubes.
In an FT formulation, the state tube is parameterized as
$\displaystyle\mathbb{X}^{F}_{l|k}$ $\displaystyle=\\{x|H_{x}(x-r_{k+l})\leq
h_{x_{l|k}}\\},\qquad\forall l\in{\mathbb{N}}_{0}^{N},$ (18)
where $H_{x}$ is a matrix chosen offline, and $h_{x_{l|k}}$ are online
optimization variables. The following proposition can then be used to
reformulate the tube propagation constraints.
###### Proposition 2
(Lu and Cannon, 2019, Lemmas 8 and 9) Let the RST be parameterized according
to (18). Then, (6) and (13) are satisfied if,
$\forall\>i{\in}\mathbb{N}_{1}^{n_{x}}$, $j{\in}\mathbb{N}_{1}^{q_{\theta}}$
and $l{\in}\mathbb{N}_{0}^{N-1}$,
$\displaystyle\begin{split}\Lambda_{1,l|k}^{F}(h_{x_{l|k}}+H_{x}r_{k+l})+G(v_{l|k}-Kr_{k+l})&\leq\mathbf{1},\\\
H_{x}(x_{k}-r_{k})&\leq h_{x_{0|k}},\\\
[1\>\>{\theta_{k}^{j}}^{T}]\Lambda_{2,l|k}^{F,i}(h_{x_{l|k}}+H_{x}r_{k+l})-[H_{x}]_{i}r_{k+l+1}&-\\\
[H_{x}]_{i}(B(\theta_{k}^{j})(Kr_{k+l}-v_{l|k}))+[\bar{w}]_{i}\leq[&h_{x_{l+1|k}}]_{i},\end{split}$
(19)
where $\Lambda_{1,l|k}^{F}{\in}\mathbb{R}_{\geq 0}^{n_{c}\times n_{x}}$ and
$\Lambda_{2,l|k}^{F,i}{\in}\mathbb{R}^{(p+1)\times n_{x}}$ are Lagrange
multipliers and are computed offline by solving $\forall
l\in\mathbb{N}_{0}^{N-1}$,
$\displaystyle\begin{split}\forall
s\in\mathbb{N}_{1}^{n_{c}},\quad[\Lambda_{1,l|k}^{F}]_{s}=&\arg\min\limits_{\lambda}\lambda(\mathbf{1}+\mu
H_{x}r_{k+l})\\\ \text{s.t.}\qquad\lambda&\geq 0,\quad\lambda
H_{x}=[F+GK]_{s},\end{split}$ (20)
$\displaystyle\begin{split}\Lambda_{2,l|k}^{F,i}=\arg&\min\limits_{\Lambda}\max\limits_{j\in\mathbb{N}_{1}^{q_{\theta}}}[1\>\>{\theta_{k}^{j}}^{T}]\Lambda(\mathbf{1}+\mu
H_{x}r_{k+l})\\\ \text{s.t.}\quad[1&\quad{\theta_{k}^{j}}^{T}]\Lambda\geq
0,\quad\forall j\in\mathbb{N}_{1}^{n_{\theta}},\\\ \Lambda
H_{x}=&\begin{bmatrix}[H_{x}]_{i}(A_{0}{+}B_{0}K)^{T}&\ldots&[H_{x}]_{i}(A_{p}{+}B_{p}K)^{T}\end{bmatrix}^{T},\end{split}$
(21)
where $\mu\in\mathbb{R}_{\geq 0}$.
The reformulation (19) is obtained by imposing (13) at each vertex of
$\Theta_{k}$, and using duality to ensure (13) for all $x\in\mathbb{X}_{l|k}$.
In addition, duality is also used to reformulate (6). The tuning factor $\mu$
facilitates the reduction of conservatism by weighing the effects of both
$h_{x_{l|k}}$ and $r_{k+l}$ on (6) and (13) while choosing the Lagrange
multipliers offline. Note that Proposition 2 is a slight modification of the
result in Lu and Cannon (2019), where the reference was the origin.
###### Remark 1
The FT formulation requires that both hyperplanes and vertices of $\Theta_{k}$
are available, and HT requires the same for $\mathbb{X}_{0}$. Moreover, the FT
formulation computes the Lagrange multipliers offline for faster online solve
times, which results in conservatism (Köhler et al. (2019)).
### 3.4 Terminal Sets
In order to ensure recursive feasibility, terminal sets are used. That is, the
last set in the RST must be an invariant set under a terminal control law of
the form
$u_{N|k}(x)=K(x-\tilde{x}_{N|k})+v_{N|k},$ (22)
where $v_{N|k}$ are MPC optimization variables and $\tilde{x}_{N|k}$ is chosen
as specified later. Then, the invariance conditions can be written as, for all
$x\in\mathbb{X}_{N|k}$, $\theta\in\Theta_{k}$ and $w\in\mathbb{W}$,
$\mathbb{X}_{N|k}\ni
A(\theta)x{+}B(\theta)u_{N|k}(x){+}w,\>\>(x,u_{N|k}(x))\in\mathbb{Z}.$ (23)
###### Proposition 3
The constraints (23) are satisfied
1. a)
under the HT parameterization (14) and $\tilde{x}_{N|k}{=}z_{N|k}$, if
$\exists\Lambda_{N|k}^{H,j}\in\mathbb{R}^{n_{x}\times n_{\theta}}_{\geq 0}$
such that $\forall j\in\mathbb{N}_{1}^{q}$
$\displaystyle\begin{split}Fz_{N|k}+Gv_{N|k}+\alpha_{N|k}\bar{f}&\leq\mathbf{1},\\\
\Lambda_{N|k}^{H,j}h_{\theta_{k}}+H_{x}d_{N|k}^{j}-\alpha_{N|k}\mathbf{1}&\leq-\bar{w},\\\
\Lambda_{N|k}^{H,j}\geq 0,\quad
H_{x}D_{N|k}^{j}&=\Lambda_{N|k}^{H,j}H_{\theta}.\end{split}$ (24)
2. b)
under the FT parameterization (18) and $\tilde{x}_{N|k}{=}r_{k{+}N}$, if
$\forall\>i{\in}\mathbb{N}_{1}^{n_{x}}$, $j{\in}\mathbb{N}_{1}^{q_{\theta}}$,
$\displaystyle\begin{split}\Lambda_{1,N|k}^{F}(h_{x_{N|k}}{+}H_{x}r_{k+N})+G(v_{N|k}-Kr_{k+N})\leq&\mathbf{1},\\\
[1\>\>{\theta_{k}^{j}}^{T}]\Lambda_{2,N|k}^{F,i}(h_{x_{N|k}}+H_{x}r_{k+N})-[H_{x}]_{i}r_{k+N}-\>\>&\\\
[H_{x}]_{i}(B(\theta_{k}^{j})(Kr_{k+N}-v_{N|k}))+\bar{w}\leq[h_{x_{N|k}}&]_{i}.\end{split}$
(25)
The proof is similar to that of Propositions 1 and 2, and is omitted. A
detailed proof can be found in Liu (2022a).
###### Assumption 2
In both HT and FT methods, the predefined state tube matrix $H_{x}$ is chosen
such that for all $\theta\in\Theta$ and $x\in\\{x|H_{x}x\leq\mathbf{1}\\}$,
$\exists\lambda_{c}\in[0,1)$ satisfying
$H_{x}\bigr{(}A(\theta)x+B(\theta)Kx\bigr{)}\leq\lambda_{c}\mathbf{1}.$ (26)
Assumption 2 imposes contractivity on the state tube shape when $w_{k}$ is
zero. Such an assumption is common in literature (Lu and Cannon (2019)). In
this work, $\lambda_{c}$ will be used to define the terminal cost in Section
4.3.
## 4 Predicted state tube for Exploration
In this section, set-membership identification is formulated as a function of
MPC control input variables. Then, a predicted state tube (PST) is constructed
to contain state trajectories as a function of predicted uncertainty sets.
Finally, the MPC cost function is defined such that performance-based
exploration is induced.
### 4.1 Predicted set-membership identification
In order to predict the effect of the future control inputs on identification,
a sequence of predicted measurements $\\{\hat{x}_{l|k}\\}_{l=0}^{N_{p}}$ is
defined such that
$\displaystyle\begin{split}\hat{x}_{0|k}&=x_{k},\quad\hat{x}_{l+1|k}=A(\bar{\theta}_{k})\hat{x}_{l|k}+B(\bar{\theta}_{k})\hat{u}_{l|k},\\\
\hat{u}_{l|k}&=K(\hat{x}_{l|k}-r_{k+l})+v_{l|k},\quad\forall
l\in\mathbb{N}_{0}^{N_{p}-1},\end{split}$ (27)
where $N_{p}{\in}\mathbb{N}_{2}^{N}$ is the user-defined look-ahead horizon. A
large $N_{p}$ improves the modeling of the dual effect inside MPC, but also
increases the computational complexity. A prediction of $\Delta_{k+l}$ can
then be constructed using (27) as
$\displaystyle\hat{\Delta}_{l|k}$
$\displaystyle:=\\{\theta\>|\hat{x}_{i+1|k}{-}A(\theta)\hat{x}_{i|k}{-}B(\theta)\hat{u}_{i|k}\in\mathbb{W}\>\forall
i\in\mathbb{N}_{l-\tau}^{l-1}\\}$
$\displaystyle=\\{\theta\>|\hat{H}_{\Delta_{l|k}}\theta\leq\hat{h}_{\Delta_{l|k}}\\},$
(28)
where $\hat{x}_{i|k}{=}x_{k+i},\hat{u}_{i|k}{=}u_{k+i}$ when $i{<}0$. Note
that $\hat{H}_{\Delta_{l|k}}$ and $\hat{h}_{\Delta_{l|k}}$ depend on future
control inputs, but the dependence is dropped for clarity of presentation.
Using (4.1), the predicted parameter sets can be defined as
$\displaystyle\hat{\Theta}_{0|k}$
$\displaystyle:=\Theta_{k},\>\hat{\Theta}_{l|k}:=\\{\theta|H_{\theta}\theta\leq\hat{h}_{\theta_{l|k}}\\},\quad
l\in\mathbb{N}_{0}^{N}.$ (29)
In (29), $\hat{h}_{\theta_{0|k}}{=}h_{\theta_{k}}$, and for
$l\in\mathbb{N}_{1}^{N_{p}}$, $\hat{h}_{\theta_{l|k}}$ is computed as
$\displaystyle\begin{split}\forall
i\in\mathbb{N}_{1}^{n_{\theta}},\>[\hat{h}_{\theta_{l|k}}]_{i}&=\mathop{\max}\limits_{\theta}\>[H_{\theta}]_{i}\theta\\\
\text{s.t.}\quad
H_{\theta}\theta&\leq\hat{h}_{\theta_{l-1|k}},\quad\hat{H}_{\Delta_{l|k}}\theta\leq\hat{h}_{\Delta_{l|k}}.\end{split}$
(30)
For $l{\in}\mathbb{N}_{N_{p}+1}^{N}$,
$\hat{h}_{\theta_{l|k}}{=}\hat{h}_{\theta_{N_{p}|k}}$, i.e., no further
identification is modeled. Because $\hat{\Delta}_{l|k}$ depends on future
control inputs, $\hat{h}_{\theta_{l|k}}$ cannot be explicitly computed before
solving the optimization problem. Instead, the following dual reformulation is
used,
$\displaystyle\begin{split}\hat{h}_{\theta_{l|k}}=\hat{\Psi}_{l|k}\begin{bmatrix}\hat{h}_{\theta_{l-1|k}}\\\
\hat{h}_{\Delta_{l|k}}\end{bmatrix},\quad\hat{\Psi}_{l|k}\begin{bmatrix}H_{\theta}\\\
\hat{H}_{\Delta_{l|k}}\end{bmatrix}=H_{\theta},\>\hat{\Psi}_{l|k}\geq
0,\end{split}$ (31)
where $\hat{\Psi}_{l|k}$ represent Lagrange multipliers. Using strong duality,
it can be seen that there exist $\hat{\Psi}_{l|k}$ such that (31) holds. Thus,
using $\hat{\Psi}_{l|k}$ as MPC optimization variables, (30) can be replaced
by (31).
###### Remark 2 (Bilinearities)
The constraints (31) include the product of $\hat{\Psi}_{l|k}$ with other
optimization variables. Thus, (31) results in bilinear constraints, and the
MPC optimization problem is nonconvex.
A similar dual adaptive MPC method has been proposed for HT in Parsi et al.
(2022), but it approximates set-membership online using an intersection of
${\Theta}_{k}$ and $\hat{\Delta}_{l|k}$. This results in MPC overestimating
the capabilities of the actual set-membership identification.
### 4.2 Predicted state tubes
The effect of the future identification on the state trajectories can be
estimated by defining a predicted state tube
$\\{\hat{\mathbb{X}}_{l|k}\\}_{l=0}^{N}$ such that
$x_{k}\in\hat{\mathbb{X}}_{0|k}$ and $\forall l{\in}\mathbb{N}_{0}^{N-1}$
$\displaystyle\hat{\mathbb{X}}_{l{+}1|k}{\ni}A(\theta)x{+}B(\theta)u(x){+}w,\>\forall
x{\in}\hat{\mathbb{X}}_{l|k},\theta{\in}\hat{\Theta}_{l|k},\forall
w{\in}\mathbb{W}.$ (32)
It can be seen from (13) that $\forall\>l\in\mathbb{N}_{0}^{N-1},$
$\hat{\mathbb{X}}_{l|k}{\subseteq}\mathbb{X}_{l|k}$ because
$\hat{\Theta}_{l|k}\subseteq\Theta_{k}$ by construction (30).
#### 4.2.1 Homothetic tubes.
In the HT parameterization, the PST is represented by polytopes of the form
$\displaystyle\hat{\mathbb{X}}^{H}_{l|k}$
$\displaystyle=\\{\hat{z}_{l|k}\\}\oplus\hat{\alpha}_{l|k}\mathbb{X}_{0},\quad\forall
l\in{\mathbb{N}}_{0}^{N},$ (33)
where $\hat{z}_{l|k}\in\mathbb{R}^{n_{x}}$ and
$\hat{\alpha}_{l|k}\in\mathbb{R}_{\geq 0}$ are translation and scaling
variables. Then, the following is a direct extension of Proposition 1.
###### Corollary 4.1 (Pr:HT_RST)
The PST dynamics (32) are satisfied under the HT parameterization (33) if
$\forall j{\in}\mathbb{N}_{1}^{q}$, $l{\in}\mathbb{N}_{0}^{N-1}$, there exist
$\hat{\Lambda}_{l|k}^{H,j}\in\mathbb{R}^{n_{x}\times n_{\theta}}_{\geq 0}$
such that
$\displaystyle\begin{split}-H_{x}\hat{z}_{0|k}-\hat{\alpha}_{0|k}\mathbf{1}\leq-
H_{x}x_{k},\quad
H_{x}\hat{D}_{l|k}^{j}&=\hat{\Lambda}_{l|k}^{H,j}H_{\theta},\\\
\hat{\Lambda}_{l|k}^{H,j}\hat{h}_{\theta_{l|k}}+H_{x}\hat{d}_{l|k}^{j}-\hat{\alpha}_{l+1|k}\mathbf{1}&\leq-\bar{w},\end{split}$
(34)
where $\hat{D}_{l|k}^{j}$ and $\hat{d}_{l|k}^{j}$ are predictions of
${D}_{l|k}^{j}$ and ${d}_{l|k}^{j}$ defined in (16).
#### 4.2.2 Flexible tubes.
In the FT parameterization the, PST is represented by polytopes of the form
$\displaystyle\hat{\mathbb{X}}^{F}_{l|k}$
$\displaystyle=\\{x|H_{x}(x-r_{k+l})\leq\hat{h}_{x_{l|k}}\\},\quad\forall
l\in{\mathbb{N}}_{0}^{N},$ (35)
where $\hat{h}_{x_{l|k}}\in\mathbb{R}^{n_{x}}$ are optimization variables.
Then, the following is a direct extension of Proposition 2.
###### Corollary 4.2 (Pr:FT_RST)
The PST dynamics (32) are satisfied under the FT parameterization (35) if
$\forall i\in\mathbb{N}_{1}^{n_{x}}$, $j\in\mathbb{N}_{1}^{q_{\theta}}$ and
$l{\in}\mathbb{N}_{0}^{N-1}$
$\displaystyle\begin{split}H_{x}(x_{k}-r_{k})&\leq\hat{h}_{x_{0|k}},\\\
[1\>\>\hat{\theta}_{l|k}^{j^{T}}]\Lambda_{2,l|k}^{F,i}(\hat{h}_{x_{l|k}}+H_{x}r_{k+l})-[H_{x}&]_{i}r_{k+l+1}-\\\
[H_{x}]_{i}(B(\hat{\theta}_{l|k}^{j})(Kr_{k+l}-v_{l|k}))+\bar{w}&\leq[\hat{h}_{x_{l+1|k}}]_{i},\end{split}$
(36)
where $\Lambda_{2,l|k}^{F,i}$ are computed offline as defined in (21).
In (36), $\hat{\theta}_{l|k}^{j}$ can be formulated as linear combination of
terms in $\hat{h}_{\theta_{l|k}}$ because $\hat{\Theta}_{l|k}$ is restricted
to hyperboxes.
### 4.3 MPC Cost function
By design, the family of trajectories defined by the PST is a function of the
input variables via the predicted parameter sets. That is, informativity of
the input can be directly related to the size of the PST. This feature is now
used to define the cost function, using the stage cost
$J_{\mathcal{S}}(\hat{\mathbb{X}}_{l|k},v)$ and terminal cost
$J_{\mathcal{T}}(\hat{\mathbb{X}}_{N|k},k)$
$\displaystyle\begin{split}&J_{\mathcal{S}}(\hat{\mathbb{X}}_{l|k},v_{l|k})=\\\
&\max_{x\in\hat{\mathbb{X}}_{l|k}}\left\lVert
Q(x{-}r_{k{+}l})\right\rVert_{\infty}{+}\left\lVert
R(u_{l|k}(x){-}\bar{u}_{l|k})\right\rVert_{\infty},\\\
&J_{\mathcal{T}}(\hat{\mathbb{X}}_{N|k},k)=\beta(k+N)\hat{J}_{\mathcal{S}}(\hat{\mathbb{X}}_{N|k},\bar{u}_{N|k}).\end{split}$
(37)
where $\beta(j)=\frac{1-\lambda_{c}^{T-j}}{1-\lambda_{c}}$ is a time dependent
scaling factor, which reduces the incentive for exploration towards the end of
the control task. Overall, the terminal cost is estimating the worst-case cost
after the prediction horizon assuming that the reference is fixed and no
disturbances act on the system. The cost (37) induces performance-based
exploration, as exploratory actions are used only when an improvement in
performance is predicted.
#### 4.3.1 Homothetic tubes.
In the HT formulation, the vertices of the state tube are formulated as linear
functions of MPC optimization variables. Thus (37) can be simplified to
$\displaystyle\begin{split}&J_{\mathcal{S}}^{H}(\hat{\mathbb{X}}_{l|k}^{H},v)=\max_{j\in\mathbb{N}_{1}^{q}}\left\lVert
Q(\hat{x}_{l|k}^{j}{-}r_{k{+}l})\right\rVert_{\infty}{+}\left\lVert
R(\hat{u}_{l|k}^{j}{-}\bar{u}_{l|k})\right\rVert_{\infty},\\\ \end{split}$
(38)
which can be written as a linear cost function using an epigraph reformulation
(Boyd and Vandenberghe (2004)).
#### 4.3.2 Flexible tubes.
In the FT formulation, (37) cannot be directly reformulated into a linear cost
function in the optimization variables. Hence, the maximization over the state
and input terms is separated, to obtain
$\displaystyle\begin{split}J_{\mathcal{S}}^{F}(\hat{\mathbb{X}}_{l|k}^{F},v)=&\max_{x\in\hat{\mathbb{X}}_{l|k}^{F}}\left\lVert
Q(x-r_{k+l})\right\rVert_{\infty}+\\\
&\max_{x\in\hat{\mathbb{X}}_{l|k}^{F}}\left\lVert
R(u_{l|k}(x)-\bar{u}_{l|k})\right\rVert_{\infty},\end{split}$ (39)
which can then be reformulated as a linear cost. Note that the (39)
overestimates the stage cost defined in (37). This can potentially lead to
suboptimal exploratory actions, but such approximations are also used in other
FT methods in literature (Lu and Cannon (2019)) to simplify the resulting
online optimization problem.
## 5 Dual adaptive MPC
In this section, HT and FT dual adaptive MPC algorithms are presented using
the ingredients described in Sections 3 and 4.
#### 5.0.1 Homothetic tubes.
In the HT formulation, the optimization variables are
$\displaystyle\begin{split}\gamma_{k}^{H}=\Big{\\{}&\big{\\{}z_{l|k},\alpha_{l|k},v_{l|k},\bar{u}_{l|k},\hat{z}_{l|k},\hat{\alpha}_{l|k},\\{\Lambda^{H,j}_{l|k}\\}_{j=1}^{q}\big{\\}}^{N}_{l=0},\\\
&\big{\\{}\\{\hat{\Lambda}^{H,j}_{l|k}\\}_{j=1}^{q}\big{\\}}^{N-1}_{l=0},\big{\\{}\hat{\Psi}_{l|k},\hat{h}_{\theta_{l|k}}\\}^{N_{p}}_{l=1}\Big{\\}},\end{split}$
(40)
and the online optimization problem is
$\displaystyle\begin{split}\min_{\gamma_{k}^{H}}\quad
J_{\mathcal{T}}^{H}(\hat{\mathbb{X}}_{N|k}^{H},k)+\sum_{l-0}^{N-1}J_{\mathcal{S}}^{H}(\hat{\mathbb{X}}_{l|k}^{H},v_{l|k})\\\
\text{ s.t.
}\quad\eqref{eq:SetpointSubspace},\eqref{eq:HT_RST},\eqref{eq:HT_TermCond},\eqref{eq:xhat},\eqref{eq:EPPS_RHS_Dual},\eqref{eq:HT_PST_propagation}.\end{split}$
(41)
The dual adaptive MPC algorithm using HT is described in Algorithm 1, where
$\hat{u}^{*}_{k,0}$ is the optimal value of $\hat{u}_{k,0}$.
Algorithm 1 Homothetic dual adaptive MPC
1:Offline:
2:Choose $\tau,N,N_{p}$, $\bar{\theta}_{0}$
3:Design $K$, $H_{x}$ satisfying Assumptions 1, 2
1:Online: At each time step $k\geq 0$:
2:Obtain the measurement $x_{k}$
3:Compute $\Theta_{k}$ and $\bar{\theta}_{k}$ using set-membership and (10)
4:Solve (41)
5:Apply $\hat{u}^{*}_{k,0}$ to the system
#### 5.0.2 Flexible tubes.
The optimization variables in the FT formulation are
$\displaystyle\gamma_{k}^{F}{=}\Big{\\{}\big{\\{}h_{x_{l|k}},\hat{h}_{x_{l|k}},\bar{u}_{l|k},v_{l|k}\big{\\}}^{N}_{l=0},\big{\\{}\hat{\Psi}_{l|k},\hat{h}_{\theta_{l|k}}\\}^{N_{p}}_{l=1}\Big{\\}},$
(42)
and the online optimization problem is
$\displaystyle\begin{split}\min_{\gamma_{k}^{F}}\quad
J_{\mathcal{T}}^{F}(\hat{\mathbb{X}}_{N|k}^{F},k)+\sum_{l-0}^{N-1}J_{\mathcal{S}}^{F}(\hat{\mathbb{X}}_{l|k}^{F},v_{l|k})\\\
\text{ s.t.
}\quad\eqref{eq:SetpointSubspace},\eqref{eq:FT_RST},\eqref{eq:FT_TermCond},\eqref{eq:xhat},\eqref{eq:EPPS_RHS_Dual},\eqref{eq:FT_PST_propagation}.\end{split}$
(43)
The dual adaptive MPC algorithm using flexible tubes is described in Algorithm
2.
Algorithm 2 Flexible dual adaptive MPC
1:Offline:
2:Choose $\tau,N,N_{p}$, $\bar{\theta}_{0}$ and $\mu$
3:Design $K$, $H_{x}$ satisfying Assumptions 1 and 2
4:Compute $\Lambda_{1,l|k}^{F,i}$, $\Lambda_{2,l|k}^{F,i}$ for
$i\in\mathbb{N}_{1}^{n_{x}},l\in\mathbb{N}_{0}^{N-1}$ from (20) and (21)
1:Online: At each time step $k\geq 0$:
2:Obtain the measurement $x_{k}$
3:Compute $\Theta_{k}$ and $\bar{\theta}_{k}$ using set-membership and (10)
4:Solve (43)
5:Apply $\hat{u}^{*}_{k,0}$ to the system
###### Remark 5.3
In numerical simulations, it was observed that (43) is difficult to solve. One
possible reason for this was the recursive computation of $\hat{\Theta}_{l|k}$
in (31) resulting in a large number of bilinearities. A relaxation which
worked well in practice was to compute $\hat{\Theta}_{l|k}$ using $\Theta_{k}$
and all the predicted measurements in $\\{\hat{x}_{i|k}\\}_{i=-\tau}^{l}$,
resulting in the following approximation of (31)
$\displaystyle\begin{split}\hat{h}_{\theta_{l|k}}=\hat{\Psi}_{l|k}\begin{bmatrix}h_{\theta_{k}}\\\
\tilde{h}_{\Delta_{l|k}}\end{bmatrix},\>\>\hat{\Psi}_{l|k}\begin{bmatrix}H_{\theta}\\\
\tilde{H}_{\Delta_{l|k}}\end{bmatrix}=H_{\theta},\>\hat{\Psi}_{l|k}\geq
0,\end{split}$ (44)
where $\tilde{h}_{\Delta_{l|k}}$ and $\tilde{H}_{\Delta_{l|k}}$ are defined
according to
$\displaystyle\begin{split}&\\{\theta\>|\>\hat{x}_{i+1|k}{-}A(\theta)\hat{x}_{i|k}{-}B(\theta)\hat{u}_{i|k}\in\mathbb{W},\>\forall
i\in\mathbb{N}_{-\tau}^{l-1}\\}\\\
&=\\{\theta\>|\>\tilde{H}_{\Delta_{l|k}}\theta\leq\tilde{h}_{\Delta_{l|k}}\\}.\end{split}$
(45)
### 5.1 Properties
###### Theorem 5.4
Let Assumptions 1, 2 hold. If the optimization problem (41) (or problem (43))
is feasible at time $k=0$, then it remains feasible for all $k>0$. Moreover,
the closed-loop system formed by any dynamics (1)-(4) and the MPC controller
satisfies (6) for all $k>0$.
The proof for the FT case is given here. The proof for the HT case is similar,
and can be found in Liu (2022a).
Let (43) be feasible at some time step $k$. Before (43) is solved at $k{+}1$,
$\bar{\theta}_{k}$ and $\Theta_{k}$ are updated to $\bar{\theta}_{k{+}1}$ and
$\Theta_{k{+}1}$ respectively, and the reference trajectory is shifted by one
time step. It will now be shown that a feasible solution to (43) can be
constructed using the solution at time $k$.
It must be noted that the modification of $\bar{\theta}_{k}$ affects only the
cost function (through the PST and the input setpoints), and does not affect
the feasibility of (43). Moreover,
$\hat{\mathbb{X}}^{F}_{l|k}\subseteq\mathbb{X}^{F}_{l|k}$ and the RST also
satisfies the PST dynamics (36). Therefore, one only needs to find a feasible
solution for the RST (defined by the variables
$\\{h_{x_{l|k+1}}\\}_{l=0}^{N-1},\\{v_{l|k+1}\\}_{l=0}^{N-1}$) and the
terminal set (defined by $v_{N|k+1}$ and $h_{x_{N|k+1}}$).
Consider the variables
$\displaystyle\begin{split}v_{l-1|k+1}&=v_{l|k},\>\forall
l\in\mathbb{N}_{1}^{N-1};\quad v_{N-1|k+1}=v_{N|k+1}=v_{N|k},\end{split}$
which ensure that $u_{l-1|k+1}(x)=u_{l|k}$ for $l\in\mathbb{N}_{1}^{N}$.
Because the updated parameter set satisfies $\Theta_{k+1}\subseteq\Theta_{k}$
from (7), a shifted RST from time $k$ also satisfies (19) at $k+1$.
Next, the invariance of the terminal set is used to define a feasible
$h_{x_{N|k+1}}$, so that the terminal set remains the same. This is achieved
by choosing
$\displaystyle h_{x_{N|k+1}}=h_{x_{N|k}}+H_{x}r_{k+N}-H_{x}r_{k+N+1},$ (46)
which is a feasible solution for the terminal conditions (25).
Therefore, (43) is feasible at time $k+1$, and by induction, it is recursively
feasible. Moreover, the constraints (6) are satisfied by design, because
$\theta^{*}\in\Theta_{k}$. To the best of the authors’ knowledge, FT methods
have been proposed only for regulation tasks (i.e., reference is the origin)
in literature. Thus, in addition to the novelty of the dual control
formulation, Algorithm 2 provides a novel design methodology for recursively
feasible tracking MPC controllers using an FT formulation.
## 6 Numerical example
In this section the performance of the proposed algorithms will be compared to
each other, and existing methods from the literature. The controller using
Algorithm 1 will be denoted as (HT-ESM) and Algorithm 2 as (FT-ESM), where ESM
stands for exact set-membership. The performance will be compared to the HT
algorithm from Parsi et al. (2022) denoted as (HT-ASM), because it
approximates set membership using set-intersections. In addition a comparison
with the FT algorithm from Lu and Cannon (2019) is also shown, and denoted as
(FT-P) because it performs passive exploration. Note that Lu and Cannon (2019)
designs a controller for a regulation task, but the tracking formulation was
added to FT-P for a meaningful comparison. The code to simulate the example is
available in a public repository (Liu (2022b)).
The model of the system is parameterized by the matrices
$\begin{array}[]{l l l}A_{0}=\begin{bmatrix}0.9&0.5\\\
0.2&0.8\end{bmatrix},&A_{1}=\begin{bmatrix}0.1&0\\\
0&0.2\end{bmatrix},&A_{2}=\begin{bmatrix}0&0\\\ 0&0\end{bmatrix},\\\
B_{0}=\begin{bmatrix}1&0.5\\\
0.2&0.775\end{bmatrix},&B_{1}=\begin{bmatrix}0&0\\\
0&0\end{bmatrix},&B_{2}=\begin{bmatrix}0&0.2\\\ 0&0.35\end{bmatrix},\\\
\end{array}$ (47)
The uncertainty set is
$\Theta=\\{\theta\in\mathbb{R}^{2}|\>||\theta||_{\infty}\leq 1.2\\}$. The
constraint set $\mathbb{Z}$ is defined as
$\displaystyle\mathbb{Z}$ $\displaystyle=\left\\{(x,u){\in}\mathbb{R}^{2\times
2}\left|\>\begin{array}[]{rl}||x||_{\infty}&\leq 3,||u||_{\infty}\leq
2\end{array}\right.\right\\},$
and the disturbance set is
$\mathbb{W}:=\\{w\in\mathbb{R}^{2}|\>||w||_{\infty}\leq 0.1\\}$. The cost
function is defined by $Q=4I_{2}$ and $R=I_{2}$.
The parameters for all the controllers are chosen to be $N=8$, $N_{p}=5$,
$\tau=1$ and $\bar{\theta}_{0}=[0.1,0.1]^{T}$. A piecewise constant reference
trajectory with five different setpoints is to be tracked, with a simulation
length of $T{=}100$ time steps. The simulations are performed for 500
different realizations of the true parameter $\theta^{*}\in\Theta$ and
disturbance sequences $w_{k}\in\mathbb{W}$. The parameters and disturbances
are chosen randomly from a uniform distribution. All the optimization problems
were implemented using YALMIP and solved using MOSEK and IPOPT on an Intel
Xeon Gold 5118 processor with 2GB RAM. The FT-P method solves a linear
program, whereas all the dual controllers solve nonconvex optimization
problems. The average computation times for the online optimization used in
FT-P and FT-ESM methods are $4.14s$ and $6.77s$, and that in HT-ASM and HT-ESM
methods are $6.05s$ and $10.54s$ respectively.
The performance of the controllers is compared in Figure 1, where the
distribution of closed loop costs achieved by each controller is shown. First,
it can be seen that the FT-P controller results in high closed loop costs
compared to the dual controllers, because the exploration is passive. Such a
result has been demonstrated for HT methods in Parsi et al. (2022), and now is
replicated for FT methods. Second, the use of exact set-membership
reformulation in HT-ESM results in lower costs compared to HT-ASM, because the
latter overestimates the information that can be obtained from dual control.
Third, it can be seen that both the HT methods perform better compared to the
FT-ESM method. This is because the FT method computes Lagrange multipliers
offline, uses the approximation in (44) and simplifies the cost function in
(39) so that the optimization problem is tractable. Thus, although the FT
parameterization allows for efficient computation, it results in a loss in
performance.
It must be noted that the large variation in closed-loop costs for a given
controller is because the true parameter affects the reference tracking
performance. Moreover, the ESM methods are susceptible to high costs in cases
when the initial guess $\bar{\theta}_{0}$ is far from the true parameter
$\theta$. It can be seen that the number of such cases is low, and future work
will focus on reducing the dependence on $\bar{\theta}_{0}$.
Figure 1: Distribution of closed loop costs over 500 random realizations of
$\theta^{*}$ and $w_{k}$. Figure 2: State and input trajectories of the
system under four different controllers, along with upper and lower bounds on
parameters. The true parameter is $\theta^{*}=[-1.16,0.96]$.
Figure 2 shows the state trajectories, input trajectories and the bounds on
the parameters for one specific realization of the true parameter and
disturbance sequence. It can be seen that that all the controllers ensure
constraint satisfaction. The FT-P controller initially does not use the inputs
due to the large uncertainty sets and lack of incentive to explore. This
results in a large tracking error until passive exploration reduces the
uncertainty. In contrast, exploratory actions are used by all the dual
controllers, as seen in the deviation of the trajectories from the setpoint
between time steps 12 and 20. This improves the overall closed loop costs. The
FT-P controller has a cost of 184.8, whereas the cost is 63.0 for FT-ESM, 47.3
for HT-ASM, and 43.5 for HT-ESM. It can also be seen that the HT-ASM
controller has worse identification performance compared to HT-ESM (see time
steps 10-25), owing to the approximation of set-membership equations using set
intersections.
## 7 Conclusion
In this work, a performance-based dual control framework is proposed to
navigate the dual control trade-off in optimal tracking problems. This
framework has been demonstrated with two popular tube MPC formulations used in
the adaptive MPC literature. Dual control is achieved by an exact
reformulation of set-membership identification within MPC optimization. The
proposed method ensures recursive feasibility and robust constraint
satisfaction, and numerical simulations demonstrate performance improvement
compared to existing methods. Future work involves reducing the dependence of
the dual effect on the initial parameter estimate, and extending the method to
nonlinear adaptive MPC algorithms.
## References
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* Soloperto et al. (2020) Soloperto, R., Köhler, J., and Allgöwer, F. (2020). Augmenting MPC schemes with active learning: Intuitive tuning and guaranteed performance. _IEEE Control Systems Letters_ , 4(3), 713–718.
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|
# Darcy’s Law for Porous Media
with Multiple Microstructures
Zhongwei Shen Supported in part by NSF grant DMS-2153585.
###### Abstract
In this paper we study the homogenization of the Dirichlet problem for the
Stokes equations in a perforated domain with multiple microstructures. First,
under the assumption that the interface between subdomains is a union of
Lipschitz surfaces, we show that the effective velocity and pressure are
governed by a Darcy law, where the permeability matrix is piecewise constant.
The key step is to prove that the effective pressure is continuous across the
interface, using Tartar’s method of test functions. Secondly, we establish the
sharp error estimates for the convergence of the velocity and pressure,
assuming the interface satisfies certain smoothness and geometric conditions.
This is achieved by constructing two correctors. One of them is used to
correct the discontinuity of the two-scale approximation on the interface,
while the other is used to correct the discrepancy between boundary values of
the solution and its approximation.
Keywords: Homogenization; Stokes Equations; Perforated Domain; Convergence
Rate.
MR (2020) Subject Classification: 35Q35; 35B27; 76D07.
## 1 Introduction
In this paper we study the homogenization of the Dirichlet problem for the
Stokes equations in a perforated domain $\Omega_{\varepsilon}$,
$\left\\{\begin{aligned} -\varepsilon^{2}\mu\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=f&\quad&\text{ in }\Omega_{\varepsilon},\\\ \text{\rm
div}(u_{\varepsilon})&=0&\quad&\text{ in }\Omega_{\varepsilon},\\\
u_{\varepsilon}&=0&\quad&\text{ on
}\partial\Omega_{\varepsilon},\end{aligned}\right.$ (1.1)
where $0<\varepsilon<1$ and $\mu>0$ is a constant. Let $\Omega$ be a bounded
Lipschitz domain in $\mathbb{R}^{d}$, $d\geq 2$. Let
$\\{\Omega^{\ell}:1\leq\ell\leq L\\}$ be a finite number of disjoint
subdomains of $\Omega$, each with a Lipschitz boundary, such that
$\overline{\Omega}=\bigcup_{\ell=1}^{L}\overline{\Omega^{\ell}}.$ (1.2)
To describe the porous domain $\Omega_{\varepsilon}$, let $Y=[0,1]^{d}$ be a
closed unit cube and $\\{Y_{s}^{\ell}:1\leq\ell\leq L\\}$ open subsets (solid
parts) of $Y$ with Lipschitz boundaries. Assume that for $1\leq\ell\leq L$,
dist$(\partial Y,\partial Y_{s}^{\ell})>0$ and
$Y^{\ell}_{f}=Y\setminus\overline{Y_{s}^{\ell}}$ (the fluid part) is
connected. For $0<\varepsilon<1$ and $1\leq\ell\leq L$, define
$\Omega_{\varepsilon}^{\ell}=\Omega^{\ell}\setminus\bigcup_{z}\varepsilon(\overline{Y_{s}^{\ell}}+z),$
(1.3)
where $z\in\mathbb{Z}^{d}$ and the union is taken over those $z$’s for which
$\varepsilon(Y+z)\subset\Omega^{\ell}$. Thus the subdomain $\Omega^{\ell}$ is
perforated periodically, using the solid obstacle $Y_{s}^{\ell}$. Let
$\Omega_{\varepsilon}=\Sigma\cup\bigcup_{\ell=1}^{L}\Omega_{\varepsilon}^{\ell}=\Omega\setminus\bigcup_{\ell=1}^{L}\bigcup_{z}\varepsilon(\overline{Y_{s}^{\ell}}+z),$
(1.4)
where $\Sigma$ is the interface between subdomains, given by
$\Sigma=\Omega\setminus\bigcup_{\ell=1}^{L}\Omega^{\ell}=\bigcup_{\ell=1}^{L}\partial\Omega^{\ell}\setminus\partial\Omega.$
(1.5)
For $f\in L^{2}(\Omega;\mathbb{R}^{d})$, let
$(u_{\varepsilon},p_{\varepsilon})\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L^{2}(\Omega_{\varepsilon})$ be the weak solution of (1.1) with
$\int_{\Omega_{\varepsilon}}p_{\varepsilon}\,dx=0$. We extend
$u_{\varepsilon}$ to the whole domain $\Omega$ by zero. Let $P_{\varepsilon}$
denote the extension of $p_{\varepsilon}$ to $\Omega$, defined by (2.21). In
the case $L=1$, where $\Omega$ is perforated periodically with small holes of
same shape, it is well known that as $\varepsilon\to 0$, $u_{\varepsilon}\to
u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and $P_{\varepsilon}\to P_{0}$
strongly in $L^{2}(\Omega)$, where the effective velocity and pressure
$(u_{0},P_{0})$ are governed by the Darcy law,
$\left\\{\begin{aligned} u_{0}&=\mu^{-1}K(f-\nabla P_{0})&\quad&\text{ in
}\Omega,\\\ \text{div}(u_{0})&=0&\quad&\text{ in }\Omega,\\\ u_{0}\cdot
n&=0&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (1.6)
with $\int_{\Omega}P_{0}\,dx=0$. Note that in (1.1) we have normalized the
velocity vector by a factor $\varepsilon^{2}$, where $\varepsilon$ is the
period. For references on the Darcy law we refer to the reader to [13, 10, 1,
3, 4].
In (1.6) the permeability matrix $K$ is a $d\times d$ positive definite,
constant and symmetric matrix and $n$ denotes the outward unit normal to
$\partial\Omega$. It was observed in [3] by G. Allaire that as $\varepsilon\to
0$,
$u_{\varepsilon}-\mu^{-1}W(x/\varepsilon)(f-\nabla P_{0})\to
0\quad\text{strongly in }L^{2}(\Omega;\mathbb{R}^{d}),$ (1.7)
where $W(y)$ is an 1-periodic $d\times d$ matrix defined by a cell problem and
$\fint_{Y}W(y)\,dy=K$. Recently, it was proved in [15] by the present author
that
$\|u_{\varepsilon}-\mu^{-1}W(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega)}+\|P_{\varepsilon}-P_{0}\|_{L^{2}(\Omega)}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)},$ (1.8)
and that
$\|\varepsilon\nabla u_{\varepsilon}-\mu^{-1}\nabla W(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega)}\leq C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)}.$
(1.9)
We point out that due to the discrepancy between boundary values of
$\mu^{-1}W(x/\varepsilon)(f-\nabla P_{0})$ and $u_{\varepsilon}$ on
$\partial\Omega$, the $O(\varepsilon^{1/2})$ convergence rates in (1.8)-(1.9)
are sharp. See [11] for an earlier partial result on solutions with periodic
boundary conditions.
The primary purpose of this paper is to study the Darcy law for the case
$L\geq 2$, where the domain $\Omega$ is divided into several subdomains and
different subdomains are perforated with small holes of different shapes.
###### Theorem 1.1.
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$, $d\geq 2$, and
$\Omega_{\varepsilon}$ be given by (1.4). Let
$(u_{\varepsilon},p_{\varepsilon})\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L^{2}(\Omega_{\varepsilon})$ be a weak solution of (1.1), where $f\in
L^{2}(\Omega;\mathbb{R}^{d})$ and
$\int_{\Omega_{\varepsilon}}p_{\varepsilon}\,dx=0$. Let $P_{\varepsilon}$ be
the extension of $p_{\varepsilon}$, defined by (2.21). Then
$u_{\varepsilon}\to u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ strongly in
$L^{2}(\Omega)$, as $\varepsilon\to 0$, where $P_{0}\in H^{1}(\Omega)$ and
$(u_{0},P_{0})$ is governed by the Darcy law (1.6) with the matrix
$K=\sum_{\ell=1}^{L}K^{\ell}\chi_{\Omega^{\ell}}\qquad\text{ in }\Omega.$
(1.10)
The matrix $K^{\ell}$ in (1.10) is the (constant) permeability matrix
associated with the solid obstacle $Y_{s}^{\ell}$. Thus, the matrix $K$ is
piecewise constant in $\Omega$, taking value $K^{\ell}$ in the subdomain
$\Omega^{\ell}$, and
$u_{0}=K^{\ell}(f-\nabla P_{0})\quad\text{ in }\Omega^{\ell}.$ (1.11)
Since $\text{\rm div}(u_{0})=0$ in $\Omega$ and $P_{0}\in H^{1}(\Omega)$, both
the normal component $u_{0}\cdot n$ and $P_{0}$ are continuous across the
interface $\Sigma$ (in the sense of trace) between subdomains. However, the
tangential components of $u_{0}$ may not be continuous across $\Sigma$.
The Dirichlet problem for the Stokes equations (1.1) is used to model fluid
flows in porous media with different microstructures in different subdomains.
The continuity of the effective pressure $P_{0}$ and the normal component
$u_{0}\cdot n$ of the effective velocity across the interface is generally
accepted in engineering [6, 9]. Theorem 1.1 is probably known to experts.
However, to the best of the author’s knowledge, the existing literatures on
rigorous proofs only treat the case of flat interfaces. In particular, the
result was proved in [9] under the assumption that $d=2$, the interface
$\Gamma=\mathbb{R}\times\\{0\\}$ and the solutions are 1-periodic in the
direction $x_{1}$. Also see related work in [5, 12]. We provide a proof here
for the general case, where the interface is a union of Lipschitz surfaces,
using Tartar’s method of test functions. We point out that the proof for
(1.11) and $P_{0}\in H^{1}(\Omega^{\ell})$ for each $\ell$ is the same as in
the classical case $L=1$. The challenge is to show that the effective pressure
$P_{0}$ is continuous across the interface and thus $P_{0}\in H^{1}(\Omega)$,
which is essential for proving the uniqueness of the limits of subsequences of
$\\{u_{\varepsilon}\\}$.
Our main contribution in this paper is on the sharp convergence rates and
error estimates for $u_{\varepsilon}$ and $P_{\varepsilon}$. We are able to
extend the results in [14] for the case $L=1$ to the case $L\geq 2$ under some
smoothness and geometric conditions on subdomains. More specifically, we
assume that each subdomain is a bounded $C^{2,1/2}$ domain, and that there
exists $r_{0}>0$ such that if
$x_{0}\in\partial\Omega^{k}\cap\partial\Omega^{m}$ for some $1\leq k,m\leq L$
and $k\neq m$, there exists a coordinate system, obtained from the standard
one by translation and rotation, such that
$\displaystyle B(x_{0},r_{0})\cap\Omega^{k}$
$\displaystyle=B(x_{0},r_{0})\cap\big{\\{}(x^{\prime},x_{d})\in\mathbb{R}^{d}:x_{d}>\psi(x^{\prime})\big{\\}},$
(1.12) $\displaystyle B(x_{0},r_{0})\cap\Omega^{m}$
$\displaystyle=B(x_{0},r_{0})\cap\big{\\{}(x^{\prime},x_{d})\in\mathbb{R}^{d}:x_{d}<\psi(x^{\prime})\big{\\}},$
where $\psi:\mathbb{R}^{d-1}\to\mathbb{R}$ is a $C^{2,1/2}$ function. Roughly
speaking, this means that inside a small ball centered on the interface
$\Sigma$, the domain $\Omega$ is divided by $\Sigma$ into exactly two
subdomains. In particular, the condition excludes the cases where the
interface intersects with each other or with the boundary of $\Omega$.
The following is the main result of the paper. The matrix $W^{\ell}(y)$ in
(1.13)-(1.14) is the 1-periodic matrix associated with the solid obstacle
$Y_{s}^{\ell}$.
###### Theorem 1.2.
Let $\Omega$ be a bounded $C^{2,1/2}$ domain and $\Omega_{\varepsilon}$ be
given by (1.4). Assume that the subdomains $\\{\Omega^{\ell}\\}$ are bounded
$C^{2,1/2}$ domains satisfying the condition (1.12). Let
$(u_{\varepsilon},P_{\varepsilon})$ and $(u_{0},P_{0})$ be the same as in
Theorem 1.1. Then, for $f\in C^{1,1/2}(\Omega;\mathbb{R}^{d})$,
$\sum_{\ell=1}^{L}\|u_{\varepsilon}-\mu^{-1}W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega^{\ell})}+\|P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}-P_{0}\|_{L^{2}(\Omega)}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)},$ (1.13)
and
$\sum_{\ell=1}^{L}\|\varepsilon\nabla u_{\varepsilon}-\mu^{-1}\nabla
W^{\ell}(x/\varepsilon)(f-\nabla P_{0})\|_{L^{2}(\Omega^{\ell})}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)},$ (1.14)
where $C$ depends on $d$, $\mu$, $\Omega$, $\\{\Omega^{\ell}\\}$ and
$\\{Y_{s}^{\ell}\\}$.
As we mentioned earlier, the sharp convergence rates in (1.13) and (1.14) were
proved in [15] for the case $L=1$. In the case of two porous media with a flat
interface, partial results were obtained in [9] for solutions with periodic
boundary conditions. Theorem 1.2 is the first result that treats the general
case of smooth interfaces.
As in [9], the basic idea in our approach to Theorem 1.2 is to use
$V_{\varepsilon}(x)=\sum_{\ell=1}^{L}W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\chi_{\Omega_{\varepsilon}^{\ell}}$ (1.15)
to approximate the solution $u_{\varepsilon}$ and obtain the error estimates
by the energy method. Observe that $V_{\varepsilon}=0$ on
$\Gamma_{\varepsilon}=\partial\Omega_{\varepsilon}\setminus\partial\Omega$.
There are three main issues with this approach: (1) the divergence of
$V_{\varepsilon}$ is not small in $L^{2}$; (2) $V_{\varepsilon}$ does not
agree with $u_{\varepsilon}$ on $\partial\Omega$; and (3) $V_{\varepsilon}$ is
not in $H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, as it is not continuous
across the interface. To overcome these difficulties, we introduce three
corresponding correctors: $\Phi_{\varepsilon}^{(1)}$,
$\Phi_{\varepsilon}^{(2)}$, and $\Phi_{\varepsilon}^{(3)}$. To correct the
divergence of $V_{\varepsilon}$, we construct $\Phi_{\varepsilon}^{(1)}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ with the property that
$\varepsilon\|\nabla\Phi_{\varepsilon}^{(1)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(1)}+V_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)}$ (1.16)
for $1\leq\ell\leq L$. The construction of $\Phi_{\varepsilon}^{(1)}$ is
similar to that in [11, 9, 15]. Next, we correct the boundary data of
$V_{\varepsilon}$ on $\partial\Omega$ by constructing
$\Phi_{\varepsilon}^{(2)}\in H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ such
that $\Phi_{\varepsilon}^{(2)}+V_{\varepsilon}=0$ on $\partial\Omega$,
$\Phi_{\varepsilon}^{(2)}=0$ on $\Gamma_{\varepsilon}$, and that
$\varepsilon\|\nabla\Phi_{\varepsilon}^{(2)}\|_{L^{2}(\Omega_{\varepsilon})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(2)})\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)}.$ (1.17)
The construction of $\Phi_{\varepsilon}^{(2)}$ is similar to that in [15] for
the case $L=1$. The key observation is that the normal component of
$V_{\varepsilon}$ on $\partial\Omega$ can be written in the form
$\varepsilon\nabla_{\tan}\left(\phi(x/\varepsilon)\right)\cdot g,$ (1.18)
where $\nabla_{\tan}$ denotes the tangential gradient on $\partial\Omega$. We
remark that a similar observation is also used in the proof of Theorem 1.1.
Finally, to correct the discontinuity of $V_{\varepsilon}$ across the
interface, we introduce
$\Phi_{\varepsilon}^{(3)}=\sum_{\ell=1}^{L}I_{\varepsilon}^{\ell}(x)(f-\nabla
P_{0})\chi_{\Omega_{\varepsilon}^{\ell}},$ (1.19)
with the properties that $V+\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, $\Phi_{\varepsilon}^{(3)}=0$ on
$\partial\Omega_{\varepsilon}$, and that
$\varepsilon\|\nabla\Phi_{\varepsilon}^{(3)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(3)})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\sqrt{\varepsilon}\|f\|_{C^{1,1/2}(\Omega)}.$ (1.20)
More specifically, for each $1\leq\ell\leq L$, the matrix-valued function
$I_{\varepsilon}^{\ell}$ is a solution of the Stokes equations in
$\Omega_{\varepsilon}^{\ell}$ with $I_{\varepsilon}^{\ell}=0$ on
$\partial\Omega_{\varepsilon}^{\ell}\setminus\partial\Omega^{\ell}$. On each
connected component $\Sigma^{k}$ of the interface $\Sigma$, the boundary value
of $I_{\varepsilon}^{\ell}$ is either $0$ or given by
$W_{j}^{-}(x/\varepsilon)-W_{j}^{+}(x/\varepsilon)-W_{i}^{-}(x/\varepsilon)(K_{mj}^{-}-K_{mj}^{+})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle},$ (1.21)
where the repeated indices $i$ and $m$ are summed from $1$ to $d$. Here the
subdomains $\Omega^{\pm}$ are separated by $\Sigma^{k}$, and
$(W^{\pm},K^{\pm})$ denote the corresponding 1-periodic matrices for
$\Omega^{\pm}$ and their averages over $Y$, respectively. To show
$V+\Phi_{\varepsilon}^{(3)}$ is continuous across $\Sigma$, we use the fact
that $(\nabla_{\tan}P_{0})^{+}=(\nabla_{\tan}P_{0})^{-}$ and
$n\cdot K^{+}(f-\nabla P_{0})^{+}=n\cdot K^{-}(f-\nabla P_{0})^{-},$ (1.22)
where $(v)^{\pm}$ denote the trace of $v$ taken from $\Omega^{\pm}$,
respectively. The proof of the estimate (1.20) again relies on the observation
that the normal component of (1.21) is of form (1.18).
Theorem 1.2 is proved under the assumption that $\\{Y_{s}^{\ell}:1\leq\ell\leq
L\\}$ are subdomains of $Y$ with Lipschitz boundaries. The $C^{2,1/2}$
condition and the geometric condition (1.12) for $\Omega$ and subdomains
$\\{\Omega^{\ell}\\}$ are dictated by the smoothness requirement in its proof
for $P_{0}$ in each subdomain. Note that $P_{0}$ is a solution of an elliptic
equation with piecewise constant coefficients in $\Omega$. Not much is known
about the boundary regularity of $P_{0}$ if the interface intersects with the
boundary $\partial\Omega$ or with each other.
The paper is organized as follows. In Section 2 we collect several useful
estimates that are more or less known. In Section 3 we establish the energy
estimates for the Dirichlet problem (1.1). Theorem 1.1 is proved in Section 4.
In Section 5 we give the proof of Theorem 1.2, assuming the existence of
suitable correctors. Finally, we construct correctors
$\Phi_{\varepsilon}^{(1)}$, $\Phi_{\varepsilon}^{(2)}$, and
$\Phi_{\varepsilon}^{(3)}$, described above, in the last three sections of the
paper. Throughout the paper we will use $C$ to denote constants that may
depend on $d$, $\mu$, $\Omega$, $\\{\Omega^{\ell}\\}$, and
$\\{Y_{s}^{\ell}\\}$. In fact, since the viscosity constant $\mu$ is
irrelevant in our study, we will assume $\mu=1$ in the rest of the paper.
Acknowledgement. The author is indebt to Professor Xiaoming Wang for raising
the question that is addressed in this paper and for several stimulating
discussions.
## 2 Preliminaries
Let $Y=[0,1]^{d}$ and $\\{Y_{s}^{\ell}:1\leq\ell\leq L\\}$ be a finite number
of open subsets of $Y$ with Lipschitz boundaries. We assume that
dist$(\partial Y,\partial Y^{\ell}_{s})>0$ and that
$Y_{f}^{\ell}=Y\setminus\overline{Y_{s}^{\ell}}$ is connected. Let
$\omega^{\ell}=\bigcup_{z\in\mathbb{Z}^{d}}(Y_{f}^{\ell}+z)$
be the periodic repetition of $Y_{f}^{\ell}$. For $1\leq j\leq d$ and
$1\leq\ell\leq L$, let
$\left(W_{j}^{\ell}(y),\pi_{j}^{\ell}(y)\right)=\left(W_{1j}^{\ell}(y),\dots,W_{dj}^{\ell}(y),\pi_{j}^{\ell}(y)\right)\in
H^{1}_{\text{loc}}(\omega^{\ell};\mathbb{R}^{d})\times
L^{2}_{\text{loc}}(\omega^{\ell})$
be the 1-periodic solution of
$\left\\{\begin{aligned} -\Delta
W_{j}^{\ell}+\nabla\pi_{j}^{\ell}&=e_{j}&\quad&\text{ in }\omega^{\ell},\\\
\text{\rm div}(W_{j}^{\ell})&=0&\quad&\text{ in }\omega^{\ell},\\\
W_{j}^{\ell}&=0&\quad&\text{ on }\partial\omega^{\ell},\end{aligned}\right.$
(2.1)
with $\int_{Y^{\ell}_{f}}\pi_{j}^{\ell}\,dy=0$, where
$e_{j}=(0,\dots,1,\dots,0)$ with $1$ in the $j^{th}$ place. We extend the
matrix $W^{\ell}=(W_{j}^{\ell})$ to $\mathbb{R}^{d}$ by zero and define
$K^{\ell}_{ij}=\int_{Y}W_{ij}^{\ell}(y)\,dy.$ (2.2)
Since
$K_{ij}^{\ell}=\int_{Y}\nabla W_{ik}^{\ell}\cdot\nabla W^{\ell}_{jk}\,dy$
(the repeated index $k$ is summed from $1$ to $d$), it follows that the
$d\times d$ matrix $K^{\ell}=(K^{\ell}_{ij})$ is symmetric and positive
definite.
The existence and uniqueness of solutions to (2.1) can be proved by applying
the Lax-Milgram Theorem on the closure of the set,
$\left\\{u\in C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d}):u\text{ is 1-periodic,
}u=0\text{ in }Y_{s}^{\ell},\text{ and }\text{\rm div}(u)=0\text{ in
}\mathbb{R}^{d}\right\\},$
in $H^{1}(Y;\mathbb{R}^{d})$. By energy estimates,
$\int_{Y}\left(|\nabla
W^{\ell}|^{2}+|W^{\ell}|^{2}+|\pi^{\ell}|^{2}\right)dy\leq C,$ (2.3)
where we have also extended $\pi^{\ell}$ to $\mathbb{R}^{d}$ by zero. By
periodicity this implies that
$\int_{D}\left(|\nabla
W^{\ell}(x/\varepsilon)|^{2}+|W^{\ell}(x/\varepsilon)|^{2}+|\pi^{\ell}(x/\varepsilon)|^{2}\right)dx\leq
C,$ (2.4)
where $D$ is a bounded domain and $C$ depends on diam($D$).
###### Lemma 2.1.
Let $D$ is a bounded Lipschitz domain in $\mathbb{R}^{d}$. Then
$\int_{\partial D}\left(|\nabla
W^{\ell}(x/\varepsilon)|^{2}+|W^{\ell}(x/\varepsilon)|^{2}+|\pi^{\ell}(x/\varepsilon)|^{2}\right)d\sigma\leq
C,$ (2.5)
where $C$ depends on $D$.
###### Proof.
If $Y_{s}^{\ell}$ is of $C^{1,\alpha}$, the inequality above follows directly
from the fact that $\nabla W^{\ell}$ and $\pi^{\ell}$ are bounded in $Y$. To
treat the case where $\partial Y_{s}^{\ell}$ is merely Lipschitz, by
periodicity, we may assume that $\varepsilon=1$ and $D$ is a subdomain of $Y$.
Note that the bound for the integral of $|W^{\ell}|^{2}$ on $\partial D$
follows from (2.3). Indeed, if $D$ is a subdomain of $Y$ with Lipschitz
boundary,
$\int_{\partial D}|W^{\ell}|^{2}\,d\sigma\leq C\int_{D}\left(|\nabla
W^{\ell}|^{2}+|W^{\ell}|^{2}\right)dy.$
The estimates for $\nabla W^{\ell}$ and $\pi^{\ell}$ are a bit more involved.
By using the fundamental solutions for the Stokes equations in
$\mathbb{R}^{d}$, we may reduce the problem to the estimate
$\|\nabla u\|_{L^{2}(\partial D)}+\|p\|_{L^{2}(\partial D)}\leq
C\left\\{\|\nabla u\|_{L^{2}(\widetilde{Y}\setminus
Y_{s}^{\ell})}+\|p\|_{L^{2}(\widetilde{Y}\setminus
Y_{s}^{\ell})}+\|h\|_{H^{1}(\partial Y_{s}^{\ell})}\right\\},$
for solutions of the Stokes equations,
$\left\\{\begin{aligned} -\Delta u+\nabla p&=0&\quad&\text{ in
}\widetilde{Y}\setminus\overline{Y_{s}^{\ell}},\\\ \text{\rm
div}(u)&=0&\quad&\text{ in }\widetilde{Y}\setminus\overline{Y_{s}^{\ell}},\\\
u&=h&\quad&\text{ on }\partial Y_{s}^{\ell},\end{aligned}\right.$
where $h\in H^{1}(\partial Y_{s}^{\ell};\mathbb{R}^{d})$ and
$\widetilde{Y}=(1+c)Y$. The desired estimates follow from the interior
estimates as well as the nontangential-maximal-function estimate,
$\|(\nabla u)^{*}\|_{L^{2}(\partial Y_{s}^{\ell})}+\|(p)^{*}\|_{L^{2}(\partial
Y_{s}^{\ell})}\leq C\left\\{\|h\|_{H^{1}(\partial
Y_{s}^{\ell})}+\|u\|_{L^{2}(\widetilde{Y}\setminus
Y_{s}^{\ell})}+\|p\|_{L^{2}(\widetilde{Y}\setminus Y_{s}^{\ell})}\right\\},$
(2.6)
where the nontangential maximal function $(v)^{*}$ is defined by
$(v)^{*}(x)=\sup\left\\{|v(y)|:\ y\in Y\setminus Y_{s}^{\ell}\text{ and
}|y-x|<C_{0}\,\text{\rm dist}(y,\partial Y_{s}^{\ell})\right\\}$
for $x\in\partial Y_{s}^{\ell}$. The estimate (2.6) is a consequence of the
nontangential-maximal-function estimates, established in [7], for solutions of
the Dirichlet problem for the Stokes equations in a bounded Lipschitz domain.
∎
###### Lemma 2.2.
Fix $1\leq j\leq d$ and $1\leq\ell\leq L$. There exist 1-periodic functions
$\phi_{kij}^{\ell}(y)$, $i,m=1,2,\dots,d$, such that $\phi_{kij}^{\ell}\in
H^{1}(Y)$, $\int_{Y}\phi_{kij}^{\ell}\,dy=0$,
$\frac{\partial}{\partial
y_{k}}\left(\phi_{kij}^{\ell}\right)=W^{\ell}_{ij}-K^{\ell}_{ij}\quad\text{
and }\quad\phi_{kij}^{\ell}=-\phi_{ikj}^{\ell},$ (2.7)
where the repeated index $k$ is summed from $1$ to $d$. Moreover,
$\int_{\partial D}|\phi^{\ell}_{kij}(x/\varepsilon)|^{2}\,d\sigma\leq C,$
(2.8)
where $D$ is a bounded Lipschitz domain in $\mathbb{R}^{d}$ and $C$ depends on
$D$.
###### Proof.
See [15, Lemma 5.3] for the proof of (2.7). The estimate (2.8) follows from
the observation,
$\displaystyle\|\nabla\phi_{kij}^{\ell}\|_{L^{2}(Y)}+\|\phi_{kij}^{\ell}\|_{L^{2}(Y)}$
$\displaystyle\leq
C\|\nabla^{2}h_{ij}^{\ell}\|_{L^{2}(Y)}+C\|\nabla^{2}h_{kj}^{\ell}\|_{L^{2}(Y)}$
$\displaystyle\leq C\|W_{j}^{\ell}\|_{L^{2}(Y)}\leq C.$
∎
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$ and
$\\{\Omega^{\ell}:1\leq\ell\leq L\\}$ be disjoint subdomains of $\Omega$, each
with Lipschitz boundary, and satisfying the condition,
$\overline{\Omega}=\cup_{\ell=1}^{L}\overline{\Omega^{\ell}}.$ (2.9)
Define
$K=\sum_{\ell=1}^{L}K^{\ell}\chi_{\Omega^{\ell}},$ (2.10)
where $K^{\ell}$ is given by (2.2) and $\chi_{\Omega^{\ell}}$ denotes the
characteristic function of $\Omega^{\ell}$.
###### Lemma 2.3.
Let $f\in L^{2}(\Omega;\mathbb{R}^{d})$. Then there exists $P_{0}\in
H^{1}(\Omega)$, unique up to constants, such that
$\left\\{\begin{aligned} \text{\rm div}\left(K(f-\nabla
P_{0})\right)&=0&\quad&\text{ in }\Omega,\\\ n\cdot K(f-\nabla
P_{0})&=0&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (2.11)
in the sense that
$\int_{\Omega}K(f-\nabla P_{0})\cdot\nabla\varphi\,dx=0$ (2.12)
for any $\varphi\in H^{1}(\Omega)$.
###### Proof.
This is standard since the coefficient matrix $K$ is positive definite in each
subdomain $\Omega^{\ell}$ and thus in $\Omega$. ∎
For each $1\leq\ell\leq L$ and $0<\varepsilon<1$, let
$\Omega^{\ell}_{\varepsilon}$ be the perforated domain defined by (1.3), using
$Y_{s}^{\ell}$. Let $\Omega_{\varepsilon}$ be given by (1.4). Note that
$\partial\Omega_{\varepsilon}=\partial\Omega\cup\Gamma_{\varepsilon},$ (2.13)
where $\Gamma_{\varepsilon}=\cup_{\ell=1}^{L}{\Gamma_{\varepsilon}^{\ell}}$
and $\Gamma_{\varepsilon}^{\ell}$ consists of the boundaries of holes
$\varepsilon(Y_{s}^{\ell}+z)$ that are removed from $\Omega^{\ell}$.
###### Lemma 2.4.
Let $u\in H^{1}(\Omega_{\varepsilon})$ with $u=0$ on $\Gamma_{\varepsilon}$.
Assume $\Gamma^{\ell}_{\varepsilon}\neq\emptyset$ for all $1\leq\ell\leq L$.
Then
$\|u\|_{L^{2}(\Omega_{\varepsilon})}\leq C\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}.$ (2.14)
###### Proof.
It follows from Lemma 2.2 in [15] that for $1\leq\ell\leq L$,
$\|u\|^{2}_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq C\varepsilon^{2}\|\nabla
u\|^{2}_{L^{2}(\Omega_{\varepsilon}^{\ell})},$
which yields (2.14) by summation. Note that we do not assume $u=0$ on
$\partial\Omega^{\ell}$. ∎
From now on we will assume that $\varepsilon>0$ is sufficiently small so that
$\Gamma_{\varepsilon}^{\ell}\neq\emptyset$ for all $1\leq\ell\leq L$. The main
results in this paper are only relevant for small $\varepsilon$.
###### Lemma 2.5.
Let $\Omega$ be a bounded Lipschitz domain and $\Omega_{\varepsilon}$ be given
by (1.4). There exists a bounded linear operator,
$R_{\varepsilon}:H^{1}(\Omega;\mathbb{R}^{d})\to
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d}),$ (2.15)
such that
$\left\\{\begin{aligned} &R_{\varepsilon}(u)=0\quad\text{ on
}\Gamma_{\varepsilon}\quad\text{ and }\quad R_{\varepsilon}(u)=u\quad\text{ on
}\partial\Omega,\\\ &R_{\varepsilon}(u)\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\quad\text{ if }\ u\in
H_{0}^{1}(\Omega;\mathbb{R}^{d}),\\\ &R_{\varepsilon}(u)=u\quad\text{ in
}\Omega\quad\text{ if }\ u=0\quad\text{ on }\Gamma_{\varepsilon},\\\
&\text{\rm div}(R_{\varepsilon}(u))=\text{\rm div}(u)\quad\text{ in
}\Omega_{\varepsilon}\ \text{ if \ \ }\text{\rm div}(u)=0\quad\text{ in
}\Omega\setminus\Omega_{\varepsilon},\end{aligned}\right.$ (2.16)
and
$\varepsilon\|\nabla
R_{\varepsilon}(u)\|_{L^{2}(\Omega_{\varepsilon})}+\|R_{\varepsilon}(u)\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\left\\{\varepsilon\|\nabla
u\|_{L^{2}(\Omega)}+\|u\|_{L^{2}(\Omega)}\right\\}.$ (2.17)
Moreover,
$\|\text{\rm div}(R_{\varepsilon}(u))\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|\text{\rm div}(u)\|_{L^{2}(\Omega)}.$ (2.18)
###### Proof.
A proof for the case $L=1$, which is similar to that of a lemma due to Tartar
(in an appendix of [13]), may be found in [15, Lemma 2.3]. Also see [10, 1].
The same proof works equally well for the case $L\geq 2$. Indeed, let $u\in
H^{1}(\Omega;\mathbb{R}^{d})$. For each $\varepsilon(Y+z)\subset\Omega^{\ell}$
with $1\leq\ell\leq L$ and $z\in\mathbb{Z}^{d}$, we define
$R_{\varepsilon}(u)$ on $\varepsilon(Y_{f}^{\ell}+z)$ by the Dirichlet
problem,
$\left\\{\begin{aligned} -\varepsilon^{2}\Delta R_{\varepsilon}(u)+\nabla
q&=-\varepsilon^{2}\Delta u&\quad&\text{ in }\varepsilon(Y_{f}^{\ell}+z),\\\
\text{\rm div}(R_{\varepsilon}(u))&=\text{\rm
div}(u)+\frac{1}{|\varepsilon(Y_{f}^{\ell}+z)|}\int_{\varepsilon(Y_{s}^{\ell}+z)}\text{\rm
div}(u)\,dx&\quad&\text{ in }\varepsilon(Y_{f}^{\ell}+z),\\\
R_{\varepsilon}(u)&=0&\quad&\text{ on
}\partial(\varepsilon(Y_{s}^{\ell}+z)),\\\ R_{\varepsilon}(u)&=u&\quad&\text{
on }\partial(\varepsilon(Y+z)).\end{aligned}\right.$ (2.19)
If $x\in\Omega_{\varepsilon}$ and $x\notin\varepsilon(Y_{f}+z)$ for any
$\varepsilon(Y+z)\subset\Omega^{\ell}$, we let $R_{\varepsilon}(u)=u$. ∎
###### Lemma 2.6.
Let $f\in L^{2}(\Omega_{\varepsilon})$ with
$\int_{\Omega_{\varepsilon}}f\,dx=0$. Then there exists $u_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ such that $\text{\rm
div}(u_{\varepsilon})=f$ in $\Omega_{\varepsilon}$ and
$\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|f\|_{L^{2}(\Omega_{\varepsilon})}.$ (2.20)
###### Proof.
Let $F$ be the zero extension of $f$ to $\Omega$. Since $F\in L^{2}(\Omega)$
and $\int_{\Omega}F\,dx=0$, there exists $u\in
H_{0}^{1}(\Omega;\mathbb{R}^{d})$ such that div$(u)=F$ in $\Omega$ and
$\|u\|_{L^{2}(\Omega)}+\|\nabla u\|_{L^{2}(\Omega)}\leq
C\|F\|_{L^{2}(\Omega)}$. Let $u_{\varepsilon}=R_{\varepsilon}(u)$. Then
$u_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon},\mathbb{R}^{d})$, and by
(2.17),
$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq C\left\\{\varepsilon\|\nabla
u\|_{L^{2}(\Omega)}+\|u\|_{L^{2}(\Omega)}\right\\}$ $\displaystyle\leq
C\|f\|_{L^{2}(\Omega_{\varepsilon})}.$
Since div$(u)=F=0$ in $\Omega\setminus\Omega_{\varepsilon}$, by the last line
in (2.16), we obtain div$(u_{\varepsilon})=\text{\rm div}(u)=f$ in
$\Omega_{\varepsilon}$. ∎
For $p\in L^{2}(\Omega_{\varepsilon})$, as in [10], we define an extension $P$
of $p$ to $L^{2}(\Omega)$ by
$P(x)=\left\\{\begin{aligned} &p(x)&\quad&\text{ if
}x\in\Omega_{\varepsilon},\\\
&\fint_{\varepsilon(Y_{f}^{\ell}+z)}p&\quad&\text{ if
}x\in\varepsilon(Y_{s}^{\ell}+z)\subset\varepsilon(Y+z)\subset\Omega^{\ell}\text{
for some }1\leq\ell\leq L\text{ and }z\in\mathbb{Z}^{d}.\end{aligned}\right.$
(2.21)
###### Lemma 2.7.
Let $p\in L^{2}(\Omega_{\varepsilon})$ and $P$ be its extension given by
(2.21). Then
$\langle\nabla p,R_{\varepsilon}(u)\rangle_{H^{-1}(\Omega_{\varepsilon})\times
H_{0}^{1}(\Omega_{\varepsilon})}=\langle\nabla
P,u\rangle_{H^{-1}(\Omega)\times H_{0}^{1}(\Omega)},$ (2.22)
where $u\in H_{0}^{1}(\Omega;\mathbb{R}^{d})$ and $R_{\varepsilon}(u)$ is
given by Lemma 2.5.
###### Proof.
We use an argument found in [10, 1, 2]. Note that if $u\in
H_{0}^{1}(\Omega;\mathbb{R}^{d})$, we have $R_{\varepsilon}(u)\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ and
$\displaystyle|\langle\nabla
p,R_{\varepsilon}(u)\rangle_{H^{-1}(\Omega_{\varepsilon})\times
H^{1}_{0}(\Omega_{\varepsilon})}|$ $\displaystyle=|\langle p,\text{\rm
div}(R_{\varepsilon}(u))\rangle_{L^{2}(\Omega_{\varepsilon})\times
L^{2}(\Omega_{\varepsilon})}|$
$\displaystyle\leq\|p\|_{L^{2}(\Omega_{\varepsilon})}\|\text{\rm
div}(R_{\varepsilon}(u))\|_{L^{2}(\Omega_{\varepsilon})}$ $\displaystyle\leq
C\|p\|_{L^{2}(\Omega_{\varepsilon})}\|\text{\rm div}(u)\|_{L^{2}(\Omega)},$
where we have used the estimate (2.18) for the last inequality. Thus there
exists $\Lambda\in H^{-1}(\Omega;\mathbb{R}^{d})$ such that
$\langle\nabla p,R_{\varepsilon}(u)\rangle_{H^{-1}(\Omega_{\varepsilon})\times
H_{0}^{1}(\Omega_{\varepsilon})}=\langle\Lambda,u\rangle_{H^{-1}(\Omega)\times
H_{0}^{1}(\Omega)}$
for any $u\in H_{0}^{1}(\Omega;\mathbb{R}^{d})$. Since
$\langle\Lambda,u\rangle=0$ if div$(u)=0$ in $\Omega$, it follows that
$\Lambda=\nabla Q$ for some $Q\in L^{2}(\Omega)$.
Next, using the fact that $R_{\varepsilon}(u)=u$ for $u\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, we obtain
$\langle\nabla p-\nabla Q,u\rangle_{H^{-1}(\Omega_{\varepsilon})\times
H_{0}^{1}(\Omega_{\varepsilon})}=0$
for any $u\in H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$. This implies
that $p-Q$ is constant in $\Omega_{\varepsilon}$. Since $Q$ is only determined
up to a constant, we may assume that $Q=p$ in $\Omega_{\varepsilon}$.
Moreover, we note that if $\varepsilon(Y+z)\subset\Omega^{\ell}$ for some
$1\leq\ell\leq L$ and $z\in\mathbb{Z}^{d}$, and $u\in
C_{0}^{1}(\varepsilon(Y_{s}^{\ell}+z),\mathbb{R}^{d})$, then
$R_{\varepsilon}(u)=0$ in $\Omega_{\varepsilon}$. It follows that $\nabla Q=0$
in $\varepsilon(Y_{s}^{\ell}+z)$. Thus $Q$ is constant in each
$\varepsilon(Y_{s}^{\ell}+z)$.
Finally, for any $u\in C_{0}^{1}(\varepsilon(Y+z);\mathbb{R}^{d})$ with
$\varepsilon(Y+z)\subset\Omega^{\ell}$, we have
$R_{\varepsilon}(u)\in H_{0}^{1}(\varepsilon(Y_{f}^{\ell}+z);\mathbb{R}^{d}),$
and by (2.19),
$\text{\rm div}(R_{\varepsilon}(u))=\text{\rm
div}(u)+\frac{1}{|\varepsilon(Y_{f}^{\ell}+z)|}\int_{\varepsilon(Y_{s}^{\ell}+z)}\text{\rm
div}(u)\,dx$
in $\varepsilon(Y_{f}^{\ell}+z)$. This, together with
$\int_{\varepsilon(Y_{f}^{\ell}+z)}p\cdot\text{\rm
div}(R_{\varepsilon}(u))\,dx=\int_{\varepsilon(Y+z)}Q\cdot\text{\rm
div}(u)\,dx$
and the fact that $Q=p$ in $\Omega_{\varepsilon}$, yields
$\int_{\varepsilon(Y_{s}^{\ell}+z)}\Big{(}Q-\fint_{\varepsilon(Y_{f}^{\ell}+z)}p\Big{)}\text{\rm
div}(u)\,dx=0.$
Consequently,
$Q=\fint_{\varepsilon(Y_{f}^{\ell}+z)}p\qquad\text{ in
}\varepsilon(Y_{s}^{\ell}+z).$
As a result, we have proved that $Q=P$, an extension of $p$ given by (2.21). ∎
## 3 Energy estimates
Let $\Omega_{\varepsilon}$ be given by (1.4). Recall that
$\partial\Omega_{\varepsilon}=\partial\Omega\cup\Gamma_{\varepsilon}$, where
$\Gamma_{\varepsilon}$ consists of the boundaries of the holes of size
$\varepsilon$ that are removed from $\Omega$. In this section we establish the
energy estimates for the Dirichlet problem,
$\left\\{\begin{aligned} -\varepsilon^{2}\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=f+\varepsilon\,\text{\rm div}(F)&\quad&\text{ in
}\Omega_{\varepsilon},\\\ \text{\rm div}(u_{\varepsilon})&=g&\quad&\text{ in
}\Omega_{\varepsilon},\\\ u_{\varepsilon}&=0&\quad&\text{ on
}\Gamma_{\varepsilon},\\\ u_{\varepsilon}&=h&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$ (3.1)
where $(g,h)$ satisfies the compatibility condition,
$\int_{\Omega_{\varepsilon}}g\,dx=\int_{\partial\Omega}h\cdot n\,d\sigma.$
(3.2)
Throughout this section we assume that $\Omega$, $\Omega^{\ell}$ and
$Y_{s}^{\ell}$ for $1\leq\ell\leq L$ are domains with Lipschitz boundaries. We
use $L^{2}_{0}(\Omega_{\varepsilon})$ to denote the subspace of functions in
$L^{2}(\Omega_{\varepsilon})$ with mean value zero.
###### Theorem 3.1.
Let $f\in L^{2}(\Omega_{\varepsilon};\mathbb{R}^{d})$ and $F\in
L^{2}(\Omega_{\varepsilon};\mathbb{R}^{d\times d})$. Let $g\in
L^{2}(\Omega_{\varepsilon})$ and $h\in H^{1/2}(\partial\Omega;\mathbb{R}^{d})$
satisfy the condition (3.2). Let $(u_{\varepsilon},p_{\varepsilon})\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L^{2}_{0}(\Omega_{\varepsilon})$ be a weak solution of (3.1). Then
$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
(3.3) $\displaystyle\leq
C\left\\{\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|H\|_{L^{2}(\Omega)}+\|\text{\rm
div}(H)\|_{L^{2}(\Omega)}+\varepsilon\|\nabla H\|_{L^{2}(\Omega)}\right\\},$
where $H$ is any function in $H^{1}(\Omega;\mathbb{R}^{d})$ with the property
$H=h$ on $\partial\Omega$.
###### Proof.
This theorem was proved in [15, Section 3] for the case $L=1$. The proof for
the case $L\geq 2$ is similar. We provide a proof here for the reader’s
convenience.
Step 1. We show that
$\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\left\\{\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}\right\\}.$
(3.4)
To this end we use Lemma 2.6 to find $v_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ such that $\text{\rm
div}(v_{\varepsilon})=p_{\varepsilon}$ in $\Omega_{\varepsilon}$ and
$\varepsilon\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}.$ (3.5)
By using $v_{\varepsilon}$ as a test function we obtain
$\displaystyle\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}^{2}$
$\displaystyle\leq\varepsilon^{2}\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}\|v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|F\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$ $\displaystyle\leq
C\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\left\\{\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}\right\\},$
where we have used (3.5) for the last inequality. This yields (3.4).
Step 2. We prove (3.3) in the case $h=0$. In this case we may use
$u_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ as a test
function to obtain
$\displaystyle\varepsilon^{2}\|\nabla
u_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}\leq\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|F\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}.$
By using the Cauchy inequality as well as the estimate
$\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq C\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$, we deduce that
$\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\left\\{\|p_{\varepsilon}\|^{1/2}_{L^{2}(\Omega_{\varepsilon})}\|g\|^{1/2}_{L^{2}(\Omega_{\varepsilon})}+\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}\right\\}.$
This, together with (3.4), gives (3.3) for the case $h=0$.
Step 3. We consider the general case $h\in
H^{1/2}(\partial\Omega;\mathbb{R}^{d})$. Let $H$ be a function in
$H^{1}(\Omega;\mathbb{R}^{d})$ such that $H=h$ on $\partial\Omega$. Let
$w_{\varepsilon}=u_{\varepsilon}-R_{\varepsilon}(H)$, where
$R_{\varepsilon}(H)$ is given by Lemma 2.5. Then $w_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ and
$\left\\{\begin{aligned} -\varepsilon^{2}\Delta w_{\varepsilon}+\nabla
p_{\varepsilon}&=f+\varepsilon\,\text{\rm div}(F)+\varepsilon^{2}\Delta
R_{\varepsilon}(H),\\\ \text{\rm div}(w_{\varepsilon})&=g-\text{\rm
div}(R_{\varepsilon}(H)),\end{aligned}\right.$
in $\Omega_{\varepsilon}$. By Step 2 we obtain
$\displaystyle\varepsilon\|\nabla
w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|w_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq
C\left\\{\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla
R_{\varepsilon}(H)\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|\text{\rm
div}(R_{\varepsilon}(H))\|_{L^{2}(\Omega_{\varepsilon})}\right\\}.$
It follows that
$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq
C\Big{\\{}\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\qquad\qquad+\varepsilon\|\nabla
R_{\varepsilon}(H)\|_{L^{2}(\Omega_{\varepsilon})}+\|R_{\varepsilon}(H)\|_{L^{2}(\Omega_{\varepsilon})}+\|\text{\rm
div}(R_{\varepsilon}(H))\|_{L^{2}(\Omega_{\varepsilon})}\Big{\\}}$
$\displaystyle\leq
C\Big{\\{}\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla
H\|_{L^{2}(\Omega)}+\|H\|_{L^{2}(\Omega)}+\|\text{\rm
div}(H)\|_{L^{2}(\Omega)}\Big{\\}},$
where we have used estimates (2.17) and (2.18) for the last inequality. ∎
###### Corollary 3.2.
Let $(u_{\varepsilon},p_{\varepsilon})$ be the same as in Theorem 3.1. Then
$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
(3.6) $\displaystyle\leq
C\Big{\\{}\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|h\|_{H^{1/2}(\partial\Omega)}\Big{\\}}.$
###### Proof.
For $h\in H^{1/2}(\partial\Omega;\mathbb{R}^{d})$, let $H$ be the weak
solution in $H^{1}(\Omega;\mathbb{R}^{d})$ of the Dirichlet problem,
$\left\\{\begin{aligned} -\Delta H+\nabla q&=0&\quad&\text{ in }\Omega,\\\
\text{\rm div}(H)&=\gamma&\quad&\text{ in }\Omega,\\\ u&=h&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$
where the constant
$\gamma=\frac{1}{|\Omega|}\int_{\partial\Omega}h\cdot n\,d\sigma$
is chosen so that the compatibility condition (3.2) is satisfied. Note that
$\|\text{\rm div}(H)\|_{L^{2}(\Omega)}=C|\gamma|\leq
C\|h\|_{L^{2}(\partial\Omega)},$
and by the standard energy estimates, $\|\nabla H\|_{L^{2}(\Omega)}\leq
C\|h\|_{H^{1/2}(\partial\Omega)}.$ In view of (3.3) we only need to show that
$\|H\|_{L^{2}(\Omega)}\leq C\|h\|_{L^{2}(\partial\Omega)}.$ (3.7)
To this end, let
$H_{1}=H-\gamma(x-x_{0})/d,$
where $x_{0}\in\Omega$. Since $-\Delta H_{1}+\nabla q=0$ and div$(H_{1})=0$ in
$\Omega$, it follows from [7] that
$\displaystyle\|H_{1}\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C\|(H_{1})^{*}\|_{L^{2}(\partial\Omega)}$ $\displaystyle\leq
C\|H_{1}\|_{L^{2}(\partial\Omega)}\leq C\|h\|_{L^{2}(\partial\Omega)},$
where $(H_{1})^{*}$ denotes the nontangential maximal function of $H_{1}$. As
a result, we obtain
$\displaystyle\|H\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|H_{1}\|_{L^{2}(\Omega)}+C|\gamma|$ $\displaystyle\leq
C\|h\|_{L^{2}(\partial\Omega)},$
which completes the proof. ∎
###### Corollary 3.3.
Let $(u_{\varepsilon},p_{\varepsilon})$ be the same as in Theorem 3.1. Let
$P_{\varepsilon}$ be the extension of $p_{\varepsilon}$, defined by (2.21).
Then
$\|P_{\varepsilon}\|_{L^{2}(\Omega)}\leq
C\left\\{\|f\|_{L^{2}(\Omega_{\varepsilon})}+\|F\|_{L^{2}(\Omega_{\varepsilon})}+\|g\|_{L^{2}(\Omega_{\varepsilon})}+\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|h\|_{H^{1/2}(\partial\Omega)}\right\\}.$
(3.8)
###### Proof.
By the definition of $P_{\varepsilon}$, we have
$\displaystyle\int_{\Omega}|P_{\varepsilon}|^{2}\,dx$
$\displaystyle=\int_{\Omega_{\varepsilon}}|p_{\varepsilon}|^{2}\,dx+\sum_{\ell=1}^{L}\sum_{z}|\varepsilon(Y_{s}^{\ell}+z)|\Big{(}\fint_{\varepsilon(Y^{\ell}_{f}+z)}p_{\varepsilon}\Big{)}^{2}$
$\displaystyle\leq\sum_{\ell=1}^{L}\frac{1}{|Y_{f}^{\ell}|}\int_{\Omega_{\varepsilon}^{\ell}}|p_{\varepsilon}|^{2}\,dx,$
which, together with (3.6), gives (3.8). ∎
## 4 Homogenization and proof of Theorem 1.1
Let $f\in L^{2}(\Omega;\mathbb{R}^{d})$ and $h\in
H^{1/2}(\partial\Omega;\mathbb{R}^{d})$ with $\int_{\partial\Omega}h\cdot
n\,d\sigma=0$, where $n$ denotes the outward unit normal to $\partial\Omega$.
Consider the Dirichlet problem,
$\left\\{\begin{aligned} -\varepsilon^{2}\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=f&\quad&\text{ in }\Omega_{\varepsilon},\\\ \text{\rm
div}(u_{\varepsilon})&=0&\quad&\text{ in }\Omega_{\varepsilon},\\\
u_{\varepsilon}&=0&\quad&\text{ on }\Gamma_{\varepsilon},\\\
u_{\varepsilon}&=h&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (4.1)
where $\Omega_{\varepsilon}$ is given by (1.4) and
$\partial\Omega_{\varepsilon}=\partial\Omega\cup\Gamma_{\varepsilon}$.
Throughout the section we assume that $\Omega$, $\Omega^{\ell}$ and
$Y_{s}^{\ell}$ for $1\leq\ell\leq L$, are domains with Lipschitz boundaries.
As before, we extend $u_{\varepsilon}$ to $\Omega$ by zero and still denote
the extension by $u_{\varepsilon}$. We use $P_{\varepsilon}$ to denote the
extension of $p_{\varepsilon}$ to $\Omega$, given by (2.21). The goal of this
section is to prove the following theorem, which contains Theorem 1.1 as a
special case $h=0$.
###### Theorem 4.1.
Let $f\in L^{2}(\Omega;\mathbb{R}^{d})$ and $h\in
H^{1/2}(\partial\Omega;\mathbb{R}^{d})$ with $\int_{\partial\Omega}h\cdot
n\,d\sigma=0$. Let $(u_{\varepsilon},p_{\varepsilon})\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L_{0}^{2}(\Omega_{\varepsilon})$ be the weak solution of (4.1). Let
$(u_{\varepsilon},P_{\varepsilon})$ be the extension of
$(u_{\varepsilon},p_{\varepsilon})$. Then $u_{\varepsilon}\to u_{0}$ weakly in
$L^{2}(\Omega;\mathbb{R}^{d})$ and
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ strongly in
$L^{2}(\Omega)$, as $\varepsilon\to 0$, where $P_{0}\in H^{1}(\Omega)$,
$\int_{\Omega}P_{0}\,dx=0$, $(u_{0},P_{0})$ is governed by a Darcy law,
$\left\\{\begin{aligned} u_{0}&=K(f-\nabla P_{0})&\quad&\text{ in }\Omega,\\\
\text{\rm div}(u_{0})&=0&\quad&\text{ in }\Omega,\\\ u_{0}\cdot n&=h\cdot
n&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (4.2)
with the permeability matrix $K$ given by (1.10).
We begin with the strong convergence of $P_{\varepsilon}$.
###### Lemma 4.2.
Let $(u_{\varepsilon_{k}},p_{\varepsilon_{k}})$ be a weak solution of (4.1)
with $\varepsilon=\varepsilon_{k}$. Suppose that as $\varepsilon_{k}\to 0$,
$P_{\varepsilon_{k}}\to P$ weakly in $L^{2}(\Omega)$ for some $P\in
L^{2}(\Omega)$. Then $P_{\varepsilon_{k}}\to P$ strongly in $L^{2}(\Omega)$.
###### Proof.
The proof is similar to that for the classical case $L=1$ (see e.g. [4]). One
argues by contradiction. Suppose that $P_{\varepsilon_{k}}$ does not converge
strongly to $P$ in $L^{2}(\Omega)$. Since
$\|\nabla P_{\varepsilon_{k}}-\nabla
P\|_{H^{-1}(\Omega)}\sim\|P_{\varepsilon_{k}}-P-\fint_{\Omega}(P_{\varepsilon_{k}}-P)\|_{L^{2}(\Omega)}$
and $\int_{\Omega}P_{\varepsilon_{k}}\,dx\to\int_{\Omega}P\,dx$, it follows
that $\nabla P_{\varepsilon_{k}}$ does not converge to $\nabla P$ strongly in
$H^{-1}(\Omega;\mathbb{R}^{d})$. By passing to a subsequence, this implies
that there exists a sequence $\\{\psi_{k}\\}\subset
H_{0}^{1}(\Omega;\mathbb{R}^{d})$ such that
$\|\psi_{k}\|_{H^{1}_{0}(\Omega)}=1$ and
$|\langle\nabla P_{\varepsilon_{k}}-\nabla
P,\psi_{k}\rangle_{H^{-1}(\Omega)\times H^{1}_{0}(\Omega)}|\geq c_{0}>0.$
By passing to another subsequence, we may assume that $\psi_{k}\to\psi_{0}$
weakly in $H_{0}^{1}(\Omega;\mathbb{R}^{d})$. Let
$\varphi_{k}=\psi_{k}-\psi_{0}$. Using $P_{\varepsilon_{k}}\to P$ weakly in
$L^{2}(\Omega)$, we obtain
$|\langle\nabla P_{\varepsilon_{k}}-\nabla
P,\varphi_{k}\rangle_{H^{-1}(\Omega)\times H^{1}_{0}(\Omega)}|\geq c_{0}/2,$
(4.3)
if $k$ is sufficiently large. Since $\varphi_{k}\to 0$ weakly in
$H^{1}_{0}(\Omega;\mathbb{R}^{d})$, we may conclude further that
$|\langle\nabla P_{\varepsilon_{k}},\varphi_{k}\rangle_{H^{-1}(\Omega)\times
H^{1}_{0}(\Omega)}|\geq c_{0}/4,$ (4.4)
if $k$ is sufficiently large. On the other hand, by (2.7), we have
$\displaystyle|\langle\nabla
P_{\varepsilon_{k}},\varphi_{k}\rangle_{H^{-1}(\Omega)\times
H_{0}^{1}(\Omega)}|=|\langle\nabla
p_{\varepsilon_{k}},R_{\varepsilon_{k}}(\varphi_{k})\rangle_{H^{-1}(\Omega_{\varepsilon_{k}})\times
H_{0}^{1}(\Omega_{\varepsilon_{k}})}|$ (4.5)
$\displaystyle=|\langle\varepsilon_{k}^{2}\Delta
u_{\varepsilon_{k}}+f,R_{\varepsilon_{k}}(\varphi_{k})\rangle_{H^{-1}(\Omega_{\varepsilon_{k}})\times
H_{0}^{1}(\Omega_{\varepsilon_{k}})}|$
$\displaystyle\leq\varepsilon_{k}^{2}\|\nabla
u_{\varepsilon_{k}}\|_{L^{2}(\Omega_{\varepsilon_{k}})}\|\nabla
R_{\varepsilon_{k}}(\varphi_{k})\|_{L^{2}(\Omega_{\varepsilon_{k}})}+\|f\|_{L^{2}(\Omega)}\|R_{\varepsilon_{k}}(\varphi_{k})\|_{L^{2}(\Omega_{\varepsilon_{k}})}$
$\displaystyle\leq
C\left(\|f\|_{L^{2}(\Omega)}+\|h\|_{H^{1/2}(\partial\Omega)}\right)\left(\varepsilon_{k}\|\nabla
R_{\varepsilon_{k}}(\varphi_{k})\|_{L^{2}(\Omega_{\varepsilon_{k}})}+\|R_{\varepsilon_{k}}(\varphi_{k})\|_{L^{2}(\Omega_{\varepsilon_{k}})}\right)$
$\displaystyle\leq
C\left(\|f\|_{L^{2}(\Omega)}+\|h\|_{H^{1/2}(\partial\Omega)}\right)\left(\varepsilon_{k}\|\nabla\varphi_{k}\|_{L^{2}(\Omega)}+\|\varphi_{k}\|_{L^{2}(\Omega)}\right),$
where we have used the estimate (3.6) for the second inequality and (2.17) for
the last. This contradicts with (4.4) as the right-hand side of (4.5) goes to
zero. ∎
By Corollaries 3.2 and 3.3, the sets $\\{u_{\varepsilon}:0<\varepsilon<1\\}$
and $\\{P_{\varepsilon}:0<\varepsilon<1\\}$ are bounded in
$L^{2}(\Omega;\mathbb{R}^{d})$ and $L^{2}(\Omega)$, respectively. It follows
that for any sequence $\varepsilon_{k}\to 0$, there exists a subsequence,
still denoted by $\\{\varepsilon_{k}\\}$, such that $u_{\varepsilon_{k}}\to u$
and $P_{\varepsilon_{k}}\to P$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and
$L^{2}(\Omega)$, respectively. By Lemma 4.2, $P_{\varepsilon_{k}}\to P$
strongly in $L^{2}(\Omega)$. Thus, as in the classical case $L=1$, to prove
Theorem 4.1, it suffices to show that if $\varepsilon_{k}\to 0$,
$u_{\varepsilon_{k}}\to u$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$, and
$P_{\varepsilon_{k}}\to P$ strongly in $L^{2}(\Omega)$, then $P\in
H^{1}(\Omega)$ and $(u,P)$ is a weak solution of (4.2). Since the solution of
(4.2) is unique under the conditions that $P_{0}\in H^{1}(\Omega)$ and
$\int_{\Omega}P_{0}\,dx=0$, one concludes that as $\varepsilon\to 0$,
$u_{\varepsilon}\to u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ strongly in
$L^{2}(\Omega)$, where $(u_{0},P_{0})$ is the unique solution of (4.2) with
the property $P_{0}\in H^{1}(\Omega)$ and $\int_{\Omega}P_{0}\,dx=0$.
###### Lemma 4.3.
Let $\\{\varepsilon_{k}\\}$ be a sequence such that $\varepsilon_{k}\to 0$.
Suppose that $u_{\varepsilon_{k}}\to u$ weakly in
$L^{2}(\Omega;\mathbb{R}^{d})$ and $P_{\varepsilon_{k}}\to P$ strongly in
$L^{2}(\Omega)$. Then $P\in H^{1}(\Omega^{\ell})$ for $1\leq\ell\leq L$ and
$(u,P)$ is a solution of $\eqref{Darcy-4}$.
###### Proof.
Since
$\int_{\Omega}u_{\varepsilon_{k}}\cdot\nabla\varphi\,dx=\int_{\partial\Omega}(h\cdot
n)\varphi\,d\sigma$
for any $\varphi\in C^{\infty}(\mathbb{R}^{d})$, by letting $k\to\infty$, we
see that
$\int_{\Omega}u\cdot\nabla\varphi\,dx=\int_{\partial\Omega}(h\cdot
n)\varphi\,d\sigma$
for any $\varphi\in C^{\infty}(\mathbb{R}^{d})$. It follows that div$(u)=0$ in
$\Omega$ and $u\cdot n=h\cdot n$ on $\partial\Omega$.
Next, we show that $P\in H^{1}(\Omega^{\ell})$ for each subdomain
$\Omega^{\ell}$ and that
$u=K^{\ell}(f-\nabla P)\quad\text{ in }\Omega^{\ell},$ (4.6)
where $K^{\ell}=(K^{\ell}_{ij})$ is defined by (2.2). The argument is the same
as that of Tartar for the case $L=1$ (see [13]). Fix $1\leq\ell\leq L$, $1\leq
j\leq d$, and $\varphi\in C_{0}^{\infty}(\Omega^{\ell})$. We assume $k>1$ is
sufficiently large that
supp$(\varphi)\subset\\{x\in\Omega^{\ell}:\text{dist}(x,\partial\Omega^{\ell})\geq
C_{d}\varepsilon_{k}\\}$. Let $(W_{j}^{\ell}(y),\pi_{j}^{\ell}(y))$ be the
1-periodic functions given by (2.1). By using
$W_{j}^{\ell}(x/\varepsilon_{k})\varphi$ as a test function, we obtain
$\displaystyle\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
u_{\varepsilon_{k}}\cdot\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\varphi\,dx+\varepsilon^{2}_{k}\int_{\Omega^{\ell}}\nabla
u_{\varepsilon_{k}}\cdot
W_{j}^{\ell}(x/\varepsilon_{k})\nabla\varphi\,dx-\int_{\Omega^{\ell}}P_{\varepsilon_{k}}W_{j}^{\ell}(x/\varepsilon_{k})\cdot\nabla\varphi\,dx$
(4.7) $\displaystyle=\int_{\Omega^{\ell}}f\cdot
W_{j}^{\ell}(x/\varepsilon_{k})\varphi\,dx,$
where we have used the facts that $\text{\rm
div}(W^{\ell}_{j}(x/\varepsilon))=0$ in $\mathbb{R}^{d}$ and
$W_{j}^{\ell}(x/\varepsilon)=0$ on $\Gamma_{\varepsilon}$. Since
$W_{ij}^{\ell}(x/\varepsilon_{k})\to K^{\ell}_{ij}$ weakly in
$L^{2}(\Omega^{\ell})$ and $P_{\varepsilon_{k}}\to P$ strongly in
$L^{2}(\Omega^{\ell})$, we deduce from (4.7) that
$\lim_{k\to\infty}\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
u_{\varepsilon_{k}}\cdot\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\varphi\,dx=\int_{\Omega^{\ell}}PK^{\ell}_{ij}\frac{\partial\varphi}{\partial
x_{i}}\,dx+\int_{\Omega^{\ell}}f_{i}K_{ij}^{\ell}\varphi\,dx,$ (4.8)
where the repeated index $i$ is summed from $1$ to $d$.
Note that
$\displaystyle-\varepsilon^{2}\Delta\left(W_{j}^{\ell}(x/\varepsilon)\right)+\nabla\left(\varepsilon\pi_{j}^{\ell}(x/\varepsilon)\right)$
$\displaystyle=e_{j}$
in the set $\\{x\in\Omega^{\ell}_{\varepsilon}:\text{\rm
dist}(x,\partial\Omega^{\ell})\geq c_{d}\varepsilon\\}$. By using
$u_{\varepsilon_{k}}\varphi$ as a test function, we see that
$\displaystyle\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\cdot(\nabla
u_{\varepsilon_{k}})\varphi\,dx+\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\cdot u_{\varepsilon_{k}}(\nabla\varphi)\,dx$
(4.9)
$\displaystyle\qquad\qquad-\varepsilon_{k}\int_{\Omega^{\ell}}\pi_{j}^{\ell}(x/\varepsilon_{k})u_{\varepsilon_{k}}(\nabla\varphi)\,dx=\int_{\Omega^{\ell}}e_{j}\cdot
u_{\varepsilon_{k}}\varphi\,dx,$
which leads to
$\lim_{k\to\infty}\varepsilon_{k}\int_{\Omega^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon_{k})\cdot(\nabla
u_{\varepsilon_{k}})\varphi\,dx.=\int_{\Omega^{\ell}}e_{j}\cdot u\varphi\,dx.$
(4.10)
In view of (4.8) and (4.10) we obtain
$\int_{\Omega^{\ell}}e_{j}\cdot
u\varphi\,dx=\int_{\Omega^{\ell}}PK^{\ell}_{ij}\frac{\partial\varphi}{\partial
x_{i}}\,dx+\int_{\Omega^{\ell}}f_{i}K_{ij}^{\ell}\varphi\,dx.$
Since $\varphi\in C_{0}^{\infty}(\Omega^{\ell})$ is arbitrary and the constant
matrix $K^{\ell}=(K_{ij}^{\ell})$ is invertible, we conclude that $P\in
H^{1}(\Omega^{\ell})$ and
$u_{j}=K_{ij}^{\ell}\Big{(}f_{i}-\frac{\partial P}{\partial x_{i}}\Big{)}$
in $\Omega^{\ell}$. Since $K^{\ell}$ is also symmetric, this gives (4.6). ∎
To prove the effective pressure in Lemma 4.3 $P\in H^{1}(\Omega)$, it remains
to show that $P$ is continuous across the interface
$\Sigma=\Omega\setminus\cup_{\ell=1}^{L}\Omega^{\ell}$ between subdomains.
###### Lemma 4.4.
Let $f\in C^{m}(B(x_{0},2c\varepsilon);\mathbb{R}^{d})$ for some
$x_{0}\in\mathbb{R}^{d}$, $m\geq 0$ and $c>0$. Suppose that
$\left\\{\begin{aligned} -\varepsilon^{2}\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=f&\quad&\text{ in }B(x_{0},2c\varepsilon),\\\ \text{\rm
div}(u_{\varepsilon})&=0&\quad&\text{ in
}B(x_{0},2c\varepsilon).\end{aligned}\right.$ (4.11)
Then
$\displaystyle\varepsilon^{m+2}\left(\fint_{B(x_{0},c\varepsilon)}|\nabla^{m+2}u_{\varepsilon}|^{2}\right)^{1/2}\leq
C\left(\fint_{B(x_{0},2c\varepsilon)}|u_{\varepsilon}|^{2}\right)^{1/2}+C\sum_{k=0}^{m}\varepsilon^{k}\|\nabla^{k}f\|_{\infty},$
(4.12)
where $C$ depends only on $d$, $m$ and $c$.
###### Proof.
The case $\varepsilon=1$ is given by interior estimates for the Stokes
equations. The general case follows by a simple rescaling argument. ∎
Define
$\gamma_{\varepsilon}=\big{\\{}x\in\Sigma:\ \text{\rm
dist}(x,\partial\Omega)\geq\varepsilon\big{\\}},$ (4.13)
where $\Sigma$ is the interface given by (1.5)
###### Lemma 4.5.
Let $(u_{\varepsilon},p_{\varepsilon})$ be a solution of (4.1) with $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ and $h\in
H^{1/2}(\partial\Omega;\mathbb{R}^{d})$. Then, for $m\geq 0$,
$\displaystyle\|\nabla^{m}u_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon})}$
$\displaystyle\leq C(f,h)\varepsilon^{-m-\frac{1}{2}},$ (4.14)
$\displaystyle\|p_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon})}$
$\displaystyle\leq C(f,h)\varepsilon^{-\frac{1}{2}},$ $\displaystyle\|\nabla
p_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon})}$ $\displaystyle\leq
C(f,h)\varepsilon^{-\frac{1}{2}},$
where $C(f,h)$ depends on $m$, $f$ and $h$, but not on $\varepsilon$.
###### Proof.
Recall that
$\Sigma=\cup_{\ell=1}^{L}\partial\Omega^{\ell}\setminus\partial\Omega.$
It follows that
$\gamma_{\varepsilon}=\cup_{\ell=1}^{L}\gamma_{\varepsilon}^{\ell}$, where
$\gamma_{\varepsilon}^{\ell}=\big{\\{}x\in\partial\Omega^{\ell}:\
\text{dist}(x,\partial\Omega)\geq\varepsilon\big{\\}}.$
Thus, it suffices to prove (4.14) with $\gamma_{\varepsilon}^{\ell}$ in the
place of $\gamma_{\varepsilon}$. Let
$D^{\ell}_{\varepsilon}=\left\\{x\in\Omega^{\ell}:\text{
dist}(x,\gamma_{\varepsilon}^{\ell})<c\,\varepsilon\right\\}.$
Using the assumption that $\Omega^{\ell}$ is a bounded Lipschitz domain, one
may show that
$\displaystyle\int_{\gamma_{\varepsilon}^{\ell}}|\nabla^{m}u_{\varepsilon}|^{2}\,d\sigma$
$\displaystyle\leq\frac{C}{\varepsilon}\int_{D_{\varepsilon}^{\ell}}|\nabla^{m}u_{\varepsilon}|^{2}\,dx+C\varepsilon\int_{D_{\varepsilon}^{\ell}}|\nabla^{m+1}u_{\varepsilon}|^{2}\,dx$
(4.15)
$\displaystyle\leq\frac{C}{\varepsilon^{1+2m}}\left\\{\int_{\Omega_{\varepsilon}}|u_{\varepsilon}|^{2}\,dx+C(f)\right\\},$
where $C(f)$ depends on $f$. We point out that the second inequality in (4.15)
follows by covering $D_{\varepsilon}^{\ell}$ with balls of radius
$c\varepsilon$ and using (4.12). This, together with the energy estimate
(3.6), yields
$\|\nabla^{m}u_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon}^{\ell})}\leq
C(f,h)\varepsilon^{-m-\frac{1}{2}},$
where $C(f,h)$ depends on $f$ and $h$. Next, using the equation
$-\varepsilon^{2}\Delta u_{\varepsilon}+\nabla p_{\varepsilon}=f$, we obtain
$\displaystyle\|\nabla p_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon}^{\ell})}$
$\displaystyle\leq\varepsilon^{2}\|\Delta
u_{\varepsilon}\|_{L^{2}(\gamma_{\varepsilon}^{\ell})}+\|f\|_{L^{2}(\gamma_{\varepsilon}^{\ell})}$
$\displaystyle\leq C(f,h)\varepsilon^{-1/2}.$
Finally, observe that
$\displaystyle\int_{\gamma_{\varepsilon}^{\ell}}|p_{\varepsilon}|^{2}\,d\sigma$
$\displaystyle\leq\frac{C}{\varepsilon}\int_{D_{\varepsilon}^{\ell}}|p_{\varepsilon}|^{2}\,dx+C\varepsilon\int_{D_{\varepsilon}^{\ell}}|\nabla
p_{\varepsilon}|^{2}\,dx$
$\displaystyle\leq\frac{C}{\varepsilon}\int_{\Omega_{\varepsilon}}|p_{\varepsilon}|^{2}\,dx+C\varepsilon^{5}\int_{D_{\varepsilon}^{\ell}}|\Delta
u_{\varepsilon}|^{2}\,dx+C(f),$
$\displaystyle\leq\frac{C}{\varepsilon}\int_{\Omega_{\varepsilon}}|p_{\varepsilon}|^{2}\,dx+C\varepsilon\int_{\Omega_{\varepsilon}}|u_{\varepsilon}|^{2}\,dx+C(f).$
This, together with the energy estimate (3.6), yields the second inequality in
(4.14). ∎
The following is the main technical lemma in the proof of Theorem 4.1.
###### Lemma 4.6.
Let $(u_{\varepsilon_{k}},p_{\varepsilon_{k}})$, $P_{\varepsilon_{k}}$, and
$(u,P)$ be the same as in Lemma 4.3. Also assume that $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$. Let $P^{\ell}$ denote the trace of
$P$, as a function in $H^{1}(\Omega^{\ell})$, on $\partial\Omega^{\ell}$.
Then, for any $\varphi\in C_{0}^{\infty}(\Omega)$,
$\int_{\partial\Omega^{\ell}}n_{j}P^{\ell}\varphi\,dx=\lim_{k\to\infty}\int_{\partial\Omega^{\ell}}n_{j}p_{\varepsilon_{k}}\varphi\,d\sigma,$
(4.16)
where $1\leq\ell\leq L$, $1\leq j\leq d$, and $n=(n_{1},n_{2},\dots,n_{d})$
denotes the outward unit normal to $\partial\Omega^{\ell}$.
###### Proof.
For notational simplicity we use $\varepsilon$ to denote $\varepsilon_{k}$.
Fix $1\leq j\leq d$ and $1\leq\ell\leq L$. Let $\varphi\in
C_{0}^{\infty}(\Omega)$. Then
$\displaystyle\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla\left(W_{j}^{\ell}(x/\varepsilon)\varphi\right)\,dx$
$\displaystyle\qquad=\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla
W_{j}^{\ell}(x/\varepsilon)\varphi\,dx+\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot W_{j}^{\ell}(x/\varepsilon)(\nabla\varphi)\,dx,$
and by integration by parts,
$\displaystyle\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla\left(W_{j}^{\ell}(x/\varepsilon)\varphi\right)\,dx$
$\displaystyle=\int_{\Omega^{\ell}}f\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,dx+\int_{\Omega^{\ell}}P_{\varepsilon}W_{j}^{\ell}(x/\varepsilon)\cdot\nabla\varphi\,dx+\int_{\partial\Omega^{\ell}}\frac{\partial
u_{\varepsilon}}{\partial\nu}\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma,$
where
$\frac{\partial u_{\varepsilon}}{\partial\nu}=\varepsilon^{2}\frac{\partial
u_{\varepsilon}}{\partial n}-p_{\varepsilon}n.$
By letting $\varepsilon\to 0$ we obtain
$\displaystyle\lim_{\varepsilon\to
0}\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla W_{j}^{\ell}(x/\varepsilon)\varphi\,dx$ (4.17)
$\displaystyle=\int_{\Omega^{\ell}}f\cdot
K_{j}^{\ell}\varphi\,dx+\int_{\Omega^{\ell}}PK_{j}^{\ell}\cdot\nabla\varphi\,dx+\lim_{\varepsilon\to
0}\int_{\partial\Omega^{\ell}}\frac{\partial
u_{\varepsilon}}{\partial\nu}\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma.$
It follows by Lemma 2.1 that
$\|W_{j}^{\ell}(x/\varepsilon)\|_{L^{2}(\partial\Omega^{\ell})}\leq C$. This,
together with the first inequality in (4.14) with $m=1$, show that
$\Big{|}\varepsilon^{2}\int_{\partial\Omega^{\ell}}\frac{\partial
u_{\varepsilon}}{\partial n}\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma\Big{|}\leq
C\varepsilon^{2}\|(\nabla
u_{\varepsilon})\varphi\|_{L^{2}(\partial\Omega^{\ell})}=O(\varepsilon^{1/2}).$
Hence, by (4.17),
$\displaystyle\lim_{\varepsilon\to
0}\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
u_{\varepsilon}\cdot\nabla W_{j}^{\ell}(x/\varepsilon)\varphi\,dx$ (4.18)
$\displaystyle=\int_{\Omega^{\ell}}f\cdot
K_{j}^{\ell}\varphi\,dx+\int_{\Omega^{\ell}}PK_{j}^{\ell}\cdot\nabla\varphi\,dx-\lim_{\varepsilon\to
0}\int_{\partial\Omega^{\ell}}p_{\varepsilon}n\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma.$
Next, note that
$\displaystyle\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla\left(W_{j}^{\ell}(x/\varepsilon)\right)\cdot\nabla(u_{\varepsilon}\varphi)\,dx$
(4.19) $\displaystyle=\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot(\nabla
u_{\varepsilon})\varphi\,dx+\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot u_{\varepsilon}(\nabla\varphi)\,dx.$
Choose a cut-off function $\eta_{\varepsilon}$ such that
supp$(\eta_{\varepsilon})\subset\\{x\in\mathbb{R}^{d}:\text{dist}(x,\partial\Omega^{\ell})\leq
2C\varepsilon\\}$, $\eta_{\varepsilon}(x)=1$ if
dist$(x,\partial\Omega^{\ell})\leq C\varepsilon$, and
$|\nabla\eta_{\varepsilon}|\leq C\varepsilon^{-1}$. Then
$\displaystyle\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla\left(W_{j}^{\ell}(x/\varepsilon)\right)\cdot\nabla(u_{\varepsilon}\varphi)\,dx$
(4.20)
$\displaystyle=\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla\left(W_{j}^{\ell}(x/\varepsilon)\right)\cdot\nabla(u_{\varepsilon}(1-\eta_{\varepsilon})\varphi)\,dx+\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla\left(W_{j}^{\ell}(x/\varepsilon)\right)\cdot\nabla(u_{\varepsilon}\eta_{\varepsilon}\varphi)\,dx$
$\displaystyle=J_{1}+J_{2}.$
Using (4.19), (4.20), and
$\displaystyle|J_{2}|$ $\displaystyle\leq
C\varepsilon\left(\int_{\\{x\in\mathbb{R}^{d}:\,\text{dist}(x,\partial\Omega^{\ell})\leq
C\varepsilon\\}}|\nabla
W_{j}^{\ell}(x/\varepsilon)|^{2}\,dx\right)^{1/2}\left\\{\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}+\varepsilon^{-1}\|u_{\varepsilon}\|_{L^{2}(\Omega)}\right\\}$
$\displaystyle\leq C\varepsilon^{3/2}\left\\{\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}+\varepsilon^{-1}\|u_{\varepsilon}\|_{L^{2}(\Omega)}\right\\}$
$\displaystyle\leq\varepsilon^{1/2}C(f,h),$
we obtain
$\lim_{\varepsilon\to 0}\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot(\nabla
u_{\varepsilon})\varphi\,dx=\lim_{\varepsilon\to 0}J_{1}$ (4.21)
To handle the term $J_{1}$, we use integration by parts as well as the fact
that
$-\varepsilon^{2}\Delta\left(W_{j}^{\ell}(x/\varepsilon)\right)+\nabla\left(\varepsilon\pi_{j}^{\ell}(x/\varepsilon)\right)=e_{j}$
in the set $\\{x\in\Omega_{\varepsilon}^{\ell}:\text{\rm
dist}(x,\partial\Omega^{\ell})\geq C\varepsilon\\}$, to obtain
$\displaystyle J_{1}$
$\displaystyle=\int_{\Omega_{\varepsilon}^{\ell}}\varepsilon\pi_{j}^{\ell}(x/\varepsilon)u_{\varepsilon}\cdot\nabla((1-\eta_{\varepsilon})\varphi)\,dx+\int_{\Omega^{\ell}}e_{j}\cdot
u_{\varepsilon}\varphi(1-\eta_{\varepsilon})\,dx$
$\displaystyle=J_{11}+J_{12},$
where we have used the fact div$(u_{\varepsilon})=0$ in
$\Omega_{\varepsilon}$. Since
$\displaystyle|J_{11}|$ $\displaystyle\leq
C\left(\int_{\\{x\in\mathbb{R}^{d}:\,\text{dist}(x,\partial\Omega^{\ell})\leq
C\varepsilon\\}}|\pi_{j}^{\ell}(x/\varepsilon)|^{2}\,dx\right)^{1/2}\|u_{\varepsilon}\|_{L^{2}(\Omega^{\ell})}+C\varepsilon\|u_{\varepsilon}\|_{L^{2}(\Omega^{\ell})}$
$\displaystyle\leq C\varepsilon^{1/2}C(f,h),$
we see that
$\lim_{\varepsilon\to 0}J_{1}=\lim_{\varepsilon\to
0}J_{12}=\int_{\Omega^{\ell}}e_{j}\cdot u\varphi\,dx.$ (4.22)
In view of (4.18), (4.21) and (4.22), we have proved that
$\lim_{\varepsilon\to 0}\int_{\partial\Omega^{\ell}}p_{\varepsilon}n\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma=\int_{\Omega^{\ell}}f\cdot
K_{j}^{\ell}\varphi\,dx+\int_{\Omega^{\ell}}PK_{j}^{\ell}\cdot\nabla\varphi\,dx-\int_{\Omega^{\ell}}e_{j}\cdot
u\varphi\,dx.$ (4.23)
Recall that $K^{\ell}=(K^{\ell}_{ij})$ is symmetric and by Lemma 4.3,
$u=K^{\ell}(f-\nabla P)\quad\text{ in }\Omega^{\ell}.$
Thus, by (4.23),
$\lim_{\varepsilon\to 0}\int_{\partial\Omega^{\ell}}p_{\varepsilon}n\cdot
W_{j}^{\ell}(x/\varepsilon)\varphi\,d\sigma=\int_{\partial\Omega^{\ell}}P^{\ell}(n\cdot
K_{j}^{\ell})\varphi\,d\sigma,$ (4.24)
where $P^{\ell}$ denotes the trace of $P$ on $\partial\Omega^{\ell}$.
Finally, we use Lemma 2.2 to obtain
$n\cdot\left(W^{\ell}_{j}(x/\varepsilon)-K^{\ell}_{j}\right)=\frac{\varepsilon}{2}\left(n_{\beta}\frac{\partial}{\partial
x_{\alpha}}-n_{\alpha}\frac{\partial}{\partial
x_{\beta}}\right)\left(\phi^{\ell}_{\alpha\beta j}(x/\varepsilon)\right),$
(4.25)
where the repeated indices $\alpha$ and $\beta$ are summed from $1$ to $d$.
Since $n_{\beta}\frac{\partial}{\partial
x_{\alpha}}-n_{\alpha}\frac{\partial}{\partial x_{\beta}}$ is a tangential
derivative on $\partial\Omega^{\ell}$, we obtain
$\displaystyle\Big{|}\int_{\partial\Omega^{\ell}}p_{\varepsilon}n\cdot\left(W_{j}^{\ell}(x/\varepsilon)-K^{\ell}_{j}\right)\varphi\,d\sigma\Big{|}$
$\displaystyle=\frac{\varepsilon}{2}\Big{|}\int_{\partial\Omega^{\ell}}\phi^{\ell}_{\alpha\beta
j}(x/\varepsilon)\left(n_{\beta}\frac{\partial}{\partial
x_{\alpha}}-n_{\alpha}\frac{\partial}{\partial
x_{\beta}}\right)(p_{\varepsilon}\varphi)\,d\sigma\Big{|}$ $\displaystyle\leq
C\varepsilon\|\nabla(p_{\varepsilon}\varphi)\|_{L^{2}(\partial\Omega^{\ell})}$
$\displaystyle\leq C(f,h)\varepsilon^{1/2},$
where we have used (2.8) for the first inequality and (4.14) for the last.
This, together with (4.24), yields
$\lim_{\varepsilon\to 0}\int_{\partial\Omega^{\ell}}p_{\varepsilon}(n\cdot
K_{j}^{\ell})\varphi\,d\sigma=\int_{\partial\Omega^{\ell}}P^{\ell}(n\cdot
K_{j}^{\ell})\varphi\,d\sigma.$ (4.26)
Since the constant matrix $K^{\ell}=(K_{ij}^{\ell})$ is invertible, the
desired equation (4.16) follows readily from (4.26). ∎
We are now in a position to give the proof of Theorem 4.1.
###### Proof of Theorem 4.1.
We first prove Theorem 4.1 under the additional assumption $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$. Let $\\{\varepsilon_{k}\\}$ be a
sequence such that $\varepsilon_{k}\to 0$, $u_{\varepsilon_{k}}\to u$ weakly
in $L^{2}(\Omega;\mathbb{R}^{d})$ and $P_{\varepsilon_{k}}\to P$ strongly in
$L^{2}(\Omega)$. By Lemma 4.3, $P\in H^{1}(\Omega^{\ell})$ and
$u=K^{\ell}(f-\nabla P)$ in $\Omega^{\ell}$ for $1\leq\ell\leq L$. It suffices
to show that $P\in H^{1}(\Omega)$. This would imply that $P$ is a weak
solution of the Neumann problem,
$\left\\{\begin{aligned} \text{\rm div}(K(f-\nabla P))&=0&\quad&\text{ in
}\Omega,\\\ n\cdot K(f-\nabla P)&=n\cdot h&\quad&\text{ on
}\partial\Omega.\end{aligned}\right.$ (4.27)
As a result, we may deduce that as $\varepsilon\to 0$, $u_{\varepsilon}\to
u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$ and
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ strongly in
$L^{2}(\Omega)$, where $u_{0}=K(f-\nabla P_{0})$ in $\Omega$ and $P_{0}$ is
the unique weak solution of (4.27) with $\int_{\Omega}P_{0}\,dx=0$.
To prove $u\in H^{1}(\Omega)$, we use the assumption $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ and Lemma 4.6 to obtain
$\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}P^{\ell}\varphi\,d\sigma=\lim_{k\to\infty}\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}p_{\varepsilon_{k}}\varphi\,d\sigma,$
for any $\varphi\in C_{0}^{\infty}(\Omega)$ and $1\leq j\leq d$, where
$P^{\ell}$ denotes the trace of $P$, as a function in $H^{1}(\Omega^{\ell})$,
on $\partial\Omega^{\ell}$. Since $p_{\varepsilon}$ is continuous in
$\Omega_{\varepsilon}$, we have
$\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}p_{\varepsilon}\varphi\,d\sigma=0.$
It follows that
$\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}P^{\ell}\varphi\,d\sigma=0$
for $1\leq j\leq d$ and for any $\varphi\in C_{0}^{\infty}(\Omega)$. This,
together with the fact that $P\in H^{1}(\Omega^{\ell})$ for $1\leq\ell\leq L$,
gives
$\displaystyle\int_{\Omega}P\frac{\partial\varphi}{\partial x_{j}}\,dx$
$\displaystyle=\sum_{\ell=1}^{L}\int_{\Omega^{\ell}}P\frac{\partial\varphi}{\partial
x_{j}}\,dx$
$\displaystyle=-\sum_{\ell=1}^{L}\int_{\Omega^{\ell}}\frac{\partial
P}{\partial
x_{j}}\varphi\,dx+\sum_{\ell=1}^{L}\int_{\partial\Omega^{\ell}}n_{j}P^{\ell}\varphi\,d\sigma$
$\displaystyle=-\sum_{\ell=1}^{L}\int_{\Omega^{\ell}}\frac{\partial
P}{\partial x_{j}}\varphi\,dx.$
As a result, we obtain $P\in H^{1}(\Omega)$.
In the general case $f\in L^{2}(\Omega;\mathbb{R}^{d})$, we choose a sequence
of functions $\\{f_{m}\\}$ in $C^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ such
that $\|f_{m}-f\|_{L^{2}(\Omega)}\to 0$ as $m\to\infty$. Let
$(u_{\varepsilon,m},p_{\varepsilon,m})$ denote the weak solution of (4.1) with
$f_{m}$ in the place of $f$ and with
$\int_{\Omega_{\varepsilon}}p_{\varepsilon,m}\,dx=0$. By the energy estimates
(3.6) and (3.8) we obtain
$\|u_{\varepsilon}-u_{\varepsilon,m}\|_{L^{2}(\Omega)}+\|P_{\varepsilon}-P_{\varepsilon,m}\|_{L^{2}(\Omega)}\leq
C\|f-f_{m}\|_{L^{2}(\Omega)},$ (4.28)
where $P_{\varepsilon,m}$ denotes the extension of $p_{\varepsilon,m}$,
defined by (2.21). Let $u_{0,m}=K(f_{m}-\nabla P_{0,m})$, where $P_{0,m}$ is
the unique solution of (4.27) with $f_{m}$ in the place of $f$ and with
$\int_{\Omega}P_{0,m}\,dx=0$. Note that
$\displaystyle\|P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}-P_{0}\|_{L^{2}(\Omega)}\leq$
$\displaystyle\|P_{\varepsilon}-P_{\varepsilon,m}-\fint_{\Omega}(P_{\varepsilon}-P_{\varepsilon,m})\|_{L^{2}(\Omega)}$
$\displaystyle\qquad+\|P_{\varepsilon,m}-\fint_{\Omega}P_{\varepsilon,m}-P_{0,m}\|_{L^{2}(\Omega)}+\|P_{0,m}-P_{0}\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|f-f_{m}\|_{L^{2}(\Omega)}+\|P_{\varepsilon,m}-\fint_{\Omega}P_{\varepsilon,m}-P_{0,m}\|_{L^{2}(\Omega)}.$
Since $P_{\varepsilon,m}-\fint_{\Omega}P_{\varepsilon,m}\to P_{0,m}$ in
$L^{2}(\Omega)$, as $\varepsilon\to 0$, we see that
$\limsup_{\varepsilon\to
0}\|P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}-P_{0}\|_{L^{2}(\Omega)}\leq
C\|f-f_{m}\|_{L^{2}(\Omega)}.$
By letting $m\to\infty$, we obtain
$P_{\varepsilon}-\fint_{\Omega}P_{\varepsilon}\to P_{0}$ in $L^{2}(\Omega)$,
as $\varepsilon\to 0$.
Finally, let $v\in L^{2}(\Omega;\mathbb{R}^{d})$. Note that
$\displaystyle\Big{|}\int_{\Omega}(u_{\varepsilon}-u_{0})v\,dx\Big{|}$
$\displaystyle\leq\Big{|}\int_{\Omega}(u_{\varepsilon}-u_{\varepsilon,m})v\,dx\Big{|}+\Big{|}\int_{\Omega}(u_{\varepsilon,m}-u_{0,m})v\,dx\Big{|}+\Big{|}\int_{\Omega}(u_{0,m}-u_{0})v\,dx\Big{|}$
$\displaystyle\leq\|u_{\varepsilon}-u_{\varepsilon,m}\|_{L^{2}(\Omega)}\|v\|_{L^{2}(\Omega)}+\Big{|}\int_{\Omega}(u_{\varepsilon,m}-u_{0,m})v\,dx\Big{|}+\|u_{0,m}-u_{0}\|_{L^{2}(\Omega)}\|v\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|f-f_{m}\|_{L^{2}(\Omega)}\|v\|_{L^{2}(\Omega)}+\Big{|}\int_{\Omega}(u_{\varepsilon,m}-u_{0,m})v\,dx\Big{|}.$
By letting $\varepsilon\to 0$ and then $m\to\infty$, we see that
$u_{\varepsilon}\to u_{0}$ weakly in $L^{2}(\Omega;\mathbb{R}^{d})$. ∎
## 5 Convergence rates and proof of Theorem 1.2
Throughout the rest of this paper, unless indicated otherwise, we will assume
that $\Omega^{\ell},1\leq\ell\leq L$, are $C^{2,1/2}$ domains satisfying the
interface condition (1.12). Given $f\in L^{2}(\Omega;\mathbb{R}^{d})$, let
$P_{0}\in H^{1}(\Omega)$ be the weak solution of
$\left\\{\begin{aligned} -\text{\rm div}\left(K(f-\nabla
P_{0})\right)&=0&\quad&\text{ in }\Omega,\\\ n\cdot K(f-\nabla
P_{0})&=0&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (5.1)
with $\int_{\Omega}P_{0}\,dx=0$, where the coefficient matrix $K$ is given by
(1.10). Since the interface $\Sigma$ and $\partial\Omega$ are of $C^{2,1/2}$,
it follows from [16, Theorem 1.1] that
$\displaystyle\|\nabla P_{0}\|_{C^{\alpha}(\Omega)}$ $\displaystyle\leq
C\|f\|_{C^{\alpha}(\Omega)},$ (5.2) $\displaystyle\|\nabla
P_{0}\|_{C^{1,\beta}(\Omega)}$ $\displaystyle\leq
C\|f\|_{C^{1,\beta}(\Omega)},$
for $0<\alpha<1$ and $0<\beta\leq 1/2$.
Let
$V_{\varepsilon}(x)=\sum_{\ell=1}^{L}W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\chi_{\Omega^{\ell}}\quad\text{ in }\Omega,$ (5.3)
where the 1-periodic matrix $W^{\ell}(y)$ is defined by (2.1) . Note that
$V_{\varepsilon}=0$ in $\Gamma_{\varepsilon}$. For each $\ell$, using
$-\varepsilon^{2}\Delta\left\\{W_{j}^{\ell}(x/\varepsilon)\right\\}+\nabla\left\\{\varepsilon\pi_{j}^{\ell}(x/\varepsilon)\right\\}=e_{j}\quad\text{
in }\bigcup_{z\in\mathbb{Z}^{d}}\varepsilon(z+{Y^{\ell}_{f}}),$ (5.4)
one may show that for any $\psi\in
H^{1}(\Omega^{\ell}_{\varepsilon};\mathbb{R}^{d})$ with $\psi=0$ on
$\Gamma^{\ell}_{\varepsilon}$,
$\displaystyle\Big{|}\varepsilon\int_{\Omega^{\ell}_{\varepsilon}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot\nabla\psi\,dx-\varepsilon\int_{\Omega_{\varepsilon}^{\ell}}\pi_{j}^{\ell}(x/\varepsilon)\,\text{div}(\psi)\,dx-\int_{\Omega_{\varepsilon}^{\ell}}\psi_{j}\,dx\Big{|}$
(5.5) $\displaystyle\leq
C\varepsilon^{3/2}\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}.$
To see (5.5), let
$\mathcal{O}_{\varepsilon}^{\ell}=\bigcup_{z}\varepsilon(z+Y_{f}^{\ell}),$
where $z\in\mathbb{Z}^{d}$ and the union is taken over those $z$’s for which
$\varepsilon(z+Y)\subset\Omega^{\ell}$. Using
$|\Omega_{\varepsilon}^{\ell}\setminus\mathcal{O}_{\varepsilon}^{\ell}|\leq
C\varepsilon^{1/2}$ and $\|\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\leq
C\varepsilon\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}$, one may show
that each integral in the left-hand side of (5.5), with
$\Omega_{\varepsilon}^{\ell}\setminus\mathcal{O}^{\ell}_{\varepsilon}$ in the
place of $\Omega_{\varepsilon}^{\ell}$, is bounded by the right-hand side of
(5.5). By using integration by parts and (5.4), it follows that the left-hand
side of (5.5) with $\mathcal{O}_{\varepsilon}^{\ell}$ in the place of
$\Omega_{\varepsilon}^{\ell}$ is bounded by
$\displaystyle
C\varepsilon\left(\int_{\partial\mathcal{O}_{\varepsilon}^{\ell}}\left(|\nabla
W^{\ell}(x/\varepsilon)|+|\pi^{\ell}(x/\varepsilon)|\right)^{2}\,d\sigma\right)^{1/2}\left(\int_{\partial\mathcal{O}_{\varepsilon}^{\ell}}|\psi|^{2}\,d\sigma\right)^{1/2}$
$\displaystyle\leq
C\varepsilon^{3/2}\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})},$
where we have used (2.5) and the observation,
$\displaystyle\|\psi\|_{L^{2}(\partial\mathcal{O}_{\varepsilon}^{\ell})}$
$\displaystyle\leq
C\varepsilon^{-1/2}\|\psi\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+C\varepsilon^{1/2}\|\nabla\psi\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq
C\varepsilon^{1/2}\|\nabla\psi\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}.$
From (5.5) we deduce further that
$\displaystyle\Big{|}\varepsilon\int_{\Omega^{\ell}_{\varepsilon}}\nabla
W_{j}^{\ell}(x/\varepsilon)\cdot\nabla\psi\,dx-\int_{\Omega_{\varepsilon}^{\ell}}\psi_{j}\,dx\Big{|}$
(5.6) $\displaystyle\qquad\leq
C\varepsilon^{1/2}\left\\{\varepsilon\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}+\varepsilon^{1/2}\|\text{\rm
div}(\psi)\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\right\\}$
for any $\psi\in H^{1}(\Omega^{\ell}_{\varepsilon};\mathbb{R}^{d})$ with
$\psi=0$ on $\Gamma^{\ell}_{\varepsilon}$.
###### Theorem 5.1.
Let $(u_{\varepsilon},p_{\varepsilon})\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L^{2}_{0}(\Omega_{\varepsilon})$ be a weak solution of (1.1). Let
$V_{\varepsilon}$ be given by (5.3). Then
$\displaystyle\Big{|}\varepsilon^{2}\sum_{\ell=1}^{L}\int_{\Omega^{\ell}_{\varepsilon}}(\nabla
u_{\varepsilon}-\nabla
V_{\varepsilon})\cdot\nabla\psi\,dx-\int_{\Omega_{\varepsilon}}(p_{\varepsilon}-P_{0})\,\text{\rm
div}(\psi)\,dx\Big{|}$ (5.7) $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}\left\\{\varepsilon\|\nabla\psi\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon^{1/2}\|\text{\rm
div}(\psi)\|_{L^{2}(\Omega_{\varepsilon})}\right\\},$
for any $\psi\in H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$.
###### Proof.
We apply (5.6) with $\psi(f_{j}-\frac{\partial P_{0}}{\partial x_{j}})$ in the
place of $\psi$. Using
$\displaystyle|\varepsilon^{2}\nabla
V_{\varepsilon}\cdot\nabla\psi-\varepsilon\nabla
W^{\ell}(x/\varepsilon)\cdot\nabla\left(\psi(f-\nabla P_{0})\right)|$
$\displaystyle\leq
C\left\\{\varepsilon^{2}|W^{\ell}(x/\varepsilon)||\nabla\psi|+C\varepsilon|\nabla
W^{\ell}(x/\varepsilon)||\psi|\right\\}|\nabla(f-\nabla P_{0})|$
in $\Omega^{\ell}_{\varepsilon}$, we obtain
$\displaystyle\Big{|}\varepsilon^{2}\int_{\Omega_{\varepsilon}^{\ell}}\nabla
V_{\varepsilon}\cdot\nabla\psi\,dx-\int_{\Omega_{\varepsilon}^{\ell}}(f-\nabla
P_{0})\cdot\psi\,dx\Big{|}$ $\displaystyle\leq
C\varepsilon^{3/2}\left(\|f\|_{\infty}+\|\nabla f\|_{\infty}+\|\nabla
P_{0}\|_{\infty}+\|\nabla^{2}P_{0}\|_{\infty}\right)\|\nabla\psi\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}$
$\displaystyle\qquad+C\varepsilon(\|f\|_{\infty}+\|\nabla
P_{0}\|_{\infty})\|\text{\rm
div}(\psi)\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}.$
This, together with
$\int_{\Omega_{\varepsilon}}(f-\nabla
P_{0})\cdot\psi\,dx=\varepsilon^{2}\int_{\Omega_{\varepsilon}}\nabla
u_{\varepsilon}\cdot\nabla\psi\,dx-\int_{\Omega_{\varepsilon}}(p_{\varepsilon}-P_{0})\,\text{\rm
div}(\psi)\,dx,$
gives (5.7). ∎
Let
$U_{\varepsilon}=V_{\varepsilon}+\Phi_{\varepsilon},$ (5.8)
where $\Phi_{\varepsilon}$ is a corrector to be constructed so that
$U_{\varepsilon}\in H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$,
$\|\text{\rm div}(U_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},$ (5.9)
and that
$\varepsilon\|\nabla\Phi_{\varepsilon}\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}$ (5.10)
for $1\leq\ell\leq L$.
Assuming that such corrector $\Phi_{\varepsilon}$ exists, we give the proof of
Theorem 1.2.
###### Proof of Theorem 1.2.
By letting
$\psi=u_{\varepsilon}-U_{\varepsilon}=u_{\varepsilon}-V_{\varepsilon}-\Phi_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ in (5.7), we obtain
$\displaystyle\varepsilon^{2}\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq\varepsilon^{2}\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\|\nabla\Phi_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}\|\text{\rm
div}(U_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\qquad+C\varepsilon^{1/2}\
\|f\|_{C^{1,1/2}(\Omega)}\left\\{\varepsilon\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon\|\nabla\Phi_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\varepsilon^{1/2}\|\text{\rm
div}(U_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon})}\right\\}$
$\displaystyle\leq C\varepsilon^{3/2}\|f\|_{C^{1,1/2}(\Omega)}\|\nabla
u_{\varepsilon}-\nabla V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\qquad+C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}+C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}^{2},$
for any $\beta\in\mathbb{R}$, where we have used (5.9) and (5.10) for the last
inequality. By the Cauchy inequality, this implies that
$\varepsilon^{2}\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon\|f\|^{2}_{C^{1,1/2}(\Omega)}+C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}.$
(5.11)
We should point out that both $V_{\varepsilon}$ and $\Phi_{\varepsilon}$ are
not in $H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$. In the estimates above
(and thereafter) we have used the convention that
$\|\
\nabla\psi\|_{L^{2}(\Omega_{\varepsilon})}=\left(\sum_{\ell=1}^{L}\|\nabla\psi\|^{2}_{L^{2}(\Omega_{\varepsilon}^{\ell})}\right)^{1/2},$
where $\psi\in H^{1}(\Omega_{\varepsilon}^{\ell})$ for $1\leq\ell\leq L$.
Next, we choose $\beta=\fint_{\Omega_{\varepsilon}}(p_{\varepsilon}-P_{0})$.
By Lemma 2.6, there exists $v_{\varepsilon}\in
H^{1}_{0}(\Omega_{\varepsilon};\mathbb{R}^{d})$ such that
$\displaystyle\text{\rm div}(v_{\varepsilon})$
$\displaystyle=p_{\varepsilon}-P_{0}-\beta\quad\text{ in
}\Omega_{\varepsilon},$ $\displaystyle\varepsilon\|\nabla
v_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$ $\displaystyle\leq
C\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}.$
By letting $\psi_{\varepsilon}=v_{\varepsilon}$ in (5.7), we obtain
$\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$
(5.12)
By combining (5.11) with (5.12), it is not hard to see that
$\varepsilon\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$ (5.13)
This, together with
$\|u_{\varepsilon}-V_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon\|\nabla u_{\varepsilon}-\nabla
V_{\varepsilon}\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}$, gives the bound for
the first term in (1.13). Also, note that
$\|\varepsilon\nabla V_{\varepsilon}-\nabla W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon\|\nabla(f-\nabla P_{0})\|_{\infty}.$
Thus,
$\|\varepsilon\nabla u_{\varepsilon}-\nabla W^{\ell}(x/\varepsilon)(f-\nabla
P_{0})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$
Finally, to estimate the pressure, we let $Q_{\varepsilon}$ be the extension
of $(P_{0}+\beta)|_{\Omega_{\varepsilon}}$ to $\Omega$, using the formula in
(2.21). Note that
$\|Q_{\varepsilon}-(P_{0}+\beta)\|^{2}_{L^{2}(\Omega)}=\sum_{\ell,z}\int_{\varepsilon(Y_{s}^{\ell}+z)}\Big{|}P_{0}-\fint_{\varepsilon(Y_{f}^{\ell}+z)}P_{0}\Big{|}^{2}\,dx,$
where the sum is taken over those $(\ell,z)$’s for which $z\in\mathbb{Z}^{d}$
and $\varepsilon(Y+z)\subset\Omega^{\ell}$. It follows that
$\displaystyle\|Q_{\varepsilon}-(P_{0}+\beta)\|_{L^{2}(\Omega)}$
$\displaystyle\leq C\varepsilon\|\nabla P_{0}\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}.$
As a result, by (5.13), we obtain
$\displaystyle\|P_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|P_{\varepsilon}-Q_{\varepsilon}\|_{L^{2}(\Omega)}+\|Q_{\varepsilon}-(P_{0}+\beta)\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|p_{\varepsilon}-P_{0}-\beta\|_{L^{2}(\Omega_{\varepsilon})}+C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}$
$\displaystyle\leq C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},$
where $\beta=-\fint_{\Omega_{\varepsilon}}P_{0}$. Clearly, we may replace
$\beta$ by
$\fint_{\Omega}(P_{\varepsilon}-P_{0})=\fint_{\Omega}P_{\varepsilon}$. This
gives the bound for the second term in (1.13). ∎
To complete the proof of Theorem 1.2, it remains to construct a corrector
$\Phi_{\varepsilon}$ such that $V_{\varepsilon}+\Phi_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ and (5.9)-(5.10) hold. This
will be done in the next three sections. More precisely, we let
$\Phi_{\varepsilon}=\Phi_{\varepsilon}^{(1)}+\Phi_{\varepsilon}^{(2)}+\Phi_{\varepsilon}^{(3)},$
(5.14)
where $\Phi_{\varepsilon}^{(1)}$ is a corrector for the divergence operator
with the properties that
$\left\\{\begin{aligned} &\Phi_{\varepsilon}^{(1)}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d}),\\\
&\varepsilon\|\nabla\Phi_{\varepsilon}^{(1)}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},\\\ &\|\text{\rm
div}(\Phi_{\varepsilon}^{(1)}+V_{\varepsilon})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},\end{aligned}\right.$ (5.15)
$\Phi_{\varepsilon}^{(2)}$ is a corrector for the boundary data of
$V_{\varepsilon}$ on $\partial\Omega$ with the properties that
$\left\\{\begin{aligned} &\Phi_{\varepsilon}^{(2)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\quad\text{ and
}\quad\Phi^{(2)}_{\varepsilon}=0\quad\text{ on }\Gamma_{\varepsilon},\\\
&\Phi_{\varepsilon}^{(2)}+V_{\varepsilon}=0\quad\text{ on }\partial\Omega,\\\
&\varepsilon\|\nabla\Phi_{\varepsilon}^{(2)}\|_{L^{2}(\Omega_{\varepsilon})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(2)})\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},\end{aligned}\right.$ (5.16)
and $\Phi_{\varepsilon}^{(3)}$ is a corrector for the interface $\Sigma$ with
the properties that
$\left\\{\begin{aligned} &\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega^{\ell}_{\varepsilon};\mathbb{R}^{d})\quad\text{ and
}\quad\Phi_{\varepsilon}^{(3)}=0\quad\text{ on
}\partial\Omega_{\varepsilon},\\\ &V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d}),\\\
&\varepsilon\|\nabla\Phi_{\varepsilon}^{(3)}\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(3)})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},\end{aligned}\right.$ (5.17)
for $1\leq\ell\leq L$. It is not hard to verify that the desired property
$V_{\varepsilon}+\Phi_{\varepsilon}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$ as well as the estimates
(5.9)-(5.10) follows from (5.15)-(5.17).
## 6 Correctors for the divergence operator
Let $V_{\varepsilon}$ be given by (5.3). Note that since
div$(W_{j}^{\ell}(x/\varepsilon))=0$ in $\mathbb{R}^{d}$,
$\text{\rm div}(V_{\varepsilon})=W^{\ell}(x/\varepsilon)\nabla(f-\nabla
P_{0})\quad\text{ in }\Omega^{\ell}_{\varepsilon}.$ (6.1)
In this section we construct a corrector $\Phi_{\varepsilon}^{(1)}$ that
satisfies (5.15). The approach is similar to that used in [11, 15].
For $1\leq\ell\leq L$ and $1\leq\,i,j\leq d$, let
$\Theta_{ij}^{\ell}=(\Theta^{\ell}_{1ij},\dots,\Theta^{\ell}_{dij})$ be a
1-periodic function in $H^{1}_{loc}(\mathbb{R}^{d};\mathbb{R}^{d})$ such that
$\left\\{\begin{aligned} \text{\rm
div}(\Theta^{\ell}_{ij})&=-W^{\ell}_{ij}+|Y_{f}|^{-1}K^{\ell}_{ij}&\quad&\text{
in }Y_{f},\\\ \Theta_{ij}^{\ell}&=0&\quad&\text{ in
}Y_{s}.\end{aligned}\right.$ (6.2)
Fix $\varphi\in C_{0}^{\infty}(B(0,1/8))$ such that $\varphi\geq 0$ and
$\int_{\mathbb{R}^{d}}\varphi\,dx=1$. Define
$S_{\varepsilon}(\psi)(x)=\psi*\varphi_{\varepsilon}(x)=\int_{\mathbb{R}^{d}}\psi(y)\varphi_{\varepsilon}(x-y)\,dy,$
(6.3)
where $\varphi_{\varepsilon}(x)=\varepsilon^{-d}\varphi(x)$. Let
$\Phi_{\varepsilon}^{(1)}=(\Phi_{\varepsilon,1}^{(1)},\dots,\Phi_{\varepsilon,d}^{(1)})$,
where, for $x\in\Omega_{\varepsilon}^{\ell}$,
$\Phi_{\varepsilon,k}^{(1)}(x)=\varepsilon\eta^{\ell}_{\varepsilon}(x)\Theta^{\ell}_{kij}(x/\varepsilon)\frac{\partial}{\partial
x_{i}}S_{\varepsilon}\left(f_{j}-\frac{\partial P_{0}}{\partial
x_{j}}\right),$ (6.4)
and $P_{0}$ is the solution of (5.1). The function $\eta_{\varepsilon}^{\ell}$
in (6.4) is a cut-off function in $C_{0}^{\infty}(\Omega^{\ell})$ with the
properties that $|\nabla\eta_{\varepsilon}^{\ell}|\leq C\varepsilon^{-1}$ and
$\left\\{\begin{aligned} &\eta_{\varepsilon}^{\ell}(x)=0&\quad&\text{ if
dist}(x,\partial\Omega^{\ell})\leq 2d\varepsilon,\\\
&\eta^{\ell}_{\varepsilon}(x)=1&\quad&\text{ if }x\in\Omega^{\ell}\text{ and
dist}(x,\partial\Omega^{\ell})\geq 3d\varepsilon.\end{aligned}\right.$
As a result, $\Phi_{\varepsilon}^{(1)}$ vanishes near $\partial\Omega^{\ell}$.
###### Theorem 6.1.
Let $\Phi_{\varepsilon}^{(1)}$ be defined by (6.4). Then (5.15) holds.
###### Proof.
Clearly, $\Phi_{\varepsilon}^{(1)}\in
H_{0}^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$. Note that
$\displaystyle\|\nabla\Phi_{\varepsilon}^{(1)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq C\varepsilon^{1/2}\|\nabla S_{\varepsilon}(f-\nabla
P_{0})\|_{L^{\infty}(N_{3d\varepsilon}\setminus N_{2d\varepsilon})}+C\|\nabla
S_{\varepsilon}(f-\nabla P_{0})\|_{L^{\infty}(\Omega^{\ell}\setminus
N_{2d\varepsilon})}$
$\displaystyle\qquad\qquad+C\varepsilon\|\nabla^{2}S_{\varepsilon}(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell}\setminus N_{2d\varepsilon})},$
where $N_{r}=\\{x\in\Omega^{\ell}:\text{\rm
dist}(x,\partial\Omega^{\ell})<r\\}$. This, together with the observation that
$\nabla S_{\varepsilon}(\psi)=S_{\varepsilon}(\nabla\psi)$ and
$|S_{\varepsilon}(\psi)(x)|+\varepsilon|\nabla S_{\varepsilon}(\psi)(x)|\leq
C\fint_{B(x,\varepsilon/8)}|\psi|,$
yields
$\displaystyle\varepsilon\|\nabla\Phi_{\varepsilon}^{(1)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq C\varepsilon\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell})}$ $\displaystyle\leq
C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}.$
Next, note that in $\Omega^{\ell}_{\varepsilon}$,
$\displaystyle\text{\rm div}(\Phi_{\varepsilon}^{(1)})$
$\displaystyle=\varepsilon(\nabla\eta_{\varepsilon}^{\ell})\Theta^{\ell}(x/\varepsilon)\nabla
S_{\varepsilon}(f-\nabla
P_{0})-\eta_{\varepsilon}^{\ell}W^{\ell}(x/\varepsilon)\nabla
S_{\varepsilon}(f-\nabla P_{0})$
$\displaystyle\qquad\qquad+\varepsilon\eta_{\varepsilon}^{\ell}\Theta^{\ell}(x/\varepsilon)\nabla^{2}S_{\varepsilon}(f-\nabla
P_{0}),$
where we have used the fact that $\text{\rm div}(K^{\ell}(f-\nabla P_{0}))=0$
in $\Omega_{\varepsilon}^{\ell}$. It follows that
$\displaystyle\|\text{\rm
div}(\Phi_{\varepsilon}^{(1)})+W^{\ell}(x/\varepsilon)\nabla(f-\nabla
P_{0})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$ $\displaystyle\leq
C\varepsilon^{1/2}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell})}+\|W^{\ell}(x/\varepsilon)\left\\{\nabla(f-\nabla
P_{0})-\eta_{\varepsilon}^{\ell}\nabla S_{\varepsilon}(f-\nabla
P_{0})\right\\}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\qquad\qquad+C\varepsilon\|\nabla^{2}S_{\varepsilon}(f-\nabla
P_{0})\|_{L^{\infty}(\Omega_{\varepsilon}^{\ell}\setminus N_{2d\varepsilon})}$
$\displaystyle\leq C\varepsilon^{1/2}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell})}+\|\nabla(f-\nabla P_{0})-\nabla
S_{\varepsilon}(f-\nabla P_{0})\|_{L^{\infty}(\Omega^{\ell}\setminus
N_{2d\varepsilon})}$
$\displaystyle\qquad\qquad+C\varepsilon\|\nabla^{2}S_{\varepsilon}(f-\nabla
P_{0})\|_{L^{\infty}(\Omega^{\ell}\setminus N_{2d\varepsilon})}$
$\displaystyle\leq C\varepsilon^{1/2}\|\nabla(f-\nabla
P_{0})\|_{C^{1/2}(\Omega^{\ell})}$ $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},$
where we have used (5.2) for the last inequality. In the third inequality
above we also used the observation that
$\nabla
S_{\varepsilon}(\psi)(x)=-\int_{\mathbb{R}^{d}}\left(\psi(x-y)-\psi(x)\right)\nabla_{y}(\varphi_{\varepsilon}(y))\,dy,$
which gives
$|\nabla S_{\varepsilon}(\psi)(x)|\leq
C\varepsilon^{\alpha-1}\|\psi\|_{C^{0,\alpha}(B(x,\varepsilon))}.$
This completes the proof of (5.15). ∎
## 7 Boundary correctors
To construct the boundary corrector $\Phi_{\varepsilon}^{(2)}$, we consider
the Dirichlet problem,
$\left\\{\begin{aligned} -\varepsilon^{2}\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=0&\quad&\text{ in }\Omega_{\varepsilon}\\\ \text{\rm
div}(u_{\varepsilon})&=\gamma&\quad&\text{ in }\Omega_{\varepsilon},\\\
u_{\varepsilon}&=0&\quad&\text{ on }\Gamma_{\varepsilon},\\\
u_{\varepsilon}&=h&\quad&\text{ on }\partial\Omega,\end{aligned}\right.$ (7.1)
where $\Omega_{\varepsilon}$ is given by (1.4) and
$\gamma=\frac{1}{|\Omega_{\varepsilon}|}\int_{\partial\Omega}h\cdot
n\,d\sigma.$ (7.2)
Let $\Phi_{\varepsilon}^{(2)}\in H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$
be the solution of (7.1) with boundary value,
$h=-V_{\varepsilon}\quad\text{ on }\partial\Omega,$ (7.3)
where $V_{\varepsilon}$ is given by (5.3). Thus, if
$\partial\Omega\cap\partial\Omega^{\ell}\neq\emptyset$ for some $1\leq\ell\leq
L$,
$\Phi_{\varepsilon}^{(2)}=-W^{\ell}(x/\varepsilon)(f-\nabla P_{0})\quad\text{
on }\partial\Omega\cap\partial\Omega^{\ell}.$ (7.4)
###### Theorem 7.1.
Let $\Phi_{\varepsilon}^{(2)}$ be defined as above. Then
$\Phi_{\varepsilon}^{(2)}$ satisfies (5.16).
To show Theorem 7.1, we first prove some general results, which will be used
also in the construction of correctors for the interface.
###### Theorem 7.2.
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$, $d\geq 2$.
Assume that $\Omega^{\ell}$ and $Y_{s}^{\ell}$ with $1\leq\ell\leq L$ are
subdomains of $\Omega$ and $Y$, respectively, with Lipschitz boundaries. Let
$(u_{\varepsilon},p_{\varepsilon})$ be a weak solution in
$H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times
L_{0}^{2}(\Omega_{\varepsilon})$ of (7.1), where $h\in
H^{1}(\partial\Omega;\mathbb{R}^{d})$ and
$u\cdot n=0\quad\text{ on }\partial\Omega.$ (7.5)
Then
$\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\sqrt{\varepsilon}\left\\{\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\nabla_{\tan}h\|_{L^{2}(\partial\Omega)}\right\\},$
(7.6)
where $\nabla_{\tan}h$ denotes the tangential gradient of $h$ on
$\partial\Omega$.
###### Proof.
This theorem was proved in [15, Theorem 4.1] for the case $L=1$. The proof
only uses the energy estimate (3.6) and the fact that
$-\varepsilon^{2}\Delta u_{\varepsilon}+\nabla p_{\varepsilon}=0\quad\text{
and }\quad\text{\rm div}(u_{\varepsilon})=0$
in the set $\\{x\in\Omega:\ \text{\rm
dist}(x,\partial\Omega)<c\,\varepsilon\\}$. As a result, the same proof works
equally well for the case $L\geq 2$. We mention that the argument relies on
the Rellich estimates in [7] for the Stokes equations in Lipschitz domains.
The condition (7.5) allows us to drop the pressure $p_{\varepsilon}$ term in
the conormal derivative $\partial u_{\varepsilon}/{\partial\nu}$ for
$u_{\varepsilon}$ on $\partial\Omega$. We omit the details. ∎
In the next theorem we consider the case where
$h\cdot n=\varepsilon\ (\nabla_{\tan}\phi_{\varepsilon})\cdot g)\quad\text{ on
}\partial\Omega.$ (7.7)
By using integration by parts on $\partial\Omega$, we see that
$\displaystyle|\gamma|$ $\displaystyle\leq C\Big{|}\int_{\partial\Omega}h\cdot
n\,d\sigma\Big{|}$ (7.8) $\displaystyle\leq
C\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}.$
###### Theorem 7.3.
Let $\Omega$ be a bounded $C^{2,\alpha}$ domain in $\mathbb{R}^{d}$, $d\geq
2$. Let $(u_{\varepsilon},p_{\varepsilon})$ be a weak solution in
$H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})\times L_{0}^{2}(\Omega)$ of (7.1),
where $h\in H^{1}(\partial\Omega)$ and $h\cdot n$ is given (7.7). Then
$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
(7.9) $\displaystyle\qquad\leq
C\sqrt{\varepsilon}\left\\{\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\nabla_{\tan}h\|_{L^{2}(\partial\Omega)}+\|\phi_{\varepsilon}g\|_{L^{2}(\partial\Omega)}+\varepsilon^{1/2}\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$
###### Proof.
A version of this theorem was proved in [15, Theorem 5.1] for the case $L=1$.
We give the proof for the general case, using a somewhat different argument.
We first note that by writing
$h=(h-(h\cdot n)n)+(h\cdot n)n$
and applying Theorem 7.2 to the solution of (7.1) with boundary data
$h-(h\cdot n)n$, we may reduce the problem to case where $h=(h\cdot n)n$ on
$\partial\Omega$.
Next, by the energy estimate (3.3) and (7.8),
$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}$
(7.10) $\displaystyle\qquad\leq C\left\\{\|H\|_{L^{2}(\Omega)}+\|\text{\rm
div}(H)\|_{L^{2}(\Omega)}+\varepsilon\|\nabla
H\|_{L^{2}(\Omega)}+\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\},$
where $H$ is any function in $H^{1}(\Omega;\mathbb{R}^{d})$ with $H=h$ on
$\partial\Omega$. We choose $H=H_{1}+\gamma(x-x_{0})/d$, where
$x_{0}\in\Omega$ and $H_{1}$ is the weak solution of
$-\Delta H_{1}+\nabla q=0\quad\text{ and }\quad\text{\rm
div}(H_{1})=0\quad\text{ in }\Omega,$
with the boundary value $H_{1}=h-\gamma(x-x_{0})/d$ on $\partial\Omega$. It
follows that
$\displaystyle\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(\Omega)}+\|u_{\varepsilon}\|_{L^{2}(\partial\Omega)}+\|p_{\varepsilon}\|_{L^{2}(\Omega)}$
(7.11) $\displaystyle\qquad\leq
C\left\\{\|H_{1}\|_{L^{2}(\Omega)}+\varepsilon\|\nabla
H_{1}\|_{L^{2}(\Omega)}+\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\},$
where we have used (7.8). By the energy estimates for the Stokes equations in
$\Omega$,
$\displaystyle\|\nabla H_{1}\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C\left\\{\|h\|_{H^{1/2}(\partial\Omega)}+|\gamma|\right\\}$ $\displaystyle\leq
C\left\\{\|h\|_{L^{2}(\partial\Omega)}^{1/2}\|h\|^{1/2}_{H^{1}(\partial\Omega)}+|\gamma|\right\\}$
$\displaystyle\leq
C\left\\{\varepsilon^{-1/2}\|h\|_{L^{2}(\partial\Omega)}+\varepsilon^{1/2}\|\nabla_{\tan}h\|_{L^{2}(\partial\Omega)}+|\gamma|\right\\}.$
It follows that
$\varepsilon\|\nabla H_{1}\|_{L^{2}(\Omega)}\leq
C\sqrt{\varepsilon}\left\\{\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\nabla_{\tan}h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$
(7.12)
To bound $\|H_{1}\|_{L^{2}(\Omega)}$, we use the following nontangential-
maximal-function estimate,
$\|(H_{1})^{*}\|_{L^{2}(\partial\Omega)}\leq
C\|H_{1}\|_{L^{2}(\partial\Omega)},$ (7.13)
where the nontangential maximal function $(H_{1})^{*}$ on $\partial\Omega$ is
defined by
$(H_{1})^{*}(x)=\sup\big{\\{}|H_{1}(y)|:\ y\in\Omega\text{ and
}|y-x|<C_{0}\,\text{dist}(y,\partial\Omega)\big{\\}}$
for $x\in\partial\Omega$. The estimate (7.13) was proved in [7] for a bounded
Lipschitz domain $\Omega$. Let
$N_{r}=\big{\\{}x\in\Omega:\ \text{dist}(x,\partial\Omega)<r\big{\\}}.$
It follows from (7.13) that
$\displaystyle\|H_{1}\|_{L^{2}(N_{\varepsilon})}$ $\displaystyle\leq
C\sqrt{\varepsilon}\|(H_{1})^{*}\|_{L^{2}(\partial\Omega)}$ (7.14)
$\displaystyle\leq
C\sqrt{\varepsilon}\left\\{\|h\|_{L^{2}(\partial\Omega)}+\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$
It remains to bound $\|H_{1}\|_{L^{2}(\Omega\setminus N_{\varepsilon})}$. To
this end, we consider the Dirichlet problem,
$\left\\{\begin{aligned} -\Delta G+\nabla\pi&=F&\quad&\text{ in }\Omega,\\\
\text{\rm div}(G)&=0&\quad&\text{ in }\Omega,\\\ G&=0&\quad&\text{ on
}\partial\Omega,\end{aligned}\right.$
where $F\in C_{0}^{\infty}(\Omega\setminus N_{\varepsilon})$ and
$\int_{\Omega}\pi\,dx=0$. Under the assumption that $\partial\Omega$ is of
$C^{2,\alpha}$, we have the $W^{2,2}$ estimates,
$\|G\|_{H^{2}(\Omega)}+\|\pi\|_{H^{1}(\Omega)}\leq C\|F\|_{L^{2}(\Omega)}.$
(7.15)
This implies that
$\|\nabla G\|_{L^{2}(\partial\Omega)}+\|\pi\|_{L^{2}(\partial\Omega)}\leq
C\|F\|_{L^{2}(\Omega)}.$ (7.16)
Moreover, since $F=0$ in $N_{\varepsilon}$, by covering $\partial\Omega$ with
balls of radius $c\varepsilon$, one may show that
$\displaystyle\int_{\partial\Omega}\left(|\nabla^{2}G|^{2}+|\nabla\pi|^{2}\right)\,d\sigma$
$\displaystyle\leq C\varepsilon^{-1}\|F\|_{L^{2}(\Omega)}^{2}.$ (7.17)
To see this, we use the Green function representation for $G$ to obtain
$|\nabla^{2}G(x)|\leq C\int_{\Omega\setminus
N_{\varepsilon}}\frac{|F(y)|}{|x-y|^{d}}\,dy$ (7.18)
for $x\in\partial\Omega$. See e.g. [8] for estimates of Green functions for
the Stokes equations. Choose $\alpha,\beta\in(0,1)$ such that
$\alpha+\beta=1$, $\alpha>(1/2)$ and $\beta>(1/2)-(1/2d)$. It follows by the
Cauchy inequality that for $x\in\partial\Omega$,
$\displaystyle|\nabla^{2}G(x)|^{2}$ $\displaystyle\leq
C\left(\int_{\Omega\setminus
N_{\varepsilon}}\frac{dy}{|x-y|^{2d\alpha}}\right)\left(\int_{\Omega\setminus
N_{\varepsilon}}\frac{|F(y)|^{2}}{|x-y|^{2d\beta}}\,dy\right)$
$\displaystyle\leq C\varepsilon^{d-2d\alpha}\int_{\Omega\setminus
N_{\varepsilon}}\frac{|F(y)|^{2}}{|x-y|^{2d\beta}}\,dy,$
where we have used the conditions $\alpha+\beta=1$ and $\alpha>(1/2)$. Hence,
$\displaystyle\int_{\partial\Omega}|\nabla^{2}G|^{2}\,d\sigma$
$\displaystyle\leq C\varepsilon^{d-2d\alpha}\int_{\Omega\setminus
N_{\varepsilon}}|F(y)|^{2}\,dy\sup_{y\in\Omega\setminus
N_{\varepsilon}}\int_{\partial\Omega}\frac{d\sigma(x)}{|x-y|^{2d\beta}}$
$\displaystyle\leq C\varepsilon^{-1}\int_{\Omega}|F(y)|^{2}\,dy,$
where we have used the condition $\beta>(1/2)-(1/2d)$. This gives the estimate
for $|\nabla^{2}G|$ in (7.17). The estimate for $\nabla\pi$ follows from the
equation $-\Delta G+\nabla\pi=0$ near $\partial\Omega$.
Finally, using integration by parts, we see that
$\displaystyle\int_{\Omega}H_{1}\cdot F\,dx$
$\displaystyle=\int_{\Omega}H_{1}\cdot(-\Delta G+\nabla\pi)\,dx$
$\displaystyle=-\int_{\partial\Omega}H_{1}\cdot\Big{(}\frac{\partial
G}{\partial n}-n\pi\Big{)}\,d\sigma$
$\displaystyle=-\int_{\partial\Omega}\Big{(}\varepsilon((\nabla_{\tan}\phi_{\varepsilon})\cdot
g)n-\gamma(x-x_{0})/d\Big{)}\cdot\Big{(}\frac{\partial G}{\partial
n}-n\pi\Big{)}\,d\sigma.$
It follows by using integration by parts on $\partial\Omega$ that
$\displaystyle\Big{|}\int_{\Omega}H_{1}\cdot F\,dx\Big{|}$ $\displaystyle\leq
C\varepsilon\int_{\partial\Omega}|\phi_{\varepsilon}|\left(|\nabla g||\nabla
G|+|g||\nabla^{2}G|+|g||\nabla G|+|\nabla
g||\pi|+|g||\nabla\pi|+|g||\pi|\right)\,d\sigma$
$\displaystyle\qquad\qquad+|\gamma|\int_{\partial\Omega}\left(|\nabla
G|+|\pi|\right)\,d\sigma$ $\displaystyle\leq
C\varepsilon\|\phi_{\varepsilon}g\|_{L^{2}(\partial\Omega)}\left\\{\|\nabla^{2}G\|_{L^{2}(\partial\Omega)}+\|\nabla
G\|_{L^{2}(\partial\Omega)}+\|\nabla\pi\|_{L^{2}(\partial\Omega)}+\|\pi\|_{L^{2}(\partial\Omega)}\right\\}$
$\displaystyle\qquad\qquad+C\varepsilon\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\left\\{\|\nabla
G\|_{L^{2}(\partial\Omega)}+\|\pi\|_{L^{2}(\partial\Omega)}\right\\},$
where we have used the Cauchy inequality and (7.8). This, together with (7.16)
and (7.17), gives
$\Big{|}\int_{\Omega}H_{1}\cdot F\,dx\Big{|}\leq
C\varepsilon^{1/2}\|F\|_{L^{2}(\Omega)}\left\\{\|\phi_{\varepsilon}g\|_{L^{2}(\partial\Omega)}+\varepsilon^{1/2}\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$
By duality we obtain
$\|H_{1}\|_{L^{2}(\Omega\setminus N_{\varepsilon})}\leq
C\varepsilon^{1/2}\left\\{\|\phi_{\varepsilon}g\|_{L^{2}(\partial\Omega)}+\varepsilon^{1/2}\|\phi_{\varepsilon}\nabla_{\tan}g\|_{L^{2}(\partial\Omega)}\right\\}.$
(7.19)
The desired estimate (7.9) follows from (7.10), (7.12), (7.14) and (7.19). ∎
###### Proof of Theorem 7.1.
Clearly, by its definition, $\Phi_{\varepsilon}^{(2)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, $\Phi_{\varepsilon}^{(2)}=0$ on
$\Gamma_{\varepsilon}$, and $\Phi_{\varepsilon}^{(2)}+V_{\varepsilon}=0$ on
$\partial\Omega$. Using the fact that $n\cdot K^{\ell}(f-\nabla P_{0})=0$ on
$\partial\Omega\cap\partial\Omega^{\ell}$, we obtain
$\displaystyle n\cdot h$ $\displaystyle=-n\cdot
W^{\ell}(x/\varepsilon)(f-\nabla P_{0})$ (7.20)
$\displaystyle=-n\cdot(W^{\ell}(x/\varepsilon)-K^{\ell})(f-\nabla P_{0})$
$\displaystyle=-\frac{\varepsilon}{2}\left(n_{i}\frac{\partial}{\partial
x_{k}}-n_{k}\frac{\partial}{\partial
x_{i}}\right)\left(\phi^{\ell}_{kij}(x/\varepsilon)\right)\left(f_{j}-\frac{\partial
P_{0}}{\partial x_{j}}\right)$
on $\partial\Omega\cap\partial\Omega^{\ell}$. It follows that
$\Big{|}\int_{\partial\Omega}n\cdot h\,d\sigma\Big{|}\leq
C\varepsilon\|\nabla(f-\nabla P_{0})\|_{L^{\infty}(\partial\Omega)}.$
Hence,
$\displaystyle\|\text{\rm
div}(\Phi_{\varepsilon}^{(2))}\|_{L^{2}(\Omega_{\varepsilon})}$
$\displaystyle\leq C|\gamma|\leq C\varepsilon\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\partial\Omega)}$ $\displaystyle\leq
C\varepsilon\|f\|_{C^{1,1/2}(\Omega)}.$
Finally, in view of (7.20), we apply Theorem 7.3 to obtain
$\displaystyle\varepsilon\|\nabla\Phi_{\varepsilon}^{(2)}\|_{L^{2}(\Omega)}$
$\displaystyle\leq C\varepsilon^{1/2}\left\\{\|f-\nabla
P_{0}\|_{L^{\infty}(\partial\Omega)}+\varepsilon^{1/2}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\partial\Omega)}\right\\}$ $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$
∎
## 8 Interface correctors
In this section we construct a corrector $\Phi_{\varepsilon}^{(3)}$ for the
interface $\Sigma$ and thus completes the proof of Theorem 1.2. Let
$D=\Omega^{\ell}$ and $D_{\varepsilon}=\Omega^{\ell}_{\varepsilon}$ for some
$1\leq\ell\leq L$. Assume that $\partial D$ has no intersection with the
boundary of the unbounded connected component of
$\mathbb{R}^{d}\setminus\overline{\Omega}$. Consider the Dirichlet problem,
$\left\\{\begin{aligned} -\Delta u_{\varepsilon}+\nabla
p_{\varepsilon}&=0&\quad&\text{ in }D_{\varepsilon},\\\ \text{\rm
div}(u_{\varepsilon})&=\gamma&\quad&\text{ in }D_{\varepsilon},\\\
u_{\varepsilon}&=0&\quad&\text{ on }\Gamma_{\varepsilon}^{\ell},\\\
u_{\varepsilon}&=h&\quad&\text{ on }\partial D,\end{aligned}\right.$ (8.1)
where $\Gamma_{\varepsilon}^{\ell}=\Gamma_{\varepsilon}\cap D$ and
$\gamma=\frac{1}{|D_{\varepsilon}|}\int_{\partial D}h\cdot n\,d\sigma.$
Let $W^{+}(y)=W^{\ell}(y)$. Fix $1\leq j\leq d$. The boundary data $h$ on
$\partial D$ in (8.1) is given as follows. Let $\partial
D=\cup_{k=1}^{k_{0}}\Sigma^{k}$, where $\Sigma^{k}$ are the connected
component of $\partial D$. On each $\Sigma^{k}$, either
$h=0\quad$ (8.2)
or
$h=W^{-}_{j}(x/\varepsilon)-W^{+}_{j}(x/\varepsilon)-W^{-}_{i}(x/\varepsilon)(K^{-}_{mj}-K^{+}_{mj})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle},$ (8.3)
where $W^{-}(y)$ denotes the 1-periodic matrix defined by (2.1) for the
subdomain on the other side of $\Sigma^{k}$, and
$K^{+}=\int_{Y}W^{+}(y)\,dy,\qquad K^{-}=\int_{Y}W^{-}(y)\,dy.$
In particular, if $\Sigma^{k}\subset\partial\Omega$, we let $h=0$ on
$\Sigma^{k}$. Note that the repeated indices $i,m$ in (8.3) are summed from
$1$ to $d$.
###### Lemma 8.1.
Let $D$ be a bounded $C^{2,\alpha}$ domain in $\mathbb{R}^{d}$, $d\geq 2$. Let
$(u_{\varepsilon},p_{\varepsilon})$ be a weak solution of (8.1) with
$\int_{D_{\varepsilon}}p_{\varepsilon}\,dx=0$, where $h$ is given by
(8.2)-(8.3). Then
$\varepsilon\|\nabla
u_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}+\|u_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}+\|p_{\varepsilon}\|_{L^{2}(D_{\varepsilon})}\leq
C\sqrt{\varepsilon},$ (8.4)
and
$\|\text{\rm div}(u_{\varepsilon})\|_{L^{2}(D_{\varepsilon})}\leq
C\varepsilon.$ (8.5)
###### Proof.
We apply Theorem 7.3 with $\Omega=D$ to establish (8.4). First, observe that
by (2.4),
$\|h\|_{L^{2}(\partial D)}+\varepsilon\|\nabla_{\tan}h\|_{L^{2}(\partial
D)}\leq C.$ (8.6)
Next, we compute $u\cdot n$ on $\Sigma^{k}$, assuming $h$ is given by (8.3).
Note that
$\displaystyle h\cdot n$
$\displaystyle=n_{t}W_{tj}^{-}(x/\varepsilon)-n_{t}W^{+}_{tj}(x/\varepsilon)-n_{t}W^{-}_{ti}(x/\varepsilon)(K_{mj}^{-}-K_{mj}^{+})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle}$ (8.7)
$\displaystyle=n_{t}\left(W_{tj}^{-}(x/\varepsilon)-K_{tj}^{-}\right)-n_{t}\left(W^{+}_{tj}(x/\varepsilon)-K_{tj}^{+}\right)$
$\displaystyle\qquad\qquad-
n_{t}\left(W^{-}_{ti}(x/\varepsilon)-K_{ti}^{-}\right)(K_{mj}^{-}-K_{mj}^{+})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle},$
where the repeated indices $t,i,m$ are summed from $1$ to $d$. We use Lemma
2.2 to write
$n_{t}\left(W_{ti}^{\pm}(x/\varepsilon)-K_{ti}^{\pm}\right)=\frac{\varepsilon}{2}\left(n_{t}\frac{\partial}{\partial
x_{s}}-n_{s}\frac{\partial}{\partial
x_{t}}\right)\left(\phi^{\pm}_{sti}(x/\varepsilon)\right).$ (8.8)
As a result, the function in the right-hand side of (8.7) may be written in
the form $\varepsilon(\nabla_{\tan}\phi_{\varepsilon})\cdot g$ with
$(\phi_{\varepsilon},g)$ satisfying
$\|\phi_{\varepsilon}\|_{L^{2}(\partial
D)}+\|g\|_{\infty}+\|\nabla_{\tan}g\|_{\infty}\leq C.$
Consequently, the estimate (8.4) follows from (7.9) in Theorem 7.3. Finally,
note that (8.7) and (8.8) yield
$\displaystyle\|\text{\rm div}(u_{\varepsilon})\|_{L^{2}(D_{\varepsilon})}$
$\displaystyle\leq C\Big{|}\int_{\partial D}h\cdot n\,d\sigma\Big{|}$
$\displaystyle\leq C\varepsilon.$
∎
Define
$\Phi^{(3)}_{\varepsilon}=\sum_{\ell=1}^{L}I_{\varepsilon}^{\ell}(x)(f-\nabla
P_{0})\chi_{\Omega_{\varepsilon}^{\ell}}\quad\text{ in }\Omega_{\varepsilon},$
(8.9)
where
$I^{\ell}_{\varepsilon}=(I_{\varepsilon,1}^{\ell},\dots,I_{\varepsilon,d}^{\ell})$
is a solution of (8.1) in $D_{\varepsilon}=\Omega^{\ell}_{\varepsilon}$ with
$h$ given by (8.2)-(8.3). To fix the boundary value $h$ for each subdomain, we
assume that the unbounded connected component of
$\mathbb{R}^{d}\setminus\overline{\Omega}$ shares boundary with $\Omega^{1}$,
and let $h=0$ on $\partial\Omega^{1}$. Thus, $I^{1}_{\varepsilon}(x)=0$ and
$\Phi_{\varepsilon}^{(3)}=0$ in $\Omega^{1}$. Next, for each subdomain
$\Omega^{\ell}$ that shares boundaries with $\partial\Omega^{1}$, we use the
boundary data (8.3) for the common boundary with $\partial\Omega^{1}$ and let
$h=0$ on other components of $\partial\Omega^{\ell}$. We continue this
process. More precisely, at each step, we use (8.3) on the connected component
$\Sigma^{k}$ of $\partial\Omega^{\ell}$ if $\Sigma^{k}$ is also the connected
component of the boundary of a subdomain considered in the previous step, and
let $h=0$ on the remaining components. We point out that at each interface
$\Sigma^{k}$, the nonzero data (8.3) is used only once. Also, $h=0$ on
$\partial\Omega$.
###### Lemma 8.2.
Let $\Phi^{(3)}_{\varepsilon}$ be given by (8.9) with $f\in
C^{1,1/2}({\Omega;\mathbb{R}^{d}})$. Then
$V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$.
###### Proof.
Let $\Psi_{\varepsilon}=V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}$. Since $f\in
C^{1,1/2}(\Omega)$ implies that $\nabla^{2}P_{0}$ is bounded in each
subdomain, it follows that $\Psi_{\varepsilon}\in
H^{1}(\Omega_{\varepsilon}^{\ell};\mathbb{R}^{d})$ for $1\leq\ell\leq L$.
Thus, to show $\Psi_{\varepsilon}\in
H^{1}(\Omega_{\varepsilon};\mathbb{R}^{d})$, it suffices to show that the
trace of $\Psi_{\varepsilon}$ is continuous across each interface
$\Sigma^{k}$.
Suppose that $\Sigma^{k}$ is the common boundary of subdomains $\Omega^{+}$
and $\Omega^{-}$. Let $\Psi_{\varepsilon}^{\pm}$ denote the trace of
$\Psi_{\varepsilon}$ on $\Sigma^{k}$, taken from $\Omega^{\pm}$ respectively.
Recall that in the definition of $\\{I_{\varepsilon}^{\ell}\\}$, the non-zero
data (8.3) is used once on each interface. Assume that the non-zero data on
$\Sigma^{k}$ is used for $\Omega^{+}$. Then
$\Psi_{\varepsilon}^{+}-\Psi_{\varepsilon}^{-}=\left(W^{+}(x/\varepsilon)+I^{+}_{\varepsilon}(x)\right)(f-\nabla
P_{0})^{+}-W^{-}(x/\varepsilon)(f-\nabla P_{0})^{-},$
where $I^{+}_{\varepsilon}$ is given by (8.3). It follows that
$\displaystyle\Psi_{\varepsilon}^{+}-\Psi_{\varepsilon}^{-}$
$\displaystyle=\left(W_{j}^{-}(x/\varepsilon)-W_{i}^{-}(x/\varepsilon)(K_{mj}^{-}-K_{mj}^{+})\frac{n_{i}n_{m}}{\langle
nK^{-},n\rangle}\right)\left(f_{j}-\frac{\partial P_{0}}{\partial
x_{j}}\right)^{+}$
$\displaystyle\qquad\qquad-W^{-}_{j}(x/\varepsilon)\left(f_{j}-\frac{\partial
P_{0}}{\partial x_{j}}\right)^{-}$
$\displaystyle=W_{j}^{-}(x/\varepsilon)\left\\{\left(\frac{\partial
P_{0}}{\partial x_{j}}\right)^{-}-\left(\frac{\partial P_{0}}{\partial
x_{j}}\right)^{+}-\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\left(f_{i}-\frac{\partial P_{0}}{\partial
x_{i}}\right)^{+}\right\\}$
$\displaystyle\qquad\qquad+W_{j}^{-}(x/\varepsilon)\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\left(f_{i}-\frac{\partial P_{0}}{\partial
x_{i}}\right)^{-},$
where we have used the observation that
$n_{m}K_{mi}^{+}\left(f_{i}-\frac{\partial P_{0}}{\partial
x_{i}}\right)^{+}=n_{m}K_{mi}^{-}\left(f_{i}-\frac{\partial P_{0}}{\partial
x_{i}}\right)^{-}$ (8.10)
on the interface. Thus,
$\displaystyle\Psi_{\varepsilon}^{+}-\Psi_{\varepsilon}^{-}$
$\displaystyle=W_{j}^{-}(x/\varepsilon)\left\\{\left(\frac{\partial
P_{0}}{\partial x_{j}}\right)^{-}-\left(\frac{\partial P_{0}}{\partial
x_{j}}\right)^{+}-\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\left(\left(\frac{\partial P_{0}}{\partial
x_{i}}\right)^{-}-\left(\frac{\partial P_{0}}{\partial
x_{i}}\right)^{+}\right)\right\\}$
$\displaystyle=W_{j}^{-}(x/\varepsilon)\left\\{\delta_{ij}-\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\right\\}\left(\left(\frac{\partial P_{0}}{\partial
x_{i}}\right)^{-}-\left(\frac{\partial P_{0}}{\partial
x_{i}}\right)^{+}\right).$
Since
$n_{i}\left\\{\delta_{ij}-\frac{n_{j}n_{m}}{\langle
nK^{-},n\rangle}K_{mi}^{-}\right\\}=0$
and $(\nabla_{\tan}P_{0})^{+}=(\nabla_{\tan}P_{0})^{-}$ on $\Sigma^{k}$, we
obtain $\Psi_{\varepsilon}^{+}=\Psi_{\varepsilon}^{-}$ on $\Sigma^{k}$. ∎
###### Theorem 8.3.
Let $\Phi_{\varepsilon}^{(3)}$ be defined by (8.9) with $f\in
C^{1,1/2}(\Omega;\mathbb{R}^{d})$. Then
$V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}\in H^{1}(\Omega;\mathbb{R}^{d})$ and
$\varepsilon\|\nabla\Phi_{\varepsilon}^{(3)}\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}+\|\text{\rm
div}(\Phi_{\varepsilon}^{(3)})\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}$ (8.11)
for $1\leq\ell\leq L$.
###### Proof.
By Lemma 8.2, we have $V_{\varepsilon}+\Phi_{\varepsilon}^{(3)}\in
H^{1}(\Omega;\mathbb{R}^{d})$. Note that by Lemma 8.1,
$\varepsilon\|\nabla
I_{\varepsilon}^{\ell}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+\|I_{\varepsilon}^{\ell}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}+\|\text{\rm
div}(I_{\varepsilon}^{\ell})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\leq
C\varepsilon^{1/2}$
for $1\leq\ell\leq L$. It follows that
$\displaystyle\varepsilon\|\nabla\Phi_{\varepsilon}^{(3)}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq\varepsilon\|\nabla
I_{\varepsilon}^{\ell}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\|f-\nabla
P_{0}\|_{L^{\infty}(\Omega_{\varepsilon}^{\ell})}+\varepsilon\|I_{\varepsilon}^{\ell}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega_{\varepsilon}^{\ell})}$ $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)},$
and
$\displaystyle\|\text{\rm
div}(\Phi_{\varepsilon}^{(3)})\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}$
$\displaystyle\leq\|\text{\rm
div}(I_{\varepsilon}^{\ell})\|_{L^{2}(\Omega^{\ell}_{\varepsilon})}\|f-\nabla
P_{0}\|_{L^{\infty}(\Omega^{\ell})}+\|I^{\ell}_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon}^{\ell})}\|\nabla(f-\nabla
P_{0})\|_{L^{\infty}(\Omega_{\varepsilon}^{\ell})}$ $\displaystyle\leq
C\varepsilon^{1/2}\|f\|_{C^{1,1/2}(\Omega)}.$
∎
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Zhongwei Shen, Department of Mathematics, University of Kentucky, Lexington,
Kentucky 40506, USA.
E-mail<EMAIL_ADDRESS>
|
# A Contextual Bandit Approach for Learning to Plan in Environments with
Probabilistic Goal Configurations
Sohan Rudra∗
Google Research &Saksham Goel∗
Google Search &Anirban Santara∗
Google Research &Claudio Gentile∗
Google Research &Laurent Perron
Google Research &Fei Xia
Robotics@Google &Vikas Sindhwani
Robotics@Google &Carolina Parada
Robotics@Google &Gaurav Aggarwal
Google Research
###### Abstract
Object-goal navigation (Object-nav) entails searching, recognizing and
navigating to a target object. Object-nav has been extensively studied by the
Embodied-AI community, but most solutions are often restricted to considering
static objects (e.g., television, fridge, etc.). We propose a modular
framework for object-nav that is able to efficiently search indoor
environments for not just static objects but also movable objects (e.g.
fruits, glasses, phones, etc.) that frequently change their positions due to
human intervention. Our contextual-bandit agent efficiently explores the
environment by showing optimism in the face of uncertainty and learns a model
of the likelihood of spotting different objects from each navigable location.
The likelihoods are used as rewards in a weighted minimum latency solver to
deduce a trajectory for the robot. We evaluate our algorithms in two simulated
environments and a real-world setting, to demonstrate high sample efficiency
and reliability.
## 1 Introduction
Personal robotics is an important domain of Embodied AI [48] that aims at
enhancing human productivity by developing physical assistants for everyday
tasks. In this paper, we study the task of Object-Goal Navigation (object-nav)
[15, 65], which is a core component of many personal robotics applications
like Mobile Manipulation [3], Embodied Question Answering [21], and Vision-
and-Language Navigation [5]111Please visit our website:
https://sites.google.com/view/find-my-glasses/home for a video presentation of
the paper and real world demonstrations.. Object-nav is defined as the task of
searching (and optionally, retrieving) a given object within a designated
space, e.g., indoor spaces like homes or offices, which are typically known
environments. The target object is often specified in natural language, e.g.,
“a blue soccer ball with stripes". This work is motivated by an everyday
_object-nav_ task: searching for our keys or cellphone at home. Analogous to
how humans search, we propose an algorithm that learns to look for objects
that change their locations due to human intervention (e.g. cellphones,
glasses and keys) in the most likely places first. Object-nav algorithms work
by learning semantic relationships between the target object and the topology
of the environment. They can be classified into two categories: map-based and
map-free [8]. Map-free algorithms [58, 52, 46, 19, 50, 60, 31] do not require
a map of the environment and can decide which way to go based directly on the
current observations and past memories without having to maintain a global
representation of the environment. This requires them to solve the problem of
localization and mapping in conjunction with the object-nav problem, making
their sample complexity very high. Another prominent challenge faced by end-
to-end learning-based algorithms in robotics is bridging the sim-to-real gap
[36]. Learning agents are usually trained in a simulator (sim) like Matterport
[13], AI2Thor [38], Gibson [64] and Habitat [53] before deploying in the real
world. However, due to limited fidelity of a simulator, observations of the
same event might be different in the simulator and in reality (domain
mismatch). Map-based algorithms [44, 43, 47, 37, 10, 9] assume that a map of
the environment is available at the time of path-planning. The map could be an
occupancy map showing the probability of obstacles being at each location. It
could also be a topological map [22] that is comprised of a graph where nodes
represent characteristic places and edges contain reachability information
(distances, times, etc.) between pairs of nodes. A few of these algorithms
construct a map of the current region before planning in that region [16, 17,
25, 15]. Map-based methods are typically modular, hence, sample-efficient [54]
and easier to deploy in the real world. However, the validity of the solution
devised by these algorithms is a strong function of the accuracy of the map
and localization of the robot. Thanks to advancements in Simultaneous
Localization and Mapping (SLAM) algorithms [23], constructing an occupancy map
of reasonable accuracy has become relatively easy in modern robotic systems.
In this paper, we aim to achieve robust Object-Goal Navigation in indoor
human-centric environments. Our algorithms are modular and map-based. They
assume access to a binary 2D occupancy map of the environment with navigable
and non-navigable parts marked out. In our day to day life, quite frequently,
we have to search portable objects that we happened to lose track of. We will
refer to such objects as “movable objects". Let us consider the example of
_cellphone_. We begin our search by trying to recall a list of places that we
are used to finding our _cellphone_ at. The algorithm presented in this paper
takes a similar approach where we aim to model the likelihood of spotting a
given object from different places in the environment. For example, a
_cellphone_ is more likely to be found on the study-table, work-desk and
bedside-table, rather than in the kitchen floor or on top of the refrigerator.
These likelihoods are learned online as the agent explores a realistic
environment. For stationary objects, we just assign probability values of $0$
or $1$. These likelihoods are then used for planning a path that minimizes the
average distance travelled by the agent to reach the target object.
Figure 1 provides an overview of our approach. (1) The robot is randomly
initialized in the environment with the task of finding a given target object.
(2) A set of reachable vantage points are sampled across the entire
environment using the current $2D$ occupancy map via farthest point sub-
sampling [24, 49]. (3) A Contextual Bandit Agent [41] estimates the likelihood
of spotting the target object from each vantage point. (4) A Weighted Minimum
Latency Problem (WMLP) solver [62] is used to generate an ordering of the
vantage points taking into account their likelihood scores, the initial
position of the robot and the geometry of the room. Finally, (5) The robot
visits the vantage points in the planned order while inspecting its
surroundings. As soon as it spots the object, it heads directly to it. The
modular design of our approach significantly reduces the sample complexity and
allows us to switch out object detectors, motion planners and point samplers
for domain-specific models when an agent is transferred between sim and real –
reducing the sim-to-real gap. The paper is organized as follows. In section 2,
we introduce the notation followed in the paper. We also provide a formal
definition of our problem and a theoretical algorithm to address it. In
section 3, we present a practical implementation of the theoretical algorithm.
In section 4, we present an empirical evaluation of the proposed methods in
two simulated and one real environments. In section 5, we conclude the paper
with a discussion on the limitations of the proposed approach and directions
of future work.
Figure 1: Overview of our approach. (a) Picture of our robot (from Everyday
Robots) and the target objects studied in our experiment. (b) The robot is
randomly initialized in the environment with the task of finding a given
target object. (c) A set of reachable vantage points (green dots) are sampled
across the entire environment using the current $2D$ occupancy map. (d) The
Contextual-Bandit agent estimates the likelihood of spotting the target object
from each vantage point. (e) A Weighted Minimum Latency Problem (WMLP) solver
is used to generate an ordering of the vantage points taking into account
their likelihood scores, the initial position of the robot and the geometry of
the room. (f) The robot visits the vantage points in the planned order while
inspecting its surroundings. As soon as it spots the object, it heads directly
to it.
## 2 Model and Algorithm
We start by explaining the mathematical model underpinning our investigation.
Model. We formalize our problem as follows. In the 2D occupancy map of the
environment, let us denote all the points by a set ${\mathcal{X}}$. Set
${\mathcal{X}}$ is partitioned into the set of feasible points ${\mathcal{F}}$
(the points the robot can freely navigate) and the set of non-feasible points
${\mathcal{N}}$ (the points occupied by obstacles like furnitures), so that
${\mathcal{X}}={\mathcal{F}}\cup{\mathcal{N}}$, and
${\mathcal{F}}\cap{\mathcal{N}}=\emptyset$. The three sets ${\mathcal{X}}$,
${\mathcal{F}}$, and ${\mathcal{N}}$ are available to us before planning. Each
$x\in{\mathcal{F}}$ comes with a visibility set $V(x)\subseteq{\mathcal{X}}$,
that is, a set of points that the robot can inspect while standing222 We are
assuming here that the robot can sample from $x$ any pose via 360 degree
rotation. at $x$. For concreteness, visibility is defined in terms of
Euclidean distance as
$V(x)=\\{x^{\prime}\in{\mathcal{X}}\,:\,||x-x^{\prime}||\leq
r_{\mathrm{vis}}\\},$ for some visibility range $r_{\mathrm{vis}}>0$, e.g.,
$r_{\mathrm{vis}}=2.5$ meters (the effective range of the object detectors on
the system).
We have $n$ movable objects of interest (glasses, keys, etc.). We use $[n]$ to
denote the set $\\{1,2,\dots,n\\}$. For each pair
$(i,x)\in[n]\times{\mathcal{F}}$, denote by $p_{i}(x)$ the probability that
the robot spots object $i$ while standing at position $x$. These probabilities
are in turn defined by $n$ (unknown) probability distributions
$\\{{\mathbb{P}}_{i}$, $i\in[n]\\}$, with support ${\mathcal{X}}$, that
determine where objects are located, so that
$p_{i}(x)={\mathbb{P}}_{y\sim{\mathbb{P}}_{i}}(y\in V(x))$. Notice that an
object can, in principle, be anywhere in the scene. The probabilities
$p_{i}(x)$ are unknown to the planner, and have to be learned through
interactions with the environment. We model them as
$p_{i}(x)=f(\phi(i,x);\theta),$ where
$\phi\,:\,[n]\times{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{D}$ is a mapping
that featurizes the pair $(i,x)$ into a $D$-dimensional real vector, for some
feature dimension $D\geq d$,
$f\,:\,{\mathbb{R}}^{D}\times{\mathbb{R}}^{m}\rightarrow[0,1]$ is a known
function, and $\theta\in{\mathbb{R}}^{m}$ is an unknown vector of parameters,
for some parameter dimension $m$. For instance,
$f(\phi;\theta)=\sigma(\theta^{\top}\phi)$, where $\sigma$ is the sigmoidal
function $\sigma(z)=\frac{e^{zs}}{1+e^{zs}}$ with slope $s$ at the origin, and
$m=D$. Our theoretical analysis uses the above generalized linear model, while
in our experimental evaluation, we compare both linear and neural models for
$f(\cdot;\theta)$.
Planning and Regret. In each navigation episode $t=1,2,\ldots$, there will be
only one object $i_{t}\in[n]$ in the scene333 This is not a strict
requirement, our analysis can be seamlessly extended to the case where
multiple instances of the same object are simultaneously present on the scene.
, and the identity of this object is known to the robot. The environment
generates the position $y_{t}$ of object $i_{t}$ by drawing $y_{t}$ from
${\mathbb{P}}_{i_{t}}$. Let $x_{0,t}\in{\mathcal{F}}$ be the starting position
of the robot in episode $t$. The algorithm begins by sampling a total of $k$
vantage points $x_{1,t},\ldots,x_{k,t}\in{\mathcal{F}}$. The planner has to
generate a path $J_{t}=\langle
x_{\pi_{t}(1),t},\ldots,x_{\pi_{t}(k),t}\rangle$ across them, where
$\pi_{t}(\cdot)$ is a permutation of the indices $\\{0\\}\cup[k]$, with
$\pi_{t}(0)=0$. The robot traverses the path in the order dictated by $J_{t}$,
and stops as soon as the object is spotted, as allowed by the visibility
structure $V(x_{\ell,t})$, $\ell\in\\{0\\}\cup[k]$. It is also reasonable to
admit that the robot may incur some detection failures during an episode, an
event we denote by ${\mathcal{E}}_{t}$, to which we shall assign an
independent probability ${\mathbb{P}}({\mathcal{E}}_{t})$ to occur. We
henceforth drop the episode subscript $t$ for notational convenience.
The path length loss $L(y,J)$ of $J$ is the actual distance traversed by the
robot over the points $x_{0},x_{1},\ldots,x_{k}$ before spotting object $i$ in
position $y$. On the other hand, if the object is not found, it is reasonable
to stipulate that the loss incurred will be a large number
$L_{M}>\sum_{\ell=1}^{k}\sum_{j=1}^{\ell},{\mathrm{dist_{\star}}}(x_{\pi(j-1)},x_{\pi(j)})$,
bigger than the total path length of any length-$k$ path $\langle
x_{\pi(1)},\ldots,x_{\pi(k)}\rangle$. Overall
$\displaystyle L(y,J)=(1-\ 1{1}{\left\\{{\mathcal{E}}\right\\}})$ (1)
$\displaystyle\times\Biggl{(}\sum_{\ell=1}^{k}\underbrace{\ 1{1}{\left\\{y\in
V(x_{\pi(\ell)})\setminus(V(x_{\pi(0)})\cup\ldots\cup
V(x_{\pi(\ell-1)}))\right\\}}}_{\mbox{\tiny$V(x_{\pi(\ell)})$ is the first
ball in the traversal order where object is located
}}\times\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{\pi(j-1)},x_{\pi(j)})\Biggl{)}+\
1{1}{\left\\{{\mathcal{E}}\right\\}}L_{M}~{},$
where $\ 1{1}{\left\\{\cdot\right\\}}$ is the indicator function of the
predicate at argument, and ${\mathrm{dist_{\star}}}(x_{1},x_{2})$ denotes the
$A^{\star}$ distance between the two vantage points $x_{1}$ and $x_{2}$ on the
scene, i.e., the collision-free shortest path length between $x_{1}$ and
$x_{2}$ for our robot (notice that
${\mathrm{dist_{\star}}}(x_{1},x_{2})\geq||x_{1}-x_{2}||$). Observe that, in
the absence of a failure, we are assuming the object will eventually be found
during each episode. Hence, a failure will be ascertained only at the end of
an episode. We approximate the above by disregarding the overlap among the
visibility balls $V(x_{\pi(\ell)})$ (hence somehow assuming these balls do not
influence one another), and then take expectation over $y\sim{\mathbb{P}}_{i}$
and the independent Bernoulli variables $\
1{1}{\left\\{{\mathcal{E}}\right\\}}$, with expectation
${\mathbb{P}}({\mathcal{E}})$. This yields the (approximate) average path
length
$\displaystyle{\mathbb{E}}_{i}[L(y,J)]={\mathbb{P}}({\mathcal{E}})\,L_{M}+(1-{\mathbb{P}}({\mathcal{E}}))\Biggl{(}\sum_{\ell=1}^{k}p_{i}(x_{\pi(\ell)})\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{\pi(j-1)},x_{\pi(j)})\Biggl{)}~{},$
(2)
where ${\mathbb{E}}_{i}[\cdot]$ is a short-hand for
${\mathbb{E}}_{y\sim{\mathbb{P}}_{i},{\mathcal{E}}}[\cdot]$. We are now in a
position to define our benchmark performance measure against which regret will
be defined. The benchmark planner knows the distributions
$\\{{\mathbb{P}}_{i}$, $i\in[n]\\}$ and the probability
${\mathbb{P}}({\mathcal{E}})$, and computes a path $J^{\star}=\langle
x^{*}_{1},\ldots,x^{\star}_{k}\rangle$ whose elements are taken from
${\mathcal{F}}$, such that $J^{\star}$ minimizes ${\mathbb{E}}_{i}[L(y,J)]$
over all length-$k$ paths (and permutations thereof) that can be constructed
out of points from ${\mathcal{F}}$. Given a sequence of episodes
$t=1,\ldots,T$ with corresponding objects $i_{1},\ldots,i_{T}$ (and starting
positions $x_{0,1},\ldots,x_{0,T}$), we define the cumulative regret
$R_{T}(J_{1},\ldots,J_{T})$ of a planner that generates paths
$J_{1},\ldots,J_{T}$ as
$R_{T}(J_{1},\ldots,J_{T})=\sum_{t=1}^{T}{\mathbb{E}}_{i_{t}}[L(y_{t},J_{t})]-{\mathbb{E}}_{i_{t}}[L(y_{t},J^{\star}_{t})]~{}.$
We would like this cumulative regret to be sublinear in $T$ with high
probability (in the random draw of position $y_{t}$ at the beginning of each
episode $t$). Notice that the last term in the RHS of (2) is independent of
$J$, hence $L_{M}$ will play no role in the regret computation.
Algorithm. Our “theoretical" algorithm is described as Algorithm 1. The
algorithm operates on an $\epsilon$-cover444 Recall that
${\mathcal{F}}_{\epsilon}$ is an $\epsilon$-cover of ${\mathcal{F}}$ if
${\mathcal{F}}_{\epsilon}\subseteq{\mathcal{F}}$ and for all
$x\in{\mathcal{F}}$ there is $x^{\prime}\in{\mathcal{F}}_{\epsilon}$ for which
${\mathrm{dist_{\star}}}(x,x^{\prime})\leq\epsilon$. It is easy to see that,
given a 2D-scene, the cardinality $|{\mathcal{F}}_{\epsilon}|$ of
${\mathcal{F}}_{\epsilon}$ is $O(1/\epsilon^{2})$. ${\mathcal{F}}_{\epsilon}$
of ${\mathcal{F}}$ and generalized linear probabilities
$p_{i_{t}}(x)=\sigma(\theta^{\top}\phi(i_{t},x))$. The algorithm replaces the
above true probabilities in the average path length (2) with lower confidence
estimations $\sigma\Bigl{(}{\widehat{\Delta}}_{t}(x)-\epsilon_{t}(x)\Bigl{)}$,
and then computes $J_{t}$ by minimizing (2) over the choice of $k$ points
within ${\mathcal{F}}_{\epsilon}$, as well as their order (permutation $\pi$).
A sequence of signals $\langle
s_{1,t},\ldots,s_{k^{\prime}_{t},t}\rangle=\langle-1,\ldots,-1,+1\rangle$
observed during episode $t$ is associated with the successful path $J_{t}$ in
that a sequence of $k_{t}^{\prime}-1$ negative signals (“$-1$" = object not
spotted from $x_{\pi_{t}(\ell),t}$, for $\ell=1,\ldots,k^{\prime}_{t}-1$)
precede a positive signal (“$+1$" = object spotted at
$x_{\pi_{t}(k^{\prime}_{t}),t}$). On the other hand, when the object is not
found throughout the length-$k$ path, the sequence of signals becomes $\langle
s_{1,t},\ldots,s_{k,t}\rangle=\langle-1,\ldots,-1,-1\rangle$ (this happens
with probability ${\mathbb{P}}({\mathcal{E}})$).
When the object is found, Algorithm 1 uses the observed signals to update over
time a $D$-dimensional weight vector ${\widehat{\theta}}$, and a $(D\times
D)$-dimensional matrix $M$. Vector ${\widehat{\theta}}_{t}$ is used to
estimate $\theta^{\top}\phi(i_{t},x)$, through ${\widehat{\Delta}}_{t}(x)$,
while matrix $M_{t}$ delivers a standard confidence bound via
$\epsilon_{t}^{2}(x)$. The update rule implements a second-order descent
method on logistic loss trying to learn the unknown vector $\theta$ out of the
signals $s_{j,t}$. In particular, the update
${\widehat{\theta}}_{c_{t}+j-1}\rightarrow{\widehat{\theta}}_{c_{t}+j}$ is
done by computing a standard online Newton step (e.g., [35]). Notice that at
each episode $t$, both matrix $M$ and vector ${\widehat{\theta}}$ get updated
$m_{t}=k^{\prime}_{t}$ times, which corresponds to the number of (valid)
signals received in that episode. Counter $c_{t}$ accumulates the number of
such updates across (non-failing) episodes. On the contrary, when the object
is not found, we know that there has been a failure, hence we disregard all
(negative) signals received and jump to the next episode with no updates
($m_{t}=0$).
From a computational standpoint, calculating $J_{t}$ as described in Algorithm
1 is hard, since the planning problem the algorithm is solving at each episode
is essentially equivalent to a Weighted Minimum Latency Problem (WMLP) [62],
also called traveling repairman problem, which, on a generic metric space is
NP-hard and also MAX-SNP-hard [7]. Fast algorithms are available only for very
special metric graphs, like paths [2, 28] edge-unweighted trees [45], trees of
diameter 3 [7], trees of constant number of leaves [39], and the like [63].
Even for weighted trees the problem remains NP-hard [56]. Approximation
algorithms are indeed available [18], but they are not practical enough for
real-world deployment. In our experiments (Sections 3–4), we implement and
compare fast planning approximations to Algorithm 1.
In both the planning and the training of the contextual bandit algorithm, we
are using the average path length as a minimization objective because it is
easier for the WMLP solver to handle. However, when it comes to evaluating
performance in our experiments, we use the Success weighted by Path Length
(SPL) metric [4] because it is a more established metric in the object-nav
literature.
Input: Learning rate $\eta>0$, exploration parameter $\alpha\geq 0$,
$\epsilon$-cover ${\mathcal{F}}_{\epsilon}$ of ${\mathcal{F}}$, $\epsilon>0$,
path length $k$.
Init: $M_{0}=kI\in{\mathbb{R}}^{D\times D}$,
${\widehat{\theta}}_{1}=0\in{\mathbb{R}}^{D}$, $c_{1}=1$.
For $t=1,2,\ldots,T$
1. 1.
Get object identity $i_{t}$ , and initial position of the robot $x_{0,t}$ ;
2. 2.
For $x\in{\mathcal{F}}$, set
${{\widehat{\Delta}}_{t}}(x)={\widehat{\theta}}_{c_{t}}^{\top}\phi(i_{t},x)\qquad{\mbox{and}}\qquad\epsilon^{2}_{t}(x)=\alpha\,\phi(i_{t},x)^{\top}M^{-1}_{c_{t}-1}\phi(i_{t},x)$
3. 3.
Compute $J_{t}=\langle x_{\pi_{t}(1),t},\ldots,x_{\pi_{t}(k),t}\rangle$ as
//solve WMLP at episode $t$
$J_{t}=\arg\min_{\stackrel{{\scriptstyle x_{1}\ldots
x_{k}\in{\mathcal{F}}_{\epsilon}}}{{{\tiny{\mbox{permutation
}}}\pi}}}\,\sum_{\ell=1}^{k}\sigma\Bigl{(}{\widehat{\Delta}}_{t}(x_{\pi(\ell)})-\epsilon_{t}(x_{\pi(\ell)})\Bigl{)}\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{\pi(j-1)},x_{\pi(j)})$
4. 4.
Observe signal $\begin{cases}\langle
s_{1,t},\ldots,s_{k^{\prime}_{t},t}\rangle=\langle-1,\ldots,-1,+1\rangle&{\mbox{set
$m_{t}=k^{\prime}_{t}$}}\\\ {\mbox{or}}\\\ \langle
s_{1,t},\ldots,s_{k,t}\rangle=\langle-1,\ldots,-1,-1\rangle&{\mbox{set
$m_{t}=0$}}\end{cases}$
5. 5.
For $j=1,\ldots,m_{t}$ (in the order of occurrence of items $x_{j}$ in
$J_{t}$) update:
$\displaystyle M_{c_{t}+j-1}$
$\displaystyle=M_{c_{t}+j-2}+\phi(i_{t},x_{j})\phi(i_{t},x_{j})^{\top},$
$\displaystyle{\widehat{\theta}}_{c_{t}+j}$
$\displaystyle={\widehat{\theta}}_{c_{t}+j-1}+\eta\,\sigma\Bigl{(}-s_{j,t}\,{\widehat{\theta}}_{c_{t}+j-1}^{\top}\phi(i_{t},x_{j})\Bigl{)}\,s_{j,t}\,M^{-1}_{c_{t}+j-1}\phi(i_{t},x_{j})$
6. 6.
$c_{t+1}\leftarrow c_{t}+m_{t}$ .
Algorithm 1 Simplified contextual bandit planning algorithm.
Regret Analysis. From a statistical standpoint, Algorithm 1 is a simplified
version of the slightly more complex Algorithm 2 which is the one our regret
analysis applies to. Please refer to Appendix A for a discussion on Algorithm
2.
###### Theorem 1
Let
$D_{M}=\max_{x,x^{\prime}\in{\mathcal{F}}}{\mathrm{dist_{\star}}}(x,x^{\prime})$
be the diameter of the scene. Also, let the feature mapping
$\phi\,:\,[n]\times{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{D}$ be such that
$||\phi(i,x)||\leq 1$ for all $i\in[n]$ and $x\in{\mathcal{F}}$, and let
constant $B$ be such that $||\theta||\leq B$. Then a variant of Algorithm 1
exists that operates in episode $t$ with an $\epsilon$-covering
${\mathcal{F}}_{\epsilon}$ of ${\mathcal{F}}$ with $\epsilon=k/\sqrt{t}$, such
that with probability at least $1-\delta$, with $\delta<1/e$, the cumulative
regret of this algorithm satisfies, on any sequence of objects
$i_{1},\ldots,i_{T}$,
$\displaystyle
R_{T}(J_{1},\ldots,J_{T})=O\Bigl{(}(1-p)k^{2}\sqrt{T}+D_{M}\,k\,\sqrt{(1-p)kT\,\alpha(k,D,T,\delta,B)\,D\log(1+kT)}\Bigl{)}~{},$
where
$\alpha(k,D,T,\delta,B)=O\left[e^{2B}\left(k+D\,\log\left(1+\frac{kDT}{\delta}\right)\right)\right]$,
and $p={\mathbb{P}}({\mathcal{E}}_{t})$ is the (constant) failure probability.
In the above, the big-oh notation hides additive and multiplicative constants
independent of $T$, $D$, $B$, $k$, $p$, and $\delta$.
In a nutshell, the above analysis provides a high-probability regret guarantee
of the form $k^{2}D\sqrt{T}$. We present the proof of this theorem in Appendix
A. The learning rate $\eta$ and the exploration parameter $\alpha$ in
Algorithm 1 are hyper-parameters.
## 3 Proposed Method
In this section, we present a practical algorithm that replicates Algorithm 1.
We have three main steps: (a) Sampling $k$ vantage-points that maximally cover
the navigable part ${\mathcal{F}}$ of the given scene; (b) Importance
assignment to the sampled points using an online learning contextual bandit
algorithm; and (c) Path planning through the vantage points. We use Model
Predictive Control (MPC) [29, 61, 42, 26] to execute the selected path. This
guarantees collision-free shortest-path trajectories between vantage points
while satisfying given safety constraints, optimality criteria, and
kinodynamic models. The resulting system is interpretable and safe. We will
elaborate on each of these.
### 3.1 Sampling vantage points
Our algorithm begins by sampling a sparse set of vantage points that are
navigable and spread all over the scene in a way that the agent’s vision is
able to cover the space to a large extent by visiting these points and looking
around. First, we extract all the navigable points (at a resolution of $0.1$
meter) from the robot’s occupancy map. Next, we select our vantage points
using the Farthest Point Sub-sampling (FPS) algorithm. FPS is a simple, yet
effective method of extracting a small number of points from a given point-
cloud to cover the extremities of the point-cloud and capture prominent point
features. Given a point cloud of size $N$ and a distance metric $d_{f}$, the
FPS algorithm works by picking a starting point (randomly or by some heuristic
criterion) from the point cloud and iteratively adding new points that are
maximally distant – according to $d_{f}$ – from the current set of selected
points, till the required number of points $k$ is reached. This method is
popularly used in image processing [24] and computer vision [49]. Since FPS
requires $O(N)$ distance calculations in each iteration, we choose the
Eucledian distance for $d_{f}$ in our experiments for computational ease.
### 3.2 Node Level Importance Assignment
Once we have the vantage points, we estimate the importance of each point in
the context of the target object. An important vantage point is one that has a
high likelihood of the robot spotting the target object standing there. For
ease of modelling, we assume that our robot is able to spot the target object
equally well from any yaw-angle if it is within its visibility range
$r_{\mathrm{viz}}$. In order to estimate the likelihood of the robot spotting
an object (which can be movable) from a given point, we need to explore the
environment. For efficient exploration, we formulate the problem as a
contextual bandit with vantage point as an arm and follow the principle of
_optimism in the face of uncertainty_ (e.g., [57]) as described in Algorithm
1. Each vantage point is described by a vector with positional, geometric and
semantic features of the point along with the identity of the target object.
We triangulate the position of the target object once the agent spots it from
a vantage point. In order to improve learning efficiency, we assign positive
training signal (“$+1$") to all the navigable points within $r_{\mathrm{vis}}$
radius from the object. Figure 2 (left) illustrates this procedure.
We study two classes of models for estimating the importance of points in our
experiments. The first is a generalized linear model (“Gen-Lin”) as presented
in Algorithm 1 with a minor modification. Since it is difficult for a single
generalized linear function to model multiple potentially non-linear decision
boundaries corresponding to different object classes, we use a disjoint set of
model parameters for each object class. The second is a simple two-layer fully
connected neural network model, approximated via Neural Tangent Kernels (NTK),
as contained in Algorithms 1-2 in [66] (“Neural") for computing uncertainties
$\epsilon_{t}(x)$ (Step 2 in Algorithm 1) and updating $\widehat{\theta}$
(Step 5 in Algorithm 1). Unlike the “GenLin” model, the same neural network
parameters are shared across all classes of objects.
Figure 2: Left: Positive sample augmentation for improved sample efficiency.
Radius $r_{\mathrm{vis}}$ is the robot’s visibility range. Right: Calculation
of ground-truth likelihood scores. $P(\mathbf{f})$ is the likelihood of the
object appearing on furniture $\mathbf{f}$.
$\sum_{\mathbf{f}\in\text{Furnitures}}P(\mathbf{f})=1$.
### 3.3 Planning
After estimating the importance-scores of the vantage points, we derive their
sequence of visitation by representing them as a graph – where each node is a
vantage point and the edges contain the $A^{\star}$ distance between vantage
points – and solving the Weighted Minimum Latency Problem (WMLP) (e.g., [7,
62]). WMLP tries to minimize the average waiting time of each node in a graph,
weighted by its importance score being reached by a travelling agent. In our
case, the solution of the WMLP minimizes the average distance traveled by the
robot to reach the target object – the average being over the position of the
object and the initial position of the robot. We consider two different ways
of solving the WMLP in our setting.
CP-SAT. The first approach directly faces the underlying optimization problem.
We relied on a satisfiability (SAT)-based constraint programming (CP) solver
[55] from Google OR-Tools [33] that uses a lazy clause generation solver on
top of a SAT solver to reach its solution conditioned on vantage-points,
starting position, and predicted relevance of the points. Although this
approach is direct and principled, the running time of this solver may
increase drastically as the number of points grows.
One-step greedy. Starting at $x$, the next point $x^{\prime}$ is chosen to
maximize a weighted combination of the estimated likelihood
$\hat{p}_{i}(x^{\prime})$ of spotting the target object $i$, and the inverse
of the $A^{\star}$ distance traveled to reach it:
$x^{\prime}=\operatorname*{arg\,max}_{x^{\prime\prime}\in\text{\\{unvisited\
points\\}}}\frac{\alpha_{p}}{{\mathrm{dist_{\star}}}(x,x^{\prime\prime})}+(1-\alpha_{p})\,\hat{p}_{i}(x^{\prime\prime})~{},$
(3)
where $\alpha_{p}\in[0,1]$ is a hyper-parameter whose value should be chosen
to achieve a good trade-off between minimizing the traveled distance in the
next step and maximizing the likelihood of spotting the target object. The
case of $\alpha_{p}=0$ corresponds to greedily choosing the unvisited point
with the highest estimated likelihood, while the case of $\alpha_{p}=1$ would
greedily choose the closest unvisited point. In the above, $\hat{p}_{i}(x)$
will be computed as suggested by Algorithm 1 via a lower confidence scheme of
the form
$\hat{p}_{i}(x)=\sigma\Bigl{(}{\widehat{\Delta}}(x)-\epsilon(x)\Bigl{)}$. The
one-step greedy approach is myopic, as it greedily optimizes for the next step
only, but is also much faster to run, hence it should be interpreted here as a
fast approximation to the CP-SAT solution.
## 4 Experiments
We run experiments in two simulated and one real office kitchen environments.
The two simulated kitchens have areas $80$ sq.m. and $120$ sq.m.,
respectively. Each experiment involves training the agent for $200$ episodes
followed by evaluation with frozen parameters in the same environment. For the
real kitchen experiment, we train the bandit model on a snapshot of the map in
our simulator and evaluate in the real environment.
Figure 3: (a) Real-kitchen sample trajectory for CP-SAT planner with the
Neural model. (b) Heat map of the estimated likelihood of spotting the goal
object (“bottle”) within a distance of $1$m along with ground truth
likelihoods (in green) of the object occurring on the surface of each
furniture.
The simulated environments have photo-realistic scenes generated from
Matterport scans [14] and Bullet [20] based physics simulation. Our robot is a
differential-drive wheeled robot from Everyday Robots
555https://everydayrobots.com/, which has a 3D LiDAR in the front, and depth
sensors mounted on its head. It is capable of accurate localization and safe
point-to-point navigation. We consider two different ways of solving the WMLP
in our setting – a) directly solving the optimization problem using CP-SAT, a
satisfiability (SAT)-based constraint programming (CP) solver [55] from Google
OR-Tools [33]; and b) a one-step greedy approach that maximizes a weighted
combination of the estimated likelihood of spotting the target object and the
inverse of the $A^{\star}$ distance traveled to reach it – to choose the next
vantage point in the visitation sequence. Although myopic, the latter is a
faster approximation of the CP-SAT algorithm that despite being direct and
more principled, suffers from drastic increases in running time with growing
number of vantage points (See Figure D.1). We use Model Predictive Control
[27, 11] to execute a path. Each vantage point $x$ is described by a feature-
vector $\phi(i,x)$ consisting of a one-hot encoding of the target object $i$
and a flattened $16\times 16$ patch of the wall-distance map centered at the
point $x$ (see also Figure C.3 in Appendix C). A sinusoidal positional
encoding [59] vector is appended to represent the location of the point in the
map. Map-resolution and positional encoding dimension are hyper-parameters.
The normalizer for the feature-vector is also a hyper-parameter that is chosen
from among: a) zero mean, unit standard-deviation, and b) unit $l^{2}$-norm.
All hyper-parameters ($\eta$ and $\alpha$ for Algorithm 1, $\alpha_{p}$ for
One-step greedy, the learning rate and batch size in Algorithms 1-2 in [66]
for Neural, the positional embededing size and the feature vector
normalization for mapping $\phi(i,x)$, and the sigmoidal scale $s$ for the
sigmoid in Algorithm 1) are tuned across suitable ranges (Table D.5) using a
Gaussian-Process Bandit based Blackbox optimizer [32] to maximize success
weighted by path length (SPL) [6] over the training episodes (Appendix D).
For training in simulation, we consider the agent has successfully reached its
goal if and when it visits a vantage point that is within $r_{\mathrm{vis}}$
radius from the target object. For learning good quality likelihood maps, we
set $r_{\mathrm{vis}}=1$m during training although the default value of
$r_{\mathrm{vis}}=2.5$m is used during evaluation in the simulated
environments. For success during evaluation in the real environment, the robot
must visually detect the target object and drive up to a grasping range of the
object. For object detection, we use our implementation of the ViLD detector
[34] that has a true positive rate of $84.6\%$ across our test objects. We
have five categories of target objects: “bottle”, “can”, “cup”, “bowl” and
“chips-bag” each of which has the same frequency of occurrence across episodes
and in any given episode we have a single instance of the target object in the
environment. We compare the performances of our proposed framework using the
generalized linear (“Gen-Lin”) and neural (“Neural”) models for training the
contextual bandit agent and “CP-SAT” and one-step greedy (“Greedy”) algorithms
for path planning to a purely geometric approach that solves the Travelling
Salesman Problem [40] (“TSP”). We assign a time budget of $30$ seconds to the
CP-SAT solver to have a realistic bound on robot response time. Each algorithm
is evaluated over $50$ episodes in real and $300$ episodes in simulated
environments. During training, whenever Algorithm 1 receives a positive signal
on a given vantage point, this signal is extended to nearby points (see
Appendix B for details). Figure 3 (a) shows a sample trajectory during the
real kitchen evaluation. Figure 3 (b) shows a sample learned likelihood map
for the simulated kitchen-1 environment.
We compare the performance of the agents on the following metrics: a) rate of
success during evaluation (“Eval. Succ.”), b) SPL [6] during training (“Train
SPL”), and c) SPL during evaluation (“Eval. SPL”). Higher “Train SPL”
indicates faster rate of convergence. Table 1 presents the results of our
first study, where we compare different learning and planning approaches for
an arbitrary spatial distribution of test objects. The columns labeled “GT-
Scores” use ground truth likelihoods of the vantage points computed as shown
in Figure 2 (right) and mapped in Figures C.1, and C.2 in Appendix C.
| Real Kitchen Env. | Kitchen Environment 1 | Kitchen Environment 2
---|---|---|---
| | Neural | | Gen-Lin | Neural | GT-Scores | | Gen-Lin | Neural | GT-Scores
| TSP | Greedy | CP-SAT | TSP | Greedy | CP-SAT | Greedy | CP-SAT | Greedy | CP-SAT | TSP | Greedy | CP-SAT | Greedy | CP-SAT | Greedy | CP-SAT
Train SPL | - | - | - | - | $0.32$ | $0.31$ | $0.30$ | $0.27$ | - | - | - | $0.26$ | $0.27$ | $0.23$ | $0.13$ | - | -
Eval. Succ. | $0.88$ | $0.80$ | $0.92$ | $0.89$ | $0.88$ | $0.89$ | $0.90$ | $0.90$ | $0.90$ | $0.90$ | $0.56$ | $0.80$ | $0.78$ | $0.74$ | $0.67$ | $0.82$ | $0.80$
Eval. SPL | $0.37$ | $0.38$ | $0.42$ | $0.38$ | $0.42$ | $0.47$ | $0.40$ | $0.39$ | $0.43$ | $0.51$ | $0.22$ | $0.26$ | $0.29$ | $0.25$ | $0.20$ | $0.33$ | $0.34$
Table 1: Empirical evaluation of our agents on $3$ environments in terms of
object-nav metrics.
For both training and evaluation, we use $25$ vantage points for the real
environment and the kitchen environment 1 and $50$ for kitchen environment 2.
Our first observation is the significant improvement in performance achieved
by our planners using “GT-Scores” over “TSP” in all the environments, and this
validates the importance of estimating the importance of the vantage points in
the context of the target object in addition to optimizing for the room
geometry. The performances for “GT-Scores” provide an upper bound for the
agents that learn the likelihood function through exploration. Although the
performance of “CP-SAT” shines in the real evaluation, under the planning time
budget of $30$ seconds, the performance of our proposed one-step greedy solver
regularly matches up and often beats the CP-SAT solver, especially in larger
and more cluttered kitchen environment 2. The performance of the “Neural”
model often seems to lag behind the “Gen-Lin” model. This is because the “Gen-
Lin” model does not have to generalize across all the object categories with
the same set of parameters and hence has a less challenging learning problem
to solve. Please visit the project website1 for videos of real world tests.
## 5 Limitations and Future Work
Our proposed approach can get adversely affected due to: 1) detection failure,
2) slowness of WMLP, and 3) early stopping of CP-SAT solver (not running the
CP-SAT solver until the end may give us feasible but poor solutions).
Inclusion of orientation along with the robot’s base position can help in
mitigating missed detections. Using richer feature embeddings can also improve
object detection from distance. Related to that, the choice of the network
architecture in the Neural model is severely limited by our usage of the NTK
approximation to compute confidence bounds. Leveraging more time-efficient
approximation schemes may allow for more complex (and potentially more
accurate) network architectures. Faster convergence is possible in training by
using a likelihood-guided sampling scheme but this may also create
opportunities for local minima. These are among the missing aspects we are
currently investigating.
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## APPENDIX
## Appendix A Theoretical Underpinning
In this section, we present Algorithm 2 and prove Theorem 1 from the main body
of the paper. We first need a couple of ancillary lemmas.
###### Lemma 1
Let Algorithm 2 be run on an $\epsilon$-cover ${\mathcal{F}}_{\epsilon}$ with
$\epsilon=O(1/\sqrt{t})$. Let episode $t$ be such that
$|\theta^{\top}\phi(i_{t},x)-\phi(i_{t},x)^{\top}{\widehat{\theta}}^{\prime}_{c_{t}}|\leq\epsilon_{t}(x)$
for all $x\in{\mathcal{F}}$. Also, let
$D_{M}=\max_{x,x^{\prime}\in{\mathcal{F}}}{\mathrm{dist_{\star}}}(x,x^{\prime})$
denote the diameter of the scene. Then
$\displaystyle{\mathbb{E}}_{i_{t}}[L(y_{t},J_{t})]-{\mathbb{E}}_{i_{t}}[L(y_{t},J^{\star})]=O\left((1-{\mathbb{P}}({\mathcal{E}}_{t}))\left(\frac{k^{2}}{\sqrt{t}}+D_{M}\,k\,\sum_{\ell=1}^{k}\epsilon_{t}(x_{\pi_{t}(\ell),t})\right)\right)~{},$
where ${\mathcal{E}}_{t}$ denotes the failure event during episode $t$.
Proof. We fix episode $t$ and remove subscript $t$ and $c_{t}$ for
convenience. As short-hands, let us denote by $J=\langle
x_{1},\ldots,x_{k}\rangle$ the path computed by Algorithm 2 in episode $t$ and
by $J^{\star}_{\epsilon}=\langle
x^{\star}_{1,\epsilon},\ldots,x^{\star}_{k,\epsilon}\rangle$ the minimizer of
(2) when the $k$ vantage points are constrained to lie in the $\epsilon$-cover
${\mathcal{F}}_{\epsilon}$. We clearly have
$\displaystyle|{\mathbb{E}}_{i}[L(y,J^{\star}_{\epsilon})]-{\mathbb{E}}_{i}[L(y,J^{\star})]|$
$\displaystyle\leq(1-{\mathbb{P}}({\mathcal{E}}_{t}))k(k+1)\epsilon$
$\displaystyle=O\left((1-{\mathbb{P}}({\mathcal{E}}_{t}))\frac{k^{2}}{\sqrt{t}}\right)~{}.$
Moreover,
$\displaystyle\frac{1}{(1-{\mathbb{P}}({\mathcal{E}}_{t}))}\,{\mathbb{E}}_{i}[L(y,J)]-{\mathbb{E}}_{i}[L(y,J^{\star}_{\epsilon})]$
$\displaystyle=\sum_{\ell=1}^{k}p_{i}(x_{\ell})\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{j-1},x_{j})-\sum_{\ell=1}^{k}p_{i}(x^{\star}_{\ell,\epsilon})\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x^{\star}_{j-1,\epsilon},x^{\star}_{j,\epsilon})$
$\displaystyle\leq\sum_{\ell=1}^{k}\sigma\Bigl{(}\theta^{\top}\phi(i,x_{\ell})\Bigl{)}\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{j-1},x_{j})-\sum_{\ell=1}^{k}\sigma\Bigl{(}{\widehat{\theta}}^{\top}\phi(i,x^{\star}_{\ell,\epsilon})-\epsilon(x^{\star}_{\ell,\epsilon})\Bigl{)}\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x^{\star}_{j-1,\epsilon},x^{\star}_{j,\epsilon})~{}.$
In turn, the above is upper bounded by
$\displaystyle\sum_{\ell=1}^{k}\sigma\Bigl{(}\theta^{\top}\phi(i,x_{\ell})\Bigl{)}\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{j-1},x_{j})-\sum_{\ell=1}^{k}\sigma\Bigl{(}{\widehat{\theta}}^{\top}\phi(i,x_{\ell})-\epsilon(x_{\ell})\Bigl{)}\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{j-1},x_{j})$
$\displaystyle\leq\sum_{\ell=1}^{k}\sigma\Bigl{(}{\widehat{\theta}}^{\top}\phi(i,x_{\ell})+\epsilon(x_{\ell})\Bigl{)}\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{j-1},x_{j})-\sum_{\ell=1}^{k}\sigma\Bigl{(}{\widehat{\theta}}^{\top}\phi(i,x_{\ell})-\epsilon(x_{\ell})\Bigl{)}\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{j-1},x_{j})$
$\displaystyle\leq\sum_{\ell=1}^{k}2\epsilon(x_{\ell})\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{j-1},x_{j})$
$\displaystyle=O\left(D_{M}\,k\,\sum_{\ell=1}^{k}\epsilon(x_{\ell})\right)~{}.$
Putting together proves the claim.
###### Lemma 2
Let $B>0$ be such that $\theta^{\top}\phi(i,x)\in[-B,B]$ for all $i\in[n]$ and
$x\in{\mathcal{F}}$. Moreover, let $c_{\sigma}$ and $c_{\sigma^{\prime}}$ be
two positive constants such that, for all $\Delta\in[-D,D]$ the conditions
$0<1-c_{\sigma}\leq\sigma(\Delta)\leq c_{\sigma}<1$ and
$\sigma^{\prime}(\Delta)\geq c_{\sigma^{\prime}}$ hold. Then with probability
at least $1-\delta$, with $\delta<1/e$, we have
$d_{c_{t}-1}(\theta,{\widehat{\theta}}^{\prime}_{c_{t}})\leq\alpha(k,D,T,\delta,B)~{},$
uniformly over $c_{t}\in[kT]$, where
$\displaystyle\alpha(k,D,T,\delta,B)$
$\displaystyle=O\Biggl{(}kB^{2}+\left(\frac{c_{\sigma}}{c_{\sigma^{\prime}}}\right)^{2}D\log\left(1+\frac{1}{k}\Bigl{(}\frac{t\,c_{\sigma}}{1-c_{\sigma}}+\log\frac{t+1}{\delta}\Bigl{)}\right)+\left(\left(\frac{c_{\sigma}}{c_{\sigma^{\prime}}}\right)^{2}+\frac{1+B}{c_{\sigma^{\prime}}}\right)\log\frac{k(t+1)}{\delta}\Biggl{)}~{}.$
Proof. The proof follows from standard concentration arguments applied to the
logistic loss, which Algorithm 2 implicitly operates on. See, e.g., [51],
Lemma 5 therein which, in turn, relies on [35] and [30]. The argument therein
can be applied to the non-failing episodes, that is, those episodes on which
state updates occur. In our bound above we are simply over-approximating the
number of non-failing episodes within the first $t$ episodes with $t$ itself.
Proof of Theorem 1 From Lemma 2 and the Cauchy-Schwarz inequality it follows
that
$\displaystyle(\theta^{\top}\phi(i,x)-\phi(i,x)^{\top}{\widehat{\theta}}^{\prime}_{c_{t}})^{2}$
$\displaystyle\leq\phi(i,x)^{\top}M_{c_{t}-1}^{-1}\phi(i,x)\,d_{c_{t}-1}(\theta,{\widehat{\theta}}^{\prime}_{c_{t}})$
$\displaystyle\leq\left(\phi(i,x)^{\top}M_{c_{t}-1}^{-1}\phi(i,x)\right)\,\alpha(k,D,T,\delta,B)$
for all $i\in[n]$ and $x\in{\mathcal{F}}$. Hence we can apply Lemma 1 with
$\epsilon^{2}_{t}(x)=\left(\phi(i,x)^{\top}M_{c_{t}-1}^{-1}\phi(i,x)\right)\,\alpha(k,D,T,\delta,B)~{}.$
Let ${\mathcal{E}}_{t}$ denote the failure event at episode $t$, with
${\mathbb{P}}({\mathcal{E}}_{t})=p$ for all $t$. Summing over $t=1,\ldots,T$,
we can write
$\displaystyle\sum_{t=1}^{T}\Bigl{(}{\mathbb{E}}_{i_{t}}[L(y_{t},J_{t})]-{\mathbb{E}}_{i_{t}}[L(y_{t},J^{\star})]\Bigl{)}$
$\displaystyle=O\left((1-p)k^{2}\sqrt{T}+D_{M}\,k\,{\mathbb{E}}\left[\sum_{t=1,\,{\mathcal{E}}_{t}=0}^{T}\sum_{\ell=1}^{k}\epsilon_{t}(x_{\pi_{t}(\ell),t})\right]\right)=O\Biggl{(}(1-p)k^{2}\sqrt{T}+D_{M}\,k\,\sqrt{\alpha(k,D,T,\delta,B)}\,\,{\mathbb{E}}\Biggl{)}~{},$
where ${\mathbb{E}}$ is a short-hand for
${\mathbb{E}}\Biggl{[}\sum_{t=1,\,{\mathcal{E}}_{t}=0}^{T}\sum_{\ell=1}^{k}\sqrt{\phi(i_{t},x_{\pi_{t}(\ell),t})^{\top}M_{c_{t}-1}^{-1}\phi(i_{t},x_{\pi_{t}(\ell),t})}\Biggl{]}~{}.$
We now follow similar arguments as in the proof of Theorem 1 in [51] by
focusing on
$\sum_{t=1,\,{\mathcal{E}}_{t}=0}^{T}\sum_{\ell=1}^{k}\phi(i_{t},x_{\pi_{t}(\ell),t})^{\top}M_{c_{t}-1}^{-1}\phi(i_{t},x_{\pi_{t}(\ell),t})~{}.$
First, by virtue of Lemma 6 in [51], we have, for each $t$,
$\displaystyle\sum_{\ell=1}^{k}$
$\displaystyle\phi(i_{t},x_{\pi_{t}(\ell),t})^{\top}M_{c_{t}-1}^{-1}\phi(i_{t},x_{\pi_{t}(\ell),t})\leq
e\,\sum_{\ell=1}^{k}\phi(i_{t},x_{\pi_{t}(\ell),t})^{\top}M_{c_{t}-1+\ell}^{-1}\,\phi(i_{t},x_{\pi_{t}(\ell),t})~{},$
so that
$\displaystyle\sum_{t=1,\,{\mathcal{E}}_{t}=0}^{T}\sum_{\ell=1}^{k}\phi(i_{t},x_{\pi_{t}(\ell),t})^{\top}M_{c_{t}-1}^{-1}\phi(i_{t},x_{\pi_{t}(\ell),t})\leq
e\,\sum_{t=1,\,{\mathcal{E}}_{t}=0}^{T}\sum_{\ell=1}^{k}\phi(i_{t},x_{\pi_{t}(\ell),t})^{\top}M_{c_{t}-1+\ell}^{-1}\,\phi(i_{t},x_{\pi_{t}(\ell),t})$
$\displaystyle\qquad=O\left(D\log(1+kT)\right)~{},$
the last inequality following from standard upper bounds (e.g., [12, 1]). As a
consequence
$\displaystyle\sum_{t=1,\,{\mathcal{E}}_{t}=0}^{T}$
$\displaystyle\sum_{\ell=1}^{k}\sqrt{\phi(i_{t},x_{\pi_{t}(\ell),t})^{\top}M_{c_{t}-1}^{-1}\phi(i_{t},x_{\pi_{t}(\ell),t})}=O\left(\sqrt{kD\log(1+kT)\sum_{t=1}^{T}(1-{\mathcal{E}}_{t})}\right)~{},$
which we plug back. Using the concavity of the square root, this allows us to
obtain
$\displaystyle\sum_{t=1}^{T}\Bigl{(}{\mathbb{E}}_{i_{t}}[L(y_{t},J_{t})]$
$\displaystyle-{\mathbb{E}}_{i_{t}}[L(y_{t},J^{\star})]\Bigl{)}$
$\displaystyle=O\Bigl{(}(1-p)k^{2}\sqrt{T}+D_{M}\,k\,\sqrt{(1-p)kT\,\alpha(k,D,T,\delta,B)\,D\log(1+kT)}\Bigl{)}~{}.$
Finally, observe that, since $\sigma(z)=\frac{\exp{(z)}}{1+\exp{(z)}}$, we
have within the expression for $\alpha(k,D,T,\delta,B)$ in Lemma 2,
$c_{\sigma}=\frac{e^{B}}{1+e^{B}}$ (hence
$\frac{c_{\sigma}}{1-c_{\sigma}}=e^{B}$), and
$c_{\sigma^{\prime}}=e^{-B}/(1+e^{-B})^{2}\geq e^{-B}/4$. Plugging back
concludes the proof.
In a nutshell, the above analysis provides a high-probability regret guarantee
of the form $k^{2}D\sqrt{T}$, when hyperparameters $\eta$ and $\alpha$ in
Algorithm 1 are assigned specific values, as detailed in Algorithm 2.
Input: $\epsilon$-cover ${\mathcal{F}}_{\epsilon}$ of ${\mathcal{F}}$,
$\epsilon>0$, path length $k$, maximal range $B>0$.
Init: $M_{0}=kI\in{\mathbb{R}}^{D\times D}$,
${\widehat{\theta}}_{1}=0\in{\mathbb{R}}^{D}$, $c_{1}=1$.
For $t=1,2,\ldots,T$
1. 1.
Get object identity $i_{t}$ , and initial position of the robot $x_{0,t}$ ;
2. 2.
For $x\in{\mathcal{F}}$, set
${{\widehat{\Delta}}_{t}}(x)=\phi(i_{t},x)^{\top}{\widehat{\theta}}^{\prime}_{c_{t}}(x)\qquad{\mbox{and}}\qquad\epsilon^{2}_{t}(x)=\alpha(k,D,T,\delta,B)\,\phi(i_{t},x)^{\top}M^{-1}_{c_{t}-1}\phi(i_{t},x)~{},$
where
$\displaystyle{\widehat{\theta}}^{\prime}_{c_{t}}(x)=\arg\min_{\theta\,:\,-B\leq\theta^{\top}\phi(i_{t},x)\leq
B}d_{c_{t}-1}(\theta,{\widehat{\theta}}_{c_{t}})~{};$
and
$\displaystyle\alpha(k,D,T,\delta,B)$
$\displaystyle=O\Biggl{(}kB^{2}+\left(\frac{c_{\sigma}}{c_{\sigma^{\prime}}}\right)^{2}D\log\left(1+\frac{1}{k}\Bigl{(}\frac{t\,c_{\sigma}}{1-c_{\sigma}}+\log\frac{t+1}{\delta}\Bigl{)}\right)$
$\displaystyle\qquad\qquad+\left(\left(\frac{c_{\sigma}}{c_{\sigma^{\prime}}}\right)^{2}+\frac{1+B}{c_{\sigma^{\prime}}}\right)\log\frac{k(t+1)}{\delta}\Biggl{)}$
3. 3.
Compute $J_{t}=\langle x_{\pi_{t}(1),t},\ldots,x_{\pi_{t}(k),t}\rangle$ as
//solve WMLP at episode $t$
$J_{t}=\arg\min_{\stackrel{{\scriptstyle x_{1}\ldots
x_{k}\in{\mathcal{F}}_{\epsilon}}}{{{\tiny{\mbox{permutation
}}}\pi}}}\,\sum_{\ell=1}^{k}\sigma\Bigl{(}{\widehat{\Delta}}_{t}(x_{\pi(\ell)})+\epsilon_{t}(x_{\pi(\ell)})\Bigl{)}\sum_{j=1}^{\ell}\,{\mathrm{dist_{\star}}}(x_{\pi(j-1)},x_{\pi(j)})$
4. 4.
Observe signal $\begin{cases}\langle
s_{1,t},\ldots,s_{k^{\prime}_{t},t}\rangle=\langle-1,\ldots,-1,+1\rangle&{\mbox{set
$m_{t}=k^{\prime}_{t}$}}\\\ {\mbox{or}}\\\ \langle
s_{1,t},\ldots,s_{k,t}\rangle=\langle-1,\ldots,-1,-1\rangle&{\mbox{set
$m_{t}=0$}}\end{cases}$
5. 5.
For $j=1,\ldots,m_{t}$ (in the order of occurrence of items $x_{j}$ in
$J_{t}$) update:
$\displaystyle M_{c_{t}+j-1}$
$\displaystyle=M_{c_{t}+j-2}+\phi(i_{t},x_{j})\phi(i_{t},x_{j})^{\top},$
$\displaystyle{\widehat{\theta}}_{c_{t}+j}$
$\displaystyle={\widehat{\theta}}^{\prime}_{c_{t}+j-1}+\frac{1}{c_{\sigma^{\prime}}}M^{-1}_{c_{t}+j-1}\nabla_{j,t}~{},$
where
$\nabla_{j,t}=\sigma(-s_{j,t}\,{\widehat{\Delta}}^{\prime}_{t}(x_{j}))\,s_{j,t}\,\phi(i_{t},x_{j})~{},$
where
${\widehat{\Delta}}^{\prime}_{t}(x_{j})=\phi(i_{t},x_{j})^{\top}{\widehat{\theta}}^{\prime}_{c_{t}+j-1}$
with
${\widehat{\theta}}^{\prime}_{c_{t}+j-1}=\arg\min_{\theta\,:\,-B\leq\theta^{\top}\phi(i_{t},x_{j})\leq
B}d_{c_{t}+j-2}(\theta,{\widehat{\theta}}_{c_{t}+j-1})~{};$
6. 6.
$c_{t+1}\leftarrow c_{t}+m_{t}$ .
Algorithm 2 Contextual bandit planning algorithm.
## Appendix B Data augmentation
Each vantage point is described by a vector with positional, geometric and
semantic features of the point along with the identity of the target object.
We triangulate the position of the target object once the agent spots it from
a vantage point. In order to improve learning efficiency, we assign positive
training signal (“$+1$”) to all the navigable points within $r_{\mathrm{vis}}$
radius from the object. Figure 2 (left) illustrates this procedure.
## Appendix C Object distributions
Figure C.1: Map of Simulated Kitchen 1 with distribution of occurrence of the
five target objects categories “cup”, “chips-bag”, “bottle”, “bowl”, and
“can”. Figure C.2: Map of Simulated Kitchen 2 with distribution of occurrence
of the five target objects categories “cup”, “chips-bag”, “bottle”, “bowl”,
and “can”.
In this section, we describe the way objects are spawned in the environment in
simulation. We only consider objects that are kept on table-tops. As shown in
Figures C.1 and C.2, each table in the environment has a certain probability
of housing the object. In each episode, for each object category, a table is
sampled from the corresponding probability distribution. A location on the
surface of the selected table is then picked uniformly at random to determine
the object location within the environment. We also experimented with a peaky
object distribution, where each object category was assigned a different table
to be spawned exclusively on. We present the results in Table C.1 and the
observations are similar to those reported in Section 4.
| Kitchen Environment 1 (Peaky distributions)
---|---
| | Gen-Lin | Neural | GT-Scores
| TSP | Greedy | CPSAT | Greedy | CP-SAT | Greedy | CP-SAT
Train SPL | - | $0.39$ | $0.37$ | $0.35$ | $0.28$ | - | -
Eval Succ Rate | $0.88$ | $0.9$ | $0.86$ | $0.85$ | $0.88$ | $0.88$ | $0.91$
Eval SPL | $0.40$ | $0.46$ | $0.55$ | $0.41$ | $0.45$ | $0.56$ | $0.65$
Table C.1: Experimental comparison of performance of our agents on a peaky
object distribution in Kitchen Environment 1 against the metrics mentioned in
Section 4. Figure C.3: The image shows grid based feature space where a local
image patch from various places in the wall distance map are shown. The dotted
white boxes indicated from which region the patch features are coming and the
patches show how the features look when scaled to a $16\times 16$ grid.
## Appendix D Hyper-parameters
Tables D.1, D.2 and D.3 contain the hyperparameters for each of the algorithms
tested in our experiments. Table D.5 gives the values searched for each
hyperparameter.
Table D.1: Hyperparameters for experiments with Non-Peaky object distributions in Kitchen Environment 1. | Gen-Lin | Neural
---|---|---
Hyperparameter | Greedy | CP-SAT | Greedy | CP-SAT
Learning Rate ($\eta$) | $0.44$ | $100$ | $0.01$ | $0.01$
Exploration parameter ($\alpha$) | $0.1$ | $0.1$ | $0.1$ | $3.44$
Number of vantage points ($k$) | $25$ | $25$ | $25$ | $25$
Alpha planner ($\alpha_{p}$) | $0.48$ | - | $0.43$ | -
Map Resolution | $75$ | $75$ | $75$ | $75$
Positional Embedding Size | $50$ | $15$ | $50$ | $10$
Feature Vector Normalization | $l^{2}$-norm | $l^{2}$-norm | $l^{2}$-norm | $l^{2}$-norm
Sigmoid Scale ($s$) | $1.0$ | $1.0$ | $20.0$ | $19.8$
Table D.2: Hyperparameters for experiments with Non-Peaky object distributions in Kitchen Environment 2. | Gen-Lin | Neural
---|---|---
Hyperparameter | Greedy | CP-SAT | Greedy | CP-SAT
Learning Rate ($\eta$) | $16.62$ | $10.44$ | $0.01$ | $0.01$
Exploration parameter ($\alpha$) | $10$ | $0.1$ | $0.1$ | $2.04$
Number of vantage points ($k$) | $50$ | $50$ | $50$ | $50$
Alpha planner ($\alpha_{p}$) | $0.49$ | - | $0.38$ | -
Map Resolution | $37$ | $37$ | $75$ | $75$
Positional Embedding Size | $20$ | $50$ | $50$ | $15$
Feature Vector Normalization | mean-var | mean-var | mean-var | mean-var
Sigmoid Scale ($s$) | - | - | $10.00$ | $17.56$
Table D.3: Hyperparameters for experiments with Peaky object distributions in Kitchen Environment 1. | Gen-Lin | Neural
---|---|---
Hyperparameter | Greedy | CP-SAT | Greedy | CP-SAT
Learning Rate ($\eta$) | $3.98$ | $75.41$ | $0.01$ | $0.01$
Exploration parameter ($\alpha$) | $7.15$ | $2.59$ | $0.10$ | $1.40$
Number of vantage points ($k$) | $25$ | $25$ | $25$ | $25$
Alpha planner ($\alpha_{p}$) | $0.59$ | - | $0.57$ | -
Map Resolution | $75$ | $75$ | $75$ | $75$
Positional Embedding Size | $10$ | $20$ | $10$ | $10$
Feature Vector Normalization | $l^{2}$-norm | $l^{2}$-norm | $l^{2}$-norm | $l^{2}$-norm
Sigmoid Scale | - | - | $20.00$ | $10.67$
Table D.4: Hyperparameter for experiments in the real world. Hyperparameter | Greedy | CP-SAT
---|---|---
Learning Rate ($\eta$) | $0.01$ | $0.01$
Exploration parameter ($\alpha$) | $0.11$ | $0.1$
Number of vantage points ($k$) | $25$ | $25$
Alpha planner ($\alpha_{p}$) | $0.49$ | -
Map Resolution | $75$ | $75$
Positional Embedding Size | $20$ | $30$
Feature Vector Normalization | $l^{2}\text{-norm}$ | $l^{2}\text{-norm}$
Sigmoid Scale | $18.38$ | $15.78$
Table D.5: Hyperparameter search ranges and scales. Hyperparameter | Values Searched | Search Scale
---|---|---
Learning Rate ($\eta$) (Gen-Lin model only) | $[0.01,100.0]$ | Log
Exploration parameter ($\alpha$) | $[0.1,10.0]$ | Linear
Number of vantage points ($k$) | $\\{25,50\\}$ | –
Alpha planner ($\alpha_{p}$) (Greedy only) | $[0.1,0.9]$ | Linear
Map Resolution | $\\{37,75,150\\}$ | –
Positional Embedding Size | $\\{5,10,15,20,30,50\\}$ | –
Feature Vector Normalization | $\\{l^{2}\text{-norm},\text{mean-var}\\}$ | –
Sigmoid Scale (for Neural model only) | $[10,20]$ | Linear
Figure D.1: Running time of CP-SAT solver on Intel Xeon 8-core CPU for
different numbers of vantage points in Kitchen Environment 2 environment.
|
Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV,
Amsterdam, The Netherlands Theoretical Chemistry, Vrije Universiteit, De
Boelelaan 1083, NL-1081 HV, Amsterdam, The Netherlands
# Towards Pair Atomic Density Fitting for Correlation Energies with Benchmark
Accuracy
Edoardo Spadetto<EMAIL_ADDRESS>Software for Chemistry and Materials NV,
NL, 1081HV, Amsterdam, The Netherlands Pier Herman Theodoor Philipsen
<EMAIL_ADDRESS>Software for Chemistry and Materials NV, NL, 1081HV,
Amsterdam, The Netherlands Arno Förster<EMAIL_ADDRESS>Software for
Chemistry and Materials NV, NL, 1081HV, Amsterdam, The Netherlands Lucas
Visscher
###### Abstract
Pair atomic density fitting (PADF) has been identified as a promising strategy
to reduce the scaling with system size of quantum chemical methods for the
calculation of the correlation energy like the direct random phase
approximation (RPA) or second-order Møller-Plesset perturbation theory (MP2).
PADF can however introduce large errors in correlation energies as the two-
electron interaction energy is not guaranteed to be bounded from below. This
issue can be partially alleviated by using very large fit sets, but this comes
at the price of reduced efficiency and having to deal with near-linear
dependencies in the fit set. One posibility is to use global density fitting
(DF), but in this work, we introduce an alternative methodology to overcome
this problem that preserves the intrinsically favourable scaling of PADF. We
first regularize the Fock matrix by projecting out parts of the basis set
which gives rise to orbital products that are hard to describe by PADF. After
having thus obtained a reliable self-consistent field solution, we then also
apply this projector to the orbital coefficient matrix to improve the
precision of PADF-MP2 and PADF-RPA. We systematically assess the accuracy of
this new approach in a numerical atomic orbital framework using Slater Type
Orbitals (STO) and correlation consistent Gaussian type basis sets up to
quintuple-$\zeta$ quality for systems with more than 200 atoms. For the small
and medium systems in the S66 database we show the maximum deviation of PADF-
MP2 and PADF-RPA relative correlation energies to DF-MP2 and DF-RPA reference
results to be 0.07 and 0.14 kcal/mol respectively. When the new projector
method is used, the errors only slightly increase for large molecules and also
when moderately sized fit sets are used the resulting errors are well under
control. Finally, we demonstrate the computational efficiency of our algorithm
by calculating the interaction energies of large, non-covalently bound
complexes with more than 1000 atoms and 20000 atomic orbitals at the
RPA@PBE/CC-pVTZ level of theory.
###### keywords:
RPA, MP2, algorithms, numerical atomic orbitals, non-covalent interactions
## 1 Introduction
There is great scientific and commercial interest to successfully predict the
electronic structure of molecules and materials. Towards this aim, Density
Functional Theory (DFT)1, 2, 3 and the Hartree–Fock (HF) method 4, 5, 6, 7, 8
are indispensable tools, but often they capture exchange-correlation effects
only insufficiently. For instance, dispersion and polarization effects which
derive mainly from mid- to long-range electron correlation9 are not accounted
for by standard DFT functionals.10, 11. While empirical dispersion corrections
have been highly successful in describing some of these aspects of electron
correlation,12, 9 methods based on coupled cluster (CC)13, 14, 15, 16, 17
theory are generally considered to be the most precise class of correlation
methods and have been the workhorse for many high precision calculations.18
However, given their large computational cost,19 their usage is in practice
still limited to relatively small systems. Using massively parallel
implementations20, 21 and/or local approximations22, 23, 24, 25 it is in
principle possible to treat larger molecules.26, 27 Massively parallel
calculations do, however, require large computational resources, while the
errors introduced by the local approximations can give rise to uncertainties
of the order of several $\nicefrac{{kcal}}{{mol}}$ in some cases.28, 29, 26
An attractive alternative is provided by double hybrid (DH) density
functionals30, 31 which usually offer a good compromise between accuracy and
computational effort.32, 33, 34, 35 DHs combine DFT with methods which treat
correlation effects explicitly, mostly second-order Møller-Plesset
perturbation theory (MP2)36 but (more recently)37, 38, 39 also the random-
phase approximation (RPA) 40, 41 has been considered for this task. While MP2
is not necessarily very accurate and limited in its applicability, RPA is
gaining popularity42, 43, 44, 45, 46, 47, 48, 49, 50. RPA is applicable to a
wider class of systems than MP2, as it is unaffected by some of the drawbacks
of MP2, namely divergences for metals, small band gap systems40, 51 and large
molecules.52. Without increasing the computational effort one can also greatly
improve upon the accuracy of RPA by using $\sigma$-functionals.53, 54, 55
Another popular alternative is the inclusion of screened exchange which comes
essentially at MP2 cost.56, 57, 58, 59
RPA and MP2 are typically implemented with $\mathcal{O}\left(N^{4}\right)$60
and $\mathcal{O}\left(N^{5}\right)$ scaling with system size using global
density fitting (DF)61, 62, 63, 64 (DF-RPA60 and DF-MP2,64, 65 respectively).
Efficient implementations of these methods enable applications to systems with
a few hundred of atoms even at the quadruple-$\zeta$ level,50 but larger
systems are out of reach on standard hardware. For this reason, more efficient
algorithms and approximate implementations have been developed to improve the
scaling of both RPA and MP2. Common strategies are the usage of localized
orbitals,66, 67 cluster-in-molecule (CIM) approaches,68, 69 or implementations
which rely on sparsity in the atomic orbital basis.70, 71, 72, 73, 74, 75, 76,
77, 78, 79, 80, 81, 82, 83, 84, 85, 86 In the latter class of methods,
implementations using local DF approximations have gained increasing
popularity.76, 82, 83 While they do not achieve linear scaling with systems
size, they typically come with a very small prefactor83 and are believed to
only introduce minor errors compared to canonical, molecular orbital based
implementations.83, 87
One particular flavour of local DF approximations is Pair Atomic Density
Fitting (PADF),62, 88, 89, 90, 83 also known also as pair atomic resolution of
the identity (PARI),91, 83 concentric atomic density fitting,92, 93 or RI-
LVL.94 PADF has originally been introduced to speed up the construction of the
Hartree contribution in non-hybrid DFT calculations but was later generalized
to accomodate also the formation of the exact exchange matrix in HF or hybrid
DFT calculations. For a comparison of PADF-HF to other approximate exact
exchange algorithms see ref. 95. PADF can also be used to reduce the
asymptotic scaling of RPA and Spin-opposite scaled (SOS)-MP296 to formally
cubic.83 However, quadratic scaling is often observed in practice since the
prefactor of the cubic steps is small.83
This speedup comes at the cost of errors which can in principle cause
variational collapse of HF calculations to solutions corresponding to
artificially low energies.90 In practice, this is usually only an issue when
insufficiently large fit sets are employed. What often is more problematic
that PADF also leads to an artificial increase of the magnitude of correlation
energies. Unless unrealistically large fit sets are used, this is difficult to
avoid these errors and this can then also affect the precision of relative
energies that are the typical target of quantum chemical calculations.
In the Amsterdam modelling suite (AMS)97, 98, the issues of PADF are mitigated
by applying a projector to the exact HF exchange matrix in order to prevent
variational collapse. The same projector can also be applied to orbital
coefficients in order to reduce errors in post SCF methods. This strategy has
already been successfully employed in the past for many-body perturbation
theory (MBPT) based calculations.59, 99 However, systematic benchmarks against
other codes using the same basis sets were not yet performed. For this
purpose, we report here an implementation of PADF-MP2 and PADF-RPA in the
numerical atomic orbital (NAO) based code BAND.100, 101, 102, 103 This
implementation allows us to use Gaussian type orbitals (GTO) as basis sets and
therefore to systematically investigate the accuracy of the PADF-MP2 and PADF-
RPA implementations in AMS for relative correlation energies with respect to
global DF based implementations (DF-MP2, DF-RPA) in Psi4 and TURBOMOLE 104,
105. Similar benchmarks of PADF based correlation energies have already been
reported by Ihrig et al. using the FHI-AIMS code106, 107, 108, 109 who found
excellent agreement of PADF-MP2 and PADF-RPA with DF-MP2 and DF-RPA94 and also
by Tew.110 However, these authors focused on small and medium sized molecules
only. To assess whether this accuracy also holds for larger systems and large
basis sets, we herein report benchmarks for non-covalently bound dimers with
up to 200 atoms and for large GTO-type basis sets up to quintuple-$\zeta$
(5Z). In our benchmarks, we focus exclusively on non-covalent interactions.
This is mostly due to the availability of accurate reference values for
datasets containing large molecules, like the L7111 or the S30L112
compilations.
This paper is organized as follows: In section 2 we review the PADF method and
introduce the projector method (PM). We also sketch how PADF can be used to
achieve low-order scaling implementations of RPA and SOS-MP2. For more
details, we refer to previous work.83, 113 After an outline of our
computational details in section 3, we assess the accuracy of relative PADF-
MP2 and PADF-RPA correlation energies in section 4. Our calculations show that
PADF-SOS-MP2 is in excellent agreement to DF-SOS-MP2. Our interaction energies
for the S66 dataset114 show maximum absolute deviations for PADF-MP2 and PADF-
RPA with respect to the reference results of 0.07 and 0.14 kcal/mol
respectively, irrespective of the chosen basis set. For much larger molecules,
we observe only a negligible loss in accuracy for SOS-MP2 and we find the PM
to be decisive to obtain accurate results. The loss in accuracy is more
pronounced for PADF-RPA, but errors are smaller than errors due to basis set
incompleteness or due to local correlation approximations for large
molecules27, 26 To showcase the efficiency of our implementation, we calculate
PADF-RPA interaction energies of eight large non-covalently bound complexes at
the triple-$\zeta$ (TZ) level, with up to 1000 atoms and more than 20000 AOs.
Finally, section 5 summarizes and concludes this work.
## 2 Theory
Throughout this paper, the indicies $\\{i,j,...\\}$ ($\\{a,b,...\\}$) refer to
occupied (virtual) orbitals, and the indices $\\{p,q,...\\}$ refer to general
molecular orbitals. Primary basis functions are labeled with
$\\{\mu,\nu,\kappa,\lambda,...\\}$ while $\\{\alpha,\beta,\gamma,...\\}$
denote fit functions. $\\{A,B,C,...\\}$ denote atoms. $o$ is a generic index
which can either denote a primary basis function or a fit function.
### 2.1 Density Fitting
We use Mulliken notation throughout this work, in which the generic form of
two-electron integrals is given by
$\mathcal{K}_{\mu\nu\kappa\lambda}=\int
d\bm{r}d\bm{r}^{\prime}\chi^{*}_{\mu}(\bm{r})\chi_{\nu}(\bm{r})\mathcal{K}(\bm{r},\bm{r}^{\prime})\chi^{*}_{\kappa}(\bm{r}^{\prime})\chi_{\lambda}(\bm{r}^{\prime})\;,$
(1)
with $\mathcal{K}(\bm{r},\bm{r}^{\prime})$ being a general (non-local) kernel.
Important examples for $\mathcal{K}$ are the electron-electron interaction,
$v_{c}(\bm{r},\bm{r}^{\prime})=\frac{1}{|\bm{r}-\bm{r}^{\prime}|}$ which is a
key ingredient in HF and post-HF methods, and the non-interacting
polarizability $P(\bm{r},\bm{r}^{\prime})$ (for a certain value of imaginary
frequency or time) which appears for instance in RPA or SOS-MP2. The symbol
$\chi_{\mu}$ refers to an atom-centered basis function which belongs to a
basis set of $N_{\text{bas}}$ functions $\\{\chi_{\mu}(\textbf{r})\in X\hskip
4.0pt\forall\hskip 4.0pt1\leq\mu\leq N_{\text{bas}}\\}$. We assume these
functions to be composed of an angular part $Y_{l}^{m}(\theta,\phi)$ and a
radial function $R_{n}(|\textbf{r}_{A}|)$,
$\chi_{\mu}(\bm{r})=\chi_{lmn,A}(\theta,\phi,|\textbf{r}_{A}|)=Y_{l}^{m}(\theta,\phi)R_{n}(|\textbf{r}_{A}|)\;.$
(2)
The radial part only depends on the distance from atom $A$, $\textbf{r}_{A}$.
The angular part $Y_{l}^{m}$ is a spherical harmonic function with angles
defined in the local coordinate system of atom A.
Representing $\mathcal{K}$ in this way, the memory required to store all the
integrals defined by (1) grows as $\mathcal{O}\left(N^{4}\right)$ with system
size. Furthermore, evaluating these electron-electron interaction integral
explicitly is difficult, when sets of STOs or NAOs are chosen as primary
basis. It is therefore convenient to look in more detail at the set of
functions $F_{p}=\\{\chi_{\mu}(\bm{r})\chi_{\nu}(\bm{r})\hskip
4.0pt\forall\hskip 4.0pt1\leq\mu\leq N_{\text{bas}}\text{ and }1\leq\nu\leq
N_{\text{bas}}\\}$ that is obtained by gathering all unique products of two
basis functions and investigate whether this function set can be represented
in a more compact form via density fitting.
We first define $\hat{\mathcal{K}}$ formally115 as a linear operator in a
Hilbert space $\mathcal{H^{K}}$: $\hat{\mathcal{K}}f(\bm{r})=\int
d\bm{r}^{\prime}\mathcal{K}(\bm{r},\bm{r}^{\prime})f(\bm{r}^{\prime})$, with
the inner product on $\mathcal{H^{K}}$ defined as
$\displaystyle(f|g)=\int f(\bm{r})\;\hat{\mathcal{K}}\;g(\bm{r})d\bm{r}.$ (3)
The set of fit functions that is defined in (PA)DF forms the basis $F_{f}$ and
spans a subspace $X_{f}$ of the full Hilbert space $\mathcal{H^{K}}$. Given
$f_{\alpha}(\bm{r})$ and $f_{\beta}(\bm{r})\in F_{f}$, the expansion of
$\hat{\mathcal{K}}$ in $F_{f}$ is
$\mathcal{K}_{\alpha\beta}=(f_{\alpha}|f_{\beta}).$ (4)
Likewise we have that the basis $F_{p}$ spans the space
$X_{p}\subset\mathcal{H^{K}}$, with the integrals taking the form (1) or more
concisely $(\chi_{\kappa}\chi_{\lambda}|\chi_{\mu}\chi_{\nu})$. If
$X_{p}\subseteq X_{f}$, $\chi_{\mu}(\bm{r})\chi_{\nu}(\bm{r})$ can be
expressed exactly in terms of fit functions and we can consequently use the
compact expression (4) instead of (1). In practice we find that part of the
product space is not spanned by the fit functions ($X_{p}\setminus
X_{f}\neq\emptyset$). To characterize errors made by fitting basis functions
products with the fit set $F_{f}$ we therefore write members of the product
basis $F_{p}$ as
$\displaystyle\chi_{\mu}(\bm{r})\chi_{\nu}(\bm{r})=\sum_{\alpha}f_{\alpha}(\bm{r})c_{\mu\nu\alpha}+e_{\mu\nu}(\bm{r})$
(5)
where $e_{\mu\nu}(\bm{r})$ is an error function which accounts for the fact
that $X_{f}$ does not completely span $X_{p}$. To keep the notation short we
only indicate explicitly the dependence of this error function of the basis
function pair indices $\mu$ and $\nu$ and omit the dependencies on the choice
of fit set and the optimization criterion used to determine the fit
coefficients $c_{\mu\nu\alpha}$. The exact representation of
$\hat{\mathcal{K}}$ in $X_{p}$ can then be written as
$\displaystyle\mathcal{K}_{\mu\nu\kappa\lambda}=\sum_{\alpha\beta}c_{\mu\nu\alpha}\mathcal{K}_{\alpha\beta}c_{\kappa\lambda\beta}+$
$\displaystyle\sum_{\alpha}c_{\mu\nu\alpha}\int
d\bm{r}f_{\alpha}(\bm{r})\hat{\mathcal{K}}e_{\kappa\lambda}(\bm{r})+$ (6)
$\displaystyle\sum_{\beta}c_{\kappa\lambda\beta}\int
d\bm{r}e_{\mu\nu}(\bm{r})\hat{\mathcal{K}}f_{\beta}(\bm{r})+\int
d\bm{r}e_{\mu\nu}(\bm{r})\hat{\mathcal{K}}e_{\kappa\lambda}(\bm{r}).$
Using the scalar product notation (3) we also have
$\displaystyle(f_{\alpha}|\chi_{\mu}\chi_{\nu})$
$\displaystyle=\sum_{\beta}\mathcal{K}_{\alpha\beta}c_{\mu\nu\beta}+(f_{\alpha}|e_{\mu\nu})$
(7)
Keeping in mind that the function $e_{\mu\nu}$ depends implicitly on the fit
space and on the kernel used to define the scalar product, we can define this
function to lie entirely in $X_{e}=X_{p}\setminus X_{f}$ so that we have:
$\displaystyle(f_{\alpha}|e_{\mu\nu})=0.$ (8)
Note that integrals over $f_{\alpha}$ and $e_{\mu\nu}$ with other kernels are
in general non-zero. Then, assuming $\mathcal{K}_{\alpha\beta}$ invertible and
considering the symmetry of scalar products,we may write
$c_{\mu\nu\alpha}=\sum_{\beta}(\chi_{\mu}\chi_{\nu}|f_{\beta})(\mathcal{K}^{-1})_{\beta\alpha}.$
(9)
This choice of fit coefficients in (9) can also be viewed as minimizing the
Lagrangian
$\displaystyle\mathcal{L}_{\mu\nu}$ $\displaystyle=(e_{\mu\nu}|e_{\mu\nu})$
(10)
$\displaystyle=({\chi_{\mu}\chi_{\nu}}|\chi_{\mu}\chi_{\nu})-2\sum_{\beta}c_{\mu\nu\beta}(\chi_{\mu}\chi_{\nu}|f_{\beta})+\sum_{\alpha\beta}c_{\mu\nu\alpha}c_{\mu\nu\beta}(f_{\alpha}|f_{\beta})$
for every $e_{\mu\nu}$. Minimizing $\mathcal{L}_{\mu\nu}$ guarantees a
reasonable precision also for off-diagonal terms in the residuals matrix since
due to the Cauchy-Schwartz inequality we have
$\displaystyle(e_{\mu\nu}|e_{\kappa\lambda})\leq\sqrt{\mathcal{L}_{\mu\nu}\mathcal{L}_{\kappa\lambda}}\;.$
(11)
With the fit coefficient definition (8), the cross error terms in (6) vanish.
This property arises naturally because the metric used in fitting is defined
through the same kernel we aim to fit. Using $\mathcal{K}=v_{c}$ and functions
of the form (2), the theoretical framework just presented is known as (global)
robust Density Fitting (DF).61, 62, 63, 116, 117, 64 It has the advantage of
reducing the storage complexity of the matrix elements of the kernel and the
amount of integrals to be evaluated from $\mathcal{O}\left(N^{4}\right)$ to
$\mathcal{O}\left(N^{3}\right)$. DF is not an approximation if the expansion
is complete, and in this case a compression would only be achieved for exact
linear dependencies in $X_{p}$. In practice the compression is obtained at the
price of an approximation since for reasons of computational efficiency
$F_{f}$ does not span the complete space of products of primary basis
functions. Considering what is left out, we note that a product set defined as
$\chi_{\mu}\chi_{\nu}$ is strongly non-orthogonal. Orthogonalization of such a
basis to span as much as possible of the full Hilbert space would result in
linear combinations of $\chi_{\mu}\chi_{\nu}$ with large coefficients. Given
the finite precision of computer operations, the calculation of matrix
representations of these parts of $\mathcal{H^{K}}$ is likely to be
numerically unstable. In addition, we can expect such combinations of pair
functions to be of minor physical relevance for a quantum chemical
calculation. For this reason it is also numerically favourable to work with a
fit set that is better behaved in terms of orthogonality than an
orthogonalized product set.
DF reduces the asymptotic scaling of the evaluation of RPA and direct MP2
correlation energies from $\mathcal{O}\left(N^{6}\right)$ and
$\mathcal{O}\left(N^{5}\right)$, respectively to
$\mathcal{O}\left(N^{4}\right)$.60 However, the asymptotic scaling of methods
involving exchange terms is not automatically reduced. For instance, using
(6), the exchange contribution to the Fock matrix can be expressed as118
$F_{\mu\nu}=\sum_{\kappa\lambda}P_{\kappa\lambda}\sum_{\alpha\beta}c_{\mu\kappa\alpha}v_{\alpha\beta}c_{\nu\lambda\beta}\;,$
(12)
which is evaluated with the same asymptotic scaling of
$\mathcal{O}\left(N^{4}\right)$ as the variant without density fitting when no
further approximations are made. This is also generally true for post-HF
methods where exchange terms profit less from the compression of 4-index
tensors.72, 83 Therefore, it is advantageous to eliminate the exchange terms
entirely and introduce empirical scaling factors, as for instance in the SOS-
MP296 or SOS-CC2 methods.119
One can however also greatly improve upon the efficiency of global DF by
constraining eq. (5) in such a way that the number of non-zero-elements in $c$
only grows linearly with system size. For instance, instead of using the
Coulomb kernel directly61, 63 one can avoid to define the scalar product on
the operator $\hat{\mathcal{K}}$ and define another metric with more suitable
properties. This could for instance be a local kernel, like the overlap
kernel63 (also known as RI-SVS) or an attenuated Coulomb kernel.64, 120 These
kernels have been used successfully to lower the complexity of for instance
$GW$121, 87, RPA76, 77, MP284, 85 and CC2122 calculations. The price to be
paid is the loss of robustness, equation (8) is then not fulfilled so that the
cross terms in (6) become non-zero. An alternative approach which introduces
the desired sparsity in the fit-coefficient tensor more directly is PADF. In
PADF the density fit is restricted to pairwise sums only and subsequently
distance cut-offs are introduced. Using Latin uppercase superscripts to denote
the atomic centers of functions, the PADF expansion of products of basis
functions is
$\displaystyle\chi^{A}_{\mu}(\bm{r})\chi^{B}_{\nu}(\bm{r})=\sum_{\alpha\in
A}c_{\nu\mu\alpha}f_{\alpha}(\bm{r})+\sum_{\beta\in
B}c_{\mu\nu\beta}f_{\beta}(\bm{r})+e_{\mu\nu}(\bm{r})\;,$ (13)
replacing the simpler expansion (5). The notation $\alpha\in A$ indicates that
the summation is restricted to fit functions centered on atom $A$.
### 2.2 Fit Set Generation
#### 2.2.1 Fit sets from products of basis functions
It is easily understood that the choice of $F$ is of key importance in a PADF
code. Ideally, the fit set should be generated on-the-fly, tailored to the
primary basis at hand and the precision of the expansion (13) should be
adjustable in a systematic way using only a single parameter. Many algorithms
for this task have been developed for global DF.123, 124, 125, 126 An
alternative way to generate fit sets on-the-fly is Cholesky decomposition,127,
128, 129 but this approach is not straightforwardly generalized to codes which
can not evaluate 3-center integrals involving the Coulomb potential
analytically. We are only aware of two algorithms which have been developed
specifically for PADF.94, 130 We here adopt the one by Ihring and coworkers94
which we recapitulate for completeness.
From all unique combinations of AOs centered on atom $A$ (denoted by $X_{A}$)
we build an atom-specific trial fit set $\tilde{F}_{A}$ of functions of form
(2),
$\displaystyle\tilde{F}_{A}=\\{\tilde{f}_{\mu\nu}(\textbf{r})\text{ such that
}$ $\displaystyle
R_{\mu\nu}(|\textbf{r}_{A}|)=R_{\mu}(|\textbf{r}_{A}|)R_{\nu}(|\textbf{r}_{A}|)\text{
and }$ (14) $\displaystyle|l_{\mu}-l_{\nu}|<l_{\mu\nu}<|l_{\mu}+l_{\nu}|\text{
}\forall\text{ }\chi_{\mu}(\textbf{r})\text{ and }\chi_{\nu}(\textbf{r})\in
X_{A}\\}$
We then regroup $F_{A}$ in subsets with same angular momentum
$\displaystyle\tilde{F_{A}}=\bigcup_{l,m}\tilde{F}_{A,lm}=\bigcup_{l,m}{Y}_{l}^{m}(\theta,\phi)\times\tilde{R}_{A,lm}$
(15)
where $\tilde{R}_{A,lm}$, the set of radial components centered on the same
atom $A$, is multiplied element-wise to the same spherical harmonic. We then
compute the eigenvectors of the matrix
$(\tilde{R}_{\alpha}|\tilde{R}_{\beta})$ for $\tilde{R}_{\alpha}\text{ and
}\tilde{R}_{\beta}\in\tilde{R}_{A,lm}$ and we keep only the ones with
eigenvalue greater than a specific threshold $\epsilon_{\text{fit}}$. This
threshold can be seen as a parameter tuning the fit quality. Setting it to
zero does not imply exact fitting, as it only solves the one-center part
exactly. In addition, as mentioned above, choosing a too small parameter will
likely introduce numerical instabilities. The set of remaining eigenvectors is
called $R_{A,lm}$. Our final fit set is then
$\displaystyle
F=\bigcup_{Alm}F_{A,lm}=\bigcup_{Alm}Y_{l}^{m}(\theta,\phi)\times R_{A,lm}$
(16)
Alg. 1 shows a pseudocode of the algorithm. In alg. 1,
$R_{\mu}(r)\times\\{Y^{m}_{l_{u}}(\theta,\phi)\\}$ denotes basis functions
with same radial part and different spherical harmonic.
Algorithm 1 Fit set generation algorithm
1:for every atom $A$ do
2: for every $R_{\mu}(r)\\{Y^{m}_{l_{u}}(\theta,\phi)\\}$ in $X_{A}$ do
3: for every $R_{\nu}(r)\\{Y^{m}_{l_{v}}(\theta,\phi)\\}$ in $X_{A}$ do
4: $\tilde{F}_{A}\texttt{ push back
}R_{\mu}(r)R_{\nu}(r)\\{Y^{-l}_{l=|l_{u}-l_{v}|},...,Y^{+l}_{l=|l_{u}+l_{v}|}\\}$
5: for every $\tilde{F}_{A,lm}$ in $\tilde{F}_{A}$ do
6: for every couple $\tilde{f}_{a},\tilde{f}_{b}$ in $\tilde{F}_{A,lm}$ do
7: $V_{ab}=(\tilde{f}_{a}(\textbf{r})|\tilde{f}_{b}(\textbf{r}))$
8: Find Spectrum of $V\Rightarrow
V_{ab}=v_{a\alpha}\varepsilon_{\alpha\beta}v^{\dagger}_{\beta b}$ (eigenvalues
in increasing order)
9: $P_{\alpha b}=\sqrt{\varepsilon_{\alpha\beta}}v^{\dagger}_{\beta b}$ ( $m$
s.t. $\varepsilon_{\alpha\alpha}>\epsilon_{a}$ $\Rightarrow\alpha=1..m$ )
10: $\tilde{F}_{A,lm}=P\cdot\tilde{F}_{A,lm}$ (sets assumed to be matrices,
same ordering used in V)
For small basis sets, the fit set generated through such a procedure does
sometimes not lead to sufficiently accurate results. To overcome this problem,
Ihrig et al.94 artificially enlarged the fit set. This is achieved by adding a
new function to each $X_{A}$. In our implementation, the function is a Slater
type orbital with angular momentum $l^{A}_{max}+1$ where $l^{A}_{max}$ is the
maximum angular momentum present in $X_{A}$, and the arbitrarily chosen
exponent $\alpha$ equal to its angular momentum.
$\displaystyle\chi(\textbf{r})=r^{\alpha}\exp(-\alpha
r)Y_{\alpha}(\theta,\phi)$ $\displaystyle\alpha=l^{A}_{max}+1$ (17)
In the following, we will refer to this procedure as $L$-enhancement ($L$-e).
#### 2.2.2 Slater type orbital fit sets
As an alternative to the algorithm just described, we herein also test the use
of hand-optimized STO type fit sets. These come with the disadvantage that
they are not systematically improvable through the adjustment of a single
parameter $\epsilon_{\text{fit}}$. They are however more compact and therefore
more suitable for large-scale applications. In this work we use three
different thresholds which we refer to as _Normal_ , _Good_ and _VeryGood_.
The former two contain STO type functions with angular momentum up to $l=4$,
while the latter one contains functions with exponents up to $l=6$. We have
described these fit sets in ref. 83 to which we refer for more details.
### 2.3 Projection Methods
The improved algorithmic scaling deriving from PADF comes with downsides which
needs to be handled carefully in order to retain sufficient numerical
precision. Therefore, we use two related projection methods which we describe
in the following:
#### 2.3.1 Projection Method for the Basis Set
To prevent instabilities due to near-linear dependencies in the AO basis set
itself, this basis set size is often reduced by modifying the Löwdin
orthornomalization131, 132 step. The Löwdin transformation matrix
$\mathbf{S}^{-1/2}$ to an orthonormal set follows from the eigensystem of the
overlap matrix
$\displaystyle\mathbf{S}=$ $\displaystyle\mathbf{U}\mathbf{D}\mathbf{U}^{T}$
(18) $\displaystyle\mathbf{S}^{-1/2}=$
$\displaystyle\mathbf{U}\mathbf{D}^{-1/2}\mathbf{U}^{T}\;.$ (19)
Here we can choose to ignore eigenvectors with eigenvalues below a threshold
$\epsilon_{\text{bas}}$. The simplest way to achieve this is to set the
corresponding eigenvectors in $U$ to zero (thus introducing artificial
states). This is done in the ADF implementation. In the BAND implementation,
we define a smaller orthonormal basis by introducing the regularized Löwdin
transformation
$\tilde{\mathbf{S}}^{-1/2}=\mathbf{S}^{-1/2}\tilde{\mathbf{U}}$ (20)
where $\tilde{\mathbf{U}}$ is the non square (tall) matrix obtained by
removing the eigenvectors columns with eigenvalues smaller than
$\epsilon_{\text{bas}}$ from $\mathbf{U}$. Doing so, fewer orbitals are
obtained and appearance of artificial states is avoided. The elimination of
problematic orbitals that are expressed in the original basis with large
coefficients also helps to prevent problems later on when considering the
product basis and can therefore be beneficial to mitigate errors resulting
from the density fitting.
#### 2.3.2 HF projection method
A known issue of PADF is that contributions to the electron repulsion energies
can become negative which can lead to variational collapses of HF SCF
calculations91, 133. As we will argue below, this problem is to a large extent
due to integrals over product functions that are difficult to describe by the
fit set. A way to avoid this problem would be to Cholesky decompose the matrix
(1) and keep only the most important Cholesky vectors, but this is not
practical in calculations with Slater or numerical type orbitals. Instead, the
PADF implementation in AMS uses a simple projector in the original AO space
$\displaystyle\mathbf{T}=\mathbf{T}\mathbf{R}\mathbf{T}^{T}$ (21)
where we use the eigensystem of the AO overlap matrix
$\displaystyle\mathbf{S}=\mathbf{U}\mathbf{D}\mathbf{U}^{T}$ (22)
with $\mathbf{D}$ being a diagonal matrix with the eigenvalues on the
diagonal, and $\mathbf{U}$ having the eigenvectors stored as columns. The
diagonal matrix $\mathbf{R}$ is obtained from $\mathbf{D}$ by taking
$\displaystyle R_{ij}=\delta_{ij}\Theta(D_{ii}-\epsilon_{K})$ (23)
Note that for $\epsilon_{K}=0$ we get $\mathbf{R}=\mathbf{1}$ and hence
$\mathbf{T}=\mathbf{1}$. Heuristically, the action of $\mathbf{T}$ on a vector
or matrix is to remove components of the space parallel to eigenvectors of the
overlap matrix with eigenvalues smaller than the specified threshold
$\epsilon_{K}$. At the SCF stage, the projector is applied both on the left
and on the right of the exact exchange matrix $\mathbf{K}$
$\displaystyle\tilde{\mathbf{K}}=\mathbf{T}^{T}\mathbf{K}\mathbf{T}$ (24)
While the regularized Löwdin orthonormalization removes a subspace for all
energy terms, the HF projector method neglects only the (small) stabilizing
action of the exchange energy, shifting energies upwards. The default value
for $\epsilon_{K}=10^{-3}$ in BAND can therefore be much higher than the one
for the Löwdin orthonormalization projector $\epsilon_{\text{bas}}=10^{-8}$.
We also use the same projector (21) to calculate correlation energies. For
this we redefine the matrix $\mathbf{b}$ which transforms from the AO to the
MO basis as
$b^{\prime}_{i\mu}=\sum_{\mu^{\prime}}b_{i\mu^{\prime}}T_{\mu^{\prime}\mu}\;.$
(25)
Here, the use of the projector is supposed to improve the accuracy of
correlation energies by removing a subspace leading to AO products which can
only be represented poorly by the fit set. The usefulness of the application
of the PM-$K$ to correlation energies can be rationalized as follows: If we
consider an eigenvector
$v_{o}(\textbf{r})=\sum_{\mu}s_{o\mu}\chi_{\mu}(\textbf{r})$ of the overlap
matrix, we can notice that $\sum_{o}s^{\dagger}_{\mu
o}s_{o\nu}=D^{-1}_{\mu\nu}$. From this we understand that the average order of
magnitude of the coefficients is at least of the order of
$s_{o\mu}\sim\frac{1}{\sqrt{D_{\mu\mu}}}$. This shows that the basis set
poorly describes eigenvectors relative to small eigenvalues of $S$, since
large coefficients are needed to expand a small orthogonal component. We then
expect that products involving such linear combinations are the most difficult
to fit. This is mostly because of the diffuse products of basis functions from
distant atoms and from the consequent difficulties in using the PADF
approximation to express such products.113
### 2.4 RPA and SOS-MP2 correlation energies
We now briefly discuss how PADF can be used to speed up the evaluation of SOS-
MP2 and RPA correlation energies, summarizing the more detailed discussions in
ref. 83 and 113. In the basis of fit functions, the RPA correlation energy can
be expressed as
$\displaystyle E^{\text{RPA}}_{c}=$
$\displaystyle\frac{1}{2\pi}\int^{\infty}_{0}d\omega\text{Tr}\left\\{\left[\log\left(\mathbf{1}-\mathbf{Z}(i\omega)\right)\right]+\mathbf{Z}(i\omega)\right\\}$
(26) $\displaystyle=$
$\displaystyle\sum^{N_{\omega}}_{k}\sigma_{k}\text{Tr}\left\\{\left[\log\left(\mathbf{1}-\mathbf{Z}(i\omega_{k})\right)\right]+\mathbf{Z}(i\omega_{k})\right\\}\;,$
which follows directly from the corresponding real-space representation of the
RPA.134 The integration is performed over the imaginary frequency axis for
which either modified Gauss-Legendre grids or, more efficiently,135 minimax
grids of size $N_{\omega}$ can be used. $\omega_{k}$, $\sigma_{k}$ denote
points and corresponding integration weights on the imaginary axis.
$\mathbf{Z}$ is
$Z_{\alpha\beta}(i\omega)=\sum_{\gamma}P^{(0)}_{\alpha\gamma,i\omega}v_{\gamma\beta}\;,$
(27)
and is obtained through the non-interacting polarizability $P^{(0)}$ and the
electron-electron interaction $v$ in the basis of fit functions. Since matrix
logarithms are difficult to calculate, we use that (assuming
$\mathbf{1}-\mathbf{Z}$ can be diagonalized with eigenvalues $\lambda_{j}$)
$\text{Tr}\left[\log(\mathbf{1}-\mathbf{Z})\right]=\sum_{j}\log(\lambda_{j})=\log\left(\prod_{j}\lambda_{j}\right)=\log|\mathbf{1}-\mathbf{Z}|\;,$
and evaluate the determinant $|\mathbf{1}-\mathbf{Z}|$ instead. The imaginary
frequency representation of $P^{(0)}$ is obtained from its discrete imaginary
time representation using nodes
$\left\\{\tau_{k}\right\\}_{j=1,\dots,N_{\tau}}$. The transformation is
achieved by the discrete cosine transform (since $P^{(0)}$ is bosonic)
$P^{(0)}(i\omega_{k})=-i\sum^{N_{\tau}}_{j}\gamma^{(c)}_{kj}\cos(\omega_{k}\tau_{j})P^{(0)}(i\tau_{j})\;,$
(28)
where the $\gamma^{(c)}_{kj}$ are the matrix elements of the kernel of the
discrete cosine transform. The imaginary time representation of $P^{(0)}$ is
given as
$P^{(0)}_{\alpha\beta}(i\tau_{j})=-ic_{\mu\nu\alpha}G^{(0),<}_{\mu\kappa}(-i\tau_{j})G^{(0),>}_{\nu\lambda}(i\tau_{j})c_{\kappa\lambda\beta}\;,$
(29)
with greater and lesser components of the time-ordered Green’s functions being
defined as
$\displaystyle G^{(0),<}_{\mu\nu}(i\tau_{j})=$
$\displaystyle-i\sum_{i}b^{\prime}_{\mu
i}e^{-\epsilon_{i}\tau_{j}}b^{\prime}_{i\nu}$ (30) $\displaystyle
G^{(0),>}_{\mu\nu}(i\tau_{j})=$ $\displaystyle-i\sum_{a}b^{\prime}_{\mu
a}e^{-\epsilon_{a}\tau_{j}}b^{\prime}_{a\nu}\;.$ (31)
Notice, that $b^{\prime}$, as defined in (25), appears in these equations so
that the the PM enters the PRA correlation energies through $G^{\lessgtr}$. As
previously discussed in some detail by us,113 the evaluation of (29) scales
asymptotically as $\mathcal{O}\left(N^{2}\right)$ with system size in the PADF
approximation, since the number of elements in $c$ only scales as
$\mathcal{O}\left(N\right)$ with system size. For detailed working equations
we refer to our previous work.83, 113 In practice, the efficiency of this
approach depends on the possibility to represent the imaginary time and
frequency dependencies of the polarizability as compactly as possible. We
follow Kaltak and Kresse135, 136 and use non-uniformly spaced minimax and
least square grids as described in ref. 137 for imaginary time, and ref. 138
for the imaginary frequency domain.
Alternatively, the imaginary frequency polarizability $P^{(0)}(i\omega_{k})$
can be evaluated directly in the MO basis as for instance described in ref.
108,
$P^{(0)}_{\alpha\beta}(i\omega_{k})=-\sum_{a}\sum_{i}c_{ia\alpha}\frac{1}{\epsilon_{a}-\epsilon_{i}-i\omega_{k}}c_{ia\beta}+c.c.$
(32)
where
$c_{ia\alpha}=\sum_{\mu\nu}b^{\prime}_{\mu i}c_{\mu\nu\alpha}b^{\prime}_{\nu
a}\;.$ (33)
Using the series expansion of the logarithm in (26) one obtains the direct
term of the MP2 correlation energy, $E_{c}^{(2)}$ as its second-order term in
$v_{c}$. $E_{c}^{(2)}$ can be evaluated directly in imaginary time and is
given by
$\displaystyle E^{(2)}_{c}=$
$\displaystyle-\frac{1}{2}\sum^{N_{\tau}}_{k}\alpha_{k}\text{Tr}\left(\sum_{\gamma}Z_{\alpha\gamma,\tau_{k}}Z_{\gamma\beta,\tau_{k}}\right)$
(34) $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum^{N_{\tau}}_{k}\alpha_{k}\sum_{\alpha\beta}Z_{\alpha\beta,\tau_{k}}Z_{\beta\alpha,\tau_{k}}\;,$
where $\left\\{\alpha_{k}\right\\}_{k=1,N_{\tau}}$ denote the integration
weights corresponding to the points
$\left\\{\tau_{k}\right\\}_{k=1,N_{\tau}}$. In the spin-polarized case we need
to calculate
$\displaystyle Z_{\alpha\beta,\tau_{k}}Z_{\beta\alpha,\tau_{k}}=$
$\displaystyle\sum_{\sigma=\alpha,\beta}\sum_{\sigma^{\prime}=\alpha,\beta}Z_{\alpha\beta,\sigma,\tau_{k}}Z_{\alpha\beta,\sigma^{\prime},\tau_{k}}\;.$
(35)
When working in the AO basis, we are only interested in the contribution to
$E_{c}^{(2)}$ from electrons with unpaired spins which is used for instance in
spin-opposite scaled (SOS) MP296 or in DOD-DHs.139 In that case, only the
terms with $\sigma\neq\sigma^{\prime}$ contribute and the resulting
correlation energy expression is scaled by an empirical factor.
While PADF can also be used to lower the time complexity of the exchange
contribution to MP2 from $\mathcal{O}\left(N^{5}\right)$ to
$\mathcal{O}\left(N^{3}\right)$83, the resulting working equations can only be
implemented with a very high prefactor and are therefore not useful in
practice. Instead, the full MP2 correlation energy is evaluated in the MO
basis
$\displaystyle
E_{\text{MP2}}=\sum_{ijab}\frac{(ia|jb)^{2}+\frac{1}{2}[(ia|jb)-(ib|ja)]^{2}}{\varepsilon_{i}+\varepsilon_{j}-\varepsilon_{a}-\varepsilon_{b}}\;.$
(36)
The expression is evaluated as in typical DF-MP2 codes.140 The necessary fit-
coefficients are again transformed to the MO basis using the relation (33).
### 2.5 Summary of the projection method thresholds
To improve the numerical accuracy we apply at various stages of the
calculation related techniques which we loosely call projector methods.
Although the eigenvectors of the overlap matrix form an orthogonal basis set,
some of the functions thus constructed are in an Orwellian sense more equal
than others: those with a small eigenvalue are less valuable, and may even be
numerically harmful. In this spirit the first threshold
$\epsilon_{\text{bas}}$ comes from the regularized Löwdin orthonormalization
transformation (20), and is about removing completely a subspace from the
basis set. Next we have the projector (21) associated with an eigenvalue
threshold $\epsilon_{K}$. We can independently apply this to the exchange
matrix $K$ through the similarity tranformation (24) for Hartree Fock
calculations, and to the orbital coefficients like in equation (25) entering
the MP2 and RPA energy expressions. We do not consider the possibility of
using different values for $\epsilon_{K}$ for Hartree Fock, MP2 and RPA, other
than completely bypassing the projector. Finally, when constructing an
automatic fit set for an atom type, again a regularized Löwdin
orthonormalization is used, based on the eigensystem of the overlap matrix of
the fit functions now in the Coulomb metric, controlled by the eigenvalue
threshold $\epsilon_{\text{fit}}$. This last threshold controls the number of
fit functions, see Alg. 1. The same threshold is also employed for the pseudo
inverse to obtain the PADF fit coefficients.
## 3 Computational Details
All calculations have been performed with modified development versions of the
ADF141 and BAND103 modules of AMS2022.97 All Psi4104 calculations have been
performed using version 1.6.1.
### 3.1 AMS calculations
For all calculations using GTOs we used correlation consistent Dunning basis
sets of double-$\zeta$ (DZ), triple-$\zeta$ (TZ), quadruple-$\zeta$ (QZ) and
5Z quality142 from the basis set exchange library.143 For comparison with ADF,
we used the triple-$\zeta$ plus double polarization (TZ2P) basis set.144
In all BAND calculations we used the PADF scheme for the Hartree-Fock exchange
operator while the Hartree potential fitting procedure is based upon a
partitioning of the density in atomic reservoirs each of which is expanded in
products of radial splines and spherical harmonics.145 This procedure does not
rely in any way on the PADF fit functions. Throughout this work we performed
tests varying the threshold $\epsilon_{\text{fit}}$ to control the size of the
fit set. The thresholds used for particular calculations will be indicated in
the next section. The same holds for the threshold for the canonical
orthogonalization of the primary basis. If not indicated otherwise, we set the
numerical quality to _VeryGood_ , which controls the accuracy of the numerical
integration,146 the quality of the ZLMfit,145 as well as of the threshold
controlling distance effects in HF, MP2 and RPA calculations.83 The same
settings have been used in all ADF calculations. In order to make the basis
functions more compact, in BAND the radial part of the basis functions is
multiplied by a Fermi-Dirac (FD) function by default. We disabled this
behavior in all calculations. In all RPA calculations for the S66 dataset we
calculated the polarizability directly in imaginary frequency using (32) and
modified Gauss–Legendre grids as described in ref. 54 with 50 integration
points.
In all calculations for the L7111 and S30L112 datasets we used the imaginary
time based algorithms for RPA and SOS-MP2 algorithm. In all SOS-MP2
calculations we used 12 imaginary time points which ensures $\mu$Hartree
convergence of correlation energies of organic systems with large HOMO-LUMO
gaps.71, 75 In all RPA calculations113, 59 we used 24 imaginary frequency and
imaginary time points each and and used PBE orbitals as input (RPA@PBE).
We calculated the interaction energies of the dimers in the CIM8 dataset68 at
the RPA@PBE level of theory. If not indicated otherwise we used correlation
consistent Dunning basis sets of DZ and TZ quality. We then extrapolated the
final correlation energies to the complete basis set limit using the
relation,147
$E_{\text{CBS}}=E_{xZ}+\frac{E_{xZ}*x^{3}-E_{(x-1)Z}*(x-1)^{3}}{x^{3}-(x-1)^{3}}\;,$
(37)
where $x=3$ for TZ, $x=4$ for QZ, and so on. We set the numerical quality to
_Good_ and set the threshold controlling distance effects for the RPA
calculation to _Normal_. Also here we used various settings for the quality of
the fit set. For details we refer to the next section. For reasons discussed
in the next section, if not indicated otherwise we set $\epsilon_{K}=10^{-2}$
and $\epsilon_{\text{bas}}=5\cdot 10^{-4}$ for calculations on the CIM8
dataset. Detailed input settings for all calculations can be found in the
supporting information.
### 3.2 Psi4 calculations
We performed Psi4 calculations for the S66 database using Dunning
double-$\zeta$ (DZ), triple-$\zeta$ (TZ), quadruple-$\zeta$ (QZ) to
quintuple-$\zeta$ (5Z) basis set, to perform global DF-MP2 calculations (in
the following simply referred to as DF-MP2). We used default settings for all
calculations, We used the default fit sets for each basis set, i.e. cc-pvxZ-
RI118, 148, 149 for the primary basis cc-pvxZ.
## 4 Results
In this section, we assess the accuracy of the algorithms described herein. We
proceed as follows: In section 4.1 we first illustrate the effect of the
threshold chosen for the regularized Löwdin orthogonaliazation (basis set
reduction, $\epsilon_{\text{bas}}$) as well as for the HF projector method
($\epsilon_{K}$), on the exchange matrix, for a simple molecule. In the
subsequent sections, we compare our results for molecules of increasing size,
starting with the S66 database in section 4.2 and moving on to the L7 and S30L
databases which contain molecules with more than 200 atoms in section 4.4.
Finally, in section 4.5 we showcase the capabilities of our PADF based
algorithms by calculating the interaction energies of 7 large non-covalently
bound complexes in the CIM8 set by Neese and coworkers with up to 910 atoms
(4500 electrons).68
### 4.1 Effect of the HF Projector Method
Figure 1: Comparison fit quality $\epsilon_{\text{fit}}$ with PM threshold
$\epsilon_{K}$, in A figure we plot bonding energy of fluorobenzene, in B the
deviation from its MP2 correlation energy computed through Psi4. Picture B
contains a subset of the grid checked in A.
We start the discussion of our results by illustrating the effect of the HF
projector method on the PADF-MP2 correlation energy of the fluorobenzene
molecule for varying size of the fit set. In the heatmaps in Fig. 1A and Fig.
1B, we show the MP2 bonding energy and MP2 correlation energy of fluorobenzene
for different thresholds for the HF PM $\epsilon_{K}$ and fit quality
$\epsilon_{\text{fit}}$. In particular, in 1A we report the bonding energy and
highlight three interesting zones in the heatmap. On the right side, the PM
threshold $\epsilon_{K}$ is large enough to completely remove the exchange
energy thus reducing to the Hartree limit of HF. In the lower left corner
where the quality of the fit set is poor, the PM threshold does only have a
small effect and the PADF-MP2 bonding energy collapses to unphysically low
values. The rest of the heatmap shows a more stable behavior, but still a
broad range of bonding energies. To check the accuracy of the PADF-MP2
implementation in BAND, in Fig. 1B we show values deviating less than 0.1 %
from the reference DF-MP2 correlation energy (Psi4), and we blur in grey the
ones outside of the range. Thus we observe that increasing the threshold
$\epsilon_{K}$, allows to use smaller fit sets while still maintaining a good
precision in the result.
### 4.2 Deviation BAND-Psi4
Figure 2: Absolute differences in non-covalent interaction energies between
Psi4 and BAND and BAND and ADF for the S66 database using different basis
sets. All values expressed in $\nicefrac{{\text{kcal}}}{{\text{mol}}}$
.
Codes | Basis Set | Max. | MAD | Worst molecule
---|---|---|---|---
$\Delta$MP2 | cc-pVTZ | 0.07 | 0.02 | Pentane-Pentane
cc-pVQZ | 0.06 | 0.02 | Pentane-Pentane
cc-pV5Z | 0.05 | 0.01 | Uracil-Cyclopentane
$\Delta$ (ADF-BAND) | TZ2P | 0.05 | 0.01 | AcOH-Uracil
$\Delta$ RPA | cc-pVTZ | 0.14 | 0.05 | Uracil-Uracil ($\pi-\pi$)
Table 1: Maximum deviations and Mean absolute deviations (MAD) between
different PADF and DF implementations of MP2 and RPA for the non-covalent
interaction energies in the S66 database for different basis sets. All values
expressed in $\nicefrac{{\text{kcal}}}{{\text{mol}}}$.: First three rows:
Deviation of DF-MP2 (Psi4) and PADF-MP2 (BAND), fourth row: Deviations of the
PADF-MP2 implementations in ADF and BAND, last row: Deviations of PADF-RPA
(BAND) to DF-RPA (TURBOMOLE) corrected by the frozen core error (see text for
explanation).
We now compare our PADF-MP2 and PADF-RPA results to DF-MP2 and DF-RPA
reference values for the non-covalent interaction energies of the S66
database. We calculated the DF-MP2 reference values using Psi4. As reference
values for PADF-RPA we use the TURBOMOLE results by Furche and coworkers50 who
used the frozen core approximation. Comparing our DF-MP2 results obtained with
Psi4 to the DF-MP2 results of TURBOMOLE, we found the error introduced by the
frozen core approximation to be between 0.0 and 0.1
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$ for relative energies. Therefore, to
allow for a better comparison of our PADF-RPA to the DF-RPA results, we
corrected the latter ones by the frozen core error for MP2, assuming the
impact of the frozen core approximation to be the same for MP2 and RPA. We
stress that this procedure might not remove the frozen core error completely
and therefore the DF-RPA reference values have certainly higher errors bars
than the DF-MP2 ones
In our calculations the relevant thresholds are the ones for the regularized
Löwdin orthonormalization $\epsilon_{\text{bas}}$, the projector method,
$\epsilon_{K}$ and for the size of the fit set, $\epsilon_{\text{fit}}$. For
the MP2 calculations, we used $\epsilon_{\text{bas}}=10^{-8}$ We set the
threshold for the the projector method (PM) to $\epsilon_{K}=10^{-3}$, and we
used $\epsilon_{\text{fit}}=10^{-12}$ except for some of the $5\zeta$
calculations for which we used $10^{-10}$ instead (see supporting information
for details). The $L$-e, was enabled only for $3\zeta$ calculations. This
corresponds to an (unrealistically sized) fit set which is around 15 times
larger than the primary basis.
The absolute deviations (AD) for all PADF-MP2 interaction energies in the S66
database with respect to DF-MP2 are shown in Fig. 2 for Dunning basis sets of
$3\zeta$ to $5\zeta$ quality. In the same plot, we also show the deviations of
the PADF-MP2 implementations in ADF and BAND using STOs. We see that in all
cases, the deviations are smaller than 0.1
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$, irrespective of the basis set. The
small deviations between BAND and Psi4 should primarily be due to errors
introduced by the PADF approximation. Given that the two implementations
differ also in other aspects, we can, however, not exclude the possibility
that differences in other technical parameters might play a role as well, for
instance differences in the definitions of the numerical integration grids.
Also incompleteness of the fit sets used in Psi4 might play a role. The
deviations of BAND to ADF are mostly due to slightly different integration
grids. In any case, for all practical purposes the agreement between the codes
is excellent. Therefore, we did not investigate the precise origin of these
small discrepancies further. The results of these calculations are summarized
in table 1.
In table 1, we show that both maximum error and MAD of PADF-RPA relative to
DF-RPA are about twice as large as for PADF-MP2. The reason for this might be
that the fit errors are more pronounced due to the presence of the higher
powers of $\mathbf{Z}$ (see (27)). There is also a slightly larger uncertainty
in the reference values due to our approach to subtract to the frozen core
errors in the reference calculations. we also assessed the effect of using a
higher threshold of $\epsilon_{\text{bas}}=10^{-5}$. This did however not
change our results at all, demonstrating the numerical stability of our
method.
Figure 3: Differences in MP2 Interaction energies of the S66 dataset with and
without $l$-e in the fit set. All values are in kcal/mol.
The $l$-e procedure is of key importance for both the GTO and STO triple zeta
basis sets. This is illustrated in Fig. 3 where we show the deviations of the
PADF-MP2 interaction energies with and without $l$-e. For the cc-pVTZ basis
set the $l$-e causes differences of the order of $\sim
1.0\nicefrac{{\text{kcal}}}{{\text{mol}}}$ and of $\sim
0.1\nicefrac{{\text{kcal}}}{{\text{mol}}}$ for TZ2P. The larger deviation in
cc-pVTZ from using $l$-e can be traced back to the fact that an STO is added
to a GTO basis set, thus adding functions with a different radial behaviour to
the fit. For the TZ2P basis set the only difference comes from the higher
angular momentum in the basis. The same comparison of PADF-RPA correlation
energies gave similar results.
### 4.3 Comparison of projector methods
Label | $\epsilon_{K}$ (HF) | $\epsilon_{K}$ (MP2) | $\epsilon_{\text{bas}}$
---|---|---|---
None | $\times$ | $\times$ | 1e-8
Bas | $\times$ | $\times$ | 1e-4
HF | 1e-3 | $\times$ | 1e-8
MP2 | $\times$ | 1e-3 | 1e-8
Both | 1e-3 | 1e-3 | 1e-8
Figure 4: Absolute differences in non-covalent MP3 interaction energies
between Psi4 and BAND for the S66 database using the cc-pVQZ basis set and
yarying parameters for the different projector methods. Each line in the
figure correspond to a different combination of the three parameters in the
table. Symbol $\times$ means that the respective method has not been used. All
values are in $\nicefrac{{\text{kcal}}}{{\text{mol}}}$.
In Fig. 4 we compare BAND results using several combinations of the above
mentioned thresholds and show absolute deviation with respect to Psi4 for the
S66 dataset. The names for the different parameters combinations are specified
in the table at the bottom of the same figure. We compare results obtained
without any projector(None), without PM-$K$ but using a tighter threshold for
the regularized Löwdin orthonormalization in the basis set (Bas), using PM-$K$
only at the HF or at the MP2 stage, and finally using PM-$K$ in both cases
(mentioned following the order of the table). The best results are obtained
applying PM-$K$ to MP2 only, followed by applying it to both. This shows that
the usage of PM-$K$ at the HF stage does not contribute much to the overall
accuracy and in fact can even worsen it. This supports our hypothesis from
section 2 since correlation energies are dependent on virtual orbitals which
have more nodes and are more diffuse than the occupied ones. Hence, they are
more difficult to represent in terms of an atomic centered basis and
consequently their products more difficult to express in terms of fit
functions. We emphasize however that we use here a very large fit set. When
fewer fit functions are used, PM-$K$ also needs to be used at the SCF stage to
prevent variational collapse. The calculations using a larger value of the
Löwdin orthonormalization threshold lead to improvements with respect to the
None and HF settings since unstable components of the basis are projected out
from all terms of the Fock matrix. This is decisive for the accuracy of MP2
and RPA correlation energies as we have already seen in the comparisons above.
### 4.4 Interaction energies for L7 and S30L
After having demonstrated the excellent agreement of our PADF-MP2/RPA results
with DF-MP2/RPA also for large basis sets up to 5Z quality, we now discuss the
accuracy of PADF-MP2 for the molecules in the L7 and S30L databases, for which
Furche and coworkers have recently published DF-MP2 reference values.50 We
focus on the direct contribution to the MP2 correlation energy only and
calculate SOS-MP2 interaction energies using our imaginary time based PADF-
SOS-MP2 implementation.83 To represent the imaginary time dependence we chose
here 12 integration points in the interval $[0,\infty)$ which is sufficient to
achieve $\mu$H precision in absolute correlation energies. The reference
values by Furche and coworkers have been calculated with the frozen core
approximation.50 We have however seen for the S66 database, that the maximum
error of this approximation is of the order of 0.1
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$ for relative energies and we do not
expect this to change for larger molecules. Finally, Furche and coworkers only
calculated correlation energies using Dunning basis sets while they used basis
sets of the Kahlsruhe type to calculate their HF or KS energies.50 Therefore,
we do not compare the non-covalent interaction energies for these sets but
only the correlation contributions to them.
#### 4.4.1 L7
$\epsilon_{K}$ | $l_{\text{max}}$ | Non-covalent interaction energy | $\Delta_{\text{TURBOMOLE}}$
---|---|---|---
$1e-3$ | 6 | | | | | | | |
System | Ref. | 1e-6 | 1e-8 | 1e-10 | 1e-12 | 1e-6 | 1e-8 | 1e-10 | 1e-12
c2c2PD | $-33.72$ | $-34.92$ | $-34.46$ | $-34.20$ | $-34.22$ | $-1.20$ | $-0.74$ | $-0.48$ | $-0.50$
c3a | $-22.07$ | $-22.96$ | $-22.65$ | $-22.37$ | $-22.53$ | $-0.89$ | $-0.58$ | $-0.30$ | $-0.46$
c3gc | $-37.81$ | $-39.37$ | $-38.81$ | $-38.38$ | $-38.85$ | $-1.56$ | $-1.00$ | $-0.57$ | $-1.04$
cbh | $-11.49$ | $-13.19$ | $-12.61$ | $-12.38$ | $-12.26$ | $-1.70$ | $-1.12$ | $-0.89$ | $-0.77$
gcgc | $-18.44$ | $-19.64$ | $-19.14$ | $-18.92$ | $-22.37$ | $-1.20$ | $-0.70$ | $-0.48$ | $-3.93$
ggg | $-7.96$ | $-8.32$ | $-8.21$ | $-8.13$ | $-8.59$ | $-0.36$ | $-0.25$ | $-0.17$ | $-0.63$
phe | $-5.15$ | $-5.87$ | $-5.63$ | $-5.51$ | – | $-0.72$ | $-0.48$ | $-0.36$ | –
$\epsilon_{K}$ | $l_{\text{max}}$ | Non-covalent interaction energy | $\Delta_{\text{TURBOMOLE}}$
$1e-3$ | 8 | | | | | | | |
System | Ref. | 1e-6 | 1e-8 | 1e-10 | 1e-12 | 1e-6 | 1e-8 | 1e-10 | 1e-12
c2c2PD | $-33.72$ | $-34.82$ | $-34.23$ | $-33.75$ | | $-1.10$ | $-0.51$ | $-0.03$ |
c3a | $-22.07$ | $-22.90$ | $-22.50$ | $-28.58$ | | $-0.83$ | $-0.43$ | $-6.51$ |
c3gc | $-37.81$ | $-39.26$ | $-38.45$ | $-38.06$ | | $-1.45$ | $-0.64$ | $-0.25$ |
cbh | $-11.49$ | $-12.96$ | $-12.25$ | $-11.94$ | | $-1.47$ | $-0.76$ | $-0.45$ |
gcgc | $-18.44$ | $-19.53$ | $-18.97$ | $-29.75$ | | $-1.09$ | $-0.53$ | $-11.31$ |
ggg | $-7.96$ | $-8.29$ | $-8.15$ | $-8.37$ | | $-0.33$ | $-0.19$ | $-0.41$ |
phe | $-5.15$ | $-5.82$ | $-5.53$ | $-6.02$ | | $-0.67$ | $-0.38$ | $-0.87$ |
$\epsilon_{K}$ | $l_{\text{max}}$ | Non-covalent interaction energy | $\Delta_{\text{TURBOMOLE}}$
$5e-3$ | 6 | | | | | | | |
System | Ref. | 1e-6 | 1e-8 | 1e-10 | 1e-12 | 1e-6 | 1e-8 | 1e-10 | 1e-12
c2c2PD | $-33.72$ | $-34.02$ | $-33.80$ | $-33.69$ | $-33.69$ | $-0.30$ | $-0.08$ | $0.03$ | $0.03$
c3a | $-22.07$ | $-22.15$ | $-22.04$ | $-21.96$ | $-21.97$ | $-0.08$ | $0.03$ | $0.11$ | $0.10$
c3gc | $-37.81$ | $-37.95$ | $-37.76$ | $-37.63$ | $-37.67$ | $-0.14$ | $0.05$ | $0.18$ | $0.14$
cbh | $-11.49$ | $-12.02$ | $-11.85$ | $-11.78$ | $-11.71$ | $-0.53$ | $-0.36$ | $-0.29$ | $-0.22$
gcgc | $-18.44$ | $-18.46$ | $-18.35$ | $-18.29$ | $-18.60$ | $-0.02$ | $0.09$ | $0.15$ | $-0.16$
ggg | $-7.96$ | $-7.95$ | $-7.92$ | $-7.91$ | $-7.93$ | $0.01$ | $0.04$ | $0.05$ | $0.03$
phe | $-5.15$ | $-5.31$ | $-5.22$ | $-5.19$ | $-8.18$ | $-0.16$ | $-0.07$ | $-0.04$ | $-3.03$
Table 2: Comparison of SOS-MP2 contributions to the non-covalent interaction
energies in the L7 database to the ones from Furche and coworkers50 for
different numerical settings using the cc-pvTZ basis set. All values are in
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$. The fit sets have been generated
from the basis products using the threshold $\epsilon_{\text{fit}}$
For the fit sets generated from the products of basis functions, the results
of this comparison for the L7 database can be found in table 2. We first focus
on the upper table. The results here have been obtained using a value of
$\epsilon_{K}=10^{-3}$ as threshold for PM-$K$ (the same value as for S66) and
without the $l$-e method. The results for the different values of
$\epsilon_{k}$ controlling the size of the fit set ranging from $10^{-6}$ to
$10^{-12}$ in table 2 show a slow convergence of the relative SOS-MP2
correlation energies to the DF-SOS-MP2 reference values. However, even with
the already very large fit set corresponding to $\epsilon_{k}=10^{-10}$ (the
number of fit functions is around 10 times larger than the number of primary
basis functions), the maximum deviation is still 0.89
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$ and only for the smallest of the
systems (GGG) in L7, the deviation to the TURBOMOLE results reaches an
acceptable value of 0.17 $\nicefrac{{\text{kcal}}}{{\text{mol}}}$. Moreover,
the results for $\epsilon_{k}=10^{-12}$ start to worsen, which can be related
to the occurrence of linear dependencies in the fit set. For the Phe system in
L7 we even obtained a completely unreasonable value for one of the correlation
energies.
Turning to the second table in table 2, we find that the $l$-e reduces the
error with respect to DF-SOS-MP2 compared to the value obtained with the same
thresholds. However, we already observe numerical instabilities for fit sets
corresponding to $\epsilon_{k}=10^{-10}$. Notice however, that the total size
of this fit set is already larger than the one corresponding to
$\epsilon_{k}=10^{-12}$ without the $l$-e. Therefore, also the $l$-e method
does not resolve the issues of PADF for the molecules in L7.
We now turn to the third table in table 2 which shows results obtained without
the $L$-e method but with a larger threshold of the HF projector method,
$\epsilon_{K}=5\cdot 10^{-3}$, instead of $\epsilon_{K}=10^{-3}$ which has
been used in the two previous tables. This changes the PADF-SOS-MP2
correlation energies drastically and bring them into much better agreement
with the TURBOMOLE results. Already for a moderate size of the fit set
corresponding to $\epsilon_{k}=10^{-6}$, the maximum deviation is reduced to
0.53 $\nicefrac{{\text{kcal}}}{{\text{mol}}}$ and for a value of
$\epsilon_{k}=10^{-8}$, for 6 out of 7 systems the agreement with the
reference values is better than 0.1 $\nicefrac{{\text{kcal}}}{{\text{mol}}}$.
Notice again, that 0.1 $\nicefrac{{\text{kcal}}}{{\text{mol}}}$ is of the
order of uncertainty in the reference values due to the frozen core
approximation. The CBH complex is the only system for which the deviation to
the reference is still relatively large. Only with $\epsilon_{k}=10^{-12}$,
the deviation to the TURBOMOLE results reduces to an acceptable value of
$0.22\nicefrac{{kcal}}{{mol}}$. However, first this fit set is very large and
therefore not very useful in applications to large molecules, and second, we
can still observe quite large deviations to TURBOMOLE data for other systems,
most drastically for PHE.
$\epsilon_{K}=5\times 10^{-3}$ | SOS-MP2 | RPA
---|---|---
System | $E^{\text{SOS-MP2}}_{\text{Ref.}}$ | normal | good | vg | $E^{\text{RPA}}_{\text{Ref.}}$ | normal | good | vg
c2c2PD | $-33.72$ | $-2.09$ | $-0.41$ | $0.09$ | $-30.29$ | $-2.17$ | $-0.32$ | $0.30$
c3a | $-22.07$ | $-1.32$ | $-0.20$ | $0.14$ | $-21.21$ | $-1.43$ | $-0.12$ | $0.32$
c3gc | $-37.81$ | $-2.16$ | $-0.30$ | $0.27$ | $-39.89$ | $-0.19$ | $2.02$ | $2.76$
cbh | $-11.49$ | $-0.83$ | $-0.10$ | $-0.03$ | $-16.84$ | $-0.94$ | $-0.03$ | $0.10$
gcgc | $-18.44$ | $-0.72$ | $-0.05$ | $0.21$ | $-22.33$ | $-0.89$ | $0.04$ | $0.37$
ggg | $-7.96$ | $-0.28$ | $-0.03$ | $0.07$ | $-9.00$ | $-0.32$ | $0.02$ | $0.15$
phe | $-5.15$ | $-0.51$ | $-0.04$ | $0.02$ | $-8.53$ | $-0.60$ | $-0.01$ | $0.10$
$\epsilon_{K}=10^{-2}$ | SOS-MP2 | RPA
System | $E^{\text{SOS-MP2}}_{\text{Ref.}}$ | normal | good | vg | $E^{\text{RPA}}_{\text{Ref.}}$ | normal | good | vg
c2c2PD | $-33.72$ | $-1.31$ | $-0.01$ | $0.09$ | $-30.29$ | $-1.24$ | $0.22$ | $0.71$
c3a | $-22.07$ | $-0.84$ | $0.06$ | $0.14$ | $-21.21$ | $-0.76$ | $0.28$ | $0.62$
c3gc | $-37.81$ | $-1.37$ | $0.11$ | $0.27$ | $-39.89$ | $0.90$ | $2.68$ | $3.25$
cbh | $-11.49$ | $-0.73$ | $-0.07$ | $-0.03$ | $-16.84$ | $-0.73$ | $0.09$ | $0.17$
gcgc | $-18.44$ | $-0.57$ | $0.07$ | $0.21$ | $-22.33$ | $-0.66$ | $0.28$ | $0.55$
ggg | $-7.96$ | $-0.17$ | $0.02$ | $0.07$ | $-9.00$ | $-0.15$ | $0.12$ | $0.22$
phe | $-5.15$ | $-0.43$ | $-0.05$ | $0.02$ | $-8.53$ | $-0.43$ | $0.04$ | $0.11$
Table 3: Deviations $\Delta_{\text{TURBOMOLE}}$ of SOS-MP2 (left) and RPA
(right) contributions to the non-covalent interaction energies in the L7
database to the ones from Furche and coworkers50 for different numerical
settings using the cc-pvTZ basis set. All values are in
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$. Standard STO type fit sets of
varying size, randing from _Basic_ to _VeryGood_ (vg) quality have been used.
##### STO type fit sets
Out of all systems in L7, CBH contains the largest number of atoms, but it is
not the system with most electrons. Therefore, it can be considered as the
most spatially extended system. As we have already discussed, such systems are
expected to be most problematic for PADF based methods since many products of
diffuse functions will occur. Since GTOs decay much faster than STOs from
their atomic centres, it is natural to ask whether GTOs are the best choice to
fit such products of diffuse atomic orbitals. Therefore, we investigate the
accuracy which can be achieved by fitting the products of GTOs with STO type
functions in table 3. In the second table table of table 3 we show some
results for $\epsilon_{K}=10^{-2}$. Additionally, we have reduced the
integration quality to _Good_ and the quality of the threshold controlling
distance effects in the SOS-MP2 method to _Normal_. Both settings together
drastically speed up the PADF-SOS-MP2 calculations.
Using the _Normal_ fit set, relatively large errors are obtained. The results
using the _Good_ and _VeryGood_ fit sets are in relatively good agreement with
TURBOMOLE. Especially for the CBH complex, the deviation is much smaller than
for the GTO type fit sets. Increasing $\epsilon_{K}$ to $10^{-1}$ improves
agreement with the TURBOMOLE results further and using the _Good_ fit set
results in perfect agreement with the reference values (given their
uncertainties due to the frozen core approximation) are obtained.
Independently of the value of $\epsilon_{K}$, the errors of the relative RPA
correlation energies are of the same order of magnitude as for SOS-MP2.
However, as already observed for the S66 benchmark, the errors tend to be
slightly larger. We also observe that relative energies tend to be less
negative, indicating that smaller values of $\epsilon_{K}$ than for PADF-SOS-
MP2 are beneficial for PADF-RPA results. Also here, using the _Normal_ fit
set, errors hardly exceed 1 $\nicefrac{{\text{kcal}}}{{\text{mol}}}$. The
system c3gc (A GC base pair absorbed on a Circumcoronene molecule) is
particularly problematic with errors of the order of 3
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$ for the _VeryGood_ fit set. We show
in the supporting information that lowering of $\epsilon_{K}$ leads to much
better agreement with experiment. This shows that the optimal settings for the
projector methods are system specific and further research is needed to
understand these patterns.
##### PADF-SOS-MP2 results using the CC-pVQZ basis set
| | Non-covalent interaction energy | $\Delta_{\text{TURBOMOLE}}$
---|---|---|---
System | Ref. | 1e-6 | 1e-8 | vg | 1e-6 | 1e-8 | vg
c2c2PD | $-35.47$ | $-35.63$ | | $-36.66$ | $-0.16$ | | $-1.19$
c3a | $-23.43$ | $-23.54$ | | $-24.15$ | $-0.11$ | | $-0.72$
c3gc | $-40.33$ | $-40.58$ | | $-41.50$ | $-0.25$ | | $-1.17$
cbh | $-12.27$ | $-12.64$ | $-12.43$ | $-12.74$ | $-0.37$ | $-0.16$ | $-0.47$
gcgc | $-20.03$ | $-20.22$ | | $-20.53$ | $-0.19$ | | $-0.50$
ggg | $-8.48$ | $-8.53$ | | $-8.67$ | $-0.05$ | | $-0.19$
phe | $-6.40$ | $-6.62$ | $-6.53$ | $-6.74$ | $-0.22$ | $-0.13$ | $-0.34$
Table 4: Comparison of SOS-MP2 contributions to the non-covalent interaction
energies in the L7 database to the ones from Furche and coworkers50 for
different numerical settings using the CC-pvQZ basis set. All values are in
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$. All values are obtained using
$\epsilon_{K}=5\cdot 10^{-3}$.
Lastly, we examine the quality of the PADF approximation at the QZ level. As
for the S66 database, the results shown in table 4 demonstrate that the
quality of the interaction energies is not deteriorated compared to the TZ
level. Already with the rather moderate threshold of $\epsilon_{k}=10^{-6}$,
the maximum deviation is -0.37 $\nicefrac{{\text{kcal}}}{{\text{mol}}}$ for
CBH. This value is reduced to -0.16 $\nicefrac{{\text{kcal}}}{{\text{mol}}}$
when the threshold is decreased to $\epsilon_{k}=10^{-8}$. Only using the
_VeryGood_ Slater type fit set leads to worse results than at the TZ
level.This is due to the fact that the _VeryGood_ fit set only contains
functions with $l\leq 6$, with the number of $l=6$ functions being rather
small, while the product basis contains functions with $l\leq 8$, with a large
number of functions with $l\geq 6$.
#### 4.4.2 S30L
Figure 5: Deviations of PADF-SOS-MP2 and PADF-RPA with different STO type fit
sets with respect to the TURBOMOLE reference values using global DF. All
calculations have been performed using the cc-pvTZ basis set and the settings
from table 3. All deviations are in kcal/mol.
To further assess the quality of PADF-SOS-MP2, we also calculate the SOS-MP2
and RPA contributions to the interaction energies in the S30L set, which
contains 30 non-covalently bound complexes with up to 205 atoms, out of which
8 are charged.We have omitted the systems 15 and 16 which contain Iodine atoms
and for which a comparison to TURBOMOLE would be difficult due to the use of
pseudopotentials in ref. 50. We also do not show results for 3 of the charged
systems due to convergence difficulties. The deviations to the respective
TURBOMOLE results are shown in figure 5 and confirm the observations for the
L7 database in table 3. When the _Good_ fit set is used, the SOS-MP2 and RPA
errors never exceed 0.5 and 1.0 kcal/mol, respectively. These results also
demonstrate that the errors in relative energies do not seem to depend much on
the sizes of the complexes any more after a certain system size is reached.
This is in fact the expected results, since AOs centered on two atoms very far
apart form each other will not overlap and therefore not contribute to the
PADF errors.
### 4.5 RPA Interaction energies of large complexes
After having demonstrated the relatively good accuracy of the PADF
approximation also for the correlation energies of large molecules, we now
calculate the interaction energies in the CIM (Cluster-in-molecule)8 set by
Neese and coworkers68 at the RPA@PBE level of theory. It comprises 8 large
non-covalently bound complexes ranging in size from 200 to 1027 atoms, with
interactions dominated by either $\sigma$-$\sigma$ dispersion or hydrogen
bonding.68
Given the size of these systems, it is clear that these interaction energies
can only be calculated if certain approximations are introduced. Neese and
coworkers used an approach in which they decomposed the complex into smaller
clusters and calculated the correlation energies using a cluster expansion of
the general form
$E^{\text{corr}}=\sum_{I}E^{\text{corr}}_{I}+\sum_{I<J}E^{\text{corr}}_{IJ}\;,$
(38)
where $I$ denotes a subset of localized occupied and virtual molecular
orbitals.68 They proposed to evaluate the correlation energy by two RI-MP2
calculations using the (aug-)cc-pvDZ and the (aug-)cc-pVTZ basis sets and by a
DLPNO-CCSD(T) correction using the smaller basis. They used the augmented
basis sets only for the two smallest systems while for the other systems they
used the non-augmented basis sets.
Basis set incompleteness errors aside, there are major sources of inaccuracies
with this approach, which could potentially lead to large errors. First, both
the CIM and the DLPNO approximations can introduce errors in relative energies
of several $\nicefrac{{\text{kcal}}}{{\text{mol}}}$ for non-covalent
interactions, even when ”tight” truncation thresholds for the DLPNO settings
are used.28, 29 Second, the chosen extrapolation scheme assumes that the CC
basis set error can be faithfully estimated at the MP2 level. This is however
not necessarily the case since the major basis set errors in CC calculations
arise from the direct MP2 contribution,151, 152, 153 a observation termed as
interference effect by Petersson et al.154 Especially since MP2 correlation
energies will be rather inaccurate for the large molecules in CIM8, the
strategy to estimate the basis set incompleteness error at the MP2 level will
be error-prone. Finally, we mention the recently observed disagreement of well
converged CCSD(T) interaction energies with quantum diffusion Monte Carlo
methods for large non-covalently bound complexes.26 For all these reasons, the
values by Neese and coworkers are certainly not of quantitative accuracy.
Despite all this, they are the most accurate reference values which are
available for these large systems and certainly serve as useful frame of
reference for our RPA@PBE calculations.
We have calculated all RPA@PBE interaction energies using (37) using cc-pvDZ
and cc-pVTZ basis sets with and without counterpoise corrections calculations.
We have extrapolated the RPA correlation energies only, while the TZ results
has been used for the remaining components of the interaction energies. We
have verified the accuracy of this strategy by comparison to results using a
(T,Q) extrapolation with the cc-pVTZ and cc-pVQZ basis sets. As shown in table
S1 in the supporting information, the results of both extrapolation schemes
differ by about 6.5 kcal/mol. This also indicates that for systems as large as
the ones in CIM8 for which QZ calculations are out of reach, basis set
incompleteness errors are typically much larger than the errors introduced by
the PADF approximation.
| | | $E_{RPA}$ (D,T) (% cp) | | $\Delta E$ (% cp)
---|---|---|---|---|---
System | $N_{\text{atom}}$ | $N_{\text{bas}}$ | 100 % | 0 % | E(ref.) | 100 % | 0 %
1 | $200$ | $5360$ | $-58.22$ | $-67.80$ | $-70.11$ | $11.89$ | $2.31$
2 | $296$ | $6576$ | $-55.76$ | $-62.79$ | $-63.61$ | $7.85$ | $0.82$
4 | $328$ | $9072$ | $-31.01$ | $-34.67$ | $-36.55$ | $5.54$ | $1.88$
3 | $381$ | $10\,806$ | $-14.09$ | $-24.90$ | $-17.83$ | $3.74$ | $-7.07$
5 | $552$ | $12\,080$ | $-34.31$ | $-48.67$ | $-40.13$ | $5.82$ | $-8.54$
6 | $750$ | $17\,316$ | $-58.97$ | $-86.81$ | $-78.80$ | $19.83$ | $-8.01$
7 | $910$ | $21\,932$ | $-336.83$ | $-414.75$ | $-416.08$ | $79.25$ | $1.33$
8 | $1027$ | $22\,778$ | $-25.58$ | $-41.27$ | $-35.70$ | $10.12$ | $-5.57$
MAD: | | | | | | $18.00$ | $4.44$
Table 5: Interaction energies for eight large-non-covalently bound complexes.
The number of basis functions refers to the full complex using cc-pVTZ. The
ref. energy in the second last column has been taken from Neese and
coworkers68 and has been calculated at the CIM-DLPNO-CCSD(T)—CIM-RI-MP2(D,T)Z
level of theory.68 All interaction energies are in
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$ and have been extrapolated using (37)
with cc-pVDZ and cc-pVTZ. The counterpoise (cp) correction used is indicated
by a percentage (0% is no cp).
The results of our RPA@PBE calculations are shown in table 5. The counterpoise
corrected results are the most accurate RPA@PBE interaction energies we can
calculate for these large systems. Given the good accuracy of RPA@PBE for
large non-covalently bound complexes50, they might serve as reference values
for more approximate methods. However, they can not be compared directly to
the values by Neese and coworkers since they did not correct for basis set
superposition errors. For this purpose, we also calculated interaction
energies which are not counterpoise corrected.
Overall, reasonable agreement of our RPA@PBE results with the reference values
by Neese and coworkers68 is observed. With a MAD of 4.44
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$, the deviations are of the same order
of magnitude than popular (dispersion corrected) density functionals155, 68,
156 (e.g. $\omega$B97-X-D with a MAD of 5.06
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$ or B3LYP-D4 with a MAD of 4.81
$\nicefrac{{\text{kcal}}}{{\text{mol}}}$) and smaller than the ones for other
_ab initio_ methods like SCS-MP2.68 We emphasize again that even though the
agreement to the CIM-DLPNO-CCSD(T) reference is satisfactory, these non-
counterpoise corrected interaction energies come with large basis set
superposition errors and are therefore likely to be incorrect (see also table
S2 in the supporting information).
## 5 Conclusion
By comparison to DF-MP2 and DF-RPA, we demonstrated the accuracy of PADF-MP2
and PADF-RPA for the S66, L7, and S30L sets of non-covalently bound complexes
ranging from 6 to more than 200 atoms in size. Especially for the small to
medium molecules in the S66 database, PADF comes with negligible loss of
accuracy compared to global density fitting.
The main advantage of PADF over global DF is that is leads to very fast
algorithms for RPA and SOS-MP2. We have shown that the PADF approach is
suitable to calculate the interaction energies of large molecules. In
particular, we calculated the PADF-RPA@PBE interaction energies of eight large
non-covalently bound complexes at the cc-pVTZ level with more than 1000 atoms
and more than 20000 AOs in size on a single compute node.
The choice of fit set is decisive for precise correlation energies with PADF.
We tested two different types of fit sets. In a first variant, the fit set is
generated directly from products of basis functions.94 In a second variant,
fit sets consisting of even tempered series of STOs are used.83 Despite being
much smaller, we found the second kind of fit set to be suitable to express
products of GTOs. This might be due to the slow decay of radial part of the
STOs making them more suitable to fit delocalized products of AOs.
To improve the precision of the PADF approach further, we introduced a
projector which acts directly on the Fock matrix and removes the attractive
component of the excact exchange. most importantly, we also use this method to
project out subspaces of AOs from the orbital coefficients matrix which can
only be represented poorly by the fit set. Especially when smaller fit sets
are used, we showed the PM-$K$ to be of key importance for accurate
interaction energies.
While the PM reduces the correlation energy errors arising from PADF it cannot
completely eliminate them. This is especially true for large molecules, for
which a compromise between accuracy and computational efficiency is required.
For very large systems like the molecules in the CIM8 dataset, it becomes
mandatory to use smaller fit sets which might introduce errors in interaction
energies which can exceed 1 $\nicefrac{{kcal}}{{mol}}$. This is however also
true for many approximations to high-level methods for the calculation of
correlation energies, for instance CC methods based on localized orbitals
using the DLPNO28, 29 or LNO approximations.26 At the moment, the HF projector
method is not used in a system specific way. A computationally efficient way
to do so could be to check the definiteness of the Hartree-exchange matrix at
runtime and to use this information to adjust $\epsilon_{K}$ during the SCF.
In practice, the errors stemming from the PADF approximation will often be of
only minor relevance. Especially MP2 is typically not used by itself but
rather in double hybrids functionals which typically use a fraction of around
30-60 % of MP2 correlation energy,35, 30 scaling the error by the same amount.
Therefore, the already small PADF errors will be negligible for those
functionals. Furthermore, when small fit sets are used, the fit set
incompleteness error always leads to too low interaction energies while basis
set incompleteness errors lead to too high interaction energies. Especially
for large molecules with several hundreds of atoms for which QZ calculations
are not feasible, the fit set incompleteness errors will be much smaller than
basis set incompleteness errors.
On a more general note, it has recently been recognized that methods which
agree well with each other for small and medium molecules might give very
different results for larger systems.26 Also, approximate (dispersion
corrected) GGAs or hybrid functionals which work well for smaller systems are
much more error-prone for large molecules.155, 156 In order to understand the
reasons for this discrepancy, it is mandatory to push the boundaries of first-
principle methods to much larger systems. At the moment, this comes at the
price of numerical errors and further research is needed to develop techniques
to mitigate these errors further.
We thank Erik van Lenthe for fruitful discussions and the idea behind the
projector methods and acknowledge the use of the computing facilities of Vrije
Universiteit Amsterdam and SCM. Edoardo Spadetto acknowledges funding from the
European Union’s Horizon 2020 research and innovation program under grant
agreement No 956813 (2Exciting).
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|
# Approximating Intersections and Differences Between Linear Statistical Shape
Models Using Markov Chain Monte Carlo
Maximilian Weiherer, Finn Klein, Bernhard Egger
Department of Computer Science
Friedrich-Alexander-Universtität Erlangen-Nürnberg
<EMAIL_ADDRESS>
###### Abstract
To date, the comparison of Statistical Shape Models (SSMs) is often solely
performance-based, carried out by means of simplistic metrics such as
compactness, generalization, or specificity. Any similarities or differences
between the actual shape spaces can neither be visualized nor quantified. In
this paper, we present a new method to qualitatively compare two linear SSMs
in dense correspondence by computing approximate intersection spaces and set-
theoretic differences between the (hyper-ellipsoidal) allowable shape domains
spanned by the models. To this end, we approximate the distribution of shapes
lying in the intersection space using Markov chain Monte Carlo and
subsequently apply Principal Component Analysis (PCA) to the posterior
samples, eventually yielding a new SSM of the intersection space. We estimate
differences between linear SSMs in a similar manner; here, however, the
resulting spaces are no longer convex and we do not apply PCA but instead use
the posterior samples for visualization. We showcase the proposed algorithm
qualitatively by computing and analyzing intersection spaces and differences
between publicly available face models, focusing on gender-specific male and
female as well as identity and expression models. Our quantitative evaluation
based on SSMs built from synthetic and real-world data sets provides detailed
evidence that the introduced method is able to recover ground-truth
intersection spaces and differences accurately.
## 1 Introduction
Statistical Shape Models (SSMs) are a popular class of generative models
providing a low-dimensional parametric representation of complex objects. SSMs
and especially 3D Morphable Models (3DMMs) [2] are widely used within the
computer vision community and often applied to model humans (faces, bodies,
bones, and organs). Their application ranges from face recognition [3],
single-shot 3D face [14, 40, 25] and body reconstruction [29], face
reenactment [37] and visual dubbing, to applications in the medical domain
[31, 17, 39], forensics, cognitive science, neuroscience, and psychology [12].
SSMs are typically built by applying Principal Component Analysis (PCA) to a
set of objects in dense correspondence. As such, an SSM is a linear model in
which shapes are represented as points in a low-dimensional, affine vector
space. Although other, non- or multi-linear models exist, PCA-based SSMs are
still most common. They are easy to interpret, convenient to visualize, have
excellent extrapolation capabilities, and are compatible with standard
computer graphics pipelines. Hence, we consider only PCA-based models in this
work. In general, however, the presented method can be used with all models
that (i) use an affine vector space as an approximation for the underlying
(manifold) shape space, and (ii) allow random sampling from and projecting
shapes onto that space, including PCA-based Point Distribution Models [8] and
Gaussian Process Morphable Models [31], multi-linear [4] and wavelet-based
models [6], non-linear models based on Principal Geodesic Analysis (by
applying our method on the tangent space) [16], but also more modern
approaches based on Variational Autoencoders [24].
To date, comparison of SSMs is typically solely performance-based and carried
out by applying metrics such as compactness, generalization, and specificity
[10, 36]. A qualitative and direct comparison of the actual shape spaces
spanned by two models is currently not possible; a visualization can only be
provided for the individual shape spaces by inspecting random samples or the
first principal modes of variation. As such, any differences or similarities
between two models’ shape spaces can not be computed nor visualized with
existing metrics.
In this paper, we present a new approach to qualitatively compare two linear
SSMs that takes into account the affine vector spaces spanned by the models.
Specifically, we aim at computing the intersection and set-theoretic
difference between the models’ (hyper-ellipsoidal) allowable shape domains,
_i.e_., those shapes that can either be explained by both SSMs or only by one
model but not the other. Given the extreme difference in dimensions between
the low-dimensional shape spaces and their high-dimensional embedding, we
propose to use a sampling-based technique to approximate the desired spaces.
Starting with a geometric motivation and formal definition, we formulate a
probability distribution over shapes lying in the intersection or set-
theoretic difference and use approximate inference based on Markov chain Monte
Carlo (MCMC) to generate samples from those spaces. Based on these samples, we
compute the mean and a basis for the intersection space using PCA, eventually
forming a new SSM that can be used for visualization, data exploration, and
analysis. Since the set-theoretic difference of two models’ subspaces is no
longer convex, we do not apply PCA but only visualize posterior samples.
Finally, note that thanks to the probabilistic nature of our method, we are
not only able to inspect differences and similarities on shapes as a whole but
also on a per-vertex level, allowing us to understand which features are
present in both models and/or what is unique.
Our method has several interesting applications beyond a general data
exploration use case. For instance, differences between race-specific models
(_e.g_., between Asian and White models) could allow for conclusions about
demographic bias in SSMs. In the medical domain, differences and similarities
between two models built from a healthy and pathological group will help to
visualize and identify novel phenotypes or shape-based clinical indicators.
Moreover, the difference between those two models or gender-specific shape
variations may be of great interest as it could be later added to other models
to increase variability, thus acting as a data augmentation strategy. Besides
medical applications, we also see opportunities in product design. When
modeling body parts, the comparison of two SSMs from distinct populations may
be helpful to improve the design for a specific market. Finally, the
difference between face identity and expression models would also reduce the
effects of the identity-expression ambiguity in 3DMMs of faces [13], allowing
for better expression neutralization or transfer.
To summarize, the key contributions of this paper are: (i) we present a new
method to qualitatively compare two SSMs by computing approximate intersection
spaces and set-theoretic differences between the low-dimensional allowable
shape domains spanned by the models, (ii) we provide an extensive,
quantitative evaluation using models built from synthetic data sets and SSMs
which are publicly available, (iii) we analyze intersections and differences
between popular face models, including gender-specific male and female and
identity and expression models, and (iv) we show how our method can be used to
approximate intersection spaces and differences between texture models of
3DMMs.
## 2 Related Work
Although tackling a different problem, most related to our work is the method
proposed by Hall et al. [22]. They present a splitting operation used to
remove one linear subspace from another and argue that this operation is seen
as the inverse of the union operation. It is important to understand that this
method yields again a linear and convex subspace; they do not compute the set-
theoretic difference between two linear models’ shape spaces. Rather, they
split one subspace from the other, effectively keeping the intersection in
that space. In contrast, the set-theoretic difference we are interested in
excludes the intersection from this space (this is why we end up with a non-
convex space). This allows us to investigate the true difference between two
models, _i.e_., those shapes that are exclusively represented in one model,
which is not possible with [22].
Performance-based metrics. The most well-known performance-based metrics to
compare SSMs are compactness, generalization, and specificity [10, 36]. Those
metrics allow to measure how well a model represents a certain population;
they enable quantitative comparison of two models of the same population and
are applied in most of the publications presenting novel SSMs. However,
although considered state-of-the-art when comparing SSMs, they do not allow
for a visual inspection of similarities or differences between actual shape
spaces. Babalola et al. [1] proposed to use the Bhattacharya distance to
measure the overlap between the two distributions implied by SSMs. Although
this might be useful in practice, it merely returns a scalar-valued distance.
Another common way to compare models is by evaluating their performance on
downstream applications [5, 18, 26, 35]. This, however, does not tell anything
about what makes one model different or superior to another from a
probabilistic point of view. In general, none of the existing approaches
enable visualization of similarities or differences between low-dimensional
shape spaces.
Computing intersections between vector spaces. Linear algebra offers an
analytic way to compute the intersection between two subspaces embedded in a
common vector space. Given bases $A,B\in\mathbb{R}^{m\times n}$ for two
subspaces, a basis of the intersection space can be calculated by computing
the null space of $C=[A,-B]\in\mathbb{R}^{m\times 2n}$. When considering SSMs,
however, we usually have $m\gg n$, _i.e_., there is an extreme difference in
dimensions between the spanned subspaces of the models and their embedding. In
this case, the linear system that we would need to solve when computing the
null space of $C$ is highly over-determined and may not have a non-trivial
solution in general.
A lot of classical mathematical theory [23, 21] is available to compute the
orthogonal projection matrix, $P$, used to project points onto the
intersection of two subspaces. Numerous formulas and methods have been
proposed to compute $P$, of which the alternating projection method [38] is
probably most well known. However, this method has a high computational
complexity and gives only the projection matrix instead of a basis for the
intersection space.
Another method to calculate a basis for the intersection between two subspaces
is by means of the Zassenhaus algorithm [30]. More recently, Fenggang et al.
[15] proposed a closed-form algorithm to compute such a basis.
## 3 Method
After fixing notations and reviewing some basic properties of linear shape
models, we first provide a formal definition of the intersection between two
SSMs. We then present the algorithm to estimate the intersection.
Additionally, we show how the same strategy can be further explored to
estimate the differences between two models.
### 3.1 Preliminaries
A linear shape model can be interpreted either by assuming a linear algebraic
or a probabilistic point of view. Firstly, we take the linear-algebraic
approach and represent an SSM as a function of the form
$f:\mathbb{R}^{q}\longrightarrow\mathbb{R}^{dn}$, $f(\alpha)=\bar{x}+U\alpha$,
where $\bar{x}\in\mathbb{R}^{dn}$ is the so-called mean shape computed over a
set of objects described with $n$ points in $d$ dimensions (for triangular
meshes, $d=3$). The orthogonal matrix
$U=(\sqrt{\lambda_{1}}u_{1},\sqrt{\lambda_{2}}u_{2},\dots,\sqrt{\lambda_{q}}u_{q})\in\mathbb{R}^{dn\times
q}$ holds the scaled eigenvectors of the dataset’s covariance matrix, where
$\lambda_{i}$ is the $i$-th eigenvalue. The vector space spanned by a linear
model is the affine subspace
$M\coloneqq\bar{x}+\operatorname{span}(U)=\\{\bar{x}+U\alpha\mid\alpha\in\mathbb{R}^{q}\\}\subseteq\mathbb{R}^{dn},$
(1)
where $\dim(M)=q$. Every linear combination of the basis vectors contained in
$U$ gives rise to a new shape.
We now describe the probabilistic perspective on linear shape models. By
assuming $\alpha\sim\mathcal{N}(0,I)$, one can verify that shapes
$x=f(\alpha)$ are distributed according to $x\sim\mathcal{N}(\bar{x},C)$,
where $C=UU^{T}\in\mathbb{R}^{dn\times dn}$ is the sample covariance matrix.
The probability of a shape is given by
$p(x)=p(\alpha)\propto\exp\left(-\frac{1}{2}\|\alpha\|_{2}^{2}\right).$ (2)
From this perspective, not every shape in $M$ is equally likely; the
probability mass is concentrated in a $q$-dimensional hyper-ellipsoid which is
centered at the mean $\bar{x}$ and whose principal axes correspond to the
eigenvectors of $C$. The eigenvalues of $C$ are the reciprocals of the squares
of the lengths of the semi-axes. The shapes lying inside the hyper-ellipsoid
can be characterized as $Q\coloneqq\\{x\in
M\mid(x-\bar{x})^{T}C^{-1}(x-\bar{x})\leq k\\}\subseteq\mathbb{R}^{dn}$ with
$k\in[0,1]$. In terms of a probabilistic interpretation, this set can be
equivalently rewritten as
$Q=\\{x\in M\mid p(x)\geq\xi\\},$ (3)
where $\xi\in[0,1]$ (see supp. material for a derivation). It contains all
shapes with a probability greater than a certain threshold. Following [7] and
for a suitable $\xi$, shapes in $Q$ are considered plausible in the sense that
they look similar to the observed (training) data; hence, $Q$ is often called
the allowable shape domain [8].
### 3.2 Computing Intersections
Given two aligned111By aligned, we mean that Euclidean similarity
transformations (rotation, translation, and scaling) between the two models
have been removed. This can always be achieved by aligning the respective mean
shapes using ordinary Procrustes analysis, see, _e.g_., [11]. SSMs in dense
correspondence, our goal is to compute their intersection. We define the
intersection $I\subseteq\Omega=M_{1}\cup M_{2}$ between two SSMs as the
intersection between their allowable shape domains, _i.e_.,
$I\coloneqq Q_{1}\cap Q_{2}.$ (4)
In this work, we consider only SSMs with non-empty $I$ (whether two models
intersect can be checked using, _e.g_., [19]). As seen from (4) and by noting
that $I=(M_{1}\cap M_{2})\cap(Q_{1}\cap Q_{2})$, a shape $x\in\Omega$ belongs
to the intersection if the following conditions are met: First, $x$ can be
represented by both models, _i.e_., $x\in M_{1}\cap M_{2}$. Assuming $x\in
M_{1}\subset\Omega$, this is equivalent to finding an $x^{\prime}\in M_{2}$
such that the (Euclidean) distance $d(x^{\prime},x)$ vanishes (or vice versa
if we assume $x\in M_{2}$). Second, $x$ is likely in both models, that is,
$p(x)\geq\xi_{1}$ and $p(x^{\prime})\geq\xi_{2}$.
Our definition is motivated by the fact that most of the shapes in the
subspaces $M_{1}$ and $M_{2}$ are unlikely and do not lead to realistic shape
instances. Hence, if we defined the intersection simply as $I=M_{1}\cap
M_{2}$, then $I$ would contain a lot of degenerated shapes that we want to
disregard. We therefore only consider the plausible regions of two models,
which are exactly identified by $Q_{1}$ and $Q_{2}$.
The problem with the definition in (4), however, is that real-world models are
usually not noise-free. As a consequence, the first condition, $x\in M_{1}\cap
M_{2}$, will never be met in practice since we can not find an $x^{\prime}\in
M_{2}$ such that $d(x^{\prime},x)=0$ exactly. To account for this, we weaken
the strong definition and formulate an approximate intersection $I_{\epsilon}$
for $\epsilon>0$ as
$I_{\epsilon}\coloneqq\left(Q_{1}\cap(Q_{2}+B_{\epsilon}(0))\right)\cup\left(Q_{2}\cap(Q_{1}+B_{\epsilon}(0))\right),$
(5)
where $B_{\epsilon}(0)=\\{y\in\mathbb{R}^{dn}:d(0,y)<\epsilon\\}$ is the
$\epsilon$-ball in $\mathbb{R}^{dn}$ centered at $0$. This allows us to also
consider shapes to be in the intersection that are almost (up to $\epsilon$)
contained in $I$. We can rewrite (5) as
$\begin{split}I_{\epsilon}=&\left\\{x\in Q_{1}\mid\exists x^{\prime}\in
Q_{2}:d(x^{\prime},x)<\epsilon\right\\}\\\ \cup&\left\\{x\in Q_{2}\mid\exists
x^{\prime}\in Q_{1}:d(x^{\prime},x)<\epsilon\right\\}.\end{split}$ (6)
Of course, $I_{\epsilon}\rightarrow I$ if $\epsilon\rightarrow 0$ and
$I\subset I_{\epsilon}$ for every $\epsilon>0$.
### 3.3 Algorithm
Instead of explicitly constructing the set $I_{\epsilon}$, we aim at
estimating the distribution of points $x\in I_{\epsilon}$. Given such
distribution, we can then effectively generate samples from the intersection,
which we use to build an SSM for the intersection by applying PCA. The
resulting linear model can then be utilized to visualize and study the
intersection space.
The advantage of estimating the distribution of $x\in I_{\epsilon}$ instead of
randomly sampling from $\Omega$ and applying hard constraints to test whether
a point $x$ belongs to the intersection is that we do not need to explicitly
set values for the parameters ($\epsilon$, $\xi_{1}$, and $\xi_{2}$) involved
in (6).
In detail, we model the posterior distribution of all $x\in Q_{1}$, given that
$x\in I_{\epsilon}$ (and vice versa for all $x\in Q_{2}$):
$p(x\,|\,x\in I_{\epsilon})=\frac{p(x\in I_{\epsilon}\,|\,x)p(x)}{\int p(x\in
I_{\epsilon}\,|\,x)p(x)\text{d}x}.$ (7)
We call $p(x\in I_{\epsilon}\,|\,x)$ the likelihood function and denote it by
$L(x;x\in I_{\epsilon})$. The probability $p(x)$ is computed according to (2)
by noting that $x=f_{1}(\alpha)$ for $\alpha\in\mathbb{R}^{q_{1}}$.
The likelihood function encodes the conditions under which $x$ belongs to the
intersection space. In the following, we show how the two conditions as stated
in Section 3.2 can be directly translated into a likelihood.
First condition. We implement the first condition, _i.e_., finding an
$x^{\prime}\in M_{2}$ such that $d(x^{\prime},x)$ becomes small, using the
orthogonal projection of $x$ onto $M_{2}$, given by
$x^{\prime}=\operatorname{proj}_{M_{2}}(x)=f_{2}(\underbrace{U_{2}^{-1}(x-\bar{x}_{2})}_{=\alpha^{\prime}\in\mathbb{R}^{q_{2}}}).$
(8)
The point $x^{\prime}$ is the closest point to $x$ contained in $M_{2}$ and
hence minimizes the distance between $x$ and $x^{\prime}$. With that, we
define a distance likelihood $L_{D}$ as
$\begin{split}L_{D}(x;x\in
I_{\epsilon})&\propto\exp\left(-\frac{1}{2}\left(\frac{d(x^{\prime},x)}{\sigma}\right)^{2}\right),\end{split}$
(9)
where we used
$d(x^{\prime},x)=\frac{1}{n}\sum_{i=1}^{n}\|x^{\prime}_{i}-x_{i}\|_{2}$ (10)
and assumed $d(x^{\prime},x)\sim\mathcal{N}(0,\sigma^{2})$. A similar
likelihood was originally proposed in [34] and recently used by [32] for
model-based surface registration.
Second condition. For the second condition, we require both $x$ and
$x^{\prime}$ to be likely. This can be implemented by taking into account the
probabilities $p(x)$ and $p(x^{\prime})$ of $x$ and $x^{\prime}$. Since the
probability of $x$ is already included in the formulation of the posterior in
(7), we only need to care about $x^{\prime}$. To enforce the projection
$x^{\prime}$ to be likely, too, we define a projection likelihood $L_{P}$
simply as
$L_{P}(x;x\in I_{\epsilon})=p(x^{\prime}),$ (11)
where the probability of $x^{\prime}=f_{2}(\alpha^{\prime})$ is given by (2).
Our final likelihood is then a combination of the distance and projection
likelihood and formulated as
$L=L_{D}L_{P}.$ (12)
The posterior distribution $p(x\,|\,x\in I_{\epsilon})$ in (7) is now fully
specified. However, it is intractable and can not be computed exactly. We
therefore make use of approximate inference as described in the following.
### 3.4 Implementation
We use Markov chain Monte Carlo (MCMC) to approximate the intractable
posterior distribution in (7). This is possible since $p(x\,|\,x\in
I_{\epsilon})\propto L(x;x\in I_{\epsilon})p(x)$.
To construct a Markov chain, we make use of the Metroplis-Hastings (MH)
algorithm. The MH algorithm requires a proposal distribution conditioned on
the current state. We use the same random walk mixture proposal as proposed in
[34]. Please refer to the supp. material for more information about the MH
algorithm and our proposal distribution. To explore the parameter space and to
reduce auto-correlation between samples as much as possible, we use an
ensemble of Markov chains instead of only one chain. Each chain has a
different starting point which is sampled from $\mathcal{N}(0,I)$. We ensured
that samples are sufficiently far apart to avoid individual chains exploring
the same part of the parameter space.
Figure 1: Schematic illustration of how we generate training data for star
models including ground-truth intersections and differences. For the first
model, we vary the first and third point of the star in
$[\theta_{j}-a,\theta_{j}+b]$, where $j\in\\{0,2\\}$. For the second model, we
vary the first and the fourth point in $[\theta_{j}-c,\theta_{j}+a]$, where
$j\in\\{0,3\\}$. The ground-truth intersection contains stars where the first
point varies from $[\theta_{0}-a,\theta_{0}+a]$, whereas the ground-truth
difference of the first and the second model contains stars where the first
point ranges from $(\theta_{0}+a,\theta_{0}+b]$ and the third between
$[\theta_{2}-a,\theta_{2}+b]$. The ground-truth difference of the second and
the first model includes stars where the first point varies from
$[\theta_{0}-c,\theta_{0}-a)$ and the fourth from
$[\theta_{3}-c,\theta_{3}+a]$.
### 3.5 Computing Differences
Keeping in mind the definition of the approximate intersection space, we
define the set-theoretic difference between two SSMs as
$D_{12}\coloneqq Q_{1}\setminus\left(M_{2}+B_{\epsilon}(0)\right).$ (13)
The set $D_{12}$ contains all $x\in Q_{1}$ that can not be represented in
$M_{2}$ (or vice versa for $D_{21}$). We can rewrite (13) into
$D_{12}=\\{x\in Q_{1}\mid\forall x^{*}\in M_{2}:d(x^{*},x)\geq\epsilon\\}.$
(14)
From a computational perspective, this formulation is rather unpleasant due to
the universal quantifier involved. Note, however, that the point
$x^{\prime}=\operatorname{proj}_{M_{2}}(x)$ minimizes the distance from $x$ to
$M_{2}$ and thus, if $d(x^{\prime},x)\geq\epsilon$, then the same holds for
all $x^{*}\in M_{2}$. With this, we can rephrase (14) as
$D_{12}=\\{x\in Q_{1}\mid d(x^{\prime},x)\geq\epsilon\\}.$ (15)
The condition of whether $x$ belongs to the difference can now be easily
checked by means of the projection operator.
To estimate the distribution of points lying in $D_{12}$ (or $D_{21}$), we
only have to make two small changes to the algorithm presented throughout the
previous sections. From the definition of the difference in (15) we observe
that the only condition a point $x\in Q_{1}$ has to fulfill in order to lie in
$D_{12}$ is that the distance to its projection onto $M_{2}$ becomes large.
This can be implemented by inverting the distance likelihood in (9) as
$\bar{L}_{D}(x;x\in I_{\epsilon})\propto 1-L_{D}(x;x\in I_{\epsilon}).$ (16)
Secondly, since we no longer require the projection to be likely, we remove
the projection likelihood from (12).
After those minor modifications, our algorithm is ready to be used to
approximate the distribution of shapes lying in the difference between two
models. However, the space in which the shapes of the difference between two
SSMs lie is no longer convex as the shapes lying in the intersection have been
removed. As such, it does not make any sense to apply PCA to the MCMC samples
as the resulting linearized space would simply interpolate the intersection.
Instead, we only visualize random samples drawn from our Markov chain in order
to inspect the difference.
## 4 Experiments and Results
We conducted several quantitative and qualitative experiments to validate our
method.
Training data Intersection (Grassmann distances, $d_{G}$ $\downarrow$)
Difference (Reconstruction errors, $d_{R}$ $\uparrow$) $Q_{1}$ $Q_{2}$ $I$
$D_{12}$ $D_{21}$ $d_{G}(\hat{I},I)$ $d_{G}(Q_{1},I)$ $d_{G}(Q_{2},I)$
$d_{G}(Q_{1},Q_{2})$ $d_{R}(\hat{D}_{12},I)$ $d_{R}(\hat{D}_{21},I)$
$d_{R}(D_{12},I)$ $d_{R}(D_{21},I)$ $[-5,40]$ $[-20,5]$ $[-5,5]$ $(5,40]$
$[-20,-5)$ $0.0260\pm 0.0086$ 1.5634 1.5634 2.1211 $0.2235\pm 0.1132$
$0.1148\pm 0.0534$ $0.1781\pm 0.2243$ $0.0911\pm 0.1122$ $[-5,20]$ $[-20,5]$
$[-5,5]$ $(5,20]$ $[-20,-5)$ $0.0370\pm 0.0110$ 1.5583 1.5583 2.1211
$0.1165\pm 0.0556$ $0.1171\pm 0.0542$ $0.0911\pm 0.1122$ $0.0911\pm 0.1122$
$[-10,40]$ $[-20,10]$ $[-10,10]$ $(10,40]$ $[-20,-10)$ $0.0140\pm 0.0052$
1.5471 1.5506 2.1211 $0.2081\pm 0.1097$ $0.1150\pm 0.0540$ $0.1867\pm 0.2110$
$0.1073\pm 0.1081$ $[-10,20]$ $[-20,10]$ $[-10,10]$ $(10,20]$ $[-20,-10)$
$0.0323\pm 0.0094$ 1.5506 1.5506 2.1211 $0.1107\pm 0.0510$ $0.1143\pm 0.0534$
$0.1126\pm 0.1080$ $0.1073\pm 0.1081$ $[-20,60]$ $[-30,20]$ $[-20,20]$
$(20,60]$ $[-30,-20)$ $0.0053\pm 0.0022$ 1.5240 1.5440 2.1211 $0.2692\pm
0.1571$ $0.1509\pm 0.0809$ $0.2834\pm 0.3170$ $0.1861\pm 0.1569$ $[-40,80]$
$[-50,40]$ $[-40,40]$ $(40,80]$ $[-50,-40)$ $0.0018\pm 0.0004$ 1.4863 1.5375
2.1211 $0.3257\pm 0.1994$ $0.2243\pm 0.1293$ $0.4200\pm 0.3993$ $0.3501\pm
0.2531$
Table 1: Quantitative results for the star models based on different training
sets, averaged over five runs. The comparison of the estimated intersection
space, $\hat{I}$, and the ground-truth intersection $I$ is based on the
Grassmann distance, denoted as $d_{G}$. We also report various baseline
distances to give an intuition for the range of the distances. To evaluate
whether or not posterior samples from the computed differences, $\hat{D}_{12}$
and $\hat{D}_{21}$, are indeed from the true difference, we calculate
reconstruction errors $d_{R}$ by projecting estimated samples and samples from
the ground-truth differences $D_{12}$ and $D_{21}$ into the ground-truth
intersection model $I$. Note that the intervals for the training data arise
from setting different values for the parameters $a$, $b$, and $c$.
### 4.1 Quantitative Analysis
Our quantitative evaluation is based on two data sets from which we generate
ground-truth intersection spaces. The first is a synthetic data set based on a
five-pointed star. Due to the rather simplistic geometry, this data set is
well suited for an initial analysis and serves as a proof-of-concept. The
second is a real data set derived from a real-world SSM and, consequently,
shares a lot of desired properties such as high dimensionality.
We employ the Grassmann distance as a natural distance between two affine
subspaces to compare ground-truth intersections with the spaces recovered by
our method. The Grassmann distance is defined as the Euclidean norm of the
affine principal angles between two subspaces and can be easily computed by
applying Singular Value Decomposition to the Stiefel coordinates of the two
spaces, see, _e.g_. [28]. Please refer to the supp. material for more
information.
Constructing ground-truth differences is far more complex due to their
topological structure. As such, the quality of the differences recovered by
our method can only be evaluated on the star models since, due to their
synthetic construction, we are able to generate samples from the ground-truth
difference (_i.e_., samples for which we definitely know they are lying in the
difference). This is not the case for real-world SSMs because the ground-truth
difference space for real-world models is unknown.
#### 4.1.1 Synthetic Data Set
For the synthetic data set, we use a five-pointed star as a simple geometric
object to generate training data by systematically varying its points, see
Figure 1. Each point of the star can be written as
$(r\cos{\theta_{i}},r\sin{\theta_{i}})$ with radius $r$ and angle
$\theta_{i}$, $i=0,\dots,4$. For the first model, we vary the first and third
point in range $[\theta_{j}-a,\theta_{j}+b]$, where $j\in\\{0,2\\}$. For the
second model, we vary the first and fourth point in range
$[\theta_{j}-c,\theta_{j}+a]$, where $j\in\\{0,3\\}$. The ground-truth
intersection model contains stars where the first point ranges from
$[\theta_{0}-a,\theta_{0}+a]$. The ground-truth difference of the first and
the second model contains stars where the first point ranges from
$(\theta_{0}+a,\theta_{0}+b]$ and the third between
$[\theta_{2}-a,\theta_{2}+b]$. The ground-truth difference of the second and
the first model includes stars where the first point varies from
$[\theta_{0}-c,\theta_{0}-a)$ and the fourth from
$[\theta_{3}-c,\theta_{3}+a]$.
We tested our method on six different values for $a,b,c$ as shown in Table 1.
An ensemble of 15 chains was used, where each chain was sampled 2,500 times.
1,000 samples were considered as burn-in. Out of all samples, 5,000 samples
were evenly chosen to build an SSM for the intersection space. For the
distance likelihood, we set $\sigma=0.003$ for all experiments (see supp.
material for an ablation on $\sigma$).
We use the Grassmann distance as described above to measure how well our
method can recover the ground-truth intersection space. To evaluate the
quality of the computed differences, we exploit the fact that we have samples
from the ground-truth difference model (clearly, we do not have a
parametrization of these models, but only a few samples thereof). To this end,
we quantify whether or not samples from the difference generated by our method
have as high a distance from their reconstruction in the ground-truth
intersection space as the samples from the ground-truth difference (note that
a sample belongs to the difference if it can not be represented in the
intersection). We, therefore, project the MCMC samples of the difference into
the ground-truth intersection model and calculate the average reconstruction
error, see supp. material for further details. We do the same for the samples
from the ground-truth difference. Similar reconstruction errors then indicate
that all samples generated by our method are indeed valid samples from the
true difference space.
Overall, the proposed method is able to recover the ground-truth intersection
space for all six models very well, see Table 1. We measure an average
Grassmann distance of 0.0194 between ground-truth intersections and the
intersection spaces recovered by our method. We observe a similar result for
the estimated differences in terms of reconstruction errors. Figure 2 provides
additional visualization of the distribution of posterior samples.
Figure 2: Visualization of (a random subset of) MCMC samples generated to
compute the estimated intersection $\hat{I}$ and differences $\hat{D}_{12}$
and $\hat{D}_{21}$ between two star models. The included polar frequency
histogram visualizes the distribution of posterior samples; it indicates that
most of the samples are indeed from the true posterior distribution (_i.e_.,
they are within the bounds shown in red). Figure 3: Random samples from the
male and female model of the LYHM data set (1st and 2nd block) as well as
samples drawn from the computed intersection model (3rd block) and the
respective differences (male without female, and female without male, 4th and
5th block). We observe stronger male and female dominance in the differences
and neutral gender in the intersection.
Training data Intersection (Grassmann distances, $d_{G}$ $\downarrow$)
$\dim(Q_{1})$ $\dim(Q_{2})$ $\dim(I)$ $d_{G}(\hat{I},I)$ $d_{G}(Q_{1},I)$
$d_{G}(Q_{2},I)$ $d_{G}(Q_{1},Q_{2})$ 3 3 1 $0.0032\pm 0.0010$ 1.5517 1.5532
2.1854 6 6 2 $0.0645\pm 0.0455$ 2.1641 2.1855 3.0835 9 9 3 $0.0877\pm 0.0178$
2.6375 2.6901 3.7747 12 12 4 $0.1683\pm 0.0106$ 3.1004 3.0849 4.3563 15 15 5
$0.1939\pm 0.0433$ 3.4606 3.4467 4.8682
Table 2: Quantitative results for the estimated intersection spaces of the
real-world data set. We tested five different configurations of individual
model dimensions, $\dim(Q_{1})$ and $\dim(Q_{2})$, and the dimension of the
ground-truth intersection space, $\dim(I)$. Results were averaged over three
different random splits per configuration.
#### 4.1.2 Real Data Set
The simple geometry and ground-truth intersection spaces of the star models
enabled an intuitive analysis of the proposed method. The geometry of real-
world SSMs, however, is oftentimes far more complex. We conduct experiments on
those models to further validate the proposed approach.
Given an SSM with a set of basis vectors, we generate two individual models
and a ground-truth intersection by randomly splitting the set of basis vectors
into three disjoint subsets, $S_{1},S_{2}$, and $S_{I}$. The first two sets,
$S_{1}$ and $S_{2}$, contain the unique basis vectors for the first and second
model, and the last set, $S_{I}$, holds the basis vectors of the ground-truth
intersection. The two models’ bases are then given by $S_{1}\cup I$ and
$S_{2}\cup I$, respectively. Since we can not generate samples from the
ground-truth difference, we present evaluations only for the computation of
intersection spaces.
We used the first 15 basis vectors of the BFM 2019 [18] for this experiment.
Moreover, an ensemble of 25 chains was used. Each chain was sampled 5,000
times with a burn-in phase of 2,000 samples. To build the intersection model,
again 5,000 samples were evenly chosen out of all samples. We empirically set
$\sigma=0.3$ for all models.
Table 2 presents the results for five different random splits. As shown, the
proposed method is able to recover all ground-truth intersection spaces quite
well with an average Grassmann distance of 0.1035. However, we observe
increasing distances as the dimension of the intersection increases. This is
most likely because MCMC does not properly sample the entire space, and may be
alleviated by choosing larger proposals or running more Markov chains. Indeed,
by doubling the number of chains, the Grassmann distance decreases by more
than 10%.
### 4.2 Qualitative Results
To showcase our method on real-world SSMs, we compute and analyze
intersections and differences between (i) a gender-specific male and female
model, and (ii) an identity and expression model. Due to the missing ground-
truth intersection spaces and differences, we can only provide qualitative
results. However, to underline that our method does not simply produce the
union of two models, or even worse, just reproduces one model, we do report
Grassmann distances to the union and to individual models. The union of two
models is computed by applying PCA to a set of random samples from both
models.
Male and female models. Both models were built using 600 faces from the LYHM
database [9]. We used a face mask to only include the frontal region of the
face and truncated both models to include the first 50 basis vectors. Each
model has 5,764 vertices. As such, both SSMs span a 50-dimensional affine
subspace which is embedded in a 17,292-dimensional vector space.
Exemplary results can be found in Figure 3. As seen, random faces sampled from
the intersection model do not show a strong preference towards male or female,
and most of the time tend to look more neutral compared to faces drawn from
the individual male or female models. Furthermore, while random faces from the
difference between male and female show extremely masculine traits, samples
from the difference between female and male look rather feminine. We report a
Grassmann distance of about 1.8 from the intersection to the union of male and
female, and a distance of 3.2 and 2.8 to individual male and female subspaces.
Computing the intersection model took about 110 minutes on a single core;
differences half the time.
Figure 4: Random samples from the original identity and expression models of
the BFM 2019 (1st and 2nd block) as well as samples of the computed
intersection model (3rd block). Also shown are samples from the difference
between identity and expression models (4th and 5th block). We observe less
expression variation in samples drawn from the difference of identity and
expression (identity without expression, 4th block) and less identity
variation in the difference of expression and identity (expression without
identity, 5th block). Figure 5: Per-vertex level visualization of similarities
and differences between the BFM 2019 [18] identity and expression models.
Depicted is the per-vertex variance of the original identity and expression
models as well as the posterior variance for the computed intersection model
and differences (red areas show high variance, blue areas low). The
intersection visualizes the source of the identity-expression ambiguity,
differences highlight features represented either by identity (nose) or
expression (opening of mouth, raising cheeks and eyebrows).
Identity and expression models. Face identity and expression are usually
modeled separately in 3DMMs. As recently demonstrated in [13], however, the
subspaces spanned by the identity and expression model are far from being
orthogonal and thus, face identities and expressions are not independent of
each other. As a consequence, it is not possible to alter a specific face
identity without leaving its expression completely unchanged (and vice versa).
Although only hardly visible with the human eye, this effect has implications
on downstream applications such as expression transfer and inverse rendering
[13].
We use our method to further study this ambiguity, also known as the identity-
expression ambiguity, and analyze the intersection as well as differences
between the identity and expression models of the BFM 2019 [18]. We again
truncated both SSMs to the first 50 basis vectors. Each model has 1,746
vertices, leading to a 5,238-dimensional vector space in which the models’
subspaces are embedded.
Results are shown in Figures 4 and 5. While there is no ground-truth for this
task, the observations match our intuition, _e.g_., shape variations of the
nose do not arise in the expression model but solely in the identity model,
and effects based on muscle movement dominate in the identity model. To still
add a quantitative element, we measured the Grassmann distances between the
intersection and original identity and expression models and report a distance
of 5.36 to identity, and 5.09 to the BFM expression model. Those distances
being close to each other show that the intersection model somehow lies
between the identity and expression model. The Grassmann distance to the union
is 3.48. It took about 30 minutes to compute the intersection model. Again,
computing differences required half the time.
For both tasks (male vs. female and identity vs. expression), we observed very
similar results for the FLAME [27] model. Please refer to the supp. material.
### 4.3 Extension to Color
Since texture in 3DMMs is modeled similarly to shape [2], our method can be
used to also compute intersections and differences between texture models
without any algorithmic changes. Results for male and female 3DMMs described
above can be found in the supp. material.
## 5 Limitations
Although the presented method yields appealing results, our approach in its
current form is not without limitations. First, with increasing dimension of
the intersection space more Markov chains are needed to explore the parameter
space; this phenomenon is known as the curse of dimensionality. It naturally
increases the run time of our method. We plan to tackle this issue by
parallelizing Markov chains and by using more effective MCMC samplers such as
the Metropolis-adjusted Langevin algorithm [20].
Second, if SSMs are not in dense correspondence, we require some form of
optimization in the projection operation, _i.e_., (8) needs to be adapted.
Clearly, while the projection can be computed in closed-form if models are in
correspondence, we require registration if they are not in correspondence.
This can be efficiently carried out prior to the application of our method
using techniques proposed in [33].
Lastly, although our method and the chosen parameters might not be trivial to
evaluate on real-world models, the result can always be well validated
empirically by estimating $\epsilon$ from the MCMC samples, see supp.
material.
## 6 Conclusion
In this paper, we have introduced a new method to compare SSMs by computing
approximate intersection spaces and set-theoretic differences between the low-
dimensional allowable shape domains spanned by two models in dense
correspondence. We showed how MCMC can be leveraged to approximate the
distribution of shapes lying inside those spaces, and, based on posterior
samples, compute a new SSM for the intersection. Building upon this algorithm,
we further demonstrated that our method can be easily adapted to also compute
set-theoretic differences. Confirmed by quantitative and qualitative
evaluation on synthetic data and real-world face models, the proposed method
is able to recover ground-truth intersection spaces and differences
accurately. Moreover, qualitative results obtained on real-world SSMs match
our intuition. In future work, we plan to study how our approach is (or may
be) correlated to performance-based metrics, such as the Bhattacharya
distance.
Acknowledgments. We thank Tinashe Mutsvangwa, Marcel Lüthi, and Tobias Meier
for interesting discussions and feedback. This work was funded by the German
Federal Ministry of Education and Research (BMBF), FKZ: 01IS22082 (IRRW), and
the FAU Emerging Talents Initiative (ETI). The authors are responsible for the
content of this publication.
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## Appendix
In this supplementary material, we provide (i) additional information on the
equivalence of the two definitions for $Q$ in Section 3.1 of the main paper,
(ii) further implementation details on our algorithm, (iii) additional
information on the evaluation metrics used in Section 4.1 of the main text,
(iv) an ablation study on the key parameter of our method, $\sigma$, (v) a
practical strategy on how to empirically choose $\sigma$, and finally (vi)
some additional qualitative results.
## Appendix A Equivalence of Q
In Section 3.1 of the main paper, the set $Q$ was defined as
$Q\coloneqq\\{x\in M\mid(x-\bar{x})^{T}C^{-1}(x-\bar{x})\leq
k\\}\subseteq\mathbb{R}^{dn},$ (17)
where $k\in[0,1]$. We claimed that, in terms of a probabilistic
interpretation, $Q$ can be equivalently rewritten as
$Q=\\{x\in M\mid p(x)\geq\xi\\}$ (18)
with $\xi\in[0,1]$, see Eq. (3) of the main paper. This can be easily verified
by
$\displaystyle(x-\bar{x})^{T}C^{-1}(x-\bar{x})$ $\displaystyle\leq k$ (19)
$\displaystyle\Longleftrightarrow$
$\displaystyle-\frac{1}{2}(x-\bar{x})^{T}C^{-1}(x-\bar{x})$
$\displaystyle\geq-\frac{1}{2}k$ $\displaystyle\Longleftrightarrow$
$\displaystyle\exp{\left(-\frac{1}{2}(x-\bar{x})^{T}C^{-1}(x-\bar{x})\right)}$
$\displaystyle\geq\exp{\left(-\frac{1}{2}k\right)}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\frac{\exp{\left(-\frac{1}{2}(x-\bar{x})^{T}C^{-1}(x-\bar{x})\right)}}{\sqrt{(2\pi)^{q}\det
C}}$
$\displaystyle\geq\frac{\exp{\left(-\frac{1}{2}k\right)}}{\sqrt{(2\pi)^{q}\det
C}}$ $\displaystyle\Longleftrightarrow$ $\displaystyle p(x)$
$\displaystyle\geq\xi.$
Hence, the hyper-ellipsoid $Q$ contains all likely shapes, _i.e_., those with
a probability greater than a certain threshold, $\xi$.
Algorithm 1 Metropolis-Hastings algorithm (symmetric proposal distribution)
1:Initialize $\alpha_{0}$; set $x_{0}=f_{1}(\alpha_{0})$.
2:for $i=0,1,\dots,m$ do
3: Draw sample $\alpha^{\prime}$ from $Q(\alpha^{\prime}\,|\,\alpha_{i})$; set
$x^{\prime}=f_{1}(\alpha^{\prime})$.
4: Compute acceptance ratio as $t=\frac{L(x^{\prime};x^{\prime}\in
I_{\epsilon})p(x^{\prime})}{L(x_{i};x_{i}\in I_{\epsilon})p(x_{i})}.$
5: Accept $\alpha^{\prime}$ with probability $t$ by drawing a sample $r$ from
$\mathcal{U}(0,1)$ and $\alpha_{i+1}=\begin{cases}\alpha^{\prime}&\text{if
}t>r,\\\ \alpha_{i}&\text{otherwise.}\end{cases}$
6:end for
7:return $\\{f_{1}(\alpha_{0}),f_{1}(\alpha_{1}),\dots,f_{1}(\alpha_{m})\\}$
## Appendix B Implementation Details
We use Markov chain Monte Carlo (MCMC) to estimate the posterior distribution
$p(x\,|\,x\in I_{\epsilon})$ for $x\in Q_{1}$ (and $Q_{2}$, respectively) as
stated in Eq. (7) of the main paper. The Markov chain is built by means of the
Metropolis-Hastings (MH) algorithm, summarized in Algorithm 1. The MH
algorithms requires a proposal distribution $Q(\alpha^{\prime}\,|\,\alpha)$,
conditioned on the current state $\alpha\in\mathbb{R}^{q_{1}}$. We use a
random walk mixture proposal of the form:
$Q(\alpha^{\prime}\,|\,\alpha)=\sum_{i=1}^{n}c_{i}Q_{i}(\alpha^{\prime}\,|\,\alpha)\quad\text{with}\quad\sum_{i=1}^{n}c_{i}=1.$
(20)
In our specific implementation, we set $n=4$ and defined:
$\begin{split}Q_{1}(\alpha^{\prime}\,|\,\alpha)&=\mathcal{N}(\alpha,0.2),\quad
c_{1}=0.1,\\\ Q_{2}(\alpha^{\prime}\,|\,\alpha)&=\mathcal{N}(\alpha,0.1),\quad
c_{2}=0.5,\\\
Q_{3}(\alpha^{\prime}\,|\,\alpha)&=\mathcal{N}(\alpha,0.025),\quad
c_{3}=0.2,\\\
Q_{4}(\alpha^{\prime}\,|\,\alpha)&=\mathcal{N}(\|\alpha\|,0.2),\quad
c_{4}=0.2.\end{split}$ (21)
This proposal distribution was originally presented in [34] and we left it
unchanged. Note that proposal is symmetric, _i.e_.,
$Q(\alpha^{\prime}\,|\,\alpha)=Q(\alpha\,|\,\alpha^{\prime})$, since all
mixture components are Gaussian.
Having the proposal distribution in mind, the MH algorithm proceeds as
follows: In every iteration, a new sample $\alpha^{\prime}$ is proposed based
only on the previous sample, $\alpha_{i}$. The proposed sample is then either
accepted or rejected with probability $t$, where $t$ is the so-called
acceptance ratio. After a sufficient number of iterations, $m$, the MH
algorithm returns a set of accepted samples from the desired posterior
distribution, $p(x\,|\,x\in I_{\epsilon})$.
## Appendix C Evaluation Metrics
We now provide some additional information on the evaluation metrics used for
quantitative analysis as briefly explained in Section 4.1 of the main paper.
Grassmann distance. The Grassmann distance is the natural distance between two
linear subspaces embedded in $\mathbb{R}^{n}$ (the set of all $k$-dimensional
linear subspaces is called the Grassmannian, usually denoted as Gr($k,n$)).
Given orthonormal bases $A,B\in\mathbb{R}^{k\times n}$ for two subspaces from
Gr($k,n$), the Grassmann distance can be calculated by means of the principal
angles $\\{\theta_{1},\theta_{2},\dots,\theta_{k}\\}$ between the two
subspaces. With slight abuse of notation but for the sake of brevity, we refer
to the Grassmann distance as
$d_{G}(A,B)=\|(\theta_{1},\theta_{2},\dots,\theta_{k})\|_{2}$.
Since SSMs span affine subspaces, the Grassmann distance as presented
previously can not directly be applied to measure distances between subspaces
spanned by linear shape models. Fortunately, as shown in [28] (Theorem 7), the
Grassmann distance can be easily extended to affine subspaces as we briefly
explain next. The key idea is to embed the affine subspace into a linear
subspace by adding one dimension. Given two affine subspaces represented by
orthonormal bases $A,B\in\mathbb{R}^{k\times n}$ and displacement vectors
$b,c\in\mathbb{R}^{n}$, their Stiefel coordinates,
$Y_{1},Y_{2}\in\mathbb{R}^{(k+1)\times n}$ are given by
$Y_{1}=\begin{pmatrix}A&b_{0}/\sqrt{1+\|b_{0}\|^{2}}\\\
0&1/\sqrt{1+\|b_{0}\|^{2}}\end{pmatrix}$ (22)
and
$Y_{2}=\begin{pmatrix}B&c_{0}/\sqrt{1+\|c_{0}\|^{2}}\\\
0&1/\sqrt{1+\|c_{0}\|^{2}}\end{pmatrix},$ (23)
where $b_{0}$ and $c_{0}$ are unit vectors orthogonal to the columns of $A$
and $B$, respectively. We compute them by
$b^{\prime}=b-\sum_{j=1}^{k}(a_{j}\cdot b)a_{j},\quad
b_{0}=\frac{b^{\prime}}{\|b^{\prime}\|_{2}},$ (24)
and analogously for $c_{0}$. Here, $a_{j}\in\mathbb{R}^{n}$, $j=1,\dots,k$,
denote the columns of $A$. Finally, the affine Grassmann distance is computed
by applying Singular Value Decomposition to $Y_{1}^{T}Y_{2}$, yielding the
$k+1$ principal angles between the respective affine subspaces. Taking the
Euclidean norm of those angles leads to the affine Grassmann distance.
Figure 6: Results of the ablation study for different values of $\sigma$,
averaged over all six star models (described in the main paper). Top:
Grassmann distance between estimated intersection space and ground-truth
intersection; center: mean squared error between MCMC samples and their
corresponding projections into the other model (averaged over all posterior
samples); bottom: acceptance rates during MCMC sampling. Figure 7: Random
samples from the male and female color model of the LYHM [9] data set (1st and
2nd block) as well as samples of the computed intersection space (3rd block),
and respective differences (male without female, and female without male, 4th
and 5th block). Similar to shape (see Figure 3 of the main paper), we see
stronger male and female dominance in the differences and neutral gender in
the intersection, especially visible in the beard region. All color samples
are visualized on the mean face.
Reconstruction error. Following the main paper, to evaluate the quality of the
computed differences, we exploit the fact that we have samples
$\\{x_{1},x_{2},\dots,x_{r}\\}$ from the ground-truth difference, $D_{12}$ (or
$D_{21}$). Denote the MCMC samples from the estimated difference
$\hat{D}_{12}$ as $\\{\hat{x}_{1},\hat{x}_{2},\dots,\hat{x}_{r}\\}$. We then
evaluate the quality of $\hat{D}_{12}$ by quantifying whether or not samples
$\hat{x}_{j}$ can be as badly represented in the ground-truth intersection,
$I$, as samples $x_{j}$ from the ground-truth difference. Here, we expect high
errors since shapes belong to the difference if they can not be represented in
the intersection. Formally, we calculate the reconstruction errors
$d_{R}(D_{12},I)=\frac{1}{r}\sum_{j=1}^{r}d(\operatorname{proj}_{M_{I}}(x_{i}),x_{i})$
(25)
and
$d_{R}(\hat{D}_{12},I)=\frac{1}{r}\sum_{j=1}^{r}d(\operatorname{proj}_{M_{I}}(\hat{x}_{i}),\hat{x}_{i})$
(26)
by projecting samples onto the subspace $M_{I}$ spanned by $I$ and evaluating
its distance using the mean squared error (MSE; see Eq. (10) of the main
paper).
## Appendix D Ablation Study
We identified the variance involved in the distance likelihood, $\sigma^{2}$,
as the most important parameter of our algorithm (see Eq. (9) of the main
paper). To study its effects, in addition to the quantitative evaluation
presented in the main text, we also provide an ablation study for different
values of $\sigma$. The ablation is carried out on the star data set as
described in Section 4.1.1 of the main paper (see also Table 1 of the main
paper). Moreover, we only investigate the estimation of ground-truth
intersection spaces as our method shows similar behavior for differences.
The results can be found in Figure 6 (top row), averaged over all six star
models. As seen, setting $\sigma$ too high leads to large Grassmann distances,
implying a worse estimation of the ground-truth intersection. Contrary, the
smaller $\sigma$, the better the estimation of the true intersection space.
Interestingly, starting from $\sigma=0.003$, its exact value seems to become
less critical as even a decrease of factor 10 does not lead to significant
changes.
## Appendix E How to Choose $\sigma$ In Practice?
Since we usually do not have ground-truth intersections for real-world SSMs,
it is natural to wonder how to empirically validate the performance of our
method and the chosen $\sigma$ in practice. To this end, we also report the
MSE between an MCMC sample, $x\in Q_{1}$ (or $x\in Q_{2}$), and its projection
into the other model, $x^{\prime}$, as well as the acceptance rates during
MCMC sampling, see Figure 6 (center and bottom row). Note that the MSE is
computed between all posterior samples and their respective projections; the
average thus serves as an empirical estimation for the mean $\epsilon$ in Eqs.
(5) and (6) of the main text.
In terms of MSE, we observe an almost similar behavior as for the Grassmann
distances. Smaller MSEs correspond to lower Grassmann distances. As a result
of this correlation, the distance between $x$ and $x^{\prime}$ can be used as
an indication to determine a suitable value for $\sigma$. It does not require
a ground-truth and can be easily monitored during the run time of MCMC. To
avoid setting $\sigma$ too small (and preventing incorrect estimation of the
desired distribution), however, one should ensure that the acceptance rates
are between 0.25 and 0.5, see [34].
In conclusion, although a suitable value for $\sigma$ might not be trivial to
determine in practice, it can be well chosen by carefully inspecting the MSE
between posterior samples and their corresponding projections as well as
acceptance rates during MCMC sampling. Both quantities can be easily computed
without the necessity of ground-truth intersections or differences.
Figure 8: Random samples from the male and female models (top) and identity
and expression models (bottom) of FLAME (1st and 2nd block). Also shown are
samples of the computed intersection model (3rd block) as well as from the
difference between male and female (top) and identity and expression models
(bottom; 4th and 5th block). We observe very similar results as for LYHM
models and BFM 2019 [18], see Figures 3 and 4 of the main paper. Figure 9:
Per-vertex level visualization of similarities and differences between the
identity and expression spaces of FLAME. Depicted is the per-vertex variance
of identity and expression models as well as the posterior variance for the
computed intersection model and differences (red high variance, blue low). We
observe a similar behavior as for BFM 2019, see Figure 5 of the main paper.
## Appendix F Additional Qualitative Results
In this section, we provide results for the extension to color experiment
(Section 4.3 of the main paper) as well as additional qualitative results on
the FLAME [27] model.
Extension to color. The results are shown in Figure 7. We observe very similar
behavior for the color as we have seen for shape, see Figure 3 of the main
paper. Male features are exaggerated in faces drawn from the difference of
male and female, and samples from the difference of female and male appear
more feminine than faces from the original female model. Similar to the effect
in shape, we also perceive neutral textures in the intersection space.
Results on FLAME. We show exemplary results on FLAME in Figures 8 and 9. They
are very similar to the findings presented in the main paper, please refer to
Figures 3–5 of the main paper.
|
# Learning and Understanding a Disentangled Feature Representation for Hidden
Parameters in Reinforcement Learning
Christopher Reale, Rebecca Russell
###### Abstract
Hidden parameters are latent variables in reinforcement learning (RL)
environments that are constant over the course of a trajectory. Understanding
what, if any, hidden parameters affect a particular environment can aid both
the development and appropriate usage of RL systems. We present an
unsupervised method to map RL trajectories into a feature space where distance
represents the relative difference in system behavior due to hidden
parameters. Our approach disentangles the effects of hidden parameters by
leveraging a recurrent neural network (RNN) world model as used in model-based
RL. First, we alter the standard world model training algorithm to isolate the
hidden parameter information in the world model memory. Then, we use a metric
learning approach to map the RNN memory into a space with a distance metric
approximating a bisimulation metric with respect to the hidden parameters. The
resulting disentangled feature space can be used to meaningfully relate
trajectories to each other and analyze the hidden parameter. We demonstrate
our approach on four hidden parameters across three RL environments. Finally
we present two methods to help identify and understand the effects of hidden
parameters on systems.
## Introduction
Figure 1: Our method maps trajectories to a feature space that represents the
hidden parameters of the system. For example, we can map the trajectory of a
UAV with an unknown payload mass (top) into a disentangled feature space
(bottom).
Many robotic skills require the ability to operate effectively in similar but
slightly-varying environments. For example, when controlling a robot to move
through an environment, it must be able to operate on surfaces with different
coefficients of friction. Like friction, many conditions that affect system
behavior are fixed over a long period of time, perhaps even the entire course
of a trajectory, but may vary from one trajectory to the next. Termed _hidden
parameters_ (Doshi-Velez and Konidaris 2016), these conditions can affect the
behavior and performance of an RL system. In many real-world applications, it
may not be clear how many hidden parameters exist or how they affect the
system dynamics. In this work, we present an unsupervised method to learn
feature representations of trajectories by their disentangled hidden parameter
values. This feature space can then be used in downstream classification or
analysis tasks to aid in interpreting the hidden parameters.
Our approach leverages world models (also termed “dynamics models”) used in
deep model-based reinforcement learning (MBRL) systems, though it is broadly
applicable dynamical systems not based on MBRL. In recent years, deep MBRL has
emerged as a data-efficient learning approach to a wide variety of tasks
(Nagabandi et al. 2018). In deep MBRL, a world model represented by a neural
network is trained to predict the next state given the current state and an
action. The world model is then used with model-predictive control or model-
free reinforcement learning to select actions given a reward function.
Partially-observable state representations are typically handled by including
a recurrent layer or layers in the world model neural network architecture,
allowing the model to incorporate a memory of state observations in its
predictions. The features learned in this world model memory should include
information on any hidden parameters mixed in with other transient and
irrelevant information.
In our previous work (Reale and Russell 2022), we presented two innovations
which enable the learning of a disentangled feature space to represent hidden
parameters. Our first innovation is to modify the world model training
algorithm to constrain part of the world model’s internal recurrent
representation (memory) to be time-invariant: the model’s predictions should
not change when the values of the time-invariant features are replaced with
their corresponding values at another time-step along the same trajectory.
This constraint essentially prohibits the world model from storing transient
information in the time-invariant features, leaving only information
pertaining to the hidden parameters.
Our second innovation is a metric learning approach to map the time-invariant
features into a space with a meaningful metric that we term a latent
bisimulation metric. While the time-invariant features contain only hidden
parameter information, it is not possible to directly relate two sets of time-
invariant features. For example, two hidden parameter representations may be
far away in time-invariant feature space but have a similar effect on the
predictions produced by the world model and thus represent a similar pair of
hidden parameter values. The only way to relate two of time-invariant
representations in a semantically meaningful way is through the usage of the
world model. To fix this, we learn a mapping from the time-invariant feature
space into a metric space with distance proportional to the difference in
system behaviors. This is akin to learning a bisimulation metric (Ferns,
Panangaden, and Precup 2011) on the latent state (_i.e._ , the hidden
parameters) as opposed to the observable state.
In this paper, we summarize our previous work and further validate our
approach by applying it to a more complex UAV application. Additionally, we
present two methods to help explain the effects of the unknown hidden
parameter on a system to a user.
## Related Work
### Hidden Parameters
While the control of systems in partially-observable environments has been a
focus of research dating back decades (Kaelbling, Littman, and Cassandra
1998), only recently did Doshi-Velez and Konidaris (2016) introduce the idea
of studying constant unobservable variables, termed “hidden parameters.” They
contend that modeling a hidden parameter system, as opposed to an arbitrary
partially-observable system, simplifies the procedure for inferring the system
dynamics. They also present an approach for modeling the dynamics of a hidden
parameter system from data. Killian, Konidaris, and Doshi-Velez (2017)
improved on their approach by incorporating a Gaussian Process latent variable
model to improve uncertainty quantification and transfer. Yao et al. (2018)
further improve transfer-ability by embedding the hidden parameter values as
an input to a policy. While many environments of interest may contain multiple
hidden parameters of different types (e.g., continuous vs. discrete) as well
as additional dynamic latent variables, we focus on the simple case of
environments with a single discrete valued hidden parameter and no other
latent variables. We leave the analysis of more complex hidden parameter
spaces to future work.
While there is relatively little work specifically analyzing hidden
parameters, there is a long and vast history of work related to the more
general scenario of Partially Observable Markov Decision Processes (POMDPs)
such as Spaan (2012); (Cassandra 1998); Oliehoek, Witwicki, and Kaelbling
(2012); and Carr, Jansen, and Topcu (2021).
### Model-Based Reinforcement Learning
Model-Based Reinforcement learning is one of the main learning-based
approaches to controlling systems. Initially, most model based reinforcement
learning systems used simple world models such as Gaussian Processes
(Boedecker et al. 2014; Deisenroth and Rasmussen 2011; Ko and Fox 2009), time-
varying linear models (Levine and Abbeel 2014; Lioutikov et al. 2014; Yip and
Camarillo 2014), and mixture of Gaussians (Khansari-Zadeh and Billard 2011).
While data efficient, these approaches have trouble modeling high-dimensional
systems with non-linear dynamics. Recently, it has been shown that more
complex deep neural networks can be used effectively as well (Akkaya et al.
2019; Nagabandi et al. 2018, 2020).
Since hidden parameter systems are partially observable, a world model cannot
accurately predict a state from the preceding state and action alone. Instead,
world models must infer information about the unobserved latent variable(s)
through interactions with the environment over time. Recurrent neural networks
(RNN) have been used as world models to enable this functionality (Schmidhuber
1990b, a, 2015; Ha and Schmidhuber 2018).
### Bisimulation Metrics
Bisimulation metrics (Ferns, Panangaden, and Precup 2011; Ferns and Precup
2014) measure the behavioral similarity of two states of a Markov decision
process (MDP). Much research has focused on methods to compute bisimulation
metrics (Taylor, Precup, and Panagaden 2008; Castro 2020). Similar to our
approach, Zhang et al. (2020) proposed using bisimulation metrics to learn a
feature representation in a reinforcement learning setting. Our approach
differs by learning a bisimulation metric on the latent portion of the state
rather than the observable portion. Additionally, we use the learned
representation to better understand hidden parameters, whereas Zhang et al.
(2020) sought to improve control by filtering out irrelevant information.
## Background
### Model-Based Reinforcement Learning
Consider an agent operating in an environment. Let $\bm{s}_{t}\in\mathcal{S}$
be the observable environmental state and $\bm{a}_{t}\in\mathcal{A}$ be the
action taken by the agent at time $t$. At every time step, the system observes
state $\bm{s}_{t}$, takes action $\bm{a}_{t}$, and obtains the reward
$r(\bm{s}_{t},\bm{a}_{t})$. The system then takes the observable state
$\bm{s}_{t+1}$ according to the system transfer function
$f:\mathcal{S}\times\mathcal{A}\rightarrow\mathcal{S}$. The goal in
reinforcement learning is to select an action sequence
$\bm{A}_{t}^{(H)}=(\bm{a}_{t},\bm{a}_{t+1},\ldots,\bm{a}_{t+H-1})$ to maximize
rewards over a horizon length $H$,
$\displaystyle\bm{A}_{t}^{(H)}=\operatorname*{arg\,max}_{\bm{A}_{t}^{(H)}}\sum_{t^{\prime}=t}^{t+H-1}$
$\displaystyle r(\bm{s}_{t^{\prime}},\bm{a}_{t^{\prime}})$ (1)
$\displaystyle\bm{s}_{t^{\prime}+1}=f(\bm{s}_{t^{\prime}},\bm{a}_{t^{\prime}}).$
In MBRL, the transfer function $f$ is explicitly modeled as $\hat{f}_{\theta}$
with parameters $\theta$. Regardless of the model chosen, its parameters are
learned from example trajectories and, at run-time, it is used select optimal
actions as follows:
$\displaystyle\bm{A}_{t}^{(H)}=\operatorname*{arg\,max}_{\bm{A}_{t}^{(H)}}\sum_{t^{\prime}=t}^{t+H-1}$
$\displaystyle r(\bm{\hat{s}}_{t^{\prime}},\bm{a}_{t^{\prime}})$ (2)
$\displaystyle\bm{\hat{s}}_{t^{\prime}+1}=\hat{f}_{\theta}(\bm{\hat{s}}_{t^{\prime}},\bm{a}_{t^{\prime}}).$
### World Models
In deep MBRL, the world model $\hat{f}_{\theta}$ is a deep neural network with
network weights $\theta$. In our case, we use a recurrent neural network
architecture to be able to operate in a partially-observable environment. We
represent the internal state of the RNN at time $t$ as $\bm{h}_{t}$. The model
parameters $\theta$ are learned by stochastic gradient descent to minimize the
prediction error over a dataset $\mathcal{D}$ of trajectories of lengths
$T_{d}$ collected with a random policy:
$\min_{\theta}\frac{1}{|\mathcal{D}|}\sum_{d=0}^{|\mathcal{D}|-1}\frac{1}{T_{d}-1}\sum_{t=1}^{T_{d}-1}\|\bm{s}_{t}-\bm{\hat{s}}_{t}\|_{1}\\\
$ (3)
where
$\displaystyle\bm{\hat{s}}_{t+1}$
$\displaystyle=\hat{f}_{\theta}(\bm{s}_{t},\bm{a}_{t},\bm{h}_{t})$
$\displaystyle\bm{h}_{t+1}$
$\displaystyle=g_{\theta}(\bm{s}_{t},\bm{a}_{t},\bm{h}_{t})$
$\displaystyle\bm{h}_{0}$ $\displaystyle=\bm{0}$
## Our Approach
Our goal is to learn a feature space that represents trajectories by the
hidden parameters of the environment in which they occurred. Our approach
consists of two main steps. First, we present a new deep reinforcement
learning world model training algorithm to isolate time-invariant information
related to the hidden parameters in the RNN memory. Then, we show how to learn
a mapping from the RNN memory to a feature space where the distance between
points approximates a bisimulation metric.
### Time-Invariant RNN Memory
Figure 2: Visualization of time-invariant training procedure shuffling the
world model RNN memory in time over a trajectory.
Figure 3: Recurrent neural network world model architecture.
When operating in an environment with hidden parameters, the world model
cannot accurately predict future states without information about the
parameters. An RNN world model acquires this information by interacting with
the environment and encoding its trajectories in memory ($\bm{h}_{t}$) to
enable better predictions in the future. While the memory is the only source
of hidden parameter information the world model has access to, it also may
contain other transient information that helps the world model make
predictions (e.g., past states and actions). In this section, we present an
approach to encode information related to hidden parameters only.
The key insight to our approach is that, since hidden parameters are constant
over the course of a trajectory, features computed at a given time step that
only contain information about hidden parameters should be usable at all times
along the trajectory. Inversely, if the features contain additional
information related to transient factors, they will not be usable at all times
along the trajectory. Thus, we seek to train the RNN to encode features in its
memory that are applicable to the rest of the trajectory. We term this time-
invariant memory.
We achieve this by adding an additional step to the training procedure
presented earlier. As before, we generate a dataset of trajectories and use
stochastic gradient descent to minimize prediction error. However, in addition
to making predictions ${\hat{s}}_{t+1}$ of the next states using the
temporally-appropriate RNN memory values $\bm{h}_{t}$, we also make
predictions ${\tilde{s}}_{t}$ using RNN memory values from other time steps as
shown in Figure 2. More formally, we define $\bm{h}_{p(t)}$ where
$p:\\{0,1,\ldots,T_{d}\\}\rightarrow\\{0,1,\ldots,T_{d}\\}$ is a random
bijection that serves to permute the hidden memory values in time. Thus, we
minimize prediction error by:
$\min_{\theta}\frac{1}{|\mathcal{D}|}\sum_{d=0}^{|\mathcal{D}|-1}\frac{1}{T_{d}-1}\sum_{t=1}^{T_{d}-1}\|\bm{s}_{t}-\bm{\hat{s}}_{t}\|_{1}+\|\bm{s}_{t}-\bm{\tilde{s}}_{t}\|_{1}$
(4)
where
$\displaystyle\bm{\hat{s}}_{t+1}$
$\displaystyle=\hat{f}_{\theta}(\bm{s}_{t},\bm{a}_{t},\bm{h}_{t})$
$\displaystyle\bm{\tilde{s}}_{t+1}$
$\displaystyle=\hat{f}_{\theta}(\bm{s}_{t},\bm{a}_{t},\bm{h}_{p(t)})$
$\displaystyle\bm{h}_{t+1}$
$\displaystyle=g_{\theta}(\bm{s}_{t},\bm{a}_{t},\bm{h}_{t})$
$\displaystyle\bm{h}_{0}$ $\displaystyle=\bm{0}$
This training procedure encourages the world model to rapidly populate its
memory from a sequence of observations at any time with information that is
useful at all times in a trajectory.
### Latent Bisimulation Metric
Ultimately, our goal is to learn a feature representation to be able to
directly compare two trajectories in terms of the hidden parameter values of
their corresponding environments. While our time-invariant RNN memory contains
information pertaining only to the hidden parameters, the RNN memory from
different time steps and/or trajectories may not be directly comparable. For
example, two points in the RNN memory feature space may have very different
values, yet have a similar effect on the world model predictions and represent
the same hidden parameter. In this section we present a metric learning
approach to map from the time-invariant RNN-memory space with distance
proportional to the differences in system behavior. Since our feature space
encodes differences in system behavior based on latent variables (hidden
parameters) only, we call our learned metric a latent bisimulation metric.
We learn the embedding in two steps. First, we compute pairwise distances
between RNN memory features as the average prediction difference over a
representative set of state-action pairs. Then, we train a neural network to
map the RNN memory features into a space that enforces those distances.
In order to compute the pairwise distances, we first create a set of state-
action pairs,
$\mathcal{P}=\\{(\bm{s}_{0},\bm{a}_{0}),(\bm{s}_{1},\bm{a}_{1}),\ldots,(\bm{s}_{P-1},\bm{a}_{P-1})\\}$
by sampling them from a large set of trajectories. We then create the input to
our training dataset: a representative set of RNN memory values
$\mathcal{H}=\\{\bm{h}_{0},\bm{h}_{1},\ldots,\bm{h}_{H-1}\\}$, again by
sampling from a large set of trajectories. Finally, we compute the distances
$d(\bm{h}_{i},\bm{h}_{j})$ between points as follows:
$d(\bm{h}_{i},\bm{h}_{j})=\frac{1}{P}\sum_{p=0}^{P-1}\|\hat{f}_{\theta}(\bm{s}_{p},\bm{a}_{p},\bm{h}_{i})-\hat{f}_{\theta}(\bm{s}_{p},\bm{a}_{p},\bm{h}_{j})\|_{1}$
(5)
The distance measures the average (over a representative set of pairs of
states and actions) difference in predictions made by the world model when
using the two memory values. Once we have computed the distances, we learn an
embedding function $e_{\phi}(\bm{h})$, modeled as a neural network with
parameters $\phi$, by solving the following optimization problem with
stochastic gradient descent:
$\min_{\phi}\sum_{i=0}^{H-1}\sum_{j=i}^{H-1}\left|\|e_{\phi}(\bm{h}_{i})-e_{\phi}(\bm{h}_{j})\|_{1}-d(\bm{h}_{i},\bm{h}_{j})\right|$
(6)
## Experiments
### Use Cases
#### Modified Mountain Car
Figure 4: Visualization of the mountain car environment.
In the mountain car (Brockman et al. 2016) RL problem, shown in Figure 4, the
goal is for the car to reach the top of the mountain. The state consists of
two variables: horizontal position and horizontal velocity. The action space
consists of three possible actions: accelerate to the right, coast, and
accelerate to the left. We modified the mountain car environment to include
variable gravity as a hidden parameter. For each trajectory, the gravity
strength is randomly chosen from a set of three values: 75%, 100%, and 125% of
the original gravity.
#### Pusher Robot
Figure 5: Visualization of the pusher robot environment with illustrated ball
trajectories for three different hidden parameter values.
The goal of the pusher robot environment, shown in Figure 5, is to control a
robotic arm to push a ball on to a target location. It is a modified version
of the reacher environment from Brockman et al. (2016). The state space
consists of 12 variables containing positions, orientations, velocities, and
rotations of the arm components and ball. The action space consists of two
torque values applied to the two arm joints (shoulder and elbow). The hidden
parameter in this environment is the relative strength of the joint torque and
can take on a value of 50%, 75%, and 100% of the original strength. Figure 5
shows the difference in ball locations of trajectories produced with the same
set of actions for different strength values.
### UAV ISR
The goal of the UAV ISR environment (Conlon et al. 2022) is to control a UAV
to fly to a target location in order to collect data. We modify the
environment to include a hidden parameter in two ways. One way is by
introducing a payload weight sampled from a continuous uniform distribution.
The other is the ambient temperature sampled from a continuous distribution.
In this work, we assume there is only a single hidden parameter, so when
analyzing one hidden parameter, we allow the other to be visible by making it
a part of the agent state.
### Implementation Details
We use a similar world model architecture (shown in Figure 3) for both of our
reinforcement learning environments. It is comprised of a gated recurrent unit
(GRU) (Chung et al. 2014), which introduces recurrence to the model, as well
as a pair of multi-layer perceptrons (MLP), which serve as an encoder and
decoder of the state. Since our goal is to filter out transient information
from the hidden memory, we introduce a skip connection around the GRU to allow
information to propagate from the input state and action to the prediction
without going through the memory. An alternative to the skip connection is to
partition the hidden memory into a time-invariant portion and a regular
portion. The time-invariant portion is trained with our algorithm to represent
hidden parameters and the regular portion is trained with the standard
training algorithm. Not only does this allow the transient information to be
passed from state and action to the prediction, it also allows the GRU to
capture information related to non-constant latent variables, which exist in
many applications.
Figure 6: Classification accuracy for mountain car hidden gravity parameter.
Figure 7: Classification accuracy for pusher hidden strength parameter.
Figure 8: Regression error for hidden UAV payload mass parameter.
Figure 9: Regression error for hidden UAV ambient temperature parameter.
### Results
#### Hidden Parameter Estimation
To demonstrate the effectiveness of our approach we show that our learned
features can be used to estimate the hidden parameter of a trajectory with
limited labeled training data. For each application, we generate 5000
trajectories and use them to train two world models: one with the baseline RNN
training algorithm and a second with our time-invariant memory algorithm.
Additionally, we use the same data set to embed the time-invariant memory in a
latent bisimulation metric space.
After training the models and learning the embeddings, we collect two smaller
data sets: one set of 30 trajectories and one set of 100 trajectories to serve
as the training and testing data for the classification task. We use the world
models and learned embeddings to generate features for the trajectories and
use a k-nearest neighbors to classify the test trajectories. Since our world
model produces a new memory output at each time step, we also can generate a
new representation of the trajectory at each time step. Therefore, we evaluate
the estimation accuracy as a function of the time since the beginning of the
trajectory. As one would expect, as the agent interacts with the environment,
it acquires more information about the hidden parameters, which increases
accuracy.
Our estimation results are shown in Figures 6, 7, 8, and 9 and compare three
feature representations. The “Baseline” results use the hidden memory of a
world model trained with a standard training algorithm, the “Time-Invariant”
results use the hidden memory of a world model trained with our time-invariant
approach, and the “Time-Invariant + Embedding” results use the latent
bisimulation metric features. It is clear that our approaches significantly
outperform the baseline for all environments and parameters, both in terms
peak accuracy and the time it takes to reach peak accuracy. The latent
bisimulation metric features provide a marginal improvement over the time-
invariant features on mountain car, while they offer a significant performance
boost on pusher and UAV. We believe this is due to the relative complexity of
the environments. Compared to the mountain car environment, the pusher and UAV
have much larger states (12+ vs 2 variables), so the world models learn
exponentially more patterns to recognize the effects of the hidden parameter.
This results in a more complicated hidden memory feature space where values
are not easily directly comparable, which benefits more from the metric
learning approach. Mountain car is simpler and so the metric learning approach
does not add as much value.
Figure 10: Visualization of latent feature spaces for pusher showing the clean
separation of the three different hidden parameter values.
Figure 11: Visualization of latent feature spaces for (continuous) UAV payload
mass.
Figure 12: Visualization of latent feature spaces for (continuous) UAV ambient
temperature.
#### Latent Space Visualization
Another way to see the effectiveness of our approach is to visualize the
feature spaces learned by the algorithms for the pusher and UAV environments.
Since they are too high-dimensional to visualize directly, we further reduce
their dimension using unsupervised techniques (t-SNE (Van der Maaten and
Hinton 2008) for pusher and UMAP (McInnes, Healy, and Melville 2018) for UAV).
Figure 10 shows the visualization of our learned features for the pusher
environment. We show the features at three times along a trajectory: after 5,
10, and 15 steps. As with the classification results, the “Baseline” method
uses the hidden memory of a world model trained with a standard training
algorithm, the “Time-Invariant” method uses the hidden memory of a world model
trained with our time-invariant approach, and the “Time-Invariant + Embedding”
method uses the latent bisimulation metric features. Each plotted marker
represents a trajectory at that time, where the marker used signifies the
value of the hidden parameter: blue circles, orange squares, and green
triangles represent torque strengths of 50%, 75%, and 100% respectively.
Ideally, all trajectories with the same hidden parameter value would be
represented by the same point in the feature space and trajectories with
different hidden parameters would be far apart from one another. At all three
time steps, the “Baseline” features perform the worst as the visualization
appears random. The “Time-Invariant” features display more structure with some
overlap between hidden parameter values. The “Time-Invariant + Embedding”
features clearly perform the best with the most structure and the least
overlap. By time-step 15, the three hidden parameter values appear to be
easily separable.
Figures 12 and 11 show the visualizations for the UAV hidden
parameters—payload mass and ambient temperature—respectively. Both feature
spaces are well organized by hidden parameter. It is especially noteworthy
that the feature space representing the payload mass hidden parameter forms a
one-dimensional manifold, as this is the dimension of the original hidden
parameter.
### Understanding Hidden Parameters
In this section, we present two methods to help understand better the effects
of hidden parameters on systems. First, we show how to determine which state
variables are most effected by the hidden parameter. Then, we show how to use
the world model to generate “imagined” trajectories for a range of hidden
parameter values.
#### Hidden Parameter State Effects
Our goal is to understand what state features are most affected by the hidden
parameters. We accomplish this by comparing the performance of a stateful
world model (an RNN) with a non-stateful world model (an MLP). While neither
world model is given explicit information about the hidden parameter, stateful
world models can learn to infer information about it from the system’s
behavior and use it to make better predictions. More specifically, we compute
the ratio (RNN / MLP) of the prediction accuracies for the two models for each
feature in the agent’s state to determine which are most affected by the
hidden parameters. We then conclude that the hidden parameters have the
greatest effect on features for which a stateful world model improves
predictions the most and have the least/no effect on features for which the
performance is unchanged.
Tables 1 and 2 show the relative prediction accuracies of the two models on
each feature in the state. For the hidden payload mass, it is clear that the
hidden parameter has the biggest effect on the velocity in the z direction as
well as the pitch. This makes sense as the increased gravitational force
should have a significant effect on those features. Additionally, the velocity
in the x and y direction, the angular roll and pitch velocities, and the
position in the z direction are all affected. For the hidden ambient
temperature, the battery level is most affected. This makes sense as the
battery life is very sensitive to the ambient temperature.
Feature | Error Ratio
---|---
Position x | 2.42
Position y | 2.36
Position z | 4.82
Roll | 1.72
Pitch | 2.16
Yaw | 1.78
Velocity x | 2.46
Velocity y | 2.39
Velocity z | 5.25
Velocity Roll | 1.71
Velocity Pitch | 1.71
Velocity Yaw | 1.35
Battery | 0.98
Table 1: Effect of payload mass on UAV state variables. Feature | Error Ratio
---|---
Position x | 1.52
Position y | 1.51
Position z | 2.71
Roll | 1.17
Pitch | 1.20
Yaw | 1.27
Velocity x | 1.59
Velocity y | 1.60
Velocity z | 2.70
Velocity Roll | 1.02
Velocity Pitch | 1.00
Velocity Yaw | 1.06
Battery | 8.26
Table 2: Effect of ambient temperature on UAV state variables.
#### Visualizing Trajectories by Hidden Parameter
In this section, we show how to visualize “imagined” trajectories to enable a
user to recognize patterns. Specifically, for a given initial state and
sequence of actions, we use the world model roll out trajectories for a
variety of hidden parameter values. We then plot the trajectories on the same
plots as shown in Figures 13 and 14. Note the color of each trajectory
indicates the corresponding color in Figures 11 and 12. Figure 11 shows the
trajectories when the payload mass is the hidden parameter. As one would
expect, different payload masses effect the position and velocity of the UAV.
Figure 12 shows the trajectories when the temperature is the hidden parameter.
The plots show that temperature has a large effect on the battery life but
otherwise does not affect the dynamics much.
Figure 13: Visualized imagined trajectories for varying UAV payload mass.
Figure 14: Visualized imagined trajectories for varying UAV ambient
temperature.
## Conclusions and Future Research
We have proposed an unsupervised learning approach to represent trajectories
by features signifying the hidden parameter of the environment. Our approach
consists of two steps. First, we use a novel world model training algorithm to
isolate hidden parameter information in an RNN-based world model memory by
enforcing a time-invariance constraint. We then use a metric learning approach
to map the the RNN-hidden memory to a latent bisimulation metric space where
the distance between points signifies the relative difference in system
dynamics due to the hidden parameter. Finally, we presented hidden parameter
estimation and visualization results to validate our approach on four hidden
parameters across three applications.
While our approach works well on the presented environments, they represent a
only a small subset of the challenges that could be faced in real world, and
there are many potential avenues for future research. One could learn how to
estimate the number of hidden parameters in an environment and ways to
decouple them to understand their individual effects. Additionally, one could
extend from static hidden parameters to dynamic latent variables that change
at different time-scales (e.g., temperature changes on the order of hours,
while the wind changes on the order of seconds). And finally, the end goal of
this problem is to better characterize the behavior of systems to inform human
users. Finding more effective ways to tether information about hidden
parameters to real-world phenomena in order to better communicate it remains
an open problem.
## Acknowledgements
This material is based upon work supported by the Defense Advanced Research
Projects Agency (DARPA) under Contract No. HR001120C0032. Any opinions,
findings and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of DARPA.
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|
# Benchmarking Evaluation Metrics for
Code-Switching Automatic Speech Recognition
###### Abstract
Code-switching poses a number of challenges and opportunities for multilingual
automatic speech recognition. In this paper, we focus on the question of
robust and fair evaluation metrics. To that end, we develop a reference
benchmark data set of code-switching speech recognition hypotheses with human
judgments. We define clear guidelines for minimal editing of automatic
hypotheses. We validate the guidelines using 4-way inter-annotator agreement.
We evaluate a large number of metrics in terms of correlation with human
judgments. The metrics we consider vary in terms of representation
(orthographic, phonological, semantic), directness (intrinsic vs extrinsic),
granularity (e.g. word, character), and similarity computation method. The
highest correlation to human judgment is achieved using transliteration
followed by text normalization. We release the first corpus for human
acceptance of code-switching speech recognition results in dialectal
Arabic/English conversation speech.
Index Terms— ASR, Code-switching, Evaluation metric
utf8
## 1 Introduction
Code-switching (CS) is the act of using more than one language within the same
discourse. The prevalence of CS across multi-cultural and multi-lingual
societies has been met with a growing interest in the NLP and speech
processing fields. Automatic Speech Recognition (ASR) for CS is a well studied
problem [1] conducted in several language pairs on acoustic modeling, language
modeling and novel system architectures.
In code-switched language pairs with different scripts, there is a tendency to
cross-transcribe words that creates a large number of homophones in the data,
leading to challenges in training and evaluation. Rendering the CS
transcription accurately is important, however, is often not straight-forward,
and can be inconsistent, leading to the same word being transcribed using
different scripts [2]. Cross-transcription is particularly challenging in
languages having a high amount of loanwords, as it is not always clear which
language a word belongs to, and hence, which script to transcribe it in.
Table 1: This example shows a hypothesis (H), the minimal edits annotation
(ME), the ArzEn transcription reference (R), and their Buckwalter
transliterations [3] (R_BW, H_BW, and ME_BW). We denote annotation decisions
(A-Decision) as substitution (S), insertion (I), deletion (D), or acceptance
(A) for words having different forms across H and ME. In this example, CER and
WER reductions are seen for minimal edits annotations over reference due to
cross-transcription (segments 1, 2, 3, and 4) and unstandardized orthography
(segments 9 and 13) issues.
Investigating the ideal ASR performance metric for CS, [2] propose a modified
WER metric based on mapping both languages into the pronunciation space, which
leads to improvements in accuracy and evaluation. [4] propose a
transliteration-based WER metric by transliterating mixed-script utterances
into a single script. The authors demonstrate the robustness of the proposed
approach on several Indic languages. However, these techniques can also lead
to false positives when there are words in the two languages that sound the
same but have a different meaning. It also has limitations in cases where
different parts of the word are written in different scripts. e.g., the word
آرت¿ficial, which is artificial in English script or آرتفيشيال¿ in Arabic
script. Thanks to word-pieces [5], we may expect partial (cross-script)
transcription in CS ASR results.
Researchers have also investigated techniques for handling the problem of non-
standardized orthography for Dialectal Arabic ASR evaluation. [6] proposed the
use of Multi-Reference Word Error Rate to allow for a wider coverage of
different spelling variants for Dialectal Arabic ASR evaluation. Whereas, [7]
investigated approaches to reduce spelling variations, which included
normalization and following the Conventional Orthography for Dialectal Arabic
(CODA) [8] guidelines to spell words, as well as relying on morphologically
abstracted forms obtained from different tokenization schemes and
lemmatization.
This work builds upon previous contributions from [4] and [2]. In this study,
we investigate various methods that go beyond orthographic transliteration
methods. We explore lexical and phonetic representations for evaluation, and
study various weighted edit-distance methods and semantic similarity
evaluation. We designed a guideline and developed a reference human
acceptability corpus (HAC) – quantifying the human judgments in terms of
minimal editing of ASR hypothesis. We evaluate a large number of metrics in
correlation with human judgments. We believe that this is the first study on
human acceptability for CS speech recognition.
The contributions of this paper are as follows:
* •
We design and develop the first corpus for human acceptance for CS speech
recognition. The corpus and its guidelines are publicly
available.111http://arzen.camel-lab.com/
* •
We introduce phone similarity edit distance (PSD) and show its correlation
with human acceptance.
* •
We propose a novel approach to use machine translation on hypotheses and
references and report results using semantic evaluation to overcome the cross-
transcription challenge.
## 2 HAC: Human Acceptability Corpus for Code-Switching
For developing the Human Acceptability Corpus for Code-switching (HAC), we use
a subset from the ArzEn Egyptian Arabic-English CS conversational speech
corpus [9] and obtain the hypotheses using different ASR systems trained on
publicly available corpora. We carefully design the human annotation task to
indicate the amount of post-editing effort needed to correct the hypotheses.
While previous work has relied on human judgments in the form of systems’
ranking [7, 10], we opt for a more fine-grained human evaluation, where each
hypothesis is evaluated independently in terms of the number of edits
performed by the annotators. Throughout the paper, we will refer to the
annotators’ post-edited text as ‘minimal edits annotations’ and to the
original ArzEn transcriptions as ‘references’.
### 2.1 Annotation Guidelines
The annotators were provided with an audio file for each utterance and its
hypothesis. The references were not provided to avoid biasing the annotations.
The annotators were asked to perform minimal edits to make the hypotheses
acceptable, obeying the following rules:
* •
Rule of Script Segregation: Words should be written in Arabic or Roman script,
and not a mix of the two. The only exception is the writing of Arabic affixes
and clitics in conjunction with English words. Arabic words should be in
Arabic script. English words can be in Arabic or Roman script (default is
Roman). For missing words, the default script of the language should be used.
English origin words that have been integrated in Arabic templatic morphology,
and phonology will be treated as Arabic words. These guidelines differ from
the guidelines used by [9] for collecting ArzEn transcriptions, where the
transcribers were suggested to use Arabic script for Arabic words and Roman
script for English words.
* •
Rule of Acceptable Readability: The transcript should be made readable enough
to allow someone to reproduce the original audio and intended meaning.
* •
Rule of Minimal Edit: Spelling variations are acceptable as long as they do
not break the rule of acceptable readability. This rule covers the cases where
certain letters can be used interchangeably, such as ت¿-ث¿, ق¿-ئ¿, ي¿-ى¿,
ا¿-إ¿-أ¿, and ة¿-ه¿. Non-verbal speech effects should not be added if absent
from the ASR output. If they are included and in the audio, they should be
accepted. If they are not in the audio, they should be deleted.
In Table 1, we provide an example demonstrating the minimal edits annotation.
While the CER and WER for the hypothesis and reference are 50.0% and 78.6%,
the error rates are dropped to 21.6% and 46.2% when comparing minimal edits
annotation and hypothesis, showing the high amount of characters and words
mispenalized by CER and WER.
### 2.2 Design Consideration
We sampled two hours of speech from ArzEn training set, consisting of seven
recordings and covering a total of 1,304 utterances. For each utterance, we
obtained the ASR hypotheses from three different ASR systems, resulting in a
total of 3,903 hypotheses to be annotated for minimal correction.222Three
utterances were excluded from the dataset as they only contain non-speech
tags, resulting in 1,301 utterances, for each of which we have three ASR
hypotheses. These hypotheses were annotated by four Arabic-English bilingual
speakers. We use three pretrained bilingual (Arabic-English) ASR systems, to
generate the hypotheses. The systems vary either in architecture or in the
decoding parameters, as described below.
##### HMM-DNN:
We used a grapheme-based model trained using a Time Delay Neural Network
(TDNN) [11] with the LF-MMI objective [12]. For training the model, we used
the alignments from the context-dependent Gaussian Mixture Model and Hidden
Markov model (GMM-HMM) system. For decoding, we opt for the $4$-gram model,
trained on the ASR transcription.
##### End-to-End ASR:
For the end-to-end (E2E) ASR [5] system we used a transformer based
architecture [13], comprised of two sub-networks: conformer encoders and the
transformer decoders [14]. The ASR system consists of 12 encoder layers and 6
decoder layers, each with 2,048 encoder/decoder units from the feed-forward
layers, and 8 attention heads with 512 transformation dimensions. Note that
E2E model uses word-piece byte-pair-encoding (BPE) [15], with size of $\sim
10k$. For the study, we adopted two variations of this model, by changing the
size of the beam search in the decoding space. The variants are: (1)
Conformer-Accurate (Conformer-A) – with a beam size of $60$ and (2) Conformer-
Fast (Conformer-F) with a beam size of $2$.
### 2.3 Inter-annotator Agreement
| A2 | A3 | A4 | H | R
---|---|---|---|---|---
A1 | 8.3 / 16.7 | 6.3 / 15.0 | 8.3 / 18.4 | 14.9 / 27.8 | 19.9 / 44.0
A2 | | 8.9 / 17.6 | 10.8 / 20.7 | 14.1 / 24.6 | 20.5 / 44.9
A3 | | | 6.9 / 15.8 | 15.1 / 27.9 | 18.6 / 41.4
A4 | | | | 16.0 / 29.0 | 20.5 / 44.9
Avg | | | 8.2 / 17.4 | 15.0 / 27.3 | 19.9 / 43.8
Table 2: Inter-annotator agreement, showing CER / WER between annotators’
minimal edits, as well as each annotator with ASR hypotheses (H) and ArzEn
references (R).
We randomly sampled 203 sentences, annotated by the four annotators, for
inter-annotator agreement (IAA).333Two sentences were excluded from the IAA
calculations as they were annotated as unclear. The IAA evaluation is
presented in Table 2, where we report the CER/WER between every two
annotators. As shown, on average, the CER and WER measured between annotators
are 8.2% and 17.4% respectively. While these figures are relatively high, it
is not surprising, and reflects the complexity of the task, where word
acceptability and choice may differ across annotators due to unstandardized
orthography. We present the CER/WER between minimal edits annotations and
hypotheses reflecting the amount of edits performed by the annotators.
Moreover, we show the CER/WER between minimal edits annotations and references
reflecting the amount of characters/words that would be mispenalized when
evaluating minimal edits annotations against references using CER/WER. These
numbers are also a good indicator of the limitation of CER and WER as accurate
evaluation metrics.
## 3 Metrics Under Evaluation
Using the developed corpus, we assess multiple evaluation metrics against the
ground truth error, measured in terms of human post-editing effort. The
metrics we consider vary in terms of representation (orthographic,
phonological, semantic), directness (intrinsic vs extrinsic), granularity
(word vs character), and similarity computation method.
### 3.1 Orthographic Metrics
In the orthographic space, we investigate the use of four performance
measures: WER, CER, Match Error Rate (MER), and Word Information Lost (WIL)
[16].
$\displaystyle WER=\frac{S+D+I}{H+S+D}$ (1) $\displaystyle
MER=\frac{S+D+I}{H+S+D+I}$ (2) $\displaystyle
WIL=1-\frac{H^{2}}{(H+S+D)(H+S+I)}$ (3)
where H, S, D and I correspond to the number of word hits, substitutions,
deletions and insertions. MER computes the probability of a given match
between reference and hypothesis being incorrect. By including S, D and I in
the denominator, the range of MER has the bound [0,1]. WIL, introduced in
[17], is an approximate measure reflecting the proportion of words lost
between hypothesis and reference.
While these measures vary in granularity, they all fall short against the
cross-transcription issue. Therefore, following the work of [2], we
investigate the effect of transliterating hypotheses and references into
primary as well as secondary language scripts on alleviating the cross-
transcription problem. We perform automatic transliteration using the
transliteration API provided by QCRI
[18]444https://transliterate.qcri.org/api. In Table 3, we present an example
showing the reduction achieved in CER and WER by mapping the texts into one
common script, handling both cross-transcription (columns 1-3) and orthography
unstandardization (column 4) challenges. One drawback of this technique though
is its dependence on the availability of a language-specific transliteration
system, and that its effectiveness is tied to the performance of that system.
Following the work of [7], in order to reduce the spelling variation resulting
from dialectal Arabic unstandardized orthography, we investigate applying
Alif/Ya normalization as a variant for the experiments relying on orthographic
evaluation metrics.
Table 3: Example for a reference (R) and hypothesis (H), showing their corresponding transliteration output to Arabic (R_trAr and H_trAr) and Roman (R_trEn and H_trEn) scripts, as well as their IPA phone mapping (R_phone and H_phone) using Epitran. In this example, we separate the phones at word boundaries for better readability, however, the spaces are not included in PER and PSD calculations. | Text | CosineSim
---|---|---
R | عادي وكان كلهم عربي¿ ناشيونال سكولز¿ كان تعليمي لغاية الجامعة كان في ¿ |
H | عادي و كان كلهم عربي¿ national schools كان تعليمي لغاية الجامعة كان في¿ | 0.906
REn | My education until university was in normal national schools, and they were all Arab |
HEn | My education until university was in a normal National School, and they were all Arab | 0.942
Table 4: Example presenting a reference (R) and hypothesis (H) along with
their translations (REn) and (HEn), showing how the cross-transcription issue
was resolved through translation. The cosine similarities between R-H and REn-
HEn are shown, where the embeddings are obtained from mBERT.
### 3.2 Phonological Metrics
The orthographic-based metrics only consider literal correction, and do not
adequately handle orthographic unstandardization. To overcome such limitation,
we propose phone similarity edit distance
(PSD),555https://github.com/JSALT2022CodeSwitchingASR/Evaluation where we
measure the edit distance between hypotheses and references in the shared
phone space using International Phonetic Alphabet (IPA) mapping. The main
advantage of this approach over the transliteration is that the IPA mapping is
model independent and deterministic based on the grapheme to phoneme (G2P)
dictionary. The IPA uses standardized representation of speech sounds across
different languages. When calculating PSD, we scale the substitution cost by
the dissimilarity between the phones based on the articulation features. As a
result, PSD adds partial substitution penalty when the phones are different
but close in pronunciation. We use the Epitran toolkit [19] that contains
massive multilingual G2P including Arabic (ara-Arab) and English (eng-Latn)
languages. PSD is calculated as:
$PSD=\frac{w_{S}\sum{S_{i}}+w_{D}\sum{D_{j}}+w_{I}\sum{I_{k}}}{N}$ (4)
where $S$, $D$, $I$ are number of phones substitutions, deletions, and
insertions, respectively; $N$ is the number of phones in the reference;
$S_{i}=1-sim(x_{i},y_{i})$ where x and y are the aligned phones. The
$w_{S},w_{D},w_{I}$ are the costs for substitution, deletion and insertion
respectively with value of $1$ by default. We investigate different values for
$w_{S}$ (${1,2,4,8}$), while keeping $w_{D}=1,w_{I}=1$. In Table 3, it can be
seen that our PSD implementation can easily handle code switching. However,
the main shortcoming of PSD approach is that it does not consider the
semantics of the words.
### 3.3 Semantic Metrics
Next, we assess the ASR output considering semantic similarity. Following
[20], we measure the semantic similarity between reference and hypothesis
pairs as the cosine similarity between their embeddings obtained from
pretrained transformer models. The embeddings are generated using mean pooling
over token embeddings.666We also investigate the use of CLS token, however the
correlations are significantly lower. While this approach has been previously
investigated for monolingual ASR evaluation [20, 10], it has not been
investigated in the scope of CS.
We also introduce a novel pipeline for semantic-based ASR evaluation, where we
translate the hypotheses and references into monolingual sentences using
Google Translate API777https://cloud.google.com/translate. The translations,
as well as the original reference-hypothesis pairs, are then evaluated in
terms of the following machine translation (MT) evaluation metrics: BLEU [21],
chrF [22], and BertScore (F1) [23], in addition to cosine similarity. We
explore translating the sentences into the primary language (Arabic),
secondary language (English), as well as a completely independent language
(Japanese), to investigate the ability of the approach to generalize on
different languages. For calculating cosine similarity, we explore the use of
several pretrained models, including mBERT and language-specific BERT models,
for obtaining the sentence embeddings. For the original texts containing CS,
we use mBERT [24] and BiBERT [25]. For Arabic translations, we use mBERT,
CAMeLBERT [26], and AraBERT [27]. For English translations, we use mBERT and
BERT-base [24]. For Japanese translations, we use mBERT, BERT-base-
japanese888https://huggingface.co/cl-tohoku/bert-base-japanese, and BERT-
large-japanese999https://huggingface.co/cl-tohoku/bert-large-japanese.101010We
only present the results for the best settings. We show the results with
lowercasing and performing Alif/Ya normalization, which have shown to improve
correlations. We do not present results for chrF++ as it gave slightly lower
correlations compared to chrF. For monolingual English translations,
BertScore(F1) using mBERT gave higher correlations than BertScore(F1) using
roberta-large. For cosine similarity, using bert-base-multilingual-cased has
also shown to give overall higher correlations over bert-base-multilingual-
uncased.
This approach provides the following advantages: (1) through translation,
words in different scripts or with spelling variations can be mapped to the
same/similar word(s), and (2) by using semantic similarity, such words can be
assigned lower errors if closely represented in the embedding space. As seen
in Table 4, the words ‘ناشيونال سكولز¿’ were successfully mapped through
translation to ‘National Schools’. One limitation of this approach, however,
is that it is highly dependent on the quality of the embeddings obtained from
the pretrained models as well as MT performance.
## 4 Experimental Results
### 4.1 Experimental Setup
We define the ground truth error for each hypothesis to be the amount of human
post-editing effort required to correct it. Accordingly, we define $GoldCER$
to be the edit distance between the hypothesis and minimal edits annotation
calculated using CER. We opt for using CER in this calculation over other
evaluation metrics as it provides higher granularity in reflecting the effort
done by annotators and is consistent with the annotation guidelines.
In order to assess the performance of the evaluation metrics, we compare their
scores against $GoldCER$ on both the sentence-level and system-level.111111We
exclude 17 utterances that are annotated as unclear. In the sentence-level
evaluation, we calculate the correlation between the scores provided by each
metric for every hypothesis-reference pair against their corresponding
$GoldCER$ values. This evaluation provides a fine-grained assessment
demonstrating the ability of each metric to distinguish the amount of errors
in each hypothesis. In the system-level evaluation, we calculate the overall
$GoldCER$ for each of the three systems (HMM-DNN, Conformer-A, and
Conformer-F) which acts as the ground truth score. We then obtain the overall
scores for the three systems using each evaluation metric, assessing its
ability to provide correct system ranking.
### 4.2 Results and Discussion
#### 4.2.1 Overall Sentence-level Evaluation
Table 5: Sentence-level correlations calculated between $GoldCER$ and the
scores of different metrics. Given that the orthographic and phonological
evaluation metrics used are error metrics, while the semantic metrics are
accuracy metrics, we present the correlations against (1-$GoldCER$) for
semantic metrics, to consistently report positive correlations for easier
readability. CosineSim[mBERT] are the results achieved using mBERT.
CosineSim[*BERT] are the results achieved using BiBERT for the original
hypothesis-reference pairs (Base), CAMeLBERT for Arabic translations (which
outperformed AraBERT), BERT-base for English translations, and BERT-base-
japanese for Japanese (which outperformed BERT-large-japanese).
In Table 5, we present the sentence-level correlations between $GoldCER$ and
different metrics’ scores. For the orthographic metrics, we demonstrate the
correlations using CER, WER, MER, and WIL applied on the original hypothesis-
reference pairs (Base) as well as their transliterations into Arabic
(Translit$>$Ar) and Roman (Translit$>$En) scripts. We also show the effect of
applying Alif/Ya normalization on the original sentences (Base+ArNorm) and the
transliterations in Arabic script (Translit$>$Ar+ArNorm). We observe that
across the different settings, the highest correlations are achieved using
CER, followed by WER and MER, then WIL. Similar to the findings in [7], with
normalization, higher correlations are achieved, which we report for all the
four metrics. When applying CER, transliterating to Arabic outperforms
transliterating to English. However, for word-level evaluation metrics (WER,
MER, and WIL), transliterating to English gives higher correlations, even
though the references are dominated by Arabic words (77%). This can be
justified by the ability to resolve Arabic unstandardized orthography issues
when transliterating into English.
Table 6: Reported sentence-level correlations per recording, with recording-
level Code-Mixing Index (CMI in percentage). We show the correlations for CER,
WER, MER, WIL, in addition to transliterating to Arabic followed by Alif/Ya
normalization (Tr+), PSD ($w_{s}=4$), and $Average(Base,MT>*)$ for the
semantic measure (Sem). Table 7: System-level overall scores and ranking
(denoted between parentheses) across the different evaluation metrics.
In the scope of phonological metrics, we find that $w_{S}=4$ provides the
highest correlation of PSD with human judgment. The efficacy of incorporating
phonological similarity in the error calculation is demonstrated, where PSD
outperforms PER (Phoneme Error Rate). For semantic metrics, the highest
correlations are achieved using cosine similarity, followed by chrF,
BertScore(F1), then BLEU. Across the different metrics, higher correlations
are achieved by applying the metrics directly on the original text rather than
on translated text. This can be foreseen, as the success of this approach is
dependent on the performance of the underlying MT system. By looking into the
translations, it is obvious that a significant amount of translation errors is
introduced, which is propagated to the metric scores. However, the translation
step still proves to be beneficial, where across the metrics, higher
correlations are achieved by applying sentence-level aggregation to the scores
achieved across the different language setups, where we have tried $Avg$ and
$Max$ functions.121212For $Avg$, the score for each reference-hypothesis pair
is calculated as the average of cosine similarity scores achieved across the
four language setups, where we choose the best-performing pretrained model for
each language. For $Max$, the score for each pair is calculated as the maximum
cosine similarity score across the four language setups using mBERT model. We
also investigate taking the average of cosine similarity scores using mBERT
model, however, it gives lower correlations. The highest correlation for
semantic metrics is achieved by aggregating the cosine similarity scores using
the $Avg$ function across the original sentences and the three translations.
While the semantic-based metrics provide lower correlations compared to
transliteration and phonetic similarity, we believe that the efficacy of the
proposed translation pipeline should be revisited with future advances in CS
MT systems.
In general, results show that transliteration and phonetic similarity
outperform conventional CER, WER, MER, and WIL evaluation metrics applied
directly on the original sentences. Despite the limitations of the semantic
measures, it outperforms WER, MER, and WIL, and performs equally well as CER.
The highest correlation is achieved by using CER over the texts transliterated
into Arabic script followed by Arabic normalization.
#### 4.2.2 Recording-based Sentence-level Evaluation
To further investigate the validity of our results, we perform the same
sentence-level evaluation across the seven different recordings in our corpus,
as presented in Table 6. Given that the recordings contain different degrees
of CS, this analysis allows us to evaluate the consistency of our results. We
measure CS levels in terms of Code-Mixing Index (CMI) [28, 29], calculated on
the utterance-level as follows:
$C_{u}(x)=100*\frac{\frac{1}{2}*(N(x)-max_{L_{i}\in\textbf{L}}\\{t_{L_{i}}\\}(x))+\frac{1}{2}P(x)}{N(x)}$
where $N$ is the number of language-dependent tokens in utterance $x$;
$L_{i}\in\textbf{L}$ the set of all languages in the corpus;
$max\\{t_{L_{i}}\\}$ represents the number of tokens in the dominating
language in $x$, with $1\leq max\\{t_{L_{i}}\\}$$\leq N$; and $P$ is the
number of code alternation points in $x$; $0\leq P<N$. We calculate the
recording-level CMI by averaging the utterance-level values. We note that
there is a degree of variety in CMI values across recordings ranging from 11
to 20. In Table 6, we show the recording-level CMI values and the correlations
for CER, WER, MER, WIL, transliterating to Arabic followed by Alif/Ya
normalization (Tr+), PSD ($w_{s}=4$), and $Average(Base,MT>*)$ for the
semantic measure (Sem). We observe that for the conventional metrics, it is
mostly the case that CER outperforms WER/MER (performing on-par), which
outperform WIL. We confirm that transliterating to Arabic followed by Alif/Ya
normalization is the best-performing metric, outperforming CER for 6/7
recordings. In the case where CER achieved highest correlation, it is only
slightly better than transliteration. By looking into the standard deviation
of the metrics’ correlation scores, we observe that CER has the highest value
(0.11), compared to transliteration (0.07), phonological (0.08) and semantic
(0.07) measures. This reflects that CER’s performance is less consistent than
other metrics. In future work, we plan to understand the relation between the
correlation scores and CS behaviour as well as other variables.
#### 4.2.3 System-level Evaluation
Results for the system-level evaluation are presented in Table 7. For
semantic-based metrics, the overall score is calculated as the average of
sentence-level scores. As indicated by the $GoldCER$, the ranking of the
systems is: Conformer-A, Conformer-F, and HMM-DNN. We show that, apart from
CER applied directly on the original sentences, all the evaluation metrics
provide the same ranking conclusion. The relative scores of the systems are
not equivalently reflected in all metrics though, which is worth further
investigations. In the future, we plan to include more systems, in order to be
able to derive correlations between systems’ overall scores.
## 5 Conclusion and Future Work
In this work, we (i) develop a corpus of human judgment – with minimal edits
of different ASR output; and (ii) benchmark the performance of different
evaluation metrics and their ability to correctly evaluate CS ASR outputs in
correlation to the ground truth human post-editing effort. We cover commonly-
used evaluation metrics, in addition to three approaches aiming at handling CS
challenges: transliteration, phonetic similarity, and semantic similarity. Our
results show that WER and CER are not adequate for evaluating CS languages
having cross-transcription and spelling variation. The highest correlation to
the post-editing effort is achieved by transliteration followed by phonetic
similarity, semantic similarity, CER, and WER, in order. In future, we plan to
evaluate the proposed methods for the MUCS2021 [30] challenge to ensure
generalization across more languages. Furthermore, we plan to create the human
acceptability corpus for language pairs sharing the same writing script.
## 6 ACKNOWLEDGMENTS
The work presented here was carried out during the 2022 Jelinek Memorial
Summer Workshop on Speech and Language Technologies at Johns Hopkins
University, which was supported with funding from Amazon, Microsoft and
Google. We also thank the anonymous reviewers for their helpful feedback.
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|
# Variational Quantum Gate Optimization at the Pulse Level
Sean Greenaway Physics Department, Blackett Laboratory, Imperial College
London, Prince Consort Road, SW7 2BW, United Kingdom Francesco Petiziol
Technische Universität Berlin, Institut für Theoretische Physik,
Hardenbergstraße 36, Berlin 10623, Germany Hongzheng Zhao Max Planck
Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187
Dresden, Germany Florian Mintert Physics Department, Blackett Laboratory,
Imperial College London, Prince Consort Road, SW7 2BW, United Kingdom
Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden,
Germany
###### Abstract
We experimentally investigate the viability of a variational quantum gate
optimization protocol informed by the underlying physical Hamiltonian of
fixed-frequency transmon qubits. Through the successful experimental
optimization of two and three qubit quantum gates the utility of the scheme
for obtaining gates based on static effective Hamiltonians is demonstrated.
The limits of such a strategy are investigated through the optimization of a
time-dependent, Floquet-engineered gate, however parameter drift is identified
as a key limiting factor preventing the implementation of such a scheme which
the variational optimization protocol is unable to overcome.
††preprint: APS/123-QED
## I Introduction
Rapid experimental progress in the development of quantum computers has led to
the realization of quantum platforms which are approaching the scales
necessary for true quantum advantage [1, 2, 3, 4, 5]. However, the inherent
noise levels of noisy intermediate-scale quantum (NISQ) devices remains a
fundamental limiting factor precluding the attainment of results that cannot
be obtained classically [6]. As a result, a large body of research has grown
around extending the utility of NISQ devices through error mitigation [7, 8,
9, 10] or circuit compilation [11, 12, 13, 14] algorithms.
These techniques are typically rooted in the gate-based approach to quantum
computation [15, 16], in which algorithms are decomposed into a finite set of
fundamental basis gates, usually a two-qubit entangling gate (e.g. cnot) and
arbitrary single qubit rotations. This paradigm is ideal for fault-tolerant
quantum computation since it is universal. However, the deep circuits
necessitated by gate-based algorithms means that implementation of gate-based
computations on NISQ devices are severely impeded by gate noise.
One method by which this limitation could potentially be overcome is through
the use of variational quantum gate optimization (VQGO) [17, 18]. VQGO seeks
to obtain the optimal gate parameters that maximize the fidelity of a target
gate through a classical optimization routine. While such a routine can be
used to increase the fidelity of standard basis gates, it can also be used to
optimize non-standard gates such as two-qubit rotation gates and gates that
act on more than two qubits, which would otherwise necessitate decomposition
into noisy cnot and single qubit rotation gates. In this way, VQGO can obtain
more efficient gate implementations, increasing the range of computations
which can be implemented on NISQ devices.
In order to implement a VQGO routine, a parameterized quantum gate is
required. A natural choice for this is to use the native operations for a
given device as the control parameters, choosing target gates that can be
realized using those operations. In this work, fixed-frequency, fixed-
interaction (FF) transmon qubits [19] as implemented by IBM Quantum [20] are
used as the basis for the VQGO routine. FF transmons are highly controllable,
with individually controllable $ZX$ and $ZY$ interactions and arbitrary single
qubit rotations natively available. However, the interaction terms are
accompanied by significant noise terms which, alongside experimental
imperfections, can severely reduce the fidelity of implemented gates. Thus,
the platform is ideally suited to optimization via VQGO. We assess the extent
to which VQGO can overcome these limitations and thereby realize high fidelity
gates on currently available hardware.
The native entangling operation in FF transmon qubits is the cross-resonance
gate [21, 22, 23], obtained by driving one qubit at the resonant frequency of
another to which it is coupled. This results in an entangling $\exp(i\theta
ZX)$ operation, with the Rabi angle $\theta$ controlled by the pulse amplitude
and duration. In addition to the desired coupling term, unwanted spurious
single qubit terms are also generated which must be controlled for high
fidelity gates to be realized.
In principle, quantum optimal control schemes [24] based on theoretical models
of this interaction can be designed to eliminate these unwanted terms.
However, while sophisticated models of the interaction have been developed
[25, 26], they cannot be used to make a priori predictions about an
experimental system given the susceptibility of experimental system parameters
to drift and the limited access to these parameters afforded to end users. As
a black-box optimization routine, VQGO can be used to obtain high-fidelity
gates without rigorously characterizing the underlying system, making it well-
suited to this application.
The choice of target gates for the VQGO routine explored in this work is
motivated by the form of the cross-resonance interaction, which we briefly
review in Sec. II. The optimization of a two-qubit $ZX$ gate is presented in
Sec. III, corresponding to the cross-resonance interaction with the error
terms eliminated, before an extension to a three qubit gate consisting of two
simultaneous cross-resonance interactions is made in Sec. IV. In both cases
the VQGO routine is very effective, resulting in the experimental realization
of high fidelity gates.
Having demonstrated the utility of VQGO for obtaining high fidelity gates
based on time-independent Hamiltonians, we try to generalize the approach to
the more challenging application of implementing time-dependent, Floquet-
engineered systems [27, 28]. A scheme for realizing a three-body $ZYZ$ gate
[29, 30, 31] at stroboscopic times is used as the testbed for such an
application, with the VQGO results presented in Sec. V. The VQGO protocol was
able to improve the fidelity of the realized gate when compared with
unoptimized gate parameters. However, significant parameter drift over the
time frame of the optimization poses a severe limitation to this method,
preventing the protocol from reaching similarly high fidelities to the other
gates.
Our results show that VQGO is effective at obtaining optimal drive routines
for novel quantum gates based on static effective Hamiltonians outside the
usual set of basis gates. We identify parameter drift as the primary limiting
factor preventing VQGO from obtaining similarly high fidelity gates based on
time-dependent Hamiltonians. In principle, this means that control schemes
that are designed to be robust to parameter drift could be amenable to
optimization through VQGO. The use of VQGO could allow for more efficient
compilation of quantum circuits than is possible using current gate
decomposition approaches, thereby increasing the utility of NISQ devices.
## II Experimental System
While the general techniques discussed in this work are applicable to any
quantum system, the specific choices of optimization targets and figures of
merit are motivated by the experimental system used to perform the
optimizations. Here the experimental platform consists of fixed-frequency,
fixed-interaction (FF) transmon qubits capacitively coupled together. This is
the experimental platform used by IBM in their IBM Quantum systems [20].
These systems may be modelled as a series of $n$ anharmonic Duffing
oscillators [32]
$\displaystyle H^{\text{Duff}}$
$\displaystyle=\sum_{i=1}^{n}(\omega_{i}a_{i}^{\dagger}a_{i}+\alpha_{i}a_{i}^{\dagger}a_{i}^{\dagger}a_{i}a_{i}$
$\displaystyle+\sum_{\langle
i,j\rangle}J_{ij}(a_{i}-a_{i}^{\dagger})(a_{j}-a_{j}^{\dagger})\ ,$ (1)
with anharmonicities $\alpha_{i}$ and resonant frequencies $\omega_{i}$. The
capacitive coupling strength $J_{ij}$ between nearest-neighbour transmons
(represented by the angled brackets) is fixed by the hardware and cannot be
externally controlled. As a result, $J_{ij}$ must be sufficiently weak such
that, in the absence of driving fields, no entanglement between coupled qubits
is generated.
In such a system, dynamics may be induced by driving the system with microwave
pulses. If these pulses have amplitudes which are significantly lower than the
transmon anharmonicities, then only the lowest two energy levels will be
populated and the dynamics can be accurately described by a simplified qubit
model. Restricting Eq. (II) to a qubit model yields
$H(t)=\sum_{i=1}^{n}\frac{\omega_{i}}{2}Z_{i}+\sum_{\langle
i,j\rangle}J_{ij}Y_{i}Y_{j}+\sum_{i}D_{i}(t)X_{i}\ ,$ (2)
where the final sum is over the subset of qubits upon which the pulses are
applied and where the driving $D_{i}(t)$ may be conveniently parameterized in
terms of dimensionless pulse envelope parameters $d^{X}_{i}(t),d^{Y}_{i}(t)$
and $d^{Z}_{i}(t)$ as
$\displaystyle D_{i}(t)=$
$\displaystyle\operatorname{Re}\Bigg{[}\frac{\Omega}{2}\Bigg{(}(d^{X}_{i}(t)-id^{Y}_{i}(t))$
$\displaystyle\times\operatorname{exp}\left(-2i\int_{0}^{t}d_{i}^{Z}(t^{\prime})dt^{\prime}\right)\Bigg{)}e^{i(\omega_{i}+\Delta_{i})t}\Bigg{]}\
,$ (3)
with $\Delta_{i}$ the detuning from the resonant qubit frequency. On resonance
driving generates single qubit dynamics. In the frame rotating with the qubit
frequencies defined by a unitary transformation using the rotation operator
$\exp(i\sum_{i}\frac{\omega_{i}}{2}Z_{i})$, the effective Hamiltonian for such
a driving (applied to the $i$th qubit) is
$\tilde{H}_{i}(t)=\frac{\Omega}{2}(d_{i}^{X}(t)X_{i}+d^{Y}_{i}(t)Y_{i}+d^{Z}_{i}(t)Z_{i})\
.$ (4)
In this way, full single qubit quantum control of individual qubits is
possible. Entangling operations between pairs of coupled qubits are also able
to be generated using off-resonant drive pulses, making transmon qubits highly
expressible as a system for quantum computation and simulation. This
interaction is outlined in the following section.
### II.1 The Cross-Resonance Gate
An entangling interaction between two coupled qubits can be generated by
driving one qubit at the resonant frequency of the other, resulting in a
cross-resonance interaction [21, 22, 23]. In the frame rotating with the qubit
frequencies, the effective Hamiltonian resulting from such a drive is given by
$H^{\text{CR}}_{ij}=\sum_{A\in\\{\mathds{1},X,Y,Z\\}}c_{\mathds{1}A}\mathds{1}_{i}A_{j}+c_{ZA}Z_{i}A_{j}\
,$ (5)
where the $i$th qubit (the control) is driven at the resonant frequency of the
$j$th (the target). The terms in Eq. (5) have different magnitudes due to the
fact that the parameters in the drive Hamiltonian Eq. (2) are of significantly
different magnitudes: in particular, $J_{ij}\ll\Omega\ll\Delta_{ij}$ with
$\Delta_{ij}=\omega_{i}-\omega_{j}$. The largest term, the single qubit
$Z\mathds{1}$ rotation on the drive qubit, is proportional to
$\Omega_{i}^{2}/\Delta_{ij}$ and arises due to a strong AC-Stark shift from
the off-resonant drive. Next largest in magnitude are the two qubit $ZX$ and
$ZY$ entangling operations and the single qubit $\mathds{1}X$ and
$\mathds{1}Y$ rotations on the target qubit, all of which are proportional to
$J_{ij}\Omega_{i}/\Delta_{ij}$. These terms arise from the interplay between
the non-commuting drive and static coupling terms in Eq. (2). Finally, the
single qubit $\mathds{1}Z$ rotation on the target qubit and the $ZZ$
interaction originate from the weak static coupling and are proportional to
$J_{ij}^{2}/\Delta_{ij}$.
For most purposes, an ideal starting point for experiments using the cross-
resonance interaction is a pure $ZX$ interaction. In such a scheme, the other
terms can be considered error terms unless stated otherwise. The weakness of
the $ZZ$ and $\mathds{1}Z$ terms arising from the qubit-qubit coupling means
that these terms can be neglected, however the rest of the terms must be
eliminated experimentally. These remaining error terms can be
straightforwardly corrected by adjusting the cross-resonant drive envelope
phase and applying additional single qubit control terms, both of which are
able to be accurately controlled experimentally. All that therefore remains is
to determine the magnitude of these corrections.
As mentioned above, our approach treats the FF transmon system as a black-box
and attempts to find optimal parameters through VQGO rather than rigorously
characterizing the experimental system to fit the theoretical models [25, 26].
This approach assumes that the experimental errors are dominated by the
unitary errors arising from miscalibrated drive parameters. Since only unitary
control terms are available, incoherent errors such as decoherence and
dephasing cannot be directly corrected by the VQGO routine and thus will
necessarily reduce the fidelity of applied gates. The two most significant
sources of incoherent errors are decoherence and measurement error. A key
advantage of FF transmon qubits is their long coherence times which, at
approximately $100\ \mu$s are much longer than the gate times investigated
here, with the longest gate times being less than $5\ \mu$s. Decoherence over
these short times is therefore minimal and can be ignored. Measurement error
is known to be a significant problem for FF transmon qubits [33]. Since all
the figures of merit for the optimizations performed here average over a
number of different expectation value measurements, the effect of measurement
error is well approximated by unbiased stochastic error. In this case, the
optimal parameters for a given gate should remain unchanged by the presence of
measurement error, and thus VQGO should still be effective. As a consequence
of neglecting measurement error, the obtained fidelities will be lower than
expected based on previously published fidelity measurements [34]. For
example, the identity gate under this assumption has an experimental fidelity
of approximately $95\%$.
## III Optimizing the Cross-Resonance Gate
Having access to a high fidelity entangling operation is highly important for
quantum computing platforms. As such, the natural starting point for a VQGO
protocol implemented on a FF transmon device is the optimization of a pure
$\exp(i\theta ZX)$ gate. Such a protocol is both inherently useful, since
applying a pure $ZX$ gate for a rotation angle of $\pi/4$ yields a maximally
entangling gate which is equivalent to a cnot up to single qubit rotations,
and highly useful as the starting point for analogue and hybrid quantum
computations [35]. Applying an analogue $ZX$ pulse for varying durations can
result in improved fidelity in quantum simulations when compared with using
cnot decompositions [36].
Control schemes implemented on FF transmon qubits typically involve
implementing an echoed cross-resonance pulse sequence in order to refocus most
of the error terms [37, 22]. However, it is possible to directly control the
Hamiltonian terms instead. This cuts down on the number of pulses which need
to be applied and additionally allows for the simultaneous application of
additional control pulses, potentially expanding the utility of the cross-
resonance interaction into the fields of quantum optimal control and analogue
quantum simulation [38, 39]. The strategy of controlling individual
Hamiltonian terms, rather than using an echoed pulse sequence, is employed in
this work.
Aside from the choice of target gate, two additional choices must be made for
the implementation of VQGO: a figure of merit quantifying the quality of the
experimental gate and a classical optimization routine. Different figures of
merit are best suited to different gates, and so the choices of figure of
merit will be discussed with respect to the different target gates in the
subsequent sections. For the classical optimization routine, Bayesian
optimization (BO) [40], a probabilistic machine learning method is utilized.
BO is well suited to applications in which evaluation of the figure of merit
incurs a significant overhead due to it requiring, for example, an experiment
to be performed and has been previously implemented successfully in various
quantum optimal control applications [41, 42, 43, 44, 45, 46, 18]. A thorough
overview of BO can be found in Refs. [47, 48, 49].
### III.1 Phase Calibration
Before working with the cross-resonance gate, it is convenient to first
calibrate the phase of the applied cross-resonant pulses such that the
experimental effective Hamiltonian matches the expected theoretical terms. For
a pure $ZX$ gate, the drive envelope in Eq. (II) should be purely real (i.e.
$h^{Y}=0$). However, systematic experimental errors such as delays in the
drive lines can induce phase shifts on the signal generated by the arbitrary
waveguide generator, resulting in a non-zero value of $h^{Y}$. The drive
envelope phase requested by the user therefore must be adjusted to eliminate
the $h^{Y}$ term and ensure the effective Hamiltonian consists only of $ZX$,
$Z\mathds{1}$ and $\mathds{1}X$ terms.
This phase can be optimized by applying the cross-resonance drive to the
$\ket{++}$ initial state and measuring in the $X$ eigenbasis on both qubits.
The optimal phase is then found by minimizing the sum of projections into the
$\ket{+-}$ and $\ket{--}$ states, which can be straightforwardly achieved
using the individual projective measurements for each qubit.
(a)
(b)
Figure 1: Quantum process matrices extracted from the application of a cross-
resonance interaction implemented on the ibmq_paris quantum device. (a) shows
results from applying a drive in which the phase of the drive envelope in Eq.
(II) requested by the user is 0, and shows significant phase error due to
drive line nonlinearities. (b) shows the same interaction with an additional
phase added to the drive envelope optimized to eliminate this error.
In order to verify that the phase optimization routine successfully eliminates
the unwanted terms, an unbiased validation method is required. A natural
choice for this is provided by quantum process tomography. Quantum process
tomography characterizes a quantum process $\Lambda$, which may be written in
terms of its action on an arbitrary input state $\rho$ as
$\Lambda(\rho)=\sum_{i,j=1}^{d^{2}}\chi_{ij}\sigma_{i}\rho\sigma_{j}\ ,$ (6)
where $\\{\sigma_{i}\\}$ is the operator basis formed form the $n-$fold tensor
products of Pauli matrices. Quantum process tomography is used to
experimentally extract the process matrix $\chi_{ij}$ which fully
characterizes $\Lambda$ [50].
Fig. 1 shows the experimental process matrices for a two qubit cross resonance
interaction before and after phase optimization. Prior to phase optimization
(Fig. 1a), the dynamics are dominated by terms generated by unwanted $ZY$ and
$\mathds{1}Y$ terms due to the phase misalignment, however by adjusting the
phase of the pulse, these can be virtually eliminated, with the final process
matrix almost entirely consisting of terms generated by $ZX$, $Z\mathds{1}$
and $\mathds{1}X$ (Fig. 1b).
The remainder of this work will exclusively use drive pulses which the drive
envelope in Eq. (II) is either purely real ($h^{Y}=0$) or purely imaginary
($h^{X}=0$) once this phase misalignment is accounted for. In principle,
however, once the calibration phase is known, any two-body interaction
consisting of a coherent mixture of $ZX$ and $ZY$ terms can be generated,
allowing for a wide array of interaction terms to be implemented, which is
another advantage of applying VQGO to FF transmon qubits.
### III.2 Reduced Process Tomography
With the phase error from the applied drive eliminated, the effective
Hamiltonian arising from the unoptimized cross-resonance interaction consists
only of terms $ZX$, $Z\mathds{1}$ and $\mathds{1}X$ terms. Since the dynamics
generated by such a limited set of Hamiltonian terms is only a small subset of
the full Hilbert space, it is possible to describe the action of the gate to a
high degree of accuracy using a significantly reduced set of measurements, in
a protocol known as reduced process tomography [51, 52, 53].
Quantum process tomography provides an effective verification that an
optimization has succeeded, since it is unbiased and captures the full action
of a particular gate. However, performing full process tomography is extremely
experimentally expensive, making it impractical for the iterative evaluation
of gate fidelity required for VQGO. For certain gates, however, most of the
elements of $\chi_{ij}$ are known to be vanishing a priori, meaning that only
a reduced subset of measurements is required to evaluate it. This is the
underlying idea behind reduced process tomography.
Explicitly, for a general two qubit cross resonance interaction of the form
$U_{ZX}(t)=\exp(-i(J_{ZX}ZX+J_{Z\mathds{1}}Z\mathds{1}+J_{\mathds{1}X}\mathds{1}X)t)\
,$ (7)
the dynamics can be entirely captured by performing only a single two-qubit
state tomography experiment. The quantum gate defined through the application
of Eq. (7) consists only of a weighted sum of operators
$\\{\mathds{1}\mathds{1},Z\mathds{1},\mathds{1}X,ZX\\}$. Each of these
operators maps the initial state $\ket{+0}$ to one of a set of mutually
orthogonal final states as
$\displaystyle\mathds{1}\mathds{1}\ket{+0}$ $\displaystyle=\ket{+0}$ (8)
$\displaystyle Z\mathds{1}\ket{+0}$ $\displaystyle=\ket{-0}$ (9)
$\displaystyle\mathds{1}X\ket{+0}$ $\displaystyle=\ket{+1}$ (10)
$\displaystyle ZX\ket{+0}$ $\displaystyle=\ket{-1}\ .$ (11)
Thus an approximation to the full process matrix can be obtained by performing
quantum state tomography on the resulting output state and obtaining the
process matrix as the matrix elements of the output density matrix. Having
extracted the reduced process matrix $\chi^{\text{red}}$, an approximation to
the quantum process fidelity can be made through the overlap between
$\chi^{\text{red}}$ and the ideal process matrix $\chi^{ZX}$
$F\approx\operatorname{Tr}[\chi^{\text{red}}(\chi^{ZX})^{\dagger}]\ .$ (12)
This approximation will be referred to as the reduced $\chi$ overlap.
Figure 2: Plot of reduced $\chi$ overlap against process fidelity for 100
experimental gates implemented with varying cross-resonance and single qubit
correction pulse amplitudes. Both figures of merit were extracted from the
same set of experimental data, with the process fidelity calculated using the
full set of 240 expectation values required for full quantum process
tomography and the reduced $\chi$ overlap using a subset of the data. The
reduced $\chi$ overlap approximates the full process fidelity very well, and
should therefore be an efficient alternative to it, requiring only 12
expectation value measurements rather than 240.
The quality of the reduced $\chi$ overlap as an approximation to the process
fidelity depends on how closely Eq. (7) captures the experimental dynamics. In
order to verify this, the reduced $\chi$ overlap and full quantum process
fidelity can be evaluated using the same experimental data, extracting the
reduced process matrix as a subset of the full tomographic expectation values.
Fig. 2 shows the experimental results of such a procedure for 100 gates
generated by varying the cross-resonance amplitude and the amplitudes of
compensating $Z\mathds{1}$ and $\mathds{1}X$ pulses. Since the range of
fidelities are obtained by varying the same parameters as are used in an
optimization protocol, the extent to which the reduced $\chi$ overlap
approximates the process fidelity is a strong indication of its viability as a
figure of merit for VQGO.
In the ideal case, the data should lie entirely along the diagonal of Fig. 2.
All of the data are indeed very close to this diagonal, which is shown as a
black line. Moreover, the deviations are consistent with the level of
measurement error in the IBM Quantum devices. Given the significant reduction
in overhead from using the reduced $\chi$ overlap over evaluations of the full
process fidelity, a reduction from 240 expectation value measurements per
evaluation to just 12, the quality of the reduced $\chi$ overlap as an
approximation makes it an appropriate choice as the figure of merit for an
optimization.
### III.3 Obtaining an Optimal Cross-Resonance Interaction
Figure 3: Convergence plot for the optimization of the cross-resonance gate,
showing the overlap between the ideal process matrix and the reduced process
matrix as a function of the Bayesian optimization iteration. The optimization
explores a range of parameter values, finding some high fidelity points before
converging to a peak value of 0.93, demonstrating the success of the
optimization. Figure 4: Plot showing the requested drive pulse amplitudes used
in the optimization of the single application of the cross-resonance drive
outlined in Sec. III.3, with the yellow plots corresponding to cross-resonance
pulses and the blue plots corresponding to the single qubit resonant pulses
needed to counteract the spurious single qubit $\mathds{1}X$ term from the
cross-resonance interaction and where the light and dark portions of the plots
correspond to the real and imaginary components of the pulses respectively.
This set of pulses corresponds to the highest fidelity observed in the
optimization. The phase of the cross-resonance drives is non-zero due to the
phase calibration outlined in Sec. III.1. The $Z\mathds{1}$ term may be
corrected using a virtual $Z$ rotation [54] and so is not represented in this
plot of physical pulses.
Having obtained a figure of merit which can efficiently evaluate the quality
of an experimental cross-resonance gate, a VQGO routine can be implemented
using BO as the classical optimizer and using the amplitudes of the cross-
resonance single qubit correction pulses as control parameters. The target
gate for this optimization is a pure $ZX$ interaction at the maximally
entangling Rabi angle of $\theta=\pi/4$. The control pulses over which the
optimization was performed were based on Eq. (II) and were parametrized as
follows:
$\displaystyle D_{12}(d^{ZX}_{1},d^{X}_{2},d^{Z}_{1};t)$
$\displaystyle=\operatorname{Re}\Bigg{[}\frac{\Omega}{2}\left(d^{ZX}_{1}e^{i\phi_{ZX}}e^{i\omega_{2}t}+d^{X}_{2}e^{i\omega_{2}t}\right)$
$\displaystyle+\underbrace{\operatorname{exp}\left(-2i\int_{0}^{t}d_{i}^{Z}(t^{\prime})dt^{\prime}\right)}_{\text{Virtual
$Z$ rotation}}\Bigg{]}\ ,$ (13)
with the control parameters being the amplitudes of the cross-resonance and
single qubit $\mathds{1}X$ pulses ($d_{1}^{ZX}$ and $d_{2}^{X}$ respectively)
and the magnitude of the single qubit $Z$ rotation on the target qubit, which
was indirectly implemented as a virtual $Z$ gate [54] by updating the qubit’s
resonant frequency in software. The pulses were also multiplied by a Gaussian
ramp up/ramp down to avoid discontinuous pulses, however the parameters for
these ramps were kept consistent throughout the optimization. Fig. 4 shows a
plot of the amplitudes $d^{ZX}_{1}$ and $d^{X}_{2}$ as a function of the pulse
envelope for the optimal set of pulse parameters obtained using VQGO, with the
top (yellow) plot showing the $d^{ZX}_{1}$ amplitude multiplied by the phase
to compensate for phase error (with the lighter shade the real part and the
darker shade the imaginary part and the bottom (blue) plot showing the
$d^{X}_{2}$ amplitude.
Fig. 3 shows the reduced process matrix overlap as a function of the BO
iterations for such an optimization. In the first $150$ iterations of the
optimization, the optimizer explores the parameter space, thus there are many
evaluations which are of poor fidelity. In the latter stage of the
optimization, however, the optimizer has enough information to converge to a
point at which the reduced process matrix overlap is maximized, with all
evaluated points being above overlaps of $90\%$.
While the quality of the channels realized by the pulse scheme can vary
significantly as shown by Fig. 3, the actual pulses used to implement them
look very similar, differing only by the magnitude of the applied pulse
amplitudes. Due to the non-trivial transformations of the signal that occur in
the physical experiments, meaning that the qubits do not experience the ideal
pulse as requested by the user, it would be extremely difficult to tell a
priori what the optimal pulse parameters should be to realize a high fidelity
gate. By using VQGO, this difficulty can be side-stepped, allowing for high
fidelity pulse schemes to be obtained without necessitating a rigorous
characterization of this transformation.
(a) Experimental process matrix
(b) Experimental process matrix (a) with largest elements dropped and rescaled
to show small elements
Figure 5: (a) Quantum process matrix extracted from the application of a
cross-resonance interaction optimized for the gate $\exp(i\pi/4ZX)$ using
Bayesian optimization and implemented on the ibmq_paris quantum device. The
ideal process matrix consists of the four terms which dominate the optimized
process matrix, each with magnitude $1/2$. (b) The same optimized process
matrix as (a), but with the four largest elements (corresponding to $\chi$
elements indexed by products of $\mathds{1}\mathds{1}$ and $ZX$) set to 0 and
the color bar rescaled to show the magnitude of the less significant terms.
The optimized process matrix has some small residual unwanted terms and the
magnitudes are slightly suppressed, but nevertheless yields a fidelity of
approximately $0.93$ even without any measurement error mitigation.
The quality of the final parameters found by the optimization routine is shown
through the full process matrix evaluated using process tomography in Fig.
5(a), with Fig. 5(b) showing the same process matrix with the largest elements
set to 0 and with the color bar rescaled to show the magnitude of the small
residual error terms. The target Rabi angle for this optimization was $\pi/4$,
corresponding to a maximally entangling gate with four equal-magnitude process
matrix elements,
$\chi_{\mathds{1}\mathds{1},\mathds{1}\mathds{1}}=\chi_{ZX,ZX}$ and
$\chi_{\mathds{1}\mathds{1},ZX}=-\chi_{ZX,\mathds{1}\mathds{1}}$, which is
realized to very high fidelity ($93\%$) in the final process matrix. As
mentioned above, while this is significantly lower than the reported cnot
error rates [20], much of this reduction can be attributed to measurement
error. In order to fairly compare this result to the state-of-the art method
in the presence of this measurement error, the process matrix for the IBM-
calibrated $ZX$ gate was experimentally extracted. This was obtained by taking
the cnot gate pulse sequence and stripping out the single qubit rotation gates
used to convert the native $ZX$ gate to a cnot. The resulting pulse sequence
yields a process fidelity of $93\%$, matching the one obtained in this work,
but achieves so at the price of using multiple pulses and longer total pulsing
time.
The fact that the final $ZX$ gate optimized through VQGO matches the fidelity
achieved by IBM indicates that the optimization routine performed as well as
it could have done – i.e. the obtained fidelity is as high as can be achieved
using the control scheme presented in this work. Additionally, the fidelity of
the IBM-calibrated cnot gate was also evaluated, which yielded a fidelity of
92.8%. This implies that the optimized $ZX$ gate could also be used to
generate a cnot gate with a comparable fidelity. However, such an application
is not the most useful application of the optimization procedure. The key
advantage of using VQGO over the IBM definition lies in its flexibility – it
can be used to implement gates which cannot be natively implemented using the
standard IBM pulse definitions and ‘hardware’ interactions. This is
illustrated in the following section by application of the protocol to a more
complicated three-qubit gate.
## IV Optimizing Non-Commuting Interactions
Having demonstrated the utility of pulse-level VQGO on the cross-resonance
gate, a natural extension is made to a three qubit quantum gate. For the $ZX$
optimization, all of the terms which generate Eq. (7) mutually commute,
meaning the control landscape can be factorized into a product of individual
control terms for the $ZX$ and single qubit $Z\mathds{1}$ and $\mathds{1}X$
pulse amplitudes, greatly simplifying the optimization. Additionally, the
favorable structure of the process matrix that permits the definition of the
reduced $\chi$ figure of merit is not generic for all gates that can be
implemented based on the cross-resonance gate. As such, it is pertinent to
evaluate the viability of pulse-level VQGO when the target gate lacks the
convenient features of the $ZX$ gate.
A natural choice for a three qubit target gate in such a setting is the
unitary evolution operator generated by a Hamiltonian of the form
$H=J\left(ZX\mathds{1}+\mathds{1}YZ\right)\ ,$ (14)
which may be implemented using only constant pulses based on the native cross-
resonance operations, with a purely real ($h^{Y}=0$ in Eq. (II)) cross-
resonance pulse on the first qubit and a purely imaginary ($h^{X}=0$ in Eq.
(II)) pulse on the third qubit. Although the implementation of gates based on
Eq. (14) appears similar to the $ZX$ gate optimized in the previous section,
the error terms generated by the two unoptimized cross-resonance drives do not
mutually commute, significantly complicating the optimization landscape.
Additionally, since this is a three qubit gate, the number of parameters over
which an optimization may be performed is doubled, providing an additional
complication.
Rather than optimizing the full gate starting from a completely unoptimized
set of pulse parameters, the constituent $ZX\mathds{1}$ and $\mathds{1}YZ$
interactions can first be optimized to find the optimal cross-resonance drive
amplitudes and correction rotations for counteracting the AC Stark shift
effects. In principle, this method should also yield the optimal single qubit
$\mathds{1}X\mathds{1}$ and $\mathds{1}Y\mathds{1}$ pulse amplitudes, however
the values obtained for the two-body interaction are not necessarily optimal
for the three qubit $ZX\mathds{1}+\mathds{1}YZ$ gate. The control parameters
used in the optimization of this gate were the amplitudes of the two cross-
resonance pulses and of the single qubit resonance pulse, the magnitudes of
the virtual $Z$ rotations [54] on the drive qubits for the cross-resonance
pulses and the phase of the single qubit resonance pulse.
For each two-body interaction, all of the error terms mutually commute. Thus,
the different terms that generate Eq. (7) can be factorized. Similarly, for
the $\mathds{1}YZ$ term an analogous factorization of the terms which generate
the unitary
$U_{YZ}(t)=\exp(-i\left(J_{YZ}YZ+J_{\mathds{1}Z}\mathds{1}Z+J_{Y\mathds{1}}Y\mathds{1}\right)t)\
,$ (15)
may be made. For each two-body interaction, there are therefore an infinite
number of solutions for each of the parameters of the form
$\theta_{\text{opt}}+m2\pi$, where $\theta_{\text{opt}}$ is the parameter
which exactly realizes the desired operation with no over or underrotation. As
the cross-resonance interaction is weak, a significant change in the drive
amplitude is required to enact a moderate change in the effective $ZX$
strength. It is therefore possible to constrain the optimization domain such
that the applied drive amplitudes only span a single Rabi oscillation.
For the single qubit correction terms, this is not straightforward to achieve.
This is due to the fact that the smallest possible amplitude may still yield a
non-zero effective $\mathds{1}X$ term. Additionally, since the resonant fields
are much stronger than the cross-resonance interaction, the required
compensating drives need to be applied at very low amplitude. At such low
amplitudes, nonlinearities in the resonance drive lines can result in unwanted
phase errors, meaning that the exact cancellation amplitude may not be
optimal.
A solution to this is to optimize the correction pulse on the central qubit
separately once the two-body interactions have been optimized – that is, the
magnitudes of the pulses which realize the $ZX\mathds{1}$ and $\mathds{1}YZ$
interactions, as well as the compensating $Z\mathds{1}\mathds{1}$ and
$\mathds{1}\mathds{1}Z$ correction rotations are individually optimized before
the optimal single qubit $\mathds{1}X\mathds{1}+\mathds{1}Y\mathds{1}$ pulse
amplitude is optimized. Only the amplitude which exactly cancels the single
qubit terms will yield maximum fidelity, thus there will only be one optimal
solution. By optimizing both the amplitude and the phase of the applied drive
pulse, the phase errors can be simultaneously corrected.
While this requires an increase in overhead, the protocol can be scaled to
large system size by optimizing blocks comprising a small number of qubits
separately and making use of parallelization to simultaneously optimize
interactions which are physically distant enough that cross-talk is unlikely.
This would necessitate an increase in quantum resources by only a constant
factor dependent on the target problem and the geometry of the experimental
system and a linear increase in classical computational resources. The latter
could also, in principle, be reduced through information-sharing protocols
[44].
### IV.1 Zero-fidelity Estimation
Unlike the two-qubit $ZX$ gate, the chosen three-qubit gate does not permit a
reduced process matrix which can be efficiently evaluated. A more general
strategy for obtaining estimates of the fidelity of a quantum gate is to use
fidelity estimation through importance sampling [55]. In this work, zero-
fidelity estimation [18] is used as a faithful approximation to the full
process fidelity as it is well suited to implementations on NISQ hardware.
The zero-fidelity between a unitary target gate $U$ and a noisy experimental
gate $\Gamma$ may be written
$F_{0}(U,\Gamma)=\frac{1}{d^{2}}\sum_{i,j=1}^{d^{2}}\operatorname{Tr}[U\rho_{i}U^{\dagger}W_{j}]\operatorname{Tr}[\Gamma(\rho_{i})W_{j}]\
,$ (16)
where the input states $\\{\rho_{i}\\}$ are formed as the tensor product of
single qubit symmetric informationally complete (SIC) states [56] and the
operators $\\{W_{j}\\}$ form an orthonormal basis.
The zero-fidelity can be efficiently approximated by sampling $l$ input
state/measurement basis pairs from the joint probability distribution
$\operatorname{Pr}(i,j)=\frac{1}{d}\operatorname{Tr}[U\rho_{i}U^{\dagger}W_{j}]\
,$ (17)
and for each experimental setting evaluating the estimator
$X(i,j)=\frac{\operatorname{Tr}[\Gamma(\rho_{i})W_{j}]}{\operatorname{Tr}[U\rho_{i}U^{\dagger}W_{j}]}\
,$ (18)
for which the expected value is the zero-fidelity. The variance of this
estimator is independent of the system size (although the individual
expectation values still need an exponentially increasing number of projective
measurements to be resolved) and converges to 0 as the zero-fidelity
approaches unity; additionally, as the zero-fidelity increases, the difference
between it and the process fidelity decreases. This makes the zero-fidelity
well suited to optimization problems.
### IV.2 Optimization Results
(a)
(b)
(c)
Figure 6: Experimental results for the optimization of an
$\exp(i\pi/4(ZX\mathds{1}+\mathds{1}YZ))$ gate: (a) shows the ideal process
matrix for the gate (showing only elements that can be generated by Eq. (5))
and (b) shows the experimental process matrix for the optimal drive parameters
obtained through BO, implemented on the ibm_oslo quantum device. (c) shows the
same experimental results as (b) but with the 9 largest terms (corresponding
to $\chi$ elements indexed by products of
$\mathds{1}\mathds{1}\mathds{1},\mathds{1}YZ$ and $ZX\mathds{1}$) set to 0
(indicated by green crosses) and the color bar rescaled to show the magnitude
of the less significant terms. Only terms that can be generated by the noisy
cross-resonance and single qubit drive pulses are shown for clarity; the full
process matrices including the dropped terms are shown in Appendix B. The
qualitative features of the process matrix are accurately obtained, with the
amplitude of the observed terms slightly reduced from the ideal matrix.
Additionally, the Rabi angle is slightly misaligned, leading to a final
process fidelity of $82\%$.
The final results for the optimization of the
$\exp(i\pi/4(ZX\mathds{1}+\mathds{1}YZ))$ gate are shown in Fig. 6, with 6(a)
showing the ideal target process matrix and 6(b) the experimental gate
following the two part optimization protocol described in the previous
section. While the optimization protocol was performed using zero-fidelity
optimization with 200 expectation value measurements per estimation, the final
result shown was obtained through full process tomography implemented on the
ibm_oslo quantum device. Fig. 6(c) shows the same data with the 9 largest
process matrix terms (corresponding to $\chi$ elements indexed by products of
$\mathds{1}\mathds{1}\mathds{1},\mathds{1}YZ$ and $ZX\mathds{1}$) set to 0
(indicated by green crosses), with the rest of the matrix elements rescaled to
allow the magnitude of the other process matrix terms to be seen. These
elements have magnitudes of at most $0.04$, showing that the dynamics are
dominated by the 9 terms seen clearly in Fig. 6(b). Moreover, the magnitude of
these terms is consistent with the level of measurement error in the device.
Only terms which are able to be generated from the application of the two
unoptimized cross-resonance Hamiltonians Eq.(5) are shown. Appendix B shows
the full process matrices for this experiment, in full form and with the 9
largest elements dropped. The full process matrix shows that all process
matrix elements which are dropped from Fig. 6 have magnitudes smaller than the
elements shown.
The final achieved process fidelity for the process matrix in Fig. 6 was
$0.82$, which is consistent with the achieved fidelities of the constituent
$ZX\mathds{1}$ and $\mathds{1}YZ$ gates, both of which were approximately
$90\%$. As above, these process fidelity values were obtained without state
preparation and measurement error mitigation, and thus are underestimates of
the quality of a gate as used in a quantum algorithm. With this in mind, the
obtained fidelity is close to the optimal fidelity that can be achieved in the
presence of measurement error – the fidelity of the three-qubit identity gate
obtained using the same experimental protocol is $88\%$.
The obtained fidelity of $0.82\%$ for this gate could represent a significant
improvement in utility for NISQ devices, since the gate-based implementation
requires substantially more pulses per two-body interaction and since the
overall three qubit gate must be composed from the two-body interactions
through Trotterization [57], further increasing the gate overhead.
## V Towards an Engineered Three-Body Gate
The previous sections demonstrate that VQGO can be used to obtain high
fidelity two and three qubit gates. This could allow for the realization of
more efficient hybrid quantum computations implemented on FF transmon devices,
with computations composed into a wider set of basis gates, each of which may
be obtained through VQGO. A natural question arises as to how far the protocol
can be pushed: can the optimization be used to obtain a high-fidelity Floquet-
engineered interaction for example [58, 59]?
Floquet engineering uses periodic driving fields to realize a time-dependent,
periodic Hamiltonian $H(t+T)=H(t)$. Using Floquet theory [27, 60, 61], the
propagator for this system can be expressed as $U(t)=U_{F}(t)\exp(-iH_{F}t)$
in terms of a time-independent effective Hamiltonian $H_{F}$ and a periodic
micromotion operator $U_{F}(t+T)=U_{F}(t)$. This is achieved by expressing
$H(t)$ in the rotating frame defined by the micromotion operator,
$H_{F}=U_{F}^{\dagger}(t)H(t)U_{F}(t)-i\dot{U}_{F}^{\dagger}(t)U_{F}(t)\ .$
(19)
At integer multiples of the driving period $t=nT$, the micromotion operator
reduces to the identity and the dynamics of the system are entirely captured
by the time-independent effective Hamiltonian as $U(nT)=\exp(-iH_{F}nT)$. By
making use of Floquet engineering and using $U(nT)$ as a target gate, the
range of gates which may be implemented on a given device can be expanded.
In this section, the results of the implementation of a Floquet-engineered
three-body $\exp(i\theta ZYZ)$ gate based on an existing theoretical drive
scheme [28, 29, 30, 31] are presented. The extension to full time-dependent
quantum control implies a significant increase in difficulty due to the
increase in parameters and due to the precision in control parameters required
to realize Floquet-engineered dynamics. As a result, the experimental
implementation of this protocol represents an evaluation of the limitations of
the VQGO routine as implemented on the IBM Quantum devices.
### V.1 Theoretical Protocol
In the theoretical driving protocol, a three-body interaction is predicted to
appear in the presence of stable pairwise interactions as a second-order
process arising from driving the central qubit in a chain of three coupled
qubits. Concretely, the protocol assumes a static drift Hamiltonian
$H_{0}=J_{ZX}\left(ZX\mathds{1}+\mathds{1}XZ\right)\ ,$ (20)
which may be modulated by a single qubit drive pulse of the form
$\frac{\Omega(t)}{2}\mathds{1}Y\mathds{1}$. Optimal parameters $\Omega_{k}$
can be numerically found such that a modulation
$\Omega(t)=\sum_{k=0}^{2}\Omega_{k}\cos(k\omega t)\ ,$ (21)
produces the desired interaction at multiples of the fundamental Floquet
driving period $T=2\pi/\omega$.
Optimal parameters for this scheme were obtained in Ref. [31] based on an
idealized Hamiltonian and are presented in Table 1 in Appendix A. To verify
that these parameters remain optimal when moving from always-on, static
interactions to a transmon system in which the interactions are dynamically
switched on, numerical simulation of the protocol is performed. For most of
the experimental parameters, realistic values based on experimental devices
are chosen as outlined in Appendix A. For the two-body coupling term $J_{ZX}$,
a compromise had to be made between obtaining a gate as quickly as possible
and avoiding transitions to unwanted energy levels. The choice of
$J_{ZX}/2\pi=0.2\ $MHz was found to be the optimal choice, which in turn fixes
$\omega/2\pi=1\ $MHz, taking the $\omega=5J_{ZX}$ case in Ref. [31], and $T=1\
\mu$s. This results in a three-body strength of $J_{ZYZ}/2\pi=0.04\ $MHz,
which yields an almost maximally entangling $ZYZ$ gate after three Floquet
periods ($6\pi/25\approx\pi/4$).
Figure 7: Evolution of the population of states $\ket{+++}$ (blue) and
$\ket{---}$ (red) evaluated numerically using a full transmon model,
demonstrating the Floquet-engineered $ZYZ$ interaction. Symbols (empty
crosses) indicate the stroboscopic evolution in steps of $T=1\mu$s, while
solid shaded lines show the micromotion. At the third Floquet period
($t=3T=3\mu$s, indicated by the dashed black line), the three-qubit
interaction realizes a beamsplitter operation between the two states,
producing a three-qubit entangled state.
The resulting characteristic dynamics for an initial state $\ket{+++}$ is
shown, as an example, in Fig. 7. The three-qubit interaction $ZYZ$
successfully induces Rabi oscillations between states $\ket{+++}$ and
$\ket{---}$ at stroboscopic times, realizing to a good approximation a
maximally entangling gate at time $t=3T$, indicated by the black dashed line
in Fig. 7. These simulations provide strong evidence that the three-qubit
interaction should be observable in the experiment.
### V.2 Optimization Results
In order to minimize the overhead of the optimization protocol, it is useful
once again to preoptimize the individual two-body $ZX\mathds{1}$ and
$\mathds{1}XZ$ interactions such that they are high-fidelity and have
interaction strengths as close to one another as possible. Once these terms
are optimized, the weights of the single qubit driving parameters
$\\{\Omega_{k}\\}$ and the amplitude of the compensating
$\mathds{1}X\mathds{1}$ pulse may then be used as the optimization parameters.
As with the previous two qubit gate, no convenient reduced process matrix can
be generated from the drive protocol, and so zero-fidelity estimation was used
as the figure of merit. As motivated in the previous section, the chosen Rabi
angle for this interaction was $6\pi/25$, which is close to a maximally
entangling gate whilst conforming to the requirement that the simulation time
be an integer multiple of the Floquet period.
Figure 8: Convergence plot for the Bayesian optimization of a three-body
$\exp(-6\pi/25ZYZ)$ interaction showing the estimated zero-fidelity as a
function of Bayesian optimization iteration. Although the optimization is able
to make modest improvements to the zero-fidelity, converging to an estimated
zero-fidelity of approximately $0.4$ (here the fluctuations are largely due to
the non-zero variance of the zero-fidelity estimation), the achieved
fidelities are significantly lower than can be achieved for the gates based on
static interactions.
As shown in Fig. 8, as the optimization progresses the observed estimated
zero-fidelities increase and fewer low fidelity results are observed,
indicating that the optimizer is adapting to the parameter landscape. The
spread of the data points even after approximately $80$ iterations can largely
be ascribed to the non-zero variance of the zero-fidelity estimation. Despite
these modest improvements, the achieved fidelity is significantly lower than
is observed for the previous gates, with the optimizer converging to an
estimated zero-fidelity of approximately $0.4$.
(a)
(b)
(c)
Figure 9: (a) ideal process matrix for the $\exp(-i\theta ZYZ)$ target unitary
gate. (b) process matrix associated with the pulse parameters that yielded the
highest estimated zero-fidelity, evaluated using full process tomography
implemented on the ibmq_jakarta quantum device. (c) process matrix submitted
to the experimental queue immediately following (b), finishing after
approximately $8$ hours, with the same pulse parameters. Only process matrix
elements generated by (5) are shown, with the full process matrices given in
Appendix B. Not only are both optimized process matrices extremely different
from the ideal case, the last two are significantly different from each other,
showing that drift in the machine is a substantial problem. It is likely that
this is the reason for the VQGO is not able to reach similarly high fidelities
to the previous gates.
In order to perform the optimizations, experimental jobs must be submitted to
a queueing system to be implemented on the physical hardware. This can
increase the necessary time for an optimization significantly. For the three-
body gate, the optimization time was approximately $12$ hours. Over this
duration, the parameters characterizing the device can drift. For the static
gates optimized in the previous section, this is unproblematic, since small
drifts induce only small changes to the effective Hamiltonian, resulting in,
for example, an over-rotation error. These errors can be effectively handled
by the optimizer and so high fidelity results are still achievable. However,
for the Floquet-engineered system a modest drift can induce significant
changes in the effective Hamiltonian since the scheme requires precise
cancellation of the nested commutators of the drive Hamiltonian terms. This
can be intuitively observed in Fig. 7: very small shifts away from the
stroboscopic drive time induce large deviations from the desired $ZYZ$
dynamics.
To investigate whether parameter drift is consistent with the experiment as an
explanation for the difference between the static and Floquet VQGO schemes,
quantum process tomography can be used. By repeating the process tomography
twice with the exact same pulse setup, the effects of parameter drift can be
observed. Fig. 9 shows the results of such a pair of experiments, with Fig. 9a
showing the ideal process matrix and where the experiments that generated Fig.
9c were completed approximately 8 hours following the completion of the
experiment presented in Fig. 9b. The drive parameters that were chosen
correspond to the optimal parameters obtained from the VQGO routine. In all
cases, only matrix elements which are generated by the Hamiltonian Eq.(5) are
shown, with the other terms being significantly lower in magnitude. The full
experimental process matrices are given in Appendix B.
Neither experimental process matrix reproduces the desired dynamics, even
qualitatively, which is to be expected from the low achieved fidelity during
the optimization. The deviation from Fig. 9b to Fig. 9c is much more
significant: the quantum gates generated by the two pulse schemes are
completely different, despite the fact that they were implemented with
identical pulse parameters with less than a day between the experiments. This
indicates that parameter drift is indeed a significant problem for obtaining
high fidelity Floquet-engineered gates using VQGO.
Even with the limitations imposed by the noise levels of current NISQ devices,
VQGO works well for optimizing gates based on static Hamiltonians. However,
for time dependent Hamiltonians the requirements for obtaining high fidelity
gates are much higher, being beyond the reach of current devices. It may be
possible to design control routines that are more robust to parameter drift
such that these requirements are relaxed enough that VQGO can effectively
realize high fidelity gates.
## VI Conclusions
Given the high noise rates characteristic of NISQ devices, finding more
efficient methods of implementing quantum algorithms could be a potentially
valuable route towards obtaining useful experimental results in the near term.
Variational quantum gate optimization (VQGO) is a protocol that can be used to
obtain high fidelity gates in the presence of experimental noise and could
therefore be used to expand the utility of NISQ devices.
In this work, we propose a VQGO protocol which uses the native operations of a
given quantum device to obtain high fidelity gates efficiently. We
experimentally evaluate the protocol through an implementation on fixed-
frequency, fixed-interaction transmon qubits.
VQGO is shown to be highly effective at obtaining high-fidelity quantum gates
based on static effective Hamiltonians. For a two-qubit maximally entangling
$ZX$ target gate, VQGO is able to obtain pulse parameters which yield a
process fidelity of $93\%$ and for a three qubit gate a fidelity of $83\%$ is
achieved. These are very promising results, since the two qubit gate fidelity
matches the IBM-optimized gates used for implementing cnot gates, using fewer
pulses and shorter total pulsing time, and since both sets of fidelities are
very close to that of the identity gate ($95\%$ and $88\%$ for the two and
three qubit experimental identity gates respectively), which indicates the
upper bound on achievable fidelity without measurement error mitigation. As
part of the optimization protocol, we derive a reduced process matrix for the
two-qubit gate which may be experimentally evaluated using only $12$
measurements and apply zero-fidelity estimation as the figure of merit for the
three qubit gate.
We assess the limitations of the scheme through an extension to the
optimization of a Floquet-engineered, time-dependent gate. While the VQGO
protocol is able to increase the fidelity of the implemented gate, the
increased requirements of the time-dependent scheme combined with significant
parameter drift over the duration of the experiment prevent the protocol from
reaching similarly high fidelities to the gates based on static Hamiltonians.
It is possible that driving schemes which are robust to this drift could be
engineered. However, currently VQGO on FF tranmon qubits is only effective for
target gates based on time-independent Hamiltonians.
## VII Outlook
In this work, VQGO is shown to be capable of obtaining high fidelity gates
based on static effective Hamiltonians. A direct application of VQGO is in
quantum simulation. For many systems of interest, it is possible to obtain
mappings to the native operations of a given device that are more efficient in
terms of hardware resources than a decomposition into cnot and single qubit
rotation gates. An example of this is the transverse field Ising model, which
can be mapped exactly to the native operations of FF transmon devices. By
using VQGO to optimize blocks of Ising-like gates, the number of Trotter steps
required to reach a given evolution time could be considerably reduced,
expanding the reach of current devices for quantum simulation. Optimal
decompositions into optimizable gates for a given system is therefore a
valuable route for future work.
The target gates and figures of merit investigated in this work are specific
to FF transmon qubits, but the general framework of VQGO is applicable to any
system. It would thus be instructive to investigate the viability of VQGO on
other NISQ systems. Zero-fidelity estimation can be applied to any quantum
platform (and may be more efficient for certain platforms such as NMR quantum
computers [18]) but the existence of reduced process matrices for systems
other than FF transmon qubits warrants further investigation.
One of the advantages of using black-box optimization protocols to obtain
optimal experimental parameters is that unknown errors in the device can be
accounted for without a rigorous characterization of the physical device.
Nevertheless, it could be valuable to complement the techniques outlined in
this work with numerical and experimental characterization techniques in order
to investigate the robustness of different pulse schemes. This could be used
to inform which classes of parametrized pulses have the most potential for use
in a VQGO scheme, expanding the utility of the protocol. The VQGO procedure
proposed here may then also be adapted to optimize for the protocol robustness
in response to variations of the optimal pulse parameters, besides the gate
fidelity, by minimizing a suitably modified cost function.
An additional route for future work lies in addressing the difficulties
associated with applying VQGO to the optimization of gates based on time-
dependent Hamiltonians on FF transmon devices. It would be interesting to
investigate whether more robust driving schemes that are stable with respect
to moderate drifts in control parameters may be derived. It is possible that
with an appropriate choice of driving routine, the utility of VQGO could be
expanded to this regime. Having this limitation in mind when designing control
schemes could lead to creative solutions which have not yet been considered –
for instance, a control scheme could be developed that approximately realizes
a given target gate over a range of parameters, as opposed to schemes which
exactly realize a target gate but only for a precise configuration of control
parameters.
## VIII Acknowledgments
We are grateful to Adam Smith and Daniel Malz for providing stimulating
discussions and to Marin Bukov for helpful comments on the manuscript. This
work is supported by Samsung GRP grant, the UK Hub in Quantum Computing and
Simulation, part of the UK National Quantum Technologies Programme with
funding from UKRI EPSRC grant EP/T001062/1 and the QuantERA ERA-NET Co-fund in
Quantum Technologies implemented within the European Union’s Horizon 2020
Programme. S.G. is supported by a studentship in the Quantum Systems
Engineering Skills and Training Hub at Imperial College London funded by EPSRC
(EP/P510257/1). F.P. acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG) via the Research Unit FOR 2414 under Project No.
277974659. We acknowledge the use of IBM Quantum services for this work. The
views expressed are those of the authors, and do not reflect the official
policy or position of IBM or the IBM Quantum team.
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## Appendix A Simulations of the device dynamics
In this appendix, we discuss how numerical simulations of the device have been
performed, and the related parameters used.
The three qubits are described by the Hamiltonian
$H(t)=\sum_{j=1}^{3}H_{Q_{j}}+H_{\mathrm{int}}+\sum_{j=1}^{3}H_{\mathrm{drive},j}(t).$
(22)
The transmon Hamiltonians $H_{Q_{j}}$ read [25, 62]
${H}_{Q_{j}}=\frac{\omega_{h,j}}{4}\left[\hat{y}^{2}_{j}-\frac{2}{\epsilon_{j}}\cos(\sqrt{\epsilon_{j}}\hat{x}_{j})\right],$
(23)
where $\hat{y}_{j}=-i(\hat{b}_{j}-\hat{b}_{j}^{\dagger})$ and
$\hat{x}_{j}=\hat{b}_{j}+\hat{b}_{j}^{\dagger}$. The bosonic operators
$\hat{b}_{j}$ describe (in unitless form) zero-point-fluctuations of transmon
flux and charge. The harmonic frequencies $\omega_{h,j}$ and the
anharmonicities parameters $\epsilon_{j}$ used are chosen to give transmon
level splittings in the range of typical IBM qubits. In particular, they are
fine tuned to give first-excitation frequencies 5.236 GHz, 5.014 GHz and 5.178
GHz and anharmonicities $-0.340$ GHz, $-0.342$ GHz and $-0.341$ GHz for qubits
$Q_{1}$, $Q_{2}$, and $Q_{3}$, respectively. The transmon-transmon interaction
Hamiltonian and the drive Hamiltonians read
$\displaystyle H_{\mathrm{int}}=\sum_{j=1,3}J_{j}\hat{y}_{j}\hat{y}_{2}\ ,$
(24) $\displaystyle
H_{\mathrm{drive},j}(t)=\Omega_{j}(t)\sin(\omega_{j}t-\phi_{j})\hat{y}_{j}\ .$
(25)
The couplings $J_{j}$ are chosen as $J_{1}/2\pi=1.955$ MHz and
$J_{3}/2\pi=2.052$ MHz. State propagation in simulations is done by including
64 lowest-energy states in the composite Hilbert space, which gives converged
results for the eight-dimensional three-qubit subspace. Choosing phases
$\phi_{j}=(\pi,\pi/2,\pi)$, the Hamiltonian (22) is predicted to yield the
following effective Hamiltonian in the qubit subspace, in a frame rotating at
the qubit frequencies,
$\displaystyle H_{\mathrm{qub}}=$ $\displaystyle
c_{Z\mathds{1}\mathds{1}}Z\mathds{1}\mathds{1}+c_{\mathds{1}\mathds{1}Z}\mathds{1}\mathds{1}Z+c_{ZZ\mathds{1}}ZZ\mathds{1}+c_{\mathds{1}ZZ}\mathds{1}ZZ$
$\displaystyle+c_{\mathds{1}X\mathds{1}}\mathds{1}X\mathds{1}+c_{ZX\mathds{1}}ZX\mathds{1}+c_{\mathds{1}XZ}\mathds{1}XZ$
$\displaystyle+\Omega_{2}(t)\mathds{1}Y\mathds{1}/2\ .$ (26)
The drive $\Omega_{2}(t)$ is chosen of the form (21) and is used to produce
the target three-body interaction, as discussed in Section V, in such a
rotating frame. The Hamiltonian (26) contains additional terms as compared to
the ideal two-body Hamiltonian $H_{0}$ (Eq. (20) of Sec. V), which could
potentially hinder the desired three-qubit effect. The terms $ZZ\mathds{1}$
and $\mathds{1}ZZ$ are weaker than other terms by an order of magnitude. They
thus contribute little to the dynamics and can be neglected. The term
$c_{\mathds{1}X\mathds{1}}\mathds{1}X\mathds{1}$ needs active compensation
instead, while aiming at attaining the same magnitude of $ZX\mathds{1}$ and
$\mathds{1}XZ$. This is done by introducing an additional drive
$-\Omega_{c}\sin(\omega_{2}t-\pi)\hat{y}_{2}$. The compensating amplitude
$\Omega_{c}$ is calibrated empirically by inspecting simulated Rabi
oscillations between states $\ket{0}$ and $\ket{1}$ of the central qubit, when
driving either only $Q_{1}$ or $Q_{3}$. Indeed, when driving $Q_{1}$, these
oscillations should occur at rate
$r_{0}=c_{\mathds{1}X\mathds{1}}+c_{ZX\mathds{1}}-\Omega_{c}/2$ if $Q_{1}$ is
in state $\ket{0}$, and at rate
$r_{1}=c_{\mathds{1}X\mathds{1}}-c_{ZX\mathds{1}}-\Omega_{c}/2$ if $Q_{1}$ is
in state $\ket{1}$. We iteratively search for a value of $\Omega_{c}$ yielding
$|r_{0}|=|r_{1}|$ for both $ZX\mathds{1}$ and $\mathds{1}XZ$, and then adapt
$\Omega_{1}$ and $\Omega_{3}$ until $c_{ZX\mathds{1}}=c_{\mathds{1}XZ}$ is
also obtained. The terms $Z\mathds{1}\mathds{1}$ and $\mathds{1}\mathds{1}Z$
commute with the drive on the central qubit and the target $ZYZ$ interaction,
and thus they do not interfere with the Floquet engineering scheme. Once their
magnitude is determined, they can be included in the rotating frame or
corrected with an initial pre-rotation of the qubits. To determine their
magnitude, we proceed similarly to the case of $\mathds{1}X\mathds{1}$, namely
we study the imbalance in the Rabi oscillations of $\ket{++0}$ and $\ket{+-0}$
for $Z\mathds{1}\mathds{1}$, and $\ket{0++}$ and $\ket{0+-}$ for
$\mathds{1}\mathds{1}Z$. All (rounded) parameter values used as an example for
the transmon Hamiltonian are summarized in Table 1.
While the parameter search for the compensating pulses is already successfully
attained ‘by hand’ in simulations, in the experiment it is done via Bayesian
optimization based on the noisy experimental data, as discussed in the main
text, according to the VQGO algorithm proposed in this work.
Table 1: Simulation parameters for Fig. 7. All dimensionfull quantities are expressed in MHz. $\omega_{h,j}/2\pi$ | $\epsilon_{j}$ | $J_{j}/2\pi$
---|---|---
$[5544,5323,5486]$ | [209, 218, 212] | [1955, 2052]
$\Omega_{1}/2\pi$ | $\Omega_{3}/2\pi$ | $\Omega_{c}/2\pi$
18.24 | 19.76 | 0.466
$\Omega_{2k}/2\pi$ | $\omega/2\pi$ |
$[0.080,2.170,2.491]$ | 1.000 |
## Appendix B Full Three Qubit Process Matrices
For the results presented in Secs.IV and V, the process matrices are more
conveniently expressed in a reduced form in which elements that cannot be
generated from error terms in the cross-resonance and resonant drives are
dropped. As evidence that this is indeed a good approximation, in this
appendix the full three qubit process matrices for these results are given.
Only the experimental gates are shown here: the ideal gates are numerically
generated and so dropped elements are 0 up to floating point error.
Figure 10: Full three qubit process matrix for the experimental data presented
in Fig. 6c. On this scale, no significant matrix elements other than those
presented in Fig. 6c(b) are observed. Figure 11: Full three qubit process
matrix for the experimental data presented in Fig. 6c with the 9 largest
elements set to 0 so that the magnitude of the smaller elements can be
observed. Similarly to Fig. 6c(c), the magnitude of the other elements is
$\lesssim 0.04$, significantly lower than the principle terms in the full
process matrix. Additionally, the terms included in Fig. 6c(c) are much larger
than the dropped terms. Figure 12: Full three qubit process matrix for the
experimental data presented in Fig. 9a(b). Terms included in Fig. 9a(b) are
much larger than the dropped terms. Figure 13: Full three qubit process matrix
for the experimental data presented in Fig. 9a(c). Terms included in Fig.
9a(c) are much larger than the dropped terms.
|
This article may be downloaded for personal use only. Any other use requires
prior permission of the author and AIP Publishing. This article appeared in
Physics of Fluids 35, 081301 (2023) and may be found at
https://doi.org/10.1063/5.0157926.
# A classification and review of cavitation models with an emphasis on
physical aspects of cavitation
Tobias Simonsen Folden Department of Mathematical Sciences, Aalborg
University, A. C. Meyers Vænge 15, 2450 Copenhagen, Denmark Fynn Jerome
Aschmoneit<EMAIL_ADDRESS>Department of Mathematical Sciences, Aalborg
University, A. C. Meyers Vænge 15, 2450 Copenhagen, Denmark
###### Abstract
This review article presents a summary of the main categories of models
developed for modelling cavitation, a multiphase phenomenon in which a fluid
locally experiences phase change due to a drop in ambient pressure. The most
common approaches to modelling cavitation along with the most common
modifications to said approaches due to other effects of cavitating flows are
identified and categorized. The application of said categorization is
demonstrated through an analysis of selected cavitation models. For each of
the models presented, the various assumptions and simplifications made by the
authors of the model are discussed, and applications of the model to
simulating various aspects of cavitating flow are also presented. The result
of the analysis is demonstrated via a visualisation of the categorizations of
the highlighted models. Using the preceding discussion of the various
cavitation models presented, the review concludes with an outlook towards
future improvements in the modelling of cavitation.
Keywords: cavitating flow, homogeneous cavitation models, multiphase flows,
phase transition, volume of fluid
## I Introduction
Cavitation is the phenomenon in which local regions of a fluid experience a
phase transition from liquid to vapor as the ambient pressure drops below the
fluid’s vapor pressureFranc and Michel (2010); Brennen (2014). Cavitating flow
describes a flow regime, where local pressure fluctuations cause the fluid to
cavitate locally. These cavities may form coherent bubbles attached to some
surface (sheet cavities), transient eddies (vortex cavities), or dispersed
bubbles (cloud cavities). Once a cavity is exposed to a higher pressure
environment, it will change phase to the liquid state again. This phase change
may be quite rapid, so that it is usually referred to as bubble implosion.
These implosions create shock waves, which carry enough momentum to damage the
confining material.
Cavitating flows are found in various industrial applications, where they
often pose problems for the respective technology. In hydrodynamic machines
such as pumps, pressure exchangers or turbines, cloud cavitation may cause
material erosion leading to machine failure, noise, vibrations or operation
instabilities, such as head losses in pumps Adamkowski, Henke, and Lewandowski
(2016); Al-Obaidi (2019); Wu _et al._ (2019); Ye _et al._ (2021); Sun, Guo,
and Luo (2020). In ship propellers, sheet cavitation leads to erosion of the
downstream side of the blades and it leads to reduced propulsion or propeller-
hull vortex cavitation Yilmaz _et al._ (2020); Wittekind and Schuster (2016);
Peters, Lantermann, and el Moctar (2018); Zhu and Fang (2012); Melissaris _et
al._ (2022). In hydrofoils, sheet cavitation on the top side lead to a
decrease in lift Saito _et al._ (2007); Watanabe, Yamaoka, and Furukawa
(2014). There are also industrial examples, where cavitating flows serve a
distinct purpose. Acoustic cavitation is the principal mechanism behind
sonochemistry, a method for surface cleaning. The generation of imploding
cavitation bubbles creates high frequency shock waves, which are used for
surface cleaning. This cleaning procedure is applied in various industrial
applications, such as in ultrafiltrationYusof _et al._ (2016), in the food
industryAzam _et al._ (2020), or in industrial-scale heat exchangers Kieser
_et al._ (2011). It is therefore of great interest to understand and control
cavitating flows, in order optimize the applications above to minimize or
exploit cavitation efficiently.
Cavitation is a microscopic effect, acting on much smaller temporal and
spacial scales, compared to representative scales of the surrounding flow.
From a macroscopic perspective, cavitation is affected by various variables:
Naturally, the static and dynamic pressure, as well as the local temperature
govern the overall the cavitating flow. However, cavitating flow is tightly
coupled to turbulent flow, as turbulent eddies cause local pressure
fluctuations O’Hern (1990), and also disperse coherent cavities into cloud
cavitiesBrandner _et al._ (2010); Huang, Zhao, and Wang (2014). As such, it
is natural to consider the effects of turbulence when attempting to model
cavitating flows; however, the task of implementing turbulent effects is
complicated by the fact that the most widely used turbulence models were
originally developed for single-phase flows, and the extension of turbulence
models to multiphase flows is still an active area of research, with many
distinct models such as the mixture $k-\varepsilon$ model by Behzadi et
al.Behzadi, Issa, and Rusche (2004) proposed for modelling turbulence in
multiphase flow. Furthermore, as cavities implode the resultant shock waves
may cause secondary cavities to appear close by, which also implode, thus
creating cascades of implosions van Rijsbergen _et al._ (2012); Dular and
Petkovšek (2015); Melissaris _et al._ (2022). Implosions in the direct
vicinity of a surface don’t create concentric shock waves, but rather produce
a directed pressure pulse towards the surface, posing the primary cause for
material damage Mihatsch, Schmidt, and Adams (2015). The implosion intensity
and the bubble interactions in the cavity cloud are dependant on the phases’
viscosities and their surface tension. Hence, cavitating flow entails the
highly complex interactions between microscopic cavitation and the macroscopic
flow.
Due to the wide range of examples of cavitating flows with highly differing
characteristics such as the geometry of the domain, the thermodynamic
variables of the cavitating fluid, and the structure of the flow, there is
currently neither a universal model for cavitation nor a general framework for
approaching the problem of developing a model. Since cavitation was first
recognized as a distinct phenomenon in fluid dynamics, various authors have
derived models that seek to explain the mechanisms behind the phase transfer
that occurs in cavitating flows by directly attempting to simulate the mass
transfer rates between the liquid phase and the vapor phase, simulating the
formation, growth, motion, and collapse of cavities within the flow, or a
combination of these approaches. Through an investigation of the various
techniques employed by the authors of the models, the various cavitation
models can broadly be categorized according to the approaches used for
simulating cavitation in the given model.
Several authors have previously reviewed various aspects of cavitation
modelling. Utturkar et al.Utturkar _et al._ (2005), Luo et al.Luo, Ji, and
Tsujimoto (2016), and Li et al.Li and Yu (2021) analysed cavitation models for
the specific applications of rocket propulsion, hydraulic machinery, and
organic Rankine cycles, respectively. Niedzwiedzka et al.Niedzwiedzka, Schnerr
Professor Dr.-Ing.habil, and Sobieski (2016) wrote a review on homogeneous
cavitation models, comparing how fundamental empirical parameters governing
the phase change are expressed in various articles. Models for bubble
implosions and their erosive potential on surfaces are reviewed in Wang et
al.Wang, Wu, and Huang (2017).
This review of cavitation modelling supplements the previous reviews mentioned
above, highlighting the major categories of cavitation models and the
methodologies and physical models used to develop them, with the discussion of
the models centered on the governing equations as well as the expressions
constructed for various source terms used in the model. As a product of this
analysis and discussion of cavitation modelling, a categorization of the
various approaches to cavitation modelling is proposed, with models based on
the same approach grouped into one category of models. This categorization is
developed with the intent of providing a tool capable of identifying an
appropriate model for the given effects of cavitation an engineer or a
researcher may wish to account for in their studies, while at the same time
illustrating the complexity of said model by identifying how many distinct
modelling approaches are employed in this model.
The structure of this review is given as follows: Section II introduces the
model categorization, with each level of the categorization presented in its
own subsection. Section III presents an application of said categorization to
a selection of cavitation models proposed by various of authors, using an
analysis of said models as justification for the proposed categorization.
Section IV presents a visual representation of the categorized models in the
form of Venn diagrams as well as observations regarding the state of
cavitation modelling based on the analysis performed in Section III. Lastly,
Section V provides an outlook towards future developments in cavitation
modelling on the basis of both the proposed categorization as well as other
recent research directions within cavitation modelling.
## II Categorization of Cavitation Models
As discussed in previous reviews of the literature, e.g. Li et al. Li and Yu
(2021), the problem of modelling cavitation has been has been treated using a
variety of approaches to simulating cavitation. Due to cavitation being a
phenomenon in multiphase flows, a model for simulating cavitation must provide
both an approximation of the flow governed by the Navier-Stokes equations
$\begin{split}\frac{\partial\rho\mathbf{u}}{\partial
t}+\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u})&=\nabla\cdot(\mu(\nabla\mathbf{u}+(\nabla\mathbf{u})^{T}))-\nabla
p,\\\ \frac{\partial\rho}{\partial
t}+\nabla\cdot(\rho\mathbf{u})&=0,\end{split}$ (1)
but also an appropriate scheme for estimating the formation, growth, motion,
shape, and collapse of the cavities present in the flow using the flow
characteristics. In order to obtain more accurate simulations that better
reflect the behavior of real cavitating flows, the model should be able to
account for effects known to influence the growth rate of cavities or events
involving mutual interaction between distinct cavities. In the remainder of
this section, we introduce the most common approaches and effects employed in
cavitation modelling,
### II.1 Categorization by modelling approach
A common method for modelling the distinct phases in cavitating flows is to
assume that the liquid phase and the vapor phase are in mechanical and thermal
equilibrium with the same velocity and pressure fields, and that the fluid
characteristics such as the density and viscosity are assumed to be specified
locally as a homogeneous mixture of the corresponding characteristics of the
two pure phases with the mixing ratio $\alpha_{v}$, i.e.
$\begin{split}\rho=\rho_{m}&=\alpha_{v}\rho_{v}+(1-\alpha_{v})\rho_{l},\\\
\mu=\mu_{m}&=\alpha_{v}\mu_{v}+(1-\alpha_{v})\mu_{l},\end{split}$ (2)
where $\alpha_{v}$ is the volume fraction field of the vaporous phase, i.e.
the ratio of the fluid volume occupied by vapor and the total fluid volume.
The vapor volume fraction $\alpha_{v}$ can be used to give an approximation of
distribution of cavities within the flow by discretizing the computational
domain into a number of computational cells, then approximating $\alpha_{v}$
locally within each cell. Models based on this approach are known as
homogeneous mixture models, and they combine a scheme for the Navier-Stokes
equations (1) with a scheme that incorporates the hypothesis (2) in some way.
Fig. 1 illustrates the limits of the homogeneous mixture approach: it is
tightly coupled with the discretization of the computational domain. The top
row of figures indicates a cavitating flow, in which a large cavity is
dispersed under the action of turbulence, creating a cavity cloud. In the
middle row, the same flow field is illustrated with the vapor volume fraction
$\alpha_{v}$. It is seen that the cavity is reasonably well resolved in the
left figure, but as the cavity is dispersed, the vapor volume fraction cannot
distinguish between individual bubbles any more. The cavity cloud on the right
is only represented as a near-homogeneous field, cavities become absolutely
indistinguishable. In the bottom row, one possible model for the bubble
density is illustrated, where the numbers per control volume increase, as the
the vapor volume fraction cannot capture the bubbles any more. It therefore
acts as a support field, when the computational grid fails to resolve bubbles.
This macroscopic view on cavitating flow highlights the importance of well-
developed cavitation models on the sub-grid scale.
Figure 1: Cavitating flows undergoing turbulent bubble dispersion. (top)
Illustration of a continuous flow, in which a large cavity is broken up into
smaller cavities, forming a cavitation could. (middle) Volume-of-Fluid
representation of the the bubble break up. The colors indicate the value of
the vapor volume fraction. (bottom) A possible number density field,
supplementing the VoF method as the computational grid resolution does not
resolve bubbles any more.
One approach to modelling cavitation using the homogeneous mixture approach as
described by (2) is to use a modified volume-of-fluid approach, which models
the evolution of the vapor volume fraction $\alpha_{v}$ and the liquid volume
fraction $\alpha_{l}=1-\alpha_{v}$ as described by the transport equations
with source terms
$\begin{split}&\frac{\partial\rho_{i}\alpha_{i}}{\partial
t}+\nabla\cdot(\rho_{i}\alpha_{i}\mathbf{u})=\dot{m}_{i}.\end{split}$ (3)
The models based on this approach are referred to as transport equation
models, or TEMs for short. With no further assumptions other than the
homogeneous mixture assumption, it is sufficient to solve (LABEL:eq:vof) for
the volume fraction of only one phase, typically the vapor phase, as the
liquid volume fraction trivially follows from the volume conservation. Based
on these observations, we have adopted the convention of always considering
the vapor phase whenever we discuss volume fractions and mass transfer rates
throughout this review. The volume fraction source term $\dot{m}_{i}$ in
(LABEL:eq:vof), hereafter denoted by $\dot{m}$, plays a crucial role in TEMs
for cavitation and is the main distinguishing feature of said models, as this
term gives an explicit rate of the phase transition. In order to account for
the different physical aspects governing growth and collapse of cavitation
bubbles, $\dot{m}$ is commonly split into two distinct terms $\dot{m}^{+}$ and
$\dot{m}^{-}$ called the evaporation term and the condensation term,
respectively. The split of the source term is defined such that at any time,
only one of $\dot{m}^{+}$ and $\dot{m}^{-}$ is active in each control volume
based on the relationship between the local pressure $p$ and the vapor
pressure $p_{v}$. When the pressure drops below the vapor pressure, the
evaporation term $\dot{m}^{+}$ is active; correspondingly, the condensation
term $\dot{m}^{-}$ is active whenever the pressure exceeds the vapor pressure.
Mathematically, the split of the source term can be expressed as
$\displaystyle\dot{m}=\begin{cases}\dot{m}^{+},&p<p_{v},\\\
\dot{m}^{-},&p>p_{v}.\end{cases}$ (4)
The main challenge in using the TEM approach is developing an appropriate
expression for the source terms $\dot{m}^{+}$ and $\dot{m}^{-}$, which is done
through an additional hypothesis relating the vapor volume fraction to some
other quantity whose growth can be estimated. The most commonly used approach
to develop an expression for the source term is to assume that within each
computational cell of the domain, the vapor volume fraction is approximately
equal to the amount of volume of the cell occupied by bubbles of vapor that
form from microscopic vapor nuclei which grow and collapse as the local
pressure experienced by these bubbles fall below and exceed the vapor pressure
$p_{v}(T)$. The growth and collapse of these bubbles is modelled by expressing
their radial growth via the Rayleigh-Plesset equationRayleigh (1917); Plesset
(1949), referred to as the RPE. This equation expresses the growth rate of a
bubble with radius $R$ immersed in an infinite domain of liquid as
$R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}\left(\frac{dR}{dt}\right)^{2}=\frac{p_{b}(t)-p_{\infty}(t)}{\rho_{l}}-\frac{4\mu_{l}}{\rho_{l}R}\frac{dR}{dt}-\frac{2\sigma}{\rho_{l}R},$
(5)
which describes the evolution of the radius $R$ of a single spherical bubble
immersed in an infinite expanse of liquid with internal pressure $p_{b}$ under
the influence of a reference pressure $p_{\infty}$, the liquid viscosity
$\mu_{l}$, and the liquid-bubble interface’s surface tension $\sigma$. The
full RPE is rarely used in cavitation modelling; instead, a simplified version
obtained by neglecting e.g. the surface tension or the inertial effects is
used in order to simplify calculations. Assuming that the cavity within a
given control volume is approximately a cluster of spherical bubbles of the
same size with radius $R$, the vapor volume fraction can be approximated via
the bubble radius $R$ as
$\alpha_{v}=n_{0}\frac{4}{3}\pi R^{3};$ (6)
in (6), $n_{0}$ is the bubble number density, which is either a constant
parameter throughout the entire domain or another flow variable governed by
its own transport equation depending on the model. The usage of the RPE as an
approach to modelling cavitation is due to the work of Kubota et al.Kubota,
Kato, and Yamaguchi (1992), who introduced these ideas in their cavitation
model. Notably, Kubota et al. tracked the motion of a cluster of bubbles
entirely without solving a transport equation for the volume fraction.
Another approach to modelling cavitation within the homogeneous mixture
framework is to develop an appropriate equation of state (EOS) that defines
the thermodynamic behavior of cavitating flows and provides an expression that
can be used to simulate the evolution of a quantity that is representative of
the distribution of the two phases, most frequently the density $\rho$. The
exact EOS used in these models highly depends on the nature of the fluid in
question as well as the properties of the surrounding domain; models for
hydrodynamic cavitation in the literature commonly employ a purely polytropic
EOS, whilst models for acoustic cavitation make use of a more general EOS such
as the stiffened gas EOS employed by Denner et al.Denner, Evrard, and van
Wachem (2020) Additionally, there have also been made efforts to include an
EOS in cavitation models primarily rooted in the TEM approach, e.g. the ”four-
equation” model by Goncalvès and Charrière Goncalvès and Charrière (2014). The
main advantage of employing an EOS is sidestepping the numerous hypotheses
associated with the RPE, e.g. cavities being approximately spherical in shape
and immersed in an infinite domain of liquid. Another benefit of introducing
an EOS is a very direct inclusion of thermodynamic effects, enabling the
development of cavitation models for cryogenic fluids.
TEMs have been applied to a wide variety of problems concerned with simulating
cavitating flows in the literature. For the problem of simulating larger
cavity structures such as sheet cavities over e.g. hydrofoil, various authors
have reported results that agree with previous experimental data, see e.g.
Gnanaskandan and Mahesh (2016); Budich, Schmidt, and Adams (2018). However,
other works have indicated that models such as TEMs, with their simplified
models to frequently assume no interactions between bubbles, can fail to
resolve the subgrid cavities when applied to the problem of simulating cloud
cavitiesAsnaghi, Svennberg, and Bensow (2018) or greatly overestimate the
bubble growth rate when the bubble-bubble interactions are accounted forYe and
Li (2016). This has lead to an interest in alternative approaches to modelling
cavitation that remedies these flaws.
One approach has been to attempt to track the interface(s) between the liquid
phase and the vapor phase by locating contours of the form $p_{v}=c$ for some
prescribed constant $c$, leading to the category of interface tracking models.
These methods models are named after their approach to modelling cavitation,
which involves estimating the shape of the cavity interface and its evolution
based on the behaviour of flow characteristics near the interface, e.g.
estimating the shape of the cavity by locating regions in which the local
temperature experiences a sharp decrease as in the model of Deshpande et
al.Deshpande, Feng, and Merkle (1997). This approach for modelling cavitation
works well when modelling sheet cavitation in two-dimensional flows over e.g.
hydrofoils, but faces difficulties associated with properly modelling the
cloud cavitation occurring in the wake region of the cavity. Additionally, the
interface tracking methods have faced difficulties in adapting their approach
to modelling three-dimensional cavitating flows, leading to these methods
being less developed than the previous two methodsLi and Yu (2021).
A more elaborate approach for resolving microscale cavitation effects within
the homogeneous mixture framework is given by the Lagrangian models, where the
liquid phase is treated in the Eulerian framework of conservation equations,
whilst the vapor phase is instead modelled by describing the Newtonian motion
of individual bubbles or parcels of bubbles within the Lagrangian framework.
These models have the capability of implementing the effects that are
neglected in models such as TEMs, e.g the influence of non-condensable gas,
the effect of turbulence at the sub-grid scales, and bubble-bubble
interactions. As a trade-off, the requirement that the dynamics of each
individual bubble be tracked limits the applicability of Lagrangian models for
large-scale industrial problems; some authors have reported success in their
efforts to apply Lagrangian models to such problemsGiannadakis, Gavaises, and
Arcoumanis (2008). For small-scale problems, the Lagrangian models have seen
more widespread usage, with several authors reporting results that agree with
known experimental dataFuster and Colonius (2011); Maeda _et al._ (2015).
Observing the difficulties of properly modelling cavitation at very small
scales using models framed in the Eulerian framework as well as the difficulty
of modelling large-scale cavitating flows with Lagrangian models, several
authors have sought to combine the two approaches into one hybrid model,
leading to the category of multiscale models. Multiscale models are cavitation
models explicitly developed to be capable of simulating the growth and
collapse of both cavities of a size sufficiently large to be visibly detected,
but also the effects of cavities existing on a scale too small to be properly
resolved by conventional methods employed in models such as e.g. TEMs. These
models track all cavities in the flow and classify them according to the local
grid’s capability to properly resolve their dynamics as either macro- or
micro-scale cavities. A multiscale model can essentially be viewed as a
combination of three models:
1. 1.
a model for the growth of the macro-scale cavities, usually a TEM,
2. 2.
a model for the growth and motion of the micro-scale bubbles, tracking each
the growth and motion of each bubble separately,
3. 3.
a scheme for determining whether a macro-scale cavity/micro-scale bubble
should transfer from being treated in its current macro-/micro-scale model to
the other model, based on the computational grid’s capacity to properly
resolve the cavity/bubble in question.
These models track all cavities within the flow, using two separate schemes
for modelling cavitation depending of the size of the cavity: for larger
singular cavities or clouds of bubbles, a model based on the homogeneous
mixture approach is employed, with most multiscale models favoring a TEM for
this purpose. At the same time, the motion and size of bubbles on a given
characteristic length scale formed from nuclei present in the flow or breakup
of larger bubbles is tracked using governing equations formulated in a
Lagrangian framework; this enables simulation of cavitation at the smallest
length scales. The two scales are related to each other through an appropriate
scheme for determining when a given cavity is not resolved properly by the
current scale and transitioning said cavity from its current scale to the
other. In recent years, different multiscale models have been proposed by
various authors, e.g. the models by Hsiao et al.Hsiao, Ma, and Chahine (2017)
and Ghahramani et al. Ghahramani, Arabnejad, and Bensow (2019), both of which
apply distinct approaches in constructing their models. Furthermore, there has
also been extensive effort devoted to studying the various aspects of
multiscale modelling such as the influence of various model parameters and
choices of discretizations on the performance of the model as well as the
capability of the model to properly resolve the micro-scale dynamicsLi _et
al._ (2021a, b, c); Wang, Cheng, and Ji (2021, 2022). Multiscale models show
promise in regards to the problem of constructing a cavitation model that more
accurately simulates the behaviour of cavitating flows, as seen from the
results of Wang et al. Wang _et al._ (2023), who performed a comparative
study in order to investigate the characteristics of both a TEM and a
multiscale model when applied to the case of simulating cavitating flow in a
funnel over a raised flat section. Wang et al. found that the multiscale model
exhibits both better agreement with experimental data for the validation case
compared to the TEM and less sensitivity to the resolution of the
computational mesh.
On the basis of the modelling approaches introduced above, a cavitation model
may be categorized according to the approaches employed within the model is
introduced. The model is categorized as belonging to a combination of the
following five categories:
* •
Category Rayleigh-Plesset Equation (RPE), where the model employs bubble
dynamics as expressed via a (simplified) RPE (5),
* •
Category Transport Equation Model (TEM), where the model employs a transport
equation to approximate the mass transfer rates and thus the growth and
collapse of cavities,
* •
Category Equation Of State (EOS), where the model employs an EOS to relate the
vapor volume fraction to thermodynamic variables,
* •
Category Interface Tracking Model (ITM), where the model employs a scheme that
directly tracks the interface between the liquid and the vapor phase,
* •
Category Multiscale (MUL), where the model employs several schemes for
simulating the growth and collapse of cavities at various length scales.
### II.2 Categorization by model effects
In addition to the categorization of models according to the employed approach
described in section II.1, another point of interest regarding cavitation
modelling is the additional effects of cavitation most frequently accounted
for in the literature. These include adjustments to the expressions used for
simulating e.g. mass transfer rates and hypotheses regarding the influence of
phenomena such as turbulence on the cavitation process.
Almost all models include additional parameters in their expressions used for
e.g. mass transfer rates whose values are not based on any physical
considerations, but instead adjusted empirically in order to provide a better
fit between the model’s predictions of cavitation and experimental data
obtained from observations of a specified system. These parameters should
ideally be chosen to provide the greatest fit under the operating conditions
of the system at hand, e.g. shape of the domain and flow rate; however, the
empirical nature of these constants makes the possibility of applying a set of
empirical parameters that provide a good fit for one system to modelling
another system questionable, and the problem of calibrating these empirical
parameters for optimal performance is still an open problem. Some authors have
proposed various approaches towards calibrating empirical constants, e.g. the
random forest-based workflow proposed by Sikirica et al. Sikirica _et al._
(2020), but as with other aspects of cavitation modelling, no uniform approach
exists as of yet.
As discussed by O’HernO’Hern (1990), Brandner et al.Brandner _et al._ (2010),
and Huang and Wang Huang, Zhao, and Wang (2014) among others, the presence of
turbulence in the system affects the cavitation process through both local
fluctuations of pressure yielding pressure drops of sufficiently high scales
to allow the inception of cavitation and through formed cavities being
bombarded with the eddies carried by the turbulent flow, causing deformations
in the surface of the cavity which may lead to the cavity rupturing and
breaking up into a cloud of smaller cavities. As such, a proper turbulence
model capable of accounting for the multiphase nature of cavitating flow is a
necessity.
In relation to the effects caused by turbulence, the population balance of
cavities in the flow undergoes rapid changes due to both existing cavities
breaking up into cavities of smaller sizes, cavities colliding and remaining
in contact for long enough in order for coalescence to occur, cavities being
formed at impurities on the surface of the domain and then entrained into the
flow, and cavities entering or exiting the flow at the inlet and outlet of the
system, respectively. In order to track these effects, cavitation models can
be extended with either a model for solving the population balance equation
associated with the bubble number density or devise schemes for detecting the
occurrence of events that may lead to changes in the population balance based
on specified criteria, then resolving these events and their impact on the
distribution of cavities.
Another possible effect to include is the possible presence of a third phase
in the cavitating flow, in the form of non-condensable gas. Should this third
phase be present, it has an effect on the approach used for modelling
cavitation. For example, the homogeneous mixture hypothesis (2) should be
reformulated to include the amount of volume occupied by the non-condensable
gas as expressed via its volume fraction $\alpha$, i.e.
$\begin{split}\rho=\rho_{m}&=\alpha_{v}\rho_{v}+\alpha_{ng}\rho_{ng}+(1-\alpha_{v}-\alpha_{ng})\rho_{l},\\\
\mu=\mu_{m}&=\alpha_{v}\mu_{v}+\alpha_{ng}\mu_{ng}+(1-\alpha_{v}-\alpha_{ng})\mu_{l}.\end{split}$
(7)
Similarly, the internal bubble pressure $p_{b}$ used in the RPE (5) also
depends on the pressure of the gas inside the bubble.
Based on these considerations, we extend the categorization presented in
section II.1 with a second categorization according to the effects accounted
for in the construction of the model. This categorization consists of four
categories, defined as follows:
* •
Category Turbulence (TUR), where the model accounts for the effects of
turbulence in their approach to modelling cavitation,
* •
Category Population Balance (POP), where the model accounts for changes in the
cavity population as described by the bubble number distribution,
* •
Category Empirical (EMP), where the model includes empirical constants
adjusted to the characteristics of a given flow,
* •
Category Non-Condensable Gas (NCG), where the model accounts for the presence
of a third phase in the form of a non-condensable gas in the flow.
## III Classification of Selected Models
Having categorized the most common approaches to cavitation modelling as well
as the effects most frequently taken into consideration in the construction of
a model for cavitation, this section is dedicated to an application of said
categorization. A collection of 20 cavitation models from across the
literature are highlighted in this section for analysis and classification.
The highlighted models include the following:
1. 1.
The Bubble Cluster Model by Kubota et al.Kubota, Kato, and Yamaguchi (1992)
2. 2.
The Thermodynamic Variable Table Model by Ventikos and TzabirasVentikos and
Tzabiras (2000)
3. 3.
The Lattice-Boltzmann EOS Model by Banerjee and SarithaBanerjee and Saritha
(2015)
4. 4.
The Polytropic Closure Model by Denner et al. Denner, Evrard, and van Wachem
(2020)
5. 5.
The Ginzburg-Landau Potential Model by Kunz et al.Kunz _et al._ (2000)
6. 6.
The Interface Mass and Normal Momentum Model by Senocak and ShyySenocak and
Shyy (2004a)
7. 7.
The Thermodynamic Interface Model by Deshpande et al.Deshpande, Feng, and
Merkle (1997)
8. 8.
The Wake Closure Model by Liu et al. Liu, Li, and Feng (2006)
9. 9.
The Bubble Density-Liquid Volume Coupling Model by Schnerr and SauerSchnerr
and Sauer (2001)
10. 10.
The Full Cavitation Model by Singhal et al.Singhal _et al._ (2002)
11. 11.
The Vapor Nuclei-Adjusted Model by Zwart et al.Zwart, Gerber, and Belamri
(2004)
12. 12.
The Viscosity-Oriented Model by Konstantinov et al.Konstantinov, Tselischev,
and Tselischev (2014)
13. 13.
The Plane Surface Evaporation Model by Saito et al.Saito, Nakamori, and
Ikohagi (2003)
14. 14.
The Four-Equation Model by Goncalvès and CharrièreGoncalvès and Charrière
(2014)
15. 15.
The Ghost-Fluid Multiscale Model by Hisao et al.Hsiao, Ma, and Chahine (2017)
16. 16.
The Density-Based Convex Combination Model by Huang and Wang Huang and Wang
(2011)
17. 17.
The Heat Balance Model by Shi et al.Shi, Wang, and Hu (2014)
18. 18.
The Population Balance Model by Li and CarricaLi and Carrica (2021)
19. 19.
The Euler-Lagrangian Multiscale Model by Ghahramani et al. Ghahramani, Ström,
and Bensow (2021)
20. 20.
The Stochastic Field Model by Dumond et al. Dumond, Magagnato, and Class
(2013)
All models discussed this section are validated against experimental data of
some sort. Since different models may be validated against the same
experimental data, a list of all experimental studies used for validation is
found in Table 1. The models are presented in order of increasing complexity
of their modelling approach in the sense that models belonging to only a
single category within the categorization presented in section II.1 are
presented first, followed by models belonging to two categories, and so forth.
As a conclusion to the analysis of each highlighted model, a summary of recent
studies applying the model and their findings are also presented. For
completeness, it is also highlighted under which discretization schemes the
model were developed. Note that physical models are ideally not limited to
specific discretization schemes. Some models have become standard methods in
commercial or open-source CFD packages. Following the analysis of all 20
models, their placement within the various categories considered in sections
II.1 and II.2 are demonstrated visually using Venn diagrams in Figures 2 and
3.
id | authors | comment
---|---|---
Hydrofoil
e1 | Izumida et al.Izumida _et al._ (1980) | -
e2 | Kubota et al.Kubota _et al._ (1989) | -
e3 | Avellan et al.Avellan, Dupont, and Ryhming (1988) | -
e7 | Shen & DimotakisShen and Dimotakis (1989) | -
e8 | Hord et al.Hord (1973) | Airfoil
e9 | Keller & ArndtKeller (2001) | -
e14 | Berntsen et al.Berntsen, Kjeldsen, and Arndt (2001) | -
e15 | Wang et al.Wang _et al._ (2001) | -
e17 | FoethFoeth (2008) | -
Venturi nozzle
e6 | Stutz & ReboudStutz and Reboud (1997a, b, 2000) | -
e12 | Barre et al.Barre _et al._ (2009) | -
e13 | Patella et al.Patella, Barre, and Reboud (2006) | -
Axis-symmetric
e5 | Rouse & McNownRouse and McNown (1948) | Solid heads, various forms
e16 | Sarósdy & AcostaSarósdy and Acosta (1961) | Axisymmetric ogive
e18 | Ghahramani et al.Ghahramani _et al._ (2020) | Semi-circular cylinder
other
e4 | Reuter & KaiserReuter and Kaiser (2019) | Bubble near solid wall
e10 | Bakir et al.Bakir _et al._ (2004) | Inducer
e11 | AcekeretAckeret (1930) | Jet element
e19 | Winklhofer et al.Winklhofer _et al._ (2001) | Throttle
Table 1: Table of experimental cases used for model validation.
### III.1 The Bubble Cluster Model
The use of bubble dynamics for modelling cavitation can be traced back to the
Bubble Cluster Cavitation Model by Kubota et al.Kubota, Kato, and Yamaguchi
(1992), which used a simplified variant of the RPE
$R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}\left(\frac{dR}{dt}\right)^{2}=\frac{p_{v}-p}{\rho_{l}},$
(8)
in which the effects of surface tension and viscosity have been neglected.
Additionally, the authors assume that the bubble number density $n_{0}$ is
constant and uniform across the entire domain and that there is no velocity
slip between the two phases, i.e. that the cluster of bubbles are convected
with the same velocity as the fluid they are immersed in. To obtain an
equation of motion the authors simplify the structure of the bubble cluster by
assuming that the relative positions of the bubbles do not change and that all
bubbles share a common radius $R$. This implies that the motion of a bubble
cluster is governed by the total velocity potential of the bubble cluster
given by
$\sum_{j}\frac{1}{r_{j}}\frac{dR_{j}}{dt}R_{j}^{2},$ (9)
where $r_{j}$ denotes the distance between the center of the $j$th bubble and
the center of the cluster. Due to this observation, the authors seek to obtain
a description of the temporal derivative of the velocity potential in order to
obtain a description of the motion of the cluster of bubbles via the
expression
$\frac{d}{dt}\left(\sum_{j}\frac{1}{r_{j}}\frac{dR_{j}}{dt}R_{j}^{2}\right)+R\frac{d^{2}R}{dt^{2}}+\frac{3}{2}\left(\frac{dR}{dt}\right)^{2}=\frac{p_{v}-p}{\rho_{l}}$
(10)
obtained by adding said derivative to the left hand side of (8). The authors
first note that the assumption of a uniform bubble number density $n_{0}$ and
common radius $R$ leads to the local approximation of the vapor volume
fraction $\alpha_{v}$ as the total volume of the bubble cluster, yielding the
relation $\alpha_{v}=n_{0}\frac{4}{3}\pi R^{3}$ previously stated in (6). This
local approximation of the vapor volume fraction is quite noteworthy, as it
forms the cornerstone of all cavitation models of the category RPE. Using
their assumption that the relative positions of the bubbles do not change and
that all bubbles share a common radius $R_{j}=R$, the authors conclude that
$\begin{split}\frac{d}{dt}\sum_{i}\frac{1}{r_{j}}\frac{dR}{dt}R^{2}&=2\pi(\Delta
r)^{2}\left(n_{0}R^{2}\frac{d^{2}R}{dt^{2}}\right.\\\
&\left.+\frac{dn_{0}}{dt}R^{2}\frac{dR}{dt}+2n_{0}R\left(\frac{dR}{dt}\right)^{2}\right),\\\
\nabla\cdot\left(\sum_{i}\frac{1}{r_{j}}\frac{dR}{dt}R^{2}\right)&=0,\end{split}$
(11)
from which (10) may be rewritten as
$\begin{split}\frac{p_{v}-p}{\rho_{l}}&=\left(\frac{3}{2}+4\pi(\Delta
r)^{2}n_{0}R\right)\left(\frac{dR}{dt}\right)^{2}+\frac{d^{2}R}{dt^{2}}\\\
&+(2\pi n_{0}(\Delta r)^{2}R)\frac{d^{2}R}{dt^{2}}+2\pi(\Delta
r)^{2}\frac{dn_{0}}{dt}R^{2}\frac{dR}{dt}.\end{split}$ (12)
To complete the model, the authors note that due to the assumptions of
incompressible flow and no velocity slip between the two phases, the temporal
derivatives in (12) may be replaced with material derivatives
$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{u\cdot\nabla}$, leading to
the local homogeneous model
$\begin{split}(1&+2\pi n_{0}(\Delta r)^{2}R)R\frac{D^{2}R}{Dt^{2}}+2\pi(\Delta
r)^{2}\frac{Dn_{0}}{Dt}R^{2}\frac{DR}{Dt}\\\ &+\left(\frac{3}{2}+4\pi(\Delta
r)^{2}n_{0}R\right)\left(\frac{DR}{Dt}\right)^{2}=\frac{p_{v}-p}{\rho_{l}}\end{split}$
(13)
describing the motion of a cluster of cavitation bubbles.
Several of the other models highlighted in this review, including the Bubble
Density-Liquid Volume Coupling ModelSchnerr and Sauer (2001) and the Vapor
Nuclei-Adjusted ModelSinghal _et al._ (2002) directly reference the idea of
using the RPE as introduced by this model as a starting point for their
approach to modelling cavitation; as such, this model is a significant
milestone in the development in cavitation modelling. Curiously, the exact
approach used by the present model, i.e. constructing a model for the momentum
of a cluster of bubbles, has not been pursued further in the literature,
instead focusing mostly on constructing various TEMs in which the vapor volume
fraction is used as a description of the approximate location of cavities and
their size along with appropriate source terms that dictate the production and
destruction of various cavities. The approach to modelling cavitation used by
the Bubble Cluster Model is very appealing when interpreted physically due to
the large number of real-world cases of cavitating flows in which the
structure of the cavitating region can qualitatively be described as a cloud
of bubbles of approximately equal size, which is exactly the type of structure
considered in the Bubble Cluster Model. However, any application of this
cavitation model must account for its deficiencies, most notably the fact that
the Bubble Cluster Model has no way to account for neither the creation of new
bubbles nor the destruction of existing bubbles, both of which are a major
part of cavitation. Additionally, the model assumes that the bubbles in the
tracked cluster all share a common radius and that their relative positions do
not change. Both of these assumptions fail to hold in a variety of
circumstances; in fact, Ida Ida (2009) has shown that the presence of other,
larger bubbles in the immediate vicinity of a bubble can greatly impact the
growth of the smaller bubble, even leading to the collapse of the smaller
bubble. Kubota et al.Kubota, Kato, and Yamaguchi (1992) implemented their
model using their in-house finite-difference solver SACT-III Kubota, Kato, and
Yamaguchi (1988) and validate their model by applying it to the task of
simulating flow over a hydrofoil using varying conditions, e.g. different
angles of attack in both cavitating and non-cavitating conditions. The results
thus obtained show good agreement with experimental data reported by Izumida
et al. Izumida _et al._ (1980) (e1) and also manage to capture the vortex-
shedding phenomena observed by Kubota et al.Kubota _et al._ (1989); Kubota,
Kato, and Yamaguchi (1989) (e2) in previous experiments and simulations.
Grandjean et al. Grandjean, Jacques, and Zaleski (2012) derived a more general
model for the dynamics of a bubble cluster that reduces to the Bubble Cluster
Momentum model in a special case, then applied this model towards simulating
shock propagation in bubbly flows.
### III.2 The Thermodynamic Variable Table Model
The Thermodynamic Variable Table Model, proposed by Ventikos and
TzabirasVentikos and Tzabiras (2000), is a model for simulating cavitation
using pre-existing information regarding the physical properties of the fluid
given information about the current pressure and enthalpy of the fluid. For
the purposes of demonstration, the authors choose water as their cavitating
fluid due to existence of reliable sources for the necessary information
regarding the physical properties of water. The authors simulate cavitation by
numerically solving both the Navier-Stokes equations and the transport
equation for the stagnation enthalpy $h$ of the mixture at a fixed Reynolds
number $Re=2000$, then using the pressure and the enthalpy to calculate fluid
properties such as the mixture density $\rho_{m}$ and the mixture viscosity
$\mu_{m}$ on a cell-by-cell basis. Within each cell, the fluid properties are
calculated using a table from the U.K. Committee on the Properties of
SteamUnited Kingdom Committee on the Properties of Steam (1970) containing the
values of various properties of a water-vapor mixture at a wide range of
possible values for the pressure and enthalpy. These properties include the
mixture viscosity $\mu_{m}$, absolute temperature $T$, thermal conductivity
$K$, specific heat at constant pressure $c_{p}$, and specific volume $V_{sp}$,
from which the mixture density $\rho_{m}$ can be calculated as
$\rho_{m}=\frac{1}{V_{sp}}$. Because of this, the fluid properties may be
viewed as functions of the pressure and the enthalpy, i.e.
$\displaystyle\rho_{m}$ $\displaystyle=\rho_{m}(p,h)=\frac{1}{V_{sp}(p,h)},$
$\displaystyle\mu_{m}$ $\displaystyle=\mu_{m}(p,h),$ $\displaystyle T$
$\displaystyle=T(p,h),$ $\displaystyle K$ $\displaystyle=K(p,h),$
$\displaystyle c_{p}$ $\displaystyle=c_{p}(p,h).$
hence the model belongs to the category EOS. Ventikos and TzabirasVentikos and
Tzabiras (2000) implemented this method in their in-house model using a finite
volume approach with staggered momentum components. The implemented model was
validated against experimental data of cavitating flow over a hydrofoil
recorded by Avellan et al.Avellan, Dupont, and Ryhming (1988) (e3), showing
moderate agreement with the trends of suction-side pressure coefficients along
the hydrofoil length. It has been reported by Li and YuLi and Yu (2021) that
the model may have difficulties in dealing with three-dimensional problems.
### III.3 The Lattice-Boltzmann EOS Model
Introduced by Banerjee and SarithaBanerjee and Saritha (2015), the Lattice-
Boltzmann EOS Model employs an appropriate EOS along with a lattice-Boltzmann
method for simulating both the flow and the development of cavitation, using
the distribution of the fluid density as a representation of the location and
size of cavities in the fluid. The authors consider two separate choices of an
EOS for use with their model, referred to as the Shan and Chen Equation of
State (SCEOS) and the Peng-Robinson Equation of State (PREOS). The SCEOS is
given byShan and Chen (1993)
$p=\frac{\rho}{3}-4435.2\exp\left(-\frac{400}{\rho}\right),$ (14)
whilst the PREOS is given byGong and Cheng (2012)
$p=\frac{\rho RT}{1-b\rho}-\frac{a\rho\alpha(T)}{1+2b\rho-b^{2}\rho^{2}},$
(15)
where
$\begin{split}\alpha(T)=\Bigg{(}1&+(0.37464+1.54226\omega-0.26992\omega^{2})\\\
&\hskip
8.61108pt\times\left(1-\sqrt{\frac{T}{T_{c}}}\right)\Bigg{)}^{2},\end{split}$
(16)
$\omega$ is the acentric factor of the fluid and the critical temperature
$T_{c}$ and the critical pressure $p_{c}$ of the fluid are related to the
attraction parameter $a$, the repulsion parameter $b$, and the gas ratio $R$
by the formulas
$\displaystyle a$ $\displaystyle=\frac{0.45724R^{2}T_{c}^{2}}{p_{c}},$ (17)
$\displaystyle b$ $\displaystyle=\frac{0.45724RT_{c}}{p_{c}}.$ (18)
The authors choose the value $\omega=0.3443$, corresponding to the acentric
factor of water and assign the values $a=\frac{2}{29},b=\frac{2}{21}$, and
$R=1$.
To validate their model, the authors attempt to simulate the growth of a
single bubble submerged in water by solving for the bubble radius growth rate
$\dot{R}$ in the RPE using the local pressure predicted by both the SCEOS and
the PREOS and comparing the predicted values to the exact values. The
predicted bubble growth rates using the SCEOS agree very well with the exact
values, but the growth rates predicted by the PREOS show large errors that
increase with time. The authors attribute this discrepancy to the influence of
a smaller computational domain in which the effects of walls are more strongly
felt. This model is implemented using the Lattice-Boltzmann method on a D2Q9
grid and employing the exact difference method. The model is validated by
simulating the saturated liquid and vapor densities at different temperatures
using both the PREOS and the SCEOS, obtaining results in agreement with
previous theoretical predictions made by Jain et al.Jain, Tentner, and Rizwan-
uddin (2009) for the PREOS and KuzminKuzmin (2010) for the SCEOS. Saritha and
BanerjeeSaritha and Banerjee (2020) applied this model for further studies of
the dynamics and deformation of bubbles in cavitating flows within a micro-
scale channel.
### III.4 The Polytropic Closure Model
Denner et al. Denner, Evrard, and van Wachem (2020) proposed the Polytropic
Closure Model as a method for simulating acoustic cavitation in polytropic
gas-liquid systems. The authors employ the Navier-Stokes equations along with
the Noble-Abel stiffened gas model, which yields a polytropic EOS given by
$\rho=\frac{K(p+\Pi)^{\Gamma}}{1+bK(p+\Pi)^{\Gamma}},$ (19)
where $\Gamma=1/\kappa$ and $\kappa$ is the polytropic exponent, $b$ is the
co-volume of the non-condensable gas, $\Pi$ is a pressure constant that
accounts for the attraction between molecules, and $K$ is a polytropic
constant defined using a reference pressure $p_{0}$ and a reference density
$\rho_{0}$ as
$K=\frac{\rho_{0}}{(p_{0}+\Pi)^{\Gamma}(1-b\rho_{0})}.$ (20)
Note that in the equations (19) and (20), the effects of surface tension and
gravity have been neglected. In order to validate their model, the authors
apply their model to three test cases which each aim to demonstrate aspects
characteristic of acoustic cavitation, using four different types of fluids
for each test. In their first validation test, the authors simulate the
propagation of acoustic waves given a small perturbation to the flow. First
the authors consider the case of propagation of acoustic waves in single-phase
flow where the predicted wavelength and pressure amplitude of the wave are
compared to their theoretical values, obtaining agreeable results for each of
the four fluids. Next, the propagation of acoustic waves in gas-liquid flows
is investigated by applying the model to predict the amplitude of the pressure
pulses in each phase of the two air-water mixtures considered by the authors
at the fluid interface and comparing to the theoretical values. As before, the
predicted values of the pressure amplitudes greatly agree for both flows
considered.
The second validation test considered by the authors aims to demonstrate the
model’s capacity to predict pressure-driven bubble dynamics properly. To this
end, the authors apply their model to simulate the collapse-expansion cycle of
a single spherical air bubble in water as described by the Gilmore equation
over a range of combinations of spatial and temporal resolutions. The results
thus obtained converge to the theoretical solution given by the Gilmore
modelGilmore (1952) for sufficiently small time steps and mesh spacings.
The final test considered by the authors aims to simulate the wall-bounded
collapse of an air bubble in water situated at various distances from a solid
wall, a process that involves complex pressure-driven interactions. The
results obtained by the authors clearly depicts both the thin liquid film
between the bubble and the wall as well as the high-velocity jet directed
towards the wall during the final steps of the bubble collapse. Additionally,
the authors compare the minimum thickness of the liquid film separating the
bubble and the wall predicted by the model for a range of dimensionless
initial stand-off distances to a set of experimental measurements of the same
values by Reuter and KaiserReuter and Kaiser (2019) (exp4), obtaining
predictions with a high coefficient of determination and thus implying the
model is capable of accurately predicting this process. Denner et al.Denner,
Evrard, and van Wachem (2020) developed their model on a finite volume
approach, using a volume-of-fluid multiphase method. The implementation is not
published. This model was applied by Denner and SchenkeDenner and Schenke
(2023) in their study of acoustic emissions and shock formation of cavitation
bubbles.
### III.5 The Ginzburg-Landau Potential Model
Kunz et al.Kunz _et al._ (2000) introduced the Ginzburg-Landau Potential
Model, in which cavitation is simulated based on expressions for the source
terms in the transport equation of the liquid volume fraction $\alpha_{l}$;
these source terms are reformulated in terms of the vapor volume fraction
$\alpha_{v}$ as mentioned in section II.1. The authors employ the split of the
source term defined in (4), and use two different approaches for deriving
expressions for the two source terms. For the evaporation term $\dot{m}^{+}$,
the authors take the production term from a cavitation model developed by
Merkle et al.Merkle, Feng, and Buelow (1998), who model their source term not
in terms of the bubble radius, but instead use dimensional arguments based on
the dynamics of large-bubble clusters. The source term derived by Merkle et
al. is given by
$\dot{m}^{+}=C_{+}\frac{\rho_{v}(1-\alpha_{v})(p_{v}-p)}{0.5\rho_{l}U_{\infty}^{2}t_{\infty}},\quad
p<p_{v},$ (21)
where $t_{\infty}=d/U_{\infty}$ is the characteristic time scale of fluid
motion and $C_{+}$ is an empirical constant. The authors express their
condensation term $\dot{m}^{-}$ based on simplified potentials obtained from
the Ginzburg-Landau theory of superconductivity, using arguments similar to
that of Kunz et al. Kunz _et al._ (1999). The idea of using these potentials
is due to the work of Hohenberg and HalperinHohenberg and Halperin (1977), who
described applications of this theory to a variety of physical systems,
including the case of two-phase fluids. The condensation term is given by
$\dot{m}^{-}=C_{-}\frac{\rho_{v}(1-\alpha_{v})^{2}\alpha_{v}}{t_{\infty}},\quad
p>p_{v},$ (22)
where $C_{-}$ is another empirical constant. Additionally, the mixture
viscosity is taken to be the turbulent eddy viscosity expressed in terms of
the turbulent kinetic energy $k$ and the turbulent dissipation rate as
$\mu_{m,t}=\frac{C_{\mu}\rho_{m}k^{2}}{\varepsilon}.$
This model is implemented using the UNCLE framework by Taylor et al.Taylor,
Arabshahi, and Whitfield (1995), employing the finite volume method with
$k-\varepsilon$ turbulence modelling. Furthermore, it is implemented as one
optional cavitation model in OpenFOAM. The authors validate their model by
simulating cavitating flow over an ogive body; the predicted values of the
pressure coefficients across a range of cavitation numbers agree with
experimental data reported by Rouse and McNownRouse and McNown (1948) (e5).
This model was recently used by Sikirica et al. Sikirica _et al._ (2020), who
proposed a machine learning framework for calibrating the empirical constants
in cavitation models. Using this model as an example, they applied the
calibrated model towards simulating cavitating flow over a five-bladed
propeller, showing good agreement with the experimental data.
### III.6 The Interface Mass and Normal Momentum Model
Senocak and Shyy’sSenocak and Shyy (2004a) model considers the interface
between the liquid phase and the vapor phase, and aims to construct a model of
the source terms for the liquid volume fraction $\alpha_{l}$ in terms of the
flow characteristics of said interface in the special case of flows at high
Reynolds numbers. Proceeding as in CareyCarey (2007), the authors obtain the
following expression for conservation of mass and normal momentum at the
bubble interface:
$\rho_{l}(u_{l,n}-u_{i,n})=\rho_{v}(u_{v,n}-u_{i,n}),$ (23)
$\displaystyle\begin{aligned}
p_{v}-p_{l}&=\sigma\frac{R_{1}+R_{2}}{R_{1}R_{2}}+2\mu_{v}\frac{\partial
u_{v,n}}{\partial n}-2\mu_{l}\frac{\partial u_{l,n}}{\partial n}\\\ &\hskip
7.10411pt+\rho_{l}(u_{l,n}-u_{i,n})^{2}-\rho_{v}(u_{v,n}-u_{i,n})^{2}.\end{aligned}$
(24)
The authors note here that thermal effects are unaccounted for in the
expressions (23), (24). Furthermore, the authors choose to neglect the effects
of both viscosity and surface tension due to their focus on flows with high
Reynolds numbers, where the effects of viscosity and surface tension on
cavitation are negligible. By considering the mixture density $\rho_{m}$,
expressed here via the liquid volume fraction $\alpha_{l}$ as
$\rho_{m}=\rho_{l}\alpha_{l}+\rho_{v}(1-\alpha_{l})$ (25)
and assuming that the phase change occurs between the vapor and the mixture
phases, the expressions (23), (24) reduce to
$\displaystyle\begin{aligned}
\rho_{m}(u_{m,n}-u_{i,n})&=\rho_{v}(u_{v,n}-u_{i,n}),\end{aligned}$ (26)
$\displaystyle\begin{aligned} p_{v}-p_{l}&=\rho_{m}(u_{m,n}-u_{i,n})^{2}\\\
&\hskip 12.91663pt-\rho_{v}(u_{v,n}-u_{i,n})^{2}.\end{aligned}$ (27)
Rewriting (26) as
$u_{m,n}-u_{i,n}=\rho_{v}\frac{u_{v,n}-u_{i,n}}{\rho_{m}},$ (28)
the reduced normal momentum expression (27) can be restated via (28) as
$p_{v}-p_{l}=\rho_{v}(u_{v,n}-u_{i,n})^{2}\left(\frac{\rho_{v}}{\rho_{m}}-1\right).$
(29)
Recalling the definition of the mixture density (25), (28) may be rewritten in
order to obtain a final expression of the liquid volume fraction $\alpha_{l}$
in terms of the liquid-vapor interface characteristics:
$\alpha_{l}=\frac{p_{l}-p_{v}}{(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})}\left(\frac{\rho_{l}}{\rho_{v}}\alpha_{l}+1-\alpha_{l}\right)$
(30)
In order to obtain expressions for the source terms in the transport equation
of $\alpha_{l}$ from the expression (30), the authors adopt an approach from
turbulence modelling described in Wilcox Wilcox (1994), in which the source
terms are derived from normalizing the expression (30) using a characteristic
time scale $t_{\infty}=L_{\text{ch}}/u_{\infty}$ that is consistent with the
definition of the Reynolds number, i.e. the rate of generation $\dot{S}$ of
$\alpha_{l}$ is expressed as
$\displaystyle\begin{aligned}
\dot{S}=\frac{\alpha_{l}}{t_{\infty}}&=\frac{\rho_{l}(p_{l}-p_{v})\alpha_{l}}{\rho_{v}(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}}\\\
&\hskip
17.22217pt+\frac{(p_{l}-p_{v})(1-\alpha_{l})}{(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}}.\end{aligned}$
(31)
With this in mind, the transport equation of $\alpha_{l}$ reduces to
$\displaystyle\begin{aligned} \frac{\partial\alpha_{l}}{\partial
t}+\nabla\cdot(\alpha_{l}\mathbf{u})&=\frac{\rho_{l}(p_{l}-p_{v})\alpha_{l}}{\rho_{v}(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}}\\\
&\hskip
4.30554pt+\frac{(p_{l}-p_{v})(1-\alpha_{l})}{(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}},\end{aligned}$
(32)
yielding the following expressions for the source terms:
$\displaystyle\dot{m}^{+}$
$\displaystyle=\frac{\rho_{l}(p_{l}-p_{v})\alpha_{l}}{\rho_{v}(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}},\quad$
(33) $\displaystyle\dot{m}^{-}$
$\displaystyle=\frac{(p_{l}-p_{v})(1-\alpha_{l})}{(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}}.$
(34)
The authors note here that as a consequence of choosing a time scale
consistent with the Reynolds number, the source terms derived in this model
express the mass transfer rate of a cluster of bubbles and not the mass
transfer rates of a single bubble. Furthermore, the expressions for the source
terms in (32) do not contain any empirical factors and are instead formulated
in terms of adjustable parameters representing physical factors such as
momentum and pressure, in contrast to most other cavitation models. To couple
the source terms (33) and (34) to the equations governing the flow
characteristics, the source terms are modified as
$\displaystyle\dot{m}^{+}$
$\displaystyle=\frac{\rho_{l}(p_{v}-p)\alpha_{l}}{\rho_{v}(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}},$
$\displaystyle p$ $\displaystyle<p_{v},$ (35) $\displaystyle\dot{m}^{-}$
$\displaystyle=\frac{(p-p_{v})(1-\alpha_{l})}{(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}},$
$\displaystyle p$ $\displaystyle>p_{v}.$ (36)
The authors applied the model to simulate the cavitating flow within two
convergent-divergent nozzles with different geometries; due to the different
geometries of the nozzles, the flow through the first nozzle forms unsteady
cavitation with prominent cloud shedding, while the flow through the second
nozzle forms a stable sheet cavity with minimal shedding. These nozzles were
previously investigated experimentally by Stutz and ReboudStutz and Reboud
(1997b, a, 2000) (exp6), who reported time-averaged velocity and vapour volume
fraction profiles within the cavity of both nozzles and provided qualitative
description of the cavity behaviour previously mentioned. The authors
performed both steady-state as well as time-dependent simulations of the
cavitating flow through the nozzles; the steady-state results were reported in
the first part of the articleSenocak and Shyy (2004a), whilst the time-
dependent calculations were reported in the second part Senocak and Shyy
(2004b). The results obtained using Interface Mass and Normal Momentum Model
to simulate the cavitating flow through both nozzles showed a trend in the
computed time-averaged velocity and vapour volume fraction profiles similar to
the trend in the experimental data reported by Stutz and Reboud. However, the
steady-state results overestimated the vapor content towards the end of the
cavity in comparison with the experimental data; this discrepancy was remedied
in the time-dependent resultsSenocak and Shyy (2004b). This model is a
departure from using the view of cavitation as the growth and collapse of
bubbles as a means of expressing the transport of vapor via the radial growth
of said bubbles, instead focusing directly on the mass exchange occurring
directly at the interface between the two phases. This avoids the problem of
approximating a solution of the equation governing the radial growth of the
bubbles, i.e. the RPE. Furthermore, the choice to use a time scale consistent
with the Reynolds number yielding the mass transfer rates of a cluster of
bubbles yields a comparison of this model to the Bubble Cluster Momentum
ModelKubota, Kato, and Yamaguchi (1992), which expressed the motion of a
cluster of bubbles using the RPE to express the radial growth of each bubble
in the cluster. The model was developed as an in-house tool, using the finite
volume method, as well as the $k-\varepsilon$ turbulence model. Utturkar et
al. Utturkar, Thakur, and Shyy (2005) expanded this model, accounting also for
thermal effects and applying it to simulate cavitation in cryogenic flows.
### III.7 The Thermodynamic Interface Model
The Thermodynamic Interface Model, introduced by Deshpande et al.Deshpande,
Feng, and Merkle (1997), models the shape of the liquid-vapor interface in
sheet cavitating flows with a single, large cavity by approximating the
thermal boundary around the cavity. This is accomplished by coupling the
Navier-Stokes equations with the energy equation and specifying appropriate
boundary conditions at the interface for the quantities of interest, namely
the pressure, the velocity, and the temperature. The authors assume that the
cavity is a region of constant pressure below the vapor pressure, with any
given point on the cavitating surface classified as either a cavitating point
or a solid wall based on the local pressure being above or below the vapor
pressure. Once the cavity points have been determined, the model then
determines the exact location and shape of the cavity from tracing the
streamlines of the flow, from which the thickness of the cavity can be
determined. To ensure a valid profile of the cavity, the authors enforce a
strict positiveness of said thickness in order to prevent the cavity from
moving inside the surface and apply a specific boundary condition for the
energy equation at the liquid-vapor interface given by
$\frac{dT}{dn}=-\frac{\rho_{v}kH_{fg}}{\rho_{l}^{2}C_{p}}\frac{dQ}{ds},$ (37)
where $H_{fg}$ is the heat of vaporization, $k$ is the fluid’s thermal
conductivity, $C_{p}$ is the specific heat of the fluid, and $dQ$ is the
volume flow rate of the vapor added to the cavity in the area $ds$ of the
interface. This condition expresses the thermal depression caused at the
interface, which forms a thermal boundary around the cavity. The authors
implemented their model using as a steady-state method, with $4^{\text{th}}$
order Runge-Kutta pseudo time stepping and a central difference spacial
derivative approximation. Turbulence was modelled using the algebraic Baldwin-
Lomax methodBaldwin and Lomax (1978). The authors validate their model by
simulating the flow of water over a hydrofoil, obtaining predicted contours of
the pressure whose shapes resemble those observed in experiments by Shen and
DimotakisShen and Dimotakis (1989) (e7). Additionally, the authors investigate
the model’s capability to simulate thermodynamics effects of cavitating flow
of both liquid hydrogen and liquid nitrogen over a two-dimensional airfoil by
comparing the predicted temperature depression across the surface of the
airfoil with experimental data by HordHord (1973) (e8), demonstrating a
moderate agreement between the predicted values and the experimental data.
### III.8 The Wake Closure Model
Introduced by Liu et al. Liu, Li, and Feng (2006), the Wake Closure Model
attempts to determine the shape of the liquid-vapor interface in cavitating
flows by using the Reynolds-averaged Navier-Stokes equations to simulate the
turbulent flow along with an appropriate model for the turbulent viscosity
$\mu_{t}$ that accounts for the fact that the area most dominated by
turbulence in cavitating flows is the wake downstream of an attached cavity,
where vapor bubbles collapse suddenly and strongly interact with any solid
wall. To this end, the authors employ the Baldwin-Lomax turbulence
modelBaldwin and Lomax (1978) to estimate $\mu_{t}$ locally. For any point
located at a normal distance $y$ from a solid wall, this model defines
$\mu_{t}$ as follows:
$\mu_{t}=\begin{cases}0.16\rho
y^{2}\left(1-\text{e}^{-y^{+}/A^{+}}\right)^{2}|\Omega|,&0\leq y\leq
y^{\prime},\\\ 0.02688\rho
F_{w}\left(1+5.5\left(\frac{0.3y}{y_{\max}}\right)^{6}\right)^{-1},&y\geq
y^{\prime},\end{cases}$ (38)
where $y^{\prime}$ is an adjustable parameter defining the separation of the
flow into an inner and outer layer, $A^{+}=26$, $y^{+}$ is defined using the
shear stress $\tau_{w}$, fluid density $\rho_{w}$ and molecular viscosity
$\mu_{w}$ at the wall as $y^{+}=\frac{\sqrt{\rho_{w}\tau_{w}}}{\mu_{w}}$,
$\Omega$ is the strength of the vortex in the flow, $y_{\max}$ is the maximum
point of the function
$F(y)=y|\Omega|\left(1-\text{e}^{-y^{+}/A^{+}}\right)$ (39)
with corresponding maximal value $F_{\max}$, and $F_{w}$ is defined using the
maximal $\mathbf{u}_{\max}$ and minimal velocity $\mathbf{u}_{\min}$ of the
flow as
$F_{w}=\min\left\\{\frac{0.25\|\mathbf{u}_{\max}-\mathbf{u}_{\min}\|^{2}}{F_{\max}},y_{\max}F_{\max}\right\\}.$
(40)
Assuming that the cavity surface is a free surface and that the pressure
inside the cavity and on the boundary of the cavity is constant and equal to
the vapor pressure, the model approximates the shape of the surface of the
cavity along the length of the solid wall by first searching for the first
point along the wall at which the local pressure is minimal and less than the
vapor pressure, then declaring each point downstream of said point to belong
to the cavity if the local pressure is below the vapor pressure. After
recalculating the flow properties with the free surface condition enforced
along the surface of the cavity, the model checks if the pressure distribution
along the surface is approximately equal to the vapor pressure and then
iteratively adjusts the local thickness $r(s)$ of the cavity at every point
$s$ along its entire length until this condition is satisfied. The updated
local thickness $r^{(n+1)}$ above a point is defined from the current local
thickness $r^{(n)}$ and a relaxation coefficient $\lambda$ with $|\lambda|\leq
1$ as $r^{(n+1)}=r^{(n)}+\lambda\Delta r^{(n)}$, with the adjustment $\Delta
r^{(n)}$ defined using the local pressure difference $p^{(n)}-p_{v}$ and the
pressure gradient $\frac{\partial p^{(n)}}{\partial s}$ along the wall at the
current iteration along with the initial local pressure difference
$p^{(0)}-p_{v}$ and pressure gradient $\frac{\partial p^{(0)}}{\partial s}$ as
$\begin{split}\Delta
r^{(n)}&=\frac{\pi}{180}\int_{s_{b}}^{s}\beta_{0}\operatorname{sign}(p_{v}-p^{n})\sqrt{\frac{|p_{v}-p^{n}|}{\|p_{v}-p^{0}\|}}\\\
&\hskip 25.83325pt+\beta_{1}\operatorname{sign}\left(\frac{\partial
p^{(n)}}{\partial s}\right)\sqrt{\frac{\left|\frac{\partial p^{(n)}}{\partial
s}\right|}{\left\|\frac{\partial p^{(0)}}{\partial
s}\right\|}}\,\text{d}s,\end{split}$ (41)
where
$\displaystyle\|p_{v}-p^{0}\|$
$\displaystyle=\sqrt{\frac{\int_{s_{b}}^{s_{e}}(p_{v}-p^{0})^{2}\,\text{d}s}{s_{e}-s_{b}}},$
$\displaystyle\left\|\frac{\partial p^{(0)}}{\partial s}\right\|$
$\displaystyle=\sqrt{\frac{\int_{s_{b}}^{s_{e}}\left(\frac{\partial
p^{(0)}}{\partial s}\right)^{2}\,\text{d}s}{s_{e}-s_{b}}},$
and $s_{b}$ and $s_{e}$ denote the inception point and the endpoint of the
cavity. Once convergence in this iterative procedure has been achieved, the
model redefines the surface of the cavity using both the previously determined
endpoints of the cavity $s_{b}$ and $s_{e}$ as well as a new point $s_{w}$,
which denotes the beginning of the wake region. The new point $s_{w}$ is
defined as the first point on the surface after $s_{b}$ at which the local
thickness $r(s_{w})$ is decreasing and less than half of the maximal local
thickness. Using the three points $s_{b}$, $s_{w}$, and $s_{e}$, the cavity
surface is defined as the cubic Hermite polynomial interpolating between the
points $s_{b}$ and $s_{e}$ on the solid wall as well as the point
$(s_{w},r(s_{w}))$. This approach to capturing the surface of the cavity was
chosen because of its efficiency and due to the lack of model available at the
time capable of accounting for the turbulence in the wake region and the
violation of the condition of constant vapor pressure along the surface of the
cavity. This model is implemented as an in-house tool, using the SIMPLEC
momentum-pressure coupling and the Baldwin-Lomax turbulence modelBaldwin and
Lomax (1978). The model was validated by simulating the pressure distributions
of cavitaing flow over an ogival headform, indicating good agreement with data
from Rouse and McNownRouse and McNown (1948) (e5). Liu et al.Li, Liu, and Feng
(2007) applied this model for simulating sheet cavitation on a cylindrical
headform, showing good agreement with experimental data. No other references
to other applications of this model were found in the literature; one possible
explanation may be the model’s difficulties in dealing with three-dimensional
problems as noted in the review of Li and YuLi and Yu (2021).
### III.9 The Bubble Density-Liquid Volume Coupling Model
Schnerr and SauerSchnerr and Sauer (2001) present a TEM which expresses the
source terms using the local bubble radius $R$ and the bubble density $n_{0}$,
assumed uniform throughout the domain. Instead of simply using the relation
(6), the authors seek an expression for the vapor volume fraction $\alpha_{v}$
that directly couples the density of vapor bubbles and the liquid volume,
corresponding to the physical observation that an increase in the density of
bubbles should correspond to a decrease in the liquid volume fraction. The
rationale given by the authors for their search for a new expression for
$\alpha_{v}$ is the fact that the standard volume-of-fluid method is capable
of accounting for convective transport of the vapor volume fraction, but not
the change in volume fraction due to phase transition. To this end, the
authors define the vapor volume fraction $\alpha_{v}$ for each cell in the
computational grid as the ratio of the volume occupied by the vapor within a
given cell, denoted $V_{v}$, and the total volume of said cell, denoted
$V_{cell}$. Letting $V_{l}=V_{cell}-V_{v}$ denote the volume occupied by the
liquid within the cell, the authors approximate $V_{v}$ using the relation (6)
instead, i.e. $V_{v}$ is approximated in terms of the bubble number $n_{0}$
and bubble radius $R$ as
$V_{v}=\frac{4}{3}n_{0}\pi R^{3}.$ (42)
Recalling that $V_{cell}=V_{l}+V_{v}$, the approximation of $V_{v}$ given by
(42) implies that the vapor volume fraction can be approximated as
$\alpha_{v}=\frac{\frac{4}{3}n_{0}\pi R^{3}}{1+\frac{4}{3}n_{0}\pi R^{3}}.$
(43)
With the expression (43), the transport equation for $\alpha_{v}$ may be
restated as
$\frac{\partial\alpha_{v}}{\partial
t}+\nabla\cdot(\alpha_{v}\mathbf{u})=\frac{\frac{4}{3}n_{0}\pi}{1+\frac{4}{3}n_{0}\pi
R^{3}}\frac{d}{dt}(R^{3});$ (44)
here, the authors note that (44) directly couples the bubble density as
expressed by $n_{0}$ with the liquid volume. In order to complete their
expression for the source term of $\alpha_{v}$ in (44), the authors
approximate the rate of change of the bubble radius $\frac{dR}{dt}$ as
$\frac{dR}{dt}=\sqrt{\frac{2|p_{v}-p|}{3\rho_{l}}}.$ (45)
This expression is obtained from the RPE (5) with the effects of the second-
order terms, viscosity, and surface tension all neglected, the pressure at the
liquid-bubble interface taken to be equal to the vapor pressure $p_{v}$, and
the reference pressure $p_{\infty}$ taken to be equal to the cell pressure
$p$. Combining (44) and (45) yields the final expressions for the source terms
$\displaystyle\dot{m}^{+}$ $\displaystyle=\frac{4n_{0}\pi
R^{2}}{1+\frac{4}{3}n_{0}\pi R^{3}}\sqrt{\frac{2(p_{v}-p)}{3\rho_{l}}},$
$\displaystyle p$ $\displaystyle<p_{v},$ (46) $\displaystyle\dot{m}^{-}$
$\displaystyle=\frac{4n_{0}\pi R^{2}}{1+\frac{4}{3}n_{0}\pi
R^{3}}\sqrt{\frac{2(p-p_{v})}{3\rho_{l}}},$ $\displaystyle p$
$\displaystyle>p_{v}.$ (47)
Notably, no empirical constants besides the uniform bubble number $n_{0}$ are
used in the expressions (46) and (47). The Schnerr and Sauer model is
implemented in a volume-of-fluid framework. It is the default model for
cavitating flow modelling in Ansys Fluent and an optional cavitation model in
OpenFOAM and STAR-CCM+. The model was validated through simulation of
cavitating flow over a hydrofoil under varying conditions, showing good
agreement with previous observations made by Keller and ArndtKeller (2001)
(e9). This model was recently applied by Moganaradjou et al.Moganaradjou _et
al._ (2023) for simulation of cavitation in a rocket pump.
### III.10 The Full Cavitation Model
Singhal et al.Singhal _et al._ (2002) introduced a cavitation model also
based on viewing cavitation as the phenomenon of existing vapor nuclei in the
liquid beginning to grow once the local pressure decreases below the saturated
vapor pressure of the liquid, then shrinking once the local pressure rises
above the vapor pressure. The present model considers the governing equation
of the vapor mass fraction, denoted $f_{v}$, and seeks to construct
expressions for the evaporation term $\dot{m}^{+}$ and the condensation term
$\dot{m}^{-}$. The authors accomplish this goal using the above mentioned
viewpoint to first restate the transport equation of the vapor mass fraction
in terms of the mixture density and the vapor volume fraction, then
reformulating the vapor volume fraction using the assumption of a uniform
number of vapor nuclei present in the liquid with a uniform radius as well as
(6). The authors incorporate the effects of bubble dynamics into their model
by using the RPE (5) to model the growth of a single bubble’s radius,
neglecting the effects of interaction between two distinct bubbles.
Furthermore, the authors simplify the RPE via neglecting the effects of
viscosity, surface tension, and the second-order derivative of the bubble
radius, justifying the latter exclusion as the second-order derivative being
”important mainly during initial bubble acceleration”. Restating their
expressions in terms of $f_{v}$, this yields the initial model
$\displaystyle\begin{aligned}
\dot{m}^{+}&=(4n_{0}\pi)^{1/3}(3\alpha_{v})^{2/3}\frac{\rho_{v}\rho_{l}}{\rho_{m}}\sqrt{\frac{2}{3}\frac{p_{b}-p_{\infty}}{\rho_{l}}},\\\
\dot{m}^{-}&=(4n_{0}\pi)^{1/3}(3\alpha_{v})^{2/3}\frac{\rho_{v}\rho_{l}}{\rho_{m}}\sqrt{\frac{2}{3}\frac{p_{\infty}-p_{b}}{\rho_{l}}}.\end{aligned}$
(48)
In order to remove the bubble number density as a parameter of the model, the
authors restate their source terms using a method introduced in the nuclear
industry cf. Markatos and SinghalMarkatos and Singhal (1982). This method
forms a correlation between the bubble radius and the local relative velocity
between the two phases as well as the surface tension, thus re-introducing
surface tension as an effect into their model. Using this correlation along
with several limiting arguments as $\alpha_{v}\rightarrow 0$ and an assumption
that the phase change rates are proportional to the local relative velocity,
the models in (48) can be rewritten as
$\displaystyle\begin{aligned}
\dot{m}^{+}&=C_{+}\frac{u_{ch}}{\sigma}\rho_{l}\rho_{v}\sqrt{\frac{2}{3}\frac{p_{b}-p_{\infty}}{\rho_{l}}}(1-f_{v}),\\\
\dot{m}^{-}&=C_{-}\frac{u_{ch}}{\sigma}\rho_{l}\rho_{l}\sqrt{\frac{2}{3}\frac{p_{\infty}-p_{b}}{\rho_{l}}}f_{v},\end{aligned}$
(49)
where $C_{+}$ and $C_{-}$ are empirical constants and $u_{ch}$ is a
characteristic velocity that reflects the impact of the local relative
velocity between the liquid phase and the vapor phase on the phase change
rates. To account for the effect of turbulence on cavitation, two further
assumptions are made. First, the local relative velocity is eliminated as a
parameter of the model through the assumption that the local relative velocity
is proportional to the square root of the local turbulent kinetic energy of
the flow. Secondly, the fluctuations of pressure in turbulent flow are
accounted for in the model by first taking the bubble pressure $p_{b}$ in the
expressions (48) and (49) to be equal to a modified vapor pressure
$\tilde{p}_{v}$ given by
$\tilde{p}_{v}=p_{v}+\frac{0.39\rho_{m}k}{2},$ (50)
in accordance with HinzeHinze (1975). Replacing the ambient pressure
$p_{\infty}$ in the models (49) with the local pressure $p$, the assumptions
stated above leads to (49) being restated as
$\begin{split}\dot{m}^{+}&=C_{+}\frac{\sqrt{k}}{\sigma}\rho_{l}\rho_{v}\sqrt{\frac{2}{3}\frac{\tilde{p}_{v}-p}{\rho_{l}}}(1-f_{v}),&p&<\tilde{p}_{v}\\\
\dot{m}^{-}&=C_{-}\frac{\sqrt{k}}{\sigma}\rho_{l}\rho_{l}\sqrt{\frac{2}{3}\frac{p-\tilde{p}_{v}}{\rho_{l}}}f_{v},&p&>\tilde{p}_{v}.\end{split}$
(51)
The final effect accounted for in this model, that of non-condensable gas, is
introduced via the assumption that the cavitating fluid also contains a finite
amount of non-condensable gas in dissolved state, i.e. a three-phase flow.
Additionally, the mass fraction of the non-condensable gas, denoted $f_{g}$,
is assumed to be constant and specified as an input parameter of the model.
The impact of this assumption on the models (51) is that the liquid mass
fraction $1-f_{v}$ is modified as $1-f_{v}-f_{g}$, leading to the final models
for the phase change rates given below:
$\begin{split}\dot{m}^{+}&=C_{+}\frac{\rho_{l}\rho_{v}}{\sigma}\sqrt{\frac{2k}{3}\frac{\tilde{p}_{v}-p}{\rho_{l}}}(1-f_{v}-f_{g}),&p&<\tilde{p}_{v},\\\
\dot{m}^{-}&=C_{-}\frac{\rho_{l}\rho_{l}}{\sigma}\sqrt{\frac{2k}{3}\frac{p-\tilde{p}_{v}}{\rho_{l}}}f_{v},&p&>\tilde{p}_{v}.\end{split}$
(52)
The authors of the model have specified recommended values for the empirical
constants $C_{+}$ and $C_{-}$ based on various validation tests, the results
of which are not disclosed in the original article. Singhal et al. implement
their model using the finite volume method with a standard $k-\varepsilon$
turbulence model. This model is one of the optional cavitation solvers in
Ansys Fluent. The model was validated by simulating cavitating flow over a
conical head as well as a hydrofoil, showing good agreement with data reported
by Rouse and McNownRouse and McNown (1948) (e5) in the former case and data
reported by Shen and DimotakisShen and Dimotakis (1989) (e7) in the latter
case. This model was recently applied by Yuan et al.Yuan _et al._ (2022) in
their development of a model for dynamic bulk modulus for aerated hydraulic
fluids.
### III.11 The Vapor Nuclei-Adjusted Model
Zwart et al.Zwart, Gerber, and Belamri (2004) present a TEM in which source
terms are derived by combining the Rayleigh equation, i.e. the RPE (5) with
the effects of radial acceleration, viscosity, and surface tension all
neglected, along with the expression (6), yielding an initial expression of
the total mass transfer rate of a single bubble during bubble growth as
$\dot{m}=\frac{3\alpha_{v}\rho_{v}}{R}\sqrt{\frac{p_{v}-p}{\rho_{l}}}.$ (53)
The expression (53) is immediately generalized to an expression for the total
mass transfer rate of a single bubble during bubble collapse as
$\dot{m}=C\frac{3\alpha_{v}\rho_{v}}{R}\sqrt{\frac{|p_{v}-p|}{\rho_{l}}}\text{sign}(p_{v}-p),$
(54)
where $C$ is an empirical constant used for calibrating the model. The authors
note here that the expression (54) works well during bubble collapse, but is
both physically incorrect and numerically unstable during bubble growth. To
remedy this, the authors seek to modify the mass transfer rate to account for
the interaction between distinct bubbles. Noting that as an increase in the
vapor volume fraction $\alpha_{v}$ corresponds to a decrease in the density of
vapor nuclei, the authors modify the mass transfer rate (54) by replacing
$\alpha_{v}$ with the expression $\alpha_{nuc}(1-\alpha_{v})$ during bubble
growth, i.e. when $p<p_{v}$, where $\alpha_{nuc}$ is a parameter of the model
specifying the vapor nuclei volume fraction. This leads to the final
expressions for the evaporation and condensation terms given by
$\begin{split}\dot{m}^{+}&=C_{+}\frac{3\alpha_{nuc}(1-\alpha_{v})\rho_{v}}{R}\sqrt{\frac{p_{v}-p}{\rho_{l}}},&p&<p_{v},\\\
\dot{m}^{-}&=C_{-}\frac{3\alpha_{v}\rho_{v}}{R}\sqrt{\frac{p-p_{v}}{\rho_{l}}},&p&>p_{v}.\end{split}$
(55)
Additionally, the authors observed that the model fails to properly predict
the oscillating behaviour of certain unsteady cavitating flow when standard
turbulence models were employed. To remedy this deficiency, the authors
approach suggested by Coutier-Delgosha et al.Coutier-Delgosha, Fortes Patella,
and Reboud (2001) that decreases the effect of turbulent viscosity in
cavitating regions. Using this approach, the standard expression for the eddy
viscosity of the mixture
$\mu_{m,t}=\frac{C_{\mu}\rho_{m}k^{2}}{\varepsilon}$
is modified as
$\mu_{m,t}=\left(\rho_{v}+\left(\frac{\rho_{v}-\rho_{m}}{\rho_{v}-\rho_{l}}\right)^{P}(\rho_{l}-\rho_{v})\right)C_{\mu}\frac{k^{2}}{\varepsilon},$
where $P>1$ is an empirical parameter. This method is an optional cavitation
method in Ansys Fluent. It is implemented using the finite volume method and
the standard $k-\varepsilon$ model. The authors validate their model by
simulating a variety of cavitating flow conditions previously reported on by
different authors: the pressure profile of cavitating flow over a hydrofoil
investigated by Shen and DimotakisShen and Dimotakis (1989) (e7), cavitating
flow in an inducer investigated by Bakir et al.Bakir _et al._ (2004) (e10),
and cavitating flow in a Venturi nozzle investigated by Stutz and ReboudStutz
and Reboud (1997b) (e6). In all cases, the model shows good agreement with the
reported data. This model was applied by Zhou et al. Zhou _et al._ (2023) to
investigate thermal properties of oil-film viscosity in squeeze film dampers.
### III.12 The Viscosity-Oriented Model
Konstantinov et al.Konstantinov, Tselischev, and Tselischev (2014) develop a
TEM with the stated intent of obtaining a model that more accurately captures
the effects of bubble dynamics on cavitation without sacrificing numerical
stability. The authors choose the source terns from the Vapor Nuclei-Adjusted
ModelZwart, Gerber, and Belamri (2004) given by
$\begin{split}\dot{m}^{+}&=C_{+}\frac{3\alpha_{nuc}(1-\alpha_{v})\rho_{v}}{R}\sqrt{\frac{p_{v}-p}{\rho_{l}}},&p&<p_{v},\\\
\dot{m}^{-}&=C_{-}\frac{3\alpha_{v}\rho_{v}}{R}\sqrt{\frac{p-p_{v}}{\rho_{l}}},&p&>p_{v}\end{split}$
(56)
as a starting point, and model the bubble dynamics using a non-dimensional RPE
$\overline{R}\>\ddot{\overline{R}}+\frac{3}{2}\dot{\overline{R}^{2}}+\frac{1}{\overline{R}}\frac{1}{\mathrm{Re}}\dot{\overline{R}}=1$
(57)
with nondimensional variables and numbers
$\overline{R}=\frac{R}{R_{0}},\,\,\mathrm{Re}=\frac{R_{0}\sqrt{|p_{v}-p|\rho_{l}}}{4\mu},\,\,\tau=\frac{t}{R_{0}}\sqrt{\frac{|p_{v}-p|}{\rho_{l}}}.$
(58)
The authors seek to express the bubble growth rate in terms of the Reynolds
number Re, neglecting the effects of surface tension and elected to use the
non-corrected expression for the eddy viscosity given by
$\mu_{m,t}=\frac{C_{\mu}\rho_{m}k^{2}}{\varepsilon}.$
Additionally, the model assumes that both the liquid and vaporous phase are
incompressible in order to facilitate simpler expressions. Through a series of
numerical calculations based on (57), the authors report the new expression
$\displaystyle\begin{aligned}
\frac{dR}{dt}&=\tanh\left[1.221\left(\frac{R_{0}\sqrt{|p_{v}-p|\rho_{l}}}{4\mu}\right)^{0.353}\right]\\\
&\hskip 17.22217pt\times\sqrt{\frac{2|p_{v}-p|}{3\rho_{l}}}.\end{aligned}$
(59)
Replacing the approximation
$\frac{dR}{dt}=\sqrt{\frac{2|p_{v}-p|}{3\rho_{l}}}$ in the source terms (56)
from the Vapor Nuclei-Adjusted ModelZwart, Gerber, and Belamri (2004) with the
right hand side of the expression (59), denoted by $\tilde{R}(p)$, the authors
obtain the new source terms
$\displaystyle\dot{m}^{+}$
$\displaystyle=C_{+}\frac{3\alpha_{nuc}(1-\alpha_{v})\rho_{v}}{R}\tilde{R}(p),$
$\displaystyle p$ $\displaystyle<p_{v},$ (60) $\displaystyle\dot{m}^{-}$
$\displaystyle=C_{-}\frac{3\alpha_{v}\rho_{v}}{R}\tilde{R}(p),$ $\displaystyle
p$ $\displaystyle>p_{v}.$ (61)
This model was developed in the finite element solver Ansys CFX, using the
standard $k-\varepsilon$ turbulence model. The model was validated against
experimental data on cavitating flow in a ”pipe-pipe” jet element, obtaining
results in good agreement with the experimental data as well as previous
observations made by AckeretAckeret (1930) (e11). This model was applied by
Konstantinov et al. Konstantinov, Tselischev, and Tselischev (2017) towards
simulating cavitating flow within a jet-cavitation fluid mass flow stabilizer.
### III.13 The Plane Surface Evaporation Model
In Saito et al.’s Saito, Nakamori, and Ikohagi (2003) model, the source terms
are derived using the theory of vaporization and condensation on a plane
surface as described by Sone and SugimotoSone and Sugimoto (1990). Due to the
different theoretical approach, the source terms expressed in this model do
not express the mass volume change, but rather the mass surface change.
Furthermore, this approach also introduces a new parameter of interest used to
express the source terms, namely the interfacial area concentration in the
liquid-vapor mixture $A$ given by $A=C_{a}\alpha_{v}(1-\alpha_{v})$. The
source terms of this model are given by
$\displaystyle\dot{m}^{+}$
$\displaystyle=C_{+}\frac{A\alpha_{v}(1-\alpha_{v})(p_{v}-p)}{\sqrt{2\pi
R_{1}T_{sat}}},$ $\displaystyle p$ $\displaystyle<p_{v},$ (62)
$\displaystyle\dot{m}^{-}$
$\displaystyle=C_{-}\frac{\rho_{l}}{\rho_{v}}\frac{A\alpha_{v}(1-\alpha_{v})(p-p_{v})}{\sqrt{2\pi
R_{1}T_{sat}}},$ $\displaystyle p$ $\displaystyle>p_{v},$ (63)
where $C_{+}$, $C_{-}$, and $C_{a}$ are empirical constants. The authors state
that the relation $C=C_{+}C_{a}=C_{-}C_{a}$ holds, but do not elaborate on the
exact nature of said relationship. Additionally, the authors use an EOS
derived for a locally homogeneous gas-liquid mixture to determine the mixture
density. This EOS was first derived by Okuda and IkohagiOkuda and Ikohagi
(1996) and is given in terms of the vapor mass fraction $f_{v}$, the pressure
$p$ and the temperature $T$ as
$\rho_{m}=\frac{p(p+p_{c})}{K(1-f_{v})p(T+T_{c})+Rf_{v}(p+p_{c})T},$ (64)
where $p_{c}$, $T_{c}$, $K$, and $R$ are the pressure, temperature, liquid,
and gas constant of the fluid, respectively. Despite the different approach,
the expressions source terms of this model are of a similar form to the source
terms obtained in other models such as the Vapor Nuclei-Adjusted ModelZwart,
Gerber, and Belamri (2004) and the Full Cavitation ModelSinghal _et al._
(2002).
The model was implemented using the finite-volume discretization with cell-
centered momentum components. Turbulence was modelled using the Baldwin-Lomax
modelBaldwin and Lomax (1978). The authors validate their model by simulating
cavitating flow over a three-dimensional cylinder; the predicted values of the
pressure coefficients across a range of cavitation numbers agree with
experimental data reported by Rouse and McNownRouse and McNown (1948) (e5).
This model was applied by Mostafa et al.Mostafa _et al._ (2016) in their
study of cavitating flow over a hydrofoil, showing good predicted values for
the drag and lift coefficient at various cavitation numbers.
### III.14 The Four-Equation Model
Goncalvès and CharrièreGoncalvès and Charrière (2014) developed a model, which
combines the Reynolds-averaged Navier-Stokes equations with expressions for
source terms for transport of the vapor volume fraction $\alpha_{v}$ using two
quantities not previously considered in the literature, namely the local speed
of sound $c$ and the propagation of acoustic waves without mass transfer
$c_{wallis}$. As the name of the model implies, the authors choose a set of
four equations as the starting point for this model, namely three conservation
laws for the mass, momentum, and total energy along with a transport equation
for the vapor volume fraction. This model is itself derived from previous work
of Goncalvès Goncalvès (2013). The system is closed by relating the pressure
to the density via a barotropic law introduced by Dellanoy and KuenyDellanoy
and Kueny (1990)
$\begin{split}p(\rho_{m})=p_{v}&+\frac{\left(\rho_{l}-\rho_{v}\right)c_{baro}^{2}}{2}\\\
&\hskip
8.61108pt\times\arcsin\left(\frac{2\rho_{m}-\rho_{l}-\rho_{v}}{\rho_{l}-\rho_{v}}\right),\end{split}$
(65)
where $c_{baro}$ is a empirical parameter representing the minimal speed of
sound in the mixture region of the fluid. The importance of this parameter and
other aspects of this EOS were investigated by Goncalvès and PatellaGoncalves
and Patella (2009). From (65), the speed of sound of the mixture may be be
calculated as
$c^{2}=\left(\frac{\partial
p}{\partial\rho_{m}}\right)_{s}=\frac{c_{baro}^{2}}{2\sqrt{\alpha_{v}(1-\alpha_{v})}}$
(66)
The new effects $c$ and $c_{wallis}$ are related to $\alpha_{v}$ via two
assumptions. First, it is assumed that the total mass transfer rate $\dot{m}$
is proportional to the divergence $\nabla\cdot\mathbf{u}$ of the velocity
field. Proceeding as in Goncalvès’ previous work, the authors conclude that
the proportional relation between $\dot{m}$ and $\nabla\cdot\mathbf{u}$ is
given by
$\dot{m}=\frac{\rho_{l}\rho_{v}}{\rho_{l}-\rho_{v}}\left(1-\frac{c^{2}}{c_{wallis}^{2}}\right)\nabla\cdot\mathbf{u}.$
(67)
Secondly, it is assumed that $c_{wallis}$ equals the weighted harmonic mean of
the local speed of sound each of the two phases, i.e. that the relation
$\frac{1}{\rho
c_{wallis}^{2}}=\frac{\alpha_{v}}{\rho_{v}c_{v}^{2}}+\frac{1-\alpha_{v}}{\rho_{l}c_{l}^{2}}$
(68)
holds. The Four-Equation Model is implemented using a cell-centred finite
volume discretization and a matrix-free, implicit time integration method due
to Luo et al. Luo, Baum, and Löhner (1998). Turbulence is modelled using the
Spalart-Allmaras modelSpalart and Allmaras (1992). The model was validated
against experimental data of cavitating flow in a Venturi nozzle reported by
Barre et al.Barre _et al._ (2009) (e12) and Patella et al.Patella, Barre, and
Reboud (2006) (e13), showing good agreement. The four equation model was used
by GoncalvèsGoncalves (2017) in a comparative study of various turbulence and
cavitation models, showing good performance when applied to the test case of
water-gas flow in an expansion tube.
### III.15 The Ghost-Fluid Multiscale Model
Introduced by Hisao et al.Hsiao, Ma, and Chahine (2017), the Ghost-Fluid
Multiscale Model is a multiscale model based on the Euler-Lagrangian approach,
i.e the model employs the Eulerian approach for simulating the growth and
collapse of cavities on the macroscale and tracks the growth and momentum of
bubbles on the microscale in a Lagrangian framework. The main distinguishing
feature of this model is the implementation of the Ghost Fluid Method, a level
set method first described by Fedkiw et al. Fedkiw _et al._ (1999) and Kang
et al. Kang, Fedkiw, and Liu (2000), to simulate larger deformations in larger
cavities such as e.g. sheet cavities as well as a different transition scheme
employed by the model to determine of a given cavity currently tracked at the
macroscale should switch to the microscale and vice versa. Recalling the
generic description mentioned at the start of this section, the schemes
comprising the Ghost-Fluid Multiscale Model can be described as follows:
1. 1.
The macro-scale cavities are resolved in an Eulerian framework by first
obtaining the flow characteristics through the Navier-Stokes equations(1),
which are solved using a finite volume method. Once the flow characteristics
have been resolved, the model approximates the vapor volume fraction locally
within each computational cell as a weighted sum of the amount of volume
occupied by all cavities currently tracked by the model which at least partly
occupy the given cell. The weights in this sum are calculated using the same
scheme as that of Ma et al. Ma, Hsiao, and Chahine (2015), who assumed that
the distribution of the bubbles is approximately a Gaussian distribution
centered at the cell of the center with a prescribed standard deviation
$R_{s}$, also referred to as the characteristic spreading radius. Having
developed the vapor volume fraction $\alpha_{v,i}$ on a per-cell basis, the
mixture density $\rho_{m}$ is then updated in each cell using (2). The final
step of the macro-scale scheme is to update the location of existing macro-
scale cavities. This is done using the Ghost Fluid Method, which identifies
the liquid-vapor interfaces as the zero level set of the smooth function
$\varphi$ and tracking its evolution via the transport equation
$\frac{D\varphi}{Dt}=\frac{\partial\varphi}{\partial
t}+\mathbf{u}_{i}\cdot\nabla\varphi=0$ (69)
coupled with the boundary conditions
$\displaystyle\begin{aligned}
\rho_{l}p&=\mathbf{n}\cdot(\bm{\tau}\cdot\mathbf{n})+gz+\sigma\kappa,&\mathbf{n}\cdot(\bm{\tau}\cdot\mathbf{t})&=0,\end{aligned}$
(70)
ensuring balance of normal stresses and zero shear, where $\mathbf{u}_{i}$
denotes the velocity of the interface,
$\mathbf{n}=\frac{\nabla\varphi}{\|\nabla\varphi\|}$ and $\mathbf{t}$ are the
surface normal and tangential vectors, $g$ is the gravitational acceleration,
$\bm{\tau}$ is the stress tensor, and
$\kappa=\frac{\nabla\cdot(\nabla\varphi)}{\|\nabla\varphi\|}$ is the surface
curvature. The authors note that integrating (69) does not guarantee that the
thickness of the interface region remains constant in space and time due to
various errors caused by numerical diffusion and distortion by the flow field,
and employ a correction developed by Sussman et al.Sussman _et al._ (1999)
2. 2.
The micro-scale cavities are resolved in the Lagrangian framework, where the
size and trajectories of bubbles either formed from nuclei present in the flow
at the start of the simulation, from nuclei created during the simulation due
to nucleation at a solid boundary, or from the breakup of larger cavities are
all tracked. The size of the bubble is modelled through its radius $R$, which
is approximated by a modified RPE given by
$\begin{split}\rho_{l}\left(R\ddot{R}+\frac{3}{2}\dot{R}^{2}\right)&=p_{v}+p_{g0}\left(\frac{R_{0}}{R}\right)^{3k}-p_{enc}\\\
&\hskip
12.91663pt-\frac{2\sigma}{R}-\frac{4\mu_{l}R}{R}+\rho_{l}\frac{\|\mathbf{u}_{s}\|^{2}}{4},\end{split}$
(71)
where $p_{enc}$ is the averaged of the liquid pressure over the surface of the
bubble, $p_{g0}$ is the initial gas pressure inside the bubble, $R_{0}$ is the
initial bubble radius, $k$ is the gas polytropic compression constant, and
$\mathbf{u}_{s}=\mathbf{u}_{enc}-\mathbf{u}_{b}$ is the velocity slip, defined
here as the difference between the liquid velocity averaged over the surface
of the bubble $\mathbf{u}_{enc}$ and the bubble’s translation velocity
$\mathbf{u}_{b}$; the last term in (71) accounts for the effects of slip
velocity between the motion of the bubble and the flow of the surrounding
liquid, and was first derived by Hsiao et alHsiao, Chahine, and Liu (2000).
Additionally, the usage of surface-averaged flow characteristics in (71) was
introduced by Hsiao et al. Hsiao, Chahine, and Liu (2003) in order to account
for the prescence of a non-uniform pressure field in the immediate vicinity of
the bubble. The bubble’s translation velocity $\mathbf{u}_{b}$ is obtained
from an equation for the bubble motion given by
$\begin{split}\frac{d\mathbf{u}_{b}}{dt}&=\left(\frac{\rho_{l}}{\rho_{b}}\right)\left[\frac{3}{8R}C_{D}\|\mathbf{u}_{s}\|\mathbf{u}_{s}+\frac{1}{2}\left(\frac{d\mathbf{u}_{enc}}{dt}-\frac{d\mathbf{u}_{b}}{dt}\right)\right.\\\
&+\left.\frac{3\dot{R}}{2R}\mathbf{u}_{s}-\frac{\nabla
p}{\rho_{l}}+\frac{\rho_{b}-\rho_{l}}{\rho_{l}}\mathbf{g}+\frac{3C_{L}\sqrt{\nu}(\mathbf{u}_{s}\times\Omega)}{4\pi
R\sqrt{\|\Omega\|}}\right].\end{split}$ (72)
The terms on the right-hand side of the bubble motion equation (72) represent
various contributions to changes in the bubble’s trajectory due to drag
forces, added mass of the bubble, the pressure gradient of the surrounding
liquid, gravitational forces, and lift forces as expressed via the vorticity
vector $\Omega$. Additionally, the expressions for the first and last terms
were first developed by Haberman and MortonHaberman and Morton (1953) and
SaffmanSaffman (1965), respectively.
3. 3.
The model features two transition schemes, one for the micro-macro transition
and one for the macro-micro transition. The micro-macro transition scheme
transitions a micro-scale bubble to the macro-scale if its current radius both
exceeds a threshold value and is greater than a specified multiple of the
local grid size as specified by the criterion
$R\geq\max(R_{thr},m_{thr}\Delta L),$ (73)
where $R_{thr}$ is the threshold radius, $\Delta L$ is the size of the local
grid hosting the bubble, and $m_{thr}$ is a threshold grid factor. When a
bubble is detected to satisfy (73), the bubble is removed from the simulation,
and a new micro-scale cavity is introduced in the cell(s) occupied by the
bubble. With this scheme, the model is capable of simulating both multiple
bubbles coalescing into a large cavity and single isolated bubbles coalescing
with a previously defined large cavity. The macro-micro transition scheme
instead determines if the shape of the macro-scale cavity, identified here as
the zero level set of $\varphi$, has developed sufficient instabilities to
breakup into a cloud of bubbles based on empirical criteria described by Ma et
al.Ma _et al._ (2011) and Hsiao et alHsiao _et al._ (2013). Once a cavity
satisfies these criteria, it is replaced with a cloud of micro-scale bubbles
of uniform size. Additionally, the fluid is initialized with no macro-scale
cavities present in the flow and with micro-scale nuclei randomly distributed
in the flow based on previous experimental measurements by Medwin Medwin
(1977), BilletBillet (1985), FranklinFranklin (1992), and Wu and ChahineWu and
Chahine (2010). Additional nuclei are added at each time step to the flow from
both the inlet using the same distribution as the initial distribution of
nuclei and from bubble entrainment occurring at the solid boundaries.
The Ghost-Fluid Multiscale Model is an in-house development employing the
finite-volume method with implicit time integration. The different phases are
modelled with the level set method, or the Lagrangian discrete bubble model.
The model was validated against experimental data of shedding frequencies and
cavity length for cavitating flow over a hydrofoil reported by Berntsen et
al.Berntsen, Kjeldsen, and Arndt (2001) (e14), showing good agreement. Ma et
al.Ma, Hsiao, and Chahine (2017) used this model in a study of bubbly flow
within a waterjet propulsion nozzle as well as unsteady sheet cavitation on a
hydrofoil.
### III.16 The Density-based Convex Combination Model
Huang and Wang’sHuang and Wang (2011) model is a TEM with source terms derived
using a convex combination of the source terms from two previous models,
namely the source terms from the Vapor Nuclei-Adjusted ModelZwart, Gerber, and
Belamri (2004) by Zwart et al.Zwart, Gerber, and Belamri (2004) given by
$\displaystyle\dot{m}_{1}^{+}$
$\displaystyle=C_{+}\frac{3\alpha_{nuc}(1-\alpha_{v})\rho_{v}}{R}\sqrt{\frac{p_{v}-p}{\rho_{l}}},$
$\displaystyle p$ $\displaystyle<p_{v},$ (74) $\displaystyle\dot{m}_{1}^{-}$
$\displaystyle=C_{-}\frac{3\alpha_{v}\rho_{v}}{R}\sqrt{\frac{p-p_{v}}{\rho_{l}}},$
$\displaystyle p$ $\displaystyle>p_{v}$ (75)
and the source terms from the Interface Mass and Normal Momentum ModelSenocak
and Shyy (2004a) given by
$\displaystyle\dot{m}_{2}^{+}$
$\displaystyle=\frac{\rho_{l}(p_{v}-p)\alpha_{l}}{\rho_{v}(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}},$
$\displaystyle p$ $\displaystyle<p_{v},$ (76) $\displaystyle\dot{m}_{2}^{-}$
$\displaystyle=\frac{(p-p_{v})(1-\alpha_{l})}{(u_{v,n}-u_{i,n})^{2}(\rho_{l}-\rho_{v})t_{\infty}},$
$\displaystyle p$ $\displaystyle>p_{v}.$ (77)
in an attempt to combine the advantages of both models. The authors introduce
their convex combination by first defining a blending function $\chi$ given by
$\chi\left(\frac{\rho_{m}}{\rho_{l}}\right)=\frac{1}{2}+\frac{\tanh\left[\frac{C_{1}(0.6\rho_{m}/\rho_{l}-C_{2})}{0.2(1-2C_{2})+C_{2}}\right]}{2\tanh(C_{1})},$
(78)
then using the source terms of both models to define
$\displaystyle\begin{aligned}
\dot{m}^{+}&=\chi\left(\frac{\rho_{m}}{\rho_{l}}\right)\dot{m}_{1}^{+}\\\
&\hskip
8.61108pt+\left(1-\chi\left(\frac{\rho_{m}}{\rho_{l}}\right)\right)\dot{m}_{2}^{+},\end{aligned}\quad
p<p_{v},$ (79) $\displaystyle\begin{aligned}
\dot{m}^{-}&=\chi\left(\frac{\rho_{m}}{\rho_{l}}\right)\dot{m}_{1}^{-}\\\
&\hskip
8.61108pt+\left(1-\chi\left(\frac{\rho_{m}}{\rho_{l}}\right)\right)\dot{m}_{2}^{-},\end{aligned}\quad
p>p_{v}.$ (80)
The weights of this convex combination are defined using a function of the
local mixture density, lending greater weight to the source terms from the
Vapor Nuclei-Adjusted Model as the mixture density approaches the pure liquid
density. Conversely, the source terms from the Interface Mass and Normal
Momentum Model are favored as the mixture density approaches the vapor
density. Huang and Wang do not report the method used for discretizing the
Navier-Stokes equations, but mention that a modified $k-\varepsilon$ model was
used for turbulence modelling. The model is validated by simulation cloud
cavitation over a hydrofoil, showing good agreement with experimental data
previously reported by Wang et al.Wang _et al._ (2001) (e15).
### III.17 The Heat Balance Model
Shi et al.’sShi, Wang, and Hu (2014) model expresses the source terms using a
revised simplification of the RPE which has been modified to include thermal
effects. The authors choose the model
$\displaystyle\dot{m}^{+}$
$\displaystyle=C_{+}\frac{3\alpha_{v}\rho_{v}}{R}\sqrt{\frac{p_{v}-p}{\rho_{l}}},$
$\displaystyle p$ $\displaystyle<p_{v}$ (81) $\displaystyle\dot{m}^{-}$
$\displaystyle=C_{-}\frac{3(1-\alpha_{v})\rho_{v}}{R}\sqrt{\frac{p-p_{v}}{\rho_{l}}},$
$\displaystyle p$ $\displaystyle>p_{v}$ (82)
with empirical constants $C_{+}$ and $C_{-}$ as a starting point, based on the
previous work of Merkle et al.Merkle, Feng, and Buelow (1998) The new
development in the Heat Balance Model is an extension of the approximation of
the bubble dynamics given by the simplified RPE
$\frac{dR}{dt}=\sqrt{\frac{|p_{v}-p|}{\rho_{l}}}$ (83)
used in the expressions (81) and (82) to the case of cryogenic cavitating
flows. To this end, the authors consider the heat flux $q$ of a transiently
evolving bubble, which at a time $t$ is given by
$q=\frac{K\Delta T}{\sqrt{at}}$ (84)
along with the heat balance across the bubble’s interface, expressed as
$4q\pi R^{2}=\frac{4\pi\rho_{v}}{3}\frac{d}{dt}(R^{3}).$ (85)
In the expressions (84) and (85), $K=a\rho_{l}c_{p}$ is the thermal
conductivity, $a$ is the thermal diffusivity, $c_{p}$ is the specific heat at
constant pressure, $L$ is the latent heat, and $\Delta T=|T_{c}-T_{\infty}|$
is the temperature drop expressed via the local temperature $T_{c}$ and the
freestream temperature $T_{\infty}$. By combining (84) and (85), the authors
obtain the relation
$\frac{dR}{dt}=\frac{\rho_{l}c_{p}\sqrt{a}\Delta T}{\rho_{v}L\sqrt{t}},$ (86)
which is added to the original approximation (83) of the bubble dynamics,
yielding the modified expressions of the source terms given by
$\displaystyle\begin{aligned}
\dot{m}^{+}&=C_{+}\frac{3\alpha_{v}\rho_{v}}{R}\left(\sqrt{\frac{\tilde{p}_{v}-p}{\rho_{l}}}\right.\\\
&\hskip
5.38193pt+\left.\frac{\rho_{l}c_{p}\sqrt{a}\max\\{T_{\infty}-T_{c},0\\}}{\rho_{v}L\sqrt{t}}\right),\quad\end{aligned}p<\tilde{p}_{v},$
(87) $\displaystyle\begin{aligned}
\dot{m}^{-}&=C_{-}\frac{3(1-\alpha_{v})\rho_{v}}{R}\left(\sqrt{\frac{p-\tilde{p}_{v}}{\rho_{l}}}\right.\\\
&\hskip
5.38193pt+\left.\frac{\rho_{l}c_{p}\sqrt{a}\max\\{T_{c}-T_{\infty},0\\}}{\rho_{v}L\sqrt{t}}\right),\quad\end{aligned}p>\tilde{p}_{v},$
(88)
where the vapor pressure $\tilde{p}_{v}$ is obtained by modifying the local
vapor pressure $p_{v}(T_{l})$ at temperature $T_{l}$ in the same manner as
that of the Full Cavitation ModelSinghal _et al._ (2002) in order to account
for the local fluctuations in pressure induced by turbulence, i.e.
$\tilde{p}_{v}=p_{v}(T_{l})+\frac{0.39\rho_{m}k}{2}.$ (89)
The authors do not mention which flow discretization method was used. The
turbulence was modelled using a shear stress model by StreletsStrelets (2001).
The model was validated by simulating cavitating flow over an axisymmetric
ogive previously investigated experimentally by Sarósdy and AcostaSarósdy and
Acosta (1961) (e16), showing good agreement with their findings.
### III.18 The Population Balance Model
Li and CarricaLi and Carrica (2021) derive the source terms using the coupling
given by (6) and a simplified version of the RPE (5). The unique feature of
this model is the treatment of the vapor volume fraction $\alpha_{v}$, which
is calculated as
$\displaystyle\alpha_{v}=\sum_{g=1}^{G}\frac{m_{g}}{\rho_{g}}N_{g},$ (90)
where the distribution of bubbles in the cavitating flow is assumed to be
distributed among $G$ groups of bubbles, in which every bubble in a group $g$
is assumed to have a uniform mass $m_{g}$ and constant density $\rho_{g}$,
with $N_{g}$ denoting the bubble number density of group $g$. The total
interphase mass transfer rate is given by
$\displaystyle\begin{aligned}
\dot{m}&=\sqrt{\frac{32}{3}}\pi\rho_{v}R^{2}\frac{p_{v}-p}{\sqrt{\rho_{l}\max\\{|p_{v}-p|,\epsilon\\}}}\\\
&\hskip 12.91663pt\times(1-0.5\tanh(20(\alpha_{v}-0.7))),\end{aligned}$ (91)
where $\epsilon>0$ is a small positive number introduced to prevent division
by zero. In order to model the evolution of the bubble number densities of the
various groups, the authors consider a Boltzmann-like transport equation
$\frac{\partial f}{\partial
t}+\nabla\cdot(f\mathbf{u}_{l})+\frac{\partial(\dot{m}f)}{\partial
t}=\beta+\chi+S,$ (92)
where $f$ is the bubble size distribution function and $\beta$, $\chi$, and
$S$ are source terms due to bubble breakup, coalescence, and entrainment,
respectively; the authors here note that $f$, $\beta$, $\chi$, and $S$ are all
functions of the mass $m$. Using a multigroup approach to discretize the
transport equation into a series of $G$ transport equations for the bubble
number densities $N_{g}$, the authors aim to obtain expressions for the
corresponding source terms $\beta_{g}$, $\chi_{g}$, and $S_{g}$ by extending
previously developed models for these source terms in the case of
noncavitating flows to cavitating flows. The source term for coalescence
$\chi=\chi^{+}-\chi^{-}$ is expressed using an extension of the model
originally developed in Prince and BlanchPrince and Blanch (1990), in which
the production and destruction of bubbles of mass $m$ are given by
$\displaystyle\begin{aligned}
\chi^{+}(m)=\frac{1}{2}\int_{0}^{m}&T(m-m^{\prime},m^{\prime})C(m-m^{\prime},m^{\prime})\\\
&\times f(m-m^{\prime})f(m^{\prime})\,dm^{\prime},\end{aligned}$ (93)
$\displaystyle\begin{aligned}
\chi^{-}(m)=\int_{0}^{\infty}&T(m,m^{\prime})C(m,m^{\prime})\\\ &\times
f(m)f(m^{\prime})\,dm^{\prime}\end{aligned}$ (94)
where $T$ and $C$ are the bubble collision rate and the coalescence efficiency
rate between bubbles of mass $m$ and $m^{\prime}$, respectively, which are
modelled as
$\displaystyle\begin{aligned}
T(m,m^{\prime})&=\frac{(R_{m}+R_{m^{\prime}})^{2}}{1-\alpha_{v}}\Big{(}\pi\|\mathbf{u}_{m}-\mathbf{u}_{m^{\prime}}\|\\\
&+1.33(R_{m}+R_{m^{\prime}})\|\nabla\mathbf{u}_{m}\|\\\
&+\left.1.41(R_{m}^{2/3}+R_{m^{\prime}}^{2/3})^{1/2}+\varepsilon^{1/3}\right),\end{aligned}$
(95) $\displaystyle\begin{aligned}
C(m,m^{\prime})&=\exp\left(-\sqrt{\frac{(R_{m}R_{m^{\prime}})^{3}\rho_{l}}{(R_{m}+R_{m^{\prime}})^{3}8\sigma}}\right.\\\
&\hskip
8.61108pt\times\left.\frac{2\ln\left(\frac{h_{0}}{h_{f}}\right)(R_{m}+R_{m^{\prime}})}{2(\varepsilon(R_{m}+R_{m^{\prime}}))^{1/3}+\|\mathbf{u}_{r}\|}\right).\end{aligned}$
(96)
To develop a model for the source term due to bubble breakup, the authors
assume that bubbles may only break up into two smaller bubbles, then consider
two different effects that induce bubble breakup: breakups induced by
turbulence and breakup due to bubble fission when a bubble collapses. In both
cases, the production term $\beta^{+}$ and the destruction term $\beta^{-}$
are given by
$\displaystyle\beta^{+}(m)$
$\displaystyle=\int_{m}^{\infty}f(m^{\prime})h_{t}(m,m^{\prime})b(m^{\prime})\,\mathrm{d}m^{\prime},$
(97) $\displaystyle\beta^{-}(m)$ $\displaystyle=b(m)f(m),$ (98)
where $b(m)$ is the bubble breakup rate for a bubble with mass $m$ and
$h(m,m^{\prime})$ is the daughter size distribution for bubbles of mass $m$
given a bubble with mass $m^{\prime}$. To model bubble breakup induced by
turbulence $\beta_{t}=\beta_{t}^{+}-\beta_{t}^{-}$, the authors use the model
introduced in Lehr et al.Lehr, Millies, and Mewes (2002), in which the
production and destruction terms are given by
$\displaystyle\beta_{t}^{+}(m)$
$\displaystyle=\int_{m}^{\infty}f(m^{\prime})h_{t}(m,m^{\prime})b_{t}(m^{\prime})\,\mathrm{d}m^{\prime},$
(99) $\displaystyle\beta_{t}^{-}(m)$ $\displaystyle=b_{t}(m)f(m),$ (100)
where $b_{t}$ and $h_{t}$ represent the bubble breakup rate and the daughter
size distribution given by
$\displaystyle b_{t}(m)$
$\displaystyle=\frac{(D_{m}^{\ast})^{5/3}}{2\tau}\exp\left(-\frac{\sqrt{2}}{(D_{m}^{\ast})^{3}}\right),$
(101) $\displaystyle h_{t}(m)$
$\displaystyle=\frac{1}{m\sqrt{\pi}}\frac{\exp\left(-\frac{9}{4}\left(\ln(2^{2/5}D_{m}^{\ast})\right)^{2}\right)}{1+\mathrm{erf}\left(\frac{3}{2}\ln\left(2^{1/15}D_{m^{\prime}}^{\ast}\right)\right)};$
(102)
here, $D_{m}^{\ast}=2R(\rho_{l}/\sigma)^{3/5}\varepsilon^{2/5}$ is a
dimensionless bubble diameter,
$\tau=(\sigma/\rho_{l})^{2/5}\varepsilon^{-3/5}$ is a time scale, and
$\mathrm{erf}$ is the error function. The model for breakup induced by bubble
fission $\beta_{bf}=\beta_{bf}^{+}-\beta_{bf}^{-}$ is based on that of
BrennenBrennen (2002), which proposes a model for bubble fission under the
assumption that the bubbles remain spherical during collapse. In this model,
the daughter size distribution $h_{bf}$ and the bubble breakup rate $b_{bf}$
are given by
$\displaystyle h_{bf}(m,m^{\prime})$
$\displaystyle=n^{\prime}\delta(m-m^{\prime}/n^{\prime}),$ (103)
$\displaystyle b_{bf}(m)$
$\displaystyle=\frac{1.1}{R}\left(\frac{p_{tot}-p_{v}}{\rho_{l}}\right)^{1/2}H(n^{\prime}-2),$
(104)
where $H$ is the Heaviside function, $\delta$ is the Dirac delta function, and
$n^{\prime}$ is an integer describing the number of daughter bubbles. The
authors do not develop a new model for the source term due to entrainment and
instead refer to the models presented in Castro et al.Castro, Li, and Carrica
(2016) and Li et al.Li, Martin, and Carrica (2019) Li and CarricaLi and
Carrica (2021) implement their model in the general purpose CFD code REX,
using a variety of RANS and LES approaches for turbulence modelling along with
a finite differences scheme for discretizing the governing equations and a
level set approach for free surface modelling. Validation on twisted
hydrofoil, showing good agreement with both previous experimental data by
Foeth Foeth (2008) (e17) as well as simulations performed by Asnaghi et
alAsnaghi, Feymark, and Bensow (2017). The model was applied by Li and
CarricaLi and Carrica (2023) in a numerical study of the cavitating flow over
a backward facing step, obtaining predictions of the bubble number densities
that illustrate the locations where smaller and larger bubbles are
concentrated as well as the important factors in the formation of the shedding
cloud.
### III.19 The Euler-Lagrangian Multiscale Model
The Euler-Lagrangian Multiscale Model developed by Ghahramani et al.
Ghahramani, Ström, and Bensow (2021) is a state-of-the-art multiscale model,
being the culmination of a series of previous iterations of cavitation models
capable of accounting for a range of different effects on cavitation. As
detailed in Ghahramani et al.Ghahramani, Arabnejad, and Bensow (2018), the
model can be split into a combination of three schemes, the first two of which
are concerned with developing appropriate source terms for the liquid volume
fraction of the cavitating flow at the two distinct scales. The liquid volume
fraction at the macro-scale and the micro-scale are treated separately and
denoted by $\alpha$ and $\beta$, respectively. At each of the scales, the
homogeneous mixture hypothesis (2) expressed in terms of the liquid volume
fraction is assumed to hold, i.e. the mixture density $\rho_{m}$ and mixture
viscosity $\mu_{m}$ are given by
$\displaystyle\rho_{m}$ $\displaystyle=\alpha\rho_{l}+(1-\alpha)\rho_{v},$
$\displaystyle\mu_{m}$ $\displaystyle=\alpha\mu_{l}+(1-\alpha)\mu_{v},$
$\displaystyle\rho_{m}$ $\displaystyle=\beta\rho_{l}+(1-\beta)\rho_{v},$
$\displaystyle\mu_{m}$ $\displaystyle=\beta\mu_{l}+(1-\beta)\mu_{v}.$
1. 1.
At the macro-scale, the flow characteristics such as velocity and pressure
along with turbulence are modelled by using large-eddy simulation, and the
source term in the transport equation (LABEL:eq:vof) for $\alpha$ is modelled
using a modified version of the Bubble Density-Liquid Volume Coupling
ModelSchnerr and Sauer (2001) based on previous results from both Schenke and
TerwisgaSchenke and Terwisga (2017) as well as Ghahramani et al.Ghahramani,
Arabnejad, and Bensow (2019).
2. 2.
At the micro-scale, the vapor phase is tracked by tracking the size and
location of parcels of (spherical) vapor bubbles, where each parcel consists
of bubbles of similar radius, and the liquid volume fraction $\beta$ is
instead obtained by estimating the corresponding vapor volume fraction
$1-\beta$ on a cell-by-cell basis. This is done by determining the number of
bubbles $n_{i}$ of a parcel of bubbles with the same radius $R_{i}$ that
occupy a given cell with index $j$, estimating the volume fraction of cell $j$
occupied by parcel $i$, and finally summing up the volume contributions from
each of the parcels of bubbles $i=1,\dotsc,N_{b,j}$ occupying cell $j$. This
approach to estimating the liquid volume fraction necessitates tracking the
motion of each parcel of bubbles as expressed by their position
$\mathbf{x}_{b}$ and velocity $\mathbf{u}_{b}$ as well as the sizes of the
bubbles as expressed by their radii $R_{i}$. Given the mass $m_{b}$ of the
bubbles, the equations governing the bubbles’ motion is expressed in the
Lagrangian framework as
$\displaystyle\frac{d\mathbf{x}_{b}}{dt}$ $\displaystyle=\mathbf{u}_{b},$
(105) $\displaystyle m_{b}\frac{d\mathbf{u}_{b}}{dt}$
$\displaystyle=\mathbf{F}_{d}+\mathbf{F}_{l}+\mathbf{F}_{a}+\mathbf{F}_{p}+\mathbf{F}_{b}+\mathbf{F}_{g},$
(106)
where the various terms on the rhs. of (106) represent force components due to
sphere drag force, lift force, added mass, pressure gradient force, buoyancy
force, and gravity, respectively. The drag forces are expressed via a drag
coefficient $C_{D}$ first derived by Amsden et al.Amsden, O’Rourke, and Butler
(1989), and the lift forces are similarly expressed using a lift coefficient
$C_{l}$ first derived by MeiMei (1992). The evolution of the bubble radius is
governed by a modified RPE including the effects of surface tension and non-
condensable gas given by
$\displaystyle\begin{aligned}
\frac{1}{2}R\ddot{R}+\frac{17}{32}\dot{R}^{2}&=\frac{p_{v}-p_{2R}}{\rho_{l}}+\frac{p_{g0}}{\rho_{l}}\left(\frac{R_{0}}{R}\right)^{3k}\\\
&\hskip
12.91663pt-\frac{4\mu_{l}\dot{R}}{\rho_{l}R}-\frac{2\sigma}{\rho_{l}R},\end{aligned}$
(107)
where $p_{2R}$ denotes the surface-average pressure of the mixture over a
concentric sphere of radius $2R$ and replaces the freestream pressure
$p_{\infty}$ in the RPE. This modification was derived in Ghahramani et
al.Ghahramani, Arabnejad, and Bensow (2019) under the assumption that the
surface-averaged pressure $p_{2R}$ gives a better representation of the
behavior of the pressure field in the immediate vicinity of the bubble.
3. 3.
The transition scheme of the model determines if a cavity tracked in the
Eulerian macro-scale scheme should transition to the Lagrangian micro-scale
scheme or a cavity tracked in the Lagrangian micro-scale scheme should
transition to the Eulerian macro-scale scheme by considering the number of
computational cells used to represent the cavity; note here that a Lagrangian
cavity refers to a cloud of any amount of micro-scale bubbles. Two threshold
values on the number of cells $N_{EL}$ and $N_{LE}$ are used to form the
criteria for the transition schemes, chosen such that $N_{LE}>N_{EL}$. If the
number of cells used to represent a macro-scale cavity is less than $N_{EL}$,
it is transitioned to a micro-scale cavity; otherwise it is kept in the
Eulerian framework. In a similar manner, if the number of cells used to
represent a (cloud of) micro-scale bubble(s) exceeds $N_{LE}$, the (cloud of)
bubble(s) is transitioned to a macro-scale cavity in the Eulerian framework;
otherwise it is kept in the Lagrangian framework. In both cases, the
micro-/macro-scale cavities that satisfy the criteria are replaced by
corresponding macro-/micro-scale cavities that occupy the same amount of
volume in the fluid.
Furthermore, collisions between two distinct cavities as well as turbulence-
induced breakage of cavities are also modelled, along with corrections to the
mixture properties and mass transfer rates due to cavities transferring from
one scheme to the other. Collisions between micro-scale bubbles and macro-
scale cavities are modelled by absorbing the micro-scale bubble into the
macro-scale cavity, whilst collisions between micro-scale cavities are
modelled in two steps using a model introduced by Breuer and AllettoBreuer and
Alletto (2012) and further extended by Vallier Vallier (2013) to first detect
incidence of collisions based on the current trajectories of the cavities,
then determining whether the colliding cavities remain in contact for long
enough to coalesce based on characteristic time scales derived by Kamp et
al.Kamp _et al._ (2001) or bounce back from each other. Finally, the
turbulence-induced breakage of cavities is modelled using a criterion
introduced by Lau et al.Lau _et al._ (2014) and Hoppe and BreuerHoppe and
Breuer (2020). This criterion declares a bubble of diameter $d_{p}$ undergoes
breakage if its Weber number
$\text{We}=\frac{\rho_{l}\overline{(u_{i}^{\prime}u_{i}^{\prime})_{d_{p}}}d_{p}}{\sigma},$
(108)
exceeds the critical value $\text{We}=15.12$, where
$\overline{(u_{i}^{\prime}u_{i}^{\prime})_{d_{p}}}$ is the mean square
velocity difference over a distance equal to the diameter of the bubble; this
choice of critical value corresponds to assuming that only binary breakups
into two equally-sized daughter bubbles of diameter $d_{s}$ occur. According
to Hoppe and BreuerHoppe and Breuer (2020), this implies that the ratio
$\frac{d_{p}}{d_{s}}$ is equal to 1.26, and this relation can be used to
determine the corresponding bubble radius $R_{s}$ of the daughter bubbles.
Ghahramani et al. Ghahramani, Ström, and Bensow (2021) implement their model
by combining their micro-scale Lagrangian model and transition scheme with the
interPhaseChangeFoam solver implemented in the open software package OpenFOAM.
Validation against exp. data of periodic cavitating flow over a bluff body
reported by Ghahramani et al. Ghahramani _et al._ (2020) (e18), showing good
agreement. Brandner et al. Brandner, Venning, and Pearce (2022) performed
experimental investigations of nucleation effects on cavitation effects about
a sphere with the stated aim of providing a high-fidelity dataset for further
improvements of the micro-scale modelling employed in this model.
### III.20 The Stochastic Field Model
Due to the close connection between cavitation and turbulence demonstrated by
several authors O’Hern (1990); Brandner _et al._ (2010); Huang, Zhao, and
Wang (2014), it is natural to seek a stochastic description of cavitation, as
turbulence is by definition a stochastic phenomenon. These considerations have
lead to the development of stochastic cavitation models, which aim to
construct models for quantities such as the volume fraction using
probabilistic models. This approach involves formulating an appropriate
stochastic process that describes the process of cavitation as well as a
stochastic partial differential equation that governs the evolution of said
process, a similar procedure to the approach by deterministic cavitation
models such as TEMs. Once this model has been established, the cavitation
process can be simulated by using tools developed for stochastic partial
differential equations to obtain a realization of the desired stochastic
process. The most prominent stochastic model developed thus far is the
Stochastic Field Model due to Dumond et al. Dumond, Magagnato, and Class
(2013), which models the probability density function $f_{Y}$ of the vapor
mass fraction, denoted here by $Y$, by applying the stochastic field method
previously developed by Valiño Valiño (1998). Within this method, the pdf
$f_{Y}$ is approximated as a sum of stochastic fields $Y^{k}$ as follows:
$f_{Y}(y;\mathbf{x},t)\approx\frac{1}{N}\sum_{k=1}^{N}\delta(y-Y^{k}(\mathbf{x},t))$
(109)
These stochastic fields $Y^{k}$ represent possible realizations of the true
vapor mass fraction $Y$, and in the limit as $N\rightarrow\infty$, the
approximation (109) converges to the true pdf $f_{Y}$. For practical
applications, the authors recommend the value $N=8$ for a good compromise
between stability and efficiency based on similar models developed for
simulating combustion. Each field $Y^{k}$ is determined in practice by solving
its associated stochastic partial differential equation, which the authors
develop in the Itô calculus using the methods of GardinerGardiner (1983) as
$\begin{split}dY^{k}&=-u_{i}\frac{\partial Y^{k}}{\partial
x_{i}}dt+S(Y^{k})dt+\frac{\partial}{\partial
x_{i}}\left(D_{Y}^{\prime}\frac{\partial Y^{k}}{\partial x_{i}}\right)dt\\\
&\hskip 12.91663pt+\sqrt{2D_{Y}^{\prime}}\frac{\partial Y^{k}}{\partial
x_{i}}dW_{i}^{k}-\frac{Y^{k}-\langle Y\rangle}{2\tau_{Y}},\end{split}$ (110)
where $\langle Y\rangle=\frac{1}{N}\sum_{k=1}^{N}Y^{k}$ is the average of the
stochastic fields, $D_{Y}^{\prime}$ and $\tau_{Y}$ are respectively the
diffusivity coefficient and the turbulent relaxation time obtained from a
turbulence model, the $W_{i}^{k}$ are Wiener processes that are independent
for each $i$ and constant in space, meaning that their time derivatives
$dW_{i}^{k}$ are independent, normally distributed variables with zero mean
and unity variance and can thus be obtained from a random number generator,
and $S(Y^{k})$ is a source term. This source term $S(Y^{k})$ is split into two
terms $S(Y^{k})=S_{+}(Y^{k})-S_{-}(Y^{k})$, each of which is modelled using
the RPE (5) to express the evolution of the bubble radius
$R^{k}=\left(\frac{3\rho Y^{k}}{4\pi\rho_{g}n}\right)^{1/3}$ of the $n$
cavities per unit volume, where $n$ is defined using the vapor volume fraction
$\alpha_{v}=\frac{\rho\langle Y\rangle}{\rho_{g}}$ and a specified initial
number of nuclei $n_{0}$ as
$n(\alpha_{v})=\frac{n_{0}+1}{2}+\frac{n_{0}-1}{2}\tanh\left(5(\alpha-0.4)\right).$
(111)
Using the bubble radius $R^{k}$, the authors derive the following expressions
for the source terms $S_{+}(Y^{k})$ and $S_{-}(Y^{k})$:
$\displaystyle S_{+}(Y^{k})$ $\displaystyle=(36n\rho_{g}\pi)^{1/3}(\rho
Y^{k})^{2/3}$ $\displaystyle\hskip
8.61108pt\times\sqrt{\frac{2}{3\rho_{l}}\max\left(p_{v}(T)-p-\frac{4\sigma}{3R^{k}},0\right)},$
$\displaystyle S_{-}(Y^{k})$
$\displaystyle=\begin{cases}S_{-}^{1}(Y^{k})&Y^{k}\leq Y_{eq}\\\
S_{-}^{2}(Y^{k})&Y^{k}>Y_{eq}\end{cases},$
where
$\displaystyle S_{-}^{1}(Y^{k})$
$\displaystyle=\frac{\rho}{\tau_{nuc}}(Y_{eq}-Y^{k}),$ $\displaystyle
S_{-}^{2}(Y^{k})$ $\displaystyle=-C_{cond}(36n\rho_{g}\pi)^{1/3}(\rho
Y^{k})^{2/3}$ $\displaystyle\hskip
8.61108pt\times\sqrt{\frac{2}{3\rho_{l}}\max\left(p_{v}(T)-p,0\right)}$
and $\tau_{nuc}$ and $C_{cond}$ are both modelling constants that ensure
vaporous cavities do not become smaller than initial nuclei after collapse and
account for inertial effects on condensation, respectively. Furthermore, the
pressure of the mixture is obtained using the EOS derived by Okuda and
IkohagiOkuda and Ikohagi (1996)
$\rho_{m}=\frac{p(p+p_{c})}{K(1-f_{v})p(T+T_{c})+Rf_{v}(p+p_{c})T},$ (112)
where $p_{c}$, $T_{c}$, $K$, and $R$ are the pressure, temperature, liquid,
and gas constant of the fluid, respectively. The authors validate their model
on test cases concerning Venturi nozzles and fluidic diodes, obtaining results
that demonstrate their model’s capability of replicating the cavitating flow
in these cases, both quantitatively in terms of the predicted velocity
profiles and vapor mass fractions that agree with experimental data obtained
from Barre et al.Barre _et al._ (2009) and Stutz and ReboudStutz and Reboud
(2000) for the Venturi nozzle case, but also demonstrate detailed qualitative
behaviour for both forwards and backwards flow in the fluidic diode case.
Dumond et al. Dumond, Magagnato, and Class (2013) implemented their model in
the large eddy simulation framework SPARC, using an explicit fourth order
Runge-Kutta scheme for temporal discretization and the SWITCH central
difference scheme with artificial dissipation for spatial discretization. Chen
and Oevermann Chen and Oevermann (2018) applied the Stochastic Field Model to
simulate cavitating flow through a throttle inside a diesel injector,
obtaining results that agree with previous numerical results by Altimira and
FuchsAltimira and Fuchs (2015), who performed their simulations using the
Bubble Density-Liquid Volume Coupling ModelSchnerr and Sauer (2001) and
obtained results in agreement with experimental results previously reported by
Winklhofer et al.Winklhofer _et al._ (2001) (e19).
## IV Conclusions
This review of cavitation modelling was developed as a supplement to the
previous reviews in the literature, focusing on the physical implications of
the approaches and assumptions employed by the authors of the models. Through
analysis of the various proposed cavitation models, five commonly used
approaches to modelling cavitation were identified, namely bubble dynamics as
expressed via the Rayleigh-Plesset equation (RPE), direct simulation of mass
transfer via transport equations (TEM), identifying cavities through an
equation of state, relating fluid characteristics to thermodynamic variables
(EOS), directly tracking the interfaces between the vapor phase and the liquid
phase (ITM), and simulating multiple scales of cavities at once with separate
schemes (MUL). Figure 2 indicates how the different models are composed of the
different modelling approaches. Starting clockwise from the upper-left of
Figure 2, the highlighted models have been labelled from 1 to 20 starting with
models only belonging to a single category, followed by labelling models
belonging to two categories clockwise from the upper-left, and so forth. In
this representation, simple, fundamental models lie on the outside, while more
complex models, which involve several modelling approaches, are found towards
the centre.
Figure 2: The cavitation models highlighted in this review, categorized by the
approach employed by the model using the categories described in section II.1.
Kubota et al.’sKubota, Kato, and Yamaguchi (1992) Bubble Cluster Model is
frequently referred to as inspiration for RPE models. The Bubble Cluster Model
exclusively models the cavitation dynamics by developing an equation for the
motion of a cluster of bubbles. While Kubota et al.’s approach is appealing
from a physical perspective, the model development has obviously favored to
embed all phase transitions and bubble dynamics in the source term of a
transport equation of the vapor volume fraction. These resultant TEM-RPE
models pose the largest group of models, see Figure 2. Based on the preceding
analysis and categorization, some general conclusions can be drawn related to
the modelling approach:
* •
The current standard for cavitation modelling is set by the combination of TEM
and RPE approaches. The most widely applied models are proposed by authors
such as Kunz et al.Kunz _et al._ (2000), Schnerr and SauerSchnerr and Sauer
(2001), Singhal et al.Singhal _et al._ (2002), and Zwart et al.Zwart, Gerber,
and Belamri (2004), which all date back to around the turn of the millennium.
* •
These standard models are all formulated within the volume-of-fluid framework,
allowing them to be readily implemented into most CFD workflows, while being
computationally efficient. As such, they have been adopted both by a large
proportion of the scientific community studying cavitation, but also within
both open-source software packages such as OpenFOAM and commercial software
such as Ansys Fluent or STAR-CCM+. Despite almost all of these models being 20
years old, they continue to be the standard for most studies involving
cavitation modelling.
* •
EOS models incorporate thermodynamic phenomena in cavitating flow modelling.
In particular, the assumption of iso-thermal flow can be liberated. EOS models
are rather applied to high resolution cases, indicating that they might be
computationally more demanding.
* •
ITM models, on the other hand, allow for high-resolution simulations of the
cavity interface dynamics. These models are utilized for studying two-
dimensional problems only; a generalization to three dimensions seems to be
problematic. The application of ITMs is therefore limited to very specific
problems only.
* •
The MUL models developed more recently, e.g. the Euler-Lagrangian Multiscale
Model by Ghahramani et al.Ghahramani, Ström, and Bensow (2021), come closer to
the realization of tracking the growth and motion of bubbles of similar size,
in the fashion of Kubota’s original Bubble Cluster Model. For resolving larger
cavities or dense bubble clouds, they still resort to the TEM approach.
Furthermore, four effects commonly accounted for in physical modelling of
cavitation and implemented through modifications to the expressions for e.g.
mass transfer rates were also identified as another way of categorizing
cavitation models; these effects include empirical adjustment of the model to
obtain a better fit (EMP), including turbulent effects on the pressure
fluctuations or the deformation of cavities (TUR), accounting for changes in
the population balance of cavities due to e.g. breakup of cavities or
coalescence (POP), and allowing for the possible presence of non-condensable
gas in the fluid (NCG). Figure 3 illustrates the effects accounted for in the
construction of the highlighted models. Based on the previous analysis,
supported by Figure 3, the following conclusions can be drawn on the model
effects:
Figure 3: The cavitation models highlighted in this review, categorized by the
effects accounted for by the model using the categories described in section
II.2.
* •
The majority of the models account for turbulence in some manner. Many
different turbulence models have been employed in the selected models:
Reynolds-averaged approaches like the algebraic Baldwin-Lomax modelBaldwin and
Lomax (1978), the one-equation Spalart-Allmaras modelSpalart and Allmaras
(1992), the two-equation $k-\varepsilon$ models, and shear stress models are
prominently featured as the turbulence models of choice, but alternatives like
large eddy simulation have also been employed successfully. No highlighted
model has employed a turbulence model developed for mutliphase flows, opting
for the established turbulence models which were developed for single-phase
flows.
* •
All of the highlighted models feature adjustable parameters which reflect
hypotheses placed on the fluid in question, e.g. the use of a prescribed
uniform bubble number density or a characteristic time scale. The vast
majority also feature empirical parameters used for calibrating the model to a
specific instance of cavitating flow, much like turbulence models such as the
$k-\varepsilon$ model. Only the Interface Mass and Normal Momentum Model
proposed by Senocak and ShyySenocak and Shyy (2004a) features no such
empirical parameters, reflected in Figure 3 by its placement outside the
category EMP.
* •
The models contained in the category MUL featured in Figure 2 are the same
models contained in the category POP featured in Figure 3, highlighting the
fact that accounting for the effects on cavitation occurring at both the
largest and the smallest scales necessitates accounting for the bubble
population in the modelling approach. Furthermore, the majority of the models
contained in this category have been developed in recent years, after the
importance of accounting for the micro-scale effects was discovered.
* •
The category NCG contains less than a quarter of the highlighted models,
indicating that most authors did not consider the effect of non-condensable
gas relevant for their purposes and chose to neglect it.
## V Outlook
As noted in the introduction, there is currently no universal cavitation
model, and the existence of such a model is predicated on obtaining a deeper
understanding of other phenomena in fluid dynamics, including turbulence. To
this end, the categorization presented in this article will assist in future
investigations concerning the nature of cavitation as a phenomenon along with
identifying which approach may be most appropriate for constructing a
cavitation model given a system of interest. The categorization proposed in
this work is neither exhaustive nor complete; future extensions and
refinements of the categorization presented in this article are also expected
in tandem with new developments in the various fields where cavitation occurs,
both in terms of categorizing the modelling approach and the included effects.
Additional aspects of cavitation modelling not considered here such as models
for cavitation erosion will be the subject for future work and may entail an
extension of the proposed categorization.
Furthermore, the present discussion on cavitation models has been restricted
to the physical modelling involved in their formulation, and attention to the
software implementation and performance of cavitation models kept to a minimum
(necessarily). A future comparative study on the performance of a selection of
cavitation models when applied to test cases such as those highlighted in
Table 1 will enrich this discussion by showcasing the adaptability of the
models and quantifying accuracy and computational performance.
It has been shown that many cavitation models are tightly coupled to
concurrent turbulence models. However, the question of how the empirical
tuning parameters relate to turbulence in the system remains open. As such, a
sensitivity analysis of the empirical parameters against different turbulent
regimes would promote the understanding of this correlation. Furthermore,
since both cavitation and turbulence affect the macroscopic flow variables, it
remains an open question if these two effects must not necessarily be modelled
together. The findings of this sensitivity study may motivate the development
of a new model focused on accounting for the inter-dependency of turbulence as
thoroughly as possible.
The formation and distribution of vapor nuclei in cavitating flows is handled
via simplifying assumptions such as a uniform bubble number density per unit
volume in many cavitation models; however, the population balance approach
proposed by Li and CarricaLi and Carrica (2021), who noted the lack of
cavitation experiments reporting bubble size data in their concluding remarks
indicate that the subject is poorly understood. Other authors have performed
experimental studies with the explicit aim of remedying this lack of data,
e.g. the work of Brandner et al. Brandner, Venning, and Pearce (2022)
detailing the influence of nucleation on cavitation about a sphere. An
investigation into the stochastic properties of the vapor nuclei distributions
typical of cavitating flows based on such data may shed further light on this
subject and motivate further development of multiscale cavitation models.
The phenomenon of cavitation in non-Newtonian fluids has been documented by
Brujan and WilliamsBrujan and Williams (2006); Brujan (2009, 2011), but little
effort towards developing cavitation models specifically for non-Newtonian
fluids appears to have been made. A future study into modelling cavitation in
non-Newtonian fluids, potentially using the framework of fractional Navier-
Stokes equations as proposed by Zhou and PengZhou and Peng (2017), would
promote greater understanding of the nature of cavitation as a phenomenon.
The increasing popularity of artificial intelligence and machine learning as a
method for solving non-linear problems has led to multiple authors
investigating the possibility of applying these tools to the problem of
modelling cavitation. Some authors have developed methods that build upon
previously established cavitation models, such as Sikirica et al. Sikirica
_et al._ (2020), who proposed a workflow for calibrating empirical constants
of cavitation models for optimal performance using a random forest method, or
Ouyang et al. Ouyang _et al._ (2023), who applied a chain of physics-informed
neural networks implementing both the Navier-Stokes equations governing fluid
flow as well as a cavitation model of choice to provide better predictions of
the changes in pressure that induce changes in the (liquid) volume fraction.
Other approaches to both modelling cavitation and predicting the onset and
intensity of cavitation using artificial intelligence have also been
investigated; Xu et al. Xu, Cheng, and Ji (2021) proposed a method for
enhancing the performance of Reynolds-averaged Navier-Stokes methods applied
to modelling cavitating flows by improving the predicted values of the
turbulent eddy viscosities using a random forest method, whilst Sha et al. Sha
_et al._ (2022) proposed a multi-task framework for detecting and classifying
the intensity of cavitation via the acoustic signals emitted by the cavitating
system using a bespoke neural network. Despite the advances made within the
studies of both machine learning and cavitation in recent years, the
application of machine learning to cavitation modelling has mostly been
limited to optimizing previous approaches to modelling cavitation; a
completely novel approach to modelling cavitation enabled by machine learning
has not yet been discovered. Future work will investigate both existing
applications of machine learning to cavitation modelling in greater detail as
well as the possibility of devising a new approach to cavitation modelling
enabled by machine learning.
## Data Availability Statement
Data sharing is not applicable to this article as no new data were created or
analyzed in this study.
## Author Declarations
The authors have no conflicts to disclose.
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|
# QuBOBS, interactive objects and a visual representation
to explain quantum computing
Sophie Laplante IRIF, Université Paris Cité Loris Perez Musician and composer
Sylvie Tissot Software engineer, Anabole Lou Vettier Designer, louvettier.com
###### Abstract
We introduce a visual representation of qubits to assist in explaining quantum
computing to a broad audience. The representation follows from physical
devices that we developed to explain superposition, entanglement, measurement,
phases, interference, and quantum gates. We describe how this representation
can be used to explain teleportation and the Bennett-Brassard quantum key
agreement protocol.
## 1 Introduction
This project is the result of a collaboration between a computer scientist, a
designer, a software developer and a composer. Together, we are developing
tools to make quantum computing accessible to a wider audience, without
sacrificing (too much) mathematical correctness. In this paper, we describe
the mathematics behind the objects that we are developing. We assume that the
reader is familiar with the basics of quantum computing. Our goal is to
provide the reader with a representation that can be used for teaching at the
undergraduate and graduate level, to students with or without mathematical
background, as well as for outreach activities.
Most researchers in quantum information and algorithms agree that explaining
quantum computing to a wide audience without relying on mathematical machinery
is a difficult task. A recent article of Aaronson lays out the difficulties
particularly well [Aar21]. Many different approaches have been used to explain
quantum computing to a wide audience, and although there are far too many to
give an exhaustive survey here, we give a few examples.
For single qubit applications such as key agreement, Charles Bennett [Ben21]
and many others use double arrows to represent photon polarization. John
Preskill [Pre16] uses colored balls to represent 0 and 1. Qubits are
represented as boxes and gates with two doors. Putting in a qubit in the top
door and taking it out of the bottom door applies a change of basis. This
representation is used for single qubit protocols as well as to explain
entanglement.
Karl Svovil explains quantum cryptography using chocolate balls [Svo06] with a
0 or 1 symbol on each ball, in either green or in red to indicate the basis.
Special glasses (red and green) are used to explain measurement in the correct
basis.
The most complete representation we are aware of is proposed by Terry Rudolph
and coauthors [Rud17, ERB05, EB22]. This representation is intended to explain
quantum circuits and computation in full generality, without using
mathematical symbols. Qubits are presented as clouds of black and white balls,
in a proportion that corresponds to the probability of getting the
corresponding outcome when measuring the qubit. Blocks represent operations on
qubits. Using this representation, they explain several advanced applications,
such as Grover search [Gro96] and CHSH games [CHSH69].
For more advanced topics in quantum information, tensor networks have been
used as a graphical language to represent quantum states and processes [WBC15,
Bia19].
Our approach has been to construct simple physical devices to help the
audience develop a very concrete mental image of concepts that can be
difficult to grasp, such as randomness which is inherent to quantum computing,
and entanglement. Special care has gone towards distinguishing randomness and
truly quantum properties. Superposition is represented by two overlapping
paper disks. Measurement is represented by spinning a window over a disk and
observing the result through a small window. Correlation is made concrete by
having two windows spinning together as cog wheels. Once the physical
representation is clear, we move on to a slightly more abstract graphical
representation, on which we can apply various operations.
## 2 The QUBOBS representation
### 2.1 Qubits
Our main contribution is a simple, robust and surprisingly versatile
representation of qubits. A qubit $\alpha|0\rangle+\beta|1\rangle$ is
represented by a two-color disc. Blue represents $|0\rangle$ and orange
represents $|1\rangle$. The proportion of each color represents the
probability of measuring the corresponding outcome, hence the blue part takes
up an $|\alpha|^{2}$ fraction of the disc’s area (and circumference) and the
orange part takes up $|\beta|^{2}$.
Most elementary quantum algorithms use signed reals as amplitudes so to
complete the description, we use a negative sign to indicate when the phase is
-1.
Our qubit representation is made of colored paper discs. Two discs can be
interleaved to obtain superpositions with arbitrary amplitudes.
Figure 1: An object that represents a quantum state $|\phi\rangle$. The
squared magnitude of $\left\langle\phi|0\right\rangle$ is blue, and the
squared magnitude of $\left\langle\phi|1\right\rangle$ is orange. On the
device on the right, the window spins around the edge of the qubit, When it
stops, the color under the window is the outcome of the measurement.
We illustrate measurements with a small device that spins around the qubit.
When it stops, a small round window reveals one of the colors. This is the
outcome of the measurement. The probability of each outcome is the part of the
disc that is the corresponding color, which is exactly the probability of
measuring the qubit in the standard basis.
Taking the modulus square of the amplitude sacrifices the phase, but we find
that this is a useful first step towards understanding the inherent randomness
of quantum measurements. When we use our devices, we show how this is also how
we could represent a biased coin. Flipping a coin and observing the outcome is
similar to measuring a qubit, but as we point out, a coin flip is not a
quantum phenomenon. This allows us to highlight the fact that the phase is a
crucial part of a quantum state, and that without phases, we are only
describing probabilistic binary states, and probabilistic computation.
Most elementary quantum algorithms use only positive and negative reals and do
not require the complex numbers in their full generality. We introduce the
phase as a sign, which is sufficient for quantum computing.
We write $|\phi\rangle\mapsto(P,Q)$ to mean that $(P,Q)$ is our representation
of $|\phi\rangle$. Using this notation, a qubit
$\alpha|0\rangle+\beta|1\rangle$ with $\alpha,\beta\in\mathbb{R}$ is
represented by $(sgn(\alpha)|\alpha|^{2},sgn(\beta)|\beta|^{2})$, and
conversely, $(A,B)$ is a representation of the qubit
$sgn(A)\sqrt{|A|}|0\rangle+sgn(B)\sqrt{|B|}|1\rangle$. When several qubits are
involved, the angle of the disk becomes important, and we will assume that,
unless indicated otherwise, the blue part starts at the top going clockwise.
### 2.2 Entanglement and partial measurements
The surprising expressiveness of our representation of qubits appears when we
consider two qubits. Separable states can be represented by two observation
windows spinning independently over two qubits. However, things become
interesting when we link the two spinning windows with a cog. so that they
spin synchronously. This allows us to represent entanglement as we explain
now.
#### 2.2.1 Entanglement
Figure 2: Two ways of representing two entangled qubits. On the left and in
the middle, a cog makes both windows spin synchronously. On the right, the
qubits are stacked and a single window lets us observe both qubits. The state
represented on the right on both devices is
$\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$.
Consider an arbitrary two-qubit pure state
$|\phi\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|11\rangle+\delta|10\rangle$.
(Notice that we have used a Gray code to order the basis elements.) If the
first qubit is measured with a standard measurement, the outcome is 0 with
probability $|\alpha|^{2}+|\beta|^{2}$ and 1 with probability
$|\gamma|^{2}+|\delta|^{2}$. Similarly, if the second qubit is measured, the
outcome is $0$ with probability $|\alpha|^{2}+|\delta|^{2}$ and 1 with
probability $|\beta|^{2}+|\gamma|^{2}$. This allows us to represent the two-
qubit state with one disc that has $|\alpha|^{2}+|\beta|^{2}$ fraction of blue
and $|\gamma|^{2}+|\delta|^{2}$ of orange, and the second disc with
$|\alpha|^{2}+|\delta|^{2}$ blue and $|\beta|^{2}+|\gamma|^{2}$ orange. They
key observation is that when the observation windows are lined up (say they
both start from the top of the disk and turn synchronously) it is always
possible to line up the two discs so that the probability of getting 00 when
both qubits are measured is $|\alpha|^{2}$, and similarly for 01, 11, and 10,
as illustrated below. To do this, line up the first qubit so that it is blue
starting from the top going clockwise, and line up the second qubit so that it
is orange starting at an angle $\theta=|\alpha|^{2}\cdot 2\pi$ from the top,
going clockwise. Then starting from the top and going clockwise, the two
qubits are blue for $|\alpha|^{2}\cdot 2\pi$, then the first qubit is blue and
the second is orange for $|\beta|^{2}\cdot 2\pi$ and so on.
We use a second device to visualize the four areas corresponding to each of
the outcome pairs 00,01, 11, and 10 (Figure 2, on the right). We stack the two
qubits on top of each other, with a single window going around the perimeter.
Going back and forth between these two representations, side by side and
stacked, gives us two complementary ways of seeing the effects of
entanglement.
When explaining protocols or algorithms, we set aside the physical devices
with the windows spinning around the perimeter, and work with the diagrams
illustrated in Figure 3.
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
Figure 3: Two representations of the same pair of entangled qubits,
$\frac{1}{\sqrt{3}}|00\rangle+\frac{1}{\sqrt{6}}|01\rangle+\frac{1}{\sqrt{3}}|11\rangle+\frac{1}{\sqrt{6}}|10\rangle$.
On top they are represented side by side like in the device with two windows
spinning synchronously. On the bottom the qubits are stacked the same way as
in the device with a single window.
Separable states can also be represented with synchronized observation
windows. For example, the state
$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$
can be represented with two half blue, half orange disks placed
perpendicularly.
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
Figure 4: Our representation of the separable state
$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$
This works well for two qubits. From three qubits onward we can no longer
represent arbitrary states with discs with two colors arranged into two
contiguous colored slices. However, it turns out that for many applications,
two qubits suffice, and furthermore, presenting qubits using more than two
color slices occurs naturally as a result of applying quantum gates.
#### 2.2.2 Partial measurements
Partial measurements of a bipartite state and the residual state after a
measurement is made is generally viewed as difficult to explain without
appealing to a formal projection operator. Our representation is surprisingly
successful in conveying the math without writing out any equations.
Returning to our two-qubit state
$|\phi\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|11\rangle+\delta|10\rangle$,
let us make a measurement on the first qubit. The blue and orange parts of the
first qubit split the second qubits into two areas. In Figure 5 we give an
example of how we can visualize the residual state of the second qubit given
that the first qubit has been measured. If the first qubit came out blue, then
the observation window could have fallen in any place where the first qubit
was blue. This part of the disk, on both qubits, since the measurement windows
are linked by a cog, is the residual state. The rest of the disks are no
longer accessible. Having this visual representation in mind makes it easier
for students to conceptualize what the projection does and what the
renormalization of the state accomplishes (not to mention what the norm
actually represents).
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
Measure 0 Measure 1
Figure 5: Visualizing the effect of a partial measurement. On the top, the
qubit before measuring the fist (outer) qubit. Below, the residual state when
the outcome is blue (on the left) or orange (on the right).
### 2.3 Phases and interference
In our basic representation, we restrict ourselves to +/-1 phases. In the
simpler quantum applications, such as key exchange or Deutsch’s and Deutsch-
Jozsa’s algorithm, when interference occurs, opposing amplitudes cancel
exactly. By this we mean that in the course of the computation, we obtain
states of the form
$\alpha|0\rangle-\alpha^{\prime}|0\rangle+\beta|1\rangle+\beta^{\prime}|1\rangle$
where $\alpha=\alpha^{\prime}$. This is a great stroke of luck since
$|\alpha-\alpha^{\prime}|^{2}=|\alpha|^{2}-|\alpha^{\prime}|^{2}$ = 0, and the
slices with opposing phases cancel exactly. Obviously this does not hold in
general.111The incorrect identity, $(x+y)^{2}=x^{2}+y^{2}$ has been coined
“freshman’s dream” following Saunders Mac Lane [Lan40] who attributes it to
Stephen Kleene. However, the fact that this happens to be correct in some key
cases of interest allows us to convey how interference plays a role in quantum
computation, in a meaningful and (almost) mathematically correct way. To more
mathematically advanced audiences, we always point out that we are taking a
sinificant shortcut here, and in a classroom situation we can return to the
more conventional mathematical representation, and address other instances of
interference later on, when discussing Grover’s algorithm, or quantum
strategies for Bell inequality violations for instance.
$|0\rangle$
$|1\rangle$
Figure 6: The qubit $\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$ represented as
$(1/2,-1/2)$.
When explaining interference, we have found that referring to colors (blue,
orange) makes it easier to grasp what is meant by interference than when using
numerical values 0 and 1. We have often been asked what it means for +0 and -0
to cancel, since $+0-0=0$. This problem of confusing the state with its label
does not occur when talking about colors. It also offers an opportunity to
discuss the fact that blue and orange can be labels for many different
propoerties, such as spin, or polarization, as long as these properties are
fully distinguishible.
To help explain phases and interference, we appeal to the audience’s knowledge
that light behaves like a wave. Positive and negative phases can be
illustrated with sine waves that can be shifted by half a wavelength.222For
more advanced audiences, one could generalize to arbitrary shifts between 0
and half of the wavelength but we have not found this to be necessary to
explain basic quantum algorithms and protocols. For younger audiences, we have
illustrated this with two people pulling and pushing on two ends of a table in
a synchronized cyclical motion. If both have the same phase, one pushing as
the other pulls, then the amplitudes add up. If they have opposite phases,
they cancel each other and the table doesn’t move. Pushing on a blue table
does not interfere with pushing on an orange table. Many audience members are
familiar with this phenomenon in the realm of sound waves and have pointed out
that this is how noise-canceling headphones work.
### 2.4 Single qubit gates and linearity
We illustrate quantum gates by flipping over a disk to reveal the effect of a
gate on each of the basis states.333Many of the gates we consider are self-
adjoint, and flipping over twice gets us back to the initial state.
In order to explain simple quantum circuits, it would be nice to apply quantum
gates on our representation of qubits by appealing to linearity. This works
well in some cases, like the X gate, or the Z gate.
However, we have to proceed with caution with other gates, such as the
Hadamard gate. Hadamard applied to zero is the state composed of half blue and
half orange, which we denote by $(1/2,1/2)$. Similarly Hadamard applied to
orange is $(1/2,-1/2)$. If we apply Hadamard on $(1/2,1/2)$, we apply it to
both parts. The blue half becomes $(1/2,1/2)$ and the orange half becomes
$(1/2,-1/2)$, so we obtain four parts, which we will write
$(1/4,1/4,1/4,-1/4)$. Up to this stage, our representation accurately reflects
the effect of the gate (up to normalization). What can get us into trouble is
interference.
In some cases we can add the parts of same sign and let parts of opposite
signs cancel. In the case of Hadamard, we are tempted to say that the orange
parts with opposite signs cancel, leaving an all-blue disk. In this case, this
correctly mimics what happens to the amplitudes, and it allows us to convey a
message that is essentially correct: the fact that applying Hadamard twice to
basis states is identity, and this occurs because of interference. It is also
all that is needed to explain how and why some simple applications, such as
the BB84 key exchange protocol, or Deutsch’s algorithm, work.
In general, however, we refrain from adding together and rearranging the
slices in our representation. The first reason is that positive and negative
interference does not usually behave well in our representation. But there is
a second very imoirtant reason which is that when multiple qubits are
entangled, moving parts around would affect the entanglement between the
qubits. We explain this is more detail in Section 2.5.
We consider the effect of single qubit gates $G$, with
$G|0\rangle=a|0\rangle+b|1\rangle$ and $G|1\rangle=c|0\rangle+d|1\rangle$ We
will say that in our representation, $G(P,Q)=(A,B,C,D)$ where
$A=P\cdot\mathrm{sgn}(a)|a|^{2}\quad B=P\cdot\mathrm{sgn}(b)|b|^{2}\quad
C=Q\cdot\mathrm{sgn}(c)|c|^{2}\quad D=Q\cdot\mathrm{sgn}(d)|d|^{2}$
In the schematic representation, we obtain four parts, starting from the top
going clockwise with blue, and alternating blue and orange parts.
$|0\rangle$
$|1\rangle$
$\stackrel{{\scriptstyle H}}{{\longrightarrow}}$
$|0\rangle$
$|1\rangle$
$|0\rangle$
$|1\rangle$
=
$|0\rangle$
$|1\rangle$
Figure 7: Applying Hadamard to $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$
behaves well since the positive and negative parts cancel completely.
$|0\rangle$
$|1\rangle$
$\stackrel{{\scriptstyle H}}{{\longrightarrow}}$
$|0\rangle$
$|1\rangle$
$|0\rangle$
$|1\rangle$
= ???
Figure 8: Hadamard applied to other states does not behave well. Applying
Hadamard to $\frac{\sqrt{2}}{\sqrt{3}}|0\rangle+\frac{1}{\sqrt{3}}|1\rangle$,
we get
$\frac{1}{\sqrt{3}}|0\rangle+\frac{1}{\sqrt{3}}|1\rangle+\frac{{1}}{\sqrt{6}}|0\rangle-\frac{{1}}{\sqrt{6}}|1\rangle$.
Our representation suggests that there should be $\frac{1}{3}-\frac{1}{6}$
orange (1) left, whereas the correct fraction is
$(\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{6}})^{2}$.
Henceforth we will use the notation $(A,B,C,D)$ to describe a disc with four
areas, whose colors are blue, orange, blue, orange. For example, if Hadamard
is applied to $(1/2,1/2)$, we get $(1/4,1/4,1/4,-1/4)$. Unless the state is
entangled with another, we start from the top with a blue slice of size $A$,
and so forth, going clockwise.
### 2.5 Applying gates on two qubits
Applying two-qubit gates works similarly to the one-qubit case.
Recall that two qubits in our representation, $(P,Q)$ and
$(P^{\prime},Q^{\prime})$, do not fully determine the two-qubit state, since
the angle at which they are placed determines how they are correlated. If the
first qubit is blue going clockwise from the top, and the second disk is
orange starting from an angle of $\theta\cdot 2\pi$, going clockwise, then
(ignoring for the sake of simplicity any possible phases) the state
represented is
$|\psi\rangle=\sqrt{\theta}|00\rangle+\sqrt{P{-}\theta}|01\rangle+\sqrt{1{-}P{-}P^{\prime}{+}\theta}|11\rangle+\sqrt{P^{\prime}{-}\theta}|10\rangle.$
We will use the notation
$|\psi\rangle\mapsto(P,Q)\theta(P^{\prime},Q^{\prime})$.
When we apply a single qubit gate to the first qubit of a two-qubit state, we
apply it while preserving the angle between the two qubits. To ensure that
entanglement is preserved, we apply it to all four areas corresponding to the
basis elements $|00\rangle,|01\rangle,|11\rangle,|10\rangle$. These four areas
are easily seen in the stacked representation. Each of the four areas will be
further subdivided by gates such as Hadamard.
Similarly, when applying a two-qubit gate, we apply it to all four areas.
Control gates are particularly easy to visualize. (See Figure 11.) In the area
where the control bit is 0, we do nothing, and in the area where the control
bit is 1, we apply the gate as usual.
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
Figure 9: The CNOT gate is applied to the qubit on the left. The control bit
is the outer bit. When the outer bit is blue, the inner bit remains the same.
When the outer bit is orange, the colors of the inner bit are flipped.
## 3 Applications
### 3.1 Key exchange
We have used our representation to explain the BB84 key agreement protocol
[BB84] to high school students and members of the general public. We
distribute to participants a kit containing
* •
Qubits with $|0\rangle$ (blue) on one side and $H|0\rangle$ on the other,
* •
Qubits with $|1\rangle$ (orange) on one side and $H|1\rangle$ on the other,
* •
Two dice, one to pick a bit, and one to apply, or not, Hadamard.
* •
Eight numbered black envelopes in which qubits can be placed and observed on
either side using small cutout flaps.
* •
A grid to mark the results: what value was selected, what basis was used, and
what measurement was made, and what the outcome was.
Figure 10: Our BB84 kit comes with eight envelopes and two dice. Alice picks 8
qubits and places each them in a numbered envelope. Qubits can be measured by
opening a small flap in the envelope. Flipping a qubit over changes the basis.
Flipping over the envelope changes the measurement basis.
Participants playing Alice can prepare qubits in the standard or diagonal
basis by placing them in envelopes with the correct side facing up. A second
participant playing Bob can measure a qubit in the standard basis by opening a
flap on the front of the envelope, or in the Hadamard basis by turning over
the envelope (and hence the qubit inside) and opening a flap on the back of
the envelope. All the results are marked on a grid. Participants can see for
themselves that any time the basis chosen for the qubit and the basis for the
measurement coincide, the outcome of the measurement made by Bob equals the
bit that was chosen by Alice. We can then explain that any discrepancy would
be due to an observation made by a third party.
### 3.2 Teleportation
We like to explain quantum teleportation [BBC+93] in two steps. First we give
a simplified and fully classical protocol to teleport a coin with an arbitraty
bias. This protocol only uses a CNOT gate, and Alice measures only one of her
qubits and sends the outcome to Bob. This is enough for Bob to obtain the bias
of Alice’s coin.
Splitting the protocol into two steps, a fully classical step with CNOT and
the truly quantum part with Hadamard demystifies how and why the protocol
works, and allows us to clearly identify what part of the magic is due to
classical correlations and what part is attributable to quantum.
#### 3.2.1 Teleporting a biased coin
At the start of the protocol, Alice has a biased coin, and Alice and Bob share
one pair of perfectly correlated unbiased coins. This is similar to Alice
having a qubit without any phase and the two players sharing an EPR pair.
In Figure 11, the first stacked pair of qubits belongs to Alice and is
composed, on the inner disk, of the qubit (biased coin) that Alice wishes to
transmit, and on the outer disk, half of an EPR pair (or correlated coins).
The rightmost qubit is Bob’s half of the EPR pair.
In Step 2 of the simplified protocol, Alice applies a CNOT operation on her
qubits, using the inner disk as the control qubit.
In Step 3, Alice measures her outer qubit. Since the qubits are entangled
(correlated), this also affects Bob’s part of the EPR pair. In Step 3a, if
Alice measures 0 (blue) on the outer disk, Bob gets the correct qubit (without
the phase). In Step 3b, if Alice measures 1 (orange) on the outer disk, Bob
needs to apply a correction to reverse the values.
Alice Bob
Step 1.
Setup
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
$|0\rangle$
$|1\rangle$
Step 2.
CNOT
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
$|0\rangle$
$|1\rangle$
Step 3a.
Measure 0
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
$|0\rangle$
$|1\rangle$
Step 3b.
Measure 1
$|00\rangle$
$|01\rangle$
$|11\rangle$
$|10\rangle$
$|0\rangle$
$|1\rangle$
Figure 11: Classical teleportation of a qubit without a phase. Alice’s two
qubits (or coins) are represented on the left, and Bob’s qubit (or coin) is on
the right.
When the measurement outcome is 0 (blue), the residual state on Bob’s side has
the correct bias. When the outcome is 1, the bias is correct provided he flips
the colors. Notice that we have given a fully classical protocol that
transmits one coin flip with an arbitrary bias, using a pair of correlated
unbiased coins plus the transmission of an unbiased coin flip.
#### 3.2.2 Full teleportation protocol
In order for Alice to transmit the phase of her qubit, she adds the truly
quantum part of the protocol by applying Hadamard to her inner qubit. This
will further split all the slices into two as we illustrate in Figure 12.
Alice Bob
Step 3’.
Hadamard
$|00\rangle$
$|01\rangle$
\- $|11\rangle$
$|10\rangle$
$|00\rangle$
\- $|01\rangle$
$|11\rangle$
$|10\rangle$
$|0\rangle$
$|1\rangle$
Figure 12: Applying the Hadamard gate to the inner qubit before measuring.
Measure 00
$|0\rangle$
$|1\rangle$
Measure 01
$|0\rangle$
\- $|1\rangle$
Measure 10
$|0\rangle$
$|1\rangle$
Measure 11
$|0\rangle$
\- $|1\rangle$
Figure 13: The residual state on Bob’s side for the four possible measurement
outcomes.
The four possible measurements oucomes and corresponding residual state on
Bob’s side are given in Figure 13. Here we can see that in two of the cases
(on top), the proportions are correct, and in the remaining two cases (on the
bottom), the colors need to be flipped. Similarly, we can see that in two of
the cases (on the left), the signs are correct and in the remaining two (on
the right), the signs need to be flipped.
## 4 Conclusions
We have presented new physical devices and a visual representation of qubits
that can be used for teaching or for outreach activities. We have found it
particularly helpful in conveying concepts such as entanglement and
interference, concepts that key to understanding quantum algorithms, but can
be difficult to grasp at an intuitive level, especially for audiences who are
not accustomed to using mathematical formalism.
We have used our devices and representation in the classroom in addition to
the usual mathematical notation and have found that it lends itself
particularly well to walking through quantum circuits to see how and why they
work.
Our representation has its drawbacks, the main one being that in general,
applying linearity in the usual way on the squared-amplitudes vector does not
yield the same result as on the amplitudes. However, by being cautious with
interference, and checking whether a particular operation behaves well on the
states that they are being applied to, it does work in sufficiently many cases
so as to allow us to explain some basic algorithms and protocols.
In forthcoming work, we plan to extend our applications to include more
algorithms as well as nonlocality with the CHSH game. Also in the works is an
interactive interface and a musical representation of qubits so that it can be
more widely accessible. More information on our project can be found on the
website https://qubobs.irif.fr.
## Acknowledgements
We would like to thank the students and participants in our outreach
activities for their feedback. We thank Maris Ozols for references to tensor
diagrams and Serge Massar for references to some of the other representations.
We are grateful to the many colleagues who served as sounding boards
throughout the different versions of this work.
This project was funded by IRIF, by a grant from IDEX Université Paris Cité,
and the ANR grant SAPS-RA-MCS QUBOBS.
## References
* [Aar21] Scott Aaronson. What makes quantum computing so hard to explain? Quanta Magazine, June 21, 2021, 2021.
* [BB84] Charles H. Bennett and Gilles Brassard. Quantum cryptography: Public key distribution and coin tossing. Theoretical Computer Science, 560:7–11, 1984.
* [BBC+93] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Physical Review Letters, 70(13):1895–1899, 1993.
* [Ben21] Charles H. Bennett. Quantum information’s revolutionary origins. https://www.youtube.com/watch?v=B5BUhzBlO-U, July 2021.
* [Bia19] Jacob Biamonte. Lectures on quantum tensor networks. Technical Report 1912.10049 (quant-ph), arXiv, 2019.
* [CHSH69] J.F. Clauser, M.A. Horne, A. Shimony, and R.A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23(15):880–884, 1969.
* [EB22] Sophia E. Economou and Edwin Barnes. Hello quantum world! a rigorous but accessible first-year university course in quantum information science. Technical Report arXiv:2210.02868, ArXiv, 2022.
* [ERB05] Sophia E. Economou, Terry Rudolph, and Edwin Barnes. Teaching quantum information science to high-school and early undergraduate students. Technical Report quant-ph 2005.07874, ArXiv, 2005.
* [Gro96] Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings, 28th Annual ACM Symposium on the Theory of Computing, pages 212–219, 1996.
* [Lan40] Saunders Mac Lane. Modular fields. The American Mathematical Monthly, 47:67–84, 1940.
* [Pre16] John Preskill. Quantum computing and the entanglement frontier, 2016 Leigh Page Prize lecture series, hosted by Yale Department of Physics and Yale Quantum Institute. https://www.youtube.com/watch?v=bPNlWTPLeqo, 2016.
* [Rud17] Terry Rudolph. Q is for Quantum. self-published, 2017.
* [Svo06] Karl Svovil. Staging quantum cryptography with chocolate balls. American Journal of Physics, 74(800), 2006.
* [WBC15] Christopher J. Wood, Jacob D. Biamonte, and David G. Cory. Tensor networks and graphical calculus for open quantum systems. Quantum Information and Computation, 15(9 and 10):0759–0811, 2015\.
|
# The Interplay between Disorder, Local Relaxation and Collective Behaviors
for an ensemble of emitters outside vs inside cavity
Zeyu Zhou Department of Chemistry, University of Pennsylvania, 231 South 34th
Street, Philadelphia, Pennsylvania 19104, United States Hsing-Ta Chen
Department of Chemistry, University of Pennsylvania, 231 South 34th Street,
Philadelphia, Pennsylvania 19104, United States Department of Chemistry and
Biochemistry, University of Notre Dame, 251 Nieuwland Science Hall, Notre
Dame, Indiana 46556, USA Joseph E. Subotnik Department of Chemistry,
University of Pennsylvania, 231 South 34th Street, Philadelphia, Pennsylvania
19104, United States Abraham Nitzan Department of Chemistry, University of
Pennsylvania, 231 South 34th Street, Philadelphia, Pennsylvania 19104, United
States Department of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel
###### Abstract
The interplay between collective optical response and molecular static and
dynamic disorder is studied using simple effective Hamiltonians for an
ensemble of two-level emitters inside and outside a single-mode cavity. We
model environmental disorder by randomly modulating the molecular transition
frequencies and the coupling between the emitters and the electromagnetic
field. We also consider effects of intermolecular interactions and
orientational disorder. We investigate how these effects lead to new features
in the steady state absorption (outside the cavity), transmission spectra
(inside the cavity) and the yield of local molecular processes such as a
unimolecular reaction. Outside the cavity, the collective behavior is
manifested in the linewidth of the steady state absorption, the emission
spectrum, and the local chemical yield. Inside the cavity, however, the
collective behavior primarily determines the Rabi splitting. Effects of
intermolecular interactions under orientational disorder are also studied. For
the most part, for all types of disorder, if we increase disorder, we find a
reduction in the collective nature of the molecular response (smaller
effective $N$), and therefore, the Rabi splitting contraction occurs with
orientational disorder. Moreover, we find that static disorder is more
destructive to collective behavior than dynamic disorder.
## I Introduction
Understanding the interplay between collective molecular optical response and
molecular disorder has drawn a great deal of interest in recent years.
Collective optical response could be seen in the time domain, for instance,
Dicke superradiance emission[1, 2, 3], superfluorescence[4, 5],
superabsorption[6] have been discussed extensively in the literature, or in
the frequency domain, e.g. in optical cavities where collective response is
manifested in the observed Rabi splitting that characterizes the strong light-
matter coupling regime in such a system. Near metal interface and in optical
cavities, the response of an individual molecule is affected also by the
confined character of the cavity as demonstrated by the Purcell effect as well
as different realizations of surface enhanced spectroscopies. [7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] Collective
response is characterized by a non-trivial dependence of experimental
observables on the number $N$ of involved molecules; for example, one finds a
rate linear in $N$ for superradiance emission and a Rabi splitting $\sqrt{N}$
in the optical response of molecular optical cavities. The dependence of these
behaviors on local disorder has been the subject of several recent
studies.[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53]
In this paper, we discuss the effect of different types of disorder on the
collective optical response of molecular systems in and out of the cavity
using two simple models: site energy disorder and coupling disorder. We also
discuss how the interplay between collective optical response (outside vs
inside a single-mode cavity) and local molecular processes might affect the
yield of molecular photoprocesses such as photochemical reactions.
## II Model and Calculation Method
### II.1 A molecular ensemble in the single exciton subspace
Consider an ensemble of 2-level molecules whose spatial extent is assumed for
shorter than the wavelength of its resonant absorption, subjected to pumping
by a near resonant classical field and further exhibiting collective emission.
In addition, each molecular excited state can decay due to interaction with
its local environment or a local chemical process. The time evolution of this
system can be described by the following Hamiltonian:
$\displaystyle\hat{H}=$
$\displaystyle\hat{H}_{0}+\hat{V}_{\text{pump}}-i\Gamma_{rad}\hat{Q}-i\Gamma_{loc}\hat{C}$
(1) $\displaystyle\hat{H}_{0}=$ $\displaystyle
E_{0}\left|0\right\rangle\left\langle
0\right|+\sum_{m}E_{m}\left|m\right\rangle\left\langle m\right|$ (2)
$\displaystyle\hat{V}_{\text{pump}}=$ $\displaystyle\sum_{m}V_{0m}\cos(\omega
t)\left|0\right\rangle\left\langle m\right|+V_{m0}\cos(\omega
t)\left|m\right\rangle\left\langle 0\right|$ (3) $\displaystyle\hat{Q}=$
$\displaystyle\sum_{m,m^{\prime}}\left|m\right\rangle\left\langle
m^{\prime}\right|$ (4) $\displaystyle\hat{C}=$
$\displaystyle\sum_{m}\left|m\right\rangle\left\langle m\right|$ (5)
In eq (1)-(5), the molecular states are
$\left\\{\left|0\right\rangle,\left|m\right\rangle\right\\}$:
$\left|0\right\rangle=\prod_{j}\left|g_{j}\right\rangle$ is the total ground
state and $\left|m\right\rangle=\left|x_{m}\right\rangle\prod_{j\neq
m}\left|g_{j}\right\rangle$ is a singly-excited molecular state. Note that we
are working in a single-exciton basis, so that our model can describe only
weak excitations where many exciton correlations can be disregarded. The
radiative emission of the molecules is treated implicitly using the effective
non-Hermitian operator $-i\Gamma_{rad}\hat{Q}$ that arises when one considers
one idealized wide band radiative continuum (with density of state
$\rho_{dos}$) coupled identically to all molecular emitters
($V_{j}=V_{rad},\forall j$), leading to the purely imaginary self-energy
$\displaystyle\Gamma_{rad}=2\pi\rho_{dos}|V_{rad}|^{2}$ (6)
If we denote the wavefunction of the molecular system by $c_{m}(t)$, then
under radiative pumping of the molecules, the outgoing radiative flux at any
given time $t$ is
$\displaystyle
J_{rad}(t)=\sum_{m,m^{\prime}}\Gamma_{rad}c_{m}(t)c^{*}_{m^{\prime}}(t)$ (7)
Finally, $-i\Gamma_{loc}\hat{C}$ represents decay through a local channel
involving a single molecule process. In contrast to the radiative decay, this
channel describes a process defined by the state of each individual molecule
(and does not depend on intermolecular coherence). The outgoing local flux at
any given time $t$ is
$\displaystyle J_{loc}(t)=\sum_{m}\Gamma_{loc}|c_{m}(t)|^{2}$ (8)
Note that in the Hamiltonian (eq 1), pumping is done by a continuous wave (CW)
field at the driving frequency $\omega$. Therefore, if we plot total steady
state outgoing flux ($J_{rad}+J_{loc}$) as a function of this frequency
$\omega$, we obtain the steady state absorption spectrum.
### II.2 A molecular ensemble in an optical cavity
To describe the same molecular ensemble inside a single-mode optical cavity,
we modify the Hamiltonian (eq (1)-(5)) in several ways.[54] First, the
external pumping does not directly couple to the emitters, but instead to the
cavity mode $\left|v\right\rangle$. Second, the molecules do not couple
directly to the far EM field, and are assumed to couple only to the cavity
mode. Hence, the term $-i\Gamma_{rad}\hat{Q}$ is replaced by the coupling term
(eq 11) below). Finally, the radiative ($\Gamma_{cav}^{(R)}$) and nonradiative
($\Gamma_{cav}^{(NR)}$) decay rates of the cavity mode to the far field and to
damping in the cavity mirrors, respectively, are taken into account. The
corresponding Hamiltonian is
$\displaystyle\hat{H}=$ $\displaystyle\hat{H}_{0}+\hat{V}_{\text{cav-
mol}}+\hat{V}_{\text{pump}}-i\Gamma_{\text{loc}}\hat{C}-i\Gamma_{\text{cav}}\left|v\right\rangle\left\langle
v\right|$ (9) $\displaystyle\hat{H}_{0}=$ $\displaystyle
E_{v}\left|v\right\rangle\left\langle
v\right|+E_{0}\left|0\right\rangle\left\langle
0\right|+\sum_{m}E_{m}\left|m\right\rangle\left\langle m\right|$ (10)
$\displaystyle\hat{V}_{\text{cav-mol}}=$
$\displaystyle\sum_{m}V_{vm}\left|v\right\rangle\left\langle
m\right|+V_{mv}\left|m\right\rangle\left\langle v\right|$ (11)
$\displaystyle\hat{V}_{\text{pump}}=$ $\displaystyle V_{0v}\cos(\omega
t)\left|0\right\rangle\left\langle v\right|+V_{v0}\cos(\omega
t)\left|v\right\rangle\left\langle 0\right|$ (12) $\displaystyle\hat{Q}=$
$\displaystyle\sum_{m,m^{\prime}}\left|m\right\rangle\left\langle
m^{\prime}\right|$ (13)
where, $v$ is the excited state of the cavity mode (and the ground state
$\left|0\right\rangle$ refers to both the cavity and the molecular systems).
Similar to many previous works, we consider only a single cavity mode and in
correspondence with the single-exciton model, we include only the ground and
first excited states of this mode.
Here, the cavity photon leaking rate
($\Gamma_{\text{cav}}=\Gamma_{\text{cav}}^{(R)}+\Gamma_{\text{cav}}^{(NR)}$)
determines the quality factor $\hbar\omega_{\text{cav}}/\Gamma_{\text{cav}}$.
In addition to the local relaxation channel (which is the same as in eq (8)),
the outgoing radiative flux through the cavity is
$\displaystyle J_{cav}(t)=\Gamma_{cav}^{(R)}|c_{v}(t)|^{2}$ (14)
This flux represents the combined transmission spectrum.
### II.3 Disorder and intermolecular couplings
Disorder can be implemented in the Hamiltonian (1) and (9) by assuming that
either the molecular transition energies $E_{m}$ or the molecular couplings
with its radiative environment ($V_{m0}$ in eq 3 or $V_{mv}$ in eq 11) have a
random component. This random component can be constant in time (static
disorder) or time-dependent (dynamic disorder)
#### II.3.1 Energy disorder
If we focus first on molecular transition energy disorder, the static disorder
limit is described by including a random component in the individual molecular
frequencies ($\omega_{m}=E_{m}/\hbar$)
$\omega_{m}=\omega_{0}+\Omega_{m}.$ (15)
In the calculations reported below, the random component $\Omega_{m}$ is
sampled from a Gaussian distribution
$P[\Omega_{m}]=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp{\left(-\Omega_{m}^{2}/2\sigma^{2}\right)}$
(16)
where different molecules are assumed to be uncorrelated,
$\langle\Omega_{m}\Omega_{m^{\prime}}\rangle=\delta_{mm^{\prime}}\sigma^{2}$
and here the variance $\sigma^{2}$ indicates the disorder strength. This
static disorder model may be viewed as the static limit of Kubo’s stochastic
modulation model where
$\omega_{m}=\omega_{0}+\Omega_{m}(t)$ (17)
and $\Omega_{m}(t)$ are stochastic random variables that change over time. We
choose $\Omega_{m}(t)$ to be a Gaussian stochastic variable that satisfies
$\langle\Omega_{m}(t)\rangle=0$ and
$\langle\Omega_{m}(t_{1})\Omega_{m^{\prime}}(t_{2})\rangle=\delta_{mm^{\prime}}\sigma^{2}e^{-|t_{1}-t_{2}|/\tau_{c}}$
(18)
where $\delta_{mm^{\prime}}$ is a Kronecker delta function. The Gaussian
stochastic variables are characterized by the amplitude
$\sigma=\langle\delta\Omega^{2}\rangle^{1/2}$ and correlation lifetime
$\tau_{c}$ of local energy fluctuations. For a detailed algorithm to generate
such a stochastic process, see Appendix A.
#### II.3.2 Orientational Disorder
The most obvious source of coupling disorder is the inherent orientational
disorder that characterizes most molecular liquids, and here we focus on this
aspect of our systems. The simplest way to model such disorder is to assign an
angle $\theta_{m}$ for each emitter $m$, where $\theta_{m}$ is oriented with
respect to the cavity mode (see Fig. 1). We assume that the field of the
cavity mode that may be excited by the external pumping is polarized in the
$z$ direction and denote $\theta_{m}$ as the angle between the $z$ axis and
the molecular transition dipole direction.
Figure 1: Schematic diagram of the molecular emitters (blue arrows) confined
in an optical cavity (grey walls) and pumped by the incoming CW field (black
line). The field of the cavity mode is assumed to be in the $z$ direction. The
transition dipole moments of the emitters have different angles $\theta_{m}$
with respect to the $z$ axis.
This disorder leads to three modifications of the model Hamiltonian: First,
the cavity-molecule coupling becomes
$\hat{V}_{\text{cav-
mol}}=\sum_{m}V_{vm}\cos(\theta_{m})\left|v\right\rangle\left\langle
m\right|+V_{mv}\cos(\theta_{m})\left|m\right\rangle\left\langle v\right|$ (19)
Similarly, for the case outside the cavity, the couplings to the external
driving EM field also depend on the emitters orientation
$\hat{V}_{\text{pump}}=\sum_{m}V_{0m}\cos(\theta_{m})\cos(\omega
t)\left|0\right\rangle\left\langle m\right|+V_{m0}\cos(\theta_{m})\cos(\omega
t)\left|m\right\rangle\left\langle 0\right|$ (20)
Also, outside the cavity, the elements of the non-Hermitian $\hat{Q}$ matrix
depend on the relative angle $\cos(\theta_{m}-\theta_{m^{\prime}})$[55]
$\hat{Q}=\sum_{m,m^{\prime}}\cos(\theta_{m}-\theta_{m^{\prime}})\left|m\right\rangle\left\langle
m^{\prime}\right|$ (21)
The disorder of $\\{\theta_{m}\\}$ can be either static or dynamic as
described in section II.3. At room temperature, the characteristic timescale
of the rotational motion in solution is of order $10ps$, which implies that in
most situations including light-matter strong coupling regime in molecular
system, orientational disorder may be assumed static. Nevertheless, for
completeness, we will consider below a full range of dynamic disorder. In the
static case, we choose a time-independent $\\{\theta_{m}\\}$ according to
random orientation angles sampled from a Gaussian distribution
$P[\theta_{m}]=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp{\left(-\theta_{m}^{2}/2\sigma^{2}\right)}$
(22)
This again is the limit $\tau_{c}\rightarrow\infty$ of a process where
orientations are dynamically modulated according to
$\langle\theta_{m}(t)\rangle=0$ and
$\langle\theta_{m}(t_{1})\theta_{m^{\prime}}(t_{2})\rangle=\delta_{mm^{\prime}}\sigma^{2}e^{-|t_{1}-t_{2}|/\tau_{c}}$
(23)
#### II.3.3 Orientational disorder with intermolecular coupling
To complete our investigation of linear response of molecules outside and
inside optical cavities as expressed by the absorption and transmission
signals, we will also consider models with intermolecular couplings (dipole-
dipole couplings), again focusing on static and dynamic orientational
disorder. The intermolecular couplings is modeled by a dipole-dipole coupling:
$\displaystyle\hat{V}_{\text{dip}}=$
$\displaystyle\sum_{m=k+1}^{N}\sum_{k=1}^{n}W_{k}f(\theta_{m},\theta_{m-k})\left|m\right\rangle\left\langle
m-k\right|+\sum_{m=k+1}^{N}\sum_{k=1}^{n}f(\theta_{m-k},\theta_{m})W_{k}\left|m-k\right\rangle\left\langle
m\right|$ (24)
Here, $W_{k}=W/k^{3}$ and in the calculations reported below we have used
$W=0.1$,111The choice of intermolecular coupling is $0.1$ in unit of Rabi
splitting. Rabi splitting is approximately on the order of $10meV$, and if the
distance between dipoles are $2nm$, the dipole moments are on the order of
$10Debye$. unless otherwise specified. In principle, this Hamiltonian
describes a chain of emitters and the interaction between them decays as one
over their distance cubed, $1/r^{3}$, as sketched in Fig 1.
The function $f(\theta_{m},\theta_{m-k})$ in eq 24 reflects the angular
dependence of the dipole-dipole coupling between emitters and is given by
$\displaystyle
f(\theta_{m},\theta_{m-k})=\cos(\theta_{m}-\theta_{m-k})-3\cos\theta_{m}\cos\theta_{m-k}$
(25)
Again, these angles are defined to be relative to the polarization of the
incoming driving electric field.
### II.4 The Steady State flux
In order to simulate cavity-molecular dynamics, we propagate the Schrödinger
equation for the Hamiltonians above under steady state boundary conditions.
The wavefunction for the driven, open quantum system can be written as
$\left|\Psi\right\rangle=c_{0}\left|0\right\rangle+c_{v}\left|v\right\rangle+\sum_{m}c_{m}\left|m\right\rangle$.
This implies propagating the Schrödinger equation
$i\hbar\frac{d}{dt}\left|\Psi\right\rangle=\hat{H}\left|\Psi\right\rangle$
under the Hamiltonians of eqs 1 and 9 keeping $|c_{0}|^{2}=1$. The long-time
evolution yields the steady state forms of the coefficients $c_{m}(t)$ and
$c_{v}(t)$ that are used to evaluate the steady state fluxes $J_{rad}$ (eq 7),
$J_{loc}$ (eq 8), and $J_{cav}$ (eq 14). Again, the total steady state flux
$J_{rad}+J_{loc}$ as a function of the incoming driving frequency yields the
steady state absorption spectrum for emitters outside optical cavities; while
inside the cavity, the steady state flux $J_{cav}$ gives
reflection/transimission spectrum and $J_{loc}$ represents the local
relaxation flux.
With this single-exciton model, there are many parameters we can adjust to
account for different physical conditions. We will discuss these different
possibilities in the following sections.
## III Results
### III.1 Energy disorder
In Fig 2, we plot the absorption spectrum for an ensemble of emitters outside
the cavity. In the limit of zero disorder, the linewidth of the absorption
spectrum is associated with the decay rate for the radiative and local non-
radiative channels. For the static disorder results (a), as the disorder
strength $\sigma$ increases, the absorption lineshape become lower and
broader. This feature reflects inhomogeneous broadening and the static
uncertainty in the molecular excitation energy. Under dynamic disorder (b), as
the correlation time $\tau_{c}$ decreases, the absorption linewidth is
smaller, which reflects motional narrowing. We can also predict the absorption
lineshape in Fig 2 by performing a Fourier transform of $\phi(t)$222This
formula is by modifying equation (7.104) in ref. 62 by taking both decay rates
$N\Gamma_{rad}+\Gamma_{loc}$ into account according to Voigt theorem.
$\displaystyle\phi(t)=\exp[-\sigma^{2}\tau_{c}^{2}(\frac{t}{\tau_{c}}-1+e^{-\frac{t}{\tau_{c}}})-(N\Gamma_{rad}+\Gamma_{loc})t].$
(26)
For the case of static disorder ($\tau_{c}\rightarrow\infty$), we can simplify
the formula by expanding the exponential of $t/\tau_{c}$
$\displaystyle\phi(t)\rightarrow\exp(-\sigma^{2}t^{2}/2-(N\Gamma_{rad}+\Gamma_{loc})t)$
(27)
which, after a Fourier transform, yields a convolution between the gaussian
associated with static disorder and the lorentzian associated with the
radiative and non-radiative decay rates. For more details and a comparison of
analytic versus numerical results, see Appendix B.
Figure 2: The steady state absorption spectrum for a molecular ensemble
outside the cavity with (a) different static energetic disorder strengths and
(b) different correlation time $\tau_{c}$ but constant disorder strength
$\sigma=0.2$. The molecular ensemble includes $N=40$ molecules. If there is no
disorder (panel (a), $\sigma=0$, blue line), the linewidth of the peak is
$2N\Gamma_{rad}+2\Gamma_{loc}$. As the static disorder strength increases, we
obtain a broader inhomogeneous peak. As the correlation time decreases under
fixed disorder strength (panel (b), $\sigma=0.2$), the peak gets narrower,
reflecting motional narrowing. As shown in eqs 26 and 27, as well as Appendix
B, these features can be explained analytically.
Next, consider the case where the same ensemble of emitters interacts with a
single-mode cavity under steady state pumping. As discussed in section II.2,
we calculate two observables: (1) the steady state flux $J_{cav}$ through the
cavity leakage channel (which gives the combined transmission spectrum) and
the steady state flux $J_{loc}$ through the local relaxation channel of
individual emitters. The results in Fig. 3 show that static disorder leads to
an inhomogeneous broadening of the two polariton peaks that are present in
both of the relevant channels. Interestingly, Fig 4, in which we suppress the
homogeneous contributions (i.e. when non-Hermitian parts $\Gamma_{rad}$,
$\Gamma_{loc}$, $\Gamma_{cav}$ are small) to the lineshape so that the width
is dominated by the inhomogeneous contribution, shows that this inhomogeneous
broadening is much weaker inside a cavity relative to outside a cavity . This
effect has already been seen in reference 58.
Returning to Fig. 3, we note that, for increasing $\sigma$, the baseline
between the two polariton peaks (around driving frequency $\omega_{d}=0$) for
the local relaxation channel becomes larger, whereas in the transmission
spectrum, the baseline around driving frequency $\omega_{d}=0$ remains
approximately zero. This different behavior arises because the dark modes
cannot contribute to the transmission spectrum but they do contribute to the
local relaxation channel. Another observation in Fig. 3 is the increase of the
effective Rabi splitting in both figures (a) and (b), as the energetic static
disorder strength increases. This increase can be captured approximately by
the following formula
$\Omega_{R}(\sigma)/\Omega_{R}(\sigma=0)=1+2\sigma^{2}$ (28)
One can obtain such a formula by using second-order perturbation theory for
the polariton states. For a simple derivation, see Appendix C.
Lastly, for the case with dynamic disorder, as shown in Fig. 5, as one might
expect, we find motional narrowing for both channels as the correlation time
$\tau_{c}$ decreases (similar to Fig. 2). Moreover, we observe the same
behavior as in Fig. 3: in between the two polariton peaks (around driving
frequency $\omega_{d}=0$), the presence of dark modes leads to a finite steady
state flux for the local relaxation channel.
Figure 3: Steady state (a) transmission spectrum and (b) local relaxation flux
spectra, as calculated for molecular ensembles with different static disorder
strengths $\sigma$ interacting with a single-mode cavity. The molecular
ensemble and the cavity mode form two polariton peaks in both figures. As the
amplitude of static disorder increases, the split peaks become broadened;
however, the baseline in between the two peaks (around driving frequency
$\omega_{d}=0$) grows larger only for the local signal. This growth arises
because the dark modes contribute only to the local relaxation flux but not to
the transmission spectrum. Lastly, note that as the energetic static disorder
increases, the effective Rabi splitting increases. This increase can be
captured approximately by perturbation theory (eq 28). Figure 4: Steady state
(a) absorption spectra for outside the cavity, (b) transmission and (c) local
relaxation flux spectra interacting with a single-mode cavity, as calculated
for molecular ensembles with different static disorder strengths $\sigma$. The
black dotted lines in panel (a) are analytical results calculated in Appendix
B. Here, the contributions to homogeneous broadening, namely the non-Hermitian
parts of the Hamiltonian matrices are taken
$\Gamma_{rad}=0.005/N,\Gamma_{cav}=0.005,\Gamma_{loc}=0.005$, which are $10$
times smaller than in Fig 3, so (except from the case without disorder) the
linewidth outside the cavity is dominated by the inhomogeneous broadening. In
agreement with earlier observations[58], the inhomogeneous broadening observed
outside the cavity is not manifested in the linewidth of the polariton peaks
inside the cavity which are therefore much narrower as seen in panel (b) and
(c). Figure 5: Steady state (a) transmission spectra and (b) local relaxation
for the molecular ensemble inside a single-mode cavity with dynamic disorder.
We fix the disorder strength ($\sigma=0.2$) and scan the correlation time
$\tau_{c}=100,10,1,0.1$. The molecular ensemble and the cavity mode forms two
polariton peaks in both figures. Note that the long correlation limit (blue
lines) recovers the results with static disorder in Fig. 3 (Yellow lines).
Note also that similar to Fig. 3, the dark modes contribute only to the local
relaxation channel. As we decrease $\tau_{c}$, the two peaks become narrower
in both figures, and the dark modes contribute less to the local relaxation
channel.
### III.2 Orientational disorder
In this subsection, we present results for angular disorder as defined in
section II.3.2. Note that the way disorder is defined in this case implies the
average contribution is nonzero ($\langle\cos\theta_{m}\rangle\neq 0$), and
therefore, some of the results shown below reflect this situation. As shown in
Fig 6, for the static angular disorder case, as the disorder strength $\sigma$
increases, both the height and the linewidth of the absorption peak decrease
(see eq 22). This decrease arises because each dipole in the ensemble of
emitters couples to the driving field differently, resulting in less
collective behavior. For the dynamic angular disorder case, in agreement with
the results in section III.1, as we decrease the correlation time of the
angular disorder, we find motional narrowing.
Figure 6: Steady state absorption spectra for the ensemble of emitters outside
the cavity. (a) static angular disorder; (b) dynamic angular disorder. As
shown in the static disorder figure, when the disorder strength $\sigma$
increases, the absorption peaks decrease. More importantly, the linewidth
decreases because the emitters act less collectively. For the case of dynamic
disorder, similar to Fig. 2, we observe motional narrowing.
Next, we consider this molecular ensemble with angular disorder inside the
cavity. The results are shown in Fig. 7. Unlike the results in Fig. 3, (where
we investigated disorder in the excitation energies), here the Rabi splitting
consistently decreases as we increase the static angular disorder strength and
lose collectivity of the response. Clearly, when interpreting the spectra from
cavity, one must consider geometry and not just the energy levels of the
molecules in the cavity. In principle, in a cavity, one might expect to have
both energy and angular disorder and so, as the temperature rises, the Rabi
splitting should decrease. Unfortunately, it remains unclear which effect
dominates. Note that the increase of Rabi splitting with increasing static
energy disorder does not suggest that more emitters are behaving collectively.
As shown both in Fig. 3 and Fig. 7, the maximal radiative steady state flux
decreases as static disorder strength $\sigma$ increases. Lastly, as in Fig.
3, the dark modes contribute to the rise of baseline between the two polariton
peaks only for the local relaxation channel.
Figure 7: Steady state transmission spectrum and local relaxation flux for
different static angular disorder strength ($\sigma=0,0.2\pi,0.4\pi,0.6\pi$)
inside a cavity. As shown in the figures, when the disorder strength $\sigma$
increases, the absorption peak height decreases and the dark modes contribute
only to the local relaxation flux but not to the transmission spectrum
(similar to Fig. 3). However, unlike Fig. 3, disorder in the angular momentum
leads to a decrease in the Rabi splitting.
The results for the dynamic angular disorder case are slightly more non-
intuitive. In Fig. 8, we fix the disorder strength ($\sigma=0.4\pi$) and
calculate transmission and local relaxation fluxes for different correlation
time $\tau_{c}$. In the long correlation time limit (blue line), we recover
the static disorder result. Two features are worth noting in Fig. 8. First, as
$\tau_{c}$ becomes smaller, we find that, as expected, the polaritonic spectra
undergo motional narrowing. However, reducing the correlation time is not
equivalent to reducing disorder insofar as the fact that the Rabi splitting in
Fig. 8 decreases as $\tau_{c}\rightarrow 0$ (whereas the Rabi splitting
increases as $\sigma\rightarrow 0$ in Fig. 7). For a simple explanation of
this Rabi splitting contraction, consider the two limiting scenarios, (i) the
static limit ($\tau_{c}=\infty$) and (ii) the no correlation limit
($\tau_{c}=0$). For the static limit ($\tau_{c}=\infty$), the averaged Rabi
splitting is calculated by estimating the eigenvalues $E_{LP/UP}$.
$\displaystyle
E_{LP/UP}=\sqrt{N|V_{mv}|^{2}\langle\cos^{2}(\theta_{m})\rangle}$ (29)
Here, $\langle\rangle$ stands for ensemble average. For the no correlation
limit ($\tau_{c}=0$), a good estimate is
$\displaystyle E_{LP/UP}=\sqrt{N|V_{mv}|^{2}}\langle\cos(\theta_{m})\rangle$
(30)
However, we are unable to predict what Rabi splittings are for finite
$\tau_{c}$. For detailed derivations of these expressions and an analysis of
how they match up with numerical results, see Appendix D. Note that in both
limits, the Rabi splittings are smaller than the case with no disorder,
reflecting the fact that the average coupling in this model is smaller than
$V_{mv}$.
Another non-intuitive result in Fig. 8 pertains to the height of the peaks
themselves. Here, we see that the height behaves in a non-monotonic fashion as
a function of $\tau_{c}$; starting from static disorder and decreasing
$\tau_{c}$, we find that the peak height and integral decreases and then
increases. Qualitatively, we believe this non-monotonic behavior can be
understood by recognizing that with moderate correlation times, introducing
disorder and slowly distorting the coherences removes the possibility of $(i)$
seeing individual configurations, each absorbing for a long time or $(ii)$
seeing many configurations averaged to one static configuration. As a result,
there is less absorption at intermediate correlation times. Pursuing a more
rigorous analysis of this effect will certainly be an important research
direction in the future. Finally, just as in Fig. 7, the dark modes contribute
to the local relaxation, and as the correlation time decreases, the Rabi
splitting decreases.
Figure 8: Steady state absorption spectra for dynamic angular disorder. We fix
the angular disorder strength ($\sigma=0.4\pi$) and scan over correlation time
$\tau_{c}=100,10,1,0.1$. We also plot the result with no disorder for
reference. As shown in both figures, when the correlation time decreases, the
collective effect diminishes, leading to a smaller Rabi splitting. Similar to
Fig. 7, the dark modes contribute only to the local relaxation signals.
### III.3 Intermolecular coupling and orientational disorder
Finally, let us address the presence of how intermolecular coupling affects
orientational disorder. Let $\left|B\right\rangle$ be the bright state of the
system outside the cavity
$\displaystyle\left|B\right\rangle$
$\displaystyle=\frac{1}{\sum_{j}\cos^{2}\theta_{j}}(\cos\theta_{1},\cos\theta_{2},...)^{T}.$
(31)
Before presenting the absorption spectra, we start by showing the density of
states of the system outside/inside a cavity, weighted by the brightness of
the states in Fig 9. For the case outside the cavity, we define the brightness
$W_{j}$ of each eigenstate $\phi_{j}$ of $\hat{H}_{sub}$ (keeping only the
real part of $m$ and $v$ elements, see eqs 36-38 in Appendix C) to be:
$\displaystyle w_{j}$ $\displaystyle=|\left\langle
B\middle|\phi_{j}\right\rangle|^{2}$ (32)
For the case inside the cavity, the bright mode must contain the cavity mode
and thus the following definition makes more sense:
$\displaystyle w_{j}^{\text{cav}}$ $\displaystyle=|\left\langle
v\middle|\phi_{j}\right\rangle|^{2}$ (33)
where $\left|v\right\rangle$ is the bare cavity state (see eqs 9-13).
Figure 9: Histogram of the brightness ($w_{j}$ in eq 32 and
$w_{j}^{\text{cav}}$ in eq 33) for the Hamiltonian eigenstates that arise with
intermolecular coupling (see eqs 24 and 25) as averaged over realizations of
$\theta_{m}$ following the distribution in eqs 22 and 23. These brightness
factors are relevant to pumping the emitters outside/inside the cavity. We
consider that the dipoles are fully aligned in the limit of zero disorder and
thus, intermolecular couplings are negative (eq 25 equals $-2$). The
eigenvalue of the bright state (symmetric superposition state) becomes
negative. Hence, for the case outside the cavity, the brightest states have
energy $E\approx-0.3\ a.u.$, which is lower than the energy of the independent
emitters (which is set to be $0$); for the case inside the cavity, the two
brightest states yield lower and upper polariton peaks at around $E=-0.6,0.3\
a.u.$ respectively. The upper polariton peak is brighter than the lower
polariton because the former has an energy eigenvalue ($0.3\ a.u.$) closer to
the energy of the bare cavity mode ($0\ a.u.$). Again, the center of the two
polariton peaks ($(E_{LP}+E_{UP})/2\approx-0.15\ a.u.$) is lower than the
energy of the independent emitters/bare cavity mode.
As shown in Fig 9, for the case outside the cavity, the brightest eigenstates
have negative energy eigenvalues (relative to the emitters energy). For the
case inside the cavity, there are two regions of bright states corresponding
to the two polariton peaks; the center of the two peaks is shifted towards the
negative direction, just as for the case outside the cavity. The upper
polariton eigenstates are brighter than the lower polariton because they are
closer to the bare cavity mode energy.
Figure 10: Steady state absorption for emitters outside cavity with
intermolecular couplings and static/dynamic orientational disorder. (a) static
orientational disorder; (b) dynamic orientational disorder. For the static
disorder case, the absorption peaks height decreases as the disorder strength
increases. As predicted in Fig. 9, intermolecular coupling reduces the energy
of the system and the peaks appear to have energy lower than that of the
individual emitters. For the dynamic disorder case, the peak gets narrower as
we decrease the correlation time. Figure 11: Steady state (a) transmission and
(b)local relaxation for an ensemble of emitters inside a single-mode cavity
with intermolecular couplings for different static orientational disorder
strength. As predicted in Fig. 9, the upper polariton contributes more
strongly than does the lower polariton peak to the transmission spectrum.
Intermolecular couplings stabilize the system and thus the center of the two
polariton peaks corresponds to an energy lower than that of an individual
emitter/bare cavity mode. As the disorder strength increases, the Rabi
splitting decreases as the collectiveness is destroyed by the disorder. The
difference in height between the upper and lower polariton peaks is small for
the local relaxation channel because the signals are not dependent on how
strong is the cavity mode contribution to the polariton (but rather are
dictated by the dynamics of the individual emitters). Figure 12: Steady state
(a) transmission and (b) local relaxation flux for an ensemble of emitters
inside the cavity with intermolecular couplings and dynamic disorder. Both
figures are for the indicated correlation times but a fixed disorder strength
($\Delta=0.4\pi$). Again, the long correlation time limit (blue lines) is
equivalent to static disorder results (yellow lines in Fig. 11). Both the Rabi
splitting and the linewidth decreases as the correlation time decreases
similar to Fig. 8. The center of the two polariton peaks shift to lower energy
by the intermolecular coupling. Same as in Fig. 11, the difference in height
between the upper and lower polariton peaks is more significant in the
transmission spectrum than in the local relaxation signal.
With these structures in mind, in Fig 10, we plot the absorption spectra for
different orientational disorder strengths. When there is no orientational
disorder, because there is finite negative intermolecular coupling, the peaks
are red shifted, as the brightest eigenstate corresponds to the smallest
eigenvalue. As we increase the disorder strength, the total absorption
decreases. For the dynamic orientational disorder case, similar to Fig. 6, as
the correlation time decreases, the absorption linewidth decreases
(corresponding to motional narrowing). The scale of the red shift is also
reduced as we increase modulation speed.
In Fig. 11, we plot the transmission and local relaxation flux for an ensemble
of emitters placed inside the cavity. First, one might guess from what was
shown in Fig. 9, the upper polariton peak height is greater than that of the
lower polariton; the center of the two polariton peaks has a lower energy than
that of the bare cavity mode. Second, as the static disorder strength
increases, the effective Rabi splitting decreases because the disorder reduces
the collectiveness of the emitters. Third, as we compare the two channels,
because the transmission spectra relies on the cavity mode and the local
relaxation relies on the emitter states, the difference in intensity between
the upper and lower polaritons for transmission spectrum is greater than that
for the local relaxation signals. Lastly, the dark modes contribute only to
the local relaxation signals, as was also seen in Fig. 7.
Finally, we show the steady state transmission and local relaxation flux
inside a single-mode cavity with intermolecular coupling and dynamic
orientational disorder in Fig. 12. As the correlation time decreases, the
effective Rabi splitting and the linewidth of each polariton peak decreases,
just as in Fig. 8. The difference between the upper and lower polariton peak
height is less significant in local relaxation flux than the transmission
spectrum, similar to what was found in Fig. 11. All of the observations are
consistent with Section III.2.
## IV cavity effect on local relaxation yield
The models defined by eqs 1-13 are minimal models that allow considerations of
three relaxation channels for the radiation absorbed from the driving field: a
radiative channel that combines transmission and reflection, nonradiative
damping of the cavity mode (heat production in the cavity walls) and reactive
relaxation. The latter represents a reaction that occurs following the
molecular excitation. It is of interest to examine the effect of the cavity
environment on the yield of the latter channel.
Outside the cavity, the system undergoes collective superradiance at a rate
$N\Gamma_{rad}$ and local relaxation at a rate $\Gamma_{loc}$. The yield of
the individual molecular relaxation (that we assume to be a reactive process)
is
$\displaystyle\text{Yield}=\frac{\Gamma_{loc}}{N\Gamma_{rad}+\Gamma_{loc}}$
(34)
By comparison, inside the cavity, the energy will leak through the cavity mode
at rate $\Gamma_{cav}$ with the same local relaxation rate as above,
$\Gamma_{loc}$. It is clear that the local yield is then
$\displaystyle\text{Yield}=\frac{\Gamma_{loc}}{\Gamma_{cav}+\Gamma_{loc}}$
(35)
Hence, without disorder, we can expect that the reactive relaxation yields
will be different outside and inside the cavity, both in terms of absolute
value and $N$ dependence. This difference is shown in Fig. 13, Note, for
simplicity, we assume $\Gamma_{cav}=10\Gamma_{rad}$. In agreement with eqs 34
and 35, the local relaxation yield depends on $N$ only when the emitters are
outside the cavity.
Consider now the effect of disorder. As discussed above and shown in Figs. 2
and 5, at reasonably high temperatures, nuclear motion and dephasing processes
lead to static and dynamic disorder, which results in a reduction of steady
state radiative fluxes. As shown in Fig. 13, we can clearly see that moderate
dynamic disorder leads to a reduction of radiative fluxes, and thus a greater
local relaxation yield both inside and outside the cavity. More importantly,
the slope of the ratio between radiative and local relaxation flux (blue-
dashed line in the subfigure (a)) is reduced in the presence of disorder
outside the cavity. In principle, this slope corresponds to the effective
number of emitters that contribute to the collective emission. For the
parameters in Fig. 13, if we focus on comparing the blue solid and dashed
lines, we find a slope of $0.06$ for moderate disorder (blue, dashed line), vs
$0.1$ for no disorder case (blue solid line), which implies that 60% molecules
are emitting collectively. That being said, for very strong dephasing and
disorder, we expect that this slope should approach zero and $N$ in eq 34
should approach unity; after all, with strong enough dephasing, we expect to
find independent spontaneous emission for emitters outside the cavity – in
contrast to the superradiant effect in eq 34. While this analysis demonstrate
a possible cavity effect on the yield of chemical reaction, a note of caution
should be added. This analysis was based on the assumption that $\Gamma_{loc}$
is identical following excitation of the bright mode outside the cavity as in
the case where a polariton is excited inside the cavity. This is not
necessarily the case because the initially prepared state differs in energy by
an amount determined by the Rabi splitting. A model where $\Gamma_{loc}$ is
sensitive to this difference was recently analyzed in ref 59
Figure 13: (a) The steady state ratio between radiative flux and local
relaxation flux outside/inside the cavity with/without energy dynamic
disorder. (b) The steady state local relaxation yield outside/inside cavity
with/without energy dynamic disorder. We keep all parameters constant and vary
only the number of emitters of the system. We assign the cavity leakage rate
to be the same as the local relaxation rate for the case of the system inside
the cavity. For the case without disorder, we choose the superradiant rate for
$N=10$ to be the same as the local relaxation rate for the case outside the
cavity. Hence, the local yield at $N=10$ is 0.5 both inside and outside the
cavity. For outside the cavity, as more emitters are collectively emitting,
the ratio between the radiative and local relaxation rate increases linearly
as $N$, and thus the local yield decreases. For inside the cavity, we assume
the cavity leakage rate remains approximately constant. Hence, the local
relaxation yield does not depend on $N$.[22] From this data, we conclude that
adding a single mode cavity can certainly change the importance of different
energy dissipation channels. Now, for the case with moderately fast energetic
dynamic disorder ($\sigma=0.2$, $\tau_{c}=1$), one clear pattern is that for
both outside/inside the cavity, the radiative channels are suppressed by the
disorder, and thus, local relaxation yields increase. Moreover, for the case
outside the cavity (blue dashed line), the slope decreases from $0.1$ (no
disorder) to $0.06$ (moderate dynamic disorder), which shows that there are
60% emitters behaving collectively. In applying this conclusion to a realistic
experiment, note that $N$ should be the effective number of emitters that
participate in the superradiant rate/Rabi splitting, which characterize the
collectiveness under the influence of disorder.
## V Conclusion
In conclusion, we investigate the influence of energetic disorder,
orientational disorder and intermolecular couplings on the absorption spectra
for an ensemble of two-level emitters outside and inside a single-mode cavity.
As far as the influence of energetic and orientational disorder are concerned,
we recover inhomogeneous broadening for static disorder and motional narrowing
for dynamic disorder. For the case inside the cavity, there are two inherently
different observables: transmission spectrum and local relaxation flux. The
transmission spectrum is dictated primarily by properties of the cavity mode
and the local relaxation flux depends mainly on the property of the individual
emitters. These two observables are complementary to each other and together
provide a comprehensive picture of the system when we introduce different
types of disorder. Introducing intermolecular coupling is essentially
equivalent to adding finite detuning between the bare cavity mode and the
emitters eigenstates. As we compare the two observables, we confirm that dark
modes do not contribute to transmission signals which requires the state to be
bright. However, the dark modes do contribute to the local relaxation signals.
Lastly, we investigate the differences between excited state local relaxation
yields outside versus inside the cavity. It is clear that the presence of a
cavity mode will modify the yield in two ways: (1) changing the radiative
energy dissipation rate (from collective spontaneous emission to cavity
leakage), and (2) changing inherently how collectiveness translates into
physical observables (from superradiance to Rabi splitting)[22].
Looking forward, we note that, in this work, we have assumed that the rate of
independent spontaneous emission towards other polarization direction is very
slow and can be completely ignored. In truth, however, the influence of other
polarization directions cannot really be modeled with a single-mode cavity. To
better isolate possible cavity effects in chemistry, one of our next steps
will be to incorporate higher dimensions and more cavity modes inside a
realistic microcavity (likely through a Maxwell-Bloch calculation).
## Appendix A Generation of Orenstein-Uhlenbeck process
In our computational procedure, for generating energy modulations, we employ
an Orenstein-Uhlenbeck process. To generate a sequence of values at time
$t_{n}=t_{0}+ndt$, we sample $\\{\Omega_{m}(t_{n})\\}$ in the following
process:[60]
1. 1.
At $t_{0}$, pick $\\{\Omega_{m}(t_{0})\\}$ according to Gaussian distribution
$P[\Omega_{m}(t_{0})]=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\Bigl{(}-\frac{\Omega_{m}(t_{0})^{2}}{2\sigma^{2}}\Bigr{)}$
2. 2.
Calculate $r_{n}$:
$r_{n}=\exp\bigl{(}-(t_{n+1}-t_{n})/\tau_{c}\bigr{)}=\exp\bigl{(}-dt/\tau_{c}\bigr{)}$
3. 3.
Pick $\\{\Omega_{m}(t_{n+1})\\}$ according to conditional probability
$P[\Omega_{m}(t_{n+1})|\Omega_{m}(t_{n})]=\frac{1}{\sqrt{2\pi\sigma^{2}(1-r_{n}^{2})}}\exp\Bigl{(}-\frac{(\Omega_{m}(t_{n+1})-r_{n}\Omega_{m}(t_{n}))^{2}}{2\sigma^{2}(1-r_{n}^{2})}\Bigr{)}$
4. 4.
Go back to 2
## Appendix B Voigt theory
In this appendix, we demonstrate that our numerical results from section III.1
for the case of a collection of molecules outside the cavity with
static/dynamic energetic disorder (Fig. 2) satisfy Voigt theory. According to
eq 26 and eq 27, we can analytically predict the Voigt linewidth for different
disorder strengths $\sigma$ and correlation time $\tau_{c}$. However, we
cannot determine the absolute peak height from eq 26 and eq 27. Nevertheless,
if we simply fit the lineshape with the numerical height, then Fig. 14
demonstrates that the anatlyic linewidths predicted by eq 27 and eq 26 (lack
dots) match perfectly with the numerical linewidths in Fig. 2.
Figure 14: Steady state absorption spectrum for a molecular ensemble with (a)
different static disorder strengths and with (b) different correlation time
$\tau_{c}$ but the same disorder strength $\sigma=0.2$, identical to Fig. 2.
The black dots are predicted by eq 27 and 26. The linewidths of numerical
results match perfectly with the prediction.
## Appendix C Derivation of the Effective Rabi Splitting under Static
Energetic Disorder
In this appendix, we use second-order perturbation theory to derive eq 28. The
total Hamiltonian of the disordered system is
$\displaystyle\hat{H}_{sub}=$ $\displaystyle H_{0}+\Lambda$ (36)
$\displaystyle H_{0}=$ $\displaystyle E_{v}\left|v\right\rangle\left\langle
v\right|+\sum_{m=1}^{N}E_{m}\left|m\right\rangle\left\langle
m\right|+V_{mv}(\left|m\right\rangle\left\langle
v\right|+\left|v\right\rangle\left\langle m\right|)$ (37)
$\displaystyle\Lambda=$
$\displaystyle\sum_{m=1}^{N}\delta{E}_{m}\left|m\right\rangle\left\langle
m\right|$ (38)
Here $N$ is the effective number of emitters in the cavity. The modulation
$\delta E_{m}$ is treated as perturbation and has zero mean and $\sigma^{2}$
variance. We assume no detuning and thus, the Rabi splitting with no disorder
is
$\Omega_{R}(\sigma=0)=2\sqrt{N}V_{mv}$ (39)
The polariton states have energy $\pm\Omega_{R}(\sigma=0)/2$ and the
corresponding eigenvectors are
$\displaystyle\left|UP/LP\right\rangle=(1/\sqrt{2},\pm 1/\sqrt{2N},...,\pm
1/\sqrt{2N})$ (40)
All remaining eigenstates are dark states and have energy $E_{dark}=0$.
Second-order perturbation theory yields the energies of the polariton states
$\displaystyle E_{LP}=E_{LP}^{0}+\left\langle
LP\right|\Lambda\left|LP\right\rangle+\sum_{k\neq LP}\frac{|\left\langle
k\right|\Lambda\left|LP\right\rangle|^{2}}{E_{LP}^{0}-E_{k}^{0}}$ (41)
The first-order term is the mean of the static disorder.
$\displaystyle\left\langle
LP\right|\Lambda\left|LP\right\rangle=\sum_{m=1}^{N}\delta{E}_{m}/2N=0$ (42)
The second term is
$\displaystyle\sum_{k\neq LP}\frac{|\left\langle
k\right|\Lambda\left|LP\right\rangle|^{2}}{E_{LP}^{0}-E_{k}^{0}}=$
$\displaystyle-\frac{\sum_{k}|\left\langle
k\right|\Lambda\left|LP\right\rangle|^{2}-|\left\langle
LP\right|\Lambda\left|LP\right\rangle|^{2}-|\left\langle
UP\right|\Lambda\left|LP\right\rangle|^{2}/2}{\Omega_{R}(\sigma_{0})/2}$ (43)
$\displaystyle\sum_{k}|\left\langle
k\right|\Lambda\left|LP\right\rangle|^{2}=$ $\displaystyle\left\langle
LP\right|\Lambda^{2}\left|LP\right\rangle=\sum_{m=1}^{N}\delta
E_{m}^{2}/2N=\sigma^{2}/2$ (44) $\displaystyle|\left\langle
UP\right|\Lambda\left|LP\right\rangle|^{2}/2=$
$\displaystyle-\sum_{m=1}^{N}\delta E_{m}/4N=0$ (45)
Hence, the perturbed energy is
$\displaystyle E_{LP}=E_{LP}^{0}-\frac{\sigma^{2}}{\Omega_{R}(\sigma=0)}$ (46)
Last, because the detuning is zero, the spectrum is completely symmetrical.
The effective Rabi splitting is simply twice the absolute value of the lower
polariton energy, which yields eq 28.
## Appendix D Derivation of the Effective Rabi Splitting under Static and
Dynamic Angular Disorder
In this appendix, we use a cumulant expansion to calculate the effective Rabi
splitting that is relevant with disorder as a function of the disorder
strength $\sigma$ and/or correlation time $\tau_{c}$. At this point, if we
ignore the pumping and non-hermitian decay from eqs 9 \- 13, the Hamiltonian
of the disordered system is
$\displaystyle\hat{H}=$ $\displaystyle H_{0}+\Lambda_{\theta}$ (47)
$\displaystyle H_{0}=$ $\displaystyle E_{v}\left|v\right\rangle\left\langle
v\right|+\sum_{m=1}^{N}E_{m}\left|m\right\rangle\left\langle m\right|$ (48)
$\displaystyle\Lambda_{\theta}=$
$\displaystyle\sum_{m=1}^{N}(V_{mv}(\theta_{m})\left|m\right\rangle\left\langle
v\right|+V_{vm}(\theta_{m})\left|v\right\rangle\left\langle m\right|$ (49)
As shown in eq 19, dynamic angular disorder enters as a modulation of $\theta$
in the coupling $V_{mv}(\theta_{m})=V\cos(\theta_{m})$ between each individual
emitter and the bare cavity mode. It is straightforward to obtain the
instantaneous eigenvalues of $H$ because the characteristic polynomial is
$\displaystyle\lambda^{N-1}(\lambda^{2}-\sum_{m=1}^{N}V^{2}\cos^{2}(\theta_{m}))=0$
(50)
Hence, the effective Rabi splitting is
$\Omega_{R}/\Omega_{R}(\sigma=0)=\sqrt{\langle\cos^{2}\theta_{m}\rangle}$,
where ($\theta_{m}(t)$) satisfies $\langle\theta_{m}(t)\rangle=0$ and
$\langle\theta_{m}(t_{j})\theta_{n}(t_{k})\rangle=\delta_{mn}\sigma^{2}\exp(-|t_{j}-t_{k}|/\tau_{c})$.
By cumulant expansion
$\displaystyle\langle\cos{\theta_{m}}\rangle=$
$\displaystyle\text{Re}(\exp(i\langle\theta_{m}\rangle-\frac{1}{2}\langle\delta\theta_{m}^{2}\rangle))=\exp(-\sigma^{2}/2)$
(51)
Hence, in the static limit ($\tau_{c}=\infty$),
$\displaystyle\langle\cos^{2}\theta_{m}\rangle=\Bigl{\langle}\frac{\cos
2\theta_{m}+1}{2}\Bigr{\rangle}=\frac{(e^{-2\sigma^{2}}+1)}{2}$ (52)
The effective Rabi splitting is
$\displaystyle\Omega_{R}(\sigma,\tau_{c}=\infty)/\Omega_{R}(\sigma=0)=\sqrt{(e^{-2\sigma^{2}}+1)/2}$
(53)
This analytical formula is verified in Fig. 15. The numerical results are
extracted from Fig. 7.
In the fast modulation limit ($\tau_{c}=0$), the external field sees an
averaged upper/lower polariton state (as defined in eq 40 in Appendix C),
which leads to average Rabi splitting
$\displaystyle\Omega_{R}(\sigma,\tau_{c}=0)/\Omega_{R}(\sigma=0)=\langle\cos\theta_{m}\rangle=e^{-\sigma^{2}/2}$
(54)
Unfortunately, we have not yet been able to derive an analytic form for the
effective Rabi splitting for finite $\tau_{c}$. For a general $\tau_{c}$, the
effective Rabi splitting is not determined exclusively by the instantaneous
eigenvalues (which allowed us to calculate the slow and fast limits (eqs. 53
and 54) above). That being said, in Fig. 15, we plot the effective Rabi
splitting for two different values of N (the number of emitters). Note that
the two lines are on top of one another. Thus, even though we do not have a
predictive theory for the Rabi splitting in the intermediate $\tau_{c}$
regime, we can at least be confident that the Rabi splitting does always scale
as $\sqrt{N}$ (which would seem to agree with the results in ref. 61).
Figure 15: Effective Rabi splitting vs static angular disorder strength and
dynamic angular disorder correlation time. As shown in figure (a), static
angular disorder leads to contraction of Rabi splitting. This contraction is
unlike the case with static energetic disorder, in which the effective Rabi
splitting increases as the disorder strength increases. The formula eq 53
obtained by cumulant expansion captures the contraction of effective Rabi
splitting induced by static angular disorder ($\tau_{c}\rightarrow\infty$).
For the case with dynamic angular disorder ($\sigma=0.4\pi$), as shown by the
Red dashed line in figure (b), we can only predict the fast modulation limit
$\tau_{c}=0$ as in eq 54. For the intermediate correlation time, the effective
Rabi splitting is not fully determined by the first and second moment. As
shown by the green dashed figure, the effective Rabi splitting is also
independent of $N$ in the system.
## Appendix E Symbols, energies/rates/frequencies and values that appear in
this paper
In this Appendix, we will list all parameters in the paper.
[b] Symbol Energy/Rate/Frequency Value $E_{v}$ Cavity mode energy 0* $E_{0}$
Total ground state energy -1* $E_{m}$ molecular singly excited state energy 0
(for no disorder system) $\hat{V}_{mv}$ coupling between cavity mode
$\left|v\right\rangle$ and molecular excited state $\left|m\right\rangle$
$0.5/\sqrt{N}$$\dagger$ $\hat{V}_{m0}$ coupling between ground state
$\left|0\right\rangle$ and molecular excited state $\left|m\right\rangle$
0.0005 $\omega$ Driving frequency [-1.2, 1.2] $\Gamma_{rad}$ Single molecule
radiative decay rate 0.05/N $\Gamma_{loc}$ Local relaxation rate 0.05
$\Gamma_{cav}(\Gamma_{cav}^{R}+\Gamma_{cav}^{NR})$ Cavity leakage rate
(including radiative and nonradiative) 0.05
Table 1: Symbols, energies/rates/frequencies, and values that appear in this
paper.
* $\dagger$
We choose to normalize the Rabi splitting for no disorder system and thus all
parameters are in unit of $\Omega_{R}(\sigma=0)$.
* *
The choice of energy difference between ground and cavity mode
energy/molecular singly excited state energy does not affect the results
because the calculation is under rotating wave approximation. We also set the
energy of the singly excited state energy/cavity mode energy for no disorder
system to be $0$.
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|
# Modulational instability in randomly dispersion-managed fiber links
Andrea Armaroli Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes
et Molécules, F-59000 Lille, France Guillaume Dujardin Univ. Lille, Inria,
CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France Alexandre
Kudlinski Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et
Molécules, F-59000 Lille, France Arnaud Mussot Univ. Lille, CNRS, UMR
8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
Stephan De Bièvre Univ. Lille, CNRS, Inria, UMR 8524 - Laboratoire Paul
Painlevé, F-59000 Lille, France Matteo Conforti Univ. Lille, CNRS, UMR
8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
###### Abstract
We study modulational instability in a dispersion-managed system where the
sign of the group-velocity dispersion is changed at uniformly distributed
random distances around a reference length. An analytical technique is
presented to estimate the instability gain from the linearized nonlinear
Schrödinger equation, which is also solved numerically. The comparison of
numerical and analytical results confirms the validity of our approach.
Modulational instability of purely stochastic origin appears. A competition
between instability bands of periodic and stochastic origin is also discussed.
We find an instability gain comparable to the conventional values found in a
homogeneous anomalous dispersion fiber.
Modulational instability (MI) is a pervasive phenomenon in the physics of
nonlinear dispersive waves. It manifests itself as the destabilization of a
uniform wavepacket by the exponential growth of small harmonic perturbations
around the carrier frequency of the wavepacket [1]. Its study originated in
hydrodynamics [2, 3], but analogous phenomena were also discovered in
electromagnetic waves [4] and optical fibers [5]. The main ingredients to
observe MI are focusing cubic nonlinearity (like the Kerr effect in silica)
and anomalous (negative) group-velocity dispersion (GVD).
Notwithstanding, MI can be found also in normal (positive) GVD, if higher-
order dispersion [6] or birefringence [7] are considered. Moreover, in single-
mode fibers, the periodic variation of GVD along the fiber length can also
give rise to MI in the normal GVD regime. This effect is similar to the
destabilization of a parametrically excited harmonic oscillator and is denoted
as parametric MI [8, 9, 10, 11, 12].
Optical fibers featuring random GVD variations were also extensively studied.
In the late 90s the exactly solvable white noise process was considered [9,
13, 14, 15]. More recently some of the present authors focused on different
processes such as localized GVD kicks [16] and coloured processes of low-pass
and band-pass type [17].
So far, both periodic and random fluctuations have been mostly assumed to
occur around an average GVD different from zero (and more often normal, to
avoid competition with conventional MI, which exhibits much higher MI gain).
The fluctuations can be large, though.
On the contrary, systems with zero average GVD have attracted a lot of
attention for the suppression of the dispersion-induced pulse broadening and
the optimization of nonlinear pulse transmission [18, 19]. This approach is
commonly denoted as dispersion-management (DM): segments of positive and
negative GVD alternate along the fiber. The study of MI in periodic DM fiber
links [20] shows a behavior different from the parametric MI: a threshold is
found for the segment lengths below which no MI appears. The amplitude of GVD
variations influences the MI spectral range, but has no effect on this
threshold.
Here we consider random fluctuations of the DM segment lengths. While pulse
propagation in a similar system was analyzed in Ref. [21], we focus here on
MI, by applying the technique developed in Ref. [14, 16]. After deriving some
analytical relations for uniformly distributed fluctuations around the
periodic arrangement, we compare them to numerical solutions. We find MI bands
of purely stochastic origin and characterize the transition from periodic to
stochastic DM.
Figure 1: Schematic representation of the GVD profile in a typical fiber
realisation.
We consider the propagation of optical pulses ruled by the nonlinear
Schrödinger equation (NLSE)[18],
$i\partial_{z}U-\frac{1}{2}\beta_{2}(z)\partial_{tt}U+\gamma|U|^{2}U=0,$ (1)
where $U(t,z)$ is the complex envelope of the optical field, ($t$,$z$) are the
physical time and propagation distance in a frame moving at the group velocity
of the fiber mode, $\gamma$ the (constant) nonlinear coefficient, and
$\beta_{2}(z)=\pm\beta_{2}^{0}$ ($\beta_{2}^{0}>0$) is the GVD, which takes
only two values. As schematically illustrated in Fig. 1, the sign changes
occur at $z_{1},z_{2},\ldots,z_{N}$, where $z_{n}=z_{n-1}+L_{n}$,
$n=1,2,\ldots,N$ and $z_{0}=0$. The lengths $L_{n}$ are independent,
identically distributed random variables with uniform probability distribution
function in $[\bar{L}(1-\varepsilon),\bar{L}(1+\varepsilon)]$, where $\bar{L}$
is the average length of one fiber segment (half of the DM period) and
$\varepsilon$ the amplitude of the fluctuation.
Equation (1) has a continuous wave ($t$-independent) solution
$U_{0}(z)=\sqrt{P}\exp(i\gamma Pz)$. In order to study its stability, we
insert in (1) the Ansatz
${U}(z,t)=\left[\sqrt{P}+\check{x}_{1}(z,t)+i\check{x}_{2}(z,t)\right]\exp(i\gamma
Pz)$, where $\check{x}_{1,2}$ are assumed to be small, linearize and Fourier-
transform the resulting equation with respect to $t$ ($\omega$ is used as the
associated angular frequency detuning from the carrier $U_{0}$). We obtain
$\frac{\mathrm{d}x}{\mathrm{d}z}=\begin{bmatrix}0&-g(z)\\\
h(z)&0\end{bmatrix}x,$ (2)
with $x\equiv(x_{1},x_{2})^{\mathrm{T}}$ (functions of $\omega$ and $z$),
$g(z)=\beta_{2}(z)\frac{\omega^{2}}{2}$ and $h(z)=g(z)+2\gamma P$. (2) is a
system of stochastic differential equations (SDEs) for each value $\omega$.
Equation (2) is solved in the interval $(z_{0},z_{N})$ as a product of random
matrices, depending on the random variables $L_{n}$ as
$\displaystyle x(z_{N})$ $\displaystyle=T_{-}(L_{N})T_{+}(L_{N-1})\ldots
T_{-}(L_{2})T_{+}(L_{1})x(z_{0}),$ (3) $\displaystyle T_{\pm}(L_{n})$
$\displaystyle=\begin{bmatrix}\cos(k_{\pm}L_{n})&-\mu_{\pm}\sin(k_{\pm}L_{n})\\\
\mu_{\pm}^{-1}\sin(k_{\pm}L_{n})&\cos(k_{\pm}L_{n})\end{bmatrix},$ (4)
with
$k_{\pm}^{2}=\pm\frac{\beta_{2}^{0}\omega^{2}}{2}\left(\pm\frac{\beta_{2}^{0}\omega^{2}}{2}+2\gamma
P\right)$, $\mu_{\pm}=\pm\frac{\beta_{2}^{0}\omega^{2}}{2k_{\pm}}$. The sign
$\pm$ is chosen according to the sign of GVD in the corresponding segment. The
wave-number $k_{+}$ is always real and positive, whereas $k_{-}$ is purely
imaginary in the conventional MI band $0\leq\omega\leq\sqrt{\frac{4\gamma
P}{\beta_{2}^{0}}}$.
If the DM link is periodic, i.e., $\varepsilon=0$, we can apply Floquet theory
[20]. The unit cell of DM to be periodically replicated is represented by one
positive and one negative GVD trait of length $\bar{L}$. The MI gain is
defined as
$G_{1}(\omega)\equiv\frac{1}{2\bar{L}}\ln\max\\{|\tilde{\lambda}|,1\\}$, where
$\tilde{\lambda}$ is the eigenvalue of the monodromy matrix associated to (2)
of largest modulus. This corresponds to $T_{2\bar{L}}\equiv
T_{-}(\bar{L})T_{+}(\bar{L})$. In [20], it was observed that a critical value
$\bar{L}\approx 1.07$ exists, below which $G_{1}=0$ identically. The MI gain
is represented as a false-color map in Fig. 2 for $\omega\geq 0$ (for
$\omega\leq 0$ we obtain its mirror image). The MI gain exhibits several
lobes, in general.
For random $L_{n}$, in Ref. [14, 16] it was shown that we have to resort to
the Lyapunov exponent of the random linear map: the sample gain
$G_{\mathrm{S}}(\omega)\equiv\lim_{z_{N}\to\infty}\frac{1}{z_{N}}\ln\lVert
T_{+}(L_{N})T_{-}(L_{n-1})\ldots
T_{+}(L_{2})T_{-}(L_{1})x(0,\omega)\rVert^{2}$ converges for almost all
realizations of the fiber and is a deterministic quantity.
By taking the average of (3) and letting $N\rightarrow\infty$, we can estimate
the MI gain as
$G_{1}(\omega)\equiv\frac{1}{2\bar{L}}\ln\max\left\\{|\lambda|,1\right\\}$,
where $\lambda$ is the largest modulus eigenvalue of
$\overline{T}\equiv\langle T_{+}\rangle\langle T_{-}\rangle$ and the angle
brackets denote the expectation operation over the random lengths $L_{n}$. We
can split the averages because $L_{n}$ are all mutually independent.
Elementary integration gives
$\displaystyle\langle\cos(mk_{\pm}n)\rangle$
$\displaystyle=\cos(mk_{\pm}\bar{L})\frac{\sin
mk_{\pm}\varepsilon}{mk_{\pm}\varepsilon}$ (5)
$\displaystyle\langle\sin(mk_{\pm}n)\rangle$
$\displaystyle=\sin(mk_{\pm}\bar{L})\frac{\sin
mk_{\pm}\varepsilon}{mk_{\pm}\varepsilon},$
thus
$\langle T_{\pm}\rangle=\frac{\sin
k_{\pm}\varepsilon}{k_{\pm}\varepsilon}\begin{bmatrix}\cos(k_{\pm}\bar{L})&-\mu_{\pm}\sin(k_{\pm}\bar{L})\\\
\mu_{\pm}^{-1}\sin(k_{\pm}\bar{L})&\cos(k_{\pm}\bar{L})\end{bmatrix}$ (6)
and $\overline{T}$ can be easily obtained as
Figure 2: MI gain for a periodic DM fiber as a function of detuning $\omega$
and period $L$. The red dashed horizontal line identifies $\bar{L}=1.07$ (MI
threshold), while the cyan dotted line denotes the reference value
$\bar{L}=1.15$, used in the inset and below in the random length fluctuation
examples.
$\overline{T}=\frac{\sin(k_{-}\varepsilon)}{k_{-}\varepsilon}\frac{\sin(k_{+}\varepsilon)}{k_{+}\varepsilon}T_{2\bar{L}}.$
(7)
As common in random dynamical systems [22], we may have $G_{1}=0$ for all
$\omega$, implying that $x_{1,2}$ decay on average. Particularly, it is
apparent that for $\bar{L}<1.07$, where both eigenvalues of $T_{2\bar{L}}$ are
on the unit circle, (7) implies that $G_{1}(\omega)=0$ identically, because
$\overline{T}$ differs from $T_{2\bar{L}}$ only by a factor no larger than
one.
Nevertheless, a different kind of instabilitity may occur, to understand which
the study of second moments is required. We let $X_{1}=x_{1}^{2}$,
$X_{2}=x_{2}^{2}$, and $X_{3}=x_{1}x_{2}$ and derive from (2)
$\frac{\mathrm{d}}{\mathrm{d}z}{X}=\begin{bmatrix}0&0&-2g(z)\\\ 0&0&2h(z)\\\
h(z)&-g(z)&0\end{bmatrix}{X},$ (8)
with $X\equiv(X_{1},X_{2},X_{3})^{\mathrm{T}}$. (8) can be again solved in
terms of transfer matrices $X(z_{n})=M_{\pm}(L_{n})X(z_{n-1})$, with
$M_{\pm}(L_{n})=\begin{bmatrix}\cos^{2}k_{\pm}L_{n}&\mu_{\pm}^{2}\sin^{2}k_{\pm}L_{n}&-\mu_{\pm}\sin
2k_{\pm}L_{n}\\\
\mu_{\pm}^{-2}\sin^{2}k_{\pm}L_{n}&\cos^{2}k_{\pm}L_{n}&\mu_{\pm}^{-1}\sin
2k_{\pm}L_{n}\\\ \frac{\mu_{\pm}^{-1}}{2}\sin
2k_{\pm}L_{n}&-\frac{\mu_{\pm}}{2}\sin 2k_{\pm}L_{n}&\cos
2k_{\pm}L_{n}\end{bmatrix}.$ (9)
The generic DM unit cell is associated to $M_{-}(L_{n})M_{+}(L_{n-1})$. After
Ref. [14, 16], we define
$G_{2}(\omega)\equiv\frac{1}{4\bar{L}}\ln\max\\{|\kappa|,1\\}$, where here
$\kappa$ is the largest modulus eigenvalue of $\overline{M}\equiv\langle
M_{+}\rangle\langle M_{-}\rangle$. In the periodic limit, the monodromy matrix
associated to (8) is $M_{2\bar{L}}\equiv M_{-}(\bar{L})M_{+}(\bar{L})$ and
$G_{2}=G_{1}$, for every $\omega$.
In the random case, we can easily find the expression of $\overline{M}$
analytically by using (5), but the expression is rather lengthy and it is not
reported here.
In contrast to $\overline{T}$, $\overline{M}$ is not, in general, trivially
proportional to $M_{2\bar{L}}$. This simple algebraic consideration implies
that we may have $G_{2}>0$ even when $G_{1}=0$. Therefore new MI sidebands of
purely stochastic origin exist. The eigenvalues $\kappa$ of $\overline{M}$ may
be found analytically, too. Their expression is very involved, though; we thus
rely on a numerical routine.
In order to assess the accuracy of our estimates, we also solve (2)
numerically, by taking a fixed number, $N=20$, of fiber segments. We take
$(x_{1}(0),x_{2}(0))^{T}=(1,0)$ [equivalently, $X(0)^{\mathrm{T}}=(1,0,0)$]
and multiply by the transfer matrix (4) $N$ times alternating the GVD sign
according to (3). We compute
$P_{\mathrm{out}}=x_{1}^{2}(z_{N})+x_{2}^{2}(z_{N})$ (obviously,
$P_{\mathrm{in}}=x_{1}^{2}(0)+x_{2}^{2}(0)=1$). We repeat this calculation
taking $N_{\mathrm{iter}}$ different fiber realizations. The mean gain is
defined as either [23, 16, 17]
$\displaystyle\overline{G}_{1}(\omega;N)$
$\displaystyle\equiv\frac{1}{N\bar{L}}\ln\left(\left|\langle
x_{1}(z_{N})\rangle\right|+\left|\langle x_{2}(z_{N})\rangle\right|\right)$
(10) $\displaystyle\overline{G}_{2}(\omega;N)$
$\displaystyle\equiv\frac{1}{2N\bar{L}}\ln\left\langle\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\right\rangle,$
(11)
which are compared to either $G_{1}$ or $G_{2}$, respectively.
Figure 3: MI gain curves for different values of $\bar{L}$ and $\varepsilon$,
for $\omega\geq 0$. (a) $\bar{L}=1$, $\varepsilon=0.2$; (b) $\bar{L}=1$,
$\varepsilon=0.5$; (c) $\bar{L}=1.15$, $\varepsilon=0.2$; (d) $\bar{L}=1.15$,
$\varepsilon=0.5$. Blue crosses (resp. purple circles) represent
$\overline{G}_{2}$ (resp. $\overline{G}_{1}$) from numerical data, the solid
yellow (resp. dash-dotted red) lines the theoretical estimates $G_{2}$ (resp.
$G_{1}$), the dashed blue (resp. dotted purple) lines the semi-analytical
estimates $\tilde{G}_{2}$ (resp. $\tilde{G}_{1}$). In (c)-(d), we include as
thin green dashed line the MI gain in the periodic case $\varepsilon=0$.
For definiteness, we take $\gamma=P=\beta_{2}^{0}=1$, which amounts to
introduce the normalized distance $z/z_{\mathrm{nl}}\rightarrow z$, time
$t/t_{0}\rightarrow t$, and field $U/\sqrt{P}\rightarrow U$ , where
$z_{\mathrm{nl}}=(\gamma P)^{-1}$ is the so-called nonlinear length and
$t_{0}=\sqrt{\beta_{2}^{0}z_{\mathrm{nl}}}$ is a characteristic time. The
conventional MI in anomalous GVD thus reaches the maximum value of
$G_{1,\mathrm{max}}=G_{2,\mathrm{max}}=1$ at $\omega=\sqrt{2}$; the MI
sidelobes are found in $0\leq|\omega|\leq 2$. We show in Fig. 3 four
illustrative examples of MI sidelobes. We notice that, in general,
$\overline{G}_{2}$ converges to $G_{2}$ for $N_{\mathrm{iter}}=1\times
10^{6}$, whereas in general a much larger $N_{\mathrm{iter}}=1\times
10^{7}-1\times 10^{8}$ is required to achieve a stable and reliable estimate
of $\overline{G}_{1}$.
We notice that $G_{1}=G_{2}=0$ is expected at $\omega=0$. $G_{1}$ exhibits a
single lobe (dash-dotted red lines), while $G_{2}$ grows monotonically to
reach a maximum value at around $\omega\approx 2$, then decays in an
oscillatory way (solid yellow lines). This is not the case in numerical
results (blue crosses and purple circles), which are affected by the limited
size of the numerical domain and present a finite $\overline{G}_{1,2}$ at
$\omega=0$. This can be quantified by deriving alternative semi-analytical
estimates of gain. We notice that $T_{-}(L_{N})\ldots
T_{+}(L_{1})\approx\overline{T}^{N/2}$ (resp. $M_{-}(L_{N})\ldots
M_{+}(L_{1})\approx\overline{M}^{N/2}$) and define
$\displaystyle\tilde{G}_{1}$
$\displaystyle\equiv\frac{1}{N\bar{L}}\ln\left[\left(\overline{T}^{\frac{N}{2}}\right)_{11}+\left(\overline{T}^{\frac{N}{2}}\right)_{21}\right],$
(12) $\displaystyle\tilde{G}_{2}$
$\displaystyle\equiv\frac{1}{2N\bar{L}}\ln\left[\left(\overline{M}^{\frac{N}{2}}\right)_{11}+\left(\overline{M}^{\frac{N}{2}}\right)_{21}\right],$
(13)
where matrix power are computed numerically and the subscripts refer to the
corresponding numerically computed matrix elements. We notice in Fig. 3 that
in every case this estimate performs very well not only for $\omega\approx 0$,
but in the whole domain.
In Fig. 3(a)-(b), $\bar{L}=1<1.07$ and $\varepsilon$ increases from $0.2$ to
$0.5$. The MI is of purely stochastic origin. The local maximum value achieved
by $\overline{G}_{2}$ at $\omega=\omega_{2,\mathrm{max}}>0$, say
$G_{2,\mathrm{max}}$, increases with $\varepsilon$ and becomes of the same
order of magnitude as the conventional MI (i.e., for constant anomalous GVD).
The width of the sidelobes increases significantly as well.
In Fig. 3(c)-(d), $\bar{L}=1.15>1.07$. There is thus a competition between the
periodic and the stochastic effects. We include also the periodic-DM MI
sidelobe for comparison (thin green dashed line).
For $\varepsilon=0.2$ [Fig. 3(c)] the random fluctuations yield a broadened MI
sidelobe. In contrast to the case of constant anomalous GVD perturbed by
white-noise, where the broadening is accompanied by a reduction of
$G_{2,\mathrm{max}}$ [9], here this value is slightly enhanced. We notice also
that $\overline{G}_{1}$ is always less than its periodic counterpart,
consistently with (7). Comparison of Fig. 3(a) and Fig. 3(c) shows that the
main sidelobe appearing in the former is located in the same region of the
periodic sidelobe. The random fluctuations facilitate the emergence of the MI
sidelobes in a range that coincides with the periodic DM.
For a larger fluctuation ($\varepsilon=0.5$), Fig. 3(d), the residual effect
of periodicity is completely erased and we obtain a single wide lobe similar
to the corresponding below-threshold example of Fig. 3(b), but with a larger
$G_{2,\mathrm{max}}$.
Figure 4: (a) Maximum gain values and (b) their corresponding $\omega$ as a
function of $\varepsilon$. The cross (resp. plus) markers correspond to the
maxima $G_{2,\mathrm{max}}$ for $\bar{L}=1$ (resp. $\bar{L}=1.15$). The dashed
(resp. solid) blue lines correspond to the maxima of $G_{2}$ for $\bar{L}=1$
(resp. $\bar{L}=1.15$). The green dash-dotted lines report for reference the
constant values found in the periodic limit, i.e., the maxima of $G_{1}=G_{2}$
for $\varepsilon=0$. Finally the red circles show the values of
$G_{1,\mathrm{max}}$ and $\omega_{1,\mathrm{max}}$ for $\bar{L}=1.15$, for
which red dotted lines illustrate the corresponding maxima of $G_{1}$. In
panel (b) the line stops at $\varepsilon\approx 0.54$ (cut-off for $G_{1}$).
In order to summarize our findings, we show in Fig. 4(a)-(b), respectively,
$G_{j,\mathrm{max}}$ and the corresponding $\omega$ value,
$\omega_{j,\mathrm{max}}$, with $j=1,2$, as a function of $\varepsilon$.
Obviously, $G_{1,\mathrm{max}}$ is the local maximum value of
$\overline{G}_{1}$ at $\omega=\omega_{1,\mathrm{max}}>0$. We compare them to
their corresponding theoretical estimates, i.e., the maxima of $G_{1}$ (resp.
$G_{2}$). Solid blue lines, crosses, red lines and circles correspond to
$\bar{L}=1.15$, while dashed lines and pluses correspond to $\bar{L}=1$. In
Fig. 4(a) we notice that $G_{2,\mathrm{max}}$ increases monotonically with
$\varepsilon$. As expected, for $\varepsilon\to 0$ the gain vanishes for
$\bar{L}=1$ and converges to the periodic $G_{1,\mathrm{max}}=0.31$ (for
$\varepsilon=0$, dash-dotted green line) for $\bar{L}=1.15$. For large
$\varepsilon$ they converge to two very similar values, which is around 30$\%$
smaller than the conventional MI value. The dotted red line in Fig. 4(a) shows
the maxima of $G_{1}$ obtained from (7). For $\varepsilon>0.54$, $G_{1}=0$ for
every $\omega$. For this value of the fluctuation amplitudes, randomness
completely overrule the effects of periodicity. This is corroborated by the
values ${G}_{1,\mathrm{max}}$ (red circles), which match very well with the
theoretical estimates (provided that a large enough $N_{\mathrm{iter}}$ is
chosen). In Fig. 4(b) we observe that for both $\bar{L}$,
$\omega_{2,\mathrm{max}}$ decreases with $\varepsilon$ and converges to a
value larger than the conventional MI value. It is always below its periodic
counterpart for $\bar{L}=1.15$, apart from numerical fluctuations. We also
notice that ${\omega}_{2,\mathrm{max}}<{\omega}_{1,\mathrm{max}}<2.18$, i.e.,
slightly below the $\varepsilon=0$ value. For $\bar{L}=1$,
$\omega_{2,\mathrm{max}}$ is above its periodic counterpart and crosses it for
$\varepsilon\approx 0.3$. The theoretical estimates work very well for every
considered value of $\varepsilon$.
To conclude, we investigated the effect of uniformly distributed random
fluctuations of the _length_ of DM fiber links. We considered the MI problem
and developed an analytical technique to estimate the instability gain. The MI
gain attains values comparable with the conventional ones in a homogeneous
anomalous GVD fiber and up to 50$\%$ larger those found for the periodic
arrangement. Comparison to the direct numerical solution in a Monte Carlo
fashion confirm the soundness of the method.
This may be of of interest for tailoring and control of MI sidebands for
telecommunications and parametric sources.
## Acknowledgments
The present research was supported by IRCICA (USR 3380 CNRS), Agence Nationale
de la Recherche (Programme Investissements d’Avenir, I-SITE VERIFICO, Labex
CEMPI); Ministry of Higher Education and Research; Hauts de France Council;
European Regional Development Fund (Photonics for Society P4S, Wavetech), CNRS
(IRP FELANI).
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|
# On the Power of Foundation Models
Yang Yuan IIIS, Tsinghua University Shanghai Artificial Intelligence
Laboratory Shanghai Qi Zhi Institute
###### Abstract
With infinitely many high-quality data points, infinite computational power,
an infinitely large foundation model with a perfect training algorithm and
guaranteed zero generalization error on the pretext task, can the model be
used for everything? This question cannot be answered by the existing theory
of representation, optimization or generalization, because the issues they
mainly investigate are assumed to be nonexistent here. In this paper, we show
that category theory provides powerful machinery to answer this question. We
have proved three results. The first one limits the power of prompt-based
learning, saying that the model can solve a downstream task with prompts if
and only if the task is representable. The second one says fine tuning does
not have this limit, as a foundation model with the minimum required power (up
to symmetry) can theoretically solve downstream tasks for the category defined
by pretext task, with fine tuning and enough resources. Our final result can
be seen as a new type of generalization theorem, showing that the foundation
model can generate unseen objects from the target category (e.g., images)
using the structural information from the source category (e.g., texts). Along
the way, we provide a categorical framework for supervised and self-supervised
learning, which might be of independent interest.
## 1 Introduction
Foundation models have recently exhibited remarkable proficiency in addressing
a myriad of complex downstream tasks that were very difficult or impossible
for the previous methods (Ramesh et al.,, 2021; Rombach et al.,, 2022; Ramesh
et al.,, 2022; Sohl-Dickstein et al.,, 2015; Brown et al.,, 2020; Radford et
al.,, 2018, 2019; He et al.,, 2022). Being different in network structure,
dataset and training algorithms, the foundation models are similar in terms of
the training process: the model is first trained with a large unlabeled
dataset on a pretext task, and then being applied to other downstream tasks
with the parameters frozen. Freezing the parameter is necessary because
training a foundation model is very expensive, and the downstream tasks
usually have limited data samples, which makes retraining less attractive.
Given a set of downstream tasks, which of them are solvable by the foundation
models, and which are not? This is a very fundamental question. The existing
theory attacks this question through various perspectives like dataset quality
and quantity, computational power, network structure, etc. However, as we
collect trillions of data points, build gigantic GPU data centers, and
optimize huge networks with billions of parameters, a simple question just
pops up:
Where are we heading to?
Specifically, with infinitely many high-quality data points, infinite
computational power, an infinitely large foundation model with a perfect
training algorithm and guaranteed zero generalization error on the pretext
task, can the model be used for every downstream task? As we will see, this
question is not about model representation, optimization or generalization,
but structural representation of the tasks. Category theory, as the theory of
mathematical structures, is the ideal tool for answering this question.
For the downstream tasks, there are mainly two types of methods: prompt tuning
and fine tuning. Prompt tuning does not train with the downstream tasks.
Instead, it only sends a task specific prompt to the model, so that the model
can “switch” its working mode for solving the task. Fine tuning trains a small
network connecting to the foundation model with the labeled dataset for the
downstream task.
In Theorem 1, we show that with prompt tuning, the model can solve the task if
and only if the task is “representable” in the category defined by the pretext
task. If the category does not have complicated structures, our theorem
indicates that the power of prompt tuning is limited. For example, in
Corollary 1, we show that by training with the pretext task of predicting the
rotation of an image Gidaris et al., (2018), the foundation model is not able
to solve complicated downstream tasks like segmentation or classification.
On the other hand, the results on fine tuning is more promising. Our Theorem 2
proves that for the foundation model with the minimum required power (up to
symmetry) for the pretext task and enough resources including training data,
it can potentially solve any downstream tasks for the category defined by
pretext task. The role of pretext task is crucial in the sense that if the
pretext task fails to extract adequate information from the unlabeled dataset,
the power of fine tuning remains restricted.
Along the way, we have provided a categorical framework for machine learning.
Interestingly, the framework injects the learning perspective to category
theory as well. Therefore, we also proved a generalization theorem for
structural learning (Theorem 3), which explains why self-supervised learning
for text-image generation tasks can generate images like avocado chair, which
do not exist in the dataset or real world. Theorem 3 can be easily generalized
to the compositional theorem of multiple categories, see Theorem 4.
Unlike most machine learning theory papers, our paper does not have any
assumptions on the data distribution or network structure. Instead, we take
the bird’s-eye view that is model oblivious, and only focuses on the structure
defined by the pretext task. It is indeed possible that by designing a special
network, one may get a more powerful model with better performance. However,
we stick with our setting because:
* •
Empirically, people do not customize network structures for different tasks.
Instead, they tend to use similar structures like ResNet He et al., (2016) or
Transformer Vaswani et al., (2017). By the no free lunch theorem Shalev-
Shwartz and Ben-David, (2014), if the model does not contain task-specific
prior information, it will not be able to completely solve all the tasks. In
other words, the standard models are not universally competent, and it is
likely that the limitation that we derived from the model oblivious setting,
also applies to the practical settings.
* •
Pretext task design is a central problem in self-supervised learning.
Currently, there are various datasets floating around, and a wide range of
diverse tasks to solve, but the practitioners do not really know what kinds of
pretext tasks to pick for solving a given task, or the limitations of each
pretrained model. They get the intuition by trial and error with experiments,
which are both expensive and noisy. Our framework will help them to think
about this problem in a more mathematical and systematic way, and provide
guidance for better pretext task design.
## 2 Related Work
### 2.1 Self-supervised learning
Self-supervised learning. Recently, researchers have proposed many self-
supervised learning algorithms for foundation models, including contrastive
methods (Chen et al.,, 2020; He et al.,, 2020; Grill et al.,, 2020; Chen and
He,, 2021; Noroozi and Favaro,, 2016; Zbontar et al.,, 2021), masked image
models (He et al.,, 2022; Dosovitskiy et al.,, 2020; Doersch et al.,, 2015;
Pathak et al.,, 2016), masked language models (Devlin et al.,, 2018; Raffel et
al.,, 2020), pure language models (Brown et al.,, 2020; Radford et al.,, 2018,
2019), and with other pretext tasks (Oord et al.,, 2018; Gidaris et al.,,
2018; Clark et al.,, 2020; Noroozi and Favaro,, 2016; Pathak et al.,, 2017).
Multimodal learning. Self-supervised learning can also be applied to
multimodal learning, including text + image (Ramesh et al.,, 2021; Rombach et
al.,, 2022; Ramesh et al.,, 2022; Sohl-Dickstein et al.,, 2015), video + audio
(Arandjelovic and Zisserman,, 2018). For generating images for the multimodal
tasks, diffusion model is the state-of-the-art approach (Rombach et al.,,
2022; Sohl-Dickstein et al.,, 2015; Dhariwal and Nichol,, 2021; Ho et al.,,
2020).
Prompt tuning. There are various prompt tuning methods, including the discrete
prompts (Brown et al.,, 2020; Jiang et al.,, 2020; Shin et al.,, 2020; Gao et
al.,, 2020) and continuous prompts (Liu et al.,, 2021; Li and Liang,, 2021).
### 2.2 Theory for deep learning and self-supervised learning
Theory for deep learning has been an active research area recently.
Optimization theory focuses on how and why first order methods like stochastic
gradient descent finds the local/global optimum of the neural networks (Du et
al., 2019b, ; Arora et al., 2019a, ; Du et al.,, 2018; Allen-Zhu et al.,
2019b, ; Allen-Zhu et al., 2019a, ; Zou et al.,, 2020; Li and Yuan,, 2017).
Generalization theory focuses on how the performance of the model in the
training set transfers to the population distribution (Allen-Zhu et al.,
2019a, ; Arora et al., 2019a, ; Bartlett et al.,, 2017, 2020; Yin et al.,,
2019). Representation theory focuses on the representation power of the neural
networks (Hornik et al.,, 1989; Cybenko,, 1989; Raghu et al.,, 2017). There
are also theoretical results on analyzing various aspects of reinforcement
learning (Du et al., 2019a, ; Du et al., 2019c, ; Jin et al.,, 2018; Cai et
al.,, 2020).
Theory for self-supervised learning. There are many interesting theory results
for self-supervised learning learning (Wen and Li,, 2021, 2022; Luo et al.,,
2022; HaoChen et al.,, 2021; Arora et al., 2019b, ; Tosh et al.,, 2021; Lee et
al.,, 2021; Zimmermann et al.,, 2021). For example, HaoChen et al., (2021)
show that SimCLR is essentially computing spectral graph clustering on the
unlabeled dataset. Liu, (2022) gives a better rectified flow algorithm with
elegant theoretical guarantees for improving the diffusion models.
Application of category theory. Category theory has been applied to many
research areas (Fong and Spivak,, 2018), including physics (Marquis,, 2008;
Kuś and Skowron,, 2019), design (Censi,, 2015), and machine learning (Shiebler
et al.,, 2021; Mahadevan, 2022a, ; Mahadevan, 2022b, ). The most relevant
paper on category theory might be Bradley et al., (2022), where they provide
an enriched category theory of natural language.
## 3 Preliminaries
Category theory is used in almost all areas of mathematics. Here we only
introduce the necessary notions for understanding the results of our paper,
and skip many important details (e.g., universe and diagram). Curious readers
may check Mac Lane, (2013); Riehl, (2017); Adámek et al., (1990) for more
comprehensive introductions.
A category $\mathcal{C}$ has a set of objects $\mathrm{Ob}(\mathcal{C})$, and
a set of morphisms $\mathrm{Hom}_{\mathcal{C}}(X,Y)$ from $X$ to $Y$ for every
$X,Y\in\mathrm{Ob}(\mathcal{C})$. Given
$f\in\mathrm{Hom}_{\mathcal{C}}(X,Y),g\in\mathrm{Hom}_{\mathcal{C}}(Y,Z)$, we
define their composition as $g\circ f\in\mathrm{Hom}_{\mathcal{C}}(X,Z)$.
Notice that $\circ$ is associative, i.e., $(h\circ g)\circ f=h\circ(g\circ
f)$. For every $X\in\mathrm{Ob}(\mathcal{C})$, there exists an unique identity
morphism $\mathrm{id}_{X}\in\mathrm{Hom}_{\mathcal{C}}(X,X)$. A morphism
$f:X\rightarrow Y$ is an isomorphism if there exists $g:X\leftarrow Y$ such
that $f\circ g=\mathrm{id}_{Y}$ and $g\circ f=\mathrm{id}_{X}$. In this case,
we say $X$ and $Y$ are isomorphic and write $X\simeq Y$.
Given a category $\mathcal{C}$, we define its opposite
$\mathcal{C}^{\mathrm{op}}$ by setting
$\mathrm{Ob}(\mathcal{C}^{\mathrm{op}})=\mathrm{Ob}(\mathcal{C})$ and
$\mathrm{Hom}_{\mathcal{C}^{\mathrm{op}}}(X,Y)=\mathrm{Hom}_{\mathcal{C}}(Y,X)$.
Moreover, given
$f\in\mathrm{Hom}_{\mathcal{C}^{\mathrm{op}}}(X,Y),g\in\mathrm{Hom}_{\mathcal{C}^{\mathrm{op}}}(Y,Z)$,
the new composition is
$g\mathrel{\overset{\makebox[0.0pt]{\mbox{op}}}{\circ}}f=f\circ
g\in\mathrm{Hom}_{\mathcal{C}^{\mathrm{op}}}(X,Z)$.
We define Set to be the category of sets, where the objects are sets, and
$\mathrm{Hom}_{\textbf{Set}}(X,Y)$ is the set of all functions with domain $X$
and codomain $Y$. Notice that we ignore the subtleties about the universe for
better presentation, so here just assume that Set does not contain strange
objects like a set containing all sets.
Functor is like a function between two categories. Given two categories
$\mathcal{C},\mathcal{C}^{\prime}$, a functor
$F:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ maps objects from $\mathcal{C}$
to $\mathcal{C}^{\prime}$ with
$F:\mathrm{Ob}(\mathcal{C})\rightarrow\mathrm{Ob}(\mathcal{C}^{\prime})$ and
morphisms from $\mathcal{C}$ to $\mathcal{C}^{\prime}$ with
$F:\mathrm{Hom}_{\mathcal{C}}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal{C}^{\prime}}(F(X),F(Y))$
for all $X,Y\in\mathcal{C}$, so that $F$ preserves identity and composition.
Formally, we have $F(\mathrm{id}_{X})=\mathrm{id}_{F(X)}$ for all
$X\in\mathcal{C}$, and $F(g\circ f)=F(g)\circ F(f)$ for all $f:X\rightarrow
Y,g:Y\rightarrow Z$. We call $F$ an embedding functor, if it is injective on
morphisms. Notice that if $F$ is an embedding, it is also injective on
objects. We call $F$ a full functor, if for any $X,Y\in\mathcal{C}$,
$\mathrm{Hom}_{\mathcal{C}}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal{C}^{\prime}}(FX,FY)$
is surjective.
The morphisms of functors, also called the natural transformation, is the way
to transform the functors while preserving the structure. Given two categories
$\mathcal{C},\mathcal{C}^{\prime}$, and two functors $F_{1},F_{2}$ from
$\mathcal{C}$ to $\mathcal{C}^{\prime}$. A morphism of functors
$\theta:F_{1}\rightarrow F_{2}$ has a morphism $\theta_{X}:F_{1}(X)\rightarrow
F_{2}(X)$ for all $X\in\mathcal{C}$ such that for all $f:X\rightarrow
Y\in\mathrm{Hom}_{\mathcal{C}}(X,Y)$, we have
$\theta_{Y}(F_{1}(f)(F_{1}(X)))=F_{2}(f)(\theta_{X}(F_{1}(X)))$. We write
$F_{1}\simeq F_{2}$ if there exists an isomorphism between $F_{1}$ and
$F_{2}$. A functor $F:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ is an
isomorphism of categories if there exists
$G:\mathcal{C}^{\prime}\rightarrow\mathcal{C}$ such that $G\circ F(X)=X$ and
$F\circ G(Y)=Y$ for all $X\in\mathcal{C},Y\in\mathcal{C}^{\prime}$, and
similarly for the morphisms. In this case, we say $\mathcal{C}$ and
$\mathcal{C}^{\prime}$ are isomorphic and write
$\mathcal{C}\simeq\mathcal{C}^{\prime}$.
A category $\mathcal{A}$ is a subcategory of a category $\mathcal{B}$, if
1. 1.
$\mathrm{Ob}(\mathcal{A})\subseteq\mathrm{Ob}(\mathcal{B})$,
2. 2.
for each
$A,A^{\prime}\in\mathrm{Ob}(\mathcal{A}),\mathrm{Hom}_{\mathcal{A}}(A,A^{\prime})\subseteq\mathrm{Hom}_{\mathcal{B}}(A,A^{\prime})$,
3. 3.
for each $\mathcal{A}$-object $A$, the $\mathcal{B}$-identity on $A$ is the
$\mathcal{A}$-identity on $A$;
4. 4.
the composition law in $\mathcal{A}$ is the restriction of the composition law
in $\mathcal{B}$ to the morphisms of $\mathcal{A}$.
Moreover, $\mathcal{A}$ is a full subcategory of $\mathcal{B}$, if it is a
subcategory of $\mathcal{B}$ and for each for each
$A,A^{\prime}\in\mathrm{Ob}(\mathcal{A}),\mathrm{Hom}_{\mathcal{A}}(A,A^{\prime})=\mathrm{Hom}_{\mathcal{B}}(A,A^{\prime})$.
Every subcategory $\mathcal{A}$ of category $\mathcal{B}$ naturally defines an
inclusion functor $E:\mathcal{A}\hookrightarrow\mathcal{B}$, which is an
embedding. For such embeddings, we have the following lemma.
###### Lemma 1 (Adámek et al., (1990)).
A functor $F:\mathcal{C}\rightarrow\mathcal{B}$ is a full embedding if and
only if there exists a full subcategory $\mathcal{A}$ of $\mathcal{B}$ with
inclusion functor $E:\mathcal{A}\hookrightarrow\mathcal{B}$ and an isomorphism
$G:\mathcal{C}\rightarrow\mathcal{A}$ with $F=E\circ G$.
### 3.1 Reproducing Kernel Hilbert Space (RKHS)
Given two objects $X,Y\in\mathcal{C}$, consider a feature map
$f:\mathcal{C}\rightarrow\mathcal{H}$, where the feature space $\mathcal{H}$
is usually much larger than $\mathcal{C}$. We may define a kernel $k$ that
measures the similarity of $X$ and $Y$ as $k(x,y)\triangleq\langle
f(x),f(y)\rangle_{\mathcal{H}}$, i.e., the inner product between the two
object after mapping them to the feature space. For any vector
$T\in\mathcal{H}$, it also corresponds to a function
$T(\cdot):\mathcal{C}\rightarrow\mathbb{R}$, defined as $T(x)=\langle
T,f(x)\rangle_{\mathcal{H}}$. Specifically, $f(y)$ as a vector in
$\mathcal{H}$ also represents the function
$k(\cdot,y):\mathcal{C}\rightarrow\mathbb{R}$, because for any
$x\in\mathcal{C}$, we have $k(x,y)=\langle f(x),f(y)\rangle_{\mathcal{H}}$.
Formally, we have:
###### Definition 1 (Reproducing kernel Hilbert space).
Let $\mathcal{H}$ be a Hilbert space of $\mathbb{R}$-valued functions defined
on a non-empty set $\mathcal{C}$. A function
$k:\mathcal{C}\times\mathcal{C}\rightarrow\mathbb{R}$ is called a reproducing
kernel of $\mathcal{H}$, and $\mathcal{H}$ is a reproducing kernel Hilbert
space (RKHS), if $k$ satisfies
* •
$\forall x\in\mathcal{C},k(\cdot,x)\in\mathcal{H}$,
* •
$\forall x\in\mathcal{C},\forall f\in\mathcal{H},\langle
f,k(\cdot,x)\rangle_{\mathcal{H}}=f(x).$
## 4 Categorical Framework of Supervised Learning
In supervised learning, we have a population distribution $D=(D_{X},D_{Y})$,
representing the ground truth distribution of the input and output data
points. The training set $(X_{\mathrm{train}},Y_{\mathrm{train}})$ and test
set $(X_{\mathrm{test}},Y_{\mathrm{test}})$ are uniformly sampled from
$(D_{X},D_{Y})$. We hope to learn a function $f:X\rightarrow Y$ so that $f(x)$
accurately predicts the label $x\in X$. We also define a loss function
$L(f,x,y)$ to measure the distance between the prediction $f(x)$ and the
correct label, which is hopefully close to $0$. The loss function on a dataset
$(X,Y)$ is $L(f,X,Y)\triangleq\mathbb{E}_{(x,y)\sim(X,Y)}L(f,x,y)$. The task
of supervised learning, is to minimize the population loss
$L_{\mathrm{population}}\triangleq\mathbb{E}_{(X,Y)\sim(D_{X},D_{Y})}L(f,X,Y)$,
with access of the training set $(X_{\mathrm{train}},Y_{\mathrm{train}})$.
Using the language of category theory, we have two categories $\mathcal{X}$
and $\mathcal{Y}$, with objects $x,y$ as input and output data points,
respectively. To avoid confusion, below we switch notation from $x,y$ to $X,Y$
to represent the objects, as later we will not use $D_{X},D_{Y}$ any more. The
population distribution can be seen as a functor $F$ from $\mathcal{X}$ to
$\mathcal{Y}$, representing the correct label $Y$ given the input $X$. Due to
the inherent noise in the real world, there may not always exist the unique
correct label for each input $X$. In other words, the Bayesian optimal
solution does not give zero population loss. We will discussion this issue in
Section 7.3, and for now let us simply assume that the unique correct labels
always exist.
With this formulation, training/test set can be seen as samples over objects
in $\mathcal{X}$ with correct labels in $\mathcal{Y}$. Therefore, supervised
learning investigates the following question: can we learn a functor $F$ with
samples of $X\in\mathcal{X}$ and $F(X)~{}\in~{}\mathcal{Y}$?
Consider the special case that both categories are discrete, meaning that the
only morphisms existed are identity morphisms like
$\mathrm{id}_{X}\in\mathrm{Hom}_{\mathcal{X}}(X,X)$ and
$\mathrm{id}_{Y}\in\mathrm{Hom}_{\mathcal{Y}}(Y,Y)$. In other words, both
categories are sets without any morphisms between different objects. In this
case, learning $F$ is impossible, because it maps a set to another set without
any prior knowledge. The no free lunch theorem tells us that unless we have
sample size larger than half of the set size, the functor we learned will have
constant generalization error with constant probability.
Generalization theory deals with this problem by assuming that $F$ is in a
predefined hypothesis class $\mathcal{H}$, or close to a function in
$\mathcal{H}$. In category theory, we do not add restrictions to the functors,
but to the structure of the categories instead. For example, if we know
$\mathcal{X}$ has a linear structure, and $F$ preserves the structure, it is
possible to learn $F$ with a few samples.
This formulation seems useless, as it does not even characterize the loss
function, which is crucial in supervised learning. This is because category
theory takes the bird’s-eye view. Our categorical framework will not replace
the classical framework, or generate better supervised learning algorithms.
Instead, it treats the existing supervised learning algorithms a subroutine or
a building block, which can be incorporated into a bigger picture. Therefore,
it does not care about the loss functions or optimization process, which are
treated as the implementation details. Instead, it focuses on the structure of
the categories and functors, and tries to understand whether certain functors
are learnable or not. It also investigates whether the ability of mastering at
one or more tasks can be generalized to other tasks.
## 5 Categorical Framework of Self-supervised Learning
### 5.1 Pretext tasks and morphisms
In self-supervised learning, we have a population distribution $D$ of data
points without labels, and try to extract useful information from $D$ by
setting up pretext tasks. As summarized in Section 2, there are different
kinds of pretext tasks, including contrastive methods, masked image/language
models, pure language models, etc. The model trained with the pretext task can
be seen as a feature extractor $f\in D\rightarrow\mathcal{H}$, mapping the
input $x$ to its feature representation $f(x)$.
Using the language of category theory, we have a category $\mathcal{C}$, where
each object in $\mathcal{C}$ is a data point $X\in D$. For
$X,Y\in\mathcal{C}$, $\mathrm{Hom}_{\mathcal{C}}(X,Y)$ is defined by the
pretext task. More specifically, all the existing pretext tasks can be seen as
building morphisms between two objects in $\mathcal{C}$:
* •
Contrastive methods. Contrastive methods like SimCLR (Chen et al.,, 2020)
modify $X\in\mathcal{C}$ to get two semantically similar copies
$X^{\prime},X^{\prime\prime}\in\mathcal{C}$ as the positive pair, and pick
another different $Y\in\mathcal{C}$ as the negative sample. The pretext task
says $f(X^{\prime}),f(X^{\prime\prime})$ should be close and both of them
should be far away from $f(Y)$. It has been proved that contrastive learning
is doing spectral clustering on the similarity graph defined by the pretext
task (Tan et al.,, 2023; HaoChen et al.,, 2021). Specifically, the similarity
graph is defined as for every two objects $X,Y\in\mathcal{C}$, they are
connected with an edge $\mathrm{Hom}_{\mathcal{C}}(X,Y)$ of weight
proportional to $\Pr(\text{$(X,Y)$ gets sampled as a positive pair})$.
Therefore, in this category, each morphism between two objects is just an edge
connecting these objects, and can be represented with a real number.
* •
Masked image/language models. For this kind of pretext tasks, we mask a
portion of the full object and ask the model to predict the masked part. We
may treat the revealed part as $X\in\mathcal{C}$, masks as $Y\in\mathcal{C}$,
and define the morphism $\mathrm{Hom}_{\mathcal{C}}(X,Y)$ as the full object.
For other pairs of objects, their morphisms are defined as empty sets.
Therefore, the pretext task defines a category of sub-objects, and the
pretrained model can find the correct complement for the given sub-object.
* •
Pure language models. Each object $X\in\mathcal{C}$ is a sentence of length at
most111Length restriction is not important as the category can contain
infinitely many objects. $N$, and $\mathrm{Hom}_{\mathcal{C}}(X,Y)$ is the
probability that $Y$ is a sentence successive to $X$ that differs by the last
token. For example, if $X$ is “I am ”, $Y$ can be “I am happy”. See the
detailed discussion in Section 7.2.
Now, we should formally define what we meant by an ideal foundation model.
### 5.2 Ideal foundation model
###### Definition 2 (Ideal foundation model).
Given a category $\mathcal{C}$ defined by a pretext task, a foundation model
$f:\mathcal{C}\rightarrow\mathcal{H}$ is ideal, if there exists a data-
oblivious function $k_{f}:\mathcal{H}\times\mathcal{H}\rightarrow\textbf{Set}$
so that for any $X,Y\in\mathcal{C}$,
$k_{f}(f(X),f(Y))=\mathrm{Hom}_{\mathcal{C}}(X,Y)$.
Here, data-oblivious means $k_{f}$ is predefined without seeing the data, but
the subscript $f$ means $k_{f}$ as a predefined function, can access $f$ and
$f^{-1}$ as oracles. In other words, $k_{f}$ is a “simple” function, but can
borrow the power of foundation model $f$ for expressing more complicated
morphisms. Take the compression algorithms for example. The compression
algorithms can be data dependent, i.e., optimized for certain data
distribution. However, as a data-oblivious function $k_{f}$, it cannot access
the data distribution, but can call the compression algorithm to decompress
the data, because the action of calling an algorithm is data-oblivious. We
define $k_{f}$ for different pretext tasks below.
* •
For contrastive methods, $\mathrm{Hom}_{\mathcal{C}}(X,Y)\in\mathbb{R}$, so
$\mathcal{H}$ can be seen as the RKHS with $k_{f}$ as a kernel function. For
example, in SimCLR (Chen et al.,, 2020) and MoCo (He et al.,, 2020), $k_{f}$
is the Gaussian kernel for the two object embeddings.
* •
For masked models, $X$ is the revealed part, $Y$ is the masks. In MAE (He et
al.,, 2022), $f(X)$ and $f(Y)$ are used for encoding both revealed part and
masks. Afterwards, $k_{f}$ concatenates $f(X)$ and $f(Y)$, then run the
decoder $f^{-1}(\cdot)$ to recover the full object.
* •
For pure language models, we want to compute $\Pr(Y|X)$ where $Y$ and $X$
differ by the last token. $k_{f}$ runs a linear function222This linear
function can be predefined with random initialization, without affect the
empirical performance of the model. and a softmax function to compute the
probability distribution of the next token. Based on this distribution,
$k_{f}$ runs $f^{-1}(f(Y))$ to extract $Y$, which determines $\Pr(Y|X)$.
In Definition 2, the ideal foundation model is not unique. Theoretically, it
is possible that the model $f$ has the general intelligence, therefore can do
everything even without learning the data in $\mathcal{C}$. In this case, we
can make no meaningful arguments about the power of $f$. Therefore, we should
look at the opposite side, and ask:
Is $f$ guaranteed to have some property, if it is ideal with $\mathcal{C}$
defined by a pretext task?
This is the main motivation of our paper. Before moving forward, we would like
to point out that Definition 2 does not rule out the possibility that $f$
simply memorizes all possible morphisms. The intuition is, the pretext task is
so difficult, that even memorizing all the morphisms in $\mathcal{C}$ is
already powerful enough. Moreover, a simple memorization function will not
have an efficient implementation empirically (although category theory does
not care about the implementation).
Below we first introduce an important notion called Yoneda embedding.
###### Definition 3 (Yoneda embedding $h_{\mathcal{C}}$).
Given any $X\in\mathcal{C}$,
$h_{\mathcal{C}}(X)\triangleq\mathrm{Hom}_{\mathcal{C}}(\cdot,X)$, which takes
input $Y\in\mathcal{C}$ and outputs $\mathrm{Hom}_{\mathcal{C}}(Y,X)$.
Define $\mathcal{C}^{\wedge}$ as the category of functors from
$\mathcal{C}^{\mathrm{op}}$ to Set, we have
$\mathrm{Hom}_{\mathcal{C}}(\cdot,X)\in\mathcal{C}^{\wedge}$ and
$h_{\mathcal{C}}:\mathcal{C}\rightarrow\mathcal{C}^{\wedge}$. Notice that we
can define $\mathrm{Hom}_{\mathcal{C}}(X,\cdot)$ similarly, which defines
another kind of Yoneda embedding.
How do we represent $\mathcal{C}^{\wedge}$ empirically? To answer this
question, we should first learn arguably the most important theorem in
category theory, as follows.
###### Lemma 2 (Yoneda lemma).
For $A\in\mathcal{C}^{\wedge}$, and $X\in\mathcal{C}$,
$\mathrm{Hom}_{\mathcal{C}^{\wedge}}(h_{\mathcal{C}}(X),A)\simeq A(X).$
In category theory, when two objects (or functors) are isomorphic, we treat
them as equal. The technical details for making them equal to each other is
omitted, as we observe that this is not a problem for modern networks
empirically. For example, if there exists an isomorphism between two objects
$X$ and $Y$, so that $\phi(X)=Y$, empirically a deep neural network can easily
fit this $\phi$. More generally, techniques like rectified flow (Liu,, 2022)
can fit very complicated isomorphisms between random noise and images, using
neural network with ODE.
Similar to the RKHS, we assume that $\mathcal{C}^{\wedge}$ is a space
$\mathcal{H}$ equipped with a function
$k_{f}:\mathcal{H}\times\mathcal{H}\rightarrow\textbf{Set}$, such that
$\mathrm{Hom}_{\mathcal{C}^{\wedge}}(h_{\mathcal{C}}(X),h_{\mathcal{C}}(Y))\triangleq
k_{f}(h_{\mathcal{C}}(X),h_{\mathcal{C}}(Y))$, where $f=h_{\mathcal{C}}$. Note
that such $\mathcal{C}^{\wedge}$ always exists when $h_{\mathcal{C}}$ is
given. For example, $k_{f}$ can first compute
$h_{\mathcal{C}}^{-1}(h_{\mathcal{C}}(X))\simeq X$, and then send it to
$h_{\mathcal{C}}(Y)$ to get $\mathrm{Hom}_{\mathcal{C}}(X,Y)$, which is
isomorphic to
$\mathrm{Hom}_{\mathcal{C}^{\wedge}}(h_{\mathcal{C}}(X),h_{\mathcal{C}}(Y))$
by Lemma 2.
Intuitively, $h_{\mathcal{C}}$ takes $X$ and outputs all relationships of $X$,
so we immediately have the following lemma.
###### Lemma 3.
$h_{\mathcal{C}}$ is ideal.
The next lemma says, $h_{\mathcal{C}}$ is not only ideal, but with “the
minimum power”, up to symmetry333We say $h_{\mathcal{C}}$ has the minimum
power up to symmetry, because there are other foundation models that are
equally powerful. .
###### Lemma 4.
Given any other ideal foundation model $f$, all the morphisms in
$h_{\mathcal{C}}$ are also contained in $f$.
###### Proof.
Since $f$ gets all the pretext tasks correctly, and $k_{f}$ is data-oblivious,
we know the morphism $\mathrm{Hom}_{\mathcal{C}}(X,Y)$ is stored in $f$ for
every $X,Y\in\mathcal{C}$. In other words, for every
$h\in\mathrm{Hom}_{\mathcal{C}}(X,Y)$, either $h\in f(X)$ or $h\in f(Y)$.
Notice that $h_{\mathcal{C}}$ contains exactly all the morphisms of
$\mathrm{Hom}_{\mathcal{C}}(\cdot,\cdot)$, we proved our lemma. ∎
The above two lemmas tell us that we should work with $h_{\mathcal{C}}$, as it
is the weakest ideal model.
### 5.3 Downstream tasks
After the model is trained, we apply it to the downstream tasks with two
different approaches: prompt tuning and fine tuning. In this section, we
investigate the power of both approaches, with different outcomes. We first
define what we mean by solving a downstream task.
###### Definition 4 (Task).
A task $T$ is a functor in $\mathcal{C}^{\wedge}$.
###### Definition 5 (Task solving).
We say the model solves a task $T$, if for any input $X\in\mathcal{C}$, the
model outputs a solution that is isomorphic to $T(X)$.
Given a task defined as a functor $T\in\mathcal{C}^{\wedge}$, prompt tuning
means we freeze the parameters of the model, and only use a task specific
prompt $P$ (usually in text or image), followed with the actual input of the
task $X$, to get the output $T(X)$. Therefore, the prompt $P$ and input $X$
are the only two inputs to the model. By Lemma 2, if we directly send
$h_{\mathcal{C}}(X)$ and $T$ as a functor in $\mathcal{C}^{\wedge}$ to
$k_{f}$, we have
$T(X)\simeq\mathrm{Hom}_{\mathcal{C}^{\wedge}}(h_{\mathcal{C}}(X),T)\triangleq
k_{f}(h_{\mathcal{C}}(X),T)$ (1)
That is, $k_{f}$ can accurately compute $T(X)$ with these two representations.
However, notice that the prompt has to be sent through $h_{\mathcal{C}}$
before sending to $k_{f}$, which means we will send $h_{\mathcal{C}}(P)$
instead of $T$ for the input of $k_{f}$. This brings up another important
definition in category theory.
###### Definition 6 (Representable functor).
A functor $T\in\mathcal{C}^{\wedge}$ is representable if there is an
isomorphism $h_{\mathcal{C}}(X)\simeq T$ for some $X\in\mathcal{C}$. Such
object $X$ is called a representative of $T$.
Based on this definition, we have the following characterization of prompt
tuning.
###### Theorem 1 (Power on prompt tuning).
$h_{\mathcal{C}}$ can solve $T$ with prompt tuning, if and only if task
$T\in\mathcal{C}^{\wedge}$ is representable. When $T$ is representable, the
optimal prompt is the representative of $T$.
###### Proof.
When the task $T$ is representable, we can use its representative $X$ as the
prompt for $T$. By (1), we know $h_{\mathcal{C}}$ can solve $T$ exactly.
On the other hand, if the task $T$ is not representable, there exists no
object $X\in\mathcal{C}$ such that $h_{\mathcal{C}}(X)\simeq T$. Therefore,
assume we use task specific prompt $P$, and get the representation
$h_{\mathcal{C}}(P)$. Since $h_{\mathcal{C}}(P)\not\simeq T$, we can always
find an object $Y\in\mathcal{C}$ such that $h_{\mathcal{C}}(P)(Y)\not\simeq
T(Y)$. In that case, our model using $k_{f}$ and $h_{\mathcal{C}}$ cannot
solve $T$ on Y. ∎
Remark. There are some interesting results on continuous prompts (Liu et al.,,
2021; Li and Liang,, 2021), which directly tunes the prompts in the feature
space of the neural network, therefore the resulting prompt is not a
representation of any real words/sentences. By doing this, it is possible to
obtain more power than the representable tasks, but the enhancement depends on
the expressive power of the feature space. Below we provide a simple
application of Theorem 1.
###### Corollary 1.
For the pretext task of predicting image rotations (Gidaris et al.,, 2018),
prompt tuning cannot solve complicated downstream tasks like segmentation or
classification.
###### Proof.
The pretext task of predicting image rotations will rotate a given image
$X\in\mathcal{C}$, by four different degrees: 0°, 90°, 180°, and 270°.
Therefore, the four objects form a simple group of $4$ elements. For any
object $X\in\mathcal{C}$, we know $\mathrm{Hom}_{\mathcal{C}}(\cdot,X)$
contains exactly $4$ morphisms. Clearly, tasks likes segmentation or
classification are not representable by objects in $\mathcal{C}$. ∎
Remark. The statement of Corollary 1 is a bit counter-intuitive, because as
presented in Gidaris et al., (2018), the model trained with the rotation
pretext task indeed works for downstream tasks like classification and
segmentation. However, our definition of solving a task in Definition 2 means
the model should generates correct output for every input, so being partially
accurate is not considered a success. This also matches with the goal of our
paper: with infinite resource, can rotation pretext task used for complicated
downstream tasks? Corollary 1 gives the negative answer.
Note that in Theorem 1 we consider the model with the minimum power, while
empirically, the network structure may contain prior knowledge like the
convolutional layers for some specific downstream tasks. However, these prior
knowledge is inherently noisy and hard to characterize, therefore cannot be
relied for completely solving the downstream tasks.
The power of prompt tuning is limited, and can be characterized by
representable functors. What about fine tuning? In the case of fine tuning, we
may use the Kan extension. However, the details of Kan extension is fairly
abstract, so here we only provide one related theorem:
###### Lemma 5 (Yoneda Extension of Functors).
Let $F:\mathcal{C}\rightarrow\textbf{Set}$ be any functor, then there exists
an extension functor
$h_{\mathcal{C}}^{{\dagger}}F:\mathcal{C}^{\wedge}\rightarrow\textbf{Set}$
such that $h_{\mathcal{C}}^{{\dagger}}F\circ h_{\mathcal{C}}\simeq F$.
The actual Yoneda extension of functors theorem is more general (see e.g.,
Proposition 2.7.1 in Masaki Kashiwara, (2006)), it can be applied to any
category $\mathcal{A}$, but here we only use Set to avoid unnecessary
technical details of inductive limits. Using Lemma 5, we immediately get the
following theorem.
###### Theorem 2 (Power on fine tuning).
Ideally, given enough resources, $h_{\mathcal{C}}$ can solve any task
$F:\mathcal{C}\rightarrow\textbf{Set}$.
###### Proof.
Applying Lemma 5, we know that with enough training data for the downstream
task, computational power, and a fine tuning model $h$ that learns
$h_{\mathcal{C}}^{{\dagger}}F$ perfectly, we can concatenate $h$ with $f$, to
solve the task $F$. ∎
The downstream tasks considered in Theorem 2 are based on the structure in
$\mathcal{C}$, not the data content in the dataset. As a result, the category
defined by Gidaris et al., (2018) still has very simple group-structure, but
with Theorem 2 it is possible to solve more diverse tasks. For example, we can
map all the objects to the same output, which cannot be achieved with prompt
tuning. Theorem 2 conveys the importance of pretext task, as more informative
pretext tasks will create more structured category $\mathcal{C}$, which
further improves the power of fine tuning on $\mathcal{C}^{\wedge}$.
Even with very informative $\mathcal{C}$, Theorem 2 only provides a raw upper
bound with strong assumptions on the resources needed. By contrast,
empirically the fine tuning step is usually done with a small network.
Therefore, instead of treating it as a strong backup of what people are doing
right now, it is more like exhibiting the future possibility. In other words,
it implies that by transforming the objects to the corresponding feature space
$\mathcal{C}^{\wedge}$, we have captured all the information encoded in
$\mathcal{C}$.
For readers who are familiar with machine learning theory, Theorem 2 may look
similar to the theory of over-parameterization at the first glance. However,
they are analyzing different steps of self-supervised learning. Over-
parameterization analyzes the pretraining step, saying that under certain
assumptions, the optimization and generalization error will be very small for
the pretext task, as long as the model is big enough and the learning rate is
small enough. By contrast, Theorem 2 analyzes the fine tuning step after
pretraining. Even if we have successfully pretrained a network on Imagenet
with contrastive learning and zero generalization error, it remains unclear
whether the model can be used for image segmentation as the downstream task,
unless someone verifies it empirically. But Theorem 2 says, as long as the
model is ideal, the feature layer contains all the information of
$\mathcal{C}$ for any downstream tasks.
## 6 Multimodal Learning
In the previous section, we have seen how the foundation models are related to
learning a category defined by a pretext task. What happens if we have
multiple different categories? We can first use the pretext tasks to learn
different foundation models separately, then connect these models together in
the embedding space. This is exactly how multimodal models like CLIP Radford
et al., (2021), Dall-E 2 Ramesh et al., (2022), Multilingual CLIP Carlsson et
al., (2022) and AltCLIP Chen et al., (2022) work.
In this section, we analyze the functors between different categories. Similar
to the previous section, we consider the case that the functors are perfectly
learned, and investigate the implications under this assumption. This setting
hides unnecessary details like how loss is defined or how network structure is
designed. However, it provides interesting insights of how to connect
different categories together, and what to expect after the functor
connection.
As the starting point, we would like to emphasize that we assume the
categories we consider are “natural”, which means they are not only self
consistent, but also consistent with each other. For example, when we use
texts like “white, red, blue” or “big, small, tiny” to describe a chair in the
language category, there are corresponding images in the image category. Such
connection can be described by functors.
### 6.1 Generalization Theorem
We first assume that our functor from the embedding space learns the object
mapping between the two categories perfectly, then show the structural
information will be preserved with this functor. Notice that below we still
use the notations of $h_{\mathcal{B}},h_{\mathcal{C}}$ and
$\mathcal{B}^{\wedge},\mathcal{C}^{\wedge}$ to denote the foundation models
and the corresponding embedding spaces, but our results hold for any other
ideal foundation models and their embedding spaces as well.
###### Definition 7 (feature-aligned functor).
Given two categories $\mathcal{B},\mathcal{C}$, a full embedding
$F:\mathcal{C}\rightarrow\mathcal{B}$, denote the corresponding foundation
model as $h_{\mathcal{B}},h_{\mathcal{C}}$. A function
$\hat{F}:\mathcal{C}^{\wedge}\rightarrow\mathcal{B}^{\wedge}$ is feature-
aligned with $F$ if for any $X\in\mathrm{Ob}(\mathcal{C})$,
$\hat{F}(h_{\mathcal{C}}(X))\simeq h_{\mathcal{B}}(F(X))$.
###### Theorem 3 (Generalization theorem for structural learning).
Consider two categories $\mathcal{B},\mathcal{C}$ and a full embedding
$F:\mathcal{C}\rightarrow\mathcal{B}$. In the learning scenario, an ideal
foundation model $h_{\mathcal{C}}$ for $\mathcal{C}$ together with a feature-
aligned functor $\hat{F}:\mathcal{C}^{\wedge}\rightarrow\mathcal{B}^{\wedge}$,
preserves the structure of $\mathcal{C}$ in a full subcategory $\mathcal{A}$
of $\mathcal{B}$: for any $X,Y\in\mathcal{C}$,
$\mathrm{Hom}_{\mathcal{C}}(X,Y)\simeq\mathrm{Hom}_{\mathcal{A}^{\wedge}}(\hat{F}(h_{\mathcal{C}}(X)),\hat{F}(h_{\mathcal{C}}(Y)))$.
Moreover, when $h_{\mathcal{B}}$ is available and invertible, we have
$F(X)\simeq h_{\mathcal{B}}^{-1}(\hat{F}(h_{\mathcal{C}}(X)))$ for any
$X\in\mathcal{C}$.
###### Proof.
By Lemma 1, there exists a full subcategory $\mathcal{A}$ of $\mathcal{B}$
with inclusion functor $E$, which is isomorphic to $\mathcal{C}$ with functor
$G$, and $F=E\circ G$. Therefore, for any $X,Y\in\mathcal{C}$,
$\displaystyle\mathrm{Hom}_{\mathcal{C}}(X,Y)\simeq\mathrm{Hom}_{\mathcal{A}}(GX,GY)=\mathrm{Hom}_{\mathcal{B}}(FX,FY)$
$\displaystyle=$
$\displaystyle\mathrm{Hom}_{\mathcal{A}}(FX,FY)\simeq\mathrm{Hom}_{\mathcal{A}^{\wedge}}(h_{\mathcal{A}}(FX),h_{\mathcal{A}}(FY))$
$\displaystyle=$
$\displaystyle\mathrm{Hom}_{\mathcal{A}^{\wedge}}(h_{\mathcal{B}}(FX),h_{\mathcal{B}}(FY))$
$\displaystyle\simeq$
$\displaystyle\mathrm{Hom}_{\mathcal{A}^{\wedge}}(\hat{F}(h_{\mathcal{C}}(X)),\hat{F}(h_{\mathcal{C}}(Y)))$
The first equality holds as $G$ is isomorphic. The second equality holds
because $F$ is full. The third equality holds as $E$ maps the objects in
$\mathcal{A}$ to the same objects in $\mathcal{B}$. The fourth equality uses
the Yoneda lemma. The fifth equality holds because $A$ is a full subcategory.
The last equality holds since $\hat{F}$ is feature-aligned. If
$h_{\mathcal{B}}$ is invertible, we have
$F(X)=h_{\mathcal{B}}^{-1}(h_{\mathcal{B}}(F(X)))\simeq
h_{\mathcal{B}}^{-1}(\hat{F}(h_{\mathcal{C}}(X)))$. ∎
Theorem 3 is much more powerful that it appears. We call it the generalization
theorem, because it provides another kind of generalization, different from
the existing generalization theory on stability or Rademacher complexity. It
tells us that, the structural information of one category can be recovered in
the feature space by the structural information of another category with a
feature-aligned functor.
### 6.2 Application of Theorem 3
In this subsection, we apply Theorem 3 to analyze two models: CLIP and Dall-E
2.
CLIP. The dataset of CLIP contains millions of (image, text) pairs. During
pretraining, for each batch of $N$ pairs of data points, CLIP uses an image
encoder and a text encoder to get $N$ pairs of embeddings, and learns a
function for matching the $N$ pairs out of $N\times N$ possible connections.
Since the network used in CLIP is invertible, Theorem 3 immediately gives us
the following corollary.
###### Corollary 2 (Creativity of CLIP).
Let $\mathcal{C}$ be a category that describes images, which is a subcategory
of language category $\mathcal{C}^{\prime}$. Let $\mathcal{B}$ be the image
category. Assume CLIP learns a feature-aligned functor
$\hat{F}:\mathcal{C}^{\wedge}\rightarrow\mathcal{B}^{\wedge}$ for a full
embedding $F$, then CLIP is able to create new images that can be described in
$\mathcal{C}$, but do not existed in the training set.
Remark. Corollary 2 explains why CLIP can draw images like “avocado chair”
that did not exist previously. The full embedding assumption is crucial,
otherwise the embedding computed in $\mathcal{A}^{\wedge}$ cannot be used for
generating images in $\mathcal{B}$.
Dall-E 2. Dall-E 2 is the combination of Clip and the diffusion model
(Rombach et al.,, 2022; Sohl-Dickstein et al.,, 2015; Dhariwal and Nichol,,
2021; Ho et al.,, 2020). If Clip can be seen as the feature-aligned functor
between two isomorphic subcategories, why do we need an extra diffusion model?
This is because these two categories are not purely isomorphic. When we type
“a photo of dog”, there exists millions of different matching images.
Therefore, Dall-E2 modifies the definition of image category, so that each
object is a probability distribution of images. From this perspective, the
diffusion model is a generator of images, while Clip learns the functor from
the category of texts to the category of probability distributions of images.
### 6.3 Compositional Theorem
By applying Theorem 3 multiples times through a list of categories, we
immediately get the following theorem.
###### Theorem 4 (Compositional Theorem).
Consider a list of categories $\\{\mathcal{B}_{i}\\}_{i=1}^{n}$, and $n-1$
full embeddings $\\{F_{i}\\}_{i=1}^{n-1}$, where
$F_{i}:\mathcal{A}_{i}\rightarrow\mathcal{B}_{i+1}$, where
$\mathcal{A}_{1}=\mathcal{B}_{1}$, and for $i>1$, $\mathcal{A}_{i}$ is the
full subcategory of $B_{i}$ induces by $F_{i-1}$. Denote the foundation model
for $\mathcal{B}_{i}$ as $h_{\mathcal{B}_{i}}$, and feature-aligned functors
as $\\{\hat{F}_{i}\\}_{i=1}^{n-1}$. Denote $F\triangleq F_{n-1}F_{n-2}\cdots
F_{1}$ and $\hat{F}\triangleq\hat{F}_{n-1}\hat{F}_{n-2}\cdots\hat{F}_{1}$. By
composing all the $\hat{F}_{i}$s together, we preserve the structure of
$\mathcal{B}_{1}$ in $\mathcal{A}_{n}$: for any $X,Y\in\mathcal{C}$,
$\mathrm{Hom}_{\mathcal{B}_{1}}(X,Y)=\mathrm{Hom}_{\mathcal{A}_{n}^{\wedge}}(\hat{F}(h_{\mathcal{B}_{1}}(X)),\hat{F}(h_{\mathcal{B}_{1}}(Y)))$.
Moreover, when $h_{\mathcal{B}_{n}}$ is invertible, we have
$F(X)=h_{\mathcal{B}_{n}}^{-1}(\hat{F}(h_{\mathcal{C}}(X)))$ for any
$X\in\mathcal{C}$.
If we want to map objects from $\mathcal{B}_{1}$ and $\mathcal{B}_{n}$, but
the training data between the two categories is limited, Theorem 4 provides an
alternative way. It suffices to find a path between the two categories, and
learn the functors for each edge of the path. This is especially useful when
some of the categories are pretrained (like GPT or CLIP). For example,
Carlsson et al., (2022) and Chen et al., (2022) use multilingual text pairs to
train the functors between various language categories and the English
category, which are further connected to the image category using CLIP. By
doing that, they easily extend CLIP to the multilingual settings.
## 7 Discussion
### 7.1 Applying to small categories
In our abstract and introduction, we raised the question about an infinitely
large model with infinite resources, which makes our theorem look unrealistic.
However, these assumptions are purely rhetoric, and our real assumption is
simply the model being ideal, i.e., the model perfectly solves the pretext
task. This assumption is much weaker than the infinite one, because it means
our theorems can be directly applied to small datasets, which correspond to
smaller categories. For instance, consider a dataset containing only 100k
sentences, and a model is trained to learn the data using a language model
where each sentence is connected to its neighboring sentences with certain
probabilities. While the sentences or words in the corresponding category C
may not be as informative as the language category that humans possess, it is
well-defined, and the ideal model is also well-defined. Training an ideal
model for a small dataset of this scale is feasible. According to our Theorem
1, such model can only perfectly solve tasks that are representable in
$\mathcal{C}$.
### 7.2 Modeling probabilities
In this subsection, we provide the full definition of the category of language
models. In our new category $\mathcal{C}$, every object $X\in\mathcal{C}$ is a
probability measure $\mu$ on the space of sentences $\mathbb{S}$ of length at
most $N$. when $\mu$ concentrates at a single point $s\in\mathbb{S}$, it means
we have a deterministic sentence $s$. Otherwise, $\mu$ represents a
probability distribution over all possible sentences.
Given any sentence $s_{1}=(w_{1},\cdots,w_{N})$, we will define a probability
distribution $\nu$ on the next token $w_{N+1}$, so that we can define an
object $Y$ with the same probability measure on the sentences of concatenating
$(w_{2},\cdots,w_{N})$ to $w_{N+1}$. By defining $X$ as the object contains
single sentence $s_{1}$, we have a morphism from $X$ to $Y$. We call $Y$ the
canonical successor of $X$, and its probability distribution (on length $N$
sentences) the canonical successive measure of $X$.
Given any object $Z\in\mathcal{C}$, assume $Z$ has nonzero probabilities on
various objects, like $X_{1},\cdots,X_{n}$, with probability measure $\mu$. We
define canonical successive distribution of $X_{i}$ as $\nu_{i}$. In this
case, we define its canonical successive distribution $\mu^{\prime}$ as:
$\mu^{\prime}(s)=\int_{X_{i}}\int_{s}d\nu_{i}(s)d\mu(X_{i})$
Intuitively, if $s_{1}$ is the first sentence of a paragraph, we may use an
object with measure concentrated at $s_{1}$ to represent it. However, due to
the nature of natural language (not the dataset noise), the next word of
$s_{1}$ is not deterministic. So we define $s_{2}$ that drops the first word
and adds the next word of $s_{1}$, which should be represented as a
probability distribution (canonical successive distribution). What about the
next word of $s_{2}$? Without the realization of $s_{1}$, we have to enumerate
all the possibilities of $s_{1}$, and use the canonical successive
distribution of each of them to define $s_{2}$. By doing so, we have defined a
category with probabilities, where the pretext task defines its morphisms.
### 7.3 Bayesian optimal classifier
The notion of Bayesian optimal classifier is widely used in the generalization
theory for supervised learning. It simply means that due to the inherent noise
in the labels, even the best model will not be able to get zero loss in the
population distribution. For example, given a blurry image of a dog, it might
be difficult to tell its exact breed.
However, in our categorical framework, we take the physicist’s view of the
world, and assume that all the objects and relationships are self consistent,
without any noises. For example, Alaskan Husky and Siberian Husky indeed look
similar from their photos, but they are fundamentally different breeds, in
terms of their origins, size, appearance, temperament and so on. Our framework
models these conceptual differences, and treat the photos of the dogs (which
have noise) as the outcome of the last step of diffusion as discussed in
Section 6. If the reader is familiar with the functional programming, the
categorical framework can be seen as the “pure functional” part, which is
predictable and precise. By contrast, the process that deals with the actual
input and output, can be seen as the “side effects”, where the notion of
Bayesian optimal classifier comes in.
### 7.4 Relationship to RKHS
Our categorical framework can be seen as a natural generalization to the RKHS
framework, where in RKHS a kernel function outputs a value in $\mathbb{R}$,
and our framework generalizes it to Set.
For example, applying the Yoneda Lemma on RKHS, we get the property: $\forall
x\in\mathcal{C},\forall f\in\mathcal{H},\langle
k(\cdot,x),f\rangle_{\mathcal{H}}=f(x)$, where $f$ can be seen as a functor in
$\mathcal{C}^{\wedge}$, $k(\cdot,x)$ is a representable functor, and
$\langle\cdot,\cdot\rangle_{\mathcal{H}}$ computes
$\mathrm{Hom}_{\mathcal{C}}$. Therefore in RKHS, the morphisms between two
objects are represented by a single real number computed by
$\langle\cdot,\cdot\rangle_{\mathcal{H}}$. This case works perfectly for
algorithms like SimCLR where the relationship between two objects is a real
number representing the similarity. However, in practice, the relationship
between two objects can be much more than a real number, especially for NLP
tasks. In that case, RKHS is insufficient, and the general Yoneda lemma is
necessary.
## Acknowledgements
This paper was greatly inspired by the fruitful interactions with the Qianfang
(functor) team at the Beijing Academy of Artificial Intelligence. Special
thanks are extended to Yue Cao and anonymous reviewers for their helpful
discussions and suggestions. This paper is supported by the Ministry of
Science and Technology of the People’s Republic of China, the 2030 Innovation
Megaprojects “Program on New Generation Artificial Intelligence” (Grant No.
2021AAA0150000).
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# Network-Iterated Prisoner’s Dilemma
Author: Martín Soto Quintanilla Facultat de Física, Universitat de Barcelona,
Diagonal 645, 08028 Barcelona, Spain. Advisor: Marián Boguñá Espinal
###### Abstract
Abstract: We introduce the stochastic Network-Iterated Prisoner’s Dilemma
(NIPD) model, a network of players playing the Prisoner’s Dilemma with their
neighbours, each with a memory-one strategy which they constantly and locally
update to improve their success. This process is non-deterministic, and
mirrors societal interactions in many relevant aspects. We use it to assess
the flexibility, noise tolerance and real-world adaptability of some well-
known strategies. Furthermore, in the model a new strategy naturally emerges
which proves way more successful than those. We also derive some theoretical
parameters that gauge the success of a strategy in this context.
## I Introduction
The Prisoner’s Dilemma is a simple two player game which abstractly models a
certain kind of competitive social interaction. It is a foundational example
in the theoretical field of Game Theory, and has found many applications in
Economics or Rational Decision Theory in helping explain natural phenomena or
empirical data.
In its original formulation, both players A and B are conceptualized as
robbers just caught in a joint theft. They are put in separate rooms (can’t
communicate) and each has to decide whether to remain silent (cooperate) or
betray the other. Of course, betraying the other yields personal gain, and so
the years each player will have to serve in prison look like follows depending
on their decisions:
A B | | B
---
cooperates
| B
---
betrays
| A
---
cooperates
1 1 | 5 0
| A
---
betrays
0 5 | 3 3
Notice thus that, if they act in a purely selfish and rational fashion, both A
and B are always better off betraying, no matter the move the other player
makes. The situation is paradoxical and problematic for our usual
understanding of rationality, since both players acting rationally inhibits
each of them from obtaining a better individual outcome through mutual
cooperation. This is where the relevance of the Dilemma resides: it’s a simple
conceptualization of many real world situations in which individual
rationality doesn’t produce the best global outcome. It’s been fruitfully
applied to many such situations pris .
In all generality, the payoffs of the table can be understood as any positive
values that the players want to minimize, such as most prominently economical
losses. Of course, the concrete numerical values can vary, and the same
paradoxical situation will arise as long as the order relation between them
remains the same (that is, it is no coincidence that we chose $5>3>1>0$). We
stick to these values through this work.
So the situation is clear when playing a single instance of this game:
rational players will inevitably defect. But the situation changes when the
game is iterated through different rounds. In this situation, closer to the
recurring interactions of the real world, the prospective gains of future
cooperation can sometimes make betrayal an irrational decision. Reference Li
presents the mathematics of this Infinitely Iterated Prisoner’s Dilemma as
played by two players.
We will study it through a different treatment: we build a network of players
repeatedly playing the Prisoner’s Dilemma with their neighbors and constantly
updating their individual strategies striving for better results. This is a
more general model closer to how societal interactions and belief update
really work. As we will see, it is more general and realistic for randomness
to affect some aspects of the simulation, and so we are dealing with a
stochastic process.
## II The NIPD model
We apply to our situation some ideas from the stochastic modelling of disease
or information spreading, as presented in Hof .
We build a network of $N=10^{3}$ nodes (players), each one connected to some
others (its neighbors) by a randomly generated adjacency vector (connections
are of course bi-directional). To simplify treatment, we consider a degree-
regular network, in which every node is connected to $k=4$ others.
Each one of the nodes has at all times a strategy. In all generality, this
strategy can take into account any one of the previous played rounds. As a
simplification, we consider those strategies depending only upon the outcome
of the previous round. As presented in Li , these memory-one strategies can be
expressed as a 4-dimensional vector, in which every one of the 4 parameters
expresses the probability the player has of cooperating in the next round,
given the previous round having a certain of the 4 possible game outcomes.
That is, if $C$ is cooperation and $B$ betrayal, each player’s strategy will
have the form
$(p_{CC},p_{CB},p_{BC},p_{BB})$
where $p_{CC}$ is the probability of the player cooperating if the previous
round resulted in double cooperation, $p_{CB}$ the probability of the player
cooperating if the previous round they cooperated and were betrayed, etc. (and
so all 4 parameters will be real numbers between 0 and 1).
As an illustrative example, a player with strategy $(1,1,1,1)$ will always
cooperate, and one with strategy $(0.5,0,0,0)$ will only cooperate with a 50%
probability when the previous round resulted in double cooperation.
Notice this makes the network and process drastically more complex than that
of mono-viral spread. There, the state of every node (healthy or infected)
could be expressed by a binary value. Now we need four continuous variables.
We initialize the network by giving an initial strategy to every node. Then we
start randomly selecting which neighboring nodes play a round. We could use
continuous time as in Hof , and give each connection between two nodes the
parameter $\lambda_{ij}$ of a Poisson process that randomly determines when a
round is played. But then we would choose all of these parameters equal as a
simplification for tractability, and this is actually equivalent to just
randomly choosing each time one connection (all connections having the same
probability of being chosen). So we more simply use discrete time steps.
Of course, when a round is played, both players will choose one of the 4
parameters of their strategy depending on the result of the last round played
between them, and then flip an unfair coin with that probability to determine
whether they cooperate. 111In the first round of the process, we choose
players to act as if the previous (non-existent) round resulted in double
cooperation, thus showing initial optimism or innocence. Choosing this initial
condition differently doesn’t drastically affect the initial dynamics.
Based on the outcomes of the rounds, each player gradually accumulates their
relative payoff $R_{i}$, a value which each player tries to minimize and
serves as a measure for how good their strategy is doing. This value resets to
0 whenever the node or any of its neighbours changes strategy, and from then
on accumulates the nodes’ payoff minus the average of the neighbour’s payoffs.
That is, if $P_{i}^{(n)}$ is the accumulated payoff of a node $i$ since round
$n$, and $i\sim j$ means the two nodes are connected, then
$R_{i}=P_{i}^{(n)}-\frac{1}{k}\sum_{j\sim i}P_{j}^{(n)}$
where $n$ is the last round in which $i$ or one of its neighbours changed
strategy.
After each played round, if the $R_{i}$ of any of the two players surpasses a
threshold $R_{max}=15$ 222We could also decide when a node changes strategy by
using any probability distribution on $R_{i}$, instead of a sharp cutoff
$R_{max}$. The results obtained by most reasonable probability distributions
don’t seem to be drastically different., $i$ changes strategy. They do so by
choosing their best performing neighbor (the one with lowest $R_{j}$) and
copying their strategy 333We also tried taking the average of their
neighbour’s strategies, but this yielded too thermalized systems rapidly
converging to the absorbent state in which all nodes have the same strategy
(which is just the average of all initial strategies).. Notice this has as a
consequence that any strategy a node has at any point of the process was
already some node’s strategy at the start of the process. The whole process is
iterated for a certain number of played rounds. 444Of course, our nodes play a
big but finite amount of rounds. Strictly speaking, if a rational agent knows
the number of rounds played will be finite, we will again be in the situation
where it will always betray. But in our model we haven’t given our players any
information about the finiteness of the game, so we might conceptualize this
as the players not knowing whether the game will end at any time.
## III Four strategies
Dealing with all possible strategies is mathematically more complicated, so we
start with a simpler situation where nodes can only have one of a small set of
representative strategies (that furthermore have a clear motivation). Consider
the following strategies Kuh :
Name | Description | Vector
---|---|---
Cooperator (C) | Always cooperates | (1, 1, 1, 1)
Traitor (T) | Always betrays | (0, 0, 0, 0)
Tit for Tat (TFT) | | Copies the opponent’s
---
last move
(1, 0, 1, 0)
Pavlov (P) | | Cooperates only when
---
last round both players
played the same move
(1, 0, 0, 1)
TFT is generally regarded as a versatile strategy, since it can protect itself
from regular betrayal, but can also benefit from cooperation in the long run
when possible. P on the other hand would not seem that smart a strategy, since
for instance cooperating after being betrayed is hardly beneficial.
We first study the process beginning with equal amounts of the previous
strategies distributed randomly across the population. This process is similar
to a multi-viral scenario, where different viruses compete for population, but
in our case the viral infection process is more complex than just accumulating
viral load (it involves playing a game), and so different pairs of virus
interact differently. The evolution of a realization of this process is
presented in Fig. 1, which we discuss later.
Even in this case where strategies are non-probabilistic, the simulation is
non-deterministic and the result does not only depend upon the initial
conditions (the adjacency vector and the initial strategy distribution), but
also on the specific realization of the simulation (that is, the random seed
used to run the program). This is seen in all the successive figures, in which
equal initial conditions yield different outcomes across different
realizations. So every initial condition will have not one definite outcome,
but an array of possible evolutions driven by different attractors, and thus a
probability distribution of different final states.
From this we conclude that round order (the only random variable of the
realization) can drastically affect the success of some strategies. This makes
sense: for instance, a TFT with some Cs and some Ts as neighbors might by
chance play some successive rounds against the Ts, with poor payoff which will
lead to it changing strategy, or on the contrary against the Cs, from which it
will benefit.
Exactly because of that, the only truly absorbent states in this situation are
those in which all population carry the exact same strategy. This is because
there’s always a chance that a node updates strategy (although maybe very
small), and if a neighbor has a different strategy this might entail an actual
change. The existence of absorbent states implies the system to be non-ergodic
(there are inescapable setups from which you can’t access all phase space).
But there may be more states of stable equilibrium, even if they are not
absorbent. Accurately assessing when these states appear would take way longer
simulations, that our computing power doesn’t allow. We could also
quantitatively measure non-ergodicity by calculating the probability
distribution of final absorbent or equilibrium states for a certain initial
condition.
But we can still obtain information about the process from its starting
dynamics and initial growths, as shown in Fig. 1, and there are some evident
ways in which the initial conditions are relevant. In this setup, T grow
rapidly at first, by taking advantage of the C and P. But T runs out of
players to betray, and TFT then starts outperforming T (thanks to clusters of
TFTs cooperating). In this newly non-betraying environment, the few P
survivors start to thrive, and eventually reach a state of apparent
oscillating equilibrium with higher concentration than TFT. This might really
be a stable equilibrium in the long run, since it is conceivable that thanks
to the topological distribution of connections two clusters of different
strategies coexist. But it might also be destined, when enough rounds pass, to
reach the absorbent state with all Ps or all TFTs (each with a certain
probability of occurring in every simulation). That situation would be
reminiscent of the Voter Model, where every realization can only end in one of
two absorbent states, but the average of both these probabilities across
realizations converges vote .
$10^{6}$ rounds played
Figure 1: Percentile of the population carrying each of the four strategies,
during three simulations of our model starting with equal (0.25)
concentrations (with a same randomly chosen adjacency vector). T grows so
rapidly to 0.4 that it’s not appreciated due to scale.
We need a complementary theoretical parameter that objectively assesses how
successful a strategy will be in a certain context. For that, we calculate the
expected payoff of a strategy after each round (averaged over all its
opponents). This will be a global variable approximating local behaviour. As
an exemplification, consider the outcomes of iterated rounds between a TFT and
a T at the start of the process:
TFT vs T: CB BB BB BB … [6]
So, neglecting the exceptional first round, the expected payoff per round in
the long run will be $E(\mbox{TFTvsT})=E(\mbox{TvsTFT})=3$ for both players.
Applying the stochastic mean-field approximation, the probability of a node
facing a certain strategy in a neighbor is just the overall concentration of
that strategy in the whole population. Since at the beginning of this process
these concentrations are all equal, the expected payoff per round of a TFT
node will then be the following (and similarly we calculate the others):
$E(\mbox{TFT})=\frac{1}{4}(E(\mbox{TFTvsTFT})+E(\mbox{TFTvsT})+\ldots)=\frac{6}{4}$
$E(\mbox{T})=\frac{7.5}{4}\>\>\>\>\>\>\>\>E(\mbox{C})=\frac{8}{4}\>\>\>\>\>\>\>\>E(\mbox{P})=\frac{7}{4}$
This theoretically explains why TFT grows at the beginning, since its expected
payoff is considerably lower (thus more successful) than all others. Still,
behind this useful general assessment lie some topological and local details
that cannot be captured by a single numerical value. How come, for instance, T
grow at the expense of P, if $E(\mbox{T})$ is slightly bigger than
$E(\mbox{P})$? This is because, even if overall T do slightly worse, any T
surrounded by Cs or Ps will drastically benefit, and very rapidly transform
its neighbors into other Ts.
We also notice that, in our model, the success of a strategy is heavily
context dependent (a good sign that it approximates real situations). For
instance, we see in Fig. 1 that the dynamic changes when Cs and Ts are almost
not present. Not only does the change in concentrations alter the previous
expected payoffs, but furthermore the outcome of previous rounds will affect
current performance. For instance, two previous Ts turned TFTs will continue
forever betraying each other, failing to reconcile, to their disadvantage. On
the other hand, PvsP will pass from any outcome to unending double
cooperation. That’s why in the aftermath, where the betraying T are no longer
present, some P outperform some (non-cooperating) TFT.
This kind of beneficial noise tolerance or self-adjusting back to cooperation
of P is not captured by the previous averaged payoff $E$, and so in some
situations it might be more appropriate to consider a different average
($\bar{E}$) which also takes into account which of the possible outcomes of a
game (CC, CB, BC, BB) might start a series of successive rounds between two
players. As an exemplification, in our situation we can have, among others,
the two sequences
P vs P: CB BB CC CC …
TFT vs TFT: BB BB BB BB …
(with longterm average payoff per round 1 and 3 respectively) and the more
accurate averages will be
$\bar{E}(\mbox{PvsP})=\frac{1}{4}(\bar{E}(\mbox{PvsP; CC})+\bar{E}(\mbox{PvsP;
CB})+\ldots)=\frac{4}{4}$
$\bar{E}(\mbox{TFT})=\frac{16}{8}\>\>\>\>\>\>\>\>\bar{E}(\mbox{P})=\frac{10}{8}$
which do show why P performs better. Notice this indicator will work best in
chaotic situations where each strategy can come across different past outcomes
(as in an ongoing process), and $E$ is more applicable when past outcomes are
less chaotic (as in the start of a process).
TFT might perform better if they were somehow capable of making amends. One
(maybe too artificial) way of ensuring this is making every node, when
adapting a new strategy, act as if the last round was CC (thus every node can
make amends). As we see in Fig. 2, this small memory reduction indeed ends the
edge P had over TFT. It also yields way more clear hints of a stable
equilibrium. The model with this simplification seems more tame.
Another (probably more natural) way is changing its strategy to be less
drastically punitive. These are the $p$TFT of the next section.
$10^{5}$ rounds played
Figure 2: Percentile of the population carrying each of the four strategies,
during three simulations of our model in which nodes act after a strategy
change as if CC had just been played.
## IV One strategy against the world
We now deal with the richer complexity of probabilistic strategies, that is,
vectors with any parameters.
One natural way of assessing the fitness of a strategy is seeing how well it
does against any possible strategy. For that, we see how well a randomly
scattered minority carrying a strategy does against a majority of individuals
with randomly chosen strategies.
We start with a quarter of the population initially carrying TFT. As seen in
Fig. 3, TFT spreads fast (the posterior slight decay is probably due to other
randomly appearing strategies more easily cooperating between them than TFT,
as discussed earlier).
$10^{5}$ rounds played
Figure 3: Percentile of the population carrying TFT, during three simulations
of our model starting with $\frac{1}{4}$ of TFTs and $\frac{3}{4}$ random
strategies (with a same randomly chosen adjacency vector and strategies).
We can study its pervasiveness (the fraction of nodes with a TFT strategy)
after a certain number of rounds, depending on its initial concentration
$\rho\in[0,1]$. This is analogous to the study of viral endemicity Hof , but
now we can’t lengthen the simulation enough to reach an equilibrium which
confirms TFT remains endemic. But we can check the initial evolution (the
concentration after a low number of rounds) for these values, as shown in Fig.
4, and theoretically reason that any $\rho$ for which the TFT don’t die out
quickly will present a stably TFT fraction of the population: the initial TFT
cluster smaller than $\rho$ which always cooperates. So this is a highly local
phenomenon. Basically, as long as there are enough TFTs so that by chance a
cluster of them is formed, we will have both rapid growth of TFT, and stable
permanence of at least some of them. We notice in Fig. 4 that this happens
even for very low concentrations, and so the concentration threshold above
which we have endemicity is $\rho_{t}(\mbox{TFT})\approx 0.005$. We also
observe that a linear relation is apparent, although we would need many more
realizations to average out the noise and provide moderately accurate
parameters for the linear regression.
As mentioned last section, a less rigid form of TFT might be better equipped
to deal with these probabilistic environments. So consider $p$TFT
$=(p,p,1-p,1-p)$. Its performance (for two values of $p$) and those of other
strategies are also presented in Fig. 4. We observe these variants of TFT
actually perform worse. The context is very different from that of section
III, and even if there TFT could have benefited from making amends, rigidity
is more beneficial when facing random strategies. P also performs almost as
well as TFT. Indeed its only relevant flaw was cooperating after CB, and
that’s not as big of a problem in a random environment where it won’t usually
encounter constant betrayers. T far outdoes all of them in initial growth. But
when dealing with adequately long simulations, we do expect T to decay rapidly
after a certain time (as in Fig. 1), since unlike P or TFT it can’t form
clusters of cooperation, and some random strategy probably will, to T’s
disadvantage.
Although not included due to lack of space, a more complex complementary
theoretical assessment of the fitness of strategies (as before with $E$ and
$\bar{E}$) can now be carried out by considering probabilistic game trees
instead of sequences, and resulting equations.
initial concentrations ($\rho_{i}$)
Figure 4: Average over three simulations of the final concentrations
($\langle\rho_{f}\rangle$) of TFT, 0.9TFT, 0.8TFT, P, E (see next section) and
T in a random environment (after $2\times 10^{5}$ rounds) for different
initial concentrations of them ($\rho_{i}$). Error bars (derived from
averaging) comparable to symbol size.
$10^{5}$ rounds played
Figure 5: Evolution of the global average (for all nodes) of each of the four
strategy parameters, during three simulations of our model starting with all
random strategies.
## V an emergent strategy
We might wonder what random strategies will prevail in a fully random
environment. In that case we can’t keep track of any one concrete strategy, so
in Fig. 5 we keep track of the global average of the strategy parameters of
all nodes. This process rewards the strategies close to (taking the average
approximately) E $=(0.5,0.2,0.35,0)$. Might this be the strategy best fit to
deal with a random environment? As we see in Fig. 4, it does indeed perform
way better than the others, except for the unequaled starting growth of T. But
we do know, as seen in Fig. 5, that this strategy performs better in the long
run (unlike T), thanks to the benefit of some cooperation. So it does seem
like an ideal all terrain strategy.
## VI Conclusions
The NIPD model proves versatile and explanatory when dealing both with
designed and random strategies. It provides a rich landscape, where many
behaviors are heavily context dependent, mirroring some real societal
dynamics. Thus it cannot always be analysed with complete mathematical
accuracy.
The interaction between some of the strategies considered depend upon local
phenomena and the topology of the network. Still, many macroscopic dynamics
are easily noticed and intuitively explained. The two averages $E$ and
$\bar{E}$ prove helpful tools (in different circumstances) for these
macroscopic explanations.
The random strategies arena of section IV is a more pragmatic computational
method to gauge a strategy’s flexibility. We obtained interesting growth
results, portraying the first moments of a strategy outbreak. Still, with more
computational power, a more thorough study taking into account the
pervasiveness of the strategy in far off stable equilibriums could be carried
out.
The emergent strategy does overall great on this context. A more thorough
sweep over the space of possible strategies (or averages taken as in Fig. 5)
might provide slightly more well-adjusted parameters for an even more
efficient strategy.
###### Acknowledgements.
I am immensely grateful to my supervisor Marián Boguñá for his help, patience,
great ideas and great mood. It was a pleasure working with him. Thank you also
to all of my colleagues and friends of the double degree, who have made these
five years the best of my life.
## References
* (1) N. Glynatsi, V. Knight, A systematic literature review of the Prisoner’s Dilemma; collaboration and influence, Cardiff University, School of Mathematics (2019)
* (2) Xavier Hoffmann, Marián Boguñá, Memory-induced complex contagion in epidemic spreading, New Journal of Physics, Vol. 21, N. 3 (2019)
* (3) Juan Fernández, Updating rules and the voter model, Master Thesis, Universitat de les Illes Balears (2011)
* (4) Steven Kuhn, Strategies for the Iterated Prisoner’s Dilemma, The Stanford Encyclopedia of Philosophy (Winter 2019 Edition), Edward N. Zalta (ed.) (2014)
* (5) Siwei Li, Strategies in the Stochastic Iterated Prisoner’s Dilemma, University of Chicago (2014)
|
# Hong-Ou-Mandel Interference between Two Hyper-Entangled Photons Enables
Observation of Symmetric and Anti-Symmetric Particle Exchange Phases
Zhi-Feng Liu (刘志峰) Chao Chen (陈超) National Laboratory of Solid State
Microstructures, School of Physics, Nanjing University, Nanjing 210093, China
Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093,
China Jia-Min Xu (徐佳敏) Hefei National Laboratory, Hefei 230088, China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China Zi-Mo
Cheng (程子默)
Zhi-Cheng Ren (任志成) Bo-Wen Dong (董博文) Yan-Chao Lou (娄严超)
Yu-Xiang Yang (杨雨翔) Shu-Tian Xue (薛舒天) Zhi-Hong Liu (刘志红)
Wen-Zheng Zhu (朱文正) National Laboratory of Solid State Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China Collaborative
Innovation Center of Advanced Microstructures, Nanjing 210093, China Xi-Lin
Wang (汪喜林<EMAIL_ADDRESS>National Laboratory of Solid State
Microstructures, School of Physics, Nanjing University, Nanjing 210093, China
Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093,
China Hefei National Laboratory, Hefei 230088, China Synergetic Innovation
Center of Quantum Information and Quantum Physics, University of Science and
Technology of China, Hefei 230026, China Hui-Tian Wang (王慧田)
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures, School
of Physics, Nanjing University, Nanjing 210093, China Collaborative
Innovation Center of Advanced Microstructures, Nanjing 210093, China
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan
030006, China
###### Abstract
Two-photon Hong-Ou-Mandel (HOM) interference is a fundamental quantum effect
with no classical counterpart. The exiting researches on two-photon
interference were mainly limited in one degree of freedom (DoF), hence it is
still a challenge to realize the quantum interference in multiple DoFs. Here
we demonstrate the HOM interference between two hyper-entangled photons in two
DoFs of polarization and orbital angular momentum (OAM) for all the sixteen
hyper-entangled Bell states. We observe hyper-entangled two-photon
interference with bunching effect for ten symmetric states (nine Boson-Boson
states, one Fermion-Fermion state) and anti-bunching effect for six anti-
symmetric states (three Boson-Fermion states, three Fermion-Boson states).
More interestingly, expanding the Hilbert space by introducing an extra DoF
for two photons enables to transfer the unmeasurable external phase in the
initial DoF to a measurable internal phase in the expanded two DoFs. We
directly measured the symmetric exchange phases being $0.012\pm 0.002$,
$0.025\pm 0.002$ and $0.027\pm 0.002$ in radian for the three Boson states in
OAM and the anti-symmetric exchange phase being $0.991\pi\pm 0.002$ in radian
for the other Fermion state, as theoretical predictions. Our work may not only
pave the way for more wide applications of quantum interference, but also
develop new technologies by expanding Hilbert space in more DoFs.
Two-photon interference, Hong-Ou-Mandel (HOM) interference Hong1987 , is a
genuine quantum effect with no classical counterpart. Since a single
experiment can unveil the quantum features of particle-wave duality and
indistinguishability simultaneously, the HOM interference has attracted
significant interest over the past decades Mandel1999 ; Bouchard2020 and has
been generalized to many other quantum particles or quasi-particles, such as
two atoms Kaufman2014 ; Lopes2015 , two deterministic collective excitations
in an atomic ensemble Li2016 , two surface plasmons Heeres2013 ; Fakonas2014 ;
Martino2014 ; Cai2014 and two phonons Toyoda2015 . Unlike the bunching effect
in the HOM interference of two indistinguishable Bosons, the anti-bunching
effect will be resulted from the interference of two indistinguishable
Fermions Liu1998 ; Bocquillon2013 .
The HOM interference has become a cornerstone of modern quantum technologies
and widely utilized to characterize the single photons from solid-state
emitter He2013 ; Ding2016 ; Somaschi2016 ; Wang2019 ; Tomm2021 and the photon
pairs from spontaneous parametric down-conversion (SPDC) Wang2016 ; Zhong2018
; Zhong2020 . The HOM interference is also an important way to combine photons
to implement controlled-NOT gate O'Brien2003 ; Bao2007 ; Li2021 and to
construct multi-photon entangled states including N00N states Walther2004 ;
Nagata2007 , GHZ states Pan2012 ; Wang2016 ; Zhong2018 , graph states Lu2007 ;
Pan2012 and multi-photon high-dimensional entangled states Malik2016 ;
Zhang2017 ; Erhard2018 . The HOM interference is also the foundation of
implementing the Bell state measurement, which is crucial for a variety of
important quantum protocols, such as quantum teleportation Bouwmeester1997 ;
Wang2015 ; Pirandola2015 ; Ren2017 , entanglement swapping Pan1998 ;
Basset2019 ; Zopf2019 and connecting quantum nodes in quantum network
Hofmann2012 ; Hensen2015 ; Yu2020 ; Liu2021 .
Two-photon interference in more general scenarios has attracted significant
interest, such as high dimensions Zhang2016 ; Hiekkamaki2021 , multiple-mode
states Walborn2003 ; Francesconi2020 , and structured light D'Ambrosio2019 .
The HOM interference can be utilized as a state filter, which can engineer
two-photon high-dimensional states Zhang2016 . The HOM interference is
determined by the exchange symmetry of two-photon states, where the symmetric
(anti-symmetric) state leads to the bunching (anti-bunching) effect. So the
key is how to harness the symmetry of two-photon state for the HOM
interference. One well-known method is to prepare the symmetric and anti-
symmetric two-photon states with the Bell states, as all the Bell states have
the symmetry and form a set of orthogonal and complete bases. Another
interesting method is to tame the symmetry of two-photon state by tailoring
the multiple mode. For example, the multi-mode HOM interference was realized
by transferring the symmetry of transverse spatial modes from the pump to the
produced two-photon state via the SPDC and the global symmetry of two-photon
state was further controlled by the polarization DoF (only one of the four
polarization Bell states is anti-symmetric) Walborn2003 . Recently, one has
reported that the symmetry of two-photon state in frequency DoF can be
controlled via the spatial mode of the pump Francesconi2020 . It has been
revealed that the hidden entanglement is in the anti-symmetric state
Fedrizzi2009 , which gives rise to the observation of anti-bunching HOM
effect.
Using the hyper-entangled photons in multiple DoFs in the Bell-state bases,
the HOM interference can be explored in a more general case, e.g., changing
the symmetry of two-photon state by selecting the different Bell states in
each DoF independently. However, most previous experiments were limited to the
Bell-state entanglement in one DoF. It is still a challenge to experimentally
explore the HOM interference in two and more DoFs. Here we demonstrate the
two-photon HOM interference in two DoFs of polarization and orbital angular
momentum (OAM) in all the sixteen hyper-entangled Bell states. Moreover, based
on the HOM interference in two DoFs, we successfully measure the exchange
phases of two photons in the OAM DoF for the four OAM Bell states including
three symmetric and one anti-symmetric exchange phases. The crux to success is
to extend the Hilbert space by introducing an extra DoF of polarization to
convert the unmeasurable external exchange phase in one OAM DoF into the
measurable internal phase in two DoFs of polarization and OAM.
In one DoF of polarization, the four Bell states perfectly characterize the
exchange symmetries of two photons Sun2007 ,
$\left\lvert{\phi^{\pm}}\right\rangle_{12}$ and
$\left\lvert{\psi^{\pm}}\right\rangle_{12}$, can be written as
$\displaystyle\left\lvert{\phi^{\pm}}\right\rangle_{12}=(\left\lvert{H}\right\rangle_{1}\left\lvert{H}\right\rangle_{2}\pm\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2})/\sqrt{2},$
(1)
$\displaystyle\left\lvert{\psi^{\pm}}\right\rangle_{12}=(\left\lvert{H}\right\rangle_{1}\left\lvert{V}\right\rangle_{2}\pm\left\lvert{V}\right\rangle_{1}\left\lvert{H}\right\rangle_{2})/\sqrt{2},$
(2)
where $\left\lvert{H}\right\rangle$ ($\left\lvert{V}\right\rangle$) refers to
horizontal (vertical) linearly polarized state. Three Bell states
$\left\lvert{\phi^{\pm}}\right\rangle_{12}$ and
$\left\lvert{\psi^{+}}\right\rangle_{12}$ (one
$\left\lvert{\psi^{-}}\right\rangle_{12}$) have exchange symmetry (anti-
symmetry) with the feature of
$\left\lvert{\varphi}\right\rangle_{21}=\left\lvert{\varphi}\right\rangle_{12}$
($\left\lvert{\varphi}\right\rangle_{21}=-\left\lvert{\varphi}\right\rangle_{12}$)
and exhibit the bunching (anti-bunching) effect, i.e. a central dip (peak) in
the HOM interference curve shown in Fig. 1(a), so the three states (the other
state) belong to Boson states (Fermion state). In fact, any identical two
photons with the state of
$\left\lvert{\varphi}\right\rangle_{12}=(\alpha\left\lvert{H}\right\rangle_{1}+\beta\left\lvert{V}\right\rangle_{1})\otimes(\alpha\left\lvert{H}\right\rangle_{2}+\beta\left\lvert{V}\right\rangle_{2})$
(where $\alpha$ and $\beta$ are normalized complex constants satisfying
$\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$) can be represented as the
superposition of the three Boson states, resulting in that two
indistinguishable photons are exchange symmetric, as the HOM interference
between two identical photons. Therefore, the four Bell states are essential
to describe the bunching and anti-bunching effects in the two-photon
interference.
Similarly, we can remark the four Bell states in the DoF of OAM,
$\left\lvert{\mu^{\pm}}\right\rangle_{12}$ and
$\left\lvert{\nu^{\pm}}\right\rangle_{12}$ as follows
$\displaystyle\left\lvert{\mu^{\pm}}\right\rangle_{12}$
$\displaystyle=(\left\lvert{+m}\right\rangle_{1}\left\lvert{+m}\right\rangle_{2}\pm\left\lvert{-m}\right\rangle_{1}\left\lvert{-m}\right\rangle_{2})/\sqrt{2},$
(3) $\displaystyle\left\lvert{\nu^{\pm}}\right\rangle_{12}$
$\displaystyle=(\left\lvert{+m}\right\rangle_{1}\left\lvert{-m}\right\rangle_{2}\pm\left\lvert{-m}\right\rangle_{1}\left\lvert{+m}\right\rangle_{2})/\sqrt{2},$
(4)
where $\left\lvert{\pm m}\right\rangle$ refer to the photon states with OAMs
of $\pm m\hbar$ per photon, respectively. The four OAM Bell states can also be
divided into two categories of Boson states
($\left\lvert{\mu^{\pm}}\right\rangle_{12}$ and
$\left\lvert{\nu^{+}}\right\rangle_{12}$) and Fermion state
($\left\lvert{\nu^{-}}\right\rangle_{12}$), leading to three central dips and
one central peak in the HOM interference curves shown in Fig. 1(a). When
exploring two-photon interference in two DoFs of polarization and OAM, a
hyper-entangled state (Boson/Fermion-Boson/Fermion state) in the form of the
product of a polarization Bell state and an OAM Bell state can be prepared for
two photons, as shown in Fig. 1(b). Interesting conclusions would be expected
for the two-photon interference with hyper-entangled state according to the
general physics law that more is different Anderson1972 .
Figure 1: Two-photon HOM interference. (a) Four Bell states in one DoF of
polarization or OAM containing three exchange symmetric Boson states and one
exchange anti-symmetric Fermion state. (b) Polarization-OAM hyper-entangled
states containing10 symmetric and 6 anti-symmetric states.
To explore the two-photon HOM interference in two DoFs, we classify sixteen
hyper-entangled states into four groups, i.e., one Fermion-Fermion state
($\left\lvert{\psi^{-}}\right\rangle\otimes\left\lvert{\nu^{-}}\right\rangle$),
three Fermion-Boson states
($\left\lvert{\psi^{-}}\right\rangle\otimes\\{\left\lvert{\mu^{+}}\right\rangle,\left\lvert{\mu^{-}}\right\rangle,\left\lvert{\nu^{+}}\right\rangle\\}$),
three Boson-Fermion states
($\\{\left\lvert{\phi^{+}}\right\rangle,\left\lvert{\phi^{-}}\right\rangle,\left\lvert{\psi^{+}}\right\rangle\\}\otimes\left\lvert{\nu^{-}}\right\rangle$),
and nine Boson-Boson states
($\\{\left\lvert{\phi^{+}}\right\rangle,\left\lvert{\phi^{-}}\right\rangle,\left\lvert{\psi^{+}}\right\rangle\\}\otimes\\{\left\lvert{\mu^{+}}\right\rangle,\left\lvert{\mu^{-}}\right\rangle,\left\lvert{\nu^{+}}\right\rangle\\}$).
The exchange symmetry governing the bunching or anti-bunching effect in the
HOM interference can be revealed by the number of Fermion states appearing in
the hyper-entangled state. When a hyper-entangled state contains odd Fermion
states, it is exchange anti-symmetric and results in the anti-bunching effect
in the HOM interference. As a comparison, even Fermion states appearing in a
hyper-entangled state will lead to the exchange symmetry and the bunching
effect. This conclusion is scalable to hyper-entangled state in the more DoFs.
Figure 2: Experimental setup for hyper-entangled two-photon interference on a
BS and direct measurement of two-photon exchange phase, where a PBS replaces
the interference element to realize two-photon exchange. We have added
mirror(s) in some optical paths to ensure that any OAM state is reflected for
even times, so as to remain its handedness unchanged. Figure 3: HOM
interference when using a pair of filters with 3-nm bandwidth for (a)
polarization Bell states, (b) orbital angular momentum Bell states, and (c)
hyper-entangled Bell states. The error bars are concealed as they are smaller
than the data spheres.
In the experimental setup (Fig. 2), the hyper-entangled photon pairs are
generated via the SPDC when a femtosecond (fs) pulsed laser passes through two
0.6-mm-thick type-I BBO ($\beta$-barium borate) crystals glued with
orthogonally configured optic axes Barreiro2005 ; Wang2015 , in which the
polarization and OAM entanglements are achieved simultaneously in the small-
angle type-I SPDC Mair2001 . The spatial and temporal walk-offs between
photons from two BBO crystals are compensated with various YVO4 crystals (see
SuppMat for details). Thus we directly prepare the hyper-entangled state
$\left\lvert{\phi^{+}}\right\rangle\otimes\left\lvert{\nu^{+}}\right\rangle$
and the other fifteen hyper-entangled states can also be easily realized with
the aid of two Dove prisms (DPs) and two half wave plates (HWPs) (see SuppMat
for details). Here the prepared OAM entanglement in the subspace has the OAMs
of $\pm\hbar$ (i.e. $m=1$). Then the two photons enter into two input ports of
a beam splitter (BS) to implement the HOM interference in two DoFs, where the
arriving time of one photon is controlled by the delay line. A state
measurement configuration by recording the two-photon coincidence counts in
the two output port of BS is utilized to obtain the HOM interference curve
between two hyper-entangled photons. For the first one
$\left\lvert{\phi^{+}}\right\rangle\otimes\left\lvert{\nu^{+}}\right\rangle$
among the hyper-entangled states, we choose
$\left\lvert{H}\right\rangle_{1}\left\lvert{+1}\right\rangle_{1}$ and
$\left\lvert{H}\right\rangle_{2}\left\lvert{-1}\right\rangle_{2}$ as the two-
photon measurement bases (see SuppMat for all the sixteen measurement bases).
When a pair of narrow-band filters with 3-nm bandwidth is utilized, the
measured HOM interference in one DoF of polarization or OAM in Fig. 3(a) or
(b) shows three dips at zero delay for the three Boson states and one peak at
zero delay for the Fermion state. As shown in Fig. 3(c), the HOM interference
in the polarization-OAM hyper-entangled states exhibits ten dips in nine
Boson-Boson states
($\\{\left\lvert{\phi^{+}}\right\rangle,\left\lvert{\phi^{-}}\right\rangle,\left\lvert{\psi^{+}}\right\rangle\\}\otimes\\{\left\lvert{\mu^{+}}\right\rangle,\left\lvert{\mu^{-}}\right\rangle,\left\lvert{\nu^{+}}\right\rangle\\}$)
and one Fermion-Fermion state
($\left\lvert{\psi^{-}}\right\rangle\otimes\left\lvert{\nu^{-}}\right\rangle$),
and six peaks in three Boson-Fermion states
($\\{\left\lvert{\phi^{+}}\right\rangle,\left\lvert{\phi^{-}}\right\rangle,\left\lvert{\psi^{+}}\right\rangle\\}\otimes\left\lvert{\nu^{-}}\right\rangle$)
and three Fermion-Boson states
($\left\lvert{\psi^{-}}\right\rangle\otimes\\{\left\lvert{\mu^{+}}\right\rangle,\left\lvert{\mu^{-}}\right\rangle,\left\lvert{\nu^{+}}\right\rangle\\}$,
as the theoretical predictions. To characterize a HOM interference, we utilize
the visibility defined as $V_{dip}=1-C_{0}/C_{\infty}$
($V_{peak}=C_{0}/C_{\infty}-1$) for the dip (peak) Wang2015 ; Gao2022 , where
$C_{0}$ and $C_{\infty}$ are fitted counts at zero and infinite delays. The
extracted visibilities of the HOM interference for the four Bell states in the
single DoF of polarization or OAM ranges from $0.975\pm 0.005$ to $0.992\pm
0.066$ or $0.957\pm 0.068$ to $0.993\pm 0.003$; while the HOM interference for
the sixteen hyper-entangled states has the extracted visibilities ranging from
$0.902\pm 0.071$ to $0.993\pm 0.002$ (see SuppMat for details and the
corresponding results with 8-nm filters).
An interesting application for the hyper-entangled two-photon interference is
the direct measurement of the two-photon exchange phases. Recently, an
approach has been in theory proposed to reveal the quantum statistics by
constructing the superposition of a reference state of two distant particles
and a physically exchanged two-particle state Roos2017 . The experiment has
also demonstrated the direct measurement of exchange phase of
indistinguishable photons, which is the symmetric exchange phase for two-
photon Boson state Tschernig2021 . In fact, expanding the Hilbert space by
introducing an extra DoF for the two photons enables to transfer the
unmeasurable external (global) phase in a single DoF to a measurable internal
phase in the expanded two DoFs.
Figure 4 shows the principle for directly measuring the exchange phases for
the two-photon OAM entangled states. The exchange process can be described as
$\left\lvert{OAM}\right\rangle_{12}\xrightarrow{\rm
Exchange}\left\lvert{OAM}\right\rangle_{21}=e^{j\Phi_{O}}\left\lvert{OAM}\right\rangle_{12}$.
In theory, the OAM exchange phase $\Phi_{O}$ is zero for a Boson state and
$\pi$ for the Fermion state. In the four OAM Bell states
$\left\lvert{\mu^{\pm}}\right\rangle_{12}$ and
$\left\lvert{\nu^{\pm}}\right\rangle_{12}$, the three Boson states of
$\left\lvert{\mu^{\pm}}\right\rangle_{12}$ and
$\left\lvert{\nu^{+}}\right\rangle_{12}$ will result in a zero exchange phase.
It is worth noting that, two identical photons is in the superposition state
of these three Boson states, therefore it is also a Boson state and has a zero
exchange symmetric phase. For two photons, only the unique Fermion OAM state
of $\left\lvert{\nu^{-}}\right\rangle_{12}$ can result in an exchange anti-
symmetric phase of $\pi$.
To directly measure these exchange phases, we introduce an extra DoF of
polarization and then prepare the initial hyper-entangled state of
$\left\lvert{\phi^{+}}\right\rangle_{12}\otimes\left\lvert{OAM}\right\rangle_{12}$
in the two DoFs of polarization and OAM as
$(\left\lvert{H}\right\rangle_{1}\left\lvert{H}\right\rangle_{2}+\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2})/\sqrt{2}\otimes\left\lvert{OAM}\right\rangle_{12},$
(5)
where
$\left\lvert{H}\right\rangle_{1}\left\lvert{H}\right\rangle_{2}\otimes\left\lvert{OAM}\right\rangle_{12}$
is considered as the reference two-photon state and the two-photon state of
$\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2}\otimes\left\lvert{OAM}\right\rangle_{12}$
will be exchanged. When such a two-photon state passes through a PBS, the
transmitted horizontal polarization component will be physically preserved on
its original propagation path as the input one, therefore, this polarization
component could be utilized as the the reference state. On the contrary, for
the vertical polarization component in the two-photon state of
$\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2}\otimes\left\lvert{OAM}\right\rangle_{12}$,
it will be reflected on the PBS and the two photons will be physically
exchanged, resulting in the exchange phase $\Phi_{O}$ with respect to its
original two-photon state, which will follow the process below
$\displaystyle\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2}\otimes\left\lvert{OAM}\right\rangle_{12}$
$\displaystyle\xrightarrow[\rm by\ PBS]{\rm
Exchange}e^{j\pi}\left\lvert{V}\right\rangle_{2}\left\lvert{V}\right\rangle_{1}\otimes\left\lvert{OAM}\right\rangle_{21}$
(6)
$\displaystyle=e^{j(\pi+\Phi_{P})}\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2}\otimes
e^{j\Phi_{O}}\left\lvert{OAM}\right\rangle_{12}$
$\displaystyle\xrightarrow{\rm
BC}\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2}\otimes
e^{j\Phi_{O}}\left\lvert{OAM}\right\rangle_{12}.$
Considering only the DoF of polarization, for the two-photon exchange process
on the PBS, the reflections will introduce a $\pi$ phase for
$\left\lvert{V}\right\rangle_{2}\left\lvert{V}\right\rangle_{1}$. In addition,
an exchange phase $\Phi_{P}$ for the polarization will be also brought as
$\left\lvert{V}\right\rangle_{2}\left\lvert{V}\right\rangle_{1}=e^{j\Phi_{P}}\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2}$.
The total phase of $(\pi+\Phi_{P})$ appears in the DoF of polarization, and
can be compensated by a Babinet compensator (BC) before measuring the OAM
exchange phase $\Phi_{O}$.
Figure 4: Principle to directly measure the exchange phases for two photons in
the OAM state by the aid of polarization as an ancillary DoF with hyper-
entangled two-photon interference. The exchange process is realized on PBS for
vertically polarized component.
After the PBS and the BC, the OAM exchange phase will appear in the two-photon
vertical polarization component, yielding the superposition state
$(\left\lvert{H}\right\rangle_{1}\left\lvert{H}\right\rangle_{2}+e^{j\Phi_{O}}\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2})/\sqrt{2}\otimes\left\lvert{OAM}\right\rangle_{12}.$
(7)
The measurement of external phase $\Phi_{O}$ for the exchanged two-photon OAM
state $e^{j\Phi_{O}}\left\lvert{OAM}\right\rangle_{12}$ has been turned into
the measurement of internal phase in the two-photon polarization state
$(\left\lvert{H}\right\rangle_{1}\left\lvert{H}\right\rangle_{2}+e^{j\Phi_{O}}\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2})/\sqrt{2}$
between $\left\lvert{H}\right\rangle_{1}\left\lvert{H}\right\rangle_{2}$ and
$\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2}$. In
experiment, the above process can be easily implemented by replacing the BS
with a PBS and inserting a BC into the quantum interference unit shown in Fig.
2.
To measure $\Phi_{O}$ in experiment, for the two-photon polarization state
$(\left\lvert{H}\right\rangle_{1}\left\lvert{H}\right\rangle_{2}+e^{j\Phi_{O}}\left\lvert{V}\right\rangle_{1}\left\lvert{V}\right\rangle_{2})/\sqrt{2}$,
we require to project the photon-1 into the basis of
$(\left\lvert{H}\right\rangle_{1}+\left\lvert{V}\right\rangle_{1})/\sqrt{2}$
and then to measure the photon-2 in the basis of
$(\left\lvert{H}\right\rangle_{2}\pm
e^{j\theta}\left\lvert{V}\right\rangle_{2})/\sqrt{2}$ that are the eigenstates
of the observable $M_{\theta}=\cos\theta\sigma_{x}+\sin\theta\sigma_{y}$ with
their eigenvalues of $+1$ and $-1$, where $\sigma_{x}$ and $\sigma_{y}$ are
the Pauli $x$ and $y$ matrices. Here $\theta$ spans from 0 to $2\pi$, in
experiment $\theta$ is determined by the orientation angle $\Theta$ of the
HWP’s fast axis with respect to the horizontal direction, where
$\Theta=3\pi/8+\theta/4$ (see SuppMat for details).
We can calculate the relative probabilities of two-photon counts from the
measurements in the bases
$(\left\lvert{H}\right\rangle_{1}+\left\lvert{V}\right\rangle_{1})/\sqrt{2}$
$(\left\lvert{H}\right\rangle_{2}\pm
e^{j\theta}\left\lvert{V}\right\rangle_{2})/\sqrt{2}$ as $P_{\pm}$, to obtain
the expectation value of $\langle M_{\theta}$ as $\langle
M_{\theta}\rangle=P_{+}-P_{-}$. In theory, $\langle M_{\theta}\rangle$ reaches
its maximum of 1 when $\theta=\Phi_{O}$, and in experiment we use the value
$\theta$ corresponding to the maximum value on the fixed sine curve. From the
experimental results shown in Fig. 5, we obtain the extracted exchange phases
$\Phi_{O}$ are $0.012\pm 0.002$, $0.025\pm 0.002$ and $0.027\pm 0.002$ in
radian for three Boson states $\left\lvert{\mu^{\pm}}\right\rangle_{12}$ and
$\left\lvert{\nu^{+}}\right\rangle_{12}$, and $0.991\pi\pm 0.002$ in radian
for the Fermion state $\left\lvert{\nu^{-}}\right\rangle_{12}$, respectively.
Supplemental Material SuppMat also shows the results for the other
superposition states.
Figure 5: Experimental results for exchange phase of two photons in the four
OAM Bell states. Two-photon counts are recorded by projecting the photon-1 in
the basis of
$(\left\lvert{H}\right\rangle+\left\lvert{V}\right\rangle)/\sqrt{2}$ and
measuring the photon-2 in the basis of $(\left\lvert{H}\right\rangle\pm
e^{j\theta}\left\lvert{V}\right\rangle)/\sqrt{2}$. The error bars are
concealed as they are smaller than the data dots.
To summarize, we have for the first time systematically realized the HOM
interference between the hyper-entangled two photons. It is greatly
interesting to extend our work to high-dimensional region by exploring the HOM
effect with hyper-entanglement in (two-dimensional) polarization and other
(high-dimensional) DoF such as OAM. Benefitted from the recently developed
technologies of high-dimensional entanglement preparation Valencia2021 ;
Valencia2020 and measurement methods Bavaresco2018 ; Kong2020 , the HOM
interference of such hyper-entangled two photons is expected in future. Our
approach enables to directly measure the exchange symmetric and anti-symmetric
phases, demonstrating an interesting and significant application of the hyper-
entangled two-photon interference. More applications of the HOM interference
in the more DoFs would be expected in quantum network Wang2015 ; Liu2021 ,
quantum computation Wang2018 and high-dimensional quantum entanglement and
processing Reimer2018 ; Kong2019 ; Erhard2020 . Our results show that
expanding the Hilbert space by introducing the extra DoFs will open new
opportunities for quantum information, such as turning the external and
unmeasurable exchange phase in one DoF to an internal and measurable phase in
another DoF. More DoFs not only contribute to high-dimensional quantum
information processing Erhard2020 , but also excite novel functions in quantum
applications such as alignment-free quantum communications D'Ambrosio2012 and
complete Bell state measurement Ecker2021 .
###### Acknowledgements.
This work was supported by National Natural Science Foundation of China (Nos.
11922406, 12234009, 12274215); National Key R&D Program of China (Nos.
2019YFA0308700, 2020YFA0309500); Innovation Program for Quantum Science and
Technology (No. 2021ZD0301400); Program for Innovative Talents and
Entrepreneurs in Jiangsu; Key R&D Program of Guangdong Province (No.
2020B0303010001).
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|
Large Deviation Probabilities for Sums of Random Variables with Heavy or
Subexponential Tails
Daren B. H. Cline∗ and Tailen Hsing∗∗
Texas A&M University and University of Michigan
Abstract. Let $S_{n}$ be the sum of independent random variables with
distribution $F$. Under the assumption that $-\log(1-F(x))$ is slowly varying,
conditions for
$\lim_{n\to\infty}\sup_{s\geq t_{n}}\left|{P[S_{n}>s]\over
n(1-F(s))}-1\right|=0$
are given. These conditions extend and strengthen a series of previous
results. Additionally, a connection with subexponential distributions is
demonstrated. That is, $F$ is subexponential if and only if the condition
above holds for some $t_{n}$ and
$\lim_{t\to\infty}{1-F(t+x)\over 1-F(t)}=1\quad\text{for each real $x$.}$
†† Key words: large deviations, regular variation, subexponential distribution
MSC classification: 60F10, 60G50 ∗ Research sponsored by National Science
Foundation Grant DMS 9101083 ∗∗Research sponsored by National Science
Foundation Grant DMS 9107507
## 1 Subexponentiality and large deviations
Let $X_{i},\ i\geq 1$, be independent and identically distributed random
variables with distribution function $F$, and $S_{n}=\sum_{i=1}^{n}X_{i}$.
Suppose $\\{t_{n}\\}$ is a sequence of positive constants such that
${S_{n}/t_{n}}\buildrel P\over{\rightarrow}0$. It is often of interest to
consider the rate at which the large deviation probabilities
$P[|S_{n}|>t_{n}]$ and $P[S_{n}>t_{n}]$ tend to zero. This topic is of
traditional importance in probability theory, and has numerous statistical
applications.
Following Cramér’s pioneering work (1938), the study of large deviation
probabilities at first was confined to distributions $F$ satisfying Cramér’s
condition, namely, for some constant $\epsilon>0$,
$\displaystyle\int_{-\infty}^{\infty}e^{cx}F(dx)<\infty\quad\text{for all
$c\in[-\epsilon,\epsilon]$}.$ (1.1)
For example, (1.1) implies
$\lim_{n\to\infty}n^{-1}\log P[S_{n}>nx]=\sup_{\lambda\geq 0}[\lambda
x-L(\lambda)]\quad\text{for all }x>E(X_{1}),$
and
$\lim_{n\to\infty}n^{-1}\log P[S_{n}<nx]=\sup_{\lambda\leq 0}[\lambda
x-L(\lambda)]\quad\text{for all }x<E(X_{1}),$
where $L(\lambda)=\log Ee^{\lambda X_{1}}$.
It was not until the 1960’s that attention was given to heavy–tailed
distributions. In that connection, the most noticeable work was by Linnik
(1961), Heyde (1967a,b,1968), A.V. Nagaev (1969a,b) and S.V. Nagaev (1973).
(See also the review paper by S.V. Nagaev (1979).) While Linnik, A. V. Nagaev,
and S. V. Nagaev mostly considered distributions with finite variance, Heyde
focused on distributions with infinite variance, including the ones that are
attracted to nonnormal stable laws. However, their results contain a common
implication for the heavy tailed distributions they considered: namely, if
$t_{n}$ tends to $\infty$ fast enough then one has
$\displaystyle
0<\liminf_{n\to\infty}\dfrac{P[S_{n}>t_{n}]}{nP[X>t_{n}]}\leq\limsup_{n\to\infty}\dfrac{P[S_{n}>t_{n}]}{nP[X>t_{n}]}<\infty$
(1.2)
and, in some cases, even
$\displaystyle\lim_{n\to\infty}\dfrac{P[S_{n}>t_{n}]}{nP[X>t_{n}]}=1.$ (1.3)
Typically the results require at least $n(1-F(t_{n}))\to 0$. Note that if this
is so then (1.2) and (1.3) state that the probabilities of large deviations
for the sum and the maximum are asymptotically comparable.
What class of distributions can be expected to have these properties? The most
natural notion is that of subexponentiality. Specifically, the subexponential
class $\cal S$ consists of those probability distributions $F$ satisfying
$\displaystyle\lim_{t\to\infty}{P[X_{1}+X_{2}>t]\over\overline{F}(t)}\quad\text{exists
finite}$ (1.4)
and
$\displaystyle\lim_{t\to\infty}{\overline{F}(t+x)\over\overline{F}(t)}=1\quad\text{for
each real}\ x,$ (1.5)
where $\overline{F}(x)=1-F(x)$ and $X_{1}$ and $X_{2}$ are independent random
variables distributed according to $F$. This class was first studied by
Chistyakov (1964) and by Chover, Ney and Wainger (1973a) in the case $P[X\geq
0]=1$, with application to branching processes. Additional work is found in
Embrechts and Goldie (1980) and Cline (1987). Extensions for the case $P[X\geq
0]<1$ are given by Willekens (1986) and numerous references and applications
are provided in Embrechts and Goldie (1982).
Two properties of $\cal S$ are of interest to us here. First, if $F\in\cal S$
then (1.1) fails to hold (cf. Embrechts and Goldie (1982)). Second, the limit
in (1.4) equals $2$ and furthermore
$\displaystyle\lim_{t\to\infty}{P[S_{n}>t]\over
n\overline{F}(t)}=1\quad\text{for every}\ n\geq 1$ (1.6)
(cf. Chistyakov (1964), with extension by Willekens (1986)). Immediately one
has that, for some sequence $\\{t_{n}\\}$, (1.3) holds for distributions in
the subexponential family. In fact, we may characterize $\cal S$ as follows.
###### Theorem 1.1.
$F\in\cal S$ if and only if (1.5) holds and there exists $t_{n}$ such that
$\displaystyle\limsup_{n\to\infty}\sup_{s\geq t_{n}}{P[S_{n}>s]\over
n\overline{F}(s)}\leq 1.$ (1.7)
Moreover, when $F\in\cal S$ there exists $t_{n}$ such that
$\displaystyle\lim_{n\to\infty}\sup_{s\geq t_{n}}\left|{P[S_{n}>s]\over
n\overline{F}(s)}-1\right|=0.$ (1.8)
Our proof of Theorem 1.1 (in Section 4), however, does not construct a
specific choice of $\\{t_{n}\\}$ for (1.8) to hold. Therefore our second goal
is to consider how to choose the constants $t_{n}$ for a particular
subexponential distribution.
The subexponential distributions that we will focus on are those $F$ for which
$-\log\overline{F}(x)$ is slowly varying as $x\to\infty$. These are the
distributions whose tails are on the heavy side in $\cal S$, and they include
distributions that are attracted to the nonnormal stable distributions as well
as distributions with finite variance, such as the lognormal. Large deviation
results for this class are few and scattered. For example, there are no
results for distributions whose tails behave like the Cauchy and lognormal
distributions. We show that for these distributions, it is possible to have a
unified treatment based on a truncation argument and Markov’s inequality. We
not only obtain non-trivial extensions of results previously proved by using
different methods, but we also fill some holes in the literature. The main
result is Theorem 2.1 in the next section which is followed by a discussion on
the conditions of the result. In Section 3 we give some examples that can be
derived directly from Theorem 2.1, or from slight modifications of Theorem
2.1. We also describe how existing results relate to the examples.
The paper does not cover “semiexponential” distributions, e.g.
$F(x)=1-e^{-x^{\rho}}$ with $0<\rho<1$, which are also in the subexponential
class (cf. Cline (1986) and Goldie and Resnick (1988)). A.V. Nagaev (1969b)
and S.V. Nagaev (1973) have considered large deviations for special cases of
semiexponential distributions. We do not know of a unified approach that will
work for all subexponential distributions.
For clarity, all the proofs and technical details are collected in Section 4.
## 2 Main Result
For what follows, let $\psi(t)=-\log\overline{F}(t)$. When (1.5) holds,
$\overline{F}(\log t)$ is slowly varying and thus $\psi$ has the
representation
$\displaystyle\psi(t)=b(t)+\int_{0}^{t}\eta(u)du,\qquad{\rm for\ all\ }t\geq
0,$ (2.1)
where $\eta(t)$ and $b(t)$ are measurable and $\eta(t)\to 0$, $b(t)\to
b\in\mathbb{R}$, as $t\to\infty$. Because of this, a represenation like (2.1)
will be the starting point for our theorems.
Also, define
$\mu_{1}(t)=\int_{|x|\leq t}xdF(x)\quad\text{and}\quad\mu_{2}(t)=\int_{|x|\leq
t}x^{2}dF(x),\quad t>0.$
We now state the main result of this paper.
###### Theorem 2.1.
Let $a=a_{n}(s)=\psi(s)-\log n$. Assume (2.1) where $\eta(t)\downarrow 0$ and
$b(t)$ is measurable, bounded and satisfies
$\displaystyle\lim_{y\to 1,t\to\infty}(b(yt)-b(t))=0.$ (2.2)
Suppose $t_{n}$ are constants increasing to $\infty$ and $\lambda\in(0,1)$
such that, as $n\to\infty$,
$\displaystyle\sup_{s\geq t_{n}}\inf_{w\geq
s/a}n\left({|\mu_{1}(w)|+a\mu_{2}(w)/s\over s\wedge\eta^{-1}(\lambda
s)}+F(-w)\right)\to 0,$ (2.3) $\displaystyle\sup_{s\geq t_{n}}1\vee
s\eta(\lambda s)\log s\eta(\lambda s)$ (2.4)
and
$\displaystyle\sup_{s\geq t_{n}}n(1\vee s\eta(\lambda s))\overline{F}(s/a)\to
0,$ (2.5)
where $\eta^{-1}(\lambda s)=1/\eta(\lambda s)$. Then
$\displaystyle\lim_{n\to\infty}\sup_{s\geq t_{n}}\left|{P[S_{n}>s]\over
n\overline{F}(s)}-1\right|=0.$ (2.6)
The conditions (2.3)-(2.5) attempt to cover a variety of situations. While
they do not offer much insight into the large deviation problem at this stage,
it is worth mentioning that they are sharp since they form the weakest
condition for (2.6) to hold in the case where the tail probabilities of $F$
are regularly varying with index in $(-1,0]$ (cf. Theorem 3.3). The following
remarks are useful.
Remark 1. Note that (2.3) is equivalent to
$\displaystyle\sup_{s\geq t_{n}}n\left({|\mu_{1}(w)|+a\mu_{2}(w)/s\over
s\wedge\eta^{-1}(\lambda s)}+F(-w)\right)\to 0\text{ for some }w=w_{n}(s)\geq
s/a.$ (2.3’)
Remark 2. The conditions require
$nP[|X|>w_{n}]\to 0,\ {n\mu_{1}(w_{n})\over t_{n}\wedge\eta^{-1}(\lambda
t_{n})}\to 0,\text{ and }{n\mu_{2}(w_{n})\over(t_{n}\wedge\eta^{-1}(\lambda
t_{n}))^{2}}\to 0,$
for some $w_{n}$ and hence
${S_{n}\over t_{n}\wedge\eta^{-1}(\lambda t_{n})}\buildrel
P\over{\rightarrow}0,$
by a standard argument using truncation and Chebyshev’s inequality.
Remark 3. Condition (2.4) is possible for some $t_{n}$ if and only if
$(t\eta(t)\log t\eta(t))/\psi(t)\to 0$, as $t\to\infty$, which implies that
$\psi$ is slowly varying. Condition (2.5) is possible for some $t_{n}$ if and
only if $t\eta(t)\overline{F}(t/\psi(t))\to 0$. The following lemma gives
sufficient conditions.
###### Lemma 2.2.
Assume (2.1) with $\eta(t)\downarrow 0$ and $b(t)\to b$, as $t\to\infty$.
(i) If $\exp(\log^{2}\psi(t))$ is slowly varying then
$\lim_{t\to\infty}{t\eta(t)\log\psi(t)\over\psi(t)}=0.$
(ii) If
$\displaystyle\lim_{t\to\infty}{t\eta(t)\over\psi(t)}=0$ (2.7)
and
$\displaystyle\liminf_{t\to\infty}{\eta(t/\psi(t))\over\eta(t)}>1$ (2.8)
then
$\lim_{t\to\infty}t\eta(t)\overline{F}(t/\psi(t))=0.$
Remark 4. In particular, (2.7) and (2.8) hold if $\eta$ is monotone and
regularly varying with index $-1$. Some examples are given in Section 3.
## 3 Examples and ramifications
For our first example we consider a class of finite variance distributions for
which the condition on $t_{n}$ is simply expressed. This class includes the
lognormal distribution and some log-Weibull distributions.
###### Theorem 3.1.
Suppose $F$ has mean 0 and a finite $2+\delta$ moment for some positive
$\delta$. Suppose also that (2.1) and (2.2) hold with $b$ bounded and
measurable, $\eta(t)\downarrow 0$, $t\eta(t)$ bounded away from 0, and
$\displaystyle\lim_{t\to\infty}{t\eta(t)\log t\eta(t)\over\psi(t)}=0.$ (3.1)
If $\lambda\in(0,1)$ and $t_{n}\uparrow\infty$ satisfy
$\displaystyle\sup_{s\geq t_{n}}{n\psi(s)\eta(\lambda s)\over s}\to 0,$ (3.2)
as $n\to\infty$, then (2.6) holds.
For example, suppose $F$ is the centered lognormal distribution
$F(x)=\Phi\left({\log(x+\beta)\over\sigma}\right),\qquad x>-\beta,$
where $\Phi$ is the standard normal distribution and $\beta=e^{\sigma^{2}/2}$.
Then
$\overline{F}(x)\sim{\sigma\over\sqrt{2\pi}}e^{-(\log(x+\beta))^{2}/2\sigma^{2}-\log\log(x+\beta)}\sim{\sigma\over\sqrt{2\pi}}e^{-(\log
x)^{2}/2\sigma^{2}-\log\log x}.$
and we can take
$\eta(x)={\log x\over\sigma^{2}x}+{1\over x\log x}\sim{\log
x\over\sigma^{2}x}.$
Hence if $t_{n}$ is such that
$\lim_{n\to\infty}{n\log^{3}t_{n}\over t_{n}^{2}}=0$
then (2.6) holds.
If $b$ is bounded in Theorem 2.1 but does not satisfy (2.2) then $F$ may not
be subexponential. Nevertheless, useful limiting bounds are available. From
the proof of Theorem 2.1 one may deduce that if all the other assumptions are
met then
$\displaystyle e^{-\gamma}\leq\liminf_{n\to\infty}\inf_{s\geq
t_{n}}{P[S_{n}>s]\over n\overline{F}(s)}\leq\limsup_{n\to\infty}\sup_{s\geq
t_{n}}{P[S_{n}>s]\over n\overline{F}(s)}\leq e^{\gamma},$ (3.3)
where
$\gamma=\lim_{y\downarrow 1}\limsup_{t\to\infty}\sup_{1\leq x\leq
y}(b(xt)-b(t)).$
In a slightly different direction, the proof of Theorem 2.1 can be specialized
to handle distributions with dominated varying tails. To that end, we recall
some basic definitions (cf. Bingham, Goldie, and Teugels (1987), p. 61–76, and
Cline (1994)).
For a distribution $F$, $\overline{F}$ is regularly varying $(\overline{F}\in
RV_{-\alpha})$ if, for some nonnegative $\alpha$,
$\lim_{t\to\infty}{\overline{F}(\lambda
t)\over\overline{F}(t)}=\lambda^{-\alpha},\quad\text{for all $\lambda\geq
1$}.$
$\overline{F}$ is intermediate regularly varying if
$\lim_{\lambda\to 1,t\to\infty}{\overline{F}(\lambda
t)\over\overline{F}(t)}=1.$
$\overline{F}$ is dominated varying if
$\liminf_{t\to\infty}{\overline{F}(\lambda
t)\over\overline{F}(t)}>0,\quad\text{for some $\lambda>1$}.$
###### Theorem 3.2.
Suppose $P[|X_{1}|>x]$ is dominated varying and $S_{n}/t_{n}\buildrel
P\over{\rightarrow}0$. If
$\displaystyle\limsup_{n\to\infty}\sup_{s\geq t_{n}}{na\mu_{2}(s)\over
s^{2}}=0$ (3.4)
then (3.3) holds with
$\gamma=-\log\lim_{\lambda\downarrow
1}\liminf_{t\to\infty}{\overline{F}(\lambda t)\over\overline{F}(t)}.$
Moreover, if $F$ is intermediate regular varying then (2.6) holds.
Theorem 3.2 shows that distributions with intermediate regularly varying tails
are subexponential. In fact it is known that the subexponential class contains
those $F$ with dominated varying right tails and satisfying (1.5) (Goldie
(1978)).
###### Theorem 3.3.
Suppose $P[|X|>x]\in RV_{-\alpha}$, where $\alpha\geq 0$, $p=\sup_{x\geq
0}\dfrac{F(-x)}{\overline{F}(x)}<\infty$ and $t_{n}$ satisfies
$n\overline{F}(t_{n})\rightarrow 0$. Then any one of the following implies
(2.6):
* (i)
$0\leq\alpha<1$.
* (ii)
$1\leq\alpha<2$ and $\displaystyle\lim_{n\to\infty}{n\over
t_{n}}\mu_{1}(t_{n})=0$.
* (iii)
$\alpha\geq 2$ and $\displaystyle\lim_{n\to\infty}{nE(X)\over
t_{n}}=\lim_{n\to\infty}{n\mu_{2}(t_{n})\log t_{n}\over t_{n}^{2}}=0$.
In connection with Theorem 3.3, Heyde (1968) considered the infinite variance
case and Nagaev (1969b) considered the finite variance case. Our method, which
is a mixture of theirs, has several advantages. First we are able to study
these two cases in a unifying way, whereas the methods in Heyde (1968) and
Nagaev (1969b) are essentially different. For the infinite variance case, our
result is new since Heyde (1968) only considered large deviations of
$|S_{n}|$. Also neither author considered the cases where $\alpha=0,1$ or 2.
## 4 Proofs
proof of Theorem 1.1. If $F\in\cal S$, then (1.5) holds by definition. To show
both (1.7) and (1.8) we simply choose $t_{n}$ by (1.6) to satisfy
$\sup_{s\geq t_{n}}\left|{P[S_{n}>s]\over
n\overline{F}(s)}-1\right|\leq{1\over n}.$
Suppose instead that (1.5) holds and such $t_{n}$ exist that (1.7) holds.
Choose $n_{0}$ so that for each $n\geq n_{0}$,
$\sup_{s\geq t_{n}}{P[S_{n}>s]\over n\overline{F}(s)}\leq 1+\epsilon.$
Let $F_{n}(x)=P[S_{n}\leq x]$. Equation (1.5) is also satisfied with $F_{n}$
replacing $F$ (cf. Embrechts and Goldie (1980); with extension by Willekens
(1986)). Using this fact and Fatou’s Lemma,
$\displaystyle\begin{split}\liminf_{x\to\infty}{\overline{F}_{2n}(x)\over\overline{F}_{2}(x)}&\geq\liminf_{x\to\infty}\int_{-\infty}^{x/2}{\overline{F}_{2n-2}(x-u)\over\overline{F}_{2}(x)}F_{2}(du)+\liminf_{x\to\infty}\int_{-\infty}^{x/2}{\overline{F}_{2}(x-u)\over\overline{F}_{2}(x)}F_{2n-2}(du)\\\
&\geq\liminf_{x\to\infty}{\overline{F}_{2n-2}(x)\over\overline{F}_{2}(x)}+1.\end{split}$
(4.1)
By induction, $\liminf_{x\to\infty}{\overline{F}_{2n}(x)\over
n\overline{F}_{2}(x)}\geq 1.$ Choose successively, therefore,
$y_{n}\geq\max(t_{2n},y_{n-1})$ so that
$\inf_{y\geq y_{n}}{\overline{F}_{2n}(y)\over n\overline{F}_{2}(y)}\geq
1-\epsilon.$
Now let $x_{k}\to\infty$ and define $m=\sup\\{n:x_{k}\geq y_{n}\\}$. Then for
$k$ large enough so that $2m>n_{0}$,
${\overline{F}_{2}(x_{k})\over
2\overline{F}(x_{k})}={m\overline{F}_{2}(x_{k})\over\overline{F}_{2m}(x_{k})}{\overline{F}_{2m}(x_{k})\over
2m\overline{F}(x_{k})}\leq{1+\epsilon\over 1-\epsilon}.$
That is,
$\limsup_{k\to\infty}{\overline{F}_{2}(x_{k})\over 2\overline{F}(x_{k})}\leq
1.$
By the same use of Fatou’s Lemma that gave (4.1),
$\liminf_{k\to\infty}{\overline{F}_{2}(x_{k})\over 2\overline{F}(x_{k})}\geq
1.$
Since the sequence $x_{k}$ is arbitrary, then (1.4) holds and $F\in\cal S$. ∎
We need the following lemma in the proof of Theorem 2.1.
###### Lemma 4.1.
Let $1\leq y_{n}\leq x_{n}$. There exists $z_{n}$ such that
$x_{n}/z_{n}\to\infty$, $z_{n}/y_{n}\to\infty$ and $x_{n}/(y_{n}\log
z_{n})\to\infty$ if and only if $x_{n}/y_{n}\to\infty$ and $x_{n}/(y_{n}\log
y_{n})\to\infty$.
###### Proof.
The sufficiency is obvious. For the necessary part, let
$v_{n}=\min\left(\left({x_{n}\log y_{n}\over
y_{n}}\right)^{1/2}\\!\\!\\!,\,\,\,{1\over 2}(\log x_{n}+\log y_{n})\right)$
and $z_{n}=e^{v_{n}}$. ∎
Proof of Theorem 2.1. We shall proceed by first proving
$\displaystyle\limsup_{n\to\infty}\sup_{s\geq t_{n}}{P[S_{n}>s]\over
n\overline{F}(s)}\leq 1$ (4.2)
and then
$\displaystyle\liminf_{n\to\infty}\inf_{s\geq t_{n}}{P[S_{n}>s]\over
n\overline{F}(s)}\geq 1.$ (4.3)
Let $\epsilon=\epsilon_{n}(s)$ and
$\epsilon^{\prime}=\epsilon^{\prime}_{n}(s)$ be functions which vanish
uniformly in $s\geq t_{n}$, as $n\to\infty$. Specific choices for $\epsilon$
and $\epsilon^{\prime}$ will be made later. Define
$m=m_{n}(s)=(1-\epsilon^{\prime})s.$
Let $S_{n}^{\prime}=\sum_{i=1}^{n}X_{i}1_{X_{i}\leq m}$. Then
$\displaystyle\begin{split}{P[S_{n}>s]\over
n\overline{F}(s)}&\leq{nP[X_{1}>m]+P[S_{n}^{\prime}>s]\over
n\overline{F}(s)}\\\
&={\overline{F}(m)\over\overline{F}(s)}+{P[S_{n}^{\prime}>s]\over
n\overline{F}(s)}.\end{split}$ (4.4)
By Markov’s inequality,
${P[S_{n}^{\prime}>s]\over
n\overline{F}(s)}\leq\exp\left(n\int_{-\infty}^{m}(e^{cx}-1)F(dx)-cs+a\right)$
for any $c>0$. In particular, the choice $c=(1+\epsilon)a/s$ gives
$\displaystyle\begin{split}{P[S_{n}^{\prime}>s]\over
n\overline{F}(s)}&\leq\exp\left(n\int_{-\infty}^{m}(e^{(1+\epsilon)ax\over
s}-1)F(dx)-\epsilon a\right)\\\
&\leq\exp\left(n\int_{-w}^{s/a}(e^{(1+\epsilon)ax\over
s}-1)F(dx)+n\int_{s/a}^{m}(e^{(1+\epsilon)ax\over s}-1)F(dx)-\epsilon
a\right)\end{split}$ (4.5)
for any $w>0$. In the following, let $w=w_{n}(s)$ be chosen by (2.3’). In view
of (4.4) and (4.5), condition (4.2) follows from choosing $\epsilon$ and
$\epsilon^{\prime}$ so that
$\displaystyle\lim_{n\to\infty}\sup_{s\geq
t_{n}}{\overline{F}(m)\over\overline{F}(s)}=1,$ (4.6)
$\displaystyle\limsup_{n\to\infty}\sup_{s\geq t_{n}}{n\over\epsilon
a}\int_{-w}^{s/a}(e^{(1+\epsilon)ax\over s}-1)F(dx)\leq\delta$ (4.7)
$\displaystyle\limsup_{n\to\infty}\sup_{s\geq t_{n}}{n\over\epsilon
a}\int_{s/a}^{m}(e^{(1+\epsilon)ax\over s}-1)F(dx)<1-\delta$ (4.8)
and
$\displaystyle\epsilon a\to\infty\text{ as }n\to\infty$ (4.9)
for some $\delta\in(0,1)$. To this end, define
$B=1+\sup_{u\geq t\geq 0}|b(t)-b(u)|$
and choose $z_{n}(s)$ according to Lemma 4.1 with $x_{n}(s)=a_{n}(s)$ and
$y_{n}(s)=1\vee s\eta(\lambda s)$. Fix $\delta\in(0,1-e^{-2})$ and define
$\displaystyle\epsilon=\max\left(z_{n}(s)^{-1},ne^{B+3}\overline{F}(s/a),{2n\over\delta}\left({|\mu_{1}(w)|\over
s}+{e^{2}\mu_{2}(w)\over s^{2}/a}\right)\right).$ (4.10)
Using (2.3’), (2.4) and (2.5) and Lemma 4.1 we have, uniformly for $s\geq
t_{n}$,
$\displaystyle\begin{split}(1\vee s\eta(\lambda
s))\epsilon&\leq\max\Bigg{(}{y_{n}(s)\over z_{n}(s)},ne^{B+3}(1\vee
s\eta(\lambda
s))\overline{F}(s/a),{2n(|\mu_{1}(w)|+e^{2}a\mu_{2}(w)/s)\over\delta(s\wedge\eta^{-1}(\lambda
s))}\Bigg{)}\\\ &\to 0,\end{split}$ (4.11) $\displaystyle\epsilon
a\geq{x_{n}(s)\over z_{n}(s)}$ $\displaystyle\to\infty$ (4.12)
and
$\displaystyle{(-\log\epsilon)(1\vee s\eta(\lambda s))\over a}$
$\displaystyle\leq{y_{n}(s)\log z_{n}(s)\over x_{n}(s)}\to 0.$ (4.13)
Now choose
$\epsilon^{\prime}={\epsilon+(2B-\log\epsilon)/a\over 1-s\eta(\lambda s)/a}.$
Thus
$\displaystyle(\epsilon^{\prime}-\epsilon)a-\epsilon^{\prime}s\eta(\lambda
s)-2B=-\log\epsilon.$ (4.14)
In addition, from (2.4), (4.11) and (4.13) we have $\epsilon^{\prime}\to 0$
and with the monotonicity of $\eta$,
$\displaystyle s\eta(m)\epsilon^{\prime}$ $\displaystyle\leq(1\vee
s\eta(\lambda s))\epsilon^{\prime}$ (4.15) $\displaystyle={(1\vee
s\eta(\lambda s))(\epsilon+(2B-\log\epsilon)/a)\over 1-s\eta(\lambda s)/a}\to
0.$ (4.16)
We now show (4.6)–(4.8), since (4.9) already follows from (4.12). From (4.15)
and the assumption on $b$,
$\displaystyle\psi(s)-\psi(m)\leq b(s)-b(m)+\epsilon^{\prime}s\eta(m)\to 0,$
(4.17)
uniformly for $s\geq t_{n}$, which is (4.6). Using $w\geq s/a$, (4.10) and a
Taylor expansion, for large enough $n$,
$n\int_{-w}^{s/a}(e^{(1+\epsilon)ax\over s}-1)F(dx)\leq 2n{\mu_{1}(w)\over
s/a}+2ne^{2}{\mu_{2}(w)\over(s/a)^{2}}\leq\delta a\epsilon,$
uniformly for $s\geq t_{n}$, which is (4.7).
Integrating by parts, we get
$\displaystyle n\int_{s/a}^{m}(e^{(1+\epsilon)ax\over s}-1)F(dx)\leq
n{(1+\epsilon)a\over s}\int_{s/a}^{m}e^{{(1+\epsilon)ax\over
s}-\psi(x)}\,dx+ne^{(1+\epsilon)}\overline{F}(s/a).$ (4.18)
Since $\eta$ is decreasing, ${(1+\epsilon)ax\over s}-(\psi(x)-b(x))$ is convex
and hence
$\displaystyle\sup_{s/a\leq x\leq m}e^{{(1+\epsilon)ax\over
s}-\psi(x)}\leq\max\left(e^{{(1+\epsilon)ma\over
s}-\psi(m)+B-1},e^{B+\epsilon}\overline{F}(s/a)\right).$ (4.19)
By (4.18), (4.19) and the fact that $(1+\epsilon)m\leq s$, we obtain the bound
$\displaystyle\begin{split}n\int_{s/a}^{m}(e^{(1+\epsilon)ax\over
s}-1)F(dx)&\leq na\max\left(e^{{(1+\epsilon)ma\over
s}-\psi(m)+B-1},e^{B+\epsilon}\overline{F}(s/a)\right)+ne^{(1+\epsilon)}\overline{F}(s/a)\\\
&\leq
a\max\left(e^{-(\epsilon^{\prime}-\epsilon)a+\epsilon^{\prime}s\eta(\lambda
s)+2B-2},e^{B+1}n\overline{F}(s/a)\right)+ne^{2}\overline{F}(s/a)\\\
&=e^{-2}\epsilon a+ne^{2}\overline{F}(s/a),\end{split}$ (4.20)
where the final equality comes from (4.10) and (4.14). The bound in (4.8)
follows from (2.5) and that completes the proof of (4.2).
Finally, we verify (4.3). Let $m^{\prime}=s+\zeta(s\wedge\eta^{-1}(\lambda
s))$ where $\zeta$ is any fixed positive constant. By Bonferroni’s inequality,
$\displaystyle\begin{split}P[S_{n}>s]&\geq P[S_{n}>s,\ \max_{1\leq i\leq
n}X_{i}>m^{\prime}]\\\ &\geq\sum_{i=1}^{n}P[S_{n}>s,\
X_{i}>m^{\prime}]\,-\\!\\!\\!\sum_{1\leq i<j\leq n}\\!\\!P[S_{n}>s,\
X_{i}>m^{\prime},\ X_{j}>m^{\prime}]\\\ &\geq
n\overline{F}(m^{\prime})\left(P[S_{n-1}>-\zeta(s\wedge\eta^{-1}(\lambda
s))]-{n\over 2}\overline{F}(s)\right).\end{split}$ (4.21)
Note that $n\overline{F}(s)\to 0$ by (2.4), and if we let
$\gamma(\zeta)=\limsup_{t\to\infty}\sup_{0\leq u\leq\zeta}(b((1+u)t)-b(t))$
then
$\displaystyle\liminf_{n\to\infty}\inf_{s\geq
t_{n}}{\overline{F}(m^{\prime})\over\overline{F}(s)}\geq
e^{-\zeta-\gamma(\zeta)}$ (4.22)
(cf. the derivation of (4.17)).
By Remark 2 of Section 2,
$\sup_{s\geq t_{n}}{S_{n-1}\over s\wedge\eta^{-1}(\lambda s)}\buildrel
P\over{\rightarrow}0.$
Hence
$\inf_{s\geq t_{n}}P[S_{n-1}\geq-\zeta(s\wedge\eta^{-1}(\lambda s))]\to 1$
and by (4.21) and (4.22),
$\displaystyle\liminf_{n\to\infty}\inf_{s\geq t_{n}}{P[S_{n}>s]\over
n\overline{F}(s)}$ $\displaystyle\geq\liminf_{n\to\infty}\inf_{s\geq
t_{n}}{\overline{F}(m^{\prime})\over\overline{F}(s)}\left(P[S_{n-1}>-\zeta(s\wedge\eta^{-1}(\lambda
s))]-{n\over 2}\overline{F}(s)\right)$ $\displaystyle\geq
e^{-\zeta-\gamma(\zeta)}.$
Condition (4.3) follows from this since $\zeta>0$ is arbitrary and
$\gamma(\zeta)\to 0$ as $\zeta\to 0$ by (2.2). ∎
Proof of Lemma 2.2. Define $B_{1}=\inf_{u\geq 0}b(u)$ and
$\psi_{1}(t)=\psi(t)-b(t)+B_{1}$. Also define $B_{2}=\sup_{u\geq 0}b(u)$ and
$\psi_{2}(t)=\psi(t)-b(t)+B_{2}$.
(i) Let $\xi(t)=\exp(\log^{2}\psi_{1}(t))$. Then, for large enough $t$,
$\displaystyle{2\eta(2t)\log\psi(2t)\over\psi(2t)}$
$\displaystyle\leq{2\over\xi(t)}\int_{t}^{2t}{\eta(u)(\log\psi_{1}(u))\xi(u)\over\psi_{1}(u)}\,du$
$\displaystyle={\xi(2t)\over\xi(t)}-1$
$\displaystyle\leq{e^{\log^{2}\psi(2t)}\over
e^{\log^{2}\psi(t)}}e^{-2\log\psi(t)\log(1-(B_{2}-B_{1})/\psi(t))}-1\to 0.$
(ii) From (2.7) and (2.8) choose $t_{0}$ so that
$\left(1-{t\eta(t)\over\psi_{2}(t)}\right){\eta(t/\psi_{2}(t))\over\eta(t)}\geq
1,\qquad\qquad\text{for\ all\ $t\geq t_{0}$.}$
Then
$\displaystyle\psi(t/\psi(t))$
$\displaystyle\geq\psi_{2}(t/\psi_{2}(t))+B_{1}-B_{2}$
$\displaystyle=\psi_{2}(t_{0}/\psi_{2}(t_{0}))+B_{1}-B_{2}+\int_{t_{0}}^{t}\left(1-{u\eta(u)\over\psi_{2}(u)}\right){\eta(u/\psi_{2}(u))\over\psi_{2}(u)}\,du$
$\displaystyle\geq\psi_{2}(t_{0}/\psi_{2}(t_{0}))+B_{1}-B_{2}+\int_{t_{0}}^{t}{\eta(u)\over\psi_{2}(u)}\,du$
$\displaystyle\geq\psi_{2}(t_{0}/\psi_{2}(t_{0}))+B_{1}-B_{2}-\log\psi_{2}(t_{0})+\log\psi(t).$
Hence
$t\eta(t)\overline{F}(t/\psi(t))={t\eta(t)\over\psi(t)}e^{\log\psi(t)-\psi(t/\psi(t))}\to
0.$
∎
Proof of Theorem 3.1. Condition (3.2), $t\eta(t)$ bounded away from 0 and
$t^{2+\delta}\overline{F}(t)\to 0$ imply
$\limsup_{n\to\infty}\sup_{s\geq t_{n}}{\log n\over\psi(s)}\leq{2\over
2+\delta}$
so that (2.4) follows from (3.1). In addition, $\psi(t)$ must be slowly
varying so
$\displaystyle\limsup_{s\geq t_{n}}ns\eta(\lambda
s)\left(\overline{F}(s/a)+F(-s/a)\right)$ $\displaystyle\leq{na\eta(\lambda
s)\over s}{\psi^{1+\delta}(s)\over
s^{\delta}}(s/a)^{2+\delta}\left(\overline{F}(s/a)+F(-s/a)\right)$
$\displaystyle\to 0.$
Furthermore,
$\sup_{s\geq t_{n}}{na\mu_{2}(s)/s\over s\wedge\eta^{-1}(\lambda s)}\to 0.$
Therefore, (2.3) and (2.5) hold. ∎
Proof of Theorem 3.2. By the dominated variation assumption, (2.1) holds with
$b$ measurable and bounded and $\eta$ measurable and satisfying
$\displaystyle 0\leq t\eta(t)\leq A\text{ for all }t>0,$ (4.23)
for some finite $A$ (cf. Bingham, Goldie and Teugels (1987), Theorem 2.2.7).
Conditions (2.3) and (2.4) are easily checked from (3.4) and the assumption
${S_{n}\over t_{n}}\buildrel P\over{\rightarrow}0$ (cf. Loève (1955), p. 317).
It is actually not necessary to check (2.5) for this proof though in fact
something stronger holds (see below). The basic structure in the argument for
Theorem 2.1 may be followed, with a slight difference. We focus on the
relevant distinction here.
Since $\eta$ is not necessarily monotone, the bound in (4.20) must be
modified. In fact, using (4.23),
$\displaystyle n\int_{s/a}^{m}(e^{(1+\epsilon)ax\over s}-1)F(dx)$
$\displaystyle\leq ne^{(1+\epsilon)am\over s}\overline{F}(s/a)$
$\displaystyle\leq e^{-(\epsilon^{\prime}-\epsilon)a+\psi(s)-\psi(s/a)}$
$\displaystyle\leq e^{-(\epsilon^{\prime}-\epsilon)a+B-1+A\log a}.$
So if we let
$\epsilon=\max\left(z_{n}(s)^{-1},{2n(|\mu_{1}(w)|+e^{2}a\mu_{2}(w)/s)\over\delta(s\wedge\eta^{-1}(\lambda
s))}\right)$
and
$\epsilon^{\prime}=\epsilon+{B-1+A\log a-\log\epsilon\over a}$
and we use the idea in (3.3), then the rest of the proof follows readily. ∎
Proof of Theorem 3.3. We need only to check the assumptions of Theorem 3.2
since $P[|X|>x]$ is regularly varying. Note that (2.2) holds. Consider the
following three cases separately.
i) If $\alpha<1$, we make use of the well known relationships (cf. Bingham,
Goldie and Teugels (1987), Theorem 8.1.2) between $\overline{F}$, $\mu_{1}$
and $\mu_{2}$ to obtain
$\displaystyle\limsup_{t\to\infty}{|\mu_{1}(t)|\over
t\overline{F}(t)}\leq\dfrac{(1+p)\alpha}{1-\alpha}<\infty$ (4.24)
and
$\limsup_{t\to\infty}{\mu_{2}(t)\over
t^{2}\overline{F}(t)}\leq\dfrac{(1+p)\alpha}{2-\alpha}<\infty.$
Hence $n\overline{F}(t_{n})\to\infty$ implies $S_{n}/t_{n}\buildrel
P\over{\rightarrow}0$ and
$\lim_{n\to\infty}\sup_{s\geq t_{n}}{na\mu_{2}(s)\over
s^{2}}\leq\dfrac{(1+p)\alpha}{2-\alpha}\lim_{n\to\infty}\sup_{s\geq
t_{n}}n\overline{F}(s)(-\log n\overline{F}(s))=0.$
ii) In case $1\leq\alpha<2$, the only difference is that (4.24) fails so the
appropriate centering of $S_{n}$ must be assumed. Otherwise, the proof is as
above.
iii) For $\alpha\geq 2$, the extra conditions imply $S_{n}/t_{n}\buildrel
P\over{\rightarrow}0$. Also,
$\lim_{t\to\infty}{-\log\overline{F}(t)\over\log t}=\alpha,$
so that for some finite $C$,
$-\log n\overline{F}(t)\leq C\log t.$
Furthermore, $\mu_{2}(x)\log x/x^{2}\in RV_{-2}$ and therefore is almost
decreasing (cf. Bingham, Goldie and Teugels (1987), p. 41). That is,
$\lim_{n\to\infty}\sup_{s\geq t_{n}}{\mu_{2}(s)\log
s/s^{2}\over\mu_{2}(t_{n})\log t_{n}/t_{n}^{2}}<\infty.$
Hence (3.4) follows readily. ∎
Acknowledgement. This paper grew out of a conversation with Professor Jozef
Teugels. We are very grateful to him for his inspiration.
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Daren B. H. Cline
Department of Statistics
Texas A&M University
College Station, TX 77843-3143
<EMAIL_ADDRESS>
Tailen Hsing
Department of Statistics
University of Michigan
Ann Arbor, MI 48109-1107
<EMAIL_ADDRESS>
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